Hidrodinâmica e Propulsão do navio

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Hidrodinaˆmica e Propulsa˜o
Engenharia de Ma´quinas Mar´ıtimas
Jorge Trindade
ENIDH
2012
I´ndice
1 Introduc¸a˜o 1
1.1 Geometria do navio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.1 Principais dimenso˜es dos navios . . . . . . . . . . . . . . . . . . . . . . . 1
1.1.2 Coeficientes de forma do navio . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Comportamento hidrodinaˆmico do navio . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Me´todos emp´ıricos . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Me´todos experimentais . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.5 Simulac¸o˜es nume´ricas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2 Resisteˆncia 13
2.1 Ana´lise dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2 Leis da semelhanc¸a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.1 Semelhanc¸a geome´trica . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.2 Semelhanc¸a cinema´tica . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.3 Semelhanc¸a dinaˆmica . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Decomposic¸a˜o da resisteˆncia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3.1 Resisteˆncia de onda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.3.2 Resisteˆncia de atrito . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.3.3 Resisteˆncia viscosa de pressa˜o . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Ensaios de resisteˆncia em tanques de reboque . . . . . . . . . . . . . . . . . . . 26
2.5 Ca´lculo da resisteˆncia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.1 Me´todos de extrapolac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.2 Resisteˆncias adicionais . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.6 Previsa˜o com dados sistema´ticos ou estat´ısticos . . . . . . . . . . . . . . . . . . 32
2.7 Ensaios a` escala real . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3 Propulsa˜o 35
3.1 Sistemas de propulsa˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 He´lices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Outros meios de propulsa˜o . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.2 He´lices propulsores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.1 Geometria do he´lice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.2.2 Valores caracter´ısticos . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.3 Teoria da quantidade de movimento . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3.1 Forc¸a propulsiva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
i
ii I´NDICE
3.3.2 Coeficiente de carga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.3.3 Rendimento ideal do he´lice . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.4 Ensaios com modelos reduzidos de he´lices . . . . . . . . . . . . . . . . . . . . . 45
3.4.1 Diagrama em a´guas livres . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.2 Rendimento . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.4.3 I´ndice de qualidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 Se´ries sistema´ticas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5.1 Se´rie sistema´tica de Wageningen . . . . . . . . . . . . . . . . . . . . . . 48
3.5.2 Outras se´ries sistema´ticas . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.5.3 Diagrama de 4 quadrantes . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.6 Cavitac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.1 Origem da cavitac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.6.2 Controle da cavitac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.6.3 Considerac¸a˜o da cavitac¸a˜o na selecc¸a˜o do he´lice . . . . . . . . . . . . . . 55
3.6.4 Ensaios experimentais . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.7 Selecc¸a˜o do he´lice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7.1 Varia´veis de optimizac¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.7.2 Tipos de problema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.8 Interacc¸a˜o entre casco e he´lice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.8.1 Ensaios de propulsa˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.8.2 Poteˆncia e velocidade . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.8.3 Extrapolac¸a˜o dos resultados do ensaio de propulsa˜o . . . . . . . . . . . 66
4 Instalac¸o˜es Propulsoras 67
4.1 Introduc¸a˜o . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.2 Propulsa˜o diesel-mecaˆnica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2.1 Accionamento de auxiliares . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.2.2 Engrenagens redutoras . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
4.2.3 Configurac¸a˜o ”pai-e-filho” . . . . . . . . . . . . . . . . . . . . . . . . . . 73
4.3 Propulsa˜o diesel-ele´ctrica . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.1 Propulsa˜o por motor ele´ctrico . . . . . . . . . . . . . . . . . . . . . . . . 74
4.3.2 Propulsores azimutais . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.4 Selecc¸a˜o do motor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.4.1 Turbinas e motores ele´ctricos . . . . . . . . . . . . . . . . . . . . . . . . 79
4.4.2 Motores diesel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
I´ndice Remissivo 83
A Previsa˜o Baseada nos Ensaios de Propulsa˜o 87
B Provas de velocidade e Poteˆncia 121
C Condic¸o˜es das Provas de Velocidade e Poteˆncia 133
D Selecc¸a˜o de Motores Propulsores 141
E Derating 175
Lista de Figuras
1.1 Plano de flutuac¸a˜o, longitudinal e transversal de um navio. . . . . . . . . . . . 2
1.2 Plano geome´trico de um navio. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Principais dimenso˜es dos navios. . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Marcac¸a˜o no costado das linhas de carga do navio. . . . . . . . . . . . . . . . . 5
1.5 Tanque de provas utilizado por W. Froude. . . . . . . . . . . . . . . . . . . . . 7
1.6 Tanque de testes actual. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.7 Bacia para testes com ondulac¸a˜o. . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.8 Bacia para testes com a´guas geladas. . . . . . . . . . . . . . . . . . . . . . . . . 8
1.9 Escoamento num he´lice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.10 Malha colocada a` esquerda e desfasada a` direita. . . . . . . . . . . . . . . . . . 9
1.11 Representac¸a˜o esquema´tica de um “PC-cluster”. . . . . . . . . . . . . . . . . . . 10
1.12 Um “PC-cluster” com 24 no´s computacionais. . . . . . . . . . . . . . . . . . . . 10
1.13 Decomposic¸a˜o 1D, 2D ou 3D do domı´nio espacial de um problema. . . . . . . . 11
1.14 Troca de valores nas fronteiras dos sub-domı´nios. . . . . . . . . . . . . . . . . . 11
2.1 Decomposic¸a˜o da resisteˆncia. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.2 Sistema de ondas gerado por um ponto de pressa˜o em movimento. . . . . . . . 20
2.3 Sistemas de ondas da proa e da popa. . . . . . . . . . . . . . . . . . . . . . . . 21
2.4 Interacc¸a˜o entre os dois sistemas de ondas. . . . . . . . . . . . . . . . . . . . . . 22
2.5 Curvada resisteˆncia de onda. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.6 Variac¸a˜o do coeficiente da resisteˆncia de atrito com o nu´mero de Reynolds e
com a rugosidade da superf´ıcie. . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.7 Distribuic¸a˜o de pressa˜o num escoamento ideal, inv´ıscido. . . . . . . . . . . . . . 26
2.8 Modelo a` escala reduzida para ensaios de resisteˆncia. . . . . . . . . . . . . . . . 27
2.9 Representac¸a˜o gra´fica da dependeˆncia de
cT
cF0
com
Fr4
cF0
. . . . . . . . . . . . . . 29
2.10 Reduc¸a˜o de velocidade (%) em a´guas pouco profundas. . . . . . . . . . . . . . . 33
3.1 He´lice com tubeira. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 He´lices de passo fixo e de passo controla´vel. . . . . . . . . . . . . . . . . . . . . 36
3.3 He´lices em contra-rotac¸a˜o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
3.4 He´lices supercavitante. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.5 Propulsa˜o por jacto de a´gua. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.6 Propulsores azimutais. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.7 Propulsores cicloidais. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
iii
iv LISTA DE FIGURAS
3.8 Geometria do he´lice. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3.9 Distribuic¸a˜o espacial de velocidade e pressa˜o para a teoria da quantidade de
movimento. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.10 Diagrama de a´guas livres. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.11 Aspecto geome´trico das pa´s da se´rie B de Wageningen . . . . . . . . . . . . . . 48
3.12 Diagrama em a´guas livres de um he´lice da se´rie sistema´tica de Wageningen. . . 50
3.13 Notac¸a˜o do diagrama com 4 quadrantes. . . . . . . . . . . . . . . . . . . . . . . 51
3.14 Diagrama em a´guas livres de 4 quadrantes para os he´lices Wageningen B-4.70. 53
3.15 Efeito da cavitac¸a˜o no valor dos paraˆmetros relativos a a´guas livres. . . . . . . 54
3.16 Pressa˜o de vapor da a´gua em func¸a˜o da temperatura. . . . . . . . . . . . . . . 55
3.17 Diagrama de Burrill. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.18 Instalac¸o˜es de ensaio do RINA. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.19 Imagem da cavitac¸a˜o num he´lice. . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.20 Modelo para ensaios de propulsa˜o. . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.21 Resultados dos ensaios de propulsa˜o. . . . . . . . . . . . . . . . . . . . . . . . . 66
4.1 Variantes de instalac¸o˜es propulsoras diesel-mecaˆnicas lentas e de me´dia veloci-
dade. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.2 Instalac¸o˜es propulsoras diesel-mecaˆnica (em cima) e diesel-ele´ctrica (em baixo). 69
4.3 Acoplamento com relac¸a˜o varia´vel de velocidades. . . . . . . . . . . . . . . . . . 71
4.4 Conversa˜o da frequeˆncia da energia ele´ctrica. . . . . . . . . . . . . . . . . . . . 72
4.5 Instalac¸a˜o propulsora com quatro motores, engrenagens redutoras e dois he´lices. 73
4.6 Instalac¸a˜o com dois motores diesel diferentes, engrenagens redutoras, embrai-
agens e geradores acoplados aos veios. . . . . . . . . . . . . . . . . . . . . . . . 74
4.7 Motor ele´ctrico de propulsa˜o. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.8 Instalac¸a˜o diesel-ele´ctrica. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.9 Representac¸a˜o esquema´tica de uma instalac¸a˜o diesel-ele´ctrica. . . . . . . . . . . 77
4.10 Propulsores azimutais. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
4.11 Diagrama de carga de um motor diesel . . . . . . . . . . . . . . . . . . . . . . . 80
Lista de Tabelas
1.1 Valores de K na fo´rmula de Alexander. . . . . . . . . . . . . . . . . . . . . . . 6
2.1 Valores do coeficiente de correcc¸a˜o cA em func¸a˜o do comprimento do navio. . . 29
3.1 Se´ries sistema´ticas de propulsores. . . . . . . . . . . . . . . . . . . . . . . . . . 48
3.2 Coeficiente para atribuic¸a˜o do diaˆmetro ma´ximo do he´lice pela Eq. (3.34). . . . 59
3.3 Constante para o ca´lculo do diaˆmetro equivalente em a´gua livres pela Eq. (3.35). 59
v
vi LISTA DE TABELAS
Cap´ıtulo 1
Introduc¸a˜o
1.1 Geometria do navio
A variac¸a˜o da proporc¸a˜o relativa das dimenso˜es principais de um navio tem um importante
efeito nas suas caracter´ısticas operacionais. Afecta as suas caracter´ısticas hidrodinaˆmicas, a
sua resisteˆncia estrutural e, naturalmente a capacidade de carga.
Os navios existentes, em particular as unidades de construc¸a˜o mais recente, constituem
uma boa “fonte de inspirac¸a˜o” para o pre´-dimensionamento de um navio novo. No que diz
respeito a` informac¸a˜o mais detalhada, estas bases de dados sa˜o, regra geral, bem resguardadas
pelos gabinetes de estudo e projecto, bem como pelos estaleiros construtores. No entanto,
alguns destes dados esta˜o dispon´ıveis nos registos publicados pelas sociedades classificadoras
e por alguns gabinetes de estudo.
Depois de um processo iterativo de dimensionamento do navio, durante o qual sa˜o tidas
em considerac¸a˜o as varia´veis de optimizac¸a˜o seleccionadas, a soluc¸a˜o final da forma do navio
constitui o plano geome´trico do navio. Na pra´tica, este plano geome´trico e´ gerado por uma
das seguintes vias:
- deformac¸a˜o de um navio de refereˆncia;
- modelo matema´tico para definic¸a˜o de forma em func¸a˜o de paraˆmetros do navio;
- utilizac¸a˜o das se´ries sistema´ticas.
1.1.1 Principais dimenso˜es dos navios
O casco de um navio e´ uma forma tridimensional, na maior parte dos casos sime´trica rela-
tivamente a um plano vertical longitudinal do navio. O contorno do casco fica definido pela
sua intersecc¸a˜o com treˆs planos ortogonais (Fig. 1.1):
- o plano de flutuac¸a˜o de projecto;
- o plano longitudinal;
- o plano transversal.
1
2 CAPI´TULO 1. INTRODUC¸A˜O
Figura 1.1: Plano de flutuac¸a˜o, longitudinal e transversal de um navio.
O plano longitudinal, u´nico plano de simetria do navio, e´ o plano de refereˆncia. A forma
do navio cortada por este plano e´ o perfil. O plano de flutuac¸a˜o de projecto e´ um plano
perpendicular ao plano longitudinal, escolhido como plano de refereˆncia. Os planos paralelos
ao plano de flutuac¸a˜o de projecto sa˜o conhecidos como planos de a´gua, ou de flutuac¸a˜o, e as
linhas de intersecc¸a˜o como linhas de a´gua. Os planos de flutuac¸a˜o sa˜o sime´tricos relativamente
ao plano longitudinal. Os planos perpendiculares ao plano longitudinal e ao plano de flutuac¸a˜o
de projecto sa˜o os planos transversais. As secc¸o˜es transversais exibem simetria relativamente
ao plano longitudinal.
A secc¸a˜o do navio equidistante das perpendiculares e normal aos planos de flutuac¸a˜o de
vera˜o e longitudinal e´ designada por secc¸a˜o de meio-navio, ou secc¸a˜o mestra. Na Fig. 1.2
esta´ representado um plano de linhas do navio, que inclui o plano do casco, no qual, por
convenc¸a˜o, sempre que o navio e´ sime´trico, se exibem metades das secc¸o˜es. Do lado direito
representam-se metades das secc¸o˜es avante de meio-navio e do lado esquerdo metades das
secc¸o˜es a re´. O plano de linhas do navio inclui ainda o plano da metade da boca, no qual sa˜o
representados os planos de flutuac¸a˜o.
Figura 1.2: Plano geome´trico de um navio.
1.1. GEOMETRIA DO NAVIO 3
Na Fig. 1.3 esta˜o representadas as dimenso˜es mais frequentemente utilizadas para definir
o navio. Quanto ao comprimento do navio, sa˜o treˆs as definic¸o˜es a considerar:
- o comprimento entre perpendiculares, Lpp, distaˆncia medida ao longo do plano de flu-
tuac¸a˜o de vera˜o entre a perpendicular da popa e a perpendicularda proa;
- o comprimento na linha de a´gua, Lwl, distaˆncia na linha de flutuac¸a˜o que se verifique, se
nada for referido devera´ entender-se a linha de flutuac¸a˜o de vera˜o, entre as intersecc¸o˜es
da proa e popa com a mesma linha de flutuac¸a˜o;
- o comprimento fora a fora, Loa, distaˆncia entre os pontos extremos a vante e a re´ do
navio, medida numa direcc¸a˜o paralela a` linha de flutuac¸a˜o de vera˜o.
Designa-se por boca, a ma´xima distaˆncia entre as faces interiores das chapas de costado
nos dois bordos do navio, na secc¸a˜o mestra, se outra secc¸a˜o na˜o for indicada. O pontal e´ a
distaˆncia na vertical, medida a meio navio, entre a face inferior do conve´s e a face superior
da chapa da quilha. O calado de um navio em qualquer ponto do seu comprimento e´ a
distaˆncia na vertical entre a quilha e a linha de a´gua. O calado varia na˜o so´ com o estado de
carregamento do navio mas tambe´m com a densidade da a´gua em que este se encontra.
A altura desde a linha de flutuac¸a˜o e o conve´s e´ designada por bordo livre. Pode ser
calculado pela diferenc¸a entre o pontal e o calado.
Um aspecto importante relativamente a` seguranc¸a de um navio mercante prende-se com
a alocac¸a˜o regulamentar de um valor mı´nimo do bordo livre, como forma de garantir uma
reserva de estabilidade suficiente para a seguranc¸a da navegac¸a˜o. Este valor mı´nimo do bordo
livre depende do local de navegac¸a˜o e da e´poca do ano. No costado do navio esta˜o marcadas
as linhas de carga por forma a permitir verificar facilmente se as condic¸o˜es de seguranc¸a sa˜o
verificadas. O valor de refereˆncia e´ a linha de Vera˜o que e´ marcada no centro de um c´ırculo,
Fig. 1.4. Ao lado deste c´ırculo, sa˜o marcadas na horizontal linhas adicionais que correspondem
ao:
- bordo livre de Inverno, superior em 1/48 avos do bordo livre de Vera˜o;
- bordo livre de Inverno no Atlaˆntico Norte, ainda superior em 50 mm;
- bordo livre tropical, inferior em 1/48 avos do bordo livre de Vera˜o;
- bordo livre em a´gua doce, inferior em ∆ / (40 t) cm, sendo ∆ o deslocamento em ton e
t as ton por cm de imersa˜o;
- bordo livre tropical em a´gua doce e´ inferior em 1/48 avos do bordo livre de Vera˜o ao
bordo livre em a´gua doce.
1.1.2 Coeficientes de forma do navio
O deslocamento do navio e´ o peso do volume de a´gua que o navio desloca quando a flutuar
em a´guas tranquilas,
∆ = ρg∇ (1.1)
em que ρ e´ a massa volu´mica da a´gua em que o navio se encontra a flutuar, g e´ a acelerac¸a˜o
da gravidade e ∇ o volume deslocado.
A partir das principais dimenso˜es da navio, definem-se os seguintes coeficientes de forma:
4 CAPI´TULO 1. INTRODUC¸A˜O
Figura 1.3: Principais dimenso˜es dos navios.
- o coeficiente de finura total (“block coeficient”):
Cb =
∇
LppBT
(1.2)
onde:
- ∇ e´ o volume do deslocamento;
- Lpp o comprimento entre perpendiculares;
- B a boca (ma´xima abaixo da linha de a´gua);
- e T e´ o calado me´dio do navio.
- o coeficiente de finura da flutuac¸a˜o:
Cwp =
Awp
LwpB
(1.3)
em que:
- Awp e´ a a´rea do plano de flutuac¸a˜o;
1.1. GEOMETRIA DO NAVIO 5
Figura 1.4: Marcac¸a˜o no costado das linhas de carga do navio.
- Lwp o comprimento na linha de flutuac¸a˜o;
- e B a boca (ma´xima na linha de flutuac¸a˜o).
- o coeficiente de finura da secc¸a˜o mestra:
Cm =
Am
BT
(1.4)
representando por:
- Am a a´rea imersa na secc¸a˜o mestra;
- B a boca na secc¸a˜o mestra;
- e T o calado a meio navio.
- o coeficiente prisma´tico longitudinal:
Cp =
∇
AmLpp
(1.5)
em que novamente:
- ∇ e´ o volume da querena;
- Am a a´rea imersa a meio navio;
- e Lpp o comprimento entre perpendiculares.
Como exemplo da utilizac¸a˜o dos coeficientes de forma no estabelecimento de relac¸o˜es
emp´ıricas para in´ıcio do projecto de um navio, pode-se indicar a fo´rmula de Alexander,
Cb = K − 0.5× V√
L
(1.6)
em que K apresenta os valores da Tab. 1.1 de acordo com o tipo de navio. A fo´rmula de
Alexander estabelece uma relac¸a˜o emp´ırica entre o coeficiente de finura total do navio, a sua
velocidade e o comprimento. Pela especificidade de cada caso, o coeficiente de finura total
Cb do navio podera´ depois desviar-se do valor inicialmente previsto durante o processo de
optimizac¸a˜o das caracter´ısticas do navio.
6 CAPI´TULO 1. INTRODUC¸A˜O
Tipo de Navio K
Petroleiro 1.13
Graneleiro 1.11
Carga geral 1.10
Navio de linha 1.05
Ferry 1.08
Rebocador 1.18
Tabela 1.1: Valores de K na fo´rmula de Alexander.
1.2 Comportamento hidrodinaˆmico do navio
A ana´lise do comportamento hidrodinaˆmico do navio pode ser decomposta em diversas a´reas,
de entre as quais se pode salientar:
- a resisteˆncia;
- a propulsa˜o;
- o comportamento do navio no mar;
- a capacidade de manobra.
O ca´lculo do escoamento e o projecto de he´lices pode ser considerado como um sub-to´pico do
tema resisteˆncia e propulsa˜o.
As metodologias para o ca´lculo ou para a previsa˜o dos paraˆmetros relevantes do compor-
tamento do navio podem ser classificadas como:
- emp´ıricas e estat´ısticas;
- experimentais em modelos a` escala reduzida, ou a` escala real;
- nume´ricas, atrave´s de soluc¸o˜es anal´ıticas ou com recurso a` mecaˆnica de fluidos compu-
tacional.
Os princ´ıpios fundamentais destas metodologias sa˜o sumariamente descritos nas secc¸o˜es
seguintes.
1.3 Me´todos emp´ıricos
Os me´todos emp´ıricos baseiam-se num modelo f´ısico relativamente simples e na ana´lise por
regressa˜o para a determinac¸a˜o dos coeficientes relevantes, a partir de um so´ navio ou de uma
se´rie de navios. Os resultados assim obtidos sa˜o depois expressos sob a forma de constantes,
fo´rmulas, tabelas, gra´ficos, etc.
Numerosos estudos realizados entre 1940 e 1960 permitiram criar se´ries de “boas” formas
de carenas. O efeito da variac¸a˜o dos principais paraˆmetros do casco, como por exemplo o
coeficiente de bloco, foi determinado por alterac¸a˜o sistema´tica daqueles paraˆmetros.
1.4. ME´TODOS EXPERIMENTAIS 7
Figura 1.5: Tanque de provas utilizado
por W. Froude. Figura 1.6: Tanque de testes actual.
1.4 Me´todos experimentais
Esta abordagem baseia-se no teste de modelos em escala reduzida para extrair informac¸a˜o que
possa ser extrapolada para a escala do navio. Apesar dos grandes esforc¸os de investigac¸a˜o e
normalizac¸a˜o, a correlac¸a˜o modelo-navio esta´ sujeita a algum grau de empirismo. Cada uma
das principais instalac¸o˜es de teste (tu´neis, bacias, etc.) tende a adoptar os me´todos de ensaio
e tratamento da informac¸a˜o que melhor se adaptam a` experieˆncia ja´ incorporada nas suas
bases de dados. Esta na˜o uniformidade de processos dificulta, se na˜o mesmo em muitos casos
impossibilita, o aproveitamento estat´ıstico dos dados de uma forma agregada.
Embora a metodologia base para a avaliac¸a˜o da resisteˆncia de um modelo num tanque de
testes se mantenha praticamente inalterada desde os tempos de Froude (1874), va´rios aspectos
te´cnicos sofreram grande evoluc¸a˜o. De entre estes, podem-se salientar:
- as te´cnicas experimentais na˜o-intrusivas, como a Laser-Doppler Velocimetry, que per-
mitem a medic¸a˜o do campo de velocidades na esteira do navio para melhorar o projecto
do he´lice;
- a ana´lise do padra˜o da formac¸a˜o ondosa gerada pelo modelo para estimar a resisteˆncia
de onda;
- nos testes de modelos com propulsa˜o auto´noma, e´ poss´ıvel agora medir grandezas rela-
cionadas com o propulsor como o impulso, bina´rio, rpm, etc.
Instalac¸o˜es com caracter´ısticas bem diferentes surgiram entretanto para possibilitar outro
tipo de estudos. Trata-se de bacias equipadas com geradores de ondas, para ensaios de modelos
com o objectivo de estudar as questo˜es de manobrabilidade e de comportamento do navio no
mar, Fig. 1.7.
Outro tipo de bacias para ensaios de modelos de navios, Fig. 1.8, dedica-se preferencial-
mente a estudos e ensaiosrelacionados com a presenc¸a de gelo no mar.
Por u´ltimo, um outro tipo de instalac¸a˜o de teste nesta a´rea dedica-se ao estudo do desem-
penho de he´lices propulsores. Neste tipo de instalac¸a˜o, que iremos abordar com um pouco
mais de detalhe no Cap. 3, para ale´m da determinac¸a˜o de va´rias caracter´ısticas de desempenho
do he´lice, pode-se vizualizar o padra˜o de cavitac¸a˜o no he´lice.
8 CAPI´TULO 1. INTRODUC¸A˜O
Figura 1.7: Bacia para testes
com ondulac¸a˜o.
Figura 1.8: Bacia para testes com
a´guas geladas.
Figura 1.9: Escoamento num he´lice.
1.5 Simulac¸o˜es nume´ricas
As simulac¸o˜es de escoamento obtidas pela mecaˆnica de fluidos computacional sa˜o ainda consi-
deradas pela indu´stria como pouco precisas para a previsa˜o da resisteˆncia de um casco ou da
forc¸a propulsiva de um he´lice. No entanto, o contributo da mecaˆnica de fluidos computacional
esta´ a tornar-se cada vez mais importante em determinados passos do processo de projecto.
Casos t´ıpicos de aplicac¸a˜o sa˜o, por exemplo:
- a simulac¸a˜o de escoamento inv´ıscido, com superf´ıcie livre, para ana´lise do comporta-
mento da proa, interacc¸a˜o com o bolbo, formac¸a˜o ondosa, etc.
- as simulac¸o˜es de escoamento viscoso na zona da popa, desprezando a formac¸a˜o ondosa
para avaliac¸a˜o do comportamento de apeˆndice ou ana´lise do escoamento de aproximac¸a˜o
ao he´lice.
No caso mais geral, o escoamento de fluidos incompress´ıveis em regime na˜o-estaciona´rio e´
modelado pelas seguintes equac¸o˜es:
1.5. SIMULAC¸O˜ES NUME´RICAS 9
- Equac¸a˜o da continuidade,
∂ui
∂xi
= 0 (1.7)
- Equac¸a˜o de conservac¸a˜o da quantidade de movimento,
∂ρui
∂t
+
∂
∂xj
(ρuiuj) = − ∂p
∂xi
+ µ
∂2ui
∂xj∂xj
+ ρbi (1.8)
- Equac¸a˜o de conservac¸a˜o da energia (forma simplificada),
∂θ
∂t
+
∂ (θuj)
∂xj
=
κ
ρc
∂2θ
∂xj∂xj
(1.9)
em que xi e´ a coordenada na direcc¸a˜o i, ui e´ a componente da velocidade na direcc¸a˜o i, ρ
e µ sa˜o a massa espec´ıfica e a viscosidade do fluido, respectivamente, p e´ a pressa˜o, κ e´ a
condutividade te´rmica, c e´ o calor espec´ıfico, θ e´ a temperatura, b e´ a componente na direcc¸a˜o
i das forc¸as exteriores por unidade de massa e t e´ o tempo.
As equac¸o˜es sa˜o discretizadas no espac¸o de acordo com uma malha colocada ou desfasada.
Na Fig. 1.10 esta´ indicada a localizac¸a˜o das varia´veis, no caso bi-dimensional, para cada
uma daqueles tipos de malhas. Cada um daqueles tipos de malha de discretizac¸a˜o apresenta
Figura 1.10: Malha colocada a` esquerda e desfasada a` direita.
algumas vantagens e desvantagens. As mais importantes esta˜o relacionadas com:
- a complexidade da programac¸a˜o;
- o tratamento das fronteiras do problema;
- a soluc¸a˜o para o acoplamento pressa˜o-velocidade (formato xadrez na soluc¸a˜o da pressa˜o).
Selecionado o tipo de malha a utilizar, outras opc¸o˜es ha´ a tomar para desenvolver o me´todo
de soluc¸a˜o. Algumas das mais comuns sa˜o:
10 CAPI´TULO 1. INTRODUC¸A˜O
Figura 1.11: Representac¸a˜o esquema´tica
de um “PC-cluster”.
Figura 1.12: Um “PC-cluster”
com 24 no´s computacionais.
- SIMPLE / me´todo de projecc¸a˜o;
- volume finito / diferenc¸as finitas;
- aproximac¸a˜o dos termos convectivos/difusivos das equac¸o˜es;
- “upwind”;
- diferenc¸as centrais de ordem 2;
- diferenc¸as centrais de ordem 4;
- o me´todo de integrac¸a˜o para a evoluc¸a˜o temporal;
- Euler;
- Crank-Nicolson;
- Adams-Bashforth;
- Runge-Kutta.
Tratando-se de ca´lculos complexos, o tempo de ca´lculo podera´ ser reduzido, sem acre´scimo
significativo de custos, com recurso de um “PC-cluster”, Fig. 1.11.
Este tipo de estruturas computacionais caracterizam-se por dispor de:
- 20 a 1000 CPU;
- 2 a 8 GB RAM por no´;
- comunicac¸a˜o em rede com velocidade superior a 1 Gbps;
- elevada capacidade para armazenamento de dados;
- sistema operativo esta´vel.
1.5. SIMULAC¸O˜ES NUME´RICAS 11
Para a soluc¸a˜o de um problema de mecaˆnica de fluidos num “PC-cluster” e´ necessa´rio pro-
ceder a` decomposic¸a˜o do domı´nio espacial do problema (Fig. 1.13) e recorrer a rotinas de uma
das va´rias bibliotecas dispon´ıveis para efectuar a troca de dados entre os no´s computacionais,
como por exemplo a biblioteca Message Passing Interface, necessa´ria para a continuac¸a˜o do
ca´lculo. Na Fig. 1.14 esta˜o representados esquematicamente aquelas comunicac¸o˜es de dados
relativos a`s fronteiras dos sub-domı´nios de ca´lculo.
Figura 1.13: Decomposic¸a˜o 1D, 2D ou 3D do domı´nio espacial de um pro-
blema.
Figura 1.14: Troca de valores nas fronteiras dos sub-domı´nios.
12 CAPI´TULO 1. INTRODUC¸A˜O
Cap´ıtulo 2
Resisteˆncia
2.1 Ana´lise dimensional
A resisteˆncia do navio a uma velocidade constante e´ a forc¸a necessa´ria para rebocar o navio
a essa velocidade em a´guas tranquilas. Se a querena na˜o tiver apeˆndices, a resisteˆncia diz-se
da querena simples. Designaremos por poteˆncia efectiva, ou poteˆncia de reboque, a poteˆncia
necessa´ria para vencer a resisteˆncia do navio a uma dada velocidade,
Pe = V RT (2.1)
em que V e´ a velocidade do navio e RT a sua resisteˆncia total.
A resisteˆncia do navio RT = f (V,L, ρ, ν, g) depende:
- da velocidade do navio V ;
- das dimenso˜es do navio, representadas aqui por uma dimensa˜o linear L;
- da massa espec´ıfica do fluido ρ;
- da viscosidade cinema´tica do fluido ν;
- da acelerac¸a˜o da gravidade g.
Assim, a resisteˆncia do navio devera´ ser uma func¸a˜o da forma
RT = V
aLbρcνdge (2.2)
Ao estudar a resisteˆncia de um navio e´ importante calcular na˜o o seu valor absoluto, mas
tambe´m a sua relac¸a˜o com outro valor, dimensionalmente semelhante, tomado como refereˆn-
cia. Vamos dar o nome de coeficientes espec´ıficos a estas relac¸o˜es. No caso da resisteˆncia
total do navio, o valor do coeficiente e´ obtido por
cT =
RT
1
2
ρSV 2
(2.3)
em que ρ e´ a massa espec´ıfica do fluido, S a superf´ıcie molhada do navio e V a sua velocidade.
13
14 CAPI´TULO 2. RESISTEˆNCIA
Resolvendo o sistema de equac¸o˜es gerado pela Eq. (2.2) em ordem a a, b e c, e considerando
a definic¸a˜o do coeficiente em 2.3 dada pela Eq. (2.3), temos
RT = ρV
2L2f
(
V L
ν
,
gL
V 2
)
(2.4)
Ou seja, a ana´lise dimensional mostra que o coeficiente de resisteˆncia total do navio,
ct = f
(
V L
ν
,
gL
V 2
)
(2.5)
depende dos grupos adimensionais designados por nu´mero de Froude,
Fr =
V√
gL
(2.6)
e por nu´mero de Reynolds,
Re =
V L
ν
(2.7)
calculados para o navio.
2.2 Leis da semelhanc¸a
No caso dos ensaios de modelos para avaliac¸a˜o da resisteˆncia de uma querena, podemos
considerar treˆs formas de semelhanc¸a:
- semelhanc¸a geome´trica;
- semelhanc¸a cinema´tica;
- semelhanc¸a dinaˆmica.
2.2.1 Semelhanc¸a geome´trica
Verificar-se semelhanc¸a geome´trica significa a existeˆncia de uma raza˜o constante entre qual-
quer dimensa˜o linear na escala real do proto´tipo (comprimento, boca, calado do navio, etc.)
Ls e o dimensa˜o linear na escala do modelo Lm. Aquela raza˜o e´ a escala geome´trica do modelo
λ,
Ls = λLm (2.8)
Consequentemente, temos para as a´reas,
As = λ
2Am (2.9)
e para os volumes,
∇s = λ3∇m (2.10)
2.2. LEIS DA SEMELHANC¸A 15
2.2.2 Semelhanc¸a cinema´tica
A semelhanc¸a cinema´tica significa a existeˆncia de uma raza˜o constante entre o “tempo” na
escala real, ts e o “tempo” na escala do modelo tm, a escala cinema´tica τ :
ts = τ · tm (2.11)
A verificac¸a˜o simultaˆnea das condic¸o˜es de semelhanc¸a geome´trica e cinema´tica resulta nos
seguintes factores de escala:
- para a velocidade:
Vs =
λ
τ
Vm (2.12)
- e para a acelerac¸a˜o:
as =
λ
τ2
am (2.13)
2.2.3 Semelhanc¸a dinaˆmica
Obter semelhanc¸a dinaˆmica significa que a raza˜o entre cada uma das forc¸as actuantes no navioa` escala real e as correspondentes forc¸as actuantes no modelo e´ constante, escala dinaˆmica do
modelo κ,
Fs = κ ·Fm (2.14)
As forc¸as presentes, actuantes sobre o navio e sobre o modelo, podem ser classificadas de
acordo com a sua natureza como:
- as forc¸as de ine´rcia;
- as forc¸as grav´ıticas;
- as forc¸as de atrito.
Forc¸as de ine´rcia
As forc¸as de ine´rcia regem-se pela lei de Newton, expressa por
F = m · a (2.15)
em que F e´ a forc¸a de ine´rcia, m a massa do corpo, e a a acelerac¸a˜o a que ele e´ sujeito.
Considerando o volume deslocado pelo navio ∇, a massa do navio e´
m = ρ · ∇ (2.16)
sendo ρ a massa volu´mica da a´gua.
Enta˜o, a raza˜o entre as forc¸as de ine´rcia e´ uma equac¸a˜o que incorpora os treˆs factores de
escala, lei da Semelhanc¸a de Newton, e´ dada por
κ =
Fs
Fm
=
ρs · ∇s · as
ρm · ∇m · am =
ρs
ρm
· λ
4
τ2
(2.17)
que pode ser re-escrita como
κ =
Fs
Fm
=
ρs
ρm
·λ2 ·
(
λ
τ
)2
=
ρs
ρm
· As
Am
·
(
Vs
Vm
)2
(2.18)
16 CAPI´TULO 2. RESISTEˆNCIA
Forc¸as de origem hidrodinaˆmica
As forc¸as de origem hidrodinaˆmica sa˜o normalmente descritas recorrendo a um coeficiente
adimensional c na seguinte forma, semelhante a` Eq. (2.3),
F = c · 1
2
ρ ·V 2 ·A (2.19)
em que V e´ uma velocidade de refereˆncia, por exemplo a velocidade do navio e A uma a´rea de
refereˆncia como, por exemplo, a a´rea das obras vivas com mar calmo. Aplicando a Eq. (2.19)
ao navio e ao modelo e combinando as duas equac¸o˜es, obtem-se
Fs
Fm
=
cs · ρs ·V 2s ·As
cm · ρm ·V 2m ·Am
=
cs
cm
ρs
ρm
· As
Am
·
(
Vs
Vm
)2
(2.20)
Daqui resulta que igualando o valor dos coeficientes no navio e no modelo, cs = cm, fica
garantida a verificac¸a˜o da lei da semelhanc¸a de Newton.
Forc¸as Grav´ıticas
As forc¸as grav´ıticas podem ser descritas de forma semelhante a`s forc¸as de ine´rcia, para o
navio
Gs = ρs · g · ∇s (2.21)
e para o modelo
Gs = ρs · g · ∇s Gm = ρm · g · ∇m (2.22)
daqui resultando uma nova escala,
κg =
Gs
Gm
=
ρs
ρm
· ∇s∇m =
ρs
ρm
·λ3 (2.23)
Para que se possa verificar a semelhanc¸a dinaˆmica, os factores de escala devem apresentar
o mesmo valor, ou seja, κ = κg. Se
κ =
ρs
ρm
· λ
4
τ2
e
κg =
ρs
ρm
·λ3
enta˜o, para que κ = κg e´ necessa´rio verificar-se
τ =
√
λ (2.24)
Esta nova relac¸a˜o permite eliminar a escala temporal em todas as relac¸o˜es apresentadas,
ficando a proporcionalidade apenas dependente de λ como, por exemplo, na Eq. (2.12), fazendo
Vs
Vm
=
√
λ (2.25)
2.2. LEIS DA SEMELHANC¸A 17
Nu´mero de Froude
A Eq. (2.25) pode ainda assumir a forma de uma relac¸a˜o entre a dimensa˜o linear e a
velocidade do modelo e do navio,
Vs√
Ls
=
Vm√
Lm
(2.26)
Adimensionalisando a raza˜o entre a velocidade V e a raiz quadrada do comprimento L
com a acelerac¸a˜o da gravidade, g = 9.81 m/s2, obtemos o nu´mero de Froude
Fr =
V√
g ·L (2.27)
Na auseˆncia de forc¸as viscosas, igual nu´mero de Froude assegura semelhanc¸a dinaˆmica.
Para igual nu´mero de Froude, as ondulac¸o˜es no modelo e a` escala real, desde que de pequena
amplitude, podem considerar-se geometricamente semelhantes.
A lei de Froude e´ verificada em todos os ensaios de modelos de navios, ensaios de resis-
teˆncia, propulsa˜o, comportamento no mar e manobrabilidade. A aplicac¸a˜o da lei de Froude
impo˜e os seguintes factores de escala para a velocidade,
Vs
Vm
=
√
λ (2.28)
forc¸a,
Fs
Fm
=
ρs
ρm
·λ3 (2.29)
e poteˆncia,
Ps
Pm
=
Fs ·Vs
Fm ·Vm =
ρs
ρm
·λ3.5 (2.30)
Forc¸as de atrito
As forc¸as viscosas R, com origem no atrito entre camadas de fluido, sa˜o modeladas por
R = µ · ∂u
∂n
·A (2.31)
em que µ e´ a viscosidade dinaˆmica do fluido, A a a´rea sujeita ao atrito e
∂u
∂n
o gradiente de
velocidade, avaliado na direcc¸a˜o normal ao escoamento.
A raza˜o das forc¸as de atrito no navio e no modelo e´ dada por
κf =
Rs
Rm
=
µs · ∂us
∂ns
·As
µm · ∂um
∂nm
·Am
=
µs
µm
λ2
τ
(2.32)
Na presenc¸a das forc¸as de atrito, para verificar a condic¸a˜o de semelhanc¸a dinaˆmica, sera´
necessa´rio que κf = κ, ou seja:
µs
µm
λ2
τ
=
ρs
ρm
λ4
τ2
(2.33)
18 CAPI´TULO 2. RESISTEˆNCIA
Se introduzirmos a viscosidade cinema´tica, como ν = µ/ρ, obte´m-se:
νs
νm
=
λ2
τ
=
Vs ·Ls
Vm ·Lm
ou seja,
Vs ·Ls
νs
=
Vm ·Lm
νm
(2.34)
Nu´mero de Reynolds
Enta˜o, de acordo com a Eq. (2.34), se apenas estiverem presentes forc¸as de ine´rcia e de
atrito, a igualdade do nu´mero de Reynolds,
Re =
V ·L
ν
(2.35)
assegura semelhanc¸a dinaˆmica entre o modelo e o navio.
Para o ca´lculo do nu´mero de Reynolds, a viscosidade cinema´tica da a´gua do mar (m2/s)
pode ser estimada, em func¸a˜o da temperatura θ (◦C) e da salinidade s (%), por
ν = (0.014 · s+ (0.000645 · θ − 0.0503) · θ + 1.75) · 10−6 (2.36)
Semelhanc¸a dinaˆmica
O nu´mero de Froude e o nu´mero de Reynolds esta˜o relacionados por,
Re
Fr
=
V ·L
ν
√
gL
V
=
√
gL3
ν
(2.37)
A semelhanc¸a de Froude e´ facilmente obtida para testes em modelos porque para modelos
mais pequenos a velocidade de teste diminui. A semelhanc¸a de Reynolds e´ mais dif´ıcil de
obter pois modelos mais pequenos exigem superior velocidade de teste para igual viscosidade
cinema´tica.
os navios de superf´ıcie esta˜o sujeitos a forc¸as grav´ıticas e de atrito. Assim, nos testes de
modelos a` escala reduzida ambas as leis, de Froude e de Reynolds, deveriam ser satisfeitas;
Res
Rem
=
νm
νs
·
√
L3s
L3m
=
νm
νs
·λ1.5 = 1 (2.38)
No entanto, na˜o existem, ou pelo menos na˜o sa˜o economicamente via´veis, fluidos que permitam
satisfazer esta condic¸a˜o. Para diminuir os erros de extrapolac¸a˜o dos efeitos viscosos, a a´gua em
que sa˜o realizados os testes pode ser aquecida para aumentar a diferenc¸a entre as viscosidades.
2.3 Decomposic¸a˜o da resisteˆncia
A resisteˆncia do navio tem origem complexa e, para facilidade de ana´lise, e´ tradicionalmente
decomposta em va´rios termos. No entanto, na˜o existe uniformidade nos diversos textos quanto
a` forma como realizar aquela decomposic¸a˜o. Uma das abordagens a este assunto consiste
em considerar as decomposic¸o˜es constantes na Fig. 2.1. De acordo com a figura, podemos
considerar a seguinte decomposic¸a˜o da resisteˆncia total:
2.3. DECOMPOSIC¸A˜O DA RESISTEˆNCIA 19
- a resisteˆncia de onda;
- a resisteˆncia de atrito;
- a resisteˆncia viscosa de pressa˜o.
Figura 2.1: Decomposic¸a˜o da resisteˆncia.
Para ale´m dos termos relativos a uma querena simples em a´guas tranquilas, outras com-
ponentes adicionais da resisteˆncia devera˜o ser consideradas:
- a resisteˆncia aerodinaˆmica, resisteˆncia ao avanc¸o no ar da parte emersa do casco e
superestruturas do navio;
- a resisteˆncia adicional em mar ondoso, resisteˆncia resultante da acc¸a˜o de ondas inciden-
tes sobre a estrutura do navio;
- a resisteˆncia adicional devida aos apeˆndices da querena.
2.3.1 Resisteˆncia de onda
Quando o navio avanc¸a na superf´ıcie tranquila do mar e´ rodeado e seguido por uma formac¸a˜o
ondosa. Esta formac¸a˜o e´ quase impercept´ıvel a baixa velocidade. No entanto, a partir de
uma dada velocidade torna-se claramente vis´ıvel e, a partir da´ı, tem dimensa˜o crescente
com a velocidade. Para ale´m da dependeˆncia com a velocidade, a formac¸a˜o ondosa depende
tambe´m da forma da querena.
20 CAPI´TULO 2. RESISTEˆNCIA
Nos estudos de resisteˆncia de onda na˜o se pode afirmar que uma dada velocidade e´ elevada
ou baixa sem conhecermos tambe´m a dimensa˜o do navio. Assim, surge frequentemente a
refereˆncia ao conceito de velocidade relativa, como raza˜o entre a velocidade do navio e um
paraˆmetro representativo da dimensa˜o do navio,
vrel =
V√
L
(2.39)
com V em no´s e L em pe´s, em substituic¸a˜o do adimensionalnu´mero de Froude.
Numa perspectiva do estudo hidrodinaˆmico do escoamanto, o navio pode ser considerado
como um campo de pressa˜o em movimento. Kelvin resolveu analiticamente o caso simplificado
do sistema de ondas criado pelo movimento de um ponto de pressa˜o. Demonstrou que o padra˜o
da formac¸a˜o ondosa inclui um sistemas de ondas divergentes e um outro sistema cujas cristas
das ondas se apresentam normais a` direcc¸a˜o do movimento, como representado na Fig. 2.2.
Ambos os sistemas de ondas viajam a` velocidade do ponto de pressa˜o.
Figura 2.2: Sistema de ondas gerado por um ponto de pressa˜o em movi-
mento.
O sistema de ondas associado ao movimento de um navio e´ bastante mais complicado.
No entanto, como primeira aproximac¸a˜o, o navio pode ser considerado com um campo de
pressa˜o em movimento composto por uma sobrepressa˜o considerada pontual na proa e uma
depressa˜o, tambe´m pontual, na popa. Assim, num navio que se desloque a uma velocidade
relativa elevada, a formac¸a˜o ondosa provocada e´ constitu´ıda por dois sistemas principais de
ondas, Fig. 2.3:
- o sistema da proa;
- o sistema da popa.
Cada um dos sistemas de ondas formados, com origem na proa e na popa do navio, e´
constitu´ıdo por dois tipos de ondas:
- as ondas transversais;
- as ondas divergentes.
Geralmente, os dois sistemas de ondas divergentes sa˜o detecta´veis apesar de o sistema da
popa ser muito mais fraco. Na˜o e´ normalmente poss´ıvel isolar o sistema transversal da popa,
sendo apenas vis´ıvel a re´ do navio a composic¸a˜o dos dois sistemas, transversal e divergente.
2.3. DECOMPOSIC¸A˜O DA RESISTEˆNCIA 21
Figura 2.3: Sistemas de ondas da proa e da popa.
A proa produz um sistema de ondas semelhante ao descrito por Kelvin para um ponto de
pressa˜o em movimento e, pelo contra´rio, na popa forma-se um sistema de ondas semelhante,
mas com uma depressa˜o localizada na popa. Conforme representado na Fig. 2.3, se a linha
que une os pontos de maior elevac¸a˜o das cristas das ondas divergentes fizer com a direcc¸a˜o
longitudinal do navio um aˆngulo α, enta˜o a direcc¸a˜o destas fara´ um aˆngulo 2α com a mesma
direcc¸a˜o.
O comprimento de onda de ambos os sistemas transversais e´ igual e dado por:
λ =
2piV 2
g
(2.40)
Existe uma interacc¸a˜o entre as formac¸o˜es ondosa transversais dos sistemas de ondas da
proa e da popa. Se os sistemas estiverem “em fase”, de tal forma que as cristas das ondas
coincidam, o sistema resultante tera´ maior altura e, consequentemente, maior energia. Se,
pelo contra´rio, a cava de um dos sistemas de ondas ficar sobreposta com uma crista do outro
sistema, a energia consumida para gerar o sistema de ondas sera´ reduzida. A velocidade V
e o comprimento do navio L sa˜o muito importantes para a determinac¸a˜o da energia total do
sistema de ondas gerado e, consequentemente, para a resisteˆncia de onda do navio.
Continuando a assumir o modelo f´ısico que aproxima o movimento do navio por um
campo de pressa˜o em movimento, a distaˆncia entre os dois pontos de pressa˜o, proa e popa,
e´ aproximada por 0, 9L. Sabendo que uma onda grav´ıtica com comprimento de onda λ se
desloca em a´guas profundas a` velocidade
C =
√
λg
2pi
(2.41)
para que haja coincideˆncia de uma crista ou cava do sistema da proa com a primeira cava
gerada na popa, devera´ verificar-se
V 2
0, 9L
=
g
Npi
(2.42)
Tomando em considerac¸a˜o a Fig. 2.4, verifica-se que as cavas va˜o coincidir para N =
1, 3, 5, ... enquanto que para N par as cristas do sistema da proa coincidem com as cavas do
sistema da popa. Se na˜o existisse esta interacc¸a˜o entre os dois sistemas de ondas a resisteˆncia
de onda apresentaria uma evoluc¸a˜o “bem comportada” crescente com a velocidade do navio,
22 CAPI´TULO 2. RESISTEˆNCIA
Figura 2.4: Interacc¸a˜o entre os dois sistemas de ondas.
conforme representado pela linha tracejada da Fig. 2.5. Na realidade, a partir de uma dada
velocidade a partir da qual esta interacc¸a˜o se torna significativa, verifica-se a existeˆncia de
elevac¸o˜es e depresso˜es na curva correspondendo aos casos extremos de interacc¸a˜o entre os
sistemas de ondas. E´ de esperar que a maior elevac¸a˜o se verifique para N = 1 porque a
velocidade e´ mais elevada para esta condic¸a˜o.
Como a curva de resisteˆncia de onda exibe estes ma´ximos e mı´nimos locais, o navio deve
ser projectado para operar num mı´nimo local da curva de resisteˆncia de onda, a velocidade
econo´mica.
Quando o comprimento de onda das ondas transversais e´ igual ao comprimento do na-
vio, o nu´mero de Froude e´ aproximadamente 0, 4. Ate´ este valor do nu´mero de Froude, as
ondas transversais sa˜o as principais responsa´veis pelas elevac¸o˜es e depresso˜es na curva da
resisteˆncia de onda. Se o nu´mero de Froude aumentar, aumentara´ tambe´m a resisteˆncia de
onda sobretudo a` custa da influeˆncia das ondas divergentes. O ma´ximo da resisteˆncia de
onda verifica-se para Fr ≈ 0, 5. A velocidade correspondente designa-se por “velocidade da
querena”. Acima da “velocidade da querena” a resisteˆncia de onda do navio decresce. Navios
ra´pidos que operem acima da velocidade de querena devera˜o naturalmente dispor de poteˆncia
instalada suficiente para vencer aquele pico de resisteˆncia.
Bolbo de proa
A finalidade da instalac¸a˜o dos bolbos de proa e´ a reduc¸a˜o da resisteˆncia de onda. O
mecanismo de reduc¸a˜o consiste na interfereˆncia dos sistemas de onda. O sistema de ondas
gerado pela pressa˜o elevada no bolbo interfere com o sistema de ondas da proa, reduzindo a
sua amplitude. A interfereˆncia favora´vel ocorre quando a cava do sistema transversal de ondas
2.3. DECOMPOSIC¸A˜O DA RESISTEˆNCIA 23
Figura 2.5: Curva da resisteˆncia de onda.
do bolbo surgir na crista do sistema de ondas da proa do navio. Esta situac¸a˜o de interfereˆncia
favora´vel sendo optimizada para uma dada velocidade, pode no entanto ser considerada como
tendo efeito favora´vel num determinado intervalo de velocidades.
Efeito da profundidade restrita
Os efeitos da profundidade finita comec¸am a fazer-se sentir quando a profundidade h e´
menor que metade do comprimento de onda da formac¸a˜o ondosa gerada pelo movimento do
navio, h < λ/2. Doutra forma, podemos considerar profundidade infinita sempre que,
h >
λ
2
(2.43)
No caso de profundidades muito pequenas, h < 0, 05λ∞, a velocidade de propagac¸a˜o deixa
de depender do comprimento de onda, Eq. (2.41) e passa a depender apenas da profundidade
C =
√
gh (2.44)
Neste caso, a velocidade de grupo e´ igual a` velocidade de propagac¸a˜o, a velocidade cr´ıtica:
Cg = C =
√
gh (2.45)
Para caracterizar o efeito da profundidade e´ usado o nu´mero de Froude baseado na pro-
fundidade h:
24 CAPI´TULO 2. RESISTEˆNCIA
- se V/
√
gh < 0, 4, o padra˜o de ondas e´ semelhante ao caso de profundidade infinita;
- se V/
√
gh se aproximar de 1, o aˆngulo da envolvente aproxima-se de 90◦;
- se V/
√
gh > 1, sinα =
√
gh/V .
2.3.2 Resisteˆncia de atrito
A resisteˆncia de atrito do navio resulta do escoamento em torno da querena com nu´mero de
Reynolds elevado. Quando um corpo se move num fluido em repouso, uma fina camada de
fluido adere ao corpo em movimento, ou seja, tem velocidade nula relativamente ao corpo.
A variac¸a˜o de velocidade e´ elevada nas proximidades da superf´ıcie do corpo e diminui com
o aumento da distaˆncia ao mesmo. E´ pra´tica habitual convencionar-se para a definic¸a˜o da
espessura da camada limite, a distaˆncia a partir da superf´ıcie do corpo ate´ que a velocidade
do fluido seja 1% da velocidade do corpo.
Desenvolve-se assim da proa para a popa do navio uma camada limite tridimensional. Esta
camada limite inicia-se em escoamento laminar e sofre transic¸a˜o para o regime turbulento.
Normalmente, esta transic¸a˜o ocorre junto a` proa do navio. Esta transic¸a˜o e´ controlada pelo
nu´mero de Reynolds do escoamento. Considerandoo caso da placa lisa plana, a transic¸a˜o
ocorre para valores entre Re = 3×105 e Re = 106. Em regime turbulento os efeitos dissipativos
de energia va˜o ale´m do atrito molecular. Com crescente nu´mero de Reynolds, verificam-se
intensas trocas de quantidade de movimento em camadas adjacentes do fluido, ou seja, maior
transporte de energia.
No caso de uma placa plana, a espessura da camada limite turbulenta pode ser aproximada
por:
δx
L
= 0, 37 (ReL)
−1/5 (2.46)
Num navio, o gradiente lontitudinal de pressa˜o na regia˜o da proa e´, em geral, favora´vel
ao escoamento. Pelo contra´rio, este gradiente e´ adverso na regia˜o da popa e a camada limite
aumenta significativamente de espessura deixando de poder ser considerada pequena quando
comparada com o comprimento ou a boca do navio. Para todos os efeitos pra´ticos, a camada
limite de um navio pode ser considerada completamente turbulenta.
A dependeˆncia da resisteˆncia de atrito com o nu´mero de Reynolds e com a rugosidade da
superf´ıcie e´ indicada pelo gra´fico da Fig. 2.6. Para uma superf´ıcie rugosa, a resisteˆncia segue
a linha da superf´ıcie lisa ate´ que, para um dado valor de Re, se separa e tem a partir da´ı
um andamento quase horizontal, ou seja, o coeficiente torna-se independente do Re. Quanto
mais rugosa for a superf´ıcie mais cedo se evidencia este comportamento.
A resisteˆncia de atrito de um navio e´ habitualmente dividida em duas componentes:
- a resisteˆncia a que ficaria sujeita uma placa plana com a´rea equivalente;
- o aumento de resisteˆncia originado pela forma do navio.
A resisteˆncia de atrito foi estimada durante de´cadas por expresso˜es emp´ıricas como, por
exemplo, a fo´rmula de Froude:
RF = 1− 0, 0043 (θ − 15) fSV 1,825 (2.47)
2.3. DECOMPOSIC¸A˜O DA RESISTEˆNCIA 25
Figura 2.6: Variac¸a˜o do coeficiente da resisteˆncia de atrito com o nu´mero
de Reynolds e com a rugosidade da superf´ıcie.
em que θ e´ a temperatura do fluido, expressa em ◦C e
f = 0, 1392 +
0, 258
2, 68 + L
(2.48)
Outra fo´rmula emp´ırica muito popular para a previsa˜o do coeficiente da resisteˆncia de atrito e´
devida a Schoenherr e conhecida como fo´rmula da ATTC (American Towing Tank Conference)
0, 242√
cF
= log (Re · cF ) (2.49)
Esta correlac¸a˜o preveˆ coeficientes de atrito excessivos quando aplicada a modelos muito
pequenos. Para ultrapassar este problema foi proposta na ITTC (International Towing Tank
Conference) de 1957 uma nova fo´rmula,
cF =
0, 075
(logRe− 2)2 (2.50)
designada por linha de correlac¸a˜o modelo-navio da ITTC 1957.
2.3.3 Resisteˆncia viscosa de pressa˜o
A componente da pressa˜o originada pelas ondas formadas pelo movimento do navio ja´ foi
considerada. Resta agora considerar a resisteˆncia originada por diferenc¸as de pressa˜o a actuar
no casco devida a efeitos viscosos do escoamento. Num escoamento ideal, ver Fig. 2.7, a
pressa˜o exercida na popa do navio seria igual a` exercida na proa, ou seja forc¸a resultante
nula. Na pra´tica, os efeitos viscosos va˜o reduzir a pressa˜o exercida na popa do navio.
Parte desta resisteˆncia sera´ devida a` gerac¸a˜o de vo´rtices nas descontinuidades do casco.
Outra parte sera´ devida a um aumento de espessura da camada limite nalguns casos po-
tenciada por feno´menos de separac¸a˜o do escoamento. Estes aspectos sa˜o fundamentalmente
condicionados pela forma do casco pelo que sa˜o normalmente considerados como uma “resis-
teˆncia de forma”.
26 CAPI´TULO 2. RESISTEˆNCIA
Figura 2.7: Distribuic¸a˜o de pressa˜o num escoamento ideal, inv´ıscido.
2.4 Ensaios de resisteˆncia em tanques de reboque
Apesar da crescente importaˆncia dos me´todos nume´ricos, os ensaios com modelos a` escala
reduzida de navios em tanques de reboque sa˜o ainda essenciais para a avaliac¸a˜o hidrodinaˆmica
dos novos projectos e para a validac¸a˜o de novas soluc¸o˜es.
Os testes devem ser realizados em condic¸o˜es que permitam considerar que o modelo e o
navio teˆm comportamentos semelhantes por forma a que os resultados obtidos para o modelo
possam ser extrapolados para a escala real do navio. Com este objectivo, os ensaios realizam-
se respeitando a igualdade do nu´mero de Froude.
Os testes sa˜o realizados em tanques de reboque, com a´gua imo´vel e o modelo rebocado por
um “carrinho” ou, em alternativa, os testes podem ser realizados em “tanques de circulac¸a˜o”,
em que o modelo esta´ imo´vel e a a´gua circula.
No primeiro caso, apo´s um percurso inicial de acelerac¸a˜o, a velocidade do “carrinho” deve
ser mantida constante para obter um regime estaciona´rio e garantir o rigor das observac¸o˜es
efectuadas. A fase final e´ de desacelerac¸a˜o e imobilizac¸a˜o do modelo. Assim, os tanques de
reboque apresentam frequentemente centenas de metros de extensa˜o.
O comprimento do modelo, como o exemplo representado esquematicamente na Fig. 2.8,
e´ escolhido de acordo com as condic¸o˜es experimentais no tanque de reboque. O modelo deve
ser ta˜o grande quanto poss´ıvel por forma a minimizar efeitos de escala relativos aos aspectos
viscosos, nomeadamente as diferenc¸as relativas a escoamentos laminares e turbulentos e as
questo˜es relacionadas com feno´menos de separac¸a˜o do escoamento. Por outro lado, a dimensa˜o
do modelo deve ainda permitir evitar deformac¸o˜es resultantes de esforc¸os no modelo e no
equipamento de teste.
A dimensa˜o do modelo deve ser suficientemente pequena para permitir que o “carrinho”
de reboque do modelo atinja a velocidade correspondente e evitar os efeitos de a´guas res-
tritas nos testes efectuados. Estes constrangimentos conduzem naturalmente a um intervalo
pra´tico de comprimentos admiss´ıveis. Os modelos para ensaios de resisteˆncia e propulsa˜o
teˆm normalmente comprimentos entre 4 m < Lm < 10 m. A escala dos modelos esta´ entre
15 < λ < 45.
2.5. CA´LCULO DA RESISTEˆNCIA 27
Figura 2.8: Modelo a` escala reduzida para ensaios de resisteˆncia.
Durante o movimento, o modelo mante´m o rumo atrave´s de fios-guia, sendo livre para
adoptar o caimento que resultar do seu movimento. Ainda de acordo com a Fig. 2.8, a
resisteˆncia total de reboque do modelo e´ dada por,
RT = G1 + sinαG2 (2.51)
Com os ensaios de resisteˆncia com o modelo a` escala reduzida pretende-se obter dados
que permitam estimar a resisteˆncia do navio sem o propulsor e apeˆndices, ou seja, dita da
querena simples. Dos ensaios no tanque de reboque obte´m-se a resisteˆncia nas condic¸o˜es do
tanque, ou seja:
- a´guas suficientemente profundas;
- auseˆncia de correntes;
- auseˆncia de vento;
- a´gua doce a` temperatura ambiente.
O nu´mero de Reynolds e´ normalmente superior duas ordens de grandeza na escala do navio
que na escala do modelo, tipicamente na ordem de 109 e 107, respectivamente. O modelo tem
frequentemente uma fita rugosa para estimular artificialmente a transic¸a˜o da camada limite
laminar para turbulenta mais perto da proa do modelo. Globalmente, o desvio originado
pelo facto de na˜o se manter constante o nu´mero de Reynolds no ensaio e´ depois compensado
atrave´s de correcc¸o˜es emp´ıricas.
2.5 Ca´lculo da resisteˆncia
2.5.1 Me´todos de extrapolac¸a˜o
A resisteˆncia do modelo tem depois de ser convertida por forma a obter-se uma estimativa
da resisteˆncia do navio na escala real. Para tal, esta˜o dispon´ıveis, entre outros, os seguintes
me´todos:
28 CAPI´TULO 2. RESISTEˆNCIA
- o me´todo ITTC 1957;
- o me´todo de Hughes/Prohaska;
- o me´todo ITTC 1978;
- o me´todo Geosim de Telfer.
Actualmente, o me´todo mais frequentemente utilizado na pra´tica e´ o me´todo ITTC 1978.
Me´todo ITTC 1957
Para a aplicac¸a˜o deste me´todo, a resisteˆncia total da querena, RT , e´ considerada decomposta
nos seguintes termos,
RT = RF +RR (2.52)
a resisteˆncia de atrito, RF , e a resisteˆncia residual, RR.
Os coeficientes de resisteˆncia, adimensionais, sa˜o genericamente calculados por,
ci =
Ri
1
2ρV2S
(2.53)
Na aplicac¸a˜o deste me´todo de previsa˜o e´ considerado igual para o modelo e para o navio
o coeficiente de resisteˆncia residual,
cR = cTm − cFm (2.54)
determinado a partir do coeficiente de resisteˆncia total do modelo,
cTm =
RTm
1
2ρmV
2
mSm
(2.55)
e da fo´rmula “ITTC 1957” (Eq. (2.50)) para o ca´lculo do coeficiente de resisteˆncia de atrito
cF ,
cF =
0.075
(log10Re− 2)2
O coeficiente de resisteˆncia total para o navio e´ enta˜o estimado por:
cTs = cFs + cR + cA = cFs + (cTm − cFm) + cA (2.56)
em que cA e´ um factor de correcc¸a˜o tradicionalmente associado a` rugosidade do casco. De
facto, embora o modelo esteja constru´ıdo a uma dada escala geome´trica, a rugosidade das
superf´ıcies do modelo e do navio na˜o respeitam esta escala. O valor de cA pode ser obtido
por correlac¸o˜es emp´ıricas como, por exemplo,
cA = 0.35× 10−3 − 2× Lpp × 10−6 (2.57)
ou a partir de valores tabelados (Tab. 2.1).
A previsa˜o da resisteˆncia total do navio e´ dada por
RTs = cTs · 1
2
ρsV
2
s Ss (2.58)
2.5. CA´LCULO DA RESISTEˆNCIA 29
Lpp(m) cA
50 - 150 0,0004-0,00035
150 - 210 0,0002
210 - 260 0,0001
260 - 300 0
300 - 350 -0,0001
350 - 400 0,00025
Tabela 2.1: Valores do coeficiente de correcc¸a˜o cA em func¸a˜o do compri-
mento do navio.
Me´todo de Hughes-Prohaska
O me´todo de Hughes-Prohaska e´ normalmente classificado como um me´todo de factor de
forma. E´ considerada a decomposic¸a˜o da resisteˆncia total em duas componentes, uma asso-
ciada a` resisteˆncia de onda e outra dependente da forma do casco. Considerando enta˜o os
coeficientes adimensionais, fica
cT = (1 + k) · cF0 + cw (2.59)
Para a determinac¸a˜o do factor de forma, presume-se aqui a relac¸a˜o
cT
cF0
= (1 + k) + α
Fr4
cF0
(2.60)
que e´ particularmente va´lida para valores reduzidos de velocidade.
Apo´s va´rios ensaios a diferentes velocidades, diferentes nu´meros de Froude, e´ poss´ıvel
construir um gra´fico semelhante ao representado na Fig. 2.9 e, com base naqueles valores,
obter o valor de k por regressa˜o linear.
Figura 2.9: Representac¸a˜o gra´fica da dependeˆncia de
cT
cF0
com
Fr4
cF0
.
Este factor de forma, (1 + k),e´ assumido como independente dos valores de Fr e de Re e
igual para o navio e modelo.
O procedimento de ca´lculo do me´todo de Hughes-Prohaska e´ o seguinte:
30 CAPI´TULO 2. RESISTEˆNCIA
- determinar o coeficiente de resisteˆncia total,
cTm =
RTm
1
2
ρmV
2
mSm
- determinar o coeficiente de resisteˆncia de onda, o mesmo para o modelo e o navio,
cw = cTm − cF0m · (1 + k) (2.61)
- determinar o coeficiente de resisteˆncia total para o navio,
cTs = cw + cF0s · (1 + k) + cA (2.62)
- determinar a resisteˆncia total para o navio, novamente por
RTs = cTs · 1
2
ρsV
2
s Ss
O coeficiente da resisteˆncia de atrito, cF0, e´ neste caso obtido pela correlac¸a˜o de Hughes,
cF0 =
0.067
(log10Re− 2)2
(2.63)
Quanto ao coeficiente de correcc¸a˜o cA, a ITTC recomenda a aplicac¸a˜o universal de
cA = 0.0004 (2.64)
na aplicac¸a˜o deste me´todo.
Me´todo ITTC 1978
E´ uma modificac¸a˜o do me´todo de Hughes-Prohaska, geralmente mais preciso que os ante-
riormente apresentados. Ao contra´rio dos me´todos anteriormente descritos, este me´todo de
extrapolac¸a˜o dos resultados obtidos nos ensaios com modelos a` escala reduzida inclui o efeito
da resisteˆncia do ar.
A previsa˜o do coeficiente de resisteˆncia total para o navio e´, tambe´m aqui, descrita em
termos do factor de forma, ou seja,
cTs = (1 + k) cFs + cw + cA + cAA (2.65)
em que:
- cw e´ o coeficiente de resisteˆncia de onda, igual para o navio e modelo;
- cA e´ o coeficiente de correcc¸a˜o;
- e cAA a resisteˆncia do ar, cAA = 0.001 · AT
S
.
2.5. CA´LCULO DA RESISTEˆNCIA 31
O coeficiente da resisteˆncia de atrito e´ determinada de forma semelhante a` preconizada
para o me´todo ITTC 57, Eq. (2.50).
Para a determinac¸a˜o da correcc¸a˜o devida pela variac¸a˜o da rugosidade da querena, e´ acon-
selhada aqui a seguinte fo´rmula:
cA · 103 = 105 3
√
ks
Loss
− 0.64 (2.66)
em que ks e´ a rugosidade do casco e Loss e´ o comprimento do navio no plano de flutuac¸a˜o.
Para navios novos ks/Loss = 10
−6 e cA = 0.00041.
Os detalhes sugeridos pela ITTC na aplicac¸a˜o deste me´todo esta˜o indicados no Apeˆndice
A.
Me´todo Geosim
Este me´todo foi proposto por Telfer em 1927. Dos me´todos aqui enunciados, e´ considerado
como o me´todo de extrapolac¸a˜o com previso˜es mais precisas da resisteˆncia do navio. A
grande vantagem do me´todo resulta de na˜o recorrer a qualquer decomposic¸a˜o, teoricamente
questiona´vel, da resisteˆncia total.
Sa˜o realizados va´rios ensaios com modelos geometricamente semelhantes mas a diferentes
escalas. Isto significa que os testes podem ser realizados, para a mesma velocidade equivalente,
com igual nu´mero de Froude e diferente nu´mero de Reynolds. O coeficiente de resisteˆncia total,
obtido naqueles ensaios, e´ representado em func¸a˜o de logRe−1/3. Para cada um dos modelos,
obte´m-se uma curva da resisteˆncia, em func¸a˜o do Fr, que permite fazer a extrapolac¸a˜o para
a escala do navio.
Pela grande quantidade de modelos a construir e ensaios a realizar, trata-se de um me´todo
muito dispendioso, utilizado sobretudo apenas para fins de investigac¸a˜o.
2.5.2 Resisteˆncias adicionais
As condic¸o˜es de ensaio dos modelos sa˜o substancialmente diferentes daquelas em que o navio
ira´ operar. As principais diferenc¸as a considerar resultam de:
- a presenc¸a de apeˆndices na querena;
- a navegac¸a˜o em a´guas pouco profundas;
- o vento;
- a crescente rugosidade do casco durante a vida do navio;
- as condic¸o˜es de mar.
Para estimar as alterac¸o˜es causadas por estes itens no comportamento do navio, usam-se
correcc¸o˜es emp´ıricas, baseadas em pressupostos f´ısicos, para correlacionar os valores obtidos
no modelo, ou no navio em provas de mar, com os estimados para as condic¸o˜es normais de
servic¸o do navio. A resisteˆncia adicional devida a apeˆndices e a resisteˆncia do navio em a´guas
pouco profundas sa˜o os to´picos sucintamente abordados nos para´grafos seguintes.
32 CAPI´TULO 2. RESISTEˆNCIA
Resisteˆncia adicional dos apeˆndices
Os modelos de navios a` escala reduzida podem ser testados com apeˆndices a` escala geome´-
trica apropriada. No entanto, nem sempre nesta altura do projecto estes esta˜o completamente
definidos. Por outro lado, o escoamento em torno dos apeˆndices e´ predominantemente go-
vernado pelas forc¸as de origem viscosa. Seria enta˜o necessa´rio, para obter resultados fia´veis,
verificarem-se condic¸o˜es de semelhanc¸a de Reynolds, o que, como ja´ referido, na˜o e´ via´vel
se, cumulativamente, pretendermos manter a igualdade do nu´mero de Froude. Consequente-
mente, a presenc¸a dos apeˆndices em condic¸o˜es de semelhanc¸a de Froude tem pouca relevaˆncia.
Em primeira ana´lise, os apeˆndices do casco contribuem para um aumento da superf´ıcie
molhada do navio. Por outro lado, da sua presenc¸a surgem tambe´m alterac¸o˜es no factor de
forma do casco. Para a determinac¸a˜o da resisteˆncia de forma dos apeˆndices pode recorrer-se a
dois ensaios, com e sem apeˆndices, a uma velocidade superior. Se admitirmos que a resisteˆncia
de onda e´ igual nos dois casos, a diferenc¸a de resisteˆncia verificada, tendo descontado a
diferenc¸a de resisteˆncia de atrito resultante da variac¸a˜o da a´rea molhada, da´-nos a resisteˆncia
de forma dos apeˆndices.
Os valores t´ıpicos de acre´scimo de resisteˆncia originados pela presenc¸a de apeˆndices sa˜o
os seguintes:
- robaletes: 1 a 2%;
- impulsores:
- de proa: 0 a 1%;
- transversais de popa: 1 a 6%;
- aranhas de veios: 5 a 12% (“twin-screw” pode chegar a 20%);
- leme: 1%.
Resisteˆncia em a´guas pouco profundas
Quando um navio navega em a´guas pouco profundas verifica-seum aumento, quer da resis-
teˆncia de atrito, quer da resisteˆncia de onda. Em particular, a resisteˆncia aumenta signifi-
cativamente para valores pro´ximos do nu´mero de Froude cr´ıtico, baseado na profundidade,
Fnh = V/
√
gH = 1.
O aumento da resisteˆncia do navio quando a navegar em a´guas pouco profundas foi es-
tudado por Schlichting. A sua hipo´tese de trabalho foi a seguinte: a resisteˆncia de onda e´ a
mesma se o comprimento de onda da ondulac¸a˜o transversal for igual.
O gra´fico da Fig. 2.10 permite prever a perda de velocidade do navio em a´guas pouco
profundas. Correcc¸o˜es simples na˜o sa˜o poss´ıveis para a´guas muito pouco profundas ja´ que os
feno´menos envolvidos sa˜o complexos. Nestes casos, so´ testes em modelos ou simulac¸o˜es por
CFD podera˜o contribuir para uma melhor previsa˜o.
2.6 Previsa˜o da resisteˆncia com dados sistema´ticos ou estat´ıs-
ticos
Na fase preliminar do projecto de um navio podem ser utilizados me´todos aproximados para a
previsa˜o da resisteˆncia baseados em ensaios de se´ries sistema´ticas de navios ou, pela regressa˜o
2.6. PREVISA˜O COM DADOS SISTEMA´TICOS OU ESTATI´STICOS 33
Figura 2.10: Reduc¸a˜o de velocidade (%) em a´guas pouco profundas.
estat´ıstica de dados experimentais relativos a modelos e a navios a` escala real.
Se´ries sistema´ticas sa˜o conjuntos de formas de querena em que se provocou a variac¸a˜o,
sistema´tica, de um ou mais dos seus paraˆmetros de forma. As variac¸o˜es sistema´ticas sa˜o
feitas em torno de uma “forma ma˜e” (“parent form”). Os resultados dos ensaios de resisteˆncia
dos modelos que constituem a se´rie permitem determinar um coeficiente adimensional de
resisteˆncia para uma forma de querena contida ou interpolada na se´rie.
Taylor mediu, entre 1907 e 1914, 80 modelos obtidos por variac¸a˜o sistema´tica de:
- a raza˜o entre o comprimento e a raiz cu´bica do deslocamento (5 valores de L/∆1/3);
- a raza˜o entre a boca e o calado (B/T = 2, 25; 3, 75);
- o coeficiente prisma´tico (8 valores de 0,48 a 0,86);
a partir de uma “forma ma˜e”: o cruzador “Leviathan”.
Estes dados foram posteriormente re-trabalhados por Gertler em 1954, disponibilizando
diagramas de resisteˆncia residual.
Outra se´rie sistema´tica, com particular interesse para os navios mercantes, e´ a se´rie 60,
devida aos trabalhos de Todd. Consta de 5 “formas ma˜e” com coeficientes de finura, 0,60,
0,65, 0,70, 0,75 e 0,80. Para cada uma daquelas “formas ma˜e” existem variac¸o˜es de L/B,
B/T , etc.
Como exemplo de um me´todo de previsa˜o da resisteˆncia de navios envolvendo dados
estat´ısticos pode-se indicar o me´todo de Holtrop e Mennen. Este me´todo pode ser aplicado
para efectuar uma ana´lise qualitativa do projecto de um navio no que diz respeito a` sua
resisteˆncia. O me´todo baseia-se na regressa˜o estat´ıstica de resultados de ensaios em modelos
e de resultados de provas de mar de navios. A base de dados e´ muito vasta cobrindo uma
gama muito alargada de tipos de navios. No entanto, para formas muito espec´ıficas de navio,
34 CAPI´TULO 2. RESISTEˆNCIA
a precisa˜o das previso˜es pode reduzir-se pelo menor nu´mero de elementos daquele tipo na
base.
2.7 Ensaios a` escala real
Os resultados obtidos nas provas de mar de um navio sa˜o talvez o mais importante requisito
para a aceitac¸a˜o deste pelo armador. A especificac¸a˜o detalhada destas provas deve estar
claramente contratualizada entre o armador e o estaleiro. Entre outros organismos, a ITTC
recomenda alguns procedimentos para a realizac¸a˜o destas provas. As recomendac¸o˜es para as
provas de velocidade e de poteˆncia esta˜o inclu´ıdas no Apeˆndice B.
Os problemas surgem normalmente em consequeˆncia de as provas se realizarem em condi-
c¸o˜es diferentes, quer das que foram consideradas como condic¸o˜es de projecto, quer daquelas
que se verificaram nos ensaios com o modelo a` escala reduzida.
O contrato de construc¸a˜o deve especificar uma velocidade contratual do navio, a` carga de
projecto, para uma dada percentagem da MCR do motor, em a´guas tranquilas e profundas e
na auseˆncia de vento. Sa˜o raras as ocasio˜es em que e´ poss´ıvel realizar as provas de mar em
condic¸o˜es pro´ximas das condic¸o˜es contratuais. As condic¸o˜es em que se realizam as provas de
mar incluem, frequentemente:
- condic¸a˜o de carga parcial ou em condic¸a˜o de lastro;
- presenc¸a de correntes e ondulac¸a˜o;
- a´guas pouco profundas;
Para prevenir maior diversidade de resultados, e´ habitual definir contratualmente valores
limite para as condic¸o˜es ambientais em que as provas de mar se realizara˜o. As condic¸o˜es
recomendadas pela ITTC para a realizac¸a˜o das provas de velocidade e poteˆncia esta˜o no
Apeˆndice C. As diferenc¸as entre as condic¸o˜es contratuais e verificadas durante a realizac¸a˜o
das provas de mar impo˜em a utilizac¸a˜o de correlac¸o˜es para corrigir os resultados obtidos para
as condic¸o˜es de contrato. Para ale´m de todas as incertezas experimentais, todo este processo
de correcc¸a˜o, com recurso a gra´ficos e tabelas, oferece muitas du´vidas de aplicac¸a˜o.
A “prova da milha” pode ser avaliada com velocidade “over ground” ou velocidade “in
water”. A velocidade na a´gua exclui o efeito das correntes. A velocidade “over ground”
era avaliada atrave´s de equipamentos de navegac¸a˜o mas, a disponibilidade de sistemas de
posicionamento por sate´lite (GPS) permitiu eliminar muitos problemas e incertezas destas
provas. Para reduzir os efeitos de ventos e correntes, as provas de velocidade, consumo, etc.
devem ser realizadas repetidamente em sentidos opostos.
De notar que as provas de mar de um navio va˜o muito para ale´m das provas de veloci-
dade e poteˆncia. Todas as funcionalidades do navio, operacionais e de seguranc¸a, devera˜o ser
demonstradas. Para as restantes provas, nomeadamente as que dizem respeito a` manobrabi-
lidade do navio, existem tambe´m recomendac¸o˜es exaustivas da ITTC para a sua realizac¸a˜o.
Cap´ıtulo 3
Propulsa˜o
3.1 Sistemas de propulsa˜o
Em qualquer tipo de navio temos presente um propulsor cuja finalidade e´ a gerac¸a˜o de uma
forc¸a propulsiva. As soluc¸o˜es propulsivas sa˜o muito diversas mas predominantemente os navios
continuam a utilizar he´lices simples como meio de propulsa˜o. Outros meios de propulsa˜o com
expressa˜o significativa em aplicac¸o˜es espec´ıficas sa˜o:
- os he´lices “especiais”, com particular destaque para os he´lices com tubeira e os he´lices
contra-rotativos;
- os sistemas de jacto de a´gua (“water-jets” ou “pump-jets”);
- os propulsores azimutais (“AziPod’s)”;
- e os propulsores cicloidais (“Voith-Schneider”).
Na escolha da soluc¸a˜o propulsiva devera´ ser sempre considerado o seu rendimento e a
interacc¸a˜o com a querena. Outro aspecto gene´rico a considerar durante o projecto da soluc¸a˜o
propulsiva e´ o feno´meno da cavitac¸a˜o originada pela velocidade elevada do movimento das
pa´s do he´lice na a´gua.
3.1.1 He´lices
O he´lice e´ colocado tradicionalmente a` popa do navio para recuperar parte da energia dis-
pendida para vencer a resisteˆncia da querena. Na forma mais tradicional da popa dos navios,
a esteira nominal e´ muito na˜o-uniforme. A uniformidade da esteira da querena e´ uma das
condic¸o˜es necessa´rias para o bom funcionamento do he´lice. A utilizac¸a˜o da popa aberta ou
de um bolbo na popa permite melhorar a esteira.
As pa´s do he´lice, animadas de velocidade de rotac¸a˜o e de avanc¸o, funcionando como
superf´ıcies sustentadoras, esta˜o distribu´ıdas simetricamente em torno do cubo. As secc¸o˜es
das pa´s funcionam como perfis alares a aˆngulo de ataque gerando uma forc¸a de sustentac¸a˜o.
Esta forc¸a de sustentac¸a˜o contribui para a forc¸a propulsiva axial e para o bina´rio resistente
ao veio.
Classificam-se com he´lices “direitos” aqueles que, quando observados de re´, rodam no
sentido hora´rio. Nos navios com dois he´lices, sa˜o normalmente utilizados:35
36 CAPI´TULO 3. PROPULSA˜O
- um he´lice direito a estibordo;
- e um he´lice esquerdo a bombordo.
Nestes navios, a popa e´ relativamente plana e os veios esta˜o expostos e suportados por
aranhas (“shaft brackets”). A presenc¸a destas aranhas provoca ainda na˜o-uniformidades na
esteira em que, devido a` forma da popa, o escoamento entra no he´lice com um certo aˆngulo.
Figura 3.1: He´lice com tubeira.
A aplicac¸a˜o de uma tubeira aceleradora, Fig. 3.1, permite aumentar o rendimento, relati-
vamente a um he´lice convencional, no caso de he´lices fortemente carregados como os aplicados
em rebocadores, arrasto˜es, petroleiros, etc. Outro objectivo da aplicac¸a˜o das tubeiras pode
ser a uniformizac¸a˜o do escoamento de entrada no he´lice. Para este fim trata-se normalmente
de tubeiras assime´tricas colocadas avante do he´lice. Frequentemente este tipo de tubeiras e´
instalada depois de o navio estar em servic¸o.
Figura 3.2: He´lices de passo fixo e de passo controla´vel.
3.1. SISTEMAS DE PROPULSA˜O 37
Para um he´lice de passo fixo, a velocidade do navio e a forc¸a propulsiva sa˜o controladas
pela velocidade de rotac¸a˜o do he´lice. Para um he´lice de passo controla´vel, a forc¸a propulsiva
pode tambe´m ser controlada por variac¸a˜o do passo do he´lice. A variac¸a˜o do passo obte´m-se
por rotac¸a˜o das pa´s em torno de um eixo, a` direita na Fig. 3.2. Utiliza-se quando a velocidade
de rotac¸a˜o e´ constante, ou varia´vel numa gama restrita, quando o he´lice tem de funcionar em
mais de uma condic¸a˜o.
Apesar de constitu´ırem uma soluc¸a˜o cara, pela complicac¸a˜o de chumaceiras e engranagens
necessa´ria, encontram-se exemplos de propulsa˜o por he´lices contrarotativos. Sa˜o dois he´lices,
em que o he´lice de tra´s tem um diaˆmetro ligeiramente menor que o he´lice da frente, a rodar
em sentidos contra´rios, permitindo ao he´lice de tra´s eliminar a perda de energia cine´tica
de rotac¸a˜o do he´lice da frente, Fig. 3.3. Em consequeˆncia, apresentam rendimentos t´ıpicos
superiores a um he´lice isolado.
Figura 3.3: He´lices em contra-rotac¸a˜o.
Outro tipo particular de he´lice e´ o he´lice supercavitante, Fig. 3.4. E´ um he´lice para
funcionar com elevada velocidade de rotac¸a˜o em que as secc¸o˜es das pa´s sa˜o concebidas para
provocar uma bolsa de cavitac¸a˜o que envolve toda a pa´. O perigo de implosa˜o e´ eliminado
porque a implosa˜o das bolhas de cavitac¸a˜o ocorre longe das faces das pa´s. Aplicam-se em
navios de alta velocidade com rendimento, em geral, fraco.
3.1.2 Outros meios de propulsa˜o
Jacto de a´gua
Nestes sistemas, a forc¸a propulsiva e´ obtida pela descarga de um jacto de a´gua a` popa do
navio. Para transmitir a energia pretendida ao jacto podem ser utilizadas bombas axiais,
como no caso da Fig. 3.5, ou bombas centr´ıfugas.
Os sistemas de jacto de a´gua constituem actualmente um soluc¸a˜o comprovada para a pro-
pulsa˜o de embarcac¸o˜es ra´pidas, com divulgac¸a˜o crescente nas embarcac¸o˜es de recreio,“ferries”,
embarcac¸o˜es de patrulha, etc. Sa˜o boas soluc¸o˜es quando os principais requisitos colocados
passam pela manobrabilidade do navio, bom rendimento propulsivo, bom comportamento em
a´guas restritas e pouca necessidade de manutenc¸a˜o. Actualmente, ja´ esta˜o dispon´ıveis no
mercado soluc¸o˜es deste tipo para poteˆncias propulsivas da ordem dos 30MW.
38 CAPI´TULO 3. PROPULSA˜O
Figura 3.4: He´lices supercavitante.
Figura 3.5: Propulsa˜o por jacto de a´gua.
Propulsores azimutais
Esta configurac¸a˜o, ver Fig. 3.6, possibilita a gerac¸a˜o de forc¸a propulsiva em qualquer direcc¸a˜o
por rotac¸a˜o do propulsor em torno do eixo vertical. No sistema tradicional de propulsa˜o
azimutal, o motor era colocado no interior do casco e um sistema mecaˆnico relativamente
complexo fazia a transmissa˜o do movimento a`s pa´s. Actualmente, o accionamento e´ feito
por um motor ele´ctrico colocado no veio de propulsor. Estes sistemas permitem combinar a
propulsa˜o e o governo do navio, dispensando a presenc¸a do leme.
Apresentam como principais vantagens um bom rendimento, justificado em grande parte
pela maior uniformidade do escoamento a` entrada do propulsor, elevada capacidade de ma-
nobra e economia de espac¸o. A sua aplicac¸a˜o, inicialmente quase que restrita a ferries, tem-se
alargado nos tempos mais recentes a praticamente quase todos os tipos de navios.
3.1. SISTEMAS DE PROPULSA˜O 39
Figura 3.6: Propulsores azimutais.
Propulsores cicloidais
Esta soluc¸a˜o propulsiva, representada na Fig. 3.7, desenvolvida pela Voight a partir duma
ideia inicial de Ernst Schneider, permite gerar impulso de magnitude varia´vel em qualquer
direcc¸a˜o. As variac¸o˜es daquele impulso sa˜o ra´pidas, cont´ınuas e precisas, combinando assim
as func¸o˜es de propulsa˜o e governo do navio.
Figura 3.7: Propulsores cicloidais.
O propulsor, colocado no fundo do navio, e´ composto por um conjunto de laˆminas paralelas
com movimento de rotac¸a˜o, segundo um eixo vertical, com velocidade varia´vel. Para gerar o
impulso, cada uma daquelas laˆminas tem um movimento oscilante em torno do seu pro´prio
eixo. O percurso das laˆminas vai determinar a forc¸a impulsiva gerada, enquanto um aˆngulo
de fase entre 0◦ e 360◦ vai definir a direcc¸a˜o do impulso. Desta forma, pode ser gerada a
mesma forc¸a propulsiva em qualquer direcc¸a˜o. A intensidade e a direcc¸a˜o da forc¸a propulsiva
40 CAPI´TULO 3. PROPULSA˜O
sa˜o controladas por um conjunto cinema´tico de transmissa˜o mecaˆnica.
Pelas suas caracter´ısticas, esta soluc¸a˜o apresenta bom desempenho na propulsa˜o de re-
bocadores, ferries, grandes iates, navios de apoio a plataformas petrol´ıferas e outros navios
especiais.
3.2 He´lices propulsores
O projecto do he´lice devera´ dar resposta a`s seguintes questo˜es:
- sera´ que o he´lice desenvolvera´ a desejada forc¸a propulsiva a` velocidade rpm de projecto?
- qual vai ser a eficieˆncia do he´lice?
- qual vai ser o desempenho do he´lice em condic¸o˜es diferentes das condic¸o˜es de projecto?
- sera´ a distribuic¸a˜o de presso˜es favora´vel a` prevenc¸a˜o da cavitac¸a˜o?
- qual sera´ o valor das forc¸as e momentos gerados pelo he´lice sobre o veio propulsor e
chumaceiras de apoio e de impulso?
- qual a pressa˜o induzida pelo funcionamento do he´lice no casco do navio, potencialmente
responsa´vel por vibrac¸o˜es e ru´ıdo?
Os principais me´todos de ca´lculo dispon´ıveis para, de alguma forma, dar resposta a`quelas
questo˜es sa˜o:
- a teoria da quantidade de movimento;
- a teoria dos elementos de pa´;
- a teoria da linha sustentadora;
- a teoria da superf´ıcie de sustentac¸a˜o;
- o me´todo de painel;
- as simulac¸o˜es RANSE.
Outro contributo importante para o projecto do he´lice vem das se´ries sistema´ticas de
he´lices, para as quais sa˜o ja´ conhecidos os principais paraˆmetros de funcionamento em a´guas
livres.
Por u´ltimo, ha´ que citar o contributo importante dos ensaios experimentais em modelos
a` escala reduzida, os ensaios do he´lice em a´guas livres e o ensaio de propulsa˜o.
3.2.1 Geometria do he´lice
Na complexa geometria do he´lice, conjunto de pa´s distribu´ıdas uniformemente em torno do
cubo montado na extremidade do veio, representada esquematicamente na Fig. 3.8, distinguem-
se as seguintes a´reas, linhas e pontos:
- o bordo de ataque (“leading edge”), a linha frontal das pa´s;
- o bordo de fuga (“trailing edge”), a aresta atra´s;
3.2. HE´LICES PROPULSORES 41
Figura 3.8: Geometria do he´lice.
- a extremidade da pa´ (“tip”) e´ o ponto linha ou secc¸a˜o de maior raio;
- o dorso (“back”) e a face da pa´ sa˜o, respectivamente, a superf´ıcie da pa´ do lado do veio,
aspirac¸a˜o, e a superf´ıcie do lado de pressa˜o;
No cubo, com uma forma axisime´trica, unem-se as pa´s pela sua raiz (“ blade root”).
A geometria do he´lice propulsor e´ caracterizada, entre outras, pelas seguintes dimenso˜es,tambe´m representadas naquela figura:
- diaˆmetro do he´lice (“propeller diameter”), D;
- diaˆmetro do cubo (“boss (or hub) diameter”), d;
- nu´mero de pa´s do he´lice (“propeller blade number”), Z;
- passo do he´lice (“propeller pitch”), P ;
- a´rea do disco, A0 = piD
2/4;
- a´rea projectada, a´rea da projecc¸a˜o das pa´s num plano normal ao eixo do he´lice, AP ;
- a´rea expandida, soma das a´reas das faces das pa´s, AE ;
- deslocamento circunferencial (“skew”);
- abatimento axial (“rake”), iG.
3.2.2 Valores caracter´ısticos
Como paraˆmetros adimensionais para caracterizac¸a˜o dos he´lices propulsores podemos apontar:
- a raza˜o entre os diaˆmetros do cubo e do he´lice, d/D;
- a raza˜o entre a a´rea expandida e a a´rea do disco, AE/A0, frequentemente designada por
“blade area ratio” (BAR);
42 CAPI´TULO 3. PROPULSA˜O
- e a raza˜o entre o passo e o diaˆmetro do he´lice, P/D.
Sa˜o valores t´ıpicos para a raza˜o de a´rea expandida 0.3 < AE/A0 < 1.5. Razo˜es superi-
ores a 1 significam que o he´lice tem pa´s sobrepostas o que o torna dispendioso. O valor de
AE/A0 e´ selecionado de tal forma que a carga das pa´s seja suficientemente baixa para evitar
uma situac¸a˜o inaceita´vel de cavitac¸a˜o. Quanto mais carregada for a condic¸a˜o de funciona-
mento prevista para o he´lice maior devera´ ser a raza˜o AE/A0 considerada na sua selecc¸a˜o. O
rendimento do he´lice diminui com o aumento da raza˜o AE/A0.
O nu´mero de pa´s Z e´ um paraˆmetro muito importante para as vibrac¸o˜es induzidas pelo
he´lice. Em geral, um nu´mero ı´mpar de pa´s Z proporciona melhores caracter´ısticas no que diz
respeito a vibrac¸o˜es. Maior nu´mero de pa´s reduz a vibrac¸a˜o, devido aos inferiores picos de
pressa˜o, mas aumenta os custos de fabrico.
Os he´lices propulsores para navios sa˜o sempre adaptados a`s caracter´ısticas espec´ıficas do
navio apo´s exaustivo estudo hidrodinaˆmico. O nu´mero de pa´s esta´ normalmente entre 4 e 7.
Os he´lices propulsores para pequenas embarcac¸o˜es, regra geral com o nu´mero de pa´s entre 2
e 4, sa˜o produzidos em massa.
3.3 Teoria da quantidade de movimento
A teoria mais simples para representar o funcionamento de um he´lice propulsor e´ a teoria da
quantidade de movimento, tambe´m designda por vezes como do disco actuante. Esta teoria
permite relacionar a forc¸a propulsiva do he´lice com as velocidades induzidas. Tem como
principais hipo´teses simplificativas:
- considerar o escoamento de fluido perfeito e incompress´ıvel;
- o nu´mero de pa´s do he´lice e´ infinito;
- o he´lice propulsor exerce uma forc¸a axial T que se distribui uniformemente sobre o disco
do he´lice de diaˆmetro D;
- o he´lice na˜o induz velocidade velocidade de rotac¸a˜o no fluido, ou seja, na˜o ha´ velocidade
circunferencia induzida.
3.3.1 Forc¸a propulsiva
Consideremos o escoamento axisime´trico atrave´s do plano do he´lice, representado na Fig. 3.9,
e denotar por VA a velocidade de aproximac¸a˜o da a´gua ao he´lice e por p∞ a pressa˜o em
pontos suficientemente afastados quer a vante quer a re´ do he´lice. Conforme representado,
sendo a a´gua incompress´ıvel, a secc¸a˜o do escoamento reduz-se pelo aumento de velocidade
transmitido pelo he´lice ao escoamento de a´gua. Na figura podemos ainda ver que no disco
existe uma descontinuidade de pressa˜o ∆p. Esta descontinuidade, como resultado do referido
“disco actuante”, gera uma forc¸a propulsiva do he´lice dada por
T = ∆pA0 (3.1)
Quanto a` distribuic¸a˜o de velocidades, vamos considerar que a velocidade no disco e´ VA+V0
e, no infinito, a velocidade e´ VA + V∞.
3.3. TEORIA DA QUANTIDADE DE MOVIMENTO 43
Figura 3.9: Distribuic¸a˜o espacial de velocidade e pressa˜o para a teoria da
quantidade de movimento.
Representando por A−∞ e A∞ as a´reas no infinito, a montante e a juzante, respectiva-
mente, do tubo de corrente que passa pelo disco actuante, para se verificar a conservac¸a˜o de
massa no escoamento sera´ necessa´rio que,
VaA−∞ = (Va + V0)A0 = (Va + V∞)A∞ (3.2)
Enta˜o, aquelas a´reas, A−∞ e A∞ esta˜o relacionadas com a a´rea do disco e com a velocidade
induzida por
A−∞ =
Va + V0
Va
A0 (3.3)
e
A∞ =
Va + V0
Va + V∞
A0 (3.4)
Aplicando agora o princ´ıpio da conservac¸a˜o da quantidade de movimento ao escoamento
de fluido no tubo de corrente, obtemos a equac¸a˜o,
T = ρ (Va + V∞)2A∞ − ρV 2a A−∞ (3.5)
Usando a equac¸a˜o de conservac¸a˜o da massa, Eq. (3.2), podemos dizer enta˜o que a forc¸a
propulsiva T e´ dada por,
T = ρ (Va + V0)V∞A0 (3.6)
e, que o “salto de pressa˜o” no disco actuante vale
∆p = ρ (Va + V0)V∞ (3.7)
44 CAPI´TULO 3. PROPULSA˜O
Por fim, vamos aplicar a equac¸a˜o de Bernoulli ao tubo de corrente. A montante do disco
temos,
p∞ +
1
2
ρV 2a = p0 +
1
2
ρ (Va + V0)
2 (3.8)
e, a juzante,
p∞ +
1
2
ρ (Va + V∞)2 = p0 + ∆p+
1
2
ρ (Va + V0)
2 (3.9)
Fazendo agora a subtracc¸a˜o das equac¸o˜es, Eq. (3.9) − Eq. (3.8), temos uma nova equac¸a˜o
para avaliar o valor de ∆p
∆p = ρ
(
Va +
1
2
V∞
)
V∞ (3.10)
Naturalmente que o “salto de pressa˜o” avaliado pela u´ltima equac¸a˜o na˜o pode ser diferente
daquele que resulta da Eq. (3.7). Logo,
ρ (Va + V0)V∞ = ρ
(
Va +
1
2
V∞
)
V∞ (3.11)
e, enta˜o, daqui resulta que a velocidade induzida no disco e´ metade da velocidade induzida
na esteira no infinito,
V0 =
1
2
V∞ (3.12)
A forc¸a propulsiva T obtida no disco actuante pode ser calculada, em func¸a˜o da velocidade
induzida no disco, por
T =
piD2
4
ρ (Va + V0) 2V0 (3.13)
3.3.2 Coeficiente de carga
Se definirmos para um he´lice propulsor como coeficiente de carga, CT ,
CT =
T
pi
4D
2 1
2ρV
2
a
(3.14)
e considerarmos a forc¸a propulsiva resultante da teoria do disco actuante, obte´m-se
CT = 4
V0
Va
(
1 +
V0
Va
)
(3.15)
ou, em termos de velocidade induzida no disco,
V0
Va
=
1
2
(
−1 +
√
1 + CT
)
(3.16)
3.4. ENSAIOS COM MODELOS REDUZIDOS DE HE´LICES 45
3.3.3 Rendimento ideal do he´lice
O rendimento ideal do he´lice e´ o rendimento ma´ximo que pode ser obtido em fluido perfeito
com um he´lice propulsor que na˜o induza velocidade de rotac¸a˜o no fluido.
Num referencial em repouso no fluido, considere-se que o he´lice avanc¸a com velocidade
Va, exercendo uma forc¸a propulsiva T . A poteˆncia efectiva do he´lice e´ dada por
PE = T Va (3.17)
A perda de energia cine´tica axial por unidade de tempo e´ o fluxo de energia por unidade
de tempo atrave´s de um plano perpendicular a` direcc¸a˜o de avanc¸o, no infinito, a juzante.
Este fluxo de energia e´ calculado pelo produto do caudal ma´ssico que se escoa pelo tubo de
corrente pela energia cine´tica espec´ıfica,
E˙p = ρ
piD2
4
(Va + V0)× 1
2
V 2∞
ou seja, considerando a relac¸a˜o conhecida entre a velocidade no disco e na esteira no infinito,
E˙p = ρ
piD2
2
(Va + V0)V
2
0 (3.18)
O rendimento ideal do he´lice propulsor sera´ enta˜o dado por
ηi =
TVa
TVa + E˙p
(3.19)
ou, considerando (3.13) e (3.18), e simplificando, ficamos com
ηi =
1
1 + V0Va
(3.20)
3.4 Ensaios com modelos reduzidos de he´lices
Apesar de o he´lice ir funcionar numa esteira na˜o-uniforme do navio, sa˜o realizados ensaios
para avaliac¸a˜o do seu desempenho numa esteira uniforme, recorrendo ao ensaio em a´guas
livres de um modelo a` escala reduzida do he´lice, em condic¸o˜es apropriadas de semelhanc¸a.
Neste ensaio, o chamado “open water test”, um modelo do he´lice e´ deslocado com a velocidade
da avanc¸o Va num fluido em repouso. O escoamento de aproximac¸a˜o deve ser ta˜o uniforme
quanto poss´ıvel. Durante o deslocamento do he´lice este e´ posto a rodar por um pequeno motor
ele´ctrico a` velocidade n (rps) pretendida. O ensaio realiza-se normalmente a uma velocidade
de rotac¸a˜o constante, ou seja, para um dado nu´mero de Reynolds.
As caracter´ısticaspropulsivas em a´guas livres, nomeadamente a forc¸a propulsiva T e o
bina´rio Q, sa˜o medidas em regime estaciona´rio de funcionamento. Depois de adimensionaliza-
dos, os valores medidos da forc¸a propulsiva e do bina´rio para va´rios regimes de funcionamento
constituem o “diagrama em a´guas livres” do he´lice em questa˜o.
A forc¸a propulsiva T e o bina´rio Q disponibilizados por um he´lice propulsor dependem de
va´rias varia´veis:
- a velocidade de avanc¸o Va;
46 CAPI´TULO 3. PROPULSA˜O
- a velocidade de rotac¸a˜o n;
- o diaˆmetro D;
- a massa espec´ıfica do fluido ρ;
- a viscosidade cinema´tica do fluido ν.
Aplicando a ana´lise dimensional, expressando a dependeˆncia dos coeficientes de forc¸a
propulsiva e de bina´rio dos seguintes grupos adimensionais:
- coeficiente de avanc¸o, J =
Va
nD
;
- e nu´mero de Reynolds, aqui definido como Re =
nD2
ν
;
ou seja,
KT = KT (J,Re) e KQ = KQ (J,Re)
obteˆm-se os seguintes expresso˜es para os referidos coeficientes adimensionais:
- coeficiente de forc¸a propulsiva KT =
T
ρn2D4
;
- coeficiente de bina´rio KQ =
Q
ρn2D5
.
3.4.1 Diagrama em a´guas livres
O diagrama em a´guas livres do he´lice integra a representac¸a˜o gra´fica da variac¸a˜o dos coefici-
entes da forc¸a propulsiva, KT , e de bina´rio, KQ, com o coeficiente de avanc¸o, Va. Um exemplo
de diagrama em a´guas livres esta´ representado na Fig. 3.10.
As curvas trac¸adas nestes diagramas servem principalmente para a optimizac¸a˜o do he´lice
e determinac¸a˜o do ponto de funcionamento. Na pra´tica, ja´ na˜o sa˜o utilizadas aquelas re-
presentac¸o˜es gra´ficas no projecto de he´lices, mas sim os polino´mios representativos daquelas
evoluc¸o˜es para permitir o ca´lculo computacional. As tabelas teˆm cerca de 50 coeficientes
para os polino´mios relativos a` se´rie sistema´tica de he´lices de Wageningen. Embora o trabalho
inicial de registo destes coeficientes seja moroso e fastidioso, os processos de ca´lculo e optimi-
zac¸a˜o posteriores ficam muito facilitados e expeditos pela utilizac¸a˜o de programas ou folhas
de ca´lculo. A importaˆncia da representac¸a˜o gra´fica esta´ actualmente restrita a` verificac¸a˜o
da tendeˆncia de variac¸a˜o do desempenho do he´lice com a alterac¸a˜o de algumas condic¸o˜es
operacionais.
3.4.2 Rendimento
Definindo o rendimento de um he´lice propulsor como sendo a raza˜o entre a poteˆncia efectiva
e a poteˆncia fornecida pelo veio ao he´lice, o rendimento em a´guas livres e´ calculado por
η0 =
PE
PD
=
VaT
2pinQ
(3.21)
3.5. SE´RIES SISTEMA´TICAS 47
Figura 3.10: Diagrama de a´guas livres.
a partir das medic¸o˜es observadas durante o ensaio.
Ou, se quisermos expressa´-lo em termos dos coeficientes adimensionais, podemos obter,
η0 =
JKT
2piKQ
(3.22)
3.4.3 I´ndice de qualidade
A qualidade de um propulsor na˜o fica bem caracterizada apenas pelo seu rendimento ma´ximo.
O ı´ndice de qualidade, que permite caracterizar melhor um he´lice para uma dada aplicac¸a˜o
espec´ıfica, e´ dado por
q =
η0
ηi
(3.23)
em que η0 e´ o rendimento em a´guas livres e ηi e´ o rendimento ideal.
Como CT =
8KT
piJ2
, substituindo em (3.23):
q =
KT
4piKQ
(
J +
√
J2 +
8
pi
KT
)
(3.24)
3.5 Se´ries sistema´ticas
Uma se´rie sistema´tica de he´lices e´ um conjunto de he´lices obtidos por variac¸a˜o sistema´tica de
paraˆmetros geome´tricos. Ao longo de de´cadas, por todo o mundo teˆm sido realizados ensaios
em se´ries sistema´ticas de propulsores para navios. As principais caracter´ısticas de alguns
exemplos de se´ries sistema´ticas de he´lices propulsores simples de passo fixo esta˜o inclu´ıdas na
Tab. 3.1.
O principal objectivo perseguido na realizac¸a˜o dos ensaios sistema´ticos nestes conjuntos
de he´lices e´ criar uma base de dados que permita ajudar o projectista a entender os principais
48 CAPI´TULO 3. PROPULSA˜O
Se´rie No Z AE/A0 P/D D(mm)
Wageningen B ≈ 120 2− 7 0, 3− 1, 05 0, 5− 1, 4 250
Au 34 4− 7 0, 4− 0, 758 0, 5− 1, 2 250
Gawn 37 3 0, 2− 1, 1 0, 4− 2, 0 508
KCA ≈ 30 3 0, 50− 1, 25 0, 6− 2, 0 406
Ma 32 3 e 5 0, 75− 1, 20 1, 0− 1, 45 250
Newton-Rader 12 3 0, 5− 1, 0 1, 05− 2, 08 254
KCD 24 3− 6 0, 44− 0, 80 0, 6− 1, 6 406
Meridian 20 6 0, 45− 1, 05 0, 4− 1, 2 305
Tabela 3.1: Se´ries sistema´ticas de propulsores.
factores que influenciam o desempenho do he´lice, bem como a ocorreˆncia de cavitac¸a˜o, em
va´rias condic¸o˜es de funcionamento. Um segundo objectivo e´ a construc¸a˜o de diagramas que
permitam ajudar a` selecc¸a˜o das caracter´ısticas mais apropriadas para uma dada aplicac¸a˜o a`
escala do navio.
3.5.1 Se´rie sistema´tica de Wageningen
Uma das se´ries sistema´ticas de he´lices propulsores mais populares e´ a se´rie B de Wageningen.
Esta se´rie, em que os trabalhos iniciais datam de 1940, sera´ talvez a mais vasta. As principais
caracter´ısticas destes he´lices sa˜o:
- ter distribuic¸a˜o radial do passo constante;
- um pequeno deslocamento circunferencial (“skew”);
- distribuic¸a˜o radial do abatimento axial (“rake”) linear 15◦;
- contorno largo da pa´ junto a` extremidade;
- secc¸a˜o das pa´s NSMB, indicada na Fig. 3.11.
Figura 3.11: Aspecto geome´trico das pa´s da se´rie B de Wageningen
Os paraˆmetros cuja variac¸a˜o sistema´tica foi considerada na realizac¸a˜o desta se´rie foram
os seguintes:
3.5. SE´RIES SISTEMA´TICAS 49
- o nu´mero de pa´s: 2 ≤ Z ≤ 7;
- a raza˜o de a´rea expandida: 0.3 ≤ AE/A0 ≤ 1.05;
- a raza˜o passo-diaˆmetro: 0.5 ≤ P/D ≤ 1.4.
A nomenclatura dos he´lices desta se´rie, considerando a t´ıtulo de exemplo um he´lice B-4.85,
e´ a seguinte:
- Se´rie B;
- Nu´mero de pa´s: 4;
- raza˜o de a´rea expandida: 0.85.
Para cada caso existe um diagrama, ou uma tabela com os ja´ referidos coeficientes po-
linomiais, com as curvas caracter´ısticas dos diagrams de a´guas livres, para diferentes razo˜es
passo-diaˆmetro, P/D. Na Fig. 3.12 esta´ representado o caso dos he´lices com duas pa´s, raza˜o
de a´rea expandida 0, 3 e raza˜o passo-diaˆmetro compreendida entre 0, 5 e 1, 4.
3.5.2 Outras se´ries sistema´ticas
A se´rie sistema´tica de he´lices propulsores Au e´ muito popular no Japa˜o mas, fora dele, na˜o
conseguiu uma divulgac¸a˜o semelhante a` se´rie de Wageningen podendo, no entanto, considerar-
se como uma se´rie complementar daquela.
A se´rie Gawn apresenta como caracter´ıstica distintiva o maior diaˆmetro dos he´lices que
a integram. Isto significa que muitos dos efeitos de escala presentes nas outras se´ries foram
aqui evitados ou, pelo menos, reduzidos. A se´rie KCA, tambe´m designada por vezes como
Gawn-Burrill, e´ complementar da se´rie de Gawn. Sa˜o 30 he´lices com 3 pa´s, tambe´m com
grande diaˆmetro, 400mm. Esta se´rie foi ensaiada num tanque de cavitac¸a˜o, e na˜o num tanque
de reboque, a diferentes nu´meros de cavitac¸a˜o e, consequentemente, permite verificar num
determinado projecto de aplicac¸a˜o os aspectos relacionados com o feno´meno da cavitac¸a˜o.
Os he´lices da se´rie de Lindgren, se´rie Ma, sa˜o mais pequenos, 250mm, e as suas pa´s
teˆm passo constante. Foram testados num tanque de reboque e num tanque de cavitac¸a˜o e,
assim, resultou dos ensaios um extenso e integrado conjunto de dados adequado para a fase
preliminar do projecto.
A se´rie de Newton-Rader compreende um conjunto limitado de 12 he´lices com treˆs pa´s
vocacionados para a propulsa˜o de embarcac¸o˜es ra´pidas.
Para ale´m destas se´ries sistema´ticas de he´lices simples, existem tambe´m alguns estudos
relativos a formas particulares de he´lices como, por exemplo, as se´ries de he´lices contra-
rotativos do MARIN e SSPA, ou a se´rie de Wageningen de he´lices com tubeira.
50 CAPI´TULO 3. PROPULSA˜O
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3.5. SE´RIES SISTEMA´TICAS 51
3.5.3 Diagrama de 4 quadrantes
No caso dos he´lices de passo fixo, a forma convencional de operac¸a˜o do he´lice, velocidade de
rotac¸a˜o positiva e velocidade de avanc¸o nula ou positiva, corresponde ao funcionamento no
primeiro quadrante do diagrama de funcionamento.
No diagrama completo, ver Fig. 3.13, necessa´rio por exemplo para estudar a manobrabi-
lidade do navio ou o seu desempenho em marcha a re´, esta˜o definidos quatro quadrantes, de
acordo o aˆngulo de avanc¸o,
β = tan−1
(
Va
0, 7 ·pi ·n ·D
)
(3.25)
Figura 3.13: Notac¸a˜o do diagrama com 4 quadrantes.
Como ja´ referido, o primeiro quadrante corresponde a:
- velocidade de rotac¸a˜o do he´lice correspondente a` marcha a vante;
- velocidade do navio a vante;
- ou seja, aˆngulo de avanc¸o 0 ≤ β ≤ 90◦.
O segundo quadrante corresponde a:
- velocidade de rotac¸a˜o do he´lice correspondente a` marcha a re´;
- velocidade do navio a vante;
- ou seja, aˆngulo de avanc¸o 90◦ < β ≤ 180◦.
No terceiro quadrante, as condic¸o˜es de operac¸a˜o do he´lice sa˜o:
- velocidade de rotac¸a˜o do he´lice correspondente a` marcha a re´;
52 CAPI´TULO 3. PROPULSA˜O
- velocidade do navio a re´;
- ou seja, aˆngulo de avanc¸o 180◦ < β ≤ 270◦.
E, por fim, no quarto quadrante temos naturalmente:
- velocidade de rotac¸a˜o do he´lice correspondente a` marcha a vante;
- velocidade do navio a re´;
- ou seja, aˆngulo de avanc¸o 270◦ < β < 360◦.
Se existirem dados experimentais suficientes torna-se poss´ıvel definir uma func¸a˜o para
estimar o desempenho do he´lice, no que diz respeito a` forc¸a propulsiva e ao bina´rio, nos
quatro quadrantes do diagrama em a´guas livres. Um exemplo de um diagrama deste tipo,
multi-quadrante, esta´ representado na Fig. 3.14, relativo aos he´lices da se´rie de Wageningen
B4-70 com relac¸a˜o P/D entre 0, 5 e 1, 4.
Justifica-se a introduc¸a˜o de uma notac¸a˜o para obter maior flexibilidade para trabalhar
nestes diagramas multi-quadrante. De notar que para β = 90◦ ou β = 270◦, situac¸o˜es em
que a velocidade de rotac¸a˜o do he´lice e´ nula, o coeficiente de avanc¸o resultaria J = ∞. De
forma semelhante, para prevenir o mesmo tipo de situac¸o˜es, sa˜o tambe´m definidos os seguintes
coeficientes:
- coeficiente de forc¸a propulsiva modificado,
C∗T =
T
1
2
ρV 2RA0
(3.26)
- coeficiente de bina´rio modificado,
C∗Q =
Q
1
2
ρV 2RA0D
(3.27)
em que VR e´ a velocidade relativa de avanc¸o para 0, 7R, ou seja,
C∗T =
T
pi
8
ρ
[
V 2a + (0, 7pinD)
2
]
D2
(3.28)
e
C∗Q =
Q
pi
8
ρ
[
V 2a + (0, 7pinD)
2
]
D3
(3.29)
Na Fig. 3.14 pode-se ver o efeito que a raza˜o P/D tem no coeficiente de bina´rio C∗Q
para praticamente toda a gama de β. Em particular, e´ nos intervalos 40◦ < β < 140◦ e
230◦ < β < 340◦ que a magnitude de C∗Q varia mais significativamente.
3.6. CAVITAC¸A˜O 53
Figura 3.14: Diagrama em a´guas livres de 4 quadrantes para os he´lices
Wageningen B-4.70.
3.6 Cavitac¸a˜o
3.6.1 Origem da cavitac¸a˜o
A velocidade elevada do escoamento de a´gua pelo he´lice provoca regio˜es com baixa pressa˜o.
Se a pressa˜o cair o suficiente, formar-se-a˜o cavidades preenchidas com vapor. Estas cavidades
desaparecera˜o quando a pressa˜o aumentar. O crescimento e o colapso destas “bolhas” e´
extremamente ra´pido.
A cavitac¸a˜o envolve feno´menos f´ısicos complexos uma vez que se trata de escoamentos a
duas fases, com modelac¸a˜o na˜o-linear. Nos he´lices dos navios, a velocidade em torno das pa´s
pode ser suficiente para reduzir a localmente a pressa˜o e desencadear a cavitac¸a˜o. Devido a`
pressa˜o hidrosta´tica, a pressa˜o total sera´ superior nas imediac¸o˜es da pa´ que se encontre com
a ma´xima imersa˜o (posic¸a˜o 06:00) do que naquela que se encontra na posic¸a˜o 12:00. Assim,
as pa´s dos he´lices em cavitac¸a˜o alternadamente passara˜o por regio˜es em que tendencialmente
se formara˜o bolhas de cavitac¸a˜o e regio˜es onde as mesmas tendera˜o a colapsar.
Esta ra´pida sucessa˜o de exploso˜es e imploso˜es nas proximidades das pa´s do he´lice tem
va´rias consequeˆncias nefastas. As principais sa˜o:
- vibrac¸a˜o;
54 CAPI´TULO 3. PROPULSA˜O
Figura 3.15: Efeito da cavitac¸a˜o no valor dos paraˆmetros relativos a a´guas
livres.
- ru´ıdo;
- erosa˜o da superf´ıcie das pa´s (sobretudo se o colapso das bolhas ocorrer na proximidade);
- reduc¸a˜o da forc¸a propulsiva.
No diagrama em a´guas livres da Fig. 3.15 esta´ assinalada a reduc¸a˜o que e´ tipicamente
provocada pela cavitac¸a˜o nos coeficientes de forc¸a propulsiva e bina´rio.
3.6.2 Controle da cavitac¸a˜o
Num meio ideal, a´gua sem impurezas ou ar dissolvido, a cavitac¸a˜o ocorrera´ quando a pressa˜o
total atingir localmente a pressa˜o de vapor a essa temperatura. Na pra´tica, a cavitac¸a˜o inicia-
se para valores de pressa˜o superiores pela presenc¸a de part´ıculas microsco´picas e da existeˆncia
de ar dissolvido na a´gua que facilitam e precipitam o in´ıcio do processo de vaporizac¸a˜o.
O nu´mero de cavitac¸a˜o σ e´ um paraˆmetro adimensional que estima a possibilidade de
aparecimento do feno´meno de cavitac¸a˜o num escoamento,
σ =
p0 − p
1
2
ρ V 20
(3.30)
em que:
- p0 e´ a pressa˜o ambiente de refereˆncia;
- p e´ a pressa˜o local;
- e V0 e´ a velocidade de refereˆncia correspondente.
3.6. CAVITAC¸A˜O 55
Figura 3.16: Pressa˜o de vapor da a´gua em func¸a˜o da temperatura.
Para σ inferior a σv, o nu´mero de cavitac¸a˜o avaliado para a pressa˜o de vapor pv, na˜o
ocorrera´ cavitac¸a˜o num fluido ideal. Na pra´tica, e´ necessa´rio considerar um coeficiente de
seguranc¸a, considerando uma pressa˜o limite superior a` pressa˜o de vapor.
Para um he´lice e´ habitual definir o nu´mero de cavitac¸a˜o σn como:
σn =
p0 − p
1
2
ρn2D2
(3.31)
adoptando-se como velocidade caracter´ıstica nD.
3.6.3 Considerac¸a˜o da cavitac¸a˜o na selecc¸a˜o do he´lice
O feno´meno da cavitac¸a˜o e´ predominantemente dominado pelo campo de pressa˜o no esco-
amento da a´gua pelo plano do he´lice. Prevenir a cavitac¸a˜o passa consequentemente pelo
controlo da mı´nima pressa˜o absoluta naquele escoamento. A possibilidade de ocorreˆncia de
cavitac¸a˜o e´ evitada pela distribuic¸a˜o da forc¸a propulsiva por uma a´rea maior, aumentando o
diaˆmetro do he´lice ou a raza˜o da a´rea expandida AE /A0. A forma mais usual de estimar,
ainda que de uma forma na˜o completamente rigorosa, o perigo de ocorreˆncia da cavitac¸a˜o
passa pela utilizac¸a˜o do diagrama de Burrill (Fig. 3.17). O diagrama indica um limite inferior
para a a´rea projectada do he´lice de um navio mercante. Nos eixos do diagrama de Burrill es-
ta˜o o nu´mero de cavitac¸a˜o, em abcissas, e o coeficiente de Burrill nas ordenadas. O coeficiente
56 CAPI´TULO 3. PROPULSA˜O
Figura 3.17: Diagrama de Burrill.
de Burrill e´ calculado por
τc =
T
q0,7RAp
(3.32)
em que, Ap e´ a a´rea projectada do he´lice, e o paraˆmetro q0,7R e´ dado por
q0,7R =
1
2
ρ V 2R
em que VR e´ o valor absoluto da velocidade local a 0, 7 do raio do he´lice, ou seja,
VR =
√
V 2a + (0, 7pi nD )
2
com Va a velocidade de entrada do escoamento no plano do he´lice.
Nos he´lices da se´rie de Wageningen, a a´rea expandida esta´ relacionada com a a´rea projec-
tada por
AE =
AP
1, 067− 0, 229P/D (3.33)
3.6.4 Ensaios experimentais
Os ensaios de cavitac¸a˜o, bem como frequentemente os ensaios em a´guas livres, realizam-se em
instalac¸o˜es que compreendem um canal fechado na qual e´ imposta a circulac¸a˜o da a´gua por
um impulsor. Na Fig. 3.18 esta´ representada esquematicamente uma instalac¸a˜o deste tipo.
Estes tu´neis sa˜o concebidos por forma a proporcionar um escoamento ta˜o uniforme quanto
poss´ıvel na secc¸a˜o de teste. A secc¸a˜o de teste, o troc¸o horizontal superior, dispo˜e de visores
parainspecc¸a˜o e vizualizac¸a˜o do escoamento. O impulsor para a circulac¸a˜o da a´gua esta´
3.6. CAVITAC¸A˜O 57
Figura 3.18: Instalac¸o˜es de ensaio do RINA.
colocado no troc¸o inferior horizontal para garantir que, mesmo quando a pressa˜o no tanque
for reduzida, a coluna hidrosta´tica vai impedir a cavitac¸a˜o neste propulsor.
Normalmente, a pressa˜o e´ reduzida por bombas de va´cuo para ajuste do nu´mero de cavi-
tac¸a˜o e a instalac¸a˜o dispo˜e de equipamento para reduzir o ar dissolvido na a´gua. Podem ser
instaladas “grelhas meta´licas” para induzir a turbuleˆncia desejada no escoamento.
Os he´lices em teste sa˜o sujeitos a iluminac¸a˜o estrobosco´pica por forma a serem “vistos”
sempre com as pa´s na mesma posic¸a˜o. Obte´m-se assim uma visualizac¸a˜o do padra˜o de cavi-
tac¸a˜o “estaciona´ria”.
O funcionamento do he´lice tem alguns pontos caracter´ısticos que se passa a identificar. A
primeira destas situac¸o˜es acontece quando o motor ele´ctrico faz rodar o veio do he´lice a uma
velocidade n mantendo-se a velocidade de avanc¸o nula, ou seja Va = 0. Nestas condic¸o˜es,
verifica-se J = 0 e η = 0, e diz-se que o he´lice funciona a ponto fixo. Se em seguida se
fizer avanc¸ar o he´lice a uma velocidade Va, mantendo a mesma velocidade de rotac¸a˜o, este
desenvolvera´ um impulso T e absorvera´ um certo momento Q. Esta fase e´ a fase propulsora,
utilizada para a propulsa˜o dos navios. Continuando a aumentar o coeficiente de impulso por
diminuic¸a˜o da velocidade de rotac¸a˜o n, o impulso vai diminuindo ate´ o he´lice chegar ao ponto
de impulso nulo. Inicia-se a fase de travagem, ate´ um ponto, no qual o he´lice trabalha em
concordaˆncia com o coeficiente de avanc¸o J , com KQ = 0, he´lice livre. Um he´lice livre opo˜e
resisteˆncia ao avanc¸o. Continuando a reduzir a velocidade de rotac¸a˜o do he´lice e mantendo
Va, entra-se na fase motora, em que o he´lice poderia fornecer energia. Quando a velocidade
do he´lice for nula, o he´lice diz-se bloqueado.
58 CAPI´TULO 3. PROPULSA˜O
Figura 3.19: Imagem da cavitac¸a˜o num he´lice.
3.7 Selecc¸a˜o do he´lice
No ca´lculo do he´lice procura-se a optimizac¸a˜o das principais varia´veis, nu´mero e a´rea das
pa´s, diaˆmetro, velocidade de rotac¸a˜o e passo, por forma a que a propulsa˜o se fac¸a com bom
rendimento em todas as condic¸o˜es de carga do navio. E´ poss´ıvel obter uma boa estimativa
das caracter´ısticas de funcionamento do he´lice utilizando uma das va´rias se´ries sistema´ticas
referenciadas. As varia´veis de optimizac¸a˜o do he´lice sa˜o descritas sucintamente nos para´grafos
seguintes.
3.7.1 Varia´veis de optimizac¸a˜o
Diaˆmetro
O rendimento do he´lice aumenta o diaˆmetro do mesmo, estando no entanto a dimensa˜o
deste limitada pela geometria da popa. Deve-se referir no entanto que o aumento do diaˆmetro
de he´lice provoca vibrac¸o˜es mais fortes e a reduc¸a˜o do rendimento do casco. As sociedades
classificadoras teˆm normas pro´prias para definir valores mı´nimos de folga entre o he´lice e o
casco do navio.
O diaˆmetro ma´ximo do he´lice e´ normalmente considerado como uma fracc¸a˜o do calado
ma´ximo do navio,
Dmax = a T (3.34)
dependente do tipo de navio, conforme indicado na Tab. 3.2.
Para compensar a na˜o uniformidade do escoamento de aproximac¸a˜o ao he´lice quando este
se encontra atra´s da querena, o diaˆmetro equivalente em a´guas livres e´ considerado como:
D0 =
D
1− b (3.35)
em que b toma os valores constantes na Tab. 3.3.
3.7. SELECC¸A˜O DO HE´LICE 59
Tipo de Navio a
Graneleiros/Petroleiros <0,65
Porta-contentores <0,74
Tabela 3.2: Coeficiente para atribuic¸a˜o do diaˆmetro ma´ximo do he´lice pela
Eq. (3.34).
He´lice b
Simples 0.05
Duplo 0.03
Tabela 3.3: Constante para o ca´lculo do diaˆmetro equivalente em a´gua
livres pela Eq. (3.35).
Velocidade de rotac¸a˜o
Em instalac¸o˜es propulsoras com transmissa˜o directa, a velocidade de rotac¸a˜o do he´lice e´
estabelecida pela velocidade do motor. Neste caso, o diaˆmetro e´ ajustado para se obter um
coeficiente de avanc¸o apropriado para a velocidade pretendida e a poteˆncia exigida. Quando
e´ utilizada uma caixa redutora, procura-se utilizar o maior diaˆmetro poss´ıvel, sendo depois
ajustada a velocidade de rotac¸a˜o do he´lice ajustada de acordo com o coeficiente de avanc¸o
pretendido. Devem evitar-se velocidades que multiplicadas pelo nu´mero de pa´s do he´lice
sejam pro´ximas das frequeˆncias de ressonaˆncia do casco e da instalac¸a˜o propulsora. Do ponto
de vista da prevenc¸a˜o da cavitac¸a˜o, sa˜o favora´veis as velocidades de rotac¸a˜o mais baixas.
Nu´mero de pa´s
O factor determinante na selecc¸a˜o do nu´mero de pa´s e´ a irregularidade das forc¸as geradas
pelo he´lice. Estas forc¸as, aplicadas com a frequeˆncia correspondente a` velocidade de rotac¸a˜o,
induzem vibrac¸o˜es no casco e instalac¸a˜o propulsora. O objectivo passa por obter um bom
compromisso entre a vibrac¸a˜o gerada e o rendimento obtido ja´ que este diminui com o aumento
do nu´mero de pa´s do he´lice.
Distribuic¸a˜o radial da pressa˜o
A distribuic¸a˜o da pressa˜o nas pa´s esta´ relacionada com a susceptibilidade de ocorreˆncia da
cavitac¸a˜o. Em particular, e´ normalmente vantajoso reduzir a pressa˜o no extremo radial das
pa´s. Esta reduc¸a˜o e´ ainda vantajosa na perspectiva do esforc¸o estrutural das pa´s e da pressa˜o
irregular induzida no casco.
Geometria das pa´s
A fo´rmula de Keller permite escolher a raza˜o de a´rea expandida para evitar o feno´meno da
cavitac¸a˜o,
Ae
A0
=
(1, 3 + 0, 3Z)T
(p0 − pv)D2 + k (3.36)
em que k e´ uma margem de seguranc¸a, que variara´ entre k = 0 para navios de guerra e
k = 0, 2 para navios mercantes com he´lices muito carregados. Quanto maior a raza˜o de a´reas,
menor sera´ o risco de cavitac¸a˜o mas, em compensac¸a˜o menor o rendimento do he´lice devido
ao atrito. A soluc¸a˜o sera´ a menor a´rea que garanta o crite´rio de cavitac¸a˜o.
60 CAPI´TULO 3. PROPULSA˜O
No entanto, a curvatura, o aˆngulo de ataque e a espessura das teˆm tambe´m uma grande
importaˆncia no controle da cavitac¸a˜o. A maior espessura das pa´s favorece a cavitac¸a˜o nas cos-
tas das pa´s enquanto que as pa´s pouco espessas teˆm maior propensa˜o para gerarem cavitac¸a˜o
no bordo de ataque.
Quanto ao rendimento, ele e´ favorecido pela diminuic¸a˜o da corda das pa´s, ou seja da sua
a´rea, mas por razo˜es estruturais, esta reduc¸a˜o tem que ser acompanhada por um aumento de
espessura que vai provocar um aumento da resisteˆncia de forma.
A utilizac¸a˜o apropriada do desvio circunferencial das pa´s do he´lice (“skew”) permite con-
trolar muito eficazmente a cavitac¸a˜o e a vibrac¸a˜o induzida tendo apenas como contrapartida
uma reduc¸a˜o do rendimento do he´lice em marcha a re´.
3.7.2 Tipos de problema
E´ poss´ıvel obter uma boa estimativa das caracter´ısticas de funcionamento do he´lice utilizando
uma das va´rias se´ries sistema´ticas referenciadas. Uma vez determinado o nu´mero e a a´rea
das pa´s, resta a determinar a combinac¸a˜o do passo e do coeficiente de avanc¸o que permite
optimizar o rendimento do he´lice. De acordo com o tipo de problema em causa, podemos
considerar va´rias situac¸o˜es. Quando a poteˆncia e a velocidade de rotac¸a˜o sa˜o conhecidas, da
eliminac¸a˜o do diaˆmetro resulta a seguinte equac¸a˜o:
KQ
J5
=
PDn
2
2piρV 5a
(3.37)
Quando a poteˆncia e o diaˆmetro do he´lice esta˜o determinados, a eliminac¸a˜o da velocidade
de rotac¸a˜o permite estabelecer:
KQ
J3
=
PD
2piρD2V 3a
(3.38)
Sendo prescritas a forc¸a propulsiva e a velocidade de rotac¸a˜o, a eliminac¸a˜o do diaˆmetro
conduz a` equac¸a˜o:
KT
J4
=
Tn2
ρV 4a
(3.39)
Por fim, quando sa˜o conhecidos o diaˆmetro do he´lice e a forc¸a propulsiva, a eliminac¸a˜o da
velocidade de rotac¸a˜o permite estabelecer a seguinte relac¸a˜o:
KT
J2
=
T
ρD2V 2a
(3.40)
3.8 Interacc¸a˜o entre cascoe he´lice
Os ensaios de he´lices a` escala reduzida em a´guas livres, conseguindo efectuar uma avaliac¸a˜o
preliminar das caracter´ısticas propulsivas de um he´lice, na˜o permitem uma previsa˜o do seu
desempenho numa dada aplicac¸a˜o espec´ıfica, porque, na realidade, o he´lice na˜o vai operar em
a´guas livres mas sim atra´s do navio.
As caracter´ısticas de um he´lice trabalhando atra´s de um navio a uma dada velocidade
diferem consideravelmente das caracter´ısticas obtidas em ensaios com modelos em a´guas livres,
a` velocidade correspondente, devido aos seguintes factores:
3.8. INTERACC¸A˜O ENTRE CASCO E HE´LICE 61
- a velocidade da a´gua na esteira do navio e´ menor que a velocidade do navio;
- a na˜o-uniformidade da esteira do navio afecta a distribuic¸a˜o das forc¸as aplicadas nas
pa´s do he´lice;
- a acelerac¸a˜o da a´gua pelo he´lice reduz a pressa˜o sobre o casco e, consequentemente
aumentando a resisteˆncia efectiva da querena.
3.8.1 Ensaios de propulsa˜o
Os ensaios de propulsa˜o teˆm por objectivo determinar, para cada velocidade de rotac¸a˜o, a
poteˆncia propulsiva e a consequente velocidade do navio. Os resultados dos ensaios permitem
tambe´m a determinac¸a˜o dos coeficientes de deduc¸a˜o da forc¸a propulsiva e da velocidade da
esteira necessa´rios para a selecc¸a˜o ou projecto do he´lice. O modelo e´ equipado com um
he´lice pre´-seleccionado de acordo com as necessidades operacionais previstas para o navio. A
optimizac¸a˜o a partir deste he´lice-base decorrera´ a partir dos resultados obtidos neste ensaio
de auto-propulsa˜o. O accionamento deste he´lice e´ normalmente realizado por um pequeno
motor ele´ctrico, conforme representado esquematicamente na Fig. 3.20.
Figura 3.20: Modelo para ensaios de propulsa˜o.
As condic¸o˜es de realizac¸a˜o do ensaio de propulsa˜o contemplam:
- semelhanc¸a geome´trica;
- semelhanc¸a cinema´tica;
- semelhanc¸a de Froude;
- igual nu´mero de cavitac¸a˜o.
Pelas razo˜es apontadas anteriormente, na˜o e´ poss´ıvel acumular com aquelas condicionan-
tes a igualdade do nu´mero de Reynolds. Assim, existem efeitos de escala a considerar na
extrapolac¸a˜o dos resultados para a escala do navio.
O primeiro efeito de escala a considerar no ensaio de propulsa˜o e´ o efeito de escala na
resisteˆncia. O coeficiente de resisteˆncia total e´ superior no teste do modelo ao que se verificara´
no navio porque o coeficiente de resisteˆncia de atrito diminui com o aumento do nu´mero
62 CAPI´TULO 3. PROPULSA˜O
de Reynolds. Este efeito resultante da variac¸a˜o do nu´mero de Reynolds e´ resolvido pela
aplicac¸a˜o de uma forc¸a de compensac¸a˜o. A intensidade da forc¸a de compensac¸a˜o necessa´ria
FD e´ determinada por,
FD =
1
2
ρ ·V 2m ·Sm · ((1 + k) (cFm − cFs)− cA − cAA) (3.41)
O he´lice tem portanto que produzir uma forc¸a propulsiva igual a` resisteˆncia total RT menos
a forc¸a de compensac¸a˜o FD.
Outro efeito de escala a considerar no ensaio de propulsa˜o diz respeito a` esteira. A
espessura da camada limite e esteira do modelo e´ relativamente maior que a correspondente
espessura no navio. Ou seja, o coeficiente de esteira do modelo e´ maior que o do navio. A
velocidade me´dia de aproximac¸a˜o ao he´lice, adimensionalizada pela velocidade do modelo, e´
menor que a correspondente velocidade adimensionalizada do navio.
Por u´ltimo, devera´ ser considerado o efeito de escala nas caracter´ısticas propulsivas do
he´lice. De facto, o nu´mero de Reynolds do he´lice no modelo e´ menor que no he´lice do navio
e os coeficientes de forc¸a propulsiva e de bina´rio sa˜o diferentes.
Na realizac¸a˜o dos ensaios de propulsa˜o e´ normalmente mantida a velocidade do “carro” de
reboque constante e e´ variada a velocidade de rotac¸a˜o do he´lice ate´ ser obtida uma condic¸a˜o
de equil´ıbrio. Sa˜o assim obtidos dados de forc¸a propulsiva e bina´rio em func¸a˜o da velocidade.
Adicionalmente, podem ainda ser registados dados sobre o calado e o caimento do modelo
durante o ensaio.
O ponto de auto-propulsa˜o do modelo e´ encontrado quando as forc¸as exteriores sobre o
modelo sa˜o nulas. O ensaio e´ realizado com o nu´mero de Froude do navio, fazendo variar a
velocidade de rotac¸a˜o do he´lice ate´ que a forc¸a de reboque se anule. Nesta situac¸a˜o, a forc¸a
propulsiva iguala a resisteˆncia da querena, alterada pela presenc¸a de he´lice. Para compensar
a diferenc¸a no coeficiente de resisteˆncia do navio e do modelo, e´ aplicada a forc¸a adicional de
reboque FD determinada pela Eq. (3.41). E´ portanto mais correcto afirmar que no ponto de
auto-propulsa˜o do modelo, a u´nica forc¸a exterior aplicada ao modelo e´ a forc¸a FD.
Para ale´m do chamado ensaio de auto-propulsa˜o, realizam-se os ensaios em sobrecarga.
Cada ensaio em sobrecarga realiza-se tambe´m com o he´lice a operar atra´s do modelo com este
a ser rebocado a velocidade constante. Faz-se variar a velocidade de rotac¸a˜o do he´lice e, para
cada uma das velocidades ensaiadas nm regista-se a forc¸a de reboque Fm, a forc¸a propulsiva
Tm e o bina´rio Qm. Pode-se encontrar tambe´m o ponto de auto-propulsa˜o do modelo por
interpolac¸a˜o nos resultados dos ensaios em sobrecarga, mais concretamente interpolando na
curva da forc¸a de reboque em func¸a˜o da velocidade de rotac¸a˜o, para o valor requerido de FD.
3.8.2 Poteˆncia e velocidade
A poteˆncia efectiva PE , poteˆncia necessa´ria para rebocar a querena, sem os apeˆndices associ-
ados a` propulsa˜o, a` velocidade Vs, e´ obtida por
PE = RT ·Vs (3.42)
em que:
- RT e´ a resisteˆncia total em a´guas livres excluindo a resisteˆncia adicional dos apeˆndices
associados a` propulsa˜o;
- e Vs e´ a velocidade do navio.
3.8. INTERACC¸A˜O ENTRE CASCO E HE´LICE 63
De forma ana´loga, a poteˆncia propulsiva PT pode ser obtida por
PT = T ·Va
em que:
- T e´ a forc¸a propulsiva calculada a partir dos ensaios de propulsa˜o;
- e Va e´ a velocidade de avanc¸o do he´lice.
A forc¸a propulsiva T e´ superior a` resisteˆncia RT avaliada a partir do ensaio de resisteˆncia
realizado sem he´lice. Isto significa, como referido antes, que a presenc¸a do he´lice induz uma
resisteˆncia adicional porque:
- a presenc¸a do he´lice aumenta a velocidade do escoamento na zona da popa do navio e,
em consequeˆncia a resisteˆncia de atrito;
- a presenc¸a do he´lice provoca uma diminuic¸a˜o da pressa˜o nos paine´is da popa do navio.
O segundo destes factores e´ normalmente o mais significativo.
O aumento da resisteˆncia devido ao efeito da presenc¸a do he´lice e´ usualmente representado
por uma reduc¸a˜o da forc¸a propulsora expressa como fracc¸a˜o dessa forc¸a. O coeficiente de
deduc¸a˜o da forc¸a propulsiva t associa enta˜o a forc¸a propulsiva e a resisteˆncia,
t = 1− RT
T
(3.43)
em que t e´ normalmente considerado igual no modelo e no navio.
Depois de realizados os ensaios de propulsa˜o e calculados os coeficientes de forc¸a propul-
siva, KTm e KQm, o coeficiente de deduc¸a˜o da forc¸a propulsiva e´ calculado por
tm =
Tm + FD −RC
Tm
(3.44)
em que RC e´ a resisteˆncia corrigida para a diferenc¸a de temperatura entre os dois ensaios,
resisteˆncia e propulsa˜o. O valor de RC sera´,
RC =
(1 + k) cFmC + cR
(1 + k) cFm + cR
RTm (3.45)
em que cFmC e´ o coeficiente da resisteˆncia de atrito avaliado a` temperatura da a´gua no ensaio
de propulsa˜o.
Para corrigir o efeito da velocidade da esteira, define-se o coeficiente de deduc¸a˜o da esteira,
w, que permite relacionar a velocidade de avanc¸o Va com a velocidade do navio V ,
w = 1− Va
V
(3.46)
Considerando o diagrama em a´guas livres do he´lice, com o valor de KTm avaliado com a
forc¸a propulsiva experimental do ensaio de propulsa˜o, pode obter-se atrave´s daquele diagrama
um valor para o coeficiente de avanc¸o J0m. O coeficiente de esteira do modelo sera´ enta˜o dado
por
wm = 1− J0mDmnm
Vm
(3.47)
64 CAPI´TULO 3. PROPULSA˜O
Ou seja, avelocidade me´dia axial no plano do he´lice atra´s do navio a` velocidade V , no
ensaio de resisteˆncia sem he´lice, e´ a velocidade da esteira nominal,
Va = (1− wn)V (3.48)
e, com o he´lice em operac¸a˜o atra´s do navio, o escoamento devido a` presenc¸a da querena e´
modificado obtendo-se a velocidade da esteira efectiva,
Ve = (1− we)V (3.49)
A velocidade total sera´ a soma da velocidade da esteira efectiva e da velocidade axial induzida
pelo he´lice.
O rendimento rotativo relativo ηR e´ calculado por
ηR =
KQ0m
KQm
(3.50)
em que KQ0m e´ obtido a partir do diagrama em a´guas livres do he´lice e o coeficiente de bina´rio
KQm e´ calculado com base nos resultados experimentais do ensaio de propulsa˜o.
Designa-se por rendimento do casco a raza˜o entre a poteˆncia efectiva e a poteˆncia propul-
siva, ou seja,
ηH =
PE
PT
=
RT ·Vs
T ·Va =
1− t
1− w (3.51)
A determinac¸a˜o de w, t e ηH e´ feita preferencialmente atrave´s de ensaios de modelos em
ensaios de auto-propulsa˜o utilizando um he´lice de “stock” com caracter´ısticas conhecidas, ta˜o
aproximadas quanto poss´ıvel do he´lice o´ptimo. Se na˜o for poss´ıvel utilizar um modelo, aqueles
paraˆmetros podera˜o ser estimados com base em dados estat´ısticos. Para navios com um ou
dois he´lices, o diagrama de Harvald permite estimar os valores de w, t e ηH em func¸a˜o do
coeficiente de finura total e da raza˜o B/L, com correcc¸o˜es associadas ao tipo de popa, cota do
veio e diaˆmetro do he´lice. Outros autores propuseram algumas expresso˜es para a estimativa
daqueles paraˆmetros. Destas, destacam-se as de Taylor, Schoenherr e Luke, para navios com
um he´lice,
w = 0, 5Cb + 0, 025 (3.52)
e,
t = 0, 5w (3.53)
com ηH = 1, 02. Para navios com dois he´lices,
w = 0, 4533Cb − 0, 114 (3.54)
e,
t = 0, 7w + 0, 01 (3.55)
com ηH = 0, 985. Podera˜o aqui ser referidas as expresso˜es mais complexas apresentadas por
Holtrop, com base em mais de duzentos ensaios de auto-propulsa˜o em modelos de navios de
diversos tipos.
3.8. INTERACC¸A˜O ENTRE CASCO E HE´LICE 65
A poteˆncia absorvida pelo he´lice pode ser expressa em termos da velocidade de rotac¸a˜o n
(em rps) e do bina´rio Q por
PD = 2pi ·n ·Q (3.56)
Devido a`s perdas mecaˆnicas no veio e chumaceiras, a poteˆncia recebida pelo he´lice PD e´
inferior a` poteˆncia efectiva do motor (’brake power’ ) PB,
PD = ηs ·PB (3.57)
em que ηs e´ o rendimento da linha de veios. A eficieˆncia do propulsor atra´s do navio, avalia
as perdas desde a poteˆncia recebida pelo he´lice PD e a poteˆncia propulsiva PT ,
PT = ηB ·PD (3.58)
Esta eficieˆncia do propulsor atra´s do navio ηB e´ diferente da eficieˆncia em a´guas livres η0
verificada experimentalmente. O rendimento rotativo relativo ηR avalia as perdas associadas
a` diferenc¸a entre o escoamento em a´guas livres e o escoamento tridimensional na˜o-uniforme
no plano do propulsor,
ηB = ηR · η0 (3.59)
Em resumo, verifica-se sempre a relac¸a˜o,
PB > PD > PT > PE
em que os valores daquelas poteˆncias sa˜o calculadas por
PE = ηH ·PT = ηH · ηB ·PD = ηH · η0 · ηR ·PD = ηH · η0 · ηR · ηS ·PB
Se o rendimento quase-propulsivo ηD espressar o conjunto de eficieˆncias hidrodinaˆmicas
consideradas,
ηD = ηH · η0 · ηR (3.60)
enta˜o, a poteˆncia efectiva pode ser dada por
PE = ηD · ηS ·PB
As leis de semelhanc¸a permitem a extrapolac¸a˜o das medic¸o˜es efectuadas para a escala do
navio,
Vs =
√
λVm , (3.61)
ns = nm/
√
λ , (3.62)
Ts = Tm · (ρs/ρm) ·λ3 (3.63)
e,
Qs = Qm · (ρs/ρm) ·λ4 (3.64)
66 CAPI´TULO 3. PROPULSA˜O
3.8.3 Extrapolac¸a˜o dos resultados do ensaio de propulsa˜o
O procedimento recomendado pela ITTC para o tratamento dos dados experimentais resul-
tantes dos ensaios de resisteˆncia e de propulsa˜o para a previsa˜o do desempenho do navio esta´
inclu´ıdo no Apeˆndice A. Para ale´m dos ja´ referidos ensaios de reboque e propulsa˜o, sa˜o ainda
necessa´rios testes do he´lice em a´guas livres. De uma forma sucinta, o referido procedimento
envolve os seguintes passos:
- prever a resisteˆncia total do navio a partir da resisteˆncia avaliada no modelo, corrigindo
de acordo com as resisteˆncias adicionais que devam ser consideradas;
- estimar as caracter´ısticas do he´lice propulsor com base nos coeficientes propulsivos de-
terminados para o modelo;
- estimar a esteira do navio e as condic¸o˜es de funcionamento do he´lice;
- estimar a velocidade de rotac¸a˜o do he´lice e poteˆncia necessa´ria com base em factores de
correlac¸a˜o entre o modelo e o navio.
Os detalhes de cada um destes passos, bem como o formula´rio de ca´lculo, devem ser
consultados no referido Apeˆndice A.
As va´rias condic¸o˜es consideradas nos ensaios do modelo servira˜o para fazer uma previsa˜o
do desempenho do navio numa gama de velocidades para as condic¸o˜es de lastro e carregado,
conforme representado na Fig. 3.21.
Figura 3.21: Resultados dos ensaios de propulsa˜o.
Cap´ıtulo 4
Instalac¸o˜es Propulsoras
4.1 Introduc¸a˜o
A escolha de uma ma´quina propulsora ou da configurac¸a˜o mais apropriada para a instalac¸a˜o
propulsora num projecto de nova construc¸a˜o ou reconversa˜o na˜o e´ actualmente uma decisa˜o
simples. E´ imperioso que esta decisa˜o seja precedida de uma ana´lise rigorosa das va´rias opc¸o˜es
dispon´ıveis para o perfil de operac¸a˜o futura definido para o navio.
Uma vez determinada a poteˆncia absorvida pelo he´lice, torna-se necessa´rio identificar as
soluc¸o˜es que satisfazem os requisitos de poteˆncia, velocidade de rotac¸a˜o, consumo e dimenso˜es.
A sua avaliac¸a˜o te´cnico-financeira sera´ enta˜o realizada por crite´rios baseados nos seguintes
factores:
- o investimento inicial;
- a fiabilidade;
- os custos de manutenc¸a˜o previstos;
- os custos de operac¸a˜o previstos;
- a margem do motor, relacionada com a diferenc¸a entre a poteˆncia ma´xima e a poteˆncia
de servic¸o do motor.
Este processo de selecc¸a˜o terminara´ sempre numa soluc¸a˜o de compromisso ja´ que nenhum
tipo de instalac¸a˜o apresentara´ apenas vantagens comparativas.
No passado, o armador ou o projectista tinha como escolha imediata um motor diesel lento
acoplado directamente a um he´lice de passo fixo, ou um motor diesel de me´dia velocidade,
a quatro tempos, accionando atrave´s de engrenagens redutoras um he´lice de passo fixo ou
controla´vel.
Actualmente, a propulsa˜o dos navios que entram em servic¸o e´ obtida com o acoplamento
directo, muito esporadicamente com engrenagens redutoras, de motores a dois tempos a he´lices
de passo fixo ou controla´vel, motores de me´dia velocidade a quatro tempos e engrenagens
redutoras ou ainda por instalac¸o˜es diesel-ele´ctricas com recurso a motores diesel, a quatro
tempos, ra´pidos ou de me´dia velocidade. Algumas variantes de instalac¸o˜es propulsoras esta˜o
representadas nas Fig. 4.1 e 4.2.
67
68 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
Figura 4.1: Variantes de instalac¸o˜es propulsoras diesel-mecaˆnicas lentas e
de me´dia velocidade.
Os motores diesel lentos predominam no sector do transporte de grane´is, l´ıquidos e so´lidos,
e contentores de longo curso. Motores de me´dia velocidade sa˜o preferidos em navios de
carga com menor dimensa˜o, ferries, turismo de passageiros, RoRo’s, bem como em nichos de
mercado muito espec´ıficos como os quebra-gelos, navios de apoio a plataformas de explorac¸a˜o
petrol´ıfera, etc.
No passado recente, estas tradicionais zonas de influeˆncia de cada um dos referidos tipos
de motores teˆm-se sobreposto. As novas gerac¸o˜es de motores a quatro tempos, com cilindros
de grande diaˆmetro e me´dia velocidade apresentam-se como soluc¸o˜es competitivas para navios
a operar em viagens de longo curso. Em contrapartida, os motores lentos a dois tempos com
cilindros de pequeno diaˆmetro tambe´m se apresentam como soluc¸o˜es va´lidas para os mercados
costeiro e fluvial.
Um aspecto fundamental a considerar no processode decisa˜o na escolha da instalac¸a˜o
propulsora sera´ necessariamente o custo. Na˜o so´ o custo inicial, o investimento a fazer na
aquisic¸a˜o do motor, mas tambe´m os custos associados a` operac¸a˜o do navio ou, de uma forma
mais geral, os custos totais do ciclo de vida do navio. Naqueles custos de operac¸a˜o devera˜o
ser tidos em conta, entre outros, os seguintes aspectos:
- o tipo de combust´ıvel que a instalac¸a˜o vai permitir consumir;
- uma previsa˜o dos custos de manutenc¸a˜o;
- os recursos humanos exigidos para a operac¸a˜o/conduc¸a˜o da instalac¸a˜o;
- a disponibilidade e quantidade/custo dos sobressalentes.
4.2. PROPULSA˜O DIESEL-MECAˆNICA 69
Figura 4.2: Instalac¸o˜es propulsoras diesel-mecaˆnica (em cima) e diesel-
ele´ctrica (em baixo).
A avaliac¸a˜o dos custos de operac¸a˜o ao fim da vida de explorac¸a˜o do navio pode variar de
forma muito significativa com o tipo de motor escolhido, e com a configurac¸a˜o da instalac¸a˜o
propulsora adoptada.
A dimensa˜o da casa da ma´quina, a cujo aumento correspondera´ uma reduc¸a˜o do espac¸o
de carga dispon´ıvel para a explorac¸a˜o do navio, e´ essencialmente condicionada pela dimensa˜o
da ma´quina principal. A pro´pria altura da casa da ma´quina e´ importante em alguns tipos de
navios como os ferries com conve´s para ve´ıculos.
4.2 Propulsa˜o diesel-mecaˆnica
Conforme ja´ referido, a propulsa˜o por um he´lice de passo fixo accionado directamente por um
motor diesel lento a dois tempos continua a ser o sistema mais frequentemente encontrado em
navios de carga de longo curso. A ligeira reduc¸a˜o no rendimento de propulsa˜o reconhecida e´
admitida face a` simplicidade da soluc¸a˜o obtida e, a introduc¸a˜o de motores de longo, super-,
e ultra-longo curso veio diminuir aquelas perdas. No entanto, a velocidade de 100/110 rpm
na˜o e´ necessariamente a mais adequada para a propulsa˜o de um grande navio. Os motores
actualmente dispon´ıveis com maior curso desenvolvem a sua poteˆncia nominal a velocidades
ta˜o baixas como 55 rpm ate´ cerca de 250 rpm. Para um dado navio, e´ enta˜o poss´ıvel prescrever
uma soluc¸a˜o de acoplamento directo motor/he´lice que permita optimizar o rendimento de
propulsa˜o.
Um outro aspecto a considerar e´ o nu´mero de cilindros do motor. Os motores lentos actu-
ais, com cilindros de grande diaˆmetro, permitem extrair a poteˆncia necessa´ria a` propulsa˜o de
um navio de um motor com um reduzido nu´mero de cilindros. Um motor com menos cilindros
influencia naturalmente de forma favora´vel a dimensa˜o da casa da ma´quina, o volume de tra-
balho afecto a` sua manutenc¸a˜o e a quantidade de sobressalentes a manter no navio. Este tipo
de soluc¸a˜o e´ portanto bem acolhida desde que daqui na˜o resultem problemas de equil´ıbrio
do motor e vibrac¸o˜es. Estes motores com cilindros de grande diaˆmetro queimam bem com-
bust´ıveis pesados de fraca qualidade e proporcionam um consumo espec´ıfico de combust´ıvel
70 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
inferior ao obtido em motores com cilindros de menor diaˆmetro.
Neste tipo de instalac¸o˜es, a energia ele´ctrica necessa´ria ao funcionamento dos equipamen-
tos auxiliares e´ normalmente fornecida por geradores accionados por motores diesel ra´pidos
ou de me´dia velocidade. A grande parte dos fabricantes de motores diesel para acciona-
mento de alternadores esta´ ja´ hoje em condic¸o˜es de oferecer soluc¸o˜es capazes de consumir o
mesmo combust´ıvel que a ma´quina principal, ou “marine diesel-oil” ou ainda uma mistura
(blended) de combust´ıveis pesado e destilado. Actualmente, sa˜o ja´ comuns instalac¸o˜es pro-
pulsoras “Unifuel”, nas quais ma´quina principal e motores auxiliares consomem o mesmo tipo
de combust´ıvel.
4.2.1 Accionamento de auxiliares
Os custos associados a` produc¸a˜o da energia ele´ctrica necessa´ria ao funcionamento dos equi-
pamentos auxiliares da instalac¸a˜o sa˜o tambe´m um factor importante na selecc¸a˜o da ma´quina
principal. O desenvolvimento das ma´quinas tem tido como principais objectivos nesta a´rea:
- maximizar o aproveitamento de energia para permitir complementar a produc¸a˜o de
energia ele´ctrica durante as viagens;
- permitir o uso de alternadores accionados pela ma´quina principal atrave´s de engrenagens
multiplicadoras ou directamente montados na linha de veio;
- possibilitar o accionamento de equipamentos auxiliares directamente pela ma´quina prin-
cipal.
A principal motivac¸a˜o para a produc¸a˜o de energia ele´ctrica a partir da ma´quina principal
resulta do seu superior rendimento te´rmico, menor consumo espec´ıfico de combust´ıvel e ca-
pacidade para consumir combust´ıveis de inferior qualidade e custo. Outra vantagem resulta
naturalmente do menor consumo de o´leo lubrificante, de menos intervenc¸o˜es de manutenc¸a˜o e
inferiores custos com sobressalentes resultantes da reduc¸a˜o do tempo de funcionamento obtida
com a paragem dos diesel-geradores durante a viagem.
No caso de uma instalac¸a˜o com he´lice de passo fixo, a utilizac¸a˜o de um acoplamento por
engrenagens, que permita manter constante a velocidade de rotac¸a˜o do alternador (Fig. 4.3),
possibilita a utilizac¸a˜o do gerador a plena carga numa gama de velocidades da ma´quina
principal que habitualmente ronda os 70 a 100% da sua velocidade nominal.
A localizac¸a˜o do alternador e´ tambe´m um aspecto importante para permitir a deseja´vel
reduc¸a˜o de espac¸o ocupado pela casa da ma´quina. Sa˜o actualmente poss´ıveis diversos arranjos
que va˜o desde a colocac¸a˜o lateral ao motor ou em qualquer uma das suas extremidades.
Em alternativa, quer no caso das instalac¸o˜es com he´lice de passo fixo, quer no caso daquelas
que dispo˜em de passo controla´vel, podem ser utilizados sistemas baseados na conversa˜o da
frequeˆncia da energia ele´ctrica produzida (Fig. 4.4).
Mais recentemente, as opc¸o˜es para a produc¸a˜o de energia ele´ctrica a bordo alargaram-
se a` utilizac¸a˜o de turbinas movimentadas pelos gases de evacuac¸a˜o do motor. O elevado
rendimento dos sobrealimentadores mais modernos torna excedenta´ria uma fracc¸a˜o dos gases
de evacuac¸a˜o. O aproveitamento destes gases de evacuac¸a˜o em pequenas turbinas podera´
integrar-se em sistemas, que contemplando ainda grupos diesel-geradores, geradores-ao-veio e
turbo-geradoras a vapor, de forma isolada ou combinada, permitira˜o a optimizac¸a˜o dos custos
de produc¸a˜o da energia ele´ctrica para os va´rios estados de operac¸a˜o do navio.
4.2. PROPULSA˜O DIESEL-MECAˆNICA 71
Figura 4.3: Acoplamento com relac¸a˜o varia´vel de velocidades.
4.2.2 Engrenagens redutoras
Em muitas instalac¸o˜es propulsoras espera-se da caixa redutora:
- a determinac¸a˜o da velocidade e do sentido de rotac¸a˜o do he´lice, e a capacidade de
inversa˜o;
- que proporcione uma forma de acoplamento, permitindo estabelecer e interromper a
transmissa˜o de poteˆncia entre o motor e o he´lice;
- que seja capaz de absorver o impulso recebido do he´lice.
O projecto de engrenagens, embraiagens ou outras formas de acoplamento usadas em ins-
talac¸o˜es navais teˆm de satisfazer va´rios, e por vezes conflituantes, requisitos quanto a` sua
flexibilidade operacional, fiabilidade, ru´ıdo emitido e espac¸o ocupado. Os desenvolvimentos
nas a´reas do projecto, dos materiais e dos sistemas de controlo contribu´ıram para soluc¸o˜es
inovadoras para instalac¸o˜es propulsoras versa´teis com um ou mais motores, envolvendo toma-
das de extrac¸a˜o de poteˆncia (“Power Take-Off’s - PTO”) para accionamento de alternadores
e tomadas para recepc¸a˜o de poteˆncia (“Power Take-In’s - PTI ”) para aumentar a poteˆncia
de propulsa˜o.
A forma mais comum do accionamento indirecto do he´lice passa pela utilizac¸a˜o de um
ou mais motores a quatro tempos de me´dia velocidade, ligados atrave´s de embraiagens e
acoplamentos a uma caixa redutora, para movimentar um he´lice de passo fixo ou controla´vel
(Fig. 4.5 e 4.6).
A utilizac¸a˜o de he´licesde passo controla´vel permite eliminar a necessidade da reversibili-
dade do motor. Por outro lado, a utilizac¸a˜o da caixa redutora permite escolher a velocidade
de funcionamento do he´lice mais apropriada. De uma forma geral, pode-se afirmar que as per-
das mecaˆnicas na transmissa˜o sa˜o compensadas por um maior rendimento propulsivo, quando
72 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
Figura 4.4: Conversa˜o da frequeˆncia da energia ele´ctrica.
comparado com um caso de acoplamento directo para a mesma poteˆncia. Os custos adicio-
nais da transmissa˜o sa˜o tambe´m, pelo menos parcialmente, compensados pelo menor custo
do motor a quatro tempos, quando comparado com um motor lento a dois tempos.
Sa˜o normalmente identificadas como principais vantagens das instalac¸o˜es propulsoras com
mais de um motor, ra´pido ou de me´dia velocidade:
- a redundaˆncia permite maior disponibilidade para a operac¸a˜o do navio:
- no caso de avaria num motor, o outro ou os outros manteˆm a navegabilidade;
- o nu´mero de motores em servic¸o para a propulsa˜o pode variar para garantir a forma
mais econo´mica para uma viagem:
- quando o navio viaja em lastro, carga parcial ou a velocidade reduzida um
dos motores pode ser utilizado a` sua poteˆncia nominal, com bom rendimento,
enquanto outro ou outros podem ser parados;
- pelo contra´rio, em condic¸o˜es operacionais semelhantes, um motor u´nico, aco-
plado directamente ao he´lice, funcionaria durante longos per´ıodos a carga par-
cial com pouco rendimento;
- A possibilidade de alterar o nu´mero de motores em servic¸o facilita o planeamento e a
execuc¸a˜o das tarefas de manutenc¸a˜o e reparac¸a˜o uma vez que estas podera˜o ser realiza-
das em viagem.
- Esta flexibilidade de operac¸a˜o e´ particularmente valorizada numa e´poca em que se
pretende uma explorac¸a˜o intensiva dos navios.
- As operac¸o˜es de manutenc¸a˜o e reparac¸a˜o podem ainda decorrer em porto sem
preocupac¸o˜es particulares relativas a` necessidade de mudanc¸a de cais ou partida
antecipada.
- As instalac¸o˜es propulsoras de uma frota de navios pode ser baseada num so´ modelo
de motor, ajustando o nu´mero de motores no navio e o nu´mero de cilindros por motor
para as necessidades de propulsa˜o de cada um dos navios, com reduc¸a˜o do custo de
4.2. PROPULSA˜O DIESEL-MECAˆNICA 73
Figura 4.5: Instalac¸a˜o propulsora com quatro motores, engrenagens redu-
toras e dois he´lices.
sobressalentes e inventa´rios, para ale´m dos benef´ıcios resultantes da familiarizac¸a˜o das
tripulac¸o˜es.
Este conceito pode ainda ser alargado aos motores auxiliares (“uniform machinery instal-
lations ”), em que os motores principais e auxiliares sa˜o do mesmo modelo.
4.2.3 Configurac¸a˜o ”pai-e-filho”
A flexibilidade de operac¸a˜o e´ potenciada pela adopc¸a˜o das instalac¸o˜es do tipo ”pai-e-filho”.
Nestas instalac¸o˜es, motores a quatro tempos do mesmo modelo, ou de dois modelos muito
semelhantes, mas com diferente nu´mero de cilindros, fazem o accionamento do veio do he´lice
acoplados a uma caixa redutora comum. Cada um daqueles motores pode ser ainda acoplado
a uma ma´quina ele´ctrica que pode funcionar como motor ou gerador.
Numa configurac¸a˜o deste tipo, a propulsa˜o pode ser assegurada:
- conjuntamente pelos dois motores diesel;
- apenas por qualquer um dos motores diesel.
Em qualquer dos casos, podem ser ainda utilizados os, nesta situac¸a˜o, motores ele´ctricos
acoplados ao veio como motores propulsores, alimentados com energia ele´ctrica produzida
pelos geradores auxiliares.
74 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
Figura 4.6: Instalac¸a˜o com dois motores diesel diferentes, engrenagens re-
dutoras, embraiagens e geradores acoplados aos veios.
4.3 Propulsa˜o diesel-ele´ctrica
4.3.1 Propulsa˜o por motor ele´ctrico
A propulsa˜o diesel-ele´ctrica, baseada em grupos electroge´neos de me´dia velocidade, e´ uma
forma de accionamento indirecto com crescente implantac¸a˜o no mercado. Apo´s um per´ıodo
em que a utilizac¸a˜o deste tipo de sistemas esteve confinada a nichos de mercado de actividades
com elevada especificidade, como por exemplo os quebra-gelos, navios de investigac¸a˜o etc.,
as mais recentes tecnologias para a conversa˜o AC/DC alargaram o potencial de utilizac¸a˜o da
propulsa˜o ele´ctrica ao mercado dos navios de passageiros, “shuttle tanker’s” no Mar do Norte.
Estando ja´ estabelecido como uma boa soluc¸a˜o neste mercados, comec¸am a surgir refe-
reˆncias da aplicac¸a˜o deste tipo de instalac¸o˜es propulsoras a navios de transporte de qu´ımicos
(costeiro e longo curso), ferries e RoRo’s. Discute-se ainda as vantagens da sua aplicac¸a˜o
pelo menos a algumas classes de porta-contentores. A propulsa˜o diesel-ele´ctrica, combinada
com motores “dual-fuel”, esta´ tambe´m bem implantada no sector do transporte de LNG.
A propulsa˜o diesel-ele´ctrica exige grandes motores ele´ctricos para accionamento dos he´-
lices (Fig. 4.7) e grupos electroge´neos para fornecer a poteˆncia ele´ctrica. Pode parecer em
primeira ana´lise algo ilo´gico usar geradores ele´ctricos, conversores e motores ele´ctricos para o
accionamento quando um acoplamento directo ou uma engrenagem redutora pode ser sufici-
ente para cumprir aquela missa˜o. As principais razo˜es que justificam a complexidade e custo
acrescidos daquele tipo de instalac¸a˜o sa˜o:
- maior flexibilidade na distribuic¸a˜o dos equipamentos na casa da ma´quina;
- maior diversidade de condic¸o˜es de fundionamento;
- funcionamento mais econo´mico a carga partial;
- facilidade de controlo;
- menor ru´ıdo;
- maior seguranc¸a de operac¸a˜o e protecc¸a˜o ambiental.
Estes aspectos sera˜o abordados nos para´grafos seguintes.
4.3. PROPULSA˜O DIESEL-ELE´CTRICA 75
Figura 4.7: Motor ele´ctrico de propulsa˜o.
Flexibilidade na distribuic¸a˜o dos equipamentos
A vantagem da transmissa˜o ele´ctrica resulta de se poder escolher a localizac¸a˜o em cada
caso mais apropriada para os grupos electroge´neos. E´ enta˜o poss´ıvel colocar os motores, bem
como os respectivos auxiliares, afastados do veio propulsor. Sempre que seja adoptado este
tipo de instalac¸a˜o, a referida flexibilidade permite aos arquitectos navais criar navios com a
casa da ma´quina muito compacta, libertando espac¸o para passageiros e/ou carga. O facto
de a casa da ma´quina ser mais compacta permite reduzir ainda a cablagem e a tubagem, em
particular a tubagem a instalar para a evacuac¸a˜o dos gases do motor (ver Fig. 4.8).
A opc¸a˜o por uma instalac¸a˜o diesel-ele´ctrica facilita tambe´m ao estaleiro de construc¸a˜o a
recepc¸a˜o de mo´dulos de grupos electroge´neos pre´-testados e prontos para serem incorporados
na instalac¸a˜o.
Deve aqui ser tambe´m referida a dificuldade de uma instalac¸a˜o diesel-ele´ctrica atingir o
rendimento obtido com um motor lento, a dois tempos, acoplado directamente ao veio do
he´lice, quando a funcionar a` sua carga ideal, tal como acontece numa viagem de longo curso
de um navio petroleiro. No entanto, alguns navios deste tipo teˆm um perfil de operac¸a˜o
que inclui tambe´m largos per´ıodos a carga parcial em lastro, navegac¸a˜o em a´guas restritas
e manobras. Numa instalac¸a˜o diesel-ele´ctrica, a elevada disponibilidade para produc¸a˜o de
energia ele´ctrica pode ser aproveitada para movimentar as bombas de carga e impulsores de
proa/popa, conforme representado esquematicamente na Fig. 4.9.
Variedade de carga
Alguns tipos de navios necessitam de quantidades significativas de energia para auxiliares
quando as necessidades de propulsa˜o sa˜o reduzidas. Uma grande instalac¸a˜o de produc¸a˜o
76 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
Figura 4.8: Instalac¸a˜o diesel-ele´ctrica.
de energia ele´ctrica nos navios de passageiros/cruzeiros e´ exigida pela carga dos servic¸os de
hotelaria e pelos propulsores tranversais de manobra. A poteˆncia ele´ctrica necessa´ria nestes
casos ronda os 30 a 40 % dapoteˆncia de propulsa˜o instalada e ainda ha´ que contar com
significativa redundaˆncia por motivos de seguranc¸a.
Estes factores teˆm promovido um novo conceito de instalac¸a˜o, a diesel-ele´ctrica ”power
station”, nas quais va´rios grupos electroge´neo movidos por motores diesel de me´dia velocidade
satisfazem as necessidades de energia para a propulsa˜o, manobra e servic¸os de hotelaria nos
grandes navios de passageiros.
Funcionamento econo´mico a carga parcial
Funcionamento econo´mico a carga parcial e´ facilmente alcanc¸ado numa instalac¸a˜o diesel-
ele´ctrica ”power station”. Uma instalac¸a˜o t´ıpica inclui quatro grupos electroge´neos, podendo
ir no entanto ate´ aos nove, e, atrave´s do funcionamento em paralelo dos grupos, e´ fa´cil ajustar
a capacidade de produc¸a˜o a`s necessidades de carga ele´ctrica. Por exemplo, no caso de quatro
geradores, aumentar o nu´mero de grupos em funcionamento de dois, a` carga ma´xima, para
treˆs a carga parcial resulta numa condic¸a˜o de carga a 67 % que, na˜o sendo ideal tambe´m na˜o
e´ problema´tica.
Os sistemas de reduc¸a˜o instantaˆnea da poteˆncia propulsora tornam desnecessa´rio colocar
em funcionamento geradores a carga parcial para prevenir a ocorreˆncia su´bita de avaria num
grupo electroge´neo. O sistema de controlo monitoriza a capacidade de produc¸a˜o de energia
ele´ctrica, e a sobrecarga de um gerador provoca um ajuste imediato no consumo dos motores
de propulsa˜o.
4.3. PROPULSA˜O DIESEL-ELE´CTRICA 77
Figura 4.9: Representac¸a˜o esquema´tica de uma instalac¸a˜o diesel-ele´ctrica.
Facilidade de controlo
Os accionamentos ele´ctricos permitem alcanc¸ar, com larga margem, as necessidades de
controlo para um sistema de propulsa˜o.
Baixo ru´ıdo
Um motor ele´ctrico proporciona um accionamento com vibrac¸o˜es reduzidas, caracter´ıstica
particularmente valorizada nalguns tipos de navios como, por exemplo, os navios para cru-
zeiros, navios de investigac¸a˜o marinha e navios de guerra. A “transmissa˜o ele´ctrica” permite
procurar a melhor localizac¸a˜o para os motores por forma a minimizar os efeitos da vibrac¸a˜o
transmitida a` estrutura do navio. A emissa˜o de vibrac¸o˜es pode ainda ser reduzida atrave´s do
recurso a` montagem de amortecedores de vibrac¸a˜o.
Protecc¸a˜o ambiental e seguranc¸a de operac¸a˜o
O controlo das emisso˜es de o´xidos de azoto pelos motores diesel dos navios favorece tambe´m
a especificac¸a˜o de instalac¸o˜es com “transmissa˜o ele´ctrica”, uma vez que o funcionamento dos
motores a velocidade constante e carga optimizada permite obter menores emisso˜es.
O aumento da seguranc¸a da navegac¸a˜o e´ tambe´m obtido nestas instalac¸o˜es pela redun-
daˆncia dos seus elementos constituintes. A redundaˆncia pode ser obtida na˜o apenas pela
existeˆncia de dois propulsores mas ainda pode ser acrescida colocando os dois, ou mais, mo-
tores de propulsa˜o em diferentes compartimentos e ligando-os por uma engrenagem redutora.
4.3.2 Propulsores azimutais
As vantagens te´cnicas e econo´micas na concepc¸a˜o, construc¸a˜o e operac¸a˜o de navios com
propulsa˜o por “azipod’s”, inicialmente restritos a navios quebra-gelos e navios de passageiros,
teˆm vindo a alargar o seu campo de aplicac¸a˜o a outro tipo de navios.
Um propulsor azimutal incorpora o motor ele´ctrico num alojamento submerso de formas
hidrodinaˆmicas optimizadas que, podendo rodar 360◦ no plano horizontal, permite extraor-
78 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
dina´ria capacidade de propulsa˜o e manobra (ver Fig. 4.10). O motor ele´ctrico e´ acoplado
directamente a um he´lice de passo fixo. A energia ele´ctrica e´ provida pelos va´rios grupos
electroge´neos do navio.
Figura 4.10: Propulsores azimutais.
Este tipo de propulsores, quando comparados com instalac¸o˜es diesel-ele´ctricas com linha(s)
de veio(s) apresentam as seguintes vantagens:
- maior liberdade para a concepc¸a˜o do casco e para o arranjo de ma´quinas no interior da
casa da ma´quina;
- o espac¸o no interior do casco destinado aos motores pode ser libertado para outras
finalidades;
- melhor capacidade de manobra quando comparado com o tradicional leme e possibidade
de eliminar propulsores transversais;
- excelente reversibilidade e capacidade de manobra com propulsa˜o a re´;
- menor ru´ıdo e vibrac¸a˜o, caracter´ısticos da propulsa˜o ele´ctrica, agora potenciados pela
posic¸a˜o mais favora´vel dos he´lices;
- na construc¸a˜o do navio, as unidades de propulsa˜o podem ser incorporadas mais tarde
reduzindo assim os custos de investimento;
- menor custo de produc¸a˜o do navio.
4.4 Selecc¸a˜o do motor
Seleccionado o tipo de instalac¸a˜o pretendido para a propulsa˜o do navio, chega-se finalmente
a` escolha do motor. Como as caracter´ısticas de funcionamento das turbinas e dos motores
4.4. SELECC¸A˜O DO MOTOR 79
ele´ctricos sa˜o bastante diferentes das caracter´ısticas dos motores diesel, a abordagem tera´ de
ser tambe´m diferente.
Em qualquer dos casos, devera´ ser tida em conta a margem de servic¸o MS. A margem
de servic¸o tem em conta a diferenc¸a entre a poteˆncia requerida para nas condic¸o˜es ideais da
prova de mar e a poteˆncia requerida pelas condic¸o˜es de servic¸o. E´ pra´tica habitual definir-se
a margem de servic¸o como uma fracc¸a˜o da poteˆncia na prova de mar, ou seja,
MS =
PDserv − PDtrial
PDtrial
(4.1)
O valor da margem de servic¸o esta´ normalmente entre os 10 e os 25%, dependendo das opc¸o˜es
estrate´gicas do armador e da importaˆncia da pontualidade do servic¸o. Em princ´ıpio, a margem
de servic¸o atribu´ıda a um navio de linha sera´ superior a` margem considerada para um navio
que vai operar no mercado do “tramping”. O valor estabelecido da margem de servic¸o deve em
conta uma estimativa da degradac¸a˜o de velocidade, para as condic¸o˜es de operac¸a˜o do navio,
bem com as condic¸o˜es habituais de mar e vento e a degradac¸a˜o do casco.
4.4.1 Turbinas e motores ele´ctricos
No caso da turbinas, de vapor ou ga´s, a poteˆncia desenvolvida depende essencialmente do
caudal de fluido em circulac¸a˜o, sendo portanto relativamente pouco sens´ıvel a` velocidade de
rotac¸a˜o.
As caracter´ısticas dos sistemas com transmissa˜o ele´ctrica sa˜o semelhantes a`s das turbinas,
independentemente de os geradores serem movidos por turbinas ou motores diesel, uma vez
que a velocidade destes pode ser mantida constante.
Neste tipo de situac¸a˜o, em que a ma´quina propulsora pode trabalhar pro´ximo da poteˆncia
ma´xima em qualquer condic¸a˜o de servic¸o, a poteˆncia instalada (PI) pode ser pro´xima da
poteˆncia de servic¸o. Na pra´tica, a turbina e´ ajustada para operar com o ma´ximo rendimento
a uma poteˆncia 10% inferior a` ma´xima poteˆncia em cont´ınuo (MCR, Maximum Continuous
Rating). Assim, a poteˆncia instalada sera´
PI(MCR) =
PDserv
0, 9ηs
= PDtrial
1 +MS
0, 9ηs
(4.2)
em que PDserv e PDtrial sa˜o as poteˆncias absorvidas pelo he´lice nas condic¸o˜es de servic¸o e na
prova de mar, respectivamente, para a velocidade de servic¸o e MS e´ a margem de servic¸o.
4.4.2 Motores diesel
Ao contra´rio das turbinas e dos motores ele´ctricos, em que a poteˆncia dispon´ıvel e´ pouco
sens´ıvel a` velocidade, os motores diesel caracterizam-se por ter uma curva do bina´rio bastante
plana. Esta caracter´ıstica faz com que a poteˆncia varie de forma aproximadamente linear com
a velocidade de rotac¸a˜o.
Para ale´m dos principais crite´rios considerados na avaliac¸a˜o dos projectos, outros aspectos
que na˜o devem ser descurados na escolha do motor sa˜o:
- a possibilidade de queimar combust´ıvel pesado de baixa qualidade sem impacto nos
componentes do motor e consequentemente nos custos previstos para sobressalentes e
operac¸o˜es de manutenc¸a˜o;
80 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
- o volume de trabalho de manutenc¸a˜o, o nu´mero de cilindros, va´lvulas, camisas, aros
e chumaceiras a necessitar de atenc¸a˜o perio´dicaem relac¸a˜o ao nu´mero de tripulantes
embarcados;
- a adequabilidade para operac¸a˜o na˜o assistida explorando sistemas de controlo automa´-
tico e sistemas de monitorizac¸a˜o;
- a dimensa˜o e o peso da instalac¸a˜o propulsora.
O valor ma´ximo da poteˆncia desenvolvida por um motor diesel e´ condicionada pela carga
te´rmica. Este limite e´ normalmente expresso em termos da pressa˜o me´dia efectiva. Depen-
dendo das caracter´ısticas do he´lice seleccionado e das condic¸o˜es operacionais, assim o valor
limite da pressa˜o me´dia efectiva sera´ atingido, ou na˜o, antes de o motor atingir a velocidade
de rotac¸a˜o correspondente a`s condic¸o˜es MCR.
Figura 4.11: Diagrama de carga de um motor diesel
Os fabricantes de motores diesel incluem diagramas de carga nos guias de selecc¸a˜o de
motores para auxiliar a escolha do ponto de funcionamento. Nestes diagramas, como o repre-
sentado na Fig. 4.11, esta˜o marcados:
- o ponto L1, que corresponde ao MCR do motor;
- a linha vertical L1 − L2, velocidade de rotac¸a˜o ma´xima do motor, que limita a zona de
funcionamento do motor;
No Apeˆndice D incluiu-se documentac¸a˜o da ”Burmeister & Wain” que permite ilustrar a
forma de selecc¸a˜o do motor para uma aplicac¸a˜o concreta, considerando va´rias hipo´teses: com
ou sem gerador acoplado ao veio, com he´lice de passo fixo ou de passo controla´vel.
4.4. SELECC¸A˜O DO MOTOR 81
Alguns fabricantes anunciam um valor de “Normal Continuous Rating” (NCR) cerca de
10% inferior ao valor MCR e a uma velocidade inferior, ao qual corresponde um desempenho
optimizado do motor em termos de consumo e de necessidades de manutenc¸a˜o. Pode ainda
definir-se uma “Service Continuous Rating” (SCR) que, dependendo da pol´ıtica do armador,
podera´ ser igual ou na˜o do NCR indicado pelo fabricante do motor.
A diferenc¸a entre a MCR e a SCR, ou, caso na˜o esteja definida, a NCR, da´ origem a`
chamada margem do motor (MM). A margem do motor e´ avaliada por,
MM =
MCR− SCR
MCR
(4.3)
Valores t´ıpicos desta margem de motor rondam os 10 a 15%. De notar que as margens de
servic¸o e de motor surgem frequentemente combinadas numa so´, a margem de servic¸o, apesar
de as suas origens serem bem distintas.
Uma vez atribu´ıdas as margens de servic¸o e de motor, a poteˆncia instalada e´ calculada
por
PI(MCR) = PDtrial
1 +MS
(1−MM) ηs (4.4)
Nas provas de mar, nas condic¸o˜es de imersa˜o e caimento contratuais, a poteˆncia absorvida
pelo he´lice, a` velocidade de rotac¸a˜o correspondente ao MCR, deve ser igual a` poteˆncia SCR,
deduzida das perdas na linha de veios. Como objectivo das provas, devera´ garantir-se que a
combinac¸a˜o motor e he´lice permite que o anvio atinja a velocidade requerida sem ultrapassar
os limites impostos pelo diagrama de carga.
Sem preju´ızo do exposto, o forte aumento do prec¸o dos combust´ıveis nos anos mais recen-
tes faz com que os custos operacionais dos navios sejam cada vez mais dominados por este
factor. Neste contexto, pode ser uma hipo´tese de trabalho interessante a opc¸a˜o por um motor
com a mesma poteˆncia, a poteˆncia calculada como necessa´ria para a propulsa˜o nas condi-
c¸o˜es contratuais, mas com um cilindro extra. Esta te´cnica, o chamado ”derating” do motor,
exigindo maior valor de investimento inicial, pode apresentar um per´ıodo de retorno atrac-
tivo. Wettstein e Brown apresentam as principais motivac¸o˜es para aplicac¸a˜o desta te´cnica e
discutem quatro casos de aplicac¸a˜o numa publicac¸a˜o da Wa¨rtsilla¨, inclu´ıda no Apeˆndice E.
82 CAPI´TULO 4. INSTALAC¸O˜ES PROPULSORAS
Bibliografia
[1] Jose´ P. Saraiva Cabral. Arquitectura Naval, estabilidade, ca´lculos, avaria e bordo livre.
Centro do Livro Brasileiro, 1979.
[2] Eric C. Tupper. Introduction to Naval Arquitecture. Elsevier, 2004.
[3] Volker Bertram. Practical Ship Hydrodynamics. Butterworth-Heinemann, 2000.
[4] Jorge d’Almeida. Arquitectura Naval - o dimensionamento do navio. Prime Books, 2009.
[5] Editor Doug Woodyard. Pounders Marine Diesel Engines and Gas Turbines. Butterworth-
Heinemann, 2004.
[6] H. Schneekluth and V. Bertram. Ship Design for Efficiency and Economy. Butterworth-
Heinemann, 1998.
83
I´ndice Remissivo
Auto-propulsa˜o, 62
Boca, 3
Bolbo de proa, 22
Bordo livre, 3
Calado, 3
Camada limite, 24
Cavitac¸a˜o, 37, 53, 60
Coeficiente
de avanc¸o, 46
de bina´rio, 46
de Burrill, 55
de carga do he´lice, 44
de deduc¸a˜o da esteira, 63
de deduc¸a˜o da forc¸a propulsiva, 63
de forc¸a propulsiva, 46
de resisteˆncia, 28
de resisteˆncia total, 13
Comprimento
entre perpendiculares, 3
fora a fora, 3
na linha de a´gua, 3
Consumo espec´ıfico de combust´ıvel, 69
Custos
de manutenc¸a˜o, 68
de operac¸a˜o, 68, 69
totais, 68
Diagrama
de Burrill, 55
em a´guas livres, 45, 46
Dual-fuel, 74
Engrenagens redutoras, 71
Ensaios
de auto-propulsa˜o, 62
de cavitac¸a˜o, 56
de he´lices em a´guas livres, 45
de propulsa˜o, 61
de resisteˆncia, 26
em sobrecarga, 62
Fo´rmula
de Alexander, 5
de atrito da ATTC, 25
de atrito da ITTC, 25
de Keller, 59
do atrito de Froude, 24
do atrito de Hugues, 30
Forc¸a
de compensac¸a˜o, 62
de ine´rcia, 15
de origem hidrodinaˆmica, 16
grav´ıtica, 16
propulsiva, 42
He´lice, 35
rendimento ideal, 45
a ponto fixo, 57
bloqueado, 57
com tubeira, 36
contrarotativo, 37
de passo controla´vel, 37, 67, 70, 71
de passo fixo, 37, 67, 70, 71, 78
diaˆmetro do, 58
distribuic¸a˜o radial de pressa˜o, 59
geometria do, 40, 59
ı´ndice de qualidade do, 47
interacc¸a˜o com o casco, 60
nu´mero de pa´s do, 59
projecto do, 40
raza˜o de a´rea expandida, 41
supercavitante, 37
Me´todo
de Hughes/Prohaska, 28
84
I´NDICE REMISSIVO 85
Geosim, 28, 31
Hughes-Prohaska, 29
ITTC 1957, 28
ITTC 1978, 28, 30
Margem
de servic¸o, 79
do motor, 81
Maximum Continuous Rating, 79
Nu´mero
de cavitac¸a˜o, 54
de Froude, 17, 23
de Reynolds, 18, 27, 46
Navio
coeficientes de forma, 3
de passageiros, 68, 74, 76, 77
deslocamento do, 3
dimenso˜es do, 3
linhas de bordo livre do, 3
planos do, 1
quebra-gelos, 68, 77
tipo ferry, 37, 38, 40, 68, 69, 74
tipo RoRo, 68, 74
tipo shuttle tanker, 74
Normal Continuous Rating, 81
PC-cluster, 10
Pontal, 3
Poteˆncia
absorvida, 65
de reboque, 13
efectiva, 13, 62
efectiva do motor, 65
propulsiva, 63
Power Take Off/In, 71
Profundidade restrita, 23, 32
Propulsa˜o
azimutal, 35, 38, 77
cicloidal, 35, 39
diesel-ele´ctrica, 74
diesel-mecaˆnica, 69
por jacto de a´gua, 35, 37
por motor ele´ctrico, 74
Provas
de mar, 34
de poteˆncia, 121, 133
de velocidade, 121, 133
Rendimento
a´guas livres, 46
da linha de veios, 65
do casco, 64
do he´lice, 46
rotativo relativo, 64
Resisteˆncia, 13
adicional, 31
aerodinaˆmica, 19
de atrito, 24
de onda, 19
decomposic¸a˜o, 18
dos apeˆndices, 32
viscosa de pressa˜o, 25
Rugosidade do casco, 28, 30, 31
Se´rie sistema´tica
60, 33
de he´lices, 47, 58
de querenas, 32
de Taylor, 33
de Wageningen, 48
Semelhanc¸a
cinema´tica, 15
dinaˆmica, 15
geome´trica, 14
leis da, 14
Service Continuous Rating, 81
Sobrealimentadores, 70
Tanque
de cavitac¸a˜o, 56
de Froude, 7
de reboque, 26
Unifuel, 70
Velocidade
da querena, 22
de aproximac¸a˜o, 42
de rotac¸a˜o do he´lice, 59
econo´mica, 22
Vibrac¸o˜es, 42, 53, 58–60, 77
86 I´NDICE REMISSIVO
Apeˆndice A
Procedimento Recomendado pela
ITTC para a Previsa˜o do
Desempenho de Navios Baseada nos
Ensaios de Propulsa˜o em Modelos
87
88 APEˆNDICE A. PREVISA˜O BASEADA NOS ENSAIOS DE PROPULSA˜O
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 1 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
Edited by 22nd ITTC QS Group 1999 Approved 
15th ITTC1978 pp388 – 402 
17th ITTC 1984 pp326 - 333 
18th ITTC 1987 pp266 - 273 
15th ITTC 1978, 17th ITTC 1984 
and 18th ITTC 1987 
Date Date 
 
CONTENTS 
 
1. PURPOSE OF PROCEDURE 
2. DESCRIPTION OF PROCEDURE 
2.1.1 Introduction for the Original 1978 ITTC Performance Prediction Method 
for Single Screw Ships 
2.1.2 Introduction for the 1978 ITTC Performance Prediction Method as Modified 
in 1984 and 1987 
2.2 Model Tests 
2.3 Analysis of the Model Test Results 
2.4 Full Scale Predictions 
2.4.1 Total Resistance of Ship 
2.4.2 Scale Effect Corrections for Propeller Characteristics. 
2.4.3 Full Scale Wake and Operating Condition of Propeller 
2.4.4 Model-Ship Correlation Factors 
2.5 Analysis of Speed Trial Results 
2.6 Input Data 
2.7 Output Data 
2.8 Test Example 
3. PARAMETERS 
3.1 Parameters to be Taken into Account 
3.2 Recommendations of ITTC for Parameters 
3.3 Input Data 
4. VALIDATION 
4.1 Uncertainty Analysis 
4.2 Comparison With Full Scale Results 
5. ITTC- 1978 PERFORMANCE PREDICTION METHOD (COMPUTER CODE) 
 
 
 
 
COMMENTS OF PROPULSION COMMITTE OF 22nd ITTC 
In its original form the ITTC 1978 Performance Prediction Method offers a valuable and rea-
sonably accurate prediction tool for reference purposes and conventional ships. 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 2 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
1978 ITTC Performance Prediction Method 
 
1. PURPOSE OF PROCEDURE 
 
The method predicts rate of revolution and 
delivered power of a ship from model results. 
 
 
2. DESCRIPTION OF PROCEDURE 
 
2.1.1 Introduction for the Original 1978 
ITTC Performance Prediction Method 
for Single Screw Ships 
 
The method predicts rate of revolution and 
delivered power of a ship from model results. 
The procedure used can be described as fol-
lows: 
 
The viscous and the residuary resistance of the 
ship are calculated from the model resistance 
tests assuming the form factor to be independ-
ent of scale and speed. 
 
The ITTC standard predictions of rate of revo-
lutions and delivered power are obtained from-
the full scale propeller characteristics. These 
characteristics have been determined by cor-
recting the model values for drag scale effects 
according to a simple formula. Individual 
corrections then give the final predictions. 
 
 
2.1.2 Introduction for the 1978 ITTC Per-
formance Prediction Method as 
Modified in 1984 and 1987 
 
The 1978 ITTC Method developed to pre-
dict the rate of propeller revolutions and deliv-
ered power of a single screw ship from the 
model test results has been extended during the 
last two terms of the ITTC for a better and 
more convenient use of the program. These 
extensions are summarized as follows. 
 
(1) Inclusion of prediction of propeller revo-
lutions on the basis of power identity. 
 
(2) Temporary measure for wTS > wTM 
 
(3) Extension to twin screw ships 
 
(4) Addition of speed trial data 
 
(5) Extension for the case of a stock propel-
ler in the self-propulsion test 
 
(6) Adaptation to the input of the non-
dimensional resistance coefficient and 
self-propulsion factors. 
 
In recent years, many member organizations 
have been asked by their customers for a gen-
eral description of the method, viz., model test 
and analysis of their results, calculation of full-
scale power and rate of propeller revolutions, 
and the model-ship correlation factors used. 
Considering the above, it was decided to pre-
pare a user's manual of the 1978 ITTC method 
which includes all of the extensions and modi-
fications made. 
 
 
2.2 Model Tests 
 
Model tests required for a full scale com-
prise the resistance test, the self-propulsion test 
and the propeller open-water test. 
In the resistance test the model is towed at 
speeds giving the same Froude numbers as for 
the full scale ship, and the total resistance of 
the model RTM is measured. The computer pro-
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 3 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
gram accepts either RTM in Newton, or in a non-
dimensional form of residuary resistance coef-
ficient CR assuming the form factor 1 + k. In 
the latter case, the friction formula used can 
then be either of the ITTC 1957, Hughes, 
Prandtl-Schlichting or Schönherr's formulae. 
 
The form factor 1 + k is usually determined 
from the resistance tests at low speed range or 
by Prohaska’s plot of CFM against Fn4 
 
The ship model is not in general fitted with 
bilge keels. In this case the total wetted surface 
area of them is recorded and their frictional 
resistance is added in calculating the full-scale 
resistance of the ship. 
 
In the self-propulsion test the model is 
towed at speeds giving the same Froude num-
bers as for the full-scale ship. Generally a tow-
ing force FD is applied to compensate for the 
difference between the model and the full-scale 
resistance coefficient. 
 
During the test, propeller thrust (TM), torque 
(OM) and rate of propeller rotation (nM) are 
measured. 
 
In many cases, stock propellers are used 
which are selected in view of the similarity in 
diameter pitch and blade area to the full-scale 
propeller. Then the diameter and the open-
water characteristics of the stock propeller 
have to be given as input data in the program. 
In the open-water test, thrust, torque and rate of 
revolutions are measured, keeping the rate of 
revolutions constant whilst the speed of ad-
vance is varied so that a loading range of the 
propeller is examined. 
 
In the case when a stock propeller is used in 
the self-propulsion test, both the stock propel-
ler and the model similar to the full-scale pro-
peller should be tested in open water. 
 
 
2.3 Analysis of the Model Test Results 
 
Resistance RTM measured in the resistance 
tests is expressed in the non-dimensional form 
 
 
2
2
1 SV
RC TMTM ρ
= 
This is reduced to residual resistance coef-
ficient CR by use of form factor k, 
viz., 
 CR = CTM - CFM (1 + k) 
 
Thrust, T, and torque Q, measured in the 
self-propulsion tests are expressed in the non-
dimensional forms 
 
24nD
TKTM ρ= and 25nD
QKQM ρ= 
 
With KTM as input data, JTM and KQTM are read 
off from the model propeller characteristics, 
and the wake fraction 
 
 
V
DJ
w MTMTM −=1 
and the relative rotative efficiency 
 
QM
QTM
R K
K=η 
are calculated. V is model speed. 
The thrust deduction is obtained from 
 
 
T
RFT
t CD
−+= 
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Procedures 
7.5 – 02 
03 – 01.4 
Page 4 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
with 
 
( )[ ]FFSFMMMMD CCCVSF ∆+−= 22
1 ρ 
 
where RC is the resistance corrected for differ-
ences in temperature between resistance and 
self-propulsion tests: 
 
 
( )
( ) TMRFM
RFMC
C RCCk
CCk
R ++
++=
.1
.1
 
 
where CFMC is the frictional resistance coeffi-
cient at the temperature of the self-propulsion 
test. 
 
 
2.4 Full Scale Predictions 
 
2.4.1 Total Resistance of Ship 
 
The total resistance coefficient of a ship 
without bilge keels is 
 
 CTS =(1+k)CFS +CR+∆ CF +CAA 
 
Where 
 
- k is the form factor determined from the 
resistance test 
 
- CFS is the frictional coefficient of the ship 
according to the ITTC-1957 ship-model 
correlation line 
 
- CR is the residual resistance calculated from 
the total and frictional coefficients of themodel in the resistance tests: 
 ( ) FMTMR CkCC +−= 1 
 
-. FC∆ is the roughness allowance 
3
3
1
1064.0105 −








−


=∆
WL
S
F L
k
C 
 
where the roughness kS=150.10-6 m and 
 
- CAA, is the air resistance 
S
A
C TAA .001.0= 
 
If the ship is fitted with bilge keels the total 
resistance is as follows: 
 
( )[ ] AARFFSBKTS CCCCkS
SS
C ++∆+++= 1 
 
 
2.4.2 Scale Effect Corrections for Propeller 
Characteristics. 
 
The characteristics of the full scale propel-
ler are calculated from the model characteris-
tics as follows 
 
TTMTS KKK ∆−= 
 
QQMQS KKK ∆−= 
where 
 
 
D
Zc
D
PCK DT
..3.0.∆−=∆ 
 
 
D
ZcCK DQ
..25.0.∆−=∆ 
 
The difference in drag coefficient DC∆ is 
 
 DSDMD CCC −=∆ 
 
where 
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Procedures 
7.5 – 02 
03 – 01.4 
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Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
( ) ( ) 





−

 +=
3
2
6
1
504.0212
nconco
DM
RRc
tC 
and 
5.2
log.62.189.1212
−



 +

 +=
p
DS k
c
c
tC 
 
In the formulae listed above c is the chord 
length, t is the maximum thickness, P/D is the 
pitch ratio and Rnco is the local Reynolds num-
ber at x=0.75. The blade roughness kp is put 
kp=30.10-6 m. Rnco must not be lower than 2.105 
at the open-water test. 
 
 
2.4.3 Full Scale Wake and Operating Con-
dition of Propeller 
 
The full scale wake is calculated from the 
model wake, wTM, and the thrust deduction, t: 
 
( ) ( ) ( )( ) FM
FFS
TMTS Ck
CCk
twtw +
∆++−−++=
1
1
04.004.0
 
where 0.04 is to take account of rudder effect. 
The load of the full scale propeller is obtained 
from 
 
 ( )( )222 11.2 TS
TST
wt
C
D
S
J
K
−−= 
 
With this 2/ JKT as input value the full 
scale advance coefficient JTS and the torque 
coefficient KQTS are read off from the full scale 
propeller characteristics and the following 
quantities are calculated 
- the rate of revolutions: 
 
( )
DJ
Vw
n
TS
STS
S
−= 1 (r/s) 
 
- the delivered power: 
 335 102 −=
R
QTS
SDS
K
nDP ηπρ (kW) 
 
- the thrust of the propeller: 
 2422 ... STS
T
S nDJJ
K
T ρ= (N) 
 
- the torque of the propeller: 
 25 S
R
QTS
S nD
K
Q ρη= : (Nm) 
 
- the effective power: 
 33 10...2/1 −= SVCP STSE ρ (kW) 
 
- the total efficiency: 
 
E
DS
D P
P=η 
 
- the hull efficiency: 
 
TS
H w
t
−
−=
1
1η 
 
 
2.4.4 Model-Ship Correlation Factors 
 
Trial prediction of rate of revolutions and de-
livered power with CP - CN corrections 
 
if CHOICE=0 the final trial predictions will be 
calculated from 
 
 nT = CN.nS (r/s) 
 
for the rate of revolutions and 
 PDT = CP.PDS (kW) 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 6 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
for the delivered power. 
 
Trial prediction with ∆CFC - ∆wC corrections 
 
If CHOICE=1 the final trial predictions are 
calculated as follows: 
 
 ( )( )222 11.2 CTS
FCTST
wwt
CC
D
S
J
K
∆+−−
∆+= 
 
With this KT/J² as input value, JTS and KQTS 
are read off from the full scale propeller char-
acteristics and 
 ( )
DJ
Vww
n
TS
SCTS
T .
1 ∆+−= (r/s) 
 
 335 10.....2 −=
RM
QTS
TDT
K
nDP ηρπ (kW) 
 
 
Trial prediction with CNP correction 
 
If CHOICE = 2 the shaft rate of rotation is pre-
dicted on the basis of power identity as fol-
lows. 
 
( )³1²..2
..1000
³ 3 TSS
DSP
T
Q
wVD
PC
J
K
−=



ρπ 
 
 RM
T
QQ
J
K
J
K η.
³
0 


= 
 
 ( ) DJwVn TSTSSS ./1−= 
 
 SNPT nCn = 
 
 
2.5 Analysis of Speed Trial Results 
 
The analysis of trials data is performed in a 
way consistent with performance prediction but 
starting PD and n backwards, i.e. from 
 
 ³10..
...2 35 RM
D
Q nD
P
K ηρπ= 
 
JS is obtained from the full-scale open-water 
characteristics KQ ≈ JS then 
 
 VDnJw ST /..1−= 
 
Further from KT ≈ JS characteristics 
 
 4².. DnKT T ρ= 
 
 ( )
SV
tTCT
²...
2
1
1.
ρ
−= 
 
Then we obtain 
 TSTFC CCC −=∆ 
 TTSC www −=∆ 
 
 
2.6 Input Data 
 
Input data sheets are given in ENCL.1 
 
 
2.7 Output Data 
 
- Output data I gives ITTC Standard Pre-
diction with CP = CN = 1.0, together with 
model and full scale propulsive coeffi-
cients (ENCL. 4). 
- Output data II gives the final ship predic-
tion (ENCL. 5). 
 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 7 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
- Output data III gives the analysis of the 
speed trial results (ENCL. 6). 
 
 
2.8 Test Example 
 
To illustrate the program a prediction was 
made for a hypothetical ship with the following 
particulars: 
length between 
 perpendiculars Lpp = 251.5m 
 breadth B = 41.5m 
 draft T = 16.5m 
 propeller diameter D = 8.2m 
 
Calculations were carried out with the 
ITTC Trial Prediction Test Program with: 
 
CP = 1.01 
CN = 1.02 
 
The input data were taken as shown in 
ENCL. 1 and the printout of the input data and 
results are given in ENCL. 4 - 6. 
 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 8 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 9 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 10 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 11 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 12 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
3. PARAMETERS 
 
3.1 Parameters to be Taken into Account 
 
Froude scaling law 
ship-model correlation line ,friction line 
kinematic viscosity 
mass density 
blockage 
form factor 
propeller loading 
hull roughness 
 
see also 3.3 Input Data 
 
 
3.2 Recommendations of ITTC for Pa-
rameters 
see 4.9-03-03-01.1 Propulsion Test 
1987 p.263 In using the 1978 ITTC Method 
it is recommended that the rudder(s) be fitted 
in hull resistance experiments for barge type 
forms where inflow velocity is relatively 
large. 
 
 
3.3 Input Data 
 
All data are either non-dimensional or 
given in SI-units. 
 
Every data card defines several parameters 
which are required by the program; each of 
these parameters must be input according to a 
specific format. 
 
"I" format means that the value is to be input 
without a decimal point and packed to the 
right of the specified field. 
 
"F" format requires the data to be input with a 
decimal point; the number can appear 
anywhere in the field indicated. 
"A" format indicates that alphanumeric char-
acters must be entered in the appropriate 
card columns.The card order of the data deck must fol-
low the order in which they are described 
below. 
 
 
Card No. 1 Identifications 
 
Card 
column 
Form 
at 
CC 
Symbol 
Definition 
1- 8 A - Project No. 
9-16 A - Ship model No 
17-24 A - 
Propeller model No. 
25-32 F SCALE Scale ratio 
 
Card No. 2 Ship particulars 
 
Card 
column 
For-
mat 
CC 
Symbol 
Definition 
9-16 F LWL Length of waterline 
17-24 F TF Draft, forward 
25-32 F TA Draft, aft 
33-40 F B Breadth 
41-48 F S Wetted surface, with-
out bilge keels 
49-56 F DISW Displacement 
157-64 F SBK Wetted surface of 
bilge keels 
65-72 F AT Transverse projected 
area of ship above 
waterline 
72-80 F C3 Form factor deter-
mined at resistance 
tests 
 
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03 – 01.4 
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Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
Card No. 3 Particulars of full scale 
Card 
column 
For-
mat 
CC 
Symbol 
Definition 
8- 8 I NOPROP Number of propellers 
should be 1 since method 
is valid only for single 
screw ships 
15-16 I NPB Number of propeller 
blades 
17-24 F DP Diameter of propeller 
25-32 F PD075 Pitch ratio at x=0.75 
33-40 F CH075 Chord length of Propeller 
blade at x=0.75 
41-48 F TMO75 Maximum blade thick-
ness of propeller at 
x=0.75 
49-56 F RNCHM Reynolds number at 
open-water test based on 
chord length and local 
velocity 275.0.
1 

+=
J
VV A
π 
at x-0.75. 
Card No. 4 General 
Card 
column 
For-
mat 
 
CC Sym-
bol 
Definition 
2.- 4 I NOJ Number of J-values in the 
open-water characteristics 
(J ≤ NOJ ≤ 10) 
7- 8 I NOSP Number of speeds in the 
self- propulsion tests 
(NOSPmax=10) 
9-16 F RHOM Density of tank water 
17-24 F RHOS Density of sea water 
25-30 F TEMM Temperature of resistance 
test 
31-36 F TEMP Temperature at self-
propulsion test - 
36-41 F TEMS Temperature of sea water 
48-48 I CHOICE CHOICE=0 NP CC − 
trial corr. 
CHOICE==1:
CFC wC ∆−∆ trial corr. 
49-56 F CP Trial correction for shaft 
power. 
57-64 F CN Trial correction for rpm 
65-72 F DELT 
CFC 
Trial correction for FC∆ 
72-80 F DELTWC Trial correction for w∆ 
Mean values of the trial correction figures, 
Cp and CN can be obtained from the trial test 
material of the individual institutions by run-
ning the ITTC Trial Prediction Test Program. 
If an institution wishes to give predictions 
with a certain margin the input CP-CN-values 
must be somewhat higher than these mean 
values. 
 
Cards Nos. 5-14 Result of resistance and self-
propulsion tests and model propeller charac-
teristics. 
 
Card 
column 
Format CC 
Symbol 
Definition 
1- 8 F VS Ship speed in knots 
9-16 F RTM Resistance of ship 
model 
17-24 F THM Thrust of propeller 
25-32 F QM Torque of propel-
ler:QM:100 
33-40 F NM Rate of revolution 
41-48 F FD Skin friction correc-
tion force 
49-56 F ADVC Advance coefficient,. 
open water 
57-64 F KT Thrust coefficient, 
open water 
65-72 F KQ Torque coefficient, 
open water 
 
The J-margin in the open-water character-
istics must be large enough to cover the 
model and full scale J-values with some mar-
gin. 
Input data sheets are given in ENCL. 1. 
 
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03 – 01.4 
Page 14 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
4. VALIDATION 
 
4.1 Uncertainty Analysis 
not yet available 
 
 
4.2 Comparison With Full Scale Results 
 
The data that led to t ITTC-78 method can 
be found in the following ITTC proceedings: 
 
1) Proposed Performance Prediction Factors 
for Single Screw Ocean Going Ships 
(13th 1972 pp.155-180) Empirical Power 
Prediction Factor ( 1+X ) 
 
2) Propeller Dynamics Comparative Tests 
(13th 1972 pp.445-446 ) 
 
3) Comparative Calculations with the ITTC 
Trial Prediction Test Programme
(14th 1975 Vol.3 pp.548-553) 
 
4) Factors Affecting Model Ship Correlation 
(17th 1984 Vol. 1, pp274-291) 
 
 
 
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Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
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Revision
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5. ITTC- 1978 PERFORMANCE PREDICTION METHOD (COMPUTER CODE) 
 
C 
C **************************************************************************************************** 
C * * 
C * 1978 ITTC PERFORMANCE PREDICTION METHOD FOR SINGLE SCREW * 
C * SHIPS * 
C * (REVISED 1983 TO INCLUDE TRIAL ANALYSIS AND TWIN SCREW SHIPS* * 
C * * 
C **************************************************************************************************** 
C 
C DECLARATIONS 
C 
 COMMON /A/ FILE(2),MODELS(2), MODELP(2), LPP,LWL,TF,TA,B,S, 
 * SCALE,RNCHM,DISW,NOPROP,NPB,DP,PD075,CH075. 
 * TM075,C3,SBK,AT,CP,CN,DELCF,DELWC,KSI,KPI, 
 * RHOM,RHOS,TEMM,TEMP,TEMS,VS(10),RTM(10),THM(10), 
 * QM(10),NM(10),ADVC(10),KT(10),KQ(10),THD(10), 
* FD(10),IC,NOJ,NOSP,PI 
C 
 COMMON /B/ ETARM(10),ETAO(10),ETAH(10),ETAD(10),AWTM(10), 
* AWTS(10),ACFM(10),ACTM(10),AVS(10),AVM(10), 
 * ATS(10),AQS(10),APDS(10),APE(10),APDT(10), 
 * ANS(10),ANT(10),BPDT(10),BNT(10),KTSJ2(10), 
* KQS(10),KTS(10),ACTS(10) 
 DIMENSION FILE1(2),MODLS1(2),MODLP1(2) 
C 
 REAL LPP, LWL, KS1, KS, KP1, KP, NM1, NM, KT, KQ, KTM, KQ0, JTM, 
* KTSJ2, JTS, NS, KQTS, KTS, KQS, KQM 
 DATA TRIAL /‘TRIA‘/ 
500 FORMAT(6A4,F8.0) 
501 FORMAT(10F8.0) 
502 FORMAT(2I4,9F8.0) 
503 FORMAT(2I4,2F8.0,3F6.0,I6,4F8.0) 
504 FORMAT(9F8.0) 
600 FORMAT(/5X,’NUMBER OF ADV,KT AND KQ POINTS =’,15/ 
 * 5X,’NUMBER OF SPEEDS =’,15/ 
 * 5X,’NUMBER OF SPEEDS OR ADVC POINTS >10’/) 
 
 
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Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
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Revision
00 
 
 
 
C 
C CONSTANTS 
C 
 G=9.81 
 PI=3.14159 
 KP1=30.0 
 KS1=150.0 
 KS=1.5E-4 
 KP=0.3E-4 
C 
C READ INPUT DATA 
C 
1000 CONTINUE 
 READ(5,500,END=999) FILE,MODELS,MODELP,SCALE 
 READ(5,501) LPP,LWL,TF,TA,B,S,DISW,SBK,AT,C3 
 READ(5,502) NOPROP,NPB,DP,PD075,CH075,TM075,RNCHM 
 READ(5,503) NOJ,NOSP,RHOM,RHOS,TEMM,TEMP,TEMS 
 * IC,CP,CN,DELCF,DELWC 
 NMAX=MAX0(NOJ,NOSP) 
 
 IF(FILE(1).EQ.TRIAL) GOTO 100 
 
 READ(5,504)(VS(I),RTM(I),THM(I),QM(I),NM(I),FD(I), 
 * ADVC(I),KT(I),KQ(I);I=1,NMAX) 
 
 
C 
C WRITE INPUT DATA 
C 
 CALL OUTPUT(1) 
C 
C CHECK 
C 
 IF(NOJ.LE.10.AND.NOSP.LE.10) GOTO 2 
 WRITE(6,600) NOJ.NOSP 
 GOTO 1000 
 2 CONTINUE 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 17 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
C RECALCULATION OF INPUT DATA 
C 
 DO 3 I=1,NOJ 
 KT(I)=KT(I)*0.1 
KQ(I)=KQ(I)*0.01 
....3 CONTINUE 
 DELCF=DELCF*0.001 
 RNCHM=RNCHM*100000. 
 VISCP=((0.585E-3*(TEMP-12.0)-0.03361)*(TEMP-12.0)+ 
* 1.2350)*1.0E-6 
VISCM=((0.585E-3*(TEMM-12.0)-0.0361)*(TEMM-12.0)+ 
* 1.2350)*1.0E-6 
VISCS=((0.659E-3*(TEMS-1.0)-0.05076)*(TEMS-1.0)+ 
* 1.7688)*1.0E-6 
C 
C CORRECTION OF PROPELLER CHARACTERISTICS 
C 
 CDM=2.0*(1.0+2.0*TM075/CH075)*(0.044/RNCHM**0.16667- 
 * 5.0/RNCHM**0.66667) 
 CDS=2.0*(1.0+2.0*TM075/CH075)/(1.89+1.62*ALOG10(CH075 
 * /KP))**2.5 
 DCD=CDM-CDS 
 DKT=-0.3*DCD*PD075*CH075*NPB/DP 
 DKQ=0.25*DCD*CH075*NPB/DP 
 DO 4 I=1,NOJ 
 KTS(I)=KT(I)-DKT 
 KQS(I)=KQ(I)-DKQ 
 KTSJ2(I)=KTS(I)/ADVC(I)**2 
 4 CONTINUE 
 DO 5 I=1,NOSP 
 VS1=VS(I)*0.15444 
 VM1=VS1/SQRT(SCALE)NM1=NM(I) 
C 
C 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 18 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C CALCULATE ROUGHNESS ALLOWANCE AND SHIP TOTAL RESISTANCE 
C 
 RNLP=LWL*VM1/(VISCP*SCALE) 
 RNLM=LWL*VM1/(VISCM*SCALE) 
 RNLS=LWL*VS1/VISCS 
 CFMC=0.075/(ALOG10(RNLP)-2)**2 
 CFM=0.075/(ALOG10(RNLM)-2)**2 
 CFS=0.075/(ALOG10(RNLS)-2)**2 
 CTM=RTM(I)*SCALE**3/(0.5*RHOM*VS1**2*S) 
 CR=CTM-(1.0+C3)*CFM 
 RTMC=RTM(I)*(1.0+C3)*CFMC+CR)/((1.0+C3)*CFM+CR) 
 THD(I)=(THM(I)+FD(I)-RTMC)/THM(I) 
 DELCF=(105.0*(KS/LWL)**0.33333-0.64)*0.001 
 CAA=0.001*AT/S 
 CTS=((1.0+C3)*CFS*DELCF)*(S+SBK)/S+CR+CAA 
C 
C MODEL PROPULSIVE COEFFICIENTS 
C 
 FNOP=NPROP 
 KTM=(THM(I)/FNOP)/(RHOM*(DP/SCALE)**4*NM1*NM1) 
 KQM=(QM(I)*0.01/FNOP)/(RHOM*(DP/SCALE)**5*NM1*NM1) 
 JTM=APOL(0,KT,ADVC,NOJ,KTM,IX) 
 KQ0=APOL(0,ADVC,KQ,NOJ,JTM,IX) 
 WTM=1.0-JTM*DP*NM1/(VM1*SCALE) 
C 
C FULL SCALE WAKE 
C 
 IF(JRUDER) 6,5,6 
 5 WTS=(THD(I)+0.04)+(WTM-THD(I)-0.04)*((1.0+C3)*CFS+DELCF)/ 
* ((1.0+C3)*CFM) 
GOTO 7 
 6 WTS=(THD(I) )+(WTM-THD(I) )*((1.0+C3)*CFS+DELCF)/ 
* ((1.0+C3)*CFM) 
 GOTO 7 
 7 IF(WTS.GT.WTM) WTS=WTM 
 ETARM(I)=KQ0/KQM 
C 
C SAVE AREAS 
C 
 ACTM(I)=CTM 
 ACFM(I)=CFM 
 AWTM(I)=WTM 
 AWTS(I)=WTS 
 ACTS(I)=CTS 
 AVS(I)=VS1 
 AVM(I)=VM1 
 8 CONTINUE 
C 
C ITTC STANDARD PREDICTION 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 19 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
 CALL IP 
C 
C RETURN FOR NEW INPUT 
C 
 DO 20 I=1,2 
 FILE1(I)=FILE(I) 
 MODLS1(I)=MODELS(I) 
 
20 MODELP1(I)=MODELP(I) 
SCALE1=SCALE 
 GOTO 1000 
C 
100 CONTINUE 
DO 110 I=1,2 
FILE(I)=FILE1(I) 
MODELS(I)=MODLS1(I) 
110 MODELP(I)=MODLP1(I) 
SCALE=SCALE1 
C 
 CALL ANLSYS 
C 
C RETURN FOR NEW INPUT 
C 
C 
 GOTO 1000 
 999 STOP 
 END 
C 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 20 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
C
 *****************************************************************************************************
*** 
C 
C OUTPUT IS USED FOR PRINTING INPUT DATA AND RESULTS 
C 
C IOUT= 1 INPUT DATA IS PRINTED 
C 2 RESULT PAGE 1 
C 3 RESULT PAGE 2 
C 
C
 *****************************************************************************************************
*** 
C 
 SUBROUTINE OUTPUT(IOUT) 
 
 
 
 COMMON /A/ FILE(2),MODELS(2),MODELP(2),LPP,LWL,TF,TA,B,S 
 * SCALE,RNCHM,DISW,NOPROP,NPB,DP,PD075,CH075, 
 * TM075,C3,SBK,AT,CP,CN,DELCFC,DELWC,KSI,KPI, 
 * RHOM,RHOS,TEMM,TEMP,TEMS,VS(10),RTM(10);THM(10), 
 * QM(10),NM(10),ADVC(10),KT(10),KQ(10),THD(10), 
 * FD(10),IC,NOJ,NOSP,PI 
C 
 COMMON /B/ ETARM(10),ETA0(10),ETAH(10),ETAD(10),AWTM(10), 
 * AWTS(10),ACFM(10),ACTM(10),AVS(10),AVM(10), 
 * ATS(10),AQS(10),APDS(10),APE(10),APDT(10), 
 * ANS(10),ANT(10),BPDT(10),BNT(10),KTSJ2(10), 
 * KQS(10),KTS(10),ACTS(10) 
C 
 REAL LPP,LWL,KS1,KS,KP1,KP,NM1,NM,KT,KQ,KTM,KQ0,JTM, 
 KTSJ2,JTS,NS,KQTS,KTS,KQS 
 DIMENSION TEXT (16) 
 DATA TEXT /’INPU’,’T DA’,’TA ‘,’ ‘, 
 * ‘OUTP’,’UT D’,’ATA ‘,’1 ‘, 
 * ‘OUTP’,’UT D’,’ATA..’,’2 ‘; 
 * `TRIA`,`L AN`,ÀLYS`,ÌS `/ 
 600 FORMAT(‘1’,19X,’1978 ITTC PERFORMANCE PREDICTION’,10X, 
 * ‘ENCL:’/ 
C?? * 20X,’METHOD ‘,8X, 
* ‘REPORT:’/20X,4A4/) 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 21 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 601 FORMAT(5X,’IDENTIFICATION :’,18X,’SHIP:’// 
 * 5X,‘PROJECT :’,2A4, 
 * 10X,’LENGTH PP :’,F8.2,’ (M)’/ 
 * 5X,’SHIP MODEL’ :’,2A4, 
 * 10X,’LENGTH WL :’,F8.2,’ (M)’/ 
 * 5X,’PROPELLER MODEL :’,2A4, 
 * 10X,’DRAFT FWD :’,F8.2,’ (M)’/ 
 * 5X,’SCALE FACTOR :’,F8.2, 
 * 10X,’DRAFT AFT :’,F8.2,’ (M)’/ 
 * 43X,’BREADTH :’,F8.2,’ (M)’/ 
 * 5X,’PROPELLER:’, 
 * 28X,’WETTED SURFACE :’,F8.0,’ (M**2)’/ 
 * 43X,’DISPLACEMENT :’,F8.0,’ (M**3)’) 
 602 FORMAT(5X,’NUMBER OF PROPELLERS:’,I8/ 
 * 5X,’NUMBER OF BLADES :’,I8, 
 * 6X,’FRICTION COEFFICIENT CF’/ 
 * 5X,’DIAMETER :’,F8.3,’ (M)’, 
 * 2X,’CALCULATED ACCORDING TO ITTC-57’/ 
 * 5X,’PITCH RATIO 0.75R :’,F8.4, 
 * 6X,’FORM FACTOR :’,F6.3,’ (BASED ON ITTC-57)’/) 
 603 FORMAT(5X,’HULL ROUGHN.*10**6 :’,F6.1,’ (M)’, 
 * 2x,’BILGE KEEL AREA :’,F6.1,’ (M**2)’, 
 * 5X,’PROPELLER BLADE ROUGHN.*10**6:’,F6.1,’ (M)’, 
 * 2X,’PROJ.AREA ABOVE WL. :’,F6.1,’ (M**2)’/) 
 604 FORMAT(5X,’CHORD LENGTH OF PROP.BLADE AT X=0.75:’, 
 * F7.4,’ (M)’/ 
 * 5X,’THICKNESS OF PROP.BLADE AT X=0.75:’, 
 * F7.4’ (M)’/) 
 605 FORMAT(5X,’DENSITY OF WATER (TANK ) :’F7.1, 
 * ‘ (KG/M**3)’/ 
 * ’DENSITY OF WATER (SEA ) :’F7.1, 
 * ‘ (KG/M**3)’/ 
 * 5X,’TEMP. OF WATER (RESISTANCE TEST) :’F7.2, 
 * ‘ (CENTIGRADES)’/ 
 * 5X,’TEMP. OF WATER (SELF PROP. TEST) :’F7.2, 
 * ‘ (CENTIGRADES)’/ 
 * 5X,’TEMP. OF WATER (SEA ) :’F7.2, 
 * ‘ (CENTIGRADES)’// 
 * 5X,’MODEL TEST RESULTS:’, 
 * 30X,’OPEN WATER CHARACT.;’/ 
 * 54X,’RNC :’’F5.2,’*10**5’/) 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 22 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 606 FORMAT(5X,’SHIP RESIS- FRICT. THRUST TORQUE RATE OF ‘, 
 * 2X,’ADVANCE THRUST TORQUE’/ 
 * 20X,’REVS. RATIO COEFF. COEFF.’/ 
 * 5X,’KNOTS N N N NM RPS ’, 
 * 7X,’J 10*KT 100*KQ’/) 
 607 FORMAT(1X) 
 608 FORMAT(‘+’,3X,F5.1,1X,F7.1,1X,F7.2,2X,2F7.1,F9.2) 
 609 FORMAT(‘+’,49X,F10.3,F7.3,F8.3) 
 610 FORMAT(5X,’SHIP MODEL:’// 
 * 8X,’SPEED RES. COEFF. FRICT. COEFF. THRUST DED.’, 
 * 2X,’MEAN REL.ROT.’/ 
 * 6X,’VS VM TOTAL’,32X, ‘WAKE EFFIC.’/ 
 * 5X,’KNOTS M/S CTM*1000 CFM*1000’,8X,’TM’, 
 * 7X,’WTM ETARM’/) 
 611 FORMAT(4X,F5.1,F7.3,F8.3,6X,F7.3,7X,F7.3,3X,F7.3,F8.3) 
 612 FORMAT(/5x,’ITTC STANDARD PREDICTION CP=CN=1.0 :’// 
 * 5X,’SPEED EFF. POWER DELIV. POWER RSATE OF REVS’, 
 * 2X,’ THRUST TORQUE’/ 
 * 6X,’VS’,7X,’PE’,10X,’PD’,12X,’N’,10X,’T’,8X,’Q’/ 
 * 5X,’KNOTS’,5X,’KW’,10X,’KW’,11X,’RPS’,9X,’KN’, 
 * 6X,’KNM’/) 
 613 FORMAT(4X,F5.1,F10.0,3X,F9.0,4X,F9.3,3X,F9.0,F8.0) 
 614 (FORMAT(/5X,’SPEED TOT. EFF. PROP.EFF. HULL EFF. SHIP WAKE’, 
 * 3X,’OPEN WATER CHAR. FULL SCALE:’/ 
 * 5X,’KNOTS ETAD ETA0 ETAH’,/X,’WTS’, 
 * 9X,’J 10*KT 100*KQ’/) 
 615 FORMAT(‘+’,3X,F5.1,F8.3,3(3X,F7.3)) 
 616 FORMAT(‘+’,50X,3F7.3) 
 617 FORMAT(/5X,’SHIP DELIVERED POWER RATE OF REVS.’/ 
 * 5X, ‘SPEED --------------------------- ---------------------‘/ 
 * 5X,’KNOTS KW HP RPS RPM’/) 
 618 FORMAT(4X,F5.1,2X,2F8.0,3X,F7.3,F8.2) 
 619 FORMAT(/5X,’SHIP TRIALS PREDICTION CP=’,F7.3,’ CN=,F7.3) 
 620 FORMAT(/5X,’SHIP TRIALS PREDICTION DELCFC*1000=’, 
 * F6.3,’ DELCW=’,F6.3) 
 ITEX=ICUT*4-4 
 WRITE(6,600) (TEXT(ITEX+1),I=1,4)WRITE(6,601) FILE,LPP,MODELS,LWL,MODELP,TF,SCALE,TA,B,S,DISW 
 WRITE(6,602) NOPROP,NPB,DP,PD075,C3 
C 
 GOTO(10,20,30,40) , IOUT 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 23 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
C INPUT DATA IS LISTED 
C 
 10 CONTINUE 
 WRITE(6,603) KS1,SBK,KP1,AT 
 WRITE(6,604) CH075,TM075 
 WRITE(6,605) RHOM,RHOS,TEMM,TEMP,TEMS,RNCHM 
 WRITE(6,606) 
 NMAX=MAX0(NOJ,NOSP) 
 DO 1 I=1,NMAX 
 WRITE(6,607) 
 IF(I. LE. NOSP) WRITE(6,608) VS(I);RTM(I);FD(I),THM(I), 
 QM(I),NM(I) 
 IF(I. LE.NOJ) WRITE(6,609) ADVC(I),KT(I),KQ(I) 
 1 CONTINUE 
 RETURN 
 
C 
C RESULTS PAGE 1 
C 
 20 CONTINUE 
 WRITE(6,610) 
 DO 21 I=1,NOSP 
 CFM=ACFM(I)*1000.0 
 CTM=ACTM(I)*1000.0 
 WRITE(6,611) VS(I),AVM(I),CTM,CFM,THD(I),AWTM(I),ETARM(I) 
 21 CONTINUE 
 WRITE(6,612) 
 DO 22 i=1,NOSP 
 WRITE(6,613) VS(I),APE(I),APDS(I),ANS(I),ATS(I),AQS(I) 
 22 CONTINUE 
 WRITE(6,614) 
 DO 23 i=1,NMAX 
 WRITE(6,607) 
 IF(I.LE.NOSP) WRITE(6,615) VS(I),ETAD(I),ETA0(I),ETAH(I); 
 AWTS(I) 
 XKTS=KTS(I)*10.0 
 XKQS=KQS(I)*100.0 
 IF(I.LE.NOSP) WRITE(6,616) ADVC(I),XKTS,XKQS 
 23 CONTINUE 
 RETURN 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 24 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
C RESULTS PAGE 3 
C 
 30 CONTINUE 
 DCFC=DELCFC*1000.0 
 IF(IC.EQ.1) WRITE(6,620) DCFC,DELWC 
 IF(IC.NE.1) WRITE (6,619) CP,CN 
 WRITE(6,617) 
 DO 31 I=1,NOSP 
 WRITE(6,618) VS(I),APDT(I),BPDT(I),ANT(I),BNT(I) 
 31 CONTINUE 
....40 RETURN 
 END 
 
C 
C
 *****************************************************************************************************
*** 
C 
C IRAT= 0 INTERPOLATION WITH A 2:ND DEGREE POLYNOMIAL 
C = 1 INTERPOLATION WITH A RATIONAL FUNCTION OF 2:ND DEGREE 
C X = ARGUMENT ARRAY 
C Y = VALUE ARRAY 
C N = NUMBER OF ARGUMENTS 
C EX = ARGUMENT 
C IFEL = ERROR RETURN CODE 
C 
C
 *****************************************************************************************************
*** 
C 
 REAL FUNCTION APOL(IRAT,X,Y,N,EX,IFEL) 
 DIMENSION X(1),Y(1) 
C 
C CHECK NUMBER OF POINTS > 2 
C 
 IFEL=0 
 IF(X(1).GT.X(N)) GOTO 2 
 IF(X(1).GT.EX.OR.X(N).LT.EX) GOTO 7 
 DO 1 I=1,N 
 L=1 
 IF(EX-X(I)) 4,4,1 
 1 CONTINUE 
 GOTO 4 
 2 CONTINUE 
 IF(X(1).LT.EX.OR.X(N).GT.EX) GOTO 7 
 DO 3 I=1,N 
 L=I 
 IF(EX-X(I)) 3,4,4 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 25 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 3 CONTINUE 
 4 CONTINUE 
 M=2 
 IF(L.EQ.1) M=1 
 IF(L.EQ.3) M=3 
 LM=L-M 
 X1=X(LM+1) 
 X2=X(LM+2) 
 X3=X(LM+3) 
 Y1=Y(LM+1) 
 Y2=Y(LM+2) 
 Y3=Y(LM+3) 
C 
C INTERPOL. 2:ND DEGREE POLYNOMIAL 
C 
 X21=X2-X1 
 X31=X3-X1 
 X32=X3-X2 
 IF(IRAT.EQ.1) GOTO 6 
 C1=Y1 
 C2=(Y2-C1)/X21 
 C3=(Y3-C1-C2*X31)/(X31*X32) 
 APOL=C1+(EX-X1)*(C2+C3*(EX-X2)) 
 RETURN 
 6 CONTINUE 
C 
C INTERPOL. RAT. FUNCTION 
C 
 Y21=Y2*X2*X2-Y1*X1*X1 
 Y32=Y3*X3*X3-Y2*X2*X2 
 A0=(Y32-X32*Y21/X21)/(X32*X31) 
 B0=(Y21/X21-A0*(X1+X2) 
 C0=((Y1-A0)*X1-B0)*X1 
 APOL=(C0/EX+B0)/EX+A0 
 RETURN 
 7 CONTINUE 
 WRITE(6,8) 
 8 FORMAT(/5X,’INCREASE THE J-RANGE’) 
 STOP 
 END 
 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 26 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
C ******************************************************************** 
C 
C ITTC PREDICTIONS 
C 
C ******************************************************************** 
C 
 SUBROUTINE IP 
 
 COMMON /A/ FILE(2),MODELS(2),MODELP(2),LPP,LWL,TF,TA,B,S, 
 * SCALE,RNCHM,DISW,NOPROP,NPB,DP,PD075,CH075, 
 * TM075,C3,SBK,AT,CP,CN,DELCFC,DELWC,KSI,KPI, 
 * RHOM,RHOS,TEMM,TEMP,TEMS,VS(10),RTM(10),THM(10), 
 * QM(10),NM(10),ADVC(10),KT(10),KQ(10),THD(10), 
 * FD(10),IC,NOJ,NOSP,PI 
C 
 COMMON /B/ ETARM(10),ETA0(10),ETAR(10),ETAD(10),AWTM(10), 
 * AWTS(10),ACFM(10),ACTM(10),AVS(10),AVM(10), 
 * ATS(10),AQS(10),APDS(10),APE(10),APDT(10), 
 * ANS(10),ANT(10),BPDT(10),BNT(10),KTSJ2(10), 
 * KQS(10),KTS(10),ACTS(10) 
C 
 REAL LPP,LWL,KS1,KS,KPI,KP,NM1,NM,KT,KQ,KTM,KQD,JTM, 
 * KTSJ2,JTS,NS,KQTS,KTJT2,KQOS,KQS,KTS 
 DO 3 I=1,NOSP 
 VS1=AVS(I) 
 CTS=ACTS(I) 
 WTS=AWTS(I) 
C 
C CALCULATE THE FULL SCALE LOAD ADVANCE COEFF: AND 
C TORQUE COEFF. 
C 
 FNOP=NOPROP 
 KTJT2=S*CTS*0.5/((DP*(1.0-WTS))**2*(1.0-THD(I))) /FNOP 
 JTS=APOL(1,KTSJ2,ADVC,NOJ,KT,KTJT2,IX) 
 KQOS=APOL(0,ADVC,KQS,NOJ,JTS,IX) 
C 
C THE RATE OF REV. AND THE DELIVERED POWER 
C 
 NS=(1.0-WTS)*VS1/(JTS*DP) 
 APDS(I)=2.0*PI*RHOS*DP**5*NS**3*KQOS/ETARM(I)*0.001 
 ANS(I)=NS 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 27 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
C THE THRUST AND TORQUE OF THE PROPELLER 
C 
 ATS(I)=KTJT2*JTS**2*RHOS*DP**4*NS*NS*0.001 
 AQS(I)=KQOS*RHOS*DP**5*NS*NS/ETARM(I)*0.001 
C 
C THE EFFECTIVE POWER, TOTAL AND HULL EFFICIENCY 
C 
 APE(I)=CTS*0.5*RHOS*VS1**3*S*0.001 
 ETAD(I)=APE(I)/APDS(I) 
 ETAH(I)=(1.0-THD(I))/(1.0-WTS) 
 IF(IC.EQ.1) GOTO 1 
C 
 IC1=IC-1 
 IF(IC1)10,11,12 
C 
C TRIAL PREDICTION WITH CP-CN CORRECTIONS (ITTC1978 ORIGINAL) 
C 
 10 ANT(I)=CN*NS 
BNT(I)=ANT(I)*60.0 
APDT(I)=CP*APDS(I) 
BPDT(I)=1.36*APDT(I) 
GOTO 100 
C 
C TRIAL PREDICTION WITH CP-CN CORRECTIONS 
C CN BASED ON POWER IDENTITY 
C 
12 APDT(I)=CP*APDS(I) 
BPDT(I)=1.36*APDT(I) 
 KQJ3T=1000.0*APDT(I)/(2.0*PI*RHOS*DP**2) /FNOP 
 KQJ3T=KQJ3T/(VS1**3*(1.0-WTS)**3) 
 KQ0J3=KQJ3T*ETARM(I) 
 JTS=APOL(1,KQSJ3,ADVC,NOJ,KQ0J3,IX) 
 NS=(1.0-WTS)*VS1/(JTS*DP) 
 ANT(I)=CN*NS 
 BNT(I)=ANT(I)*60.0 
 GOTO 100 
 11 CONTINUE 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 28 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
C 
C TRIAL PREDICTION WITH DELCF-DELWC CORRECTIONS 
C 
 KTJT2=S*(CTS+DELCFC)/(2.0*(1.0-THD(I))*(DP* 
* (1.0-(WTS-DELWC)))**2) 
 JTS=APOL(1,KTSJ2,ADVC,NOJ,KTJT2,IX) 
 KQOS=APOL(0,ADVC,KQS,NOJ,JTS,IX) 
 ANT(I)=(1.0-WTS+DELWC)*VS1/(JTS*DP) 
 BNT(I)=ANT(I)*60.0 
 APDT(I)=2.0*PI*RHOS*DP**5*ANT(I)**3*KQOS/ETARM(I)*0.001 
 BPDT(I)=1.36*APDT(I) 
 2 CONTINUE 
 ETAD(I)=KTJT2*JTS**3/(2.0*PI*KQOS) 
 3 CONTINUE 
C 
C WRITE OUTPUT 
C 
 CALL OUTPUT(2) 
 CALL OUTPUT(3) 
 RETURN 
 
 
 
SUBROUTINE ANLSYS 
C 
C***********************************************************************************************************
**** 
C * * 
C * ANALYSIS ACCORD1NG TO 1978 ITTC PREDICTION METHOD * 
C * * 
C***********************************************************************************************************
**** 
C 
C 
 DIMENSION VST(10),XNT(10),XPD(10), 
 * THDT(10),WTMT(10),WTST(10),ETART(10),CRWT(10), 
 * YNT(10),YPD(10),CPT(10),CNT(10),CNPT(10),ZNT(10) 
 * DCFT(10),WTSS(10),DWT(10),DCFM(10),DWM(I0), 
 * KQJ3(10) 
C 
COMMON /A/ FILE(2),MODELS(2),MODELP(2),LPP,LWL,TF,TA,B,S, 
 * SCALE,RNCHM,DISW,NOPROP,NPB,DP,PD075,CH075, 
 * TM075,C3,SBK,AT,CP,CN,DELCFC,DELWC,KS1,KP1, 
 * RHOM, RHOS,TEMM,TEMP,TEMS,VS(10),RTM(10),THM(10), 
 * QM(10),NM(10),ADVC(10),KT(10),KQ(10),THD(10), 
 * RA(10),IC,NOJ,NOSP,PI 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 29 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
MethodEffective Date 
1999 
Revision
00 
 
 
 
C 
COMMON /B/ ETARM(10), ETA0(10),ETAH(10),ETAD(10),AWTM(10), 
 * AWTS(10),ACFM(10),ACTM(10),AVS(10),AVM(10), 
 * ATS(10),AQS(10),APDS(10),APE(10),APDT(10), 
 * ANS(10),ANT(10),BPDT(10),BNT(10),KTSJ2(10), 
 * KQS(10),KTS(10),ACTS(10) 
 
C 
REAL LPP,LWL,KS1,KS,KP1,KP,NM1,NM,KT,KQ,KTM,KQ0,JTM, 
 * KTSJ2,JTS,NS,KQTS,KTJT2,KQOS,KTS,KQS,KQM, 
 * KQJ3,KQJ3T 
C 
C 
DO 5 I = 1,NOJ 
 5 KQJ3(I) = KQS(I) /ADVC(I)**3 
C 
 NOST=10 
 
READ(5,510) (VST(I), I=1,NOST) 
READ(5,510) (XNT(I), I=1,NOST) 
 READ(5,510) (XPD(I), .I=1,NOST) 
 
 510 FORMAT (10F8.0) 
C 
C COUNT NO. OF TRIAL RUNS 
 NOST = 0 
 DO 8 I = 1, 10 
 IF (VST(I).GT.0. ) NOST=NOST+1 
 8 CONTINUE 
 IF(XNT(1).GT.20.) GOTO 20 
 DO 10 I=1, NOST 
 XNT(I) = XNT(I)*60.0 
 10 XPD(I) = XPD(I)*1.36 
 20 CONTINUE 
 DO 50 I=1, NOST 
 VST1=VST(I)*1852.0/3600.0 
 CTST = APOL(0,AVS, ACTS, NOSP,VST1, IX) 
 THDT(I)= APOL(0,AVS, THD, NOSP,VST1, IX) 
 WTMT(I)= APOL(0,AVS, AWTM, NOSP,VST1, IX) 
 WTST(I)= APOL(0,AVS, AWTS, NOSP,VST1, IX) 
 ETART(I)= APOL(0,AVS, ETARM,NOSP,VST1, IX) 
 CF =APOL(0,AVS, ACFM, NOSP,VST1, IX) 
 CT =APOL(0,AVS, ACTM, NOSP,VST1, X) 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 30 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
 CRWT(I)= CT - (1.0+C3)*CF 
 FNOP =NOPROP 
 KTJT2 =S*(CTST/FNOP )*0.5 / ((DP*(1.0-WTST(I)))**2*(1.0-THDT(I))) 
 JTS =APOL(1, KTSJ2, ADVC, NOJ, KTJT2, IX) 
 KQOS=APOL (0, ADVC, KQS, NOJ, JTS, IX) 
 NS=(1.0-WTST(I))*VST1/(JTS*DP) 
 PDS = 2.0*PI*RHOS*DP**5*NS**3*KQ0S/ETART(I)*0.001*FNOP 
 YNT(I)= NS*60.0 
 YPD(I) = PDS*1.36 
 CPT(I)= XPD(I)/YPD(I) 
 CNT(l)=XNT(I)/YNT(I) 
 PDT1 = XPD(I) /1.36 
 XNT1 = XNT(I) / 60.0 
 FKQ = PDT1*START(I)*1000.0 / (2.0*PI*RHOS*DP**5*XNT1**3) / FNOP 
 FJT = APOL(0,KQS,ADVC,NOJ,FKQ,IX) 
 FKT = APOL(0,ADVC, KTS,NOJ,FJT,IX) 
 KQJ3T=FKQ * (DP*XNT1)**3 / ((1-WTST(I))*VST1)**3 
 FJQ= APOL( 1,KQJ3,ADVC,NOJ,KQJ3T,IX) 
 ZNT(I)=(1.0 -WTST(I)) * VST1 / (FJQ*DP) * 60.0 
 CNPT(I)=XNT(I) / ZNT(I) 
 THS= FKT * RHOS * DP**4*XNT1**2 
 CTS=THS*(1.0 - THDT(I)) / (0.5*RHOS*VST1**2*S) * FNOP 
 DCFT(I)=(CTS - CTST)*1000.0 
 WTSS(I)= 1.0 - FJT*DP*XNT1/VST1 
 DWT(I) = WTST(I) - WTSS(I) 
 DWM(I) = WTMT(I) - WTSS(I) 
 
C 
C CALCULATION OF FRICTIONAL RESISTANCE ~COEFF. OF SHIP 
C 
 T = TEMS 
 FNU = ((0.659E-3*(T-l.0)-0.05076)*(T-1)+1.7688)*1.0E-6 
 RNLS= ALOG10(LWL*VST1/FNU) 
 CFS = 0.075 / (RNLS-2.0)**2 
C 
 DCFM(I) = CTS - (l.0+C3)*CFS - ( CRWT(I)+0.001*AT / S )*S / (S+SBK) 
 DCFM(I) = DCFM(I) * 1000.0 
 CRWT(I) = CRWT(I) * 1000.0 
50 CONTINUE 
C 
 CALL OUTPUT(4) 
 WRITE(6,600) 
ITTC – Recommended 
Procedures 
7.5 – 02 
03 – 01.4 
Page 31 of 31 
 
Performance, Propulsion 
1978 ITTC Performance Prediction 
Method 
Effective Date 
1999 
Revision
00 
 
 
 
 
 600 FORMAT(' ',19X,'TRIAL ANALYSIS ACCORDING TO ITTC 1978 METHOD',///) 
 WRITE(6,610) ( VST(I), I=1, NOST) 
 610 FORMAT(5X.. ' SHIP SPEED - TRTAL',7(F10.2, 2X) /) 
 WRITE(6,620) ( XNT(I), I=1, NOST) 
 620 FORMAT(5X, ‘ PROP, RPM –TRTAL ',7(F10.2, 2X) /) 
 WRITE(6,630) ( XPD(I), I=1, NOST) 
 630 FORMAT(4X, 'DELIV.POWER-TRIAL ',7(F11.0,1X) //) 
 WRITE(6,640) ( YNT(I), I=1, NOST) 
 640 FORMAT(/5X, ‘ PROP. RPM -CN=1 ',7(F10.2,2X) /) 
 WRITE~(6,650) ( ~YPD(I), I=1,NOST) 
 650 FORMAT(4X, ' DELIV. POWER -CP =1',7(F11.0,1X) /) 
 WRITE(6,660) ( ZNT(I), I=1, NOST) 
 660 FORMAT(5X, ‘ PROP. RPM -CNP=1 ',7(F10.2,2X), //) 
 WRITE(6,670) ( CPT(I), I=1, NOST) 
 670 FORMAT(/5X, ‘ CP ‘,7(F10.3,2X) /) 
 WRITE(6,680) (CNT(I), I=1, NOST) 
 680 FORMAT(5X, ‘CN ‘,7(F10.3,2X) /) 
 WRITE(6,690) (CNPT(I), I=1,NOST) 
 690 FORMAT(5X, ‘CNP ',7(F10.3,2X) //) 
WRITE(6,700) (DCFT(I), I=1,NOST) 
 700 FORMAT(/5X, ‘DCFC*1000 -CP=CN=1’,7(F10.3,2x) /) 
WRITE(6,710) ( DWT(I), I=1, NOST) 
 710 FORMAT(5X, ' DWC CP=CN=1’,7(F10.3,2X) //) 
 WRITE(6,715) ( DCFM(I), I=1, NOST) 
 715 FORMAT(/5X, 'DCF *1000 ITTC-57’,7(F10.3,2x) /) 
 WRITE(6,717) ( DWM(I), I=1,NOST) 
 717 FORMAT(5X, ‘DW = WM-WTRIAL ',7(F10.3,2X) //) 
 WRITE(6,720) ( CRWT(I) ,I=1, NOST) 
 720 FORMAT(/5X, ‘ CR*1000 ‘,7(F10.3,2X) /) 
 WRITE (6,730) ( THDT(I), I=1, NOST) 
 730 FORMAT(5X, ‘ THDM ',7(F10.3,2X) /) 
 WRITE(6,740) ( WTMT(I), I=1, NOST) 
 740 FORMAT(5X, ’ WTM ',7(F10.3,2X) /) 
 WRITE(6,750) ( WTST(I), I=1, NOST) 
 750 FORMAT(5X, ‘ WTS CP=CN=1 ’,7(F10.3,2x) /) 
 WRITE(6,760) ( WTSS(I), I=1, NOST) 
 760 FORMAT(5X, ’ WTS TRIAL ’,7(F10.3,2X) /) 
 WRITE(6,770) ( ETART(I), I=1, NOST) 
 770 FORMAT(5X, ‘ ETARM ‘ ,7(F10.3,2X) /) 
RETURN 
END 
 
 
 
120 APEˆNDICE A. PREVISA˜O BASEADA NOS ENSAIOS DE PROPULSA˜O
Apeˆndice B
Procedimentos Recomendados pela
ITTC para a Preparac¸a˜o e
Realizac¸a˜o das Provas de Velocidade
e Poteˆncia
121
122 APEˆNDICE B. PROVAS DE VELOCIDADE E POTEˆNCIA
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 1 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
Updated / Edited by Approved 
Specialist Committee on Powering Perform-
ance of 24th ITTC 
 
24th ITTC 2005 
 
Date 2005 Date 2005 
 
Table of Contents 
 
1. PURPOSE ..............................................2 
2. DEFINITIONS.......................................2 
3. RESPONSIBILITIES............................3 
3.1 Shipbuilders Responsibilities............3 
3.2 The Trial Team ..................................4 
4. PROCEDURES......................................4 
4.1 Trial Preparation...............................4 
4.1.1 Shipbuilder’s Support Requirement:4 
4.1.2 Space Requirements ........................4 
4.2 Ship Inspection...................................5 
4.2.1 Preparation for the trials ..................5 
4.2.2 Ship Inspection ................................5 
4.2.3 Reporting of Results and 
Distribution of Information .............5 
4.3 Hull- and Propulsor Survey..............5 
4.4 Instrumentation Installation and 
Calibration .........................................5 
4.4.1 Instrumentation Installation.............5 
4.4.2 Instrumentation Calibration Check .6 
4.5 Trial Conditions.................................6 
4.5.1 Wind: ...............................................8 
4.5.2 Sea State: .........................................8 
4.5.3 Current:............................................8 
4.6 Trial Conduct: ...................................8 
5. REFERENCES ....................................10 
 
 
 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 2 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
Preparation and Conduct of Speed/Power Trials 
 
1. PURPOSE 
The general purpose of this procedure is to 
define basic requirements for the preparation 
and conduct of speed trials. 
The primary purpose of speed trials is to 
determine ship performancein terms of speed, 
power and propeller revolutions under pre-
scribed ship conditions, and thereby verify the 
satisfactory attainment of the contractually 
stipulated ship speed. 
The applicability of this procedure is lim-
ited to commercial ships of the displacement 
type. 
The procedure is 
• to provide guidelines to document the 
trial preparation prior to the conduct of 
a full scale Speed/Power trial, 
• to define the responsibility sharing 
among the parties who take part in the 
sea trial for the smooth preparation and 
execution of the speed trial 
• to establish a guideline for conducting 
inspections for the purpose of installing 
instrumentation prior to the conduct of 
a full scale Speed/Power trial, 
• to establish a baseline of the ship hull 
and propulsor condition prior to the 
conduct of a full-scale Speed/Power 
trial;(hull and propulsor surveys are 
recommended to allow an evaluation of 
the trial results for scientific purposes), 
• to install and calibrate trial instrumenta-
tion for full scale Speed/Power trials, 
• to define acceptable limits for trial con-
ditions needed to validate hydrody-
namic design and/or satisfy contractual 
requirements, 
for acceptable conduct of each speed trial. 
2. DEFINITIONS 
 
• Ship Speed is that realized under the con-
tractually stipulated conditions. Ideal condi-
tions to which the speed would be corrected 
would be 
• no wind (or maximum wind speed ac-
cording to Beaufort 2) 
• no waves (or waves with maximum 
wave heights and wave periods accord-
ing to Beaufort 1) 
• no current 
• deep water 
• smooth hull and propeller surfaces 
• Docking Report: Report that documents 
the condition of the ship hull and propul-
sors (available from the most recent dry - 
docking). 
• Trial Agenda: Document outlining the 
scope of a particular Speed/Power trial. 
This document contains the procedures on 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 3 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
how to conduct the trial and table(s) por-
traying the runs to be conducted. 
• Trial Log: For each run, the log contains 
the run number, type of maneuver, ap-
proach speed by log, approach shaft speed, 
times when the maneuvers start and stop, 
and any comments about the run. 
• Propeller Pitch: the design pitch also for 
controllable pitch propellers. 
• Running Pitch: the operating pitch of a 
CPP 
• Brake Power: Power delivered by the out-
put coupling of the propulsion machinery 
before passing through any speed reducing 
and transmission devices and with all con-
tinuously operating engine auxiliaries in 
use. 
• Shaft Power: Net power supplied by the 
propulsion machinery to the propulsion 
shafting after passing through all speed-
reducing and other transmission devices 
and after power for all attached auxiliaries 
has been taken off. 
3. RESPONSIBILITIES 
3.1 Shipbuilders Responsibilities 
 
• The Shipbuilder has the responsibility for 
planning, conducting and evaluating the tri-
als. 
• Speed – Power - Trials may be conducted 
by institutions acknowledged as competent 
to perform those trials, as agreed between 
the Shipbuilder and the Ship owner 
• The Shipbuilder has to provide all permits 
and certificates needed to go to sea. 
• The Shipbuilder is responsible to ensure 
that all qualified personnel, needed for op-
erating the ship and all engines, systems 
and equipment during the trials have been 
ordered. 
• The Shipbuilder is responsible to ensure 
that all regulatory bodies, Classification 
Society, Ship Owner, ship agents, suppliers, 
subcontractors, harbor facilities, delivering 
departments of provisions, fuel, water, tow-
ing, etc., needed for conducting the sea tri-
als, have been informed and are available 
and on board, if required. 
• It is the Shipbuilder’s responsibility that all 
safety measures have been checked and all 
fixed, portable and individual material (for 
crew, trial personnel and guests) is on 
board and operative. 
• It is the Shipbuilder’s responsibility that 
dock trials of all systems have been exe-
cuted as well as all alarms, warning and 
safety systems. 
• It is the Shipbuilder’s responsibility that an 
inclining test has been performed and/or at 
least a preliminary stability booklet has 
been approved, covering the sea trial condi-
tion, in accordance with the 1974 SOLAS 
Convention. 
• The Shipbuilder is responsible for the over-
all trial coordination between the ship's 
crew, trial personnel, and the owner repre-
sentative. A pre-trial meeting between the 
trial team, owner and the ship’s crew will 
be held to discuss the various trial events 
and to resolve any outstanding issues. 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 4 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
• The Shipbuilder has, if necessary, to ar-
range for divers to inspect the ship’s hull 
and propellers. 
• The Trial Leader is the duly authorized 
(shipbuilder’s representative) person re-
sponsible for the execution of all phases of 
the Speed/Power trials including the pre-
trial preparation. 
3.2 The Trial Team 
The trial team is responsible for correct 
measurements and analysis of the measured 
data according to the state of the art. 
The trial team is responsible for the follow-
ing: 
a. Conduct ship inspection, if possible or 
necessary. 
b. Provide, install and operate all required 
trial instrumentation and temporary ca-
bling. 
c. If previously arranged, provide the ship 
master and owner’s representative with 
a preliminary data package before de-
barking. The contents of the data pack-
age will be determined in consultation 
with the owner’s representative at the 
initial pre-trial briefing. 
d. Provide a final report after completion 
of the trials in accordance with any 
agreement between the shipbuilder and 
the ship owner. 
4. PROCEDURES 
4.1 Trial Preparation 
4.1.1 Shipbuilder’s Support Requirement: 
 
Prior to the trials the required instrumenta-
tion has to be installed. The assistance of the 
ship’s or shipbuilder’s crew will be required 
when making electrical connections to the 
ship's systems and circuits such as heading, 
wind speed, wind direction, and rudder angle 
synchronous repeaters. The following support 
is requested from the Shipbuilder to properly 
prepare for the trials: 
a. Provide access to the ship for trial in-
strumentation. 
b. Assistance is required for the following 
electrical connections: 
• Gyrocompass 
• Wind meter 
• Rudder angle indicator 
• Log Speed 
• Propeller Pitch 
c. Vary the output level of each of the 
above measurement sources to ensure 
the proper operation and alignment of 
the test instrumentation 
4.1.2 Space Requirements 
Spaces and an electric supply adequate for 
the trial equipment will be required for the trial 
instrumentation and computers. 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 5 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
4.2 Ship Inspection 
There are three stages of a ship inspection: 
in-house preparation, the actual inspection, and 
the reporting of results and distribution of in-
formation to the various parties involved in the 
trial. 
4.2.1 Preparation for the trials 
• Review shafting dimensions, propulsion 
plant specifications, etc. 
• Review trialsagenda, if available. 
 
4.2.2 Ship Inspection 
• Inspect hull- and propeller surface con-
dition, if possible. 
• Inspect ship’s instrumentation for ac-
cessibility. 
• Determine routes for cable runs/data 
transfer conduits between trial room 
and bridge or control area. 
• Contact the Engineer on duty to discuss 
trial instrumentation requirements. In-
spect machinery spaces as applicable. 
 
4.2.3 Reporting of Results and Distribution 
of Information 
Document all pertinent information related 
to the ship inspection 
 a) Last date of cleaning. 
 b) Means of cleaning. 
 c) Propeller roughness measurement, if 
available, which should include aver-
age, standard deviation, distribution 
along the blades, and existing physical 
damage. 
 d) For a clean hull; documentation indi-
cating manufacturer and kind of paint 
used, paint layer thickness and, if avail-
able, roughness measurements (average, 
standard deviation, and distribution 
along the hull) should be provided. The 
majority of this information may be 
contained in the docking report. 
 e) For a dirty hull, documentation indi-
cating visual observations of any foul-
ing and date of last dry-docking should 
be provided. 
4.3 Hull- and Propulsor Survey 
A roughness survey is recommended to 
document the conditions of the ship hull, ap-
pendages, and propulsor(s) prior to the start of 
the full-scale speed/ power trial. Cleaning may 
be required if fouling is found to be such that it 
would bias the trial data. 
Ideally, roughness surveys should be con-
ducted prior to the trials. The average hull 
roughness should not exceed 250 µm (µ = 
1x10-6 m) (6.35 mils) and the average propul-
sor roughness level should not be greater than 
150 µm (3.81 mils). 
4.4 Instrumentation Installation and Cali-
bration 
4.4.1 Instrumentation Installation 
The installation of instrumentation should 
be scheduled at a time of minimal conflict with 
ship operations. 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 6 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
The bias limits of the instrumentation used 
for the measurements should be known and as-
sessed. 
The instrumentation used for the on-board-
measurements must be calibrated before appli-
cation on board. If this is not possible, for some 
reason, the consequences of this should be 
highlighted in the final trial report. Electrical 
calibration is recommended for the torque 
measurement device and, in case of use during 
the sea trials, for the thrust measurement device. 
Further a calibration should be done for the 
pick ups and the respective amplifiers used for 
the measurement of the rate of revolutions. A 
“calibration” of a (differential) GPS-System is 
not possible without excessive measures, but at 
least the function of the device should be 
checked before use on board. 
If portable radar tracking or (differential) 
GPS is utilized, a Receiver/Transmitter (R/T) 
unit or GPS antenna is to be installed. In case 
the soft ware program used for the evaluation 
of the data received does not allow for varying 
positions on the uppermost deck of the ship the 
antenna should be placed in a location along 
the ship’s centerline as close to the ship’s CG 
as possible. This location will ideally be lo-
cated on a mast or site that is clear of obstruc-
tions, such as the ship’s superstructure. 
4.4.2 Instrumentation Calibration Check 
All shipboard signals to be recorded during 
the trials must be adjusted to zero or should 
have their zero value checked (e.g. for a (D) 
GPS-device) after the instrumentation installa-
tion is completed and prior to the trials. The 
zero values of the torsiometers, the thrust 
measurement devices and the devices for the 
measurement of the rates of revolutions must 
be checked before the trial runs start and after 
they have been finished. 
As part of the pre-trial calibration, the tor-
sion meters zero torque readings must be de-
termined since there is a residual torque in the 
shaft, which is resting on the line shaft bearings. 
This might be done in different ways; one pos-
sible way is to use the jacking motors. The 
shaft is jacked both ahead and astern and the 
average of the readings noted. The zeroes are 
set at the midpoint of the torque required to 
jack each shaft ahead and the torque required to 
jack each shaft astern. An allowance is nor-
mally made for frictional losses in the stern 
tube bearings. 
As part of the pre-trial calibration for a ship 
equipped with controllable pitch propellers, 
maximum ahead pitch, the design pitch and the 
maximum astern pitch should be determined 
and then the ship indicators should be adjusted 
to reflect the measurement. 
4.5 Trial Conditions 
Speed/Power trials require accurate position 
data. The use of (D) GPS provides great lati-
tude in choosing a trial site. Regardless of the 
instrumentation utilized for obtaining posi-
tional data, the operational area should be free 
from substantial small boat traffic. 
The tracking range should be agreed be-
tween the Trial Director and the ship’s master. 
Draft, trim and displacement of the ship on 
trials should be obtained by averaging the ship 
draft mark readings. The ship should be 
brought into a condition that is as close as pos-
sible to the contract condition and/or the condi-
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 7 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
tion on which model tests have been carried out. 
This will allow for the correction of the dis-
placement and trim with respect to the trials 
that were conducted and will be applicable to 
the suggestions outlined in the ITTC Procedure 
for the Analysis of Speed/Power Trial Data. 
Draft, trim and displacement should be ob-
tained at the beginning and at the end of the 
trial. This may be accomplished using a load-
ing computer or by taking a second draft read-
ing. The accuracy of the draft readings and the 
method used to establish draft and displace-
ment underway will be compared in port by di-
rect draft readings both port and starboard in 
conjunction with a liquid load calculation. 
Displacement should be derived from the 
hydrostatic curves by utilizing the draft data 
and the density of the water. 
Environmental factors may significantly in-
fluence the data obtained during sea trials; con-
sequently, these factors should be monitored 
and documented to the greatest extent possible: 
• High wind and sea states can force the 
use of excessive rudder to maintain 
heading, and thus cause excessive fluc-
tuations in shaft torque, shaft speed and 
ship speed. 
• Sea states of 3 or less and a true wind 
speed below Beaufort 6 (20 Kn) are the 
desired conditions for sea trials. When 
working under the time constraints of a 
contract, corrections to the trials data 
can be made in accordance with the rec-
ommendations provided in the ITTC 
Procedure for the Analysis of 
Speed/Power Trial Data for sea states 
less than or equal to 5. For sea states 
greater than 5, corrections to the trials 
data can be applied but are not consid-
ered reliable from a scientific stand-
point. 
• The local seawater temperature and spe-
cific gravity at the trial site are recorded 
to enable the calculation of ship's dis-
placement. 
• An acceptable minimum water depth 
for the trials where the data do not need 
to be corrected for shallow water can be 
calculated using: 
h > 6.0(Am)0.5 and h > 0.5 V2 (1) 
with 
Am= midship sectionarea, [m2] 
V= ship speed, [m/s] 
The larger of the 2 values obtained 
from the two equations should be used. 
• Current speed and direction should be 
determined in the test area by prognos-
tic analysis. When current speed and di-
rection is unknown, the ship’s global 
drift (also including wind effect) in 
some cases might be determined by a 
360° turning test conducted at low 
ahead speed to magnify any environ-
mental effect. 
• The runs should be conducted into and 
against the waves; i.e., head and follow-
ing seas, respectively. To ensure that 
tests are performed in comparable con-
ditions, the data between reciprocal 
runs should be reviewed for consistency 
and/or anomalies. Individual speed runs 
conducted in the same conditions 
should be averaged with their reciprocal 
runs to take into account global drift. 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 8 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
In accordance with ISO 15016 the follow-
ing, general recommendations can be given: 
4.5.1 Wind: 
Wind speed and direction shall be measured 
as relative wind; continuous recording of rela-
tive wind during each run is recommended. 
Care has to be taken whether the data derived 
from the wind indicator are reliable; checks, 
such as parallel measurements with a portable 
instrument, comparison of the data received 
from the wind indicator with wind speeds and 
directions received from local weather stations 
sufficiently close to the actual position of the 
ship or, if possible, calibration of the wind in-
dicator (taking into consideration the effects of 
boundary layers of the superstructure on the 
measured values) in a wind tunnel are recom-
mended. 
It is suggested that wind force during the 
trial runs under no conditions should be higher 
than 
• Beaufort 6 for ships with lengths equal 
or exceeding 100m and 
• Beaufort 5 for ships shorter than 100m. 
 
4.5.2 Sea State: 
If possible, instruments such as buoys or in-
struments onboard ships (e.g. seaway analysis 
radar) should be used to determine the wave 
height, wave period and direction of seas and 
swell. Considering usual practice the wave 
heights may be determined from observations 
by multiple, experienced observers, including 
the nautical staff on board. 
During the trial runs the total wave height 
(double amplitude), which allows for the wave 
heights of seas and swell (see ISO 15016), 
should not exceed 
• 3m for ships of 100m length and more 
and 
• 1,5m for ships with lengths smaller than 
100m 
4.5.3 Current: 
Current speed and direction shall be ob-
tained either as part of the evaluation of run 
and counter-run of each double run, by direct 
measurement with a current gauge buoy or by 
use of nautical charts of the respective trial area. 
It is recommended to compare measured data 
with those included on the nautical charts. 
4.6 Trial Conduct: 
All speed trials shall be carried out using 
double runs, i.e. each run is followed by a re-
turn run in the opposite direction, performed 
with the same engine settings. 
The number of such double runs should not 
be less than three. This three runs should be at 
different engine settings. 
The time necessary for a speed run depends 
on the ship’s speed, size and power. Steady 
state conditions should be achieved before the 
speed runs start. It is recommended that the 
time of one run should be as long as possible 
but should at least be 10 min. 
The ideal path of a ship in a typical 
speed/power maneuver is shown in Figure 1: 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 9 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
 
Steady Approach
Min 10 min
Steady Approach
 
Min. 10 min
 
Figure 1 
 
Prior to the trial, the data specified below 
shall be recorded, based on measurements 
where relevant: 
• Date 
• Trial area 
• Weather conditions 
• Air temperature 
• Mean water depth in the trial area 
• Water temperature and density 
• Draughts 
• Corresponding displacement 
• Propeller pitch in the case of a CPP 
It is recommended to retain a record of the 
following factors, which should prove useful 
for verifying the condition of the ship at the 
time of the speed trial: 
• Time elapsed since last hull and propel-
ler cleaning 
• Surface condition of hull and propeller. 
The following data should be monitored 
and recorded on each run: 
Clock time at commencement 
• Time elapsed over the measured dis-
tance 
• Ship heading 
• Ship’s speed over ground 
• Propeller rate of revolutions 
• Propeller shaft torque and/or brake 
power 
• Water depth 
• Relative wind velocity and direction 
• Air temperature 
• Observed wave height (or: wave height 
corresponding to observed and/or 
agreed wind conditions) 
• Rudder angle 
• Ship position and track 
ITTC – Recommended 7.5-04 
-01-01.1 
Procedures and Guidelines Page 10 of 10 
 
Full Scale Measurements 
Speed and Power Trials 
Preparation and Conduct of 
Speed/Power Trials 
Effective Date 
2005 
Revision
03 
 
 
 
Data such as ship’s speed, rate of revolu-
tions of the propeller, torque, rudder angle and 
drift angle to be used for the analyses shall be 
the average values derived on the measured 
distance. 
 
 
5. REFERENCES 
 (1) ISO 15016, Ships and marine technology – 
Guidelines for the assessment of speed and 
power performance by analysis of speed 
trial data 
(2) ITTC Procedure for the Analysis of 
Speed/Power Trial Data 
(3) ISO 19019 
 
 
Apeˆndice C
Condic¸o˜es de Realizac¸a˜o das Provas
de Velocidade e Poteˆncia
Recomendadas pela ITTC
133
134 APEˆNDICE C. CONDIC¸O˜ES DAS PROVAS DE VELOCIDADE E POTEˆNCIA
ITTC –Recommended 
Procedures 
7.5 – 0.4 
01 – 01.5 
Page 1 of 6 
 
Full Scale Measurements 
Speed and Power Trials 
Trial Conditions 
Effective Date 
2002 
Revision 
01 
 
 
 
 
Updated by Approved 
Specialist Committee of 23rd ITTC on 
Speed and Powering 
 
23rd ITTC 2002 
Date Date 2002 
 
CONTENTS 
 
1. PURPOSE 
2. SCOPE 
3. RESPONSIBILITIES 
4. DEFINITIONS 
5. PROCEDURE 
6. REFERENCES 
7. RECORDS 
8. ATTACHMENTS 
ITTC –Recommended 
Procedures 
7.5 – 0.4 
01 – 01.5 
Page 2 of 6 
 
Full Scale Measurements 
Speed and Power Trials 
Trial Conditions 
Effective Date 
2002 
Revision 
01 
 
 
 
 
 
Trial Conditions 
1. PURPOSE 
The purpose of this procedure is to estab-
lish guidelines for the definition of acceptable 
limits for trial conditions needed to validate 
hydrodynamic design and/or satisfy contractual 
requirements. 
2. SCOPE 
This procedure applies to the documenta-
tion of trial conditions (environmental and 
ship) in which the full-scale Speed/Power trial 
are performed. 
3. RESPONSIBILITIES 
• The Trial Director is the duly authorized 
shipbuilder’s representative responsible for 
the execution of all phases of the 
Speed/Power trials. When unforeseen prob-
lems, such as weather or technical difficul-
ties require that the trial schedule or trial 
logistics be modified, the Trial Director 
shall make the final decision, subject to the 
concurrence of the ship’s master and the 
owner’s representative. 
• The shipbuilder is responsible for the over-
all trial coordination between the ship's 
crew, trial personnel, and the owner repre-
sentative. A pre-trial meeting between the 
trial team, owner and the ship’screw will 
be held to discuss the various trial events 
and to resolve any outstanding issues. 
• The trial team is responsible for the follow-
ing: 
a. Operate and maintain all required trial 
instrumentation and temporary cabling. 
b. Collect and record seawater tempera-
ture and specific gravity during trial, 
daily. 
4. DEFINITIONS 
None 
5. PROCEDURE 
1. Speed/Power trials require accurate posi-
tion data and therefore will ideally be con-
ducted at an instrumented tracking range 
located in a sheltered body of water. Lack-
ing availability of an instrumented tracking 
range, the use of DGPS provides great lati-
tude in choosing a trial site. Regardless of 
the instrumentation utilized for obtaining 
positional data, the operational area should 
be free from substantial small boat traffic. 
2. If an instrumented tracking range is util-
ized, the ship’s master will receive a formal 
briefing on tracking range procedures by 
the Trial Director prior to the conduct of 
the trials. During the briefing, specific trial 
runs will be reviewed. The trial team will 
provide an on-shore observer to monitor 
data collection by the tracking range facil-
ity. If DGPS is utilized, the Trial Director 
will brief the ship’s master on specific trial 
runs and procedures. 
3. Ship characteristics and environmental fac-
tors are carefully monitored and docu-
mented throughout the trials (see Table 1). 
Accurate quantification of these conditions 
is necessary because a ship's speed and 
powering characteristics are extremely sen-
sitive to conditions such as ship and propel-
ler condition, ship displacement, shallow 
water effects, sea state and wind velocity. 
ITTC –Recommended 
Procedures 
7.5 – 0.4 
01 – 01.5 
Page 3 of 6 
 
Full Scale Measurements 
Speed and Power Trials 
Trial Conditions 
Effective Date 
2002 
Revision 
01 
 
 
 
 
 
4. Speed/Power Trials are normally scheduled 
within 30 days of undocking to minimize 
the adverse effects of hull and propulsor 
fouling and provide a more "standard" con-
dition for testing. In situations where the 
ship has become fouled after undocking, a 
hull cleaning, propeller polishing and hull 
and propeller roughness survey should be 
performed within 30 days of the 
Speed/Power trial date. Guidance may be 
found in Hull and Propulsor Survey Proce-
dure 7.5-04-01-01.3. At a minimum, the 
ship’s latest docking report and diver in-
spection should be provided to fulfill this 
requirement. Guidance may be found in 
Speed/Power Trial Ship Inspection Proce-
dure 7.5-04-01-01.2. 
5. Draft, trim and displacement of the trials 
must be obtained by averaging the ship 
draft mark readings. The ship should be 
brought into a condition that is as close as 
possible to the contract condition and/or the 
condition by which model tests have been 
carried out. This will allow for the correc-
tion of the displacement and trim with re-
spect to the trials that were conducted and 
will be applicable to the suggestions out-
lined in the 23rd ITTC Speed and Powering 
Trials Specialist Committee final report. 
a. Draft, trim and displacement must be 
obtained at the beginning and at the end 
of the trial. This may be accomplished 
using a loading computer or by taking a 
second draft reading. The accuracy of 
the ship's draft marks and the method 
used to calculate draft and displacement 
underway will be compared in port by 
direct draft readings both port and star-
board in conjunction with a liquid load 
calculation. The trial team will verify 
and document the results prior to the 
Speed/Power trials. 
b. Displacement must be derived from the 
hydrostatic curves by utilizing the draft 
data and the density of the water. When 
dealing with Froude numbers higher 
than 0.5 (e.g. a Fast Ferry with 100 m 
length and speed over 30 kn) intermedi-
ate ship loading conditions must be 
documented. This is better accom-
plished through tank soundings. 
6. Environmental factors can significantly in-
fluence the data obtained during sea trials. 
Consequently, these factors must be moni-
tored and documented to the greatest extent 
possible. 
a. High wind and sea states can force the 
use of excessive rudder to maintain 
heading, and thus cause excessive fluc-
tuations in shaft torque, shaft speed and 
ship speed. 
b. Sea states of 3 or less and a true wind 
speed below Beaufort 6 (20 kn) are the 
desired conditions for sea trials. When 
working under the time constraints of a 
contract, corrections to the trials data 
can be made in accordance with the rec-
ommendations provided in the 23rd 
ITTC Speed and Powering Trials Spe-
cialist Committee final report for sea 
states less than or equal to 5. For sea 
states greater than 5, corrections to the 
trials data can be applied but are not 
considered reliable from a scientific 
standpoint. 
c. The local seawater temperature and spe-
cific gravity at the trial site are recorded 
to enable the calculation of ship's dis-
placement. 
d. Air temperature and atmospheric pres-
sure should be measured at the trial lo-
cation using a calibrated thermometer 
and barometer. 
e. An acceptable minimum water depth for the 
trials where the data do not need to be cor-
ITTC –Recommended 
Procedures 
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01 – 01.5 
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Full Scale Measurements 
Speed and Power Trials 
Trial Conditions 
Effective Date 
2002 
Revision 
01 
 
 
 
 
 
rected for shallow water can be calculated 
using: 
 
h > 6.0(Am)0.5 and h > 0.5 V2 (1) 
 
Use the larger of the 2 values obtained from 
the two equations. 
 
Other accepted formulae are: 
 
1. SNAME 1973/21st ITTC Powering 
Performance Committee 
 
d ≥ 10TV/(L)0.5 (2) 
 
d = water depth, ft 
T =´trial draft, ft 
V = speed, kn 
L = length between perpendicu-
lars, ft 
 
2. SNAME 1989 from Det Norske 
Veritas 
Nautical Safety- Additional Classes 
NAUT-A, NAUT-B AND NAUT-
C, July 1986 
 
h > 5.0(Am)0.5 and h > 0.4 V2 (3) 
 
Use the larger of the 2 values ob-
tained from the two equations. 
 
h = water depth, m 
Am = midship section area, m2 
V = ship speed, m/s 
 
or 
 
h > 5 (T) (4) 
 
T = Mean draft, m 
 
3. 22nd ITTC Trials & Monitoring 
Specialist Committee/12th ITTC 
based on ship section and Froude 
Number. 
 
h > 3.0(BT)0.5 and h > 2.75 V2/g 
 (5) 
Use the larger of the 2 values ob-
tained from the two equations. 
 
h = depth in appropriate length 
units 
B = beam in appropriate length 
units 
T = draft in appropriate length 
units 
V = speed in system of units con-
sistent with the above dimension 
g = acceleration due to gravity in 
units consistent with the above di-
mension 
 
4. ISO/FDIS 15016:(E) based on Lack-
enby’s Formula 
 
∆V
V
= 0.1242 A m
h2
− 0.05    +1 − tanh(
gh
V2
)    
0.5
 
for h ≤ (Am/0.05)0.5 (6) ∆V
V
≤ 0.02 
 
h = water depth, m 
Am = midship section area under 
water, m2 
V = ship speed, m/s 
∆V = speed loss due to shallow wa-
ter effect, m/s 
g = acceleration due to gravity, 
m/s2 
 
 
ITTC –Recommended 
Procedures 
7.5 – 0.4 
01 – 01.5 
Page 5 of 6 
 
Full Scale Measurements 
Speed and Power Trials 
Trial Conditions 
Effective Date 
2002 
Revision 
01 
 
 
 
 
 
f. Current speed and direction should be 
determined in the test area by prognos-
tic analysis. When current speed is sus-
pected to be varying and direction is 
unknown, the ship’s global drift (also 
including wind effect) should be deter-
mined by a 360° turning test conducted 
at low ahead speed to magnify any en-
vironmental effect. Test runs should be 
conducted against and with global drift. 
It should be notedthat this method of 
determining the direction of the trial 
runs is extremely important in the case 
of small ships whose performance is 
strongly effected by environmental con-
ditions. For large ships, such as ULCCs, 
performance is not impacted as greatly 
by environmental conditions. If time is 
a critical factor, then the runs can be 
conducted into and against the waves; 
i.e., head and following seas, re-
spectively. To ensure that tests are per-
formed in comparable conditions, the 
data between reciprocal runs should be 
reviewed for consistency and/or anoma-
lies. Individual speed runs conducted in 
the same conditions should be averaged 
with their reciprocal runs to take into 
account global drift. 
6. REFERENCES 
1. SNAME 1973/21st ITTC Powering Per-
formance Committee Final Report 
2. 22nd ITTC Trials & Monitoring Specialist 
Committee Final Report 
3. Ships and marine technology – Guidelines 
for the assessment of speed and power per-
formance analysis of speed trial data, Final 
Draft International Standard ISO/FDIS 
15016: (E), ISO/TC 8/SC 9/WG 2 of 2001 
4. 23rd ITTC Speed and Powering Trials Spe-
cialist Committee Final Report 
5. Speed/Power Trial Ship Inspection Proce-
dure 7.5-04-01-01.2 
6. Hull and Propulsor Survey Procedure 7.5-
04-01-01.3 
7. RECORDS 
1. Ship conditions – displacement, draft, pro-
pulsor and hull roughness 
2. Environmental conditions – water depth, 
water temperature, wind direction and 
speed, sea state, specific gravity, air tem-
perature, atmospheric pressure, current 
speed and direction 
8. ATTACHMENTS 
1. Table 1. Documented Ship and Trial Con-
ditions Reported 
 
 
ITTC –Recommended 
Procedures 
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01 – 01.5 
Page 6 of 6 
 
Full Scale Measurements 
Speed and Power Trials 
Trial Conditions 
Effective Date 
2002 
Revision 
01 
 
 
 
 
 
Table 1. Documented Ship and Trial Conditions Reported 
Description
Ship Hull
Draft
Trim
Displacement and Load
Hull Condition
Roughness of shell and bottom paint
Height of welding beads
Waviness of hull
Size, number and position of zinc anodes
Size, number and position of openings of sea water inlets and outlets
Paint system
Hull Appendages and Rudder
Geometry, deviations, roughness
Type
Rate of movement
Propeller(s)
Geometry, deviations, roughness
Pitch
Direction of rotation
Number of blades
Propeller Shaft(s)
Geometry
Material
Trial Site
Water depth
Water temperature
Air temperature
Sea State
Specific gravity of water
Environmental Conditions
Wind
Waves
Current
Atmospheric pressure
Apeˆndice D
Utilizac¸a˜o dos Diagramas na
Selecc¸a˜o de Motores Propulsores
141
142 APEˆNDICE D. SELECC¸A˜O DE MOTORES PROPULSORES
Contents:
Basic Principles of Ship Propulsion
Page
Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Scope of this Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
Chapter 1
Ship Definitions and Hull Resistance . . . . . . . . . . . . . . . . . . 4
• Ship types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
• A ship’s load lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
• Indication of a ship’s size . . . . . . . . . . . . . . . . . . . . . . . . 5
• Description of hull forms . . . . . . . . . . . . . . . . . . . . . . . . 5
• Ship’s resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
Chapter 2
Propeller Propulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
• Propeller types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
• Flow conditions around the propeller . . . . . . . . . . . . . . . . . . 11
• Efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . . · · · · 11
• Propeller dimensions . . . . . . . . . . . . . . . . . . . . . . · · · · 13
• Operating conditions of a propeller . . . . . . . . . . . . . . . . . . . 15
Chapter 3
Engine Layout and Load Diagrams . . . . . . . . . . . . . . . . . . 20
• Power functions and logarithmic scales . . . . . . . . . . . . . . . . . 20
• Propulsion and engine running points . . . . . . . . . . . . . . . . . . 20
• Engine layout diagram . . . . . . . . . . . . . . . . . . . . . . . . . 22
• Load diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
• Use of layout and load diagrams – examples . . . . . . . . . . . . . . 25
• Influence on engine running of different types
of ship resistance – plant with FP�propeller . . . . . . . . . . . . . . . 27
• Influence of ship resistance
on combinator curves – plant with CP�propeller . . . . . . . . . . . . 29
Closing Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
Introduction
For the purpose of this paper, the term
“ship” is used to denote a vehicle em�
ployed to transport goods and persons
from one point to another over water.
Ship propulsion normally occurs with
the help of a propeller, which is the
term most widely used in English, al�
though the word “screw” is sometimes
seen, inter alia in combinations such as
a “twin�screw” propulsion plant.
Today, the primary source of propeller
power is the diesel engine, and the power
requirement and rate of revolution very
much depend on the ship’s hull form
and the propeller design. Therefore, in
order to arrive at a solution that is as
optimal as possible, some general
knowledge is essential as to the princi�
pal ship and diesel engine parameters
that influence the propulsion system.
This paper will, in particular, attempt to
explain some of the most elementary
terms used regarding ship types,
ship’s dimensions and hull forms and
clarify some of the parameters pertain�
ing to hull resistance, propeller condi�
tions and the diesel engine’s load
diagram.
On the other hand, it is considered be�
yond the scope of this publication to
give an explanation of how propulsion
calculations as such are carried out, as
the calculation procedure is extremely
complex. The reader is referred to the
specialised literature on this subject, for
example as stated in “References”.
Scope of this Paper
This paper is divided into three chapters
which, in principle, may be considered as
three separate papers but which also,
with advantage, may be read in close
connection to each other. Therefore,
some important information mentioned in
one chapter may well appear in another
chapter, too.
Chapter 1, describes the most elemen�
tary terms used to define ship sizes
and hull forms such as, for example,
the ship’s displacement, deadweight,
design draught, length between per�
pendiculars, block coefficient, etc.
Other ship terms described include the
effective towing resistance, consisting
of frictional, residual and air resistance,
and the influence of these resistances
in service.
Chapter 2, deals with ship propulsion
and the flow conditions around the pro�
peller(s). In this connection, the wake
fraction coefficient and thrust deduc�
tion coefficient, etc. are mentioned.
The total power needed for the propel�
ler is found based on the above effec�
tive towing resistance and various
propeller and hull dependent efficien�
cies which are also described. A sum�mary of the propulsion theory is shown
in Fig. 6.
The operating conditions of a propeller
according to the propeller law valid for
a propeller with fixed pitch are described
for free sailing in calm weather, and
followed up by the relative heavy/light
running conditions which apply when
the ship is sailing and subject to different
types of extra resistance, like fouling,
heavy sea against, etc.
Chapter 3, elucidates the importance
of choosing the correct specified MCR
and optimising point of the main engine,
and thereby the engine’s load diagram
in consideration to the propeller’s design
point. The construction of the relevant
load diagram lines is described in detail
by means of several examples. Fig. 24
shows, for a ship with fixed pitch pro�
peller, by means of a load diagram, the
important influence of different types of
ship resistance on the engine’s contin�
uous service rating.
3
Basic Principles of Ship Propulsion
Ship Definitions and Hull
Resistance
Ship types
Depending on the nature of their cargo,
and sometimes also the way the cargo
is loaded/unloaded, ships can be divided
into different categories, classes, and
types, some of which are mentioned in
Table 1.
The three largest categories of ships
are container ships, bulk carriers (for
bulk goods such as grain, coal, ores,
etc.) and tankers, which again can be
divided into more precisely defined
classes and types. Thus, tankers can
be divided into oil tankers, gas tankers
and chemical tankers, but there are
also combinations, e.g. oil/chemical
tankers.
Table 1 provides only a rough outline.
In reality there are many other combi�
nations, such as “Multi�purpose bulk
container carriers”, to mention just one
example.
A ship’s load lines
Painted halfway along the ship’s side
is the “Plimsoll Mark”, see Fig. 1. The
lines and letters of the Plimsoll Mark,
which conform to the freeboard rules
laid down by the IMO (International
Maritime Organisation) and local au�
thorities, indicate the depth to which
the vessel may be safely loaded (the
depth varies according to the season
and the salinity of the water).
There are, e.g. load lines for sailing in
freshwater and seawater, respectively,
with further divisions for tropical condi�
tions and summer and winter sailing.
According to the international freeboard
rules, the summer freeboard draught
for seawater is equal to the “Scantling
draught”, which is the term applied to
the ship’s draught when dimensioning
the hull.
The winter freeboard draught is less
than that valid for summer because of
the risk of bad weather whereas, on the
other hand, the freeboard draught for
tropical seas is somewhat higher than
the summer freeboard draught.
4
Category Class Type
Tanker
Oil tanker
Gas tanker
Chemical tanker
OBO
Crude (oil) Carrier
Very Large Crude Carrier
Ultra Large Crude Carrier
Product Tanker
Liquefied Natural Gas carrier
Liquefied Petroleum Gas carrier
Oil/Bulk/Ore carrier
CC
VLCC
ULCC
LNG
LPG
OBO
Bulk carrier Bulk carrier
Container ship Container ship
Container carrier
Roll On�Roll Off Ro�Ro
General cargo ship
General cargo
Coaster
Reefer Reefer Refrigerated cargo vessel
Passenger ship
Ferry
Cruise vessel
Table 1: Examples of ship types
T Tropical
S Summer
W Winter
WNA Winter - the North Atlantic
D L
D: Freeboard draught
SeawaterFreshwater
Danish load mark
TF
F
D
Freeboard deck
Fig. 1: Load lines – freeboard draught
Indication of a ship’s size
Displacement and deadweight
When a ship in loaded condition floats at
an arbitrary water line, its displacement is
equal to the relevant mass of water dis�
placed by the ship. Displacement is thus
equal to the total weight, all told, of the
relevant loaded ship, normally in seawa�
ter with a mass density of 1.025 t/m3.
Displacement comprises the ship’s
light weight and its deadweight, where
the deadweight is equal to the ship’s
loaded capacity, including bunkers and
other supplies necessary for the ship’s
propulsion. The deadweight at any time
thus represents the difference between
the actual displacement and the ship’s
light weight, all given in tons:
deadweight = displacement – light weight.
Incidentally, the word “ton” does not
always express the same amount of
weight. Besides the metric ton (1,000
kg), there is the English ton (1,016 kg),
which is also called the “long ton”. A
“short ton” is approx. 907 kg.
The light weight of a ship is not normally
used to indicate the size of a ship,
whereas the deadweight tonnage
(dwt), based on the ship’s loading ca�
pacity, including fuel and lube oils etc.
for operation of the ship, measured in
tons at scantling draught, often is.
Sometimes, the deadweight tonnage
may also refer to the design draught of
the ship but, if so, this will be mentioned.
Table 2 indicates the rule�of�thumb rela�
tionship between the ship’s displacement,
deadweight tonnage (summer freeboard/
scantling draught) and light weight.
A ship’s displacement can also be ex�
pressed as the volume of displaced
water ∇, i.e. in m3.
Gross register tons
Without going into detail, it should be
mentioned that there are also such
measurements as Gross Register Tons
(GRT), and Net Register Tons (NRT)
where 1 register ton = 100 English cubic
feet, or 2.83 m3.
These measurements express the size
of the internal volume of the ship in ac�
cordance with the given rules for such
measurements, and are extensively
used for calculating harbour and canal
dues/charges.
Description of hull forms
It is evident that the part of the ship
which is of significance for its propulsion
is the part of the ship’s hull which is
under the water line. The dimensions
below describing the hull form refer
to the design draught, which is less
than, or equal to, the scantling
draught. The choice of the design
draught depends on the degree of
load, i.e. whether, in service, the ship
will be lightly or heavily loaded. Gen�
erally, the most frequently occurring
draught between the fully�loaded and
the ballast draught is used.
Ship’s lengths LOA, LWL, and LPP
The overall length of the ship LOA is
normally of no consequence when
calculating the hull’s water resistance.
The factors used are the length of the
waterline LWL and the so�called length
between perpendiculars LPP. The di�
mensions referred to are shown in
Fig. 2.
5
Ship type dwt/light
weight ratio
Displ./dwt
ratio
Tanker and
Bulk carrier 6 1.17
Container ship 2.5�3.0 1.33�1.4
Table 2: Examples of relationship between dis�
placement, deadweight tonnage and light weight
L
L
L
PP
WL
OA
AM
DA
BWL
DF
D
Length between perpendiculars: L
Length on waterline: L
Length o : L
Breadth on waterline: B
Draught: D = 1/2 (D +D )
Midship section area: A
PP
WL
OA
WL
m
F A
verall
Fig. 2: Hull dimensions
The length between perpendiculars is
the length between the foremost per�
pendicular, i.e. usually a vertical line
through the stem’s intersection with
the waterline, and the aftmost perpen�
dicular which, normally, coincides with
the rudder axis. Generally, this length is
slightly less than the waterline length,
and is often expressed as:
LPP = 0.97 × LWL
Draught D
The ship’s draught D (often T is used in
literature) is defined as the vertical dis�
tance from the waterline to that point of
the hull which is deepest in the water,
see Figs. 2 and 3. The foremost draught
DF and aftmost draught DA are normally
the same when the ship is in the loaded
condition.
Breadth on waterline BWL
Another important factor is the hull’s
largest breadth on the waterline BWL,
see Figs. 2 and 3.
Block coefficient CB
Various form coefficients are used to
express the shape of the hull.The most
important of these coefficients is the
block coefficient CB, which is defined
as the ratio between the displacement
volume ∇ and the volume of a box with
dimensions LWL × BWL × D, see Fig. 3, i.e.:
C
L B DB WL WL
=
∇
× ×
In the case cited above, the block co�
efficient refers to the length on water�
line LWL. However, shipbuilders often use
block coefficient CB, PP based on the
length between perpendiculars, LPP, in
which case the block coefficient will, as a
rule, be slightly larger because, as previ�
ously mentioned, LPP is normally slightly
less than LWL.
C
L B DB PP PP WL
, =
∇
× ×
A small block coefficient means less re�
sistance and, consequently, the possibil�
ity of attaining higher speeds.
Table 3 shows some examples of block
coefficient sizes, and the pertaining
service speeds, on different types of
ships. It shows that large block coeffi�
cients correspond to low speeds and
vice versa.
Ship type
Block
coefficient
CB
Approxi�
mate ship
speed V
in knots
Lighter 0.90 5 – 10
Bulk carrier 0.80 – 0.85 12 – 17
Tanker 0.80 – 0.85 12 –16
General cargo 0.55 – 0.75 13 – 22
Container ship 0.50 – 0.70 14 – 26
Ferry boat 0.50 – 0.70 15 – 26
Table 3: Examples of block coefficients
Water plane area coefficient CWL
The water plane area coefficient CWL
expresses the ratio between the ves�
sel’s waterline area AWL and the product
of the length LWL and the breadth BWL of
the ship on the waterline, see Fig. 3, i.e.:
C
A
L BWL
WL
WL WL
=
×
Generally, the waterplane area coeffi�
cient is some 0.10 higher than the block
coefficient, i.e.:
CWL ≅ CB + 0.10.
This difference will be slightly larger on
fast vessels with small block coefficients
where the stern is also partly immersed
in the water and thus becomes part of
the ”waterplane” area.
Midship section coefficient CM
A further description of the hull form is
provided by the midship section coeffi�
cient CM, which expresses the ratio be�
tween the immersed midship section
area AM (midway between the foremost
and the aftmost perpendiculars) and the
product of the ship’s breadth BWL and
draught D, see Fig. 3, i.e.:
C
A
B DM
M
WL
=
×
6
LWL
AWL
B
WL
DAMWaterline plane
LPP
,
Midship section coefficient
Volume of displacement
Waterline area
Block coefficient L based
Waterplane area coefficient
WL
Longitudinal prismatic coefficient
:
:
: =
: =
: =
: =
A
C
C
C
C
B
WL
M
P
L BWL x x DWL
B x DWL
A x LM WL
L BWL WLx
AWL
AM
WL
Fig. 3: Hull coefficients of a ship
For bulkers and tankers, this coefficient
is in the order of 0.98�0.99, and for
container ships in the order of 0.97�0.98.
Longitudinal prismatic coefficient CP
The longitudinal prismatic coefficient
CP expresses the ratio between dis�
placement volume ∇ and the product
of the midship frame section area AM
and the length of the waterline LWL,
see also Fig. 3, i.e.:
C
A L C B D L
C
Cp
M WL M WL WL
B
M
=
∇
×
=
∇
× × ×
=
As can be seen, CP is not an independ�
ent form coefficient, but is entirely de�
pendent on the block coefficient CB
and the midship section coefficient CM.
Longitudinal Centre of Buoyancy LCB
The Longitudinal Centre of Buoyancy
(LCB) expresses the position of the
centre of buoyancy and is defined as
the distance between the centre of
buoyancy and the mid�point between
the ship’s foremost and aftmost perpen�
diculars. The distance is normally stated
as a percentage of the length between
the perpendiculars, and is positive if
the centre of buoyancy is located to
the fore of the mid�point between the
perpendiculars, and negative if located
to the aft of the mid�point. For a ship
designed for high speeds, e.g. container
ships, the LCB will, normally, be nega�
tive, whereas for slow�speed ships,
such as tankers and bulk carriers, it will
normally be positive. The LCB is gener�
ally between �3% and +3%.
Fineness ratio CLD
The length/displacement ratio or fine�
ness ratio, CLD, is defined as the ratio
between the ship’s waterline length LWL,
and the length of a cube with a volume
equal to the displacement volume, i.e.:
C
L
LD
WL
=
∇3
Ship’s resistance
To move a ship, it is first necessary to
overcome resistance, i.e. the force work�
ing against its propulsion. The calculation
of this resistance R plays a significant role
in the selection of the correct propeller and
in the subsequent choice of main engine.
General
A ship’s resistance is particularly influ�
enced by its speed, displacement, and
hull form. The total resistance RT, con�
sists of many source�resistances R
which can be divided into three main
groups, viz.:
1) Frictional resistance
2) Residual resistance
3) Air resistance
The influence of frictional and residual
resistances depends on how much of
the hull is below the waterline, while the
influence of air resistance depends on
how much of the ship is above the wa�
terline. In view of this, air resistance will
have a certain effect on container ships
which carry a large number of contain�
ers on the deck.
Water with a speed of V and a density
of � has a dynamic pressure of:
½ × � × V 2 (Bernoulli’s law)
Thus, if water is being completely
stopped by a body, the water will react
on the surface of the body with the dy�
namic pressure, resulting in a dynamic
force on the body.
This relationship is used as a basis
when calculating or measuring the
source�resistances R of a ship’s hull,
by means of dimensionless resistance
coefficients C. Thus, C is related to the
reference force K, defined as the force
which the dynamic pressure of water
with the ship’s speed V exerts on a
surface which is equal to the hull’s wet�
ted area AS. The rudder’s surface is
also included in the wetted area. The
general data for resistance calculations
is thus:
Reference force: K = ½ × � × V 2 × AS
and source resistances: R = C × K
On the basis of many experimental
tank tests, and with the help of pertain�
ing dimensionless hull parameters,
some of which have already been dis�
cussed, methods have been estab�
lished for calculating all the necessary
resistance coefficients C and, thus, the
pertaining source�resistances R. In
practice, the calculation of a particular
ship’s resistance can be verified by
testing a model of the relevant ship in
a towing tank.
Frictional resistance RF
The frictional resistance RF of the hull
depends on the size of the hull’s wet�
ted area AS, and on the specific fric�
tional resistance coefficient CF. The
friction increases with fouling of the
hull, i.e. by the growth of, i.a. algae,
sea grass and barnacles.
An attempt to avoid fouling is made by
the use of anti�fouling hull paints to
prevent the hull from becoming
“long�haired”, i.e. these paints reduce
the possibility of the hull becoming
fouled by living organisms. The paints
containing TBT (tributyl tin) as their
principal biocide, which is very toxic,
have dominated the market for decades,
but the IMO ban of TBT for new appli�
cations from 1 January, 2003, and a
full ban from 1 January, 2008, may in�
volve the use of new (and maybe not
as effective) alternatives, probably cop�
per�based anti�fouling paints.
When the ship is propelled through the
water, the frictional resistance increases
at a rate that is virtually equal to the
square of the vessel’s speed.
Frictional resistance represents a con�
siderable part of the ship’s resistance,
often some 70�90% of the ship’s total
resistance for low�speed ships (bulk
carriers and tankers), and sometimes
less than 40% for high�speed ships
(cruise liners and passenger ships) [1]. The
frictional resistance is found as follows:
RF = CF × K
Residual resistance RR
Residual resistance RR compriseswave
resistance and eddy resistance. Wave
resistance refers to the energy loss
caused by waves created by the vessel
during its propulsion through the water,
while eddy resistance refers to the loss
caused by flow separation which cre�
ates eddies, particularly at the aft end
of the ship.
7
Wave resistance at low speeds is pro�
portional to the square of the speed,
but increases much faster at higher
speeds. In principle, this means that a
speed barrier is imposed, so that a fur�
ther increase of the ship’s propulsion
power will not result in a higher speed
as all the power will be converted into
wave energy. The residual resistance
normally represents 8�25% of the total
resistance for low�speed ships, and up
to 40�60% for high�speed ships [1].
Incidentally, shallow waters can also
have great influence on the residual
resistance, as the displaced water un�
der the ship will have greater difficulty
in moving aftwards.
The procedure for calculating the spe�
cific residual resistance coefficient CR is
described in specialised literature [2]
and the residual resistance is found as
follows:
RR = CR × K
Air resistance RA
In calm weather, air resistance is, in prin�
ciple, proportional to the square of the
ship’s speed, and proportional to the
cross�sectional area of the ship above the
waterline. Air resistance normally repre�
sents about 2% of the total resistance.
For container ships in head wind, the
air resistance can be as much as 10%.
The air resistance can, similar to the
foregoing resistances, be expressed as
RA = CA × K, but is sometimes based
on 90% of the dynamic pressure of air
with a speed of V, i.e.:
RA = 0.90 × ½ × �air × V
2 × Aair
where �air is the density of the air, and
Aair is the cross�sectional area of the
vessel above the water [1].
Towing resistance RT
and effective (towing) power PE
The ship’s total towing resistance RT is
thus found as:
RT = RF + RR + RA
The corresponding effective (towing)
power, PE, necessary to move the ship
through the water, i.e. to tow the ship
at the speed V, is then:
PE = V × RT
The power delivered to the propeller,
PD, in order to move the ship at speed
V is, however, somewhat larger. This is
due, in particular, to the flow conditions
around the propeller and the propeller
efficiency itself, the influences of which
are discussed in the next chapter
which deals with Propeller Propulsion.
Total ship resistance in general
When dividing the residual resistance
into wave and eddy resistance, as earlier
described, the distribution of the total ship
towing resistance RT could also, as a
guideline, be stated as shown in Fig. 4.
The right column is valid for low�speed
ships like bulk carriers and tankers, and
the left column is valid for very high�speed
ships like cruise liners and ferries. Con�
tainer ships may be placed in between
the two columns.
The main reason for the difference
between the two columns is, as earlier
mentioned, the wave resistance. Thus,
in general all the resistances are pro�
portional to the square of the speed,
but for higher speeds the wave resis�
tance increases much faster, involving
a higher part of the total resistance.
This tendency is also shown in Fig. 5
for a 600 teu container ship, originally
designed for the ship speed of 15 knots.
Without any change to the hull design,
8
RF
V
RA
V
RW
RE
Ship speed V
% of RTType of resistance
RA
RW
RE
RF = Friction
= Wave
= Eddy
= Air
High
speed
ship
Low
speed
ship
45 � 90
40 � 5
5 � 3
10 � 2
Fig. 4: Total ship towing resistance RT = RF + RW + RE + RA
the ship speed for a sister ship was re�
quested to be increased to about 17.6
knots. However, this would lead to a
relatively high wave resistance, requir�
ing a doubling of the necessary propul�
sion power.
A further increase of the propulsion
power may only result in a minor ship
speed increase, as most of the extra
power will be converted into wave en�
ergy, i.e. a ship speed barrier valid for
the given hull design is imposed by
what we could call a “wave wall”, see
Fig. 5. A modification of the hull lines,
suiting the higher ship speed, is neces�
sary.
Increase of ship resistance in service,
Ref. [3], page 244
During the operation of the ship, the
paint film on the hull will break down.
Erosion will start, and marine plants
and barnacles, etc. will grow on the
surface of the hull. Bad weather, per�
haps in connection with an inappropri�
ate distribution of the cargo, can be a
reason for buckled bottom plates. The
hull has been fouled and will no longer
have a “technically smooth” surface,
which means that the frictional resist�
ance will be greater. It must also be
considered that the propeller surface
can become rough and fouled. The to�
tal resistance, caused by fouling, may
increase by 25�50% throughout the
lifetime of a ship.
Experience [4] shows that hull fouling
with barnacles and tube worms may
cause an increase in drag (ship resis�
tance) of up to 40%, with a drastical
reduction of the ship speed as the con�
sequence.
Furthermore, in general [4] for every 25
µm (25/1000 mm) increase of the aver�
age hull roughness, the result will be a
power increase of 2�3%, or a ship
speed reduction of about 1%.
Resistance will also increase because
of sea, wind and current, as shown in
Table 4 for different main routes of
ships. The resistance when navigating
in head�on sea could, in general, in�
crease by as much as 50�100% of the
total ship resistance in calm weather.
Estimates of average increase in
resistance for ships navigating the
main routes:
North Atlantic route,
navigation westward 25�35%
North Atlantic route,
navigation eastward 20�25%
Europe�Australia 20�25%
Europe�East Asia 20�25%
The Pacific routes 20�30%
Table 4: Main routes of ships
On the North Atlantic routes, the first
percentage corresponds to summer
navigation and the second percentage
to winter navigation.
However, analysis of trading conditions
for a typical 140,000 dwt bulk carrier
shows that on some routes, especially
Japan�Canada when loaded, the in�
creased resistance (sea margin) can
reach extreme values up to 220%, with
an average of about 100%.
Unfortunately, no data have been pub�
lished on increased resistance as a fun
ction of type and size of vessel. The
larger the ship, the less the relative in�
crease of resistance due to the sea.
On the other hand, the frictional resis�
tance of the large, full�bodied ships will
very easily be changed in the course of
time because of fouling.
In practice, the increase of resistance
caused by heavy weather depends on
the current, the wind, as well as the
wave size, where the latter factor may
have great influence. Thus, if the wave
size is relatively high, the ship speed
will be somewhat reduced even when
sailing in fair seas.
In principle, the increased resistance
caused by heavy weather could be
related to:
a) wind and current against, and
b) heavy waves,
but in practice it will be difficult to dis�
tinguish between these factors.
9
Power and speed relationship for a 600 TEU container ship
20 knots
6,000
Normal service point
Ship speed
Propulsion power
10 15
4,000
2,000
0
8,000
"Wave wall"
New service point
kW
Fig. 5: The “wave wall” ship speed barrier
Chapter 2
Propeller Propulsion
The traditional agent employed to
move a ship is a propeller, sometimes
two and, in very rare cases, more than
two. The necessary propeller thrust T
required to move the ship at speed V
is normally greater than the pertaining
towing resistance RT, and the flow�related
reasons are, amongst other reasons,
explained in this chapter. See also Fig. 6,
where all relevant velocity, force, power
and efficiency parameters are shown.
Propellertypes
Propellers may be divided into the follow�
ing two main groups, see also Fig. 7:
• Fixed pitch propeller (FP�propeller)
• Controllable pitch propeller
(CP�propeller)
Propellers of the FP�type are cast in
one block and normally made of a copper
alloy. The position of the blades, and
thereby the propeller pitch, is once and
for all fixed, with a given pitch that can�
not be changed in operation. This
means that when operating in, for ex�
ample, heavy weather conditions, the
propeller performance curves, i.e. the
combination of power and speed
(r/min) points, will change according to
the physical laws, and the actual pro�
peller curve cannot be changed by the
crew. Most ships which do not need a
particularly good manoeuvrability are
equipped with an FP�propeller.
Propellers of the CP�type have a rela�
tively larger hub compared with the
FP�propellers because the hub has to
have space for a hydraulically activated
mechanism for control of the pitch (an�
gle) of the blades. The CP�propeller is
relatively expensive, maybe up to 3�4
times as expensive as a corresponding
FP�propeller. Furthermore, because of
the relatively larger hub, the propeller
efficiency is slightly lower.
CP�propellers are mostly used for
Ro�Ro ships, shuttle tankers and simi�
lar ships that require a high degree of
10
Efficiencies
1 t
1 w
Relative rotative efficiency :
Propeller efficiency � open water :
Propeller efficiency � behind hull : =
Propulsive efficiency : =
Shaft efficiency :
Total efficiency :
_
_
x
x
Velocities
Ship’s speed : V
Arriving water velocity to propeller : V
Effective wake velocity : V = V _ V
A
W A
Forces
Towing resistance : R
Thrust force : T
Thrust deduction fraction : F = T _ R
T _ R
T
T
T
T
Power
Effective (Towing) power : P = R V
by the propeller to water : P = P /
Power delivered to propeller : P = P /
Brake power of main engine : P = P /
E T
T E
D T
B D
x
Thrust deduction coefficient : t =
Hull efficiency : =
Thrust power delivered
H
H
B
D
S
T B
S
B
0 R
T
���� ���� x ���� x ���� x x x x x= = = =
P P P PB T D B
P P P PE E T D
H
S H 0 R SH
VA
V
VW
PD PEPT PB
V
T
RTF
Wake fraction coefficient : w =
R
0
B
(Speed of advance of propeller)
V _ V
V
A
Fig. 6: The propulsion of a ship – theory
Controllable pitch propeller (CP�Propeller)Fixed pitch propeller (FP�Propeller)
Monobloc with fixed
propeller blades
(copper alloy)
Hub with a mechanism
for control of the pitch
of the blades
(hydraulically activated)
Fig. 7: Propeller types
manoeuvrability. For ordinary ships like
container ships, bulk carriers and crude
oil tankers sailing for a long time in nor�
mal sea service at a given ship speed,
it will, in general, be a waste of money
to install an expensive CP�propeller in�
stead of an FP�propeller. Furthermore, a
CP�propeller is more complicated, invol�
ving a higher risk of problems in service.
Flow conditions around the propeller
Wake fraction coefficient w
When the ship is moving, the friction of
the hull will create a so�called friction
belt or boundary layer of water around
the hull. In this friction belt the velocity
of the water on the surface of the hull is
equal to that of the ship, but is reduced
with its distance from the surface of the
hull. At a certain distance from the hull
and, per definition, equal to the outer
“surface” of the friction belt, the water
velocity is equal to zero.
The thickness of the friction belt increases
with its distance from the fore end of
the hull. The friction belt is therefore
thickest at the aft end of the hull and
this thickness is nearly proportional to
the length of the ship, Ref. [5]. This
means that there will be a certain wake
velocity caused by the friction along the
sides of the hull. Additionally, the ship’s
displacement of water will also cause
wake waves both fore and aft. All this
involves that the propeller behind the
hull will be working in a wake field.
Therefore, and mainly originating from
the friction wake, the water at the pro�
peller will have an effective wake veloc�
ity Vw which has the same direction as
the ship’s speed V, see Fig. 6. This
means that the velocity of arriving water
VA at the propeller, (equal to the speed
of advance of the propeller) given as
the average velocity over the propeller’s
disk area is Vw lower than the ship’s
speed V.
The effective wake velocity at the pro�
peller is therefore equal to Vw = V – VA
and may be expressed in dimensionless
form by means of the wake fraction
coefficient w. The normally used wake
fraction coefficient w given by Taylor is
defined as:
w
V
V
V V
V
you get
V
V
w
W A
A
= =
−
= −( )1
The value of the wake fraction coefficient
depends largely on the shape of the
hull, but also on the propeller’s location
and size, and has great influence on
the propeller’s efficiency.
The propeller diameter or, even better,
the ratio between the propeller diameter
d and the ship’s length LWL has some
influence on the wake fraction coeffi�
cient, as d/LWL gives a rough indication
of the degree to which the propeller
works in the hull’s wake field. Thus, the
larger the ratio d/LWL, the lower w will
be. The wake fraction coefficient w in�
creases when the hull is fouled.
For ships with one propeller, the wake
fraction coefficient w is normally in the
region of 0.20 to 0.45, corresponding
to a flow velocity to the propeller VA of
0.80 to 0.55 of the ship’s speed V. The
larger the block coefficient, the larger is
the wake fraction coefficient. On ships
with two propellers and a conventional
aftbody form of the hull, the propellers
will normally be positioned outside the
friction belt, for which reason the wake
fraction coefficient w will, in this case,
be a great deal lower. However, for a
twin�skeg ship with two propellers, the
coefficient w will be almost unchanged
(or maybe slightly lower) compared
with the single�propeller case.
Incidentally, a large wake fraction co�
efficient increases the risk of propeller
cavitation, as the distribution of the
water velocity around the propeller is
generally very inhomogeneous under
such conditions.
A more homogeneous wake field for
the propeller, also involving a higher
speed of advance VA of the propeller,
may sometimes be needed and can be
obtained in several ways, e.g. by hav�
ing the propellers arranged in nozzles,
below shields, etc. Obviously, the best
method is to ensure, already at the de�
sign stage, that the aft end of the hull is
shaped in such a way that the opti�
mum wake field is obtained.
Thrust deduction coefficient t
The rotation of the propeller causes the
water in front of it to be “sucked” back
towards the propeller. This results in an
extra resistance on the hull normally
called “augment of resistance” or, if re�
lated to the total required thrust force T
on the propeller, “thrust deduction frac�
tion” F, see Fig. 6. This means that the
thrust force T on the propeller has to
overcome both the ship’s resistance RT
and this “loss of thrust” F.
The thrust deduction fraction F may be
expressed in dimensionless form by
means of the thrust deduction coeffi�
cient t, which is defined as:
t
F
T
T R
T
you get
R
T
t
T
T
= =
−
= −( )1
The thrust deduction coefficient t can
be calculated by using calculation
models set up on the basis of research
carried out on different models.
In general, the size of the thrust deduc�
tion coefficient t increases when the
wake fraction coefficient w increases.
The shape of the hull may have a sig�
nificantinfluence, e.g. a bulbous stem
can, under certain circumstances (low
ship speeds), reduce t.
The size of the thrust deduction coeffi�
cient t for a ship with one propeller is,
normally, in the range of 0.12 to 0.30,
as a ship with a large block coefficient
has a large thrust deduction coefficient.
For ships with two propellers and a
conventional aftbody form of the hull,
the thrust deduction coefficient t will be
much less as the propellers’ “sucking”
occurs further away from the hull.
However, for a twin�skeg ship with two
propellers, the coefficient t will be almost
unchanged (or maybe slightly lower)
compared with the single�propeller case.
Efficiencies
Hull efficiency �H
The hull efficiency �H is defined as the
ratio between the effective (towing)
power PE = RT × V, and the thrust power
11
which the propeller delivers to the water
PT = T × VA, i.e.:
�H = =
×
×
= =
−
−
P
P
R V
T V
R T
V V
t
w
E
T
T
A
T
A
/
/
1
1
For a ship with one propeller, the hull
efficiency ηH is usually in the range of
1.1 to 1.4, with the high value for ships
with high block coefficients. For ships
with two propellers and a conventional
aftbody form of the hull, the hull effi�
ciency ηH is approx. 0.95 to 1.05, again
with the high value for a high block co�
efficient. However, for a twin�skeg ship
with two propellers, the hull coefficient
ηH will be almost unchanged compared
with the single�propeller case.
Open water propeller efficiency ηO
Propeller efficiency ηO is related to
working in open water, i.e. the propel�
ler works in a homogeneous wake field
with no hull in front of it.
The propeller efficiency depends, es�
pecially, on the speed of advance VA,
thrust force T, rate of revolution n, di�
ameter d and, moreover, i.a. on the de�
sign of the propeller, i.e. the number of
blades, disk area ratio, and pitch/diam�
eter ratio – which will be discussed
later in this chapter. The propeller effi�
ciency ηO can vary between approx.
0.35 and 0.75, with the high value be�
ing valid for propellers with a high
speed of advance VA, Ref. [3].
Fig. 8 shows the obtainable propeller
efficiency ηO shown as a function of the
speed of advance VA, which is given in
dimensionless form as:
J
V
n d
A
=
×
where J is the advance number of the
propeller.
Relative rotative efficiency ηR
The actual velocity of the water flowing
to the propeller behind the hull is nei�
ther constant nor at right angles to the
propeller’s disk area, but has a kind of
rotational flow. Therefore, compared
with when the propeller is working in
open water, the propeller’s efficiency is
affected by the ηR factor – called the
propeller’s relative rotative efficiency.
On ships with a single propeller the
rotative efficiency ηR is, normally, around
1.0 to 1.07, in other words, the rotation
of the water has a beneficial effect. The
rotative efficiency ηR on a ship with a
conventional hull shape and with two
propellers will normally be less, approx.
0.98, whereas for a twin�skeg ship with
two propellers, the rotative efficiency ηR
will be almost unchanged.
In combination with w and t, ηR is prob�
ably often being used to adjust the re�
sults of model tank tests to the theory.
Propeller efficiency ηB working behind
the ship
The ratio between the thrust power PT,
which the propeller delivers to the wa�
ter, and the power PD, which is deliv�
ered to the propeller, i.e. the propeller
efficiency ηB for a propeller working
behind the ship, is defined as:
� � �B
T
D
o R
P
P
= = ×
Propulsive efficiency ηD
The propulsive efficiency ηD, which
must not be confused with the open
water propeller efficiency ηO, is equal to
the ratio between the effective (towing)
power PE and the necessary power
delivered to the propeller PD, i.e.:
�
D
E
D
E
T
T
D
P
P
P
P
P
P
= = ×
= ηH × ηB = ηH × ηO × ηR
12
0.3
0.2
0.4
0
0.6
0.1
0.6
0.5
V
n x d
AAdvance number J =
0.4 0.50.2 0.30 0.1
o
0.7
Propeller
efficiency
Reefers
Container ships
Small tankers
20,000 DWT
Large tankers
>150,000 DWT
n ( revs./s )
1.66
2.00
0.7
Fig. 8: Obtainable propeller efficiency – open water, Ref. [3], page 213
As can be seen, the propulsive efficiency
ηD is equal to the product of the hull
efficiency ηH, the open water propeller
efficiency ηO, and the relative rotative
efficiency ηR, although the latter has
less significance.
In this connection, one can be led to
believe that a hull form giving a high
wake fraction coefficient w, and hence
a high hull efficiency ηH, will also provide
the best propulsive efficiency ηD.
However, as the open water propeller
efficiency ηO is also greatly dependent
on the speed of advance VA, cf. Fig. 8,
that is decreasing with increased w,
the propulsive efficiency ηD will not,
generally, improve with increasing w,
quite often the opposite effect is obtained.
Generally, the best propulsive efficiency
is achieved when the propeller works in
a homogeneous wake field.
Shaft efficiency ηS
The shaft efficiency ηS depends, i.a. on
the alignment and lubrication of the
shaft bearings, and on the reduction
gear, if installed.
Shaft efficiency is equal to the ratio be�
tween the power PD delivered to the
propeller and the brake power PB deliv�
ered by the main engine, i.e.
� �
S
D
B
P
P
The shaft efficiency is normally around
0.985, but can vary between 0.96 and
0.995.
Total efficiency ηT
The total efficiency ηT, which is equal to
the ratio between the effective (towing)
power PE, and the necessary brake
power PB delivered by the main engine,
can be expressed thus:
�
T
P
P
P
P
P
P
E
B
E
D
D
B
= = ×
= η
D
× η
S
= η
H
× η
O
× η
R
× η
S
Propeller dimensions
Propeller diameter d
With a view to obtaining the highest
possible propulsive efficiency ηD, the
largest possible propeller diameter d
will, normally, be preferred. There are,
however, special conditions to be con�
sidered. For one thing, the aftbody form
of the hull can vary greatly depending on
type of ship and ship design, for another,
the necessary clearance between the
tip of the propeller and the hull will de�
pend on the type of propeller.
For bulkers and tankers, which are often
sailing in ballast condition, there are
frequent demands that the propeller
shall be fully immersed also in this con�
dition, giving some limitation to the pro�
peller size. This propeller size limitation
is not particularly valid for container
ships as they rarely sail in ballast condi�
tion. All the above factors mean that an
exact propeller diameter/design draught
ratio d/D cannot be given here but, as
a rule�of�thumb, the below mentioned
approximations of the diameter/design
draught ratio d/D can be presented,
and a large diameter d will, normally,
result in a low rate of revolution n.
Bulk carrier and tanker:
d/D < approximately 0.65
Container ship:
d/D < approximately 0.74
For strength and production reasons,
the propeller diameter will generally not
exceed 10.0 metres and a power out�
put of about 90,000 kW. The largest�
diameter propeller manufactured so far
is of 11.0 metres and has four propeller
blades.
Number of propeller blades
Propellers can be manufactured with 2,
3, 4, 5 or 6 blades. The fewer the num�
ber of blades, the higher the propeller
efficiency will be. However, for reasons
of strength, propellers which are to be
subjected to heavy loads cannot be
manufactured with only two or three
blades.
Two�bladed propellers are used on
small ships, and 4, 5 and 6�bladed
propellers are used on large ships.
Ships using the MAN B&W two�stroke
engines are normally large�type vessels
which use 4�bladed propellers.Ships
with a relatively large power requirement
and heavily loaded propellers, e.g. con�
tainer ships, may need 5 or 6�bladed
propellers. For vibrational reasons, pro�
pellers with certain numbers of blades
may be avoided in individual cases in
order not to give rise to the excitation
of natural frequencies in the ship’s hull
or superstructure, Ref. [5].
Disk area coefficient
The disk area coefficient – referred to in
older literature as expanded blade area
ratio – defines the developed surface
area of the propeller in relation to its
disk area. A factor of 0.55 is considered
as being good. The disk area coefficient
of traditional 4�bladed propellers is of
little significance, as a higher value will
only lead to extra resistance on the
propeller itself and, thus, have little ef�
fect on the final result.
For ships with particularly heavy�loaded
propellers, often 5 and 6�bladed pro�
pellers, the coefficient may have a
higher value. On warships it can be as
high as 1.2.
Pitch diameter ratio p/d
The pitch diameter ratio p/d, expresses
the ratio between the propeller’s pitch
p and its diameter d, see Fig. 10. The
pitch p is the distance the propeller
“screws” itself forward through the wa�
ter per revolution, providing that there
is no slip – see also the next section
and Fig. 10. As the pitch can vary
along the blade’s radius, the ratio is
normally related to the pitch at 0.7 × r,
where r = d/2 is the propeller’s radius.
To achieve the best propulsive efficiency
for a given propeller diameter, an optimum
pitch/diameter ratio is to be found,
which again corresponds to a particu�
lar design rate of revolution. If, for
instance, a lower design rate of revolution
is desired, the pitch/diameter ratio has
to be increased, and vice versa, at the
cost of efficiency. On the other hand, if
a lower design rate of revolution is de�
sired, and the ship’s draught permits,
the choice of a larger propeller diame�
13
ter may permit such a lower design rate
of revolution and even, at the same time,
increase the propulsive efficiency.
Propeller coefficients J, KT and KQ
Propeller theory is based on models,
but to facilitate the general use of this
theory, certain dimensionless propeller
coefficients have been introduced in re�
lation to the diameter d, the rate of rev�
olution n, and the water’s mass density
�. The three most important of these
coefficients are mentioned below.
The advance number of the propeller J
is, as earlier mentioned, a dimensionless
expression of the propeller’s speed of
advance VA.
J
V
n d
A
=
×
The thrust force T, is expressed
dimensionless, with the help of the
thrust coefficient KT, as
K
T
n dT
=
× ×� 2 4
and the propeller torque
Q
P
n
D
=
×2�
is expressed dimensionless with the
help of the torque coefficient KQ, as
K
Q
n dQ
=
× ×� 2 5
The propeller efficiency �O can be cal�
culated with the help of the above�men�
tioned coefficients, because, as previously
mentioned, the propeller efficiency �O is
defined as:
�
� �
�
= =
×
× ×
= ×
P
P
T V
Q n
K
K
JT
D
A T
Q2 2
With the help of special and very com�
plicated propeller diagrams, which
contain, i.a. J, KT and KQ curves, it is
possible to find/calculate the propeller’s
dimensions, efficiency, thrust, power, etc.
Manufacturing accuracy of the propeller
Before the manufacturing of the propeller,
the desired accuracy class standard of
the propeller must be chosen by the
customer. Such a standard is, for ex�
ample, ISO 484/1 – 1981 (CE), which
has four different “Accuracy classes”,
see Table 5.
Each of the classes, among other de�
tails, specifies the maximum allowable
tolerance on the mean design pitch of
the manufactured propeller, and
thereby the tolerance on the correspond�
ing propeller speed (rate of revolution).
The price of the propeller, of course,
depends on the selected accuracy
class, with the lowest price for class III.
However, it is not recommended to
use class III, as this class has a too
high tolerance. This again means that
the mean pitch tolerance should nor�
mally be less than +/– 1.0 %.
The manufacturing accuracy tolerance
corresponds to a propeller speed toler�
ance of max. +/– 1.0 %. When also in�
corporating the influence of the tolerance
on the wake field of the hull, the total
propeller tolerance on the rate of revo�
lution can be up to +/– 2.0 %. This tol�
erance has also to be borne in mind
when considering the operating condi�
tions of the propeller in heavy weather.
Influence of propeller diameter and
pitch/diameter ratio on propulsive
efficiency D.
As already mentioned, the highest pos�
sible propulsive efficiency required to
provide a given ship speed is obtained
with the largest possible propeller dia�
meter d, in combination with the corre�
sponding, optimum pitch/diameter ra�
tio p/d.
14
ISO 484/1 – 1981 (CE)
Class
Manufacturing
accuracy
Mean pitch
for propel�
ler
S
I
II
III
Very high accuracy
High accuracy
Medium accuracy
Wide tolerances
+/– 0.5 %
+/– 0.75 %
+/– 1.00 %
+/– 3.00 %
Table 5: Manufacturing accuracy classes
of a propeller
110 120100 r/min130
Shaft power
80 90
8,800
70
8,700
8,900
9,100
8,600
8,500
9,400
0.95
9,200
9,300
9,000
d = Propeller diameter
p/d = Pitch/diameter ratio
Power and speed curve
for various propeller
diameters d with
optimum p/d Propeller speed
Power and speed curve
for the given propeller
diameter d = 7.2 m with
different p/d
80,000 dwt crude oil tanker
Design draught = 12.2 m
Ship speed = 14.5 kn9,500
0.90
0.85
0.80
0.71
1.00
0.60
0.75
d
0.65
0.55
6.8 m
p/d
0.67
7.2 m
6.6 m
7.4 m
7.0 m
0.70
p/d
0.68
0.69
0.50
p/d
p/d
d
kW
Fig. 9: Propeller design – influence of diameter and pitch
As an example for an 80,000 dwt crude
oil tanker, with a service ship speed of
14.5 knots and a maximum possible
propeller diameter of 7.2 m, this influence
is shown in Fig. 9.
According to the blue curve, the maxi�
mum possible propeller diameter of 7.2
m may have the optimum pitch/diame�
ter ratio of 0.70, and the lowest possi�
ble shaft power of 8,820 kW at 100
r/min. If the pitch for this diameter is
changed, the propulsive efficiency will
be reduced, i.e. the necessary shaft
power will increase, see the red curve.
The blue curve shows that if a bigger
propeller diameter of 7.4 m is possible,
the necessary shaft power will be re�
duced to 8,690 kW at 94 r/min, i.e. the
bigger the propeller, the lower the opti�
mum propeller speed.
The red curve also shows that propul�
sion�wise it will always be an advan�
tage to choose the largest possible
propeller diameter, even though the
optimum pitch/diameter ratio would
involve a too low propeller speed (in rela�
tion to the required main engine speed).
Thus, when using a somewhat lower
pitch/diameter ratio, compared with the
optimum ratio, the propeller/ engine
speed may be increased and will only
cause a minor extra power increase.
Operating conditions of a propeller
Slip ratio S
If the propeller had no slip, i.e. if the
water which the propeller “screws”
itself through did not yield (i.e. if the
water did not accelerate aft), the pro�
peller would move forward at a speed
of V = p × n, where n is the propeller’s
rate of revolution, see Fig. 10.
The similar situation is shown in Fig. 11
for a cork screw, and because the cork
is a solid material, the slip is zero and,
therefore, the cork screw always moves
forward at a speed of V = p × n. How�
ever, as the water is a fluid and does
yield (i.e. accelerate aft), the propeller’s
apparent speed forward decreases
with its slip and becomes equal to the
ship’s speed V, and its apparent slip
can thus be expressed asp × n – V.
The apparent slip ratio SA, which is
dimensionless, is defined as:
S
p n V
p n
V
p nA
=
× −
×
= −
×
1
The apparent slip ratio SA, which is cal�
culated by the crew, provides useful
knowledge as it gives an impression of
the loads applied to the propeller under
different operating conditions. The ap�
parent slip ratio increases when the
15
Velocity of corkscrew: V = p x n Pitch p
Wine bottleCorkscrew Cork
V
n
Fig. 11: Movement of a corkscrew, without slip
S x p x nV or VA
Pitch p
n
0.7 x r
r
d
p x n
Slip
The apparent slip ratio : S = = 1A
_
The real slip ratio : S = = 1R
_p x n _ V V
p x n p x n
A A
p x n _ V V
p x n p x n
Fig. 10: Movement of a ship´s propeller, with pitch p and slip ratio S
vessel sails against the wind or waves,
in shallow waters, when the hull is
fouled, and when the ship accelerates.
Under increased resistance, this in�
volves that the propeller speed (rate of
revolution) has to be increased in order
to maintain the required ship speed.
The real slip ratio will be greater than
the apparent slip ratio because the real
speed of advance VA of the propeller is,
as previously mentioned, less than the
ship’s speed V.
The real slip ratio SR, which gives a truer
picture of the propeller’s function, is:
S
V
p n
V w
p nR
A
= −
×
= −
× −
×
1 1
1( )
At quay trials where the ship’s speed is
V = 0, both slip ratios are 1.0. Incidentally,
slip ratios are often given in percentages.
Propeller law in general
As discussed in Chapter 1, the resis�
tance R for lower ship speeds is pro�
portional to the square of the ship’s
speed V, i.e.:
R = c × V2
where c is a constant. The necessary
power requirement P is thus propor�
tional to the speed V to the power of
three, thus:
P = R × V = c × V3
For a ship equipped with a fixed pitch
propeller, i.e. a propeller with unchange�
able pitch, the ship speed V will be pro�
portional to the rate of revolution n, thus:
P = c × n3
which precisely expresses the propeller
law, which states that “the necessary
power delivered to the propeller is pro�
portional to the rate of revolution to the
power of three”.
Actual measurements show that the
power and engine speed relationship
for a given weather condition is fairly
reasonable, whereas the power and
ship speed relationship is often seen
with a higher power than three. A rea�
sonable relationship to be used for esti�
mations in the normal ship speed range
could be as follows:
• For large high�speed ships like con�
tainer vessels: P = c × V 4.5
• For medium�sized, medium�speed
ships like feeder container ships,
reefers, RoRo ships, etc.: P = c × V 4.0
• For low�speed ships like tankers and
bulk carriers, and small feeder con�
tainer ships, etc.: P = c × V 3.5
Propeller law for heavy running propeller
The propeller law, of course, can only
be applied to identical ship running
conditions. When, for example, the
ship’s hull after some time in service
has become fouled and thus become
more rough, the wake field will be different
from that of the smooth ship (clean hull)
valid at trial trip conditions.
A ship with a fouled hull will, conse�
quently, be subject to extra resistance
which will give rise to a “heavy propeller
condition”, i.e. at the same propeller
power, the rate of revolution will be lower.
The propeller law now applies to an�
other and “heavier” propeller curve
than that applying to the clean hull,
propeller curve, Ref. [3], page 243.
The same relative considerations apply
when the ship is sailing in a heavy sea
against the current, a strong wind, and
heavy waves, where also the heavy
waves in tail wind may give rise to a
heavier propeller running than when
running in calm weather. On the other
hand, if the ship is sailing in ballast
condition, i.e. with a lower displace�
ment, the propeller law now applies to
a “lighter” propeller curve, i.e. at the
same propeller power, the propeller
rate of revolution will be higher.
As mentioned previously, for ships with
a fixed pitch propeller, the propeller law
is extensively used at part load running.
It is therefore also used in MAN B&W
Diesel’s engine layout and load diagrams
to specify the engine’s operational
curves for light running conditions (i.e.
clean hull and calm weather) and heavy
running conditions (i.e. for fouled hull
and heavy weather). These diagrams us�
ing logarithmic scales and straight lines
are described in detail in Chapter 3.
Propeller performance in general at
increased ship resistance
The difference between the above�men�
tioned light and heavy running propeller
curves may be explained by an exam�
ple, see Fig. 12, for a ship using, as ref�
erence, 15 knots and 100% propulsion
power when running with a clean hull in
calm weather conditions. With 15% more
power, the corresponding ship speed
may increase from 15.0 to 15.6 knots.
As described in Chapter 3, and com�
pared with the calm weather conditions,
it is normal to incorporate an extra
power margin, the so�called sea mar�
gin, which is often chosen to be 15%.
This power margin corresponds to ex�
tra resistance on the ship caused by
the weather conditions. However, for
very rough weather conditions the influ�
ence may be much greater, as de�
scribed in Chapter 1.
In Fig. 12a, the propulsion power is
shown as a function of the ship speed.
When the resistance increases to a
level which requires 15% extra power
to maintain a ship speed of 15 knots,
the operating point A will move towards
point B.
In Fig. 12b the propulsion power is
now shown as a function of the propeller
speed. As a first guess it will often be as�
sumed that point A will move towards B’
because an unchanged propeller speed
implies that, with unchanged pitch, the
propeller will move through the water
at an unchanged speed.
If the propeller was a corkscrew moving
through cork, this assumption would
be correct. However, water is not solid
as cork but will yield, and the propeller
will have a slip that will increase with in�
creased thrust caused by increased
hull resistance. Therefore, point A will
move towards B which, in fact, is very
close to the propeller curve through A.
Point B will now be positioned on a
propeller curve which is slightly heavy
running compared with the clean hull
and calm weather propeller curve.
16
Sometimes, for instance when the hull
is fouled and the ship is sailing in heavy
seas in a head wind, the increase in
resistance may be much greater, cor�
responding to an extra power demand
of the magnitude of 100% or even higher.
An example is shown in Fig. 12c.
In this example, where 100% power
will give a ship speed of 15.0 knots,
point A, a ship speed of, for instance,
12.3 knots at clean hull and in calm
weather conditions, point C, will require
about 50% propulsion power but, at
the above�mentioned heavy running
conditions, it might only be possible to
obtain the 12.3 knots by 100% propulsion
power, i.e. for 100% power going from
point A to D. Running point D may now
be placed relatively far to the left of point
A, i.e. very heavy running. Such a situ�
ation must be considered when laying�
out the main engine in relation to the
layout of the propeller, as described in
Chapter 3.
A scewed propeller (with bent blade
tips) is more sensitive to heavy running
than a normal propeller, because the
propeller is able to absorb a higher
torque in heavy running conditions. For
a ducted propeller, the opposite effect
is obtained.
Heavy waves and sea and wind against
When sailing in heavy sea against, with
heavy wave resistance, the propeller
can be up to 7�8% heavier running
than in calm weather, i.e. at the same
propeller power,the rate of revolution
may be 7�8% lower. An example valid
for a smaller container ship is shown in
Fig. 13. The service data is measured
17
(Logarithmic scales)
Power
Propeller speed
15.0 knots
100% power
Propeller curve
for clean hull and
calm weather
Propeller
curve for
fouled hull
and heavy
seas
LR
Slip
HR HR = Heavy running
LR = Light running
D´ A
C
12.3 knots
100% power
12.3 knots
50% power
10.0 knots
50% power
D
Fig. 12c: Propeller speed performance at
large extra ship resistance
(Logarithmic scales)
Power
Propeller speed
15.6 knots
115% power
15.0 knots
100% power
15%
Sea
margin
Slip
Propeller curve for clean
hull and calm weather
15.0 knots
115% power
B´
A
B
Fig. 12b: Propeller speed performance at
15% sea margin
BHP
21,000
18,000
15,000
12,000
9,000
6,000
76 80 9284 9688 100
Ship speedknots
Shaft power
Clean hull and draught D
D = 6.50 m
D = 5.25 m
D = 7.75 m
Source: Lloyd's Register
MEAN
F
A
Average weather 3%
Extremely good weather 0%
Extremely bad weather 6%
Propeller speed
Apparent slip
10%
6%
Heavy
running
2%
�2%
13
16
19
22
C
B
A
C
B
A
r/min
Fig. 13: Service data over a period of a year returned from a single screw container ship
(Logarithmic scales)
15.6 knots
115% power
15.0 knots
100% power
15%
Sea
margin
Power
Ship speed
Propeller curve for clean
hull and calm weather
15.0 knots
115% power
B
A
Fig. 12a: Ship speed performance at 15%
sea margin
over a period of one year and only
includes the influence of weather con�
ditions! The measuring points have
been reduced to three average weather
conditions and show, for extremely bad
weather conditions, an average heavy
running of 6%, and therefore, in prac�
tice, the heavy running has proved to
be even greater.
In order to avoid slamming of the ship,
and thereby damage to the stem and
racing of the propeller, the ship speed
will normally be reduced by the navigat�
ing officer on watch.
Another measured example is shown
in Fig. 14, and is valid for a reefer ship
during its sea trial. Even though the
wind velocity is relatively low, only 2.5
m/s, and the wave height is 4 m, the
measurements indicate approx. 1.5%
heavy running when sailing in head
wind out, compared with when sailing
in tail wind on return.
Ship acceleration
When the ship accelerates, the propel�
ler will be subjected to an even larger
load than during free sailing. The power
required for the propeller, therefore, will
be relatively higher than for free sailing,
and the engine’s operating point will be
heavy running, as it takes some time
before the propeller speed has reached
its new and higher level. An example
with two different accelerations, for an
engine without electronic governor and
scavenge air pressure limiter, is shown
in Fig. 15. The load diagram and scav�
enge air pressure limiter are is described
in Chapter 3.
Shallow waters
When sailing in shallow waters, the re�
sidual resistance of the ship may be in�
creased and, in the same way as when
the ship accelerates, the propeller will
be subjected to a larger load than dur�
ing free sailing, and the propeller will be
heavy running.
Influence of displacement
When the ship is sailing in the loaded
condition, the ship’s displacement vol�
ume may, for example, be 10% higher
or lower than for the displacement valid
for the average loaded condition. This,
of course, has an influence on the ship’s
resistance, and the required propeller
power, but only a minor influence on
the propeller curve.
On the other hand, when the ship is
sailing in the ballast condition, the dis�
placement volume, compared to the
loaded condition, can be much lower,
and the corresponding propeller curve
may apply to, for example, a 2% “lighter”
propeller curve, i.e. for the same power
to the propeller, the rate of revolution
will be 2% higher.
Parameters causing heavy running
propeller
Together with the previously described
operating parameters which cause a
heavy running propeller, the parame�
ters summarised below may give an in�
dication of the risk/sensitivity of getting
a heavy running propeller when sailing
in heavy weather and rough seas:
1 Relatively small ships (<70,000 dwt)
such as reefers and small container
ships are sensitive whereas large ships,
such as large tankers and container
ships, are less sensitive because the
waves are relatively small compared
to the ship size.
2 Small ships (Lpp < 135 m≈ 20,000 dwt)
have low directional stability and,
therefore, require frequent rudder
corrections, which increase the ship
resistance (a self�controlled rudder
will reduce such resistance).
3 High�speed ships
are more sensitive than low�speed
ships because the waves will act on
the fast�going ship with a relatively
18
Pr
op
ell
er 
cu
rve
SMCR: 13,000 kW x 105 r/min
Wind velocity : 2.5 m/s
Wave height : 4 m
Propeller/engine speed
100
90
105
85
100
95
80
10199 103 105 % SMCR10297 9896 104
Heavy
running
En
gin
e "
pr
op
ell
er 
cu
rve
"
Pr
op
ell
er 
cu
rve
Propeller design
light running
* 20.5
21.5
*
20.5
*
*20.8
*21.2
*22.0
21.1 *
7
5
1
3
4
Shaft power
% SMCR
22.3 *
21.8
*
SMCR
* 21.1
Head wind
Tail wind
(Logarithmic scales)
Fig. 14: Measured relationship between power, propeller and ship speed during seatrial of
a reefer ship
larger force than on the slow�going
ship.
4 Ships with a “flat” stem
may be slowed down faster by waves
than a ship with a “sharp” stem.
Thus an axe�shaped upper bow may
better cut the waves and thereby
reduce the heavy running tendency.
5 Fouling of the hull and propeller
will increase both hull resistance and
propeller torque. Polishing the pro�
peller (especially the tips) as often as
possible (also when in water) has a
positive effect. The use of effective
anti�fouling paints will prevent fouling
caused by living organisms.
6 Ship acceleration
will increase the propeller torque,
and thus give a temporarily heavy
running propeller.
7 Sailing in shallow waters
increases the hull resistance and re�
duces the ship’s directional stability.
8 Ships with scewed propeller
are able to absorb a higher torque
under heavy running conditions.
Manoeuvring speed
Below a certain ship speed, called the
manoeuvring speed, the manoeuvra�
bility of the rudder is insufficient be�
cause of a too low velocity of the water
arriving at the rudder. It is rather difficult
to give an exact figure for an adequate
manoeuvring speed of the ship as the
velocity of the water arriving at the rud�
der depends on the propeller’s slip
stream.
Often a manoeuvring speed of the
magnitude of 3.5�4.5 knots is men�
tioned. According to the propeller law,
a correspondingly low propulsion
power will be needed but, of course,
this will be higher for running in heavy
weather with increased resistance on
the ship.
Direction of propeller rotation (side thrust)
When a ship is sailing, the propeller
blades bite more in their lowermost po�
sition than in their uppermost position.
The resulting side�thrust effect is larger
the more shallow the water is as, for
example, during harbour manoeuvres.
Therefore, a clockwise (looking from aft
to fore) rotating propeller will tend to
push the ship’s stern in the starboard
direction, i.e. pushing the ship’s stem
to port, during normal ahead running.
This has to be counteracted by the
rudder.
When reversing the propeller to astern
running as, for example, when berthing
alongside the quay, the side�thrust ef�
fect is also reversed and becomes fur�
ther pronounced as the ship’s speed
decreases. Awareness of thisbehav�
iour is very important in critical situa�
tions and during harbour manoeuvres.
According to Ref. [5], page 15�3, the
real reason for the appearance of the
side thrust during reversing of the pro�
peller is that the upper part of the pro�
peller’s slip stream, which is rotative,
strikes the aftbody of the ship.
Thus, also the pilot has to know pre�
cisely how the ship reacts in a given
situation. It is therefore an unwritten
law that on a ship fitted with a fixed
pitch propeller, the propeller is always
designed for clockwise rotation when
sailing ahead. A direct coupled main
engine, of course, will have the same
rotation.
In order to obtain the same side�thrust
effect, when reversing to astern, on
ships fitted with a controllable pitch
propeller, CP�propellers are designed
for anti�clockwise rotation when sailing
ahead.
19
80 100 10585
50
7565 90 9560
60
70
80
90
mep
110%
Engine speed, % A
40
A=M
100
Engine shaft power, % A
100%
90%
80%
70%
60%
O
A 100% reference point
M Specified engine MCR
O Optimising point
110
(Logarithmic scales)
70
Fig. 15: Load diagram – acceleration
Engine Layout and
Load Diagrams
Power functions and logarithmic
scales
As is well�known, the effective brake
power PB of a diesel engine is propor�
tional to the mean effective pressure
(mep) pe and engine speed (rate of rev�
olution) n. When using c as a constant,
PB may then be expressed as follows:
PB = c × pe × n
or, in other words, for constant mep
the power is proportional to the speed:
PB = c × n
1 (for constant mep)
As already mentioned – when running
with a fixed pitch propeller – the power
may, according to the propeller law, be
expressed as:
PB = c × n
3 (propeller law)
Thus, for the above examples, the brake
power PB may be expressed as a func�
tion of the speed n to the power of i, i.e.
PB = c × n
i
Fig. 16 shows the relationship between
the linear functions, y = ax + b, see (A),
using linear scales and the power func�
tions PB = c × n
i, see (B), using logarith�
mic scales.
The power functions will be linear when
using logarithmic scales, as:
log (PB) = i × log (n) + log (c)
which is equivalent to: y = ax + b
Thus, propeller curves will be parallel to
lines having the inclination i = 3, and
lines with constant mep will be parallel
to lines with the inclination i = 1.
Therefore, in the layout and load diagrams
for diesel engines, as described in the
following, logarithmic scales are used,
making simple diagrams with straight
lines.
Propulsion and engine running
points
Propeller design point PD
Normally, estimations of the necessary
propeller power and speed are based
on theoretical calculations for loaded
ship, and often experimental tank tests,
both assuming optimum operating
conditions, i.e. a clean hull and good
weather. The combination of speed
and power obtained may be called the
ship’s propeller design point PD placed
on the light running propeller curve 6,
see Fig. 17. On the other hand, some
shipyards and/or propeller manufactur�
ers sometimes use a propeller design
point PD´ that incorporates all or part of
the so�called sea margin described be�
low.
Fouled hull
When the ship has been sailing for
some time, the hull and propeller be�
come fouled and the hull’s resistance
will increase. Consequently, the ship
speed will be reduced unless the engine
delivers more power to the propeller, i.e.
the propeller will be further loaded and
will become heavy running HR.
Furthermore, newer high�efficiency ship
types have a relatively high ship speed,
and a very smooth hull and propeller
surface (at sea trial) when the ship is
delivered. This means that the inevitable
build�up of the surface roughness on
the hull and propeller during sea service
after seatrial may result in a relatively
heavier running propeller, compared
with older ships born with a more rough
hull surface.
Heavy weather and sea margin used
for layout of engine
If, at the same time, the weather is
bad, with head winds, the ship’s resis�
tance may increase much more, and
lead to even heavier running.
When determining the necessary en�
gine power, it is normal practice to add
an extra power margin, the so�called
sea margin, which is traditionally about
15% of the propeller design PD power.
However, for large container ships,
20�30% may sometimes be used.
When determining the necessary en�
gine speed, for layout of the engine, it
is recommended – compared with the
clean hull and calm weather propeller
curve 6 – to choose the heavier propel�
ler curve 2, see Fig. 17, corresponding
to curve 6 having a 3�7% higher rate of
revolution than curve 2, and in general
with 5% as a good choice.
Note that the chosen sea power mar�
gin does not equalise the chosen
heavy engine propeller curve.
20
A. Straight lines in linear scales
a
2
X
1 20
1
0
b
y = ax + b
B. Power function curves
in logarithmic scales
P
P
B
B
= engine brake power
c = constant
n = engine speed
log( ) = i x log(n) + log(c)
y = ax + b
P = c x ni
i = 1
i = 2
i = 3
y = log (P )B
i = 0
x = log (n)
y = log (P ) = log (c x n )B
i
y
B
Fig. 16: Relationship between linear functions
using linear scales and power functions
using logarithmic scales
Continuous service propulsion point SP
The resulting speed and power combi�
nation – when including heavy propeller
running and sea margin – is called the
“continuous service rating for propulsion”
SP for fouled hull and heavy weather.
The heavy propeller curve, curve 2, for
fouled hull and heavy weather will nor�
mally be used as the basis for the en�
gine operating curve in service, and the
propeller curve for clean hull and calm
weather, curve 6, is said to represent a
“light running” LR propeller.
Continuous service rating S
The continuous service rating is the
power at which the engine, including
the sea margin, is assumed to operate,
and point S is identical to the service
propulsion point SP unless a main en�
gine driven shaft generator is installed.
Light running factor fLR
The heavy propeller curve for a fouled
hull and heavy weather, and if no shaft
generator is installed may, as mentioned
above, be used as the design basis for
the engine operating curve in service,
curve 2, whereas the light propeller
curve for clean hull and calm weather,
curve 6, may be valid for running con�
ditions with new ships, and equal to
the layout/design curve of the propel�
ler. Therefore, the light propeller curve
for clean hull and calm weather is said
to represent a “light running” LR pro�
peller and will be related to the heavy
propeller curve for fouled hull and
heavy weather condition by means of a
light running factor fLR, which, for the
same power to the propeller, is defined
as the percentage increase of the rate
of revolution n, compared to the rate of
revolution for heavy running, i.e.
f
n n
nLR
light heavy
heavy
=
−
×100%
Engine margin
Besides the sea margin, a so�called
“engine margin” of some 10�15% is
frequently added as an operational
margin for the engine. The correspond�
ing point is called the “specified MCR
for propulsion” MP, see Fig. 17, and
refers to the fact that the power for
point SP is 10�15% lower than for
point MP, i.e. equal to 90�85% of MP.
Specified MCR M
The engine’s specified MCR point M is
the maximum rating required by the
yard or owner for continuous operation
of the engine. Point M is identical to the
specified propulsion MCR point MP un�
less a main engine driven shaft genera�
tor is installed. In such a case, the extra
power demand of the shaft generator
must also be considered.
Note:
Light/heavy running, fouling and sea
margin are overlapping terms.
Light/heavy runningof the propeller re�
fers to hull and propeller deterioration,
and bad weather, and sea margin, i.e.
extra power to the propeller, refers to
the influence of the wind and the sea.
Based on feedback from service, it
seems reasonable to design the pro�
peller for 3�7% light running. The de�
gree of light running must be decided
upon, based on experience from the
actual trade and hull design, but 5%
is often a good choice.
21
Fig. 17: Ship propulsion running points and engine layout
LR(5%)
Engine speed
Power
MP
Sea margin
(15% of PD)
2 6
SP
HR
PD´
PD
Engine margin
(10% of MP)
2 Heavy propeller curve fouled hull and heavy weather
6 Light propeller curve clean hull and calm weather
MP: Specified propulsion point
SP: Service propulsion point
PD: Propeller design point
Alternative propeller design point
LR: Light running factor
HR: Heavy running
_
_
Pd´:
Engine layout diagram
An engine’s layout diagram is limited by
two constant mean effective pressure
(mep) lines L1�L3 and L2�L4, and by two
constant engine speed lines L1�L2 and
L3�L4, see Fig. 17. The L1 point refers to
the engine’s nominal maximum contin�
uous rating. Within the layout area
there is full freedom to select the en�
gines specified MCR point M and rele�
vant optimising point O, see below,
which is optimum for the ship and the
operating profile. Please note that the
lowest specific fuel oil consumption for
a given optimising point O will be ob�
tained at 70% and 80% of point O’s
power, for electronically (ME) and me�
chanically (MC) controlled engines,
respectively.
Based on the propulsion and engine
running points, as previously found, the
layout diagram of a relevant main en�
gine may be drawn�in. The specified
MCR point M must be inside the limita�
tion lines of the layout diagram; if it is
not, the propeller speed will have to be
changed or another main engine type
must be chosen. Yet, in special cases,
point M may be located to the right of
line L1�L2, see “Optimising/Matching
Point” below.
Optimising point O
The “Optimising (MC)/Matching (ME)
point” O – or, better, the layout point of
the engine – is the rating at which the
engine (timing and) compression ratio
are adjusted, with consideration to the
scavenge air pressure of the turbocharger.
As mentioned below, under “Load dia�
gram”, the optimising point O (later on
in this paper also used in general
where matching point for ME engines
was the correct one) is placed on line 1
(layout curve of engine) of the load dia�
gram, and the optimised power can be
from 85 to 100% of point M‘s power.
Overload running will still be possible
(110% of M‘s power), as long as consid�
eration to the scavenge air pressure has
been taken.
The optimising point O is to be placed
inside the layout diagram. In fact, the
specified MCR point M can be placed
outside the layout diagram, but only by
exceeding line L1�L2, and, of course,
only provided that the optimising point
O is located inside the layout diagram.
It should be noted that MC/MC�C en�
gines without VIT (variable injection tim�
ing) fuel pumps cannot be optimised at
part�load. Therefore, these engines are
always optimised in point A, i.e. having
point M‘s power.
Load diagram
Definitions
The load diagram (Fig. 18) defines the
power and speed limits for continuous
as well as overload operation of an in�
stalled engine which has an optimising
point O and a specified MCR point M
that conforms to the ship’s specification.
Point A is a 100% speed and power
reference point of the load diagram,
and is defined as the point on the pro�
22
Line 1: Propeller curve through optimising point (O) layout curve for engine
Line 2: Heavy propeller curve fouled hull and heavy seas
Line 3: Speed limit
Line 4: Torque/speed limit
Line 5: Mean effective pressure limit
Line 6: Light propeller curve clean hull and calm weather layout curve for propeller
Line 7: Power limit for continuous running
Line 8: Overload limit
Line 9: Sea trial speed limit
Line 10: Constant mean effective pressure (mep) lines
_
_
_ _
80 100 10585
50
70 7565 90 9560
60
70
80
90
mep
110%
Engine speed, % A
40
2
4
A=M
9
7
8
5
100
Engine shaft power, % A
6
100%
90%
80%
70%
60%
1
10
3
O
A 100% reference point
M Specified engine MCR
O Optimising point
110
Fig. 18: Engine load diagram
peller curve (line 1) – the layout curve of
the engine – through the optimising point
O, having the specified MCR power.
Normally, point M is equal to point A,
but in special cases, for example if a
shaft generator is installed, point M
may be placed to the right of point A
on line 7. The service points of the in�
stalled engine incorporate the engine
power required for ship propulsion and
for the shaft generator, if installed.
During shoptest running, the engine will
always operate along curve 1, with
point A as 100% MCR. If CP�propeller
and constant speed operation is re�
quired, the delivery test may be fin�
ished with a constant speed test.
Limits to continuous operation
The continuous service range is limited
by the four lines 4, 5, 7 and 3 (9), see
Fig. 18:
Line 3 and line 9
Line 3 represents the maximum accept�
able speed for continuous operation, i.e.
105% of A, however, maximum 105%
of L1. During sea trial conditions the
maximum speed may be extended to
107% of A, see line 9.
The above limits may, in general, be
extended to 105% and, during sea trial
conditions, to 107% of the nominal L1
speed of the engine, provided the tor�
sional vibration conditions permit.
The overspeed set�point is 109% of
the speed in A, however, it may be
moved to 109% of the nominal speed
in L1, provided that torsional vibration
conditions permit.
Running at low load above 100% of
the nominal L1 speed of the engine is,
however, to be avoided for extended
periods.
Line 4:
Represents the limit at which an ample
air supply is available for combustion and
imposes a limitation on the maximum
combination of torque and speed.
Line 5:
Represents the maximum mean effec�
tive pressure level (mep) which can be
accepted for continuous operation.
Line 7:
Represents the maximum power for
continuous operation.
Line 10:
Represents the mean effective pressure
(mep) lines. Line 5 is equal to the 100%
mep�line. The mep�lines are also an
expression of the corresponding fuel
index of the engine.
Limits for overload operation
The overload service range is limited as
follows, see Fig. 18:
Line 8:
Represents the overload operation limi�
tations.
The area between lines 4, 5, 7 and the
dashed line 8 in Fig. 18 is available for
overload running for limited periods
only (1 hour per 12 hours).
23
Point A of load diagram
Line 1: Propeller curve through optimising point (O)
Line 7: Constant power line through specified MCR (M)
Point A: Intersection between lines 1 and 7
Engine speed
Power
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
A=M=MP
Propulsion and
engine service curve
for heavy running
7
S=SP
O
2
1
6
Fig. 19a: Example 1 with FPP – engine layout without SG (normal case)
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Engine speed
A=M
Propulsion and engine service
curve for heavy running
3.3% A
5
62
4
Power 1
S
7
O
5
6
3
4 1
7
2
5% A
5% L1
Fig. 19b: Example 1 with FPP – load diagram without SG (normal case)
Electronic governor with load limitation
In order to safeguard the diesel engine
against thermal and mechanical overload,
the approved electronicgovernors include
the following two limiter functions:
• Torque limiter
The purpose of the torque limiter is
to ensure that the limitation lines of
the load diagram are always observed.
The torque limiter algorithm compares
the calculated fuel pump index (fuel
amount) and the actually measured
engine speed with a reference limiter
curve giving the maximum allowable
fuel pump index at a given engine
speed. If the calculated fuel pump
index is above this curve, the result�
ing fuel pump index will be reduced
correspondingly.
The reference limiter curve is to be
adjusted so that it corresponds to the
limitation lines of the load diagram.
• Scavenge air pressure limiter
The purpose of the scavenge air
pressure limiter is to ensure that the
engine is not being overfuelled during
acceleration, as for example during
manoeuvring.
The scavenge air pressure limiter
algorithm compares the calculated
fuel pump index and measured
scavenge air pressure with a refer�
ence limiter curve giving the maxi�
mum allowable fuel pump index at a
given scavenge air pressure. If the
calculated fuel pump index is above
this curve, the resulting fuel pump
index will be reduced correspondingly.
The reference limiter curve is to be
adjusted to ensure that sufficient air
will always be available for a good
combustion process.
Recommendation
Continuous operation without a time
limitation is allowed only within the area
limited by lines 4, 5, 7 and 3 of the
load diagram. For fixed pitch propeller
operation in calm weather with loaded
ship and clean hull, the propeller/engine
may run along or close to the propeller
design curve 6.
After some time in operation, the ship’s
hull and propeller will become fouled,
resulting in heavier running of the pro�
peller, i.e. the propeller curve will move
to the left from line 6 towards line 2, and
extra power will be required for propulsion
in order to maintain the ship speed.
At calm weather conditions the extent
of heavy running of the propeller will
indicate the need for cleaning the hull
and, possibly, polishing the propeller.
The area between lines 4 and 1 is avail�
able for operation in shallow water,
heavy weather and during acceleration,
i.e. for non�steady operation without
any actual time limitation.
24
Point A of load diagram
Line 1: Propeller curve through optimising point (O)
Line 7: Constant power line through specified MCR (M)
Point A: Intersection between lines 1 and 7
Engine speed
M=MP
Propulsion and
engine service curve
for heavy running
7
S=SP
621
O
Power
A
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Fig. 20a: Example 2 with FPP – engine layout without SG (special case)
M
Propulsion and engine service
curve for heavy running
3.3% A5
62
4
1
S
7
5
6
3
4 1
7
2
5% A
5% L1
A
O
Engine speed
Power
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Fig. 20b: Example 2 with FPP – load diagram without SG (special case)
The recommended use of a relatively
high light running factor for design of
the propeller will involve that a relatively
higher propeller speed will be used for
layout design of the propeller. This, in
turn, may involve a minor reduction of
the propeller efficiency, and may possi�
bly cause the propeller manufacturer to
abstain from using a large light running
margin. However, this reduction of the
propeller efficiency caused by the large
light running factor is actually relatively
insignificant compared with the improved
engine performance obtained when
sailing in heavy weather and/or with
fouled hull and propeller.
Use of layout and load
diagrams � examples
In the following, four different examples
based on fixed pitch propeller (FPP)
and one example based on controllable
pitch propeller (CPP) are given in order
to illustrate the flexibility of the layout
and load diagrams.
In this respect the choice of the optimi�
sing point O has a significant influence.
Examples with fixed pitch propeller
Example 1:
Normal running conditions, without
shaft generator
Normally, the optimising point O, and
thereby the engine layout curve 1, will
be selected on the engine service
curve 2 (for heavy running), as shown
in Fig. 19a.
Point A is then found at the intersection
between propeller curve 1 (2) and the
constant power curve through M, line
7. In this case, point A will be equal to
point M.
Once point A has been found in the
layout diagram, the load diagram can
be drawn, as shown in Fig. 19b, and
hence the actual load limitation lines
of the diesel engine may be found.
Example 2:
Special running conditions, without
shaft generator
When the ship accelerates, the propel�
ler will be subjected to an even larger
load than during free sailing. The same
applies when the ship is subjected to
an extra resistance as, for example,
when sailing against heavy wind and
sea with large wave resistance.
In both cases, the engine’s operating
point will be to the left of the normal
operating curve, as the propeller will
run heavily.
In order to avoid exceeding the
left�hand limitation line 4 of the load
diagram, it may, in certain cases, be
necessary to limit the acceleration
and/or the propulsion power.
If the expected trade pattern of the
ship is to be in an area with frequently
appearing heavy wind and sea and
25
Sh
af
t g
en
er
at
or
Propulsion curve
for heavy running
O
7
1 2
Engine service curve
for heavy running
MP
SG
6
S
SG
SP
Engine speed
Power A=M
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Point A of load diagram
Line 1: Propeller curve through optimising point (O)
Line 7: Constant power line through specified MCR (M)
Point A: Intersection between lines 1 and 7
Fig. 21a: Example 3 with FPP – engine layout with SG (normal case)
Sh
af
t g
en
er
at
or
Propulsion curve
for heavy running
3.3% A
S
5
62
4
1
7
O
5
6
3
4 1
7
2
5% A
5% L1
A=M
Engine service curve
for heavy running
MP
SP
Engine speed
Power
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Fig. 21b: Example 3 with FPP – load diagram with SG (normal case)
large wave resistance, it can, therefore,
be an advantage to design/move the
load diagram more towards the left.
The latter can be done by moving the
engine’s optimising point O – and thus
the propeller curve 1 through the opti�
mising point – towards the left. How�
ever, this will be at the expense of a
slightly increased specific fuel oil con�
sumption.
An example is shown in Figs. 20a and
20b. As will be seen in Fig. 20b, and
compared with the normal case shown
in Example 1 (Fig. 19b), the left�hand
limitation line 4 is moved to the left, giv�
ing a wider margin between lines 2 and
4, i.e. a larger light running factor has
been used in this example.
Example 3:
Normal case, with shaft generator
In this example a shaft generator (SG)
is installed, and therefore the service
power of the engine also has to incor�
porate the extra shaft power required
for the shaft generator’s electrical
power production.
In Fig. 21a, the engine service curve
shown for heavy running incorporates
this extra power.
The optimising point O, and thereby the
engine layout curve 1, will normally be
chosen on the propeller curve (~ en�
gine service curve) through point M.
Point A is then found in the same way
as in example 1, and theload diagram
can be drawn as shown in Fig. 21b.
Example 4:
Special case, with shaft generator
Also in this special case, a shaft gener�
ator is installed but, unlike in Example
3, now the specified MCR for propul�
sion MP is placed at the top of the lay�
out diagram, see Fig. 22a. This involves
that the intended specified MCR of the
engine (Point M’) will be placed outside
the top of the layout diagram.
One solution could be to choose a
diesel engine with an extra cylinder,
but another and cheaper solution is to
reduce the electrical power production
of the shaft generator when running in
the upper propulsion power range.
If choosing the latter solution, the re�
quired specified MCR power of the en�
gine can be reduced from point M’ to
point M as shown in Fig. 22a. Therefore,
when running in the upper propulsion
power range, a diesel generator has to
take over all or part of the electrical
power production.
However, such a situation will seldom
occur, as ships rather infrequently op�
erate in the upper propulsion power
range. In the example, the optimising
point O has been chosen equal to
point S, and line 1 may be found.
Point A, having the highest possible
power, is then found at the intersection
of line L1�L3 with line 1, see Fig. 22a,
and the corresponding load diagram is
26
Sh
af
t g
en
er
at
or
Propulsion curve for heavy running
SG
O=S
61
7
2
M
Engine service curve
for heavy running
MP
SP
A
M´
Engine speed
Power
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Point A and M of load diagram
Line 1: Propeller curve through optimising point (O)
M: Located on constant power line 7 through point A
and at MP’s speed
Point A: Intersection between line 1 and line L � L
Point
1 3
Fig. 22a: Example 4 with FPP – engine layout with SG (special case)
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Sh
af
t g
en
er
at
or
Propulsion curve
for heavy running
3.3% A
SG
5
62
4
1
7
O=S
6
3
1
7
2
5% A
5% L1
M
Engine service curve
for heavy running
MP
SP
A
M´
5
4
Engine speed
Power
Fig. 22b: Example 4 with FPP – load diagram with SG (special case)
drawn in Fig. 22b. Point M is found on
line 7 at MP’s speed.
Example with controllable pitch propeller
Example 5:
With or without shaft generator
Layout diagram – without shaft generator
If a controllable pitch propeller (CPP)
is applied, the combinator curve (of
the propeller with optimum propeller
efficiency) will normally be selected for
loaded ship including sea margin.
For a given propeller speed, the com�
binator curve may have a given propeller
pitch, and this means that, like for a fixed
pitch propeller, the propeller may be
heavy running in heavy weather.
Therefore, it is recommended to use a
light running combinator curve (the dotted
curve), as shown in Fig. 23, to obtain an
increased operating margin for the diesel
engine in heavy weather to the load limits
indicated by curves 4 and 5.
Layout diagram – with shaft generator
The hatched area in Fig. 23 shows the
recommended speed range between
100% and 96.7% of the specified MCR
speed for an engine with shaft generator
running at constant speed.
The service point S can be located at
any point within the hatched area.
The procedure shown in Examples 3
and 4 for engines with FPP can also be
applied for engines with CPP running
on a combinator curve.
The optimising point O for engines with
VIT can be chosen on the propeller curve
1 through point A = M with an optimised
power from 85 to 100% of the specified
MCR as mentioned before in the section
dealing with optimising point O.
Load diagram
Therefore, when the engine’s specified
MCR point M has been chosen including
engine margin, sea margin and the
power for a shaft generator, if installed,
point M can be used as point A of the
load diagram, which can then be drawn.
The position of the combinator curve
ensures the maximum load range
within the permitted speed range for
engine operation, and it still leaves a
reasonable margin to the load limits
indicated by curves 4 and 5.
Influence on engine running of
different types of ship resistance
– plant with FP�propeller
In order to give a brief summary regard�
ing the influence on the fixed pitch
propeller running and main engine opera�
tion of different types of ship resistance,
an arbitrary example has been chosen,
see the load diagram in Fig. 24.
The influence of the different types of
resistance is illustrated by means of
corresponding service points for propul�
sion having the same propulsion power,
using as basis the propeller design
point PD, plus 15% extra power.
Propeller design point PD
The propeller will, as previously described,
normally be designed according to a
specified ship speed V valid for loaded
ship with clean hull and calm weather
conditions. The corresponding engine
speed and power combination is
shown as point PD on propeller curve
6 in the load diagram, Fig. 24.
Increased ship speed, point S0
If the engine power is increased by, for
example, 15%, and the loaded ship is
still operating with a clean hull and in
calm weather, point S0, the ship speed
27
Min
speed
A=M
S
3.3%A
5
4
1
7
5%A
5%L
Recommended range
for shaft generator
operation with
constant speed
3
1
5
1
O
7
Combinator curve
for loaded ship
and incl. sea margin
Max
speed
M: Specified MCR of engine
S: Continuous service rating of engine
O: Optimising point of engine
A: Reference point of load diagram
Engine speed
Power
4
Fig. 23: Example 5 with CPP – with or without shaft generator
V and engine speed n will increase in
accordance with the propeller law (more
or less valid for the normal speed range):
V V V
n n n
S
S
0
3 5
0
3 0
115 1041
115 1048
= × = ×
= × = ×
. .
. .
.
.
Point S0 will be placed on the same
propeller curve as point PD.
Sea running with clean hull and 15%
sea margin, point S2
Conversely, if still operating with loaded
ship and clean hull, but now with extra
resistance from heavy seas, an extra
power of, for example, 15% is needed
in order to maintain the ship speed V
(15% sea margin).
As the ship speed VS2 = V, and if the
propeller had no slip, it would be expected
that the engine (propeller) speed would
also be constant. However, as the water
does yield, i.e. the propeller has a slip,
the engine speed will increase and the
running point S2 will be placed on a
propeller curve 6.2 very close to S0, on
propeller curve 6. Propeller curve 6.2
will possibly represent an approximate
0.5% heavier running propeller than
curve 6.
Depending on the ship type and size,
the heavy running factor of 0.5% may
be slightly higher or lower.
For a resistance corresponding to
about 30% extra power (30% sea mar�
gin), the corresponding relative heavy
running factor will be about 1%.
Sea running with fouled hull, and
heavy weather, point SP
When, after some time in service, the
ship’s hull has been fouled, and thus
becomes more rough, the wake field
will be different from that of a smooth
ship (clean hull).
A ship with a fouled hull will, conse�
quently, be subject to an extra resis�
tance which, due to the changed
wake field, will give rise to a heavier
running propeller than experienced
during bad weather conditions alone.
When also incorporating some aver�
age influence of heavy weather, the
propeller curve for loaded ship will
move to the left, see propeller curve
2 in the load diagram in Fig. 24. This
propeller curve, denotedfouled hull
and heavy weather for a loaded ship,
is about 5% heavy running compared
to the clean hull and calm weather
propeller curve 6.
In order to maintain an ample air
supply for the diesel engine’s com�
bustion, which imposes a limitation
on the maximum combination of
torque and speed, see curve 4 of the
load diagram, it is normal practice to
match the diesel engine and turbo�
28
8
5
932
1
7
4
90
85
95
75
70
85
105
100
110
6.3
80
90 95 100 105
S3
80 110
SP
100% ref. point (A)
Specified MCR (M)
6
6.2 6.1
S2
S1
S0
PD
Engine speed, % of A
A=M
Engine shaft power % of A
PD: Propeller design point, clean hull and calm weather
Continuous service rating for propulsion with
a power equal to 90% specified MCR, based on:
S0: Clean hull and calm weather, loaded ship
S1: Clean hull and calm weather, ballast (trial)
S2: Clean hull and 15% sea margin, loaded ship
SP: Fouled hull and heavy weather, loaded ship
S3: Very heavy sea and wave resistance
Line 1: Propeller curve through point A=M, layout curve for engine
Line 2: Heavy propeller curve, fouled hull and heavy weather, loaded ship
Line 6: Light propeller curve, clean hull and calm weather,
loaded ship, layout curve for propeller
Line 6.1: Propeller curve, clean hull and calm weather, ballast (trial)
Line 6.2: Propeller curve, clean hull and 15% sea margin, loaded ship
Line 6.3: Propeller curve, very heavy sea and wave resistance
Fig. 24: Influence of different types of ship resistance on the continuous service rating
charger etc. according to a propeller
curve 1 of the load diagram, equal to
the heavy propeller curve 2.
Instead of point S2, therefore, point SP
will normally be used for the engine lay�
out by referring this service propulsion
rating to, for example, 90% of the engine’s
specified MCR, which corresponds to
choosing a 10% engine margin.
In other words, in the example the pro�
peller’s design curve is about 5% light
running compared with the propeller
curve used for layout of the main engine.
Running in very heavy seas with
heavy waves, point S3
When sailing in very heavy sea against,
with heavy waves, the propeller can be
7�8% heavier running (and even more)
than in calm weather, i.e. at the same
propeller power, the rate of revolution
may be 7�8% lower.
For a propeller power equal to 90% of
specified MCR, point S3 in the load
diagram in Fig. 24 shows an example
of such a running condition.
In some cases in practice with strong
wind against, the heavy running has
proved to be even greater and even to
be found to the left of the limitation line
4 of the load diagram.
In such situations, to avoid slamming of
the ship and thus damage to the stem
and racing of the propeller, the ship
speed will normally be reduced by the
navigating officers on watch.
Ship acceleration and operation in
shallow waters
When the ship accelerates and the
propeller is being subjected to a larger
load than during free sailing, the effect
on the propeller may be similar to that
illustrated by means of point S3 in the
load diagram, Fig. 24. In some cases in
practice, the influence of acceleration
on the heavy running has proved to be
even greater. The same conditions are
valid for running in shallow waters.
Sea running at trial conditions, point S1
Normally, the clean hull propeller curve
6 will be referred to as the trial trip pro�
peller curve. However, as the ship is
seldom loaded during sea trials and
more often is sailing in ballast, the ac�
tual propeller curve 6.1 will be more
light running than curve 6.
For a power to the propeller equal to
90% specified MCR, point S1 on the
load diagram, in Fig. 24, indicates an
example of such a running condition. In
order to be able to demonstrate opera�
tion at 100% power, if required, during
sea trial conditions, it may in some
cases be necessary to exceed the pro�
peller speed restriction, line 3, which
during trial conditions may be allowed
to be extended to 107%, i.e. to line 9
of the load diagram.
Influence of ship resistance on
combinator curves – plant with
CP�propeller
This case is rather similar with the FP�
propeller case described above, and
therefore only briefly described here.
The CP�propeller will normally operate
on a given combinator curve, i.e. for a
given propeller speed the propeller
pitch is given (not valid for constant
propeller speed). This means that
heavy running operation on a given
propeller speed will result in a higher
power operation, as shown in the ex�
ample in Fig. 25.
29
S=PD Propeller design point incl. sea margins, and continuous service rating of engine
Line 1 Propeller curve for layout of engine
Line 1 Combinator curve for propeller design, clean hull and 15% sea margin, loaded ship
Line 6.1 Light combinator curve, fouled hull and calm weather, loaded ship
Line 2 Heavy combinator curve, fouled hull and heavy weather, loaded ship
Line 2.1 Very heavy combinator curve, very heavy sea and wave resistance
Fig. 25: Influence of ship resistance on combinator curves for CP�propeller
Engine speed, % of A
Engine shaft power % of A
50
55
60
65
70
75
80
85
90
95
100
105
110
65 70 75 80 85 90 95 100 105 110
8 4 1 6 3
75
6.1
2.1
2
A=M
S=PD
100% ref. point (A)
Specified MCR (M)
Closing Remarks
In practice, the ship’s resistance will
frequently be checked against the results
obtained by testing a model of the ship
in a towing tank. The experimental tank
test measurements are also used for
optimising the propeller and hull design.
When the ship’s necessary power re�
quirement, including margins, and the
propeller’s speed (rate of revolution)
have been determined, the correct
main engine can then be selected, e.g.
with the help of MAN B&W Diesel’s
computer�based engine selection
programme.
In this connection the interaction between
ship and main engine is extremely im�
portant, and the placing of the engine’s
load diagram, i.e. the choice of engine
layout in relation to the engine’s (ship’s)
operational curve, must be made care�
fully in order to achieve the optimum
propulsion plant. In order to avoid over�
loading of the main engine for excessive
running conditions, the installation of an
electronic governor with load control may
be useful.
If a main engine driven shaft generator –
producing electricity for the ship – is in�
stalled, the interaction between ship and
main engine will be even more complex.
However, thanks to the flexibility of the
layout and load diagrams for the MAN
B&W engines, a suitable solution will
nearly always be readily at hand.
References
[1] Technical discussion with
Keld Kofoed Nielsen,
Burmeister & Wain Shipyard,
Copenhagen
[2] Ship Resistance
H.E. Guldhammer and
Sv. Aa. Harvald, 1974
[3] Resistance and Propulsion of Ships,
Sv. Aa. Harvald, 1983
[4] Paint supplier “International
Coatings Ltd.”, 2003
[5] Fartygspropellrar och Fartygs Framdrift,
Jan Tornblad, KaMeWa Publication,
1985
Furthermore, we recommend:
[6] Prediction of Power of Ships
Sv. Aa. Harvald, 1977 and 1986
[7] Propulsion of Single�Screw Ships
Sv. Aa. Harvald & J.M. Hee, 1981
1
1
174 APEˆNDICE D. SELECC¸A˜O DE MOTORES PROPULSORES
Apeˆndice E
Derating para Reduzir Consumo de
Combust´ıvel
175
176 APEˆNDICE E. DERATING
 — 1 — © Wärtsilä Corporation, June 2008
Rudolf Wettstein1 & David Brown2
Wärtsilä Switzerland Ltd, Winterthur
Summary
This paper sets out ways to achieve worthwhile reductions in the fuel consumption of Wärtsilä low-speed engines 
when designing newbuildings. The key approach is to use the flexibility offered by the full power/speed layout field to 
select a better layout point at a deratedpower with a lower BSFC and also possibly a higher propeller efficiency.
Derating: a solution for 
high fuel savings and lower emissions
Introduction
Fuel efficiency and environmental friendliness are 
high on the list of requirements for ship propulsion 
engines from today’s shipping- and shipbuilding 
industries. Thus Wärtsilä is committed to creating 
better technology in these areas that will benefit both 
the customers and the environment.
Yet it is often forgotten by many ship designers 
and those specifying low-speed main engines that 
advantage can be taken of the power/speed layout 
field of Wärtsilä low-speed engines to select an engine 
rating point with a still lower fuel consumption.
The concept of the power/speed layout field for 
low-speed marine diesel engines originated in the 
1970s. The layout options were step-by-step widened 
until, in 1984, our low-speed engines began to be 
offered with a broad power/speed layout field. An 
engine’s contracted maximum continuous rating 
(CMCR) can be selected at any point in the power/
speed field defined by the four corner points: R1, 
R2, R3 and R4 (Fig. 1). The rating point R1 is the 
maximum continuous rating (MCR) of the engine.
Most recently, the layout fields for certain 
engines, the RT-flex82C, RTA82C, RT-flex82T and 
RTA82T, are extended to increased speeds for the 
R1+ and R2+ points (Fig. 2). The extended fields 
offer widened flexibility to select the most efficient 
propeller speed for lowest daily fuel consumption, 
and the most economic propulsion equipment, 
Fig. 1: Typical layout field for RTA and RT-flex engines. The 
contracted maximum continuous rating (CMCR) can be 
selected at any point, such as Rx, within the layout field. The 
∆BSFC is the reduction in full-load BSFC for any rating 
point Rx relative to that at the R1 rating.
[08#044]
Engine power, %R1
Engine speed, %R1
100
90
100
80
70
60
908070
R1
R2
R3
R4
Co
ns
tan
t t
or
qu
e l
ine
0
-1
-2
-3
-4
-5
-6
-7
∆BSFC
g/kWh
Rx
Higher propulsive
efficiency
Lower
specific
fuel
consumption
namely the propeller, shafting, etc.
One basic principle of the engine layout field is 
that the same maximum cylinder pressure (Pmax) 
is employed at all CMCR points within the layout 
field. Thus the reduced brake mean effective pressure 
(BMEP) obtained at the reduced power outputs in 
the field results in an increased ratio of Pmax/BMEP 
and thus lower brake specific fuel consumption 
(BSFC).
The other principle behind the layout field is 
1 Rudolf Wettstein is Director, Marketing & 
Application Development, Ship Power, Wärtsilä 
Switzerland Ltd.
2 David Brown is Manager, Marketing Support, 
Wärtsilä Switzerland Ltd.
 — 2 — © Wärtsilä Corporation, June 2008
that the lower CMCR speeds allow flexibility in 
selection of the optimum propeller with consequent 
benefits in propulsion efficiency and thus lower fuel 
consumption in terms of tonnes per day.
One feature to be borne in mind when selecting 
the rating point for the derated engine is the rating 
Engine power, %R1
Engine speed, %R1
R1+
R2+
R3
R4
100
100
90
80
9080
R1
R2
Engine power, %R1
Engine speed, %R1
100
90
100
80
70
60
908070
R1
R2
R3
R4
Rx2
Rx1
Rating line
slope = α
line (Fig. 3). This is the line through a CMCR rating 
point such that any point on the line represents 
a new power/speed combination that will give 
the same ship speed in knots. The points on the 
rating line all require the same propeller type but 
with different adaptations to suit the power/speed 
combination. In general, lower speeds of rotation 
require larger propeller diameters and thereby 
increase the total propulsive efficiency. Usually the 
selected propeller speed depends on the maximum 
permissible propeller diameter. The maximum 
diameter is often determined by operational 
requirements, such as design draught and ballast 
draught limitations, as well as class recommendations 
concerning propeller–hull clearance (pressure 
impulse induced by the propeller on the hull).
The slope of the rating line (α) depends broadly 
upon the ship type. It can range from 0.15 for 
tankers, bulk carriers and general cargo ships up to 
about 10,000 tdw to 0.22 for container ships larger 
than 3000 TEU and 0.25 for tankers and bulk 
carriers larger than 30,000 tdw.
Changing engine selection strategies
When the broad layout field was introduced in 
RTA engines in 1984 it was widely welcomed by 
shipowners and shipbuilders. Afterwards RTA 
engines were frequently selected at ratings in the 
lower part of the layout field to gain the benefits of 
Fig. 2: For the RT-flex82C, RTA82C, RT-flex82T and 
RTA82T engines the layout fields are extended to the ratings 
R1+ and R2+ at the same powers as R1 and R2 respectively 
but with increased shaft speed.
[08#049]
Fig. 3: For a given ship, a rating line (slope α) can be applied 
to the layout field so that all rating points on that line would 
give the same ship speed with a suitably optimized propeller. 
Rating points at lower speeds on the rating line require 
a larger propeller diameter and give a greater propulsive 
efficiency.
Fig. 4: Since the 1980s engine ratings have been selected over 
a steadily smaller area of the layout field.
[08#051]
Engine power, %R1
Engine speed, %R1
100
90
100
80
70
60
908070
R1
R2
R3
R4
Area of recent
CMCR selection
Area of CMCR
selection in
the 1980s
 — 3 — © Wärtsilä Corporation, June 2008
100
200
300
400
500
2004 2005 2006 2007 2008
Bunker price, US$/tonne
380cSt HFO
Fig. 5: Bunker prices have considerably increased in recent times. The chart shows the average price of 380 cSt heavy fuel oil (HFO) 
from various ports around the world from 2004 to 2008. The green bars indicate the mean price for each year.
[08#045]
lower fuel consumption.
However, under the pressure of first costs and 
softening bunker prices the strategy was changed and 
the selected power/speed combination has, during 
the past 15 years or so, been selected to be closer to 
the R1 rating (Fig. 4).
Yet, more recently, bunker prices have steadily 
climbed, rising by some 85 per cent in the course of 
2007 from US$ 270 to US$ 500 per tonne (Fig. 5). 
The result is that bunkers are now the dominant part 
of ship operating costs.
Such drastic increases in bunker prices give a 
strong impetus to reduce fuel costs. They can also 
justify additional investment cost such as selecting 
an engine with an extra cylinder. The consequent 
fuel saving may make for an acceptable payback time 
on the additional investment cost. It would justify 
any efforts to increase the overall efficiency of the 
complete propulsion system.
Further impetus to implementing such changes 
in engine selection strategy will come from a future 
need to cut CO
2
 emissions. If a carbon trading 
scheme is imposed on shipping it would give further 
economic advantage to reducing fuel consumption 
and further help to pay for any necessary extra 
investment costs.
In addition it is important to bear in mind that 
the fuel savings measures discussed here will also 
result in lower NO
X
 emissions in absolute terms.
Derating engines for greater fuel savings
In the following pages are some case studies for ship 
installations for which an engine is selected with an 
extra cylinder without increasing the engine’s power. 
These cases demonstrate that such engine derating 
can be an advantageous solution with remarkable 
saving potential. Depending on bunker costs, such a 
strategy can have a very attractive pay-back time.
The four case studies are for a Suezmax tanker, 
a Capesize bulk carrier, a Panamax container ship 
and a Post-Panamax container ship. Theyinclude 
estimations of the respective pay-back times for the 
additional engine costs.
 — 4 — © Wärtsilä Corporation, June 2008
In this case, a typical Suezmax tanker might be 
specified with a six-cylinder Wärtsilä RT-flex68-D 
main engine. However, if a seven-cylinder engine is 
employed instead, the daily fuel consumption can be 
reduced by some 3.4 per cent.
In the engine/propeller layout for this ship as 
shown in figure 6, the CMCR points for the two 
alternative engines are on the same rating line 
(α = 0.3) through a common design point for the 
same ship service speed (knots).
The calculation of annual fuel costs given in table 
2 is based on 6000 hours running with heavy fuel oil 
costing US$ 500 per tonne.
The resulting payback time for the extra cost 
associated with the additional engine cylinder is 
estimated to be between 3.5 and six years depending 
on the bunker price of US$ 600–400 per tonne 
respectively (Fig. 7). The calculations of the payback 
are based on an interest rate of eight per cent.
A similar case may be made for a Capesize bulk 
carrier as it would be similar in size and speed to a 
Suezmax tanker and would thus require a similar 
engine.
Table 1: Typical ship parameters for a Suezmax tanker
Length overall: about 274 m
Beam: 46–50 m
Design draught: 16 m
Scantling draught: 17 m
Sea margin: 15 %
Engine service load: 90 %
Table 2: Main engine options
Alternative engines: 6RT-flex68-D 7RT-flex68-D
Cylinder bore, mm: 680 680
Piston stroke, mm: 2720 2720
Stroke/bore ratio: 4:1 4:1
MCR, kW / rpm: 18,780/95 21,910/95
CMCR, kW / rpm: 18,780/95 18,460/89.7
BMEP at CMCR, bar: 20.0 17.9
CSR at 90% CMCR, kW/rpm: 16,902/91.7 16,614/86.6
BSFC at CMCR, g/kWh:
– 100% load: 169.0 164.8
– 90% load: 165.6 162.6
Daily fuel consumption, tonnes/day:
– ISO fuel, LCV 42.7 MJ/kg: 67.2 64.8
– LCV 40.5 MJ/kg: 70.8 68.4
– As percentage, %: 100 96.6 –3.4%
Annual fuel costs, US$: 8,853,000 8,544,000
Fuel saving, US$: 0 – 309,000
Engine length, mm: 8690 9870
Engine mass, tonnes: 472 533
Case 1: Suezmax tanker & Capesize bulk carrier
 — 5 — © Wärtsilä Corporation, June 2008
Engine speed, rpm
8075 85 90 95 100
22,000
20,000
18,000
16,000
Engine power, kW
6RT-flex68-D
7RT-flex68-D
α = 0.3
Constant ship speed
CSR
16,614 kW
86.6 rpm
CSR
16,902 kW
91.7 rpm
CMCR
18,460 kW
89.7 rpm
Design point
CMCR = R1
18,780 kW, 95 rpm
Case 1: Suezmax tanker & Capesize bulk carrier
Fig. 6: Engine/propeller layouts for 
a typical Suezmax tanker with a 
derated seven-cylinder RT-flex68-D 
engine compared with a six-cylinder 
engine at the full MCR power and 
speed.
[08#052]
Fig. 7: Variation of payback times 
from fuel savings according to 
bunker costs for the derated engine 
with an extra cylinder for a typical 
Suezmax tanker.
[08#144]
3.0
2.0
1.0
0
Millions US$
2 4 6 8 10 12 14
Years
Bunker price, HFO:
$600/tonne
$500/tonne
$400/tonne
Investment approx. ($)
 — 6 — © Wärtsilä Corporation, June 2008
In this case, a typical Panamax container ship with 
a container capacity of up to 5000 TEU might be 
specified with an eight-cylinder Wärtsilä RT-flex82C 
main engine. However, if a nine-cylinder engine is 
employed instead, the daily fuel consumption can be 
reduced by some two per cent.
In the engine/propeller layout for this ship as 
shown in figure 8, the CMCR points for the two 
alternative engines are on the same rating line 
(α = 0.2) through a common design point for the 
same ship service speed (knots).
Case 2: Panamax container ship
The calculation of annual fuel costs given in table 
4 is based on 6000 hours running with heavy fuel oil 
costing US$ 500 per tonne.
The resulting payback time for the extra cost 
associated with the additional engine cylinder 
is estimated to be between four and seven years 
depending on the bunker price of US$ 600–400 per 
tonne respectively (Fig. 9). The calculations of the 
payback are based on an interest rate of eight per 
cent.
Table 3: Typical ship parameters for a Panamax 
container ship
Length overall: about 295 m
Beam: 32.2 m
Design draught: 12 m
Scantling draught: 13.5 m
Sea margin: 15 %
Engine service load: 90 %
Table 4: Main engine options
Alternative engines: 8RT-flex82C 9RT-flex82C
Cylinder bore, mm: 820 820
Piston stroke, mm: 2646 2646
Stroke/bore ratio: 3.2:1 3.2:1
MCR, kW / rpm: 36,160/102 40,680/102
CMCR, kW / rpm: 36,160/102 35,480/97.5
BMEP at CMCR, bar: 19.0 17.5
CSR at 90% CMCR, kW / rpm: 32,544/98.5 32,250/94.3
BSFC at CMCR, g/kWh:
– 100% load: 169.0 166.6
– 90% load: 166.5 164.6
Daily fuel consumption, tonnes/day:
– ISO fuel, LCV 42.7 MJ/kg: 130.0 127.4
– LCV 40.5 MJ/kg: 137.1 134.3
– As percentage, %: 100 98 – 2.0%
Annual fuel costs, US$: 17,138,000 16,790,000
Fuel saving, US$: 0 – 348,000
Engine length, mm: 14,055 16,500
Engine mass, tonnes: 1020 1140
 — 7 — © Wärtsilä Corporation, June 2008
Case 2: Panamax container ship
Fig. 8: Engine/propeller layouts for a 
typical Panamax container ship with 
a derated nine-cylinder RT-flex82C 
engine compared with an eight-
cylinder engine at the full MCR 
power and speed.
[08#062]
Fig. 9: Variation of payback times 
from fuel savings according to 
bunker costs for the derated engine 
with an extra cylinder for a typical 
Panamax container ship.
[08#145]
Engine power, kW
8RT-flex82C
9RT-flex82C
α = 0.2
Constant ship speed
42,000
40,000
38,000
36,000
34,000
32,000
85 90 95 100 105
Engine speed, rpm
CMCR
35,850 kW
97.5 rpm
Design point
CMCR = R1+
36,160 kW, 102 rpm
CSR
32,544 kW
98.5 rpm
CSR
32,250 kW
94.3 rpm
3.0
2.0
1.0
0
Millions US$
2 4 6 8 10 12 14
Years
Bunker price, HFO:
$600/tonne
$500/tonne
$400/tonne
Investment approx. ($)
4.0
 — 8 — © Wärtsilä Corporation, June 2008
In this case, a typical Post-Panamax container 
ship with a container capacity of around 7000 
TEU might be specified with an eleven-cylinder 
Wärtsilä RT-flex96C main engine. However, if a 
12-cylinder engine is employed instead, the daily fuel 
consumption can be reduced by some 2.4 per cent.
In the engine/propeller layout for this ship as 
shown in figure 10, the CMCR points for the two 
alternative engines are on the same rating line 
(α = 0.2) through a common design point for the 
same ship service speed (knots).
Case 3: Post-Panamax container ship
The calculation of annual fuel costs given in table 
6 is based on 6000 hours running with heavy fuel oil 
costing US$ 500 per tonne.
The resulting payback time for the extra cost 
associated with the additional engine cylinder is 
estimated to be between two-and-a-half and four 
years depending on the bunker price of US$ 600–
400 per tonne respectively (Fig. 11). The calculations 
of the payback are based on an interest rate of eight 
per cent.
Table 5: Typical ship parameters for a Post-Panamax 
container ship
Length overall: about 325 m
Beam: 42.8 m
Design draught: 13 m
Scantling draught: 14.5 m
Sea margin: 15 %
Engine service load: 90 %
Table 6: Main engine options
Alternative engines: 11RT-flex96C 12RT-flex96C
Cylinder bore, mm: 960 960
Piston stroke, mm: 2500 2500
Stroke/bore ratio: 2.6:1 2.6:1
MCR, kW / rpm: 66,330/102 72,360/102
CMCR, kW / rpm: 66,330/102 65,919/98.9
BMEP at CMCR, bar: 19.6 18.4
CSR at 90% CMCR, kW / rpm: 59,697/98.5 59,327/95.5
BSFC at CMCR, g/kWh:
– 100% load: 171.0 168.0
– 90% load: 166.8 163.8
Daily fuel consumption, tonnes/day:
– ISO fuel, LCV 42.7 MJ/kg: 239 233.2
– LCV 40.5 MJ/kg: 252 245.9
– As percentage, %: 100 97.6 – 2.4%
Annual fuel costs, US$: 31,500,000 30,738,000
Fuel saving, US$: 0 – 762,000
Engine length, mm: 21,550 23,230
Engine mass, tonnes: 1910 2050
 — 9 — © Wärtsilä Corporation, June2008
Case 3: Post-Panamax container ship
Fig. 10: Engine/propeller layouts for 
a typical Post-Panamax container 
ship with a derated 12-cylinder RT-
flex96C engine compared with an 
11-cylinder engine at the full MCR 
power and speed.
[08#127]
Fig. 11: Variation of payback times 
from fuel savings according to 
bunker costs for the derated engine 
with an extra cylinder for the typical 
Post-Panamax container ship.
[08#146]
Engine speed, rpm
90 95 100 105
72,000
70,000
66,000
62,000
Engine power, kW
11RT-flex96C
12RT-flex96C
α = 0.2
Constant ship speed
CSR
59,697 kW
98.5 rpm
CMCR
65,919 kW
98.9 rpm
Design point
CMCR = R1
66,330 kW, 102 rpm
68,000
64,000
60,000
58,000
CSR
59,327 kW
95.5 rpm
8.0
4.0
2.0
0
Millions US$
2 4 6 8 10 12 14
Years
Bunker price, HFO:
$600/tonne
$500/tonne
$400/tonne
Investment approx. ($)
6.0
 — 10 — © Wärtsilä Corporation, June 2008
Case 4: Derating without adding an 
engine cylinder
It is also feasible to apply a derated engine to obtain 
fuel savings in such a way that an additional engine 
cylinder is not required.
An example of this can be seen with the Wärtsilä 
RT-flex50 engine. In October 2007, the D version 
of this engine was announced, in which the engine 
power was increased by 5.1 per cent and the BSFC 
at full-load was reduced by 2 g/kWh compared with 
the B version.
Thus if a ‘-D’ engine is derated to the same 
cylinder power output as the original version of the 
RT-flex50, then the BSFC at full load is reduced 
by 4.5 g/kWh, or 2.7 per cent (see Table 7). For a 
typical bulk carrier with a six-cylinder RT-flex50 
engine this can translate into annual savings of 
US$ 124,000 when operating for 6000 running 
hours a year with heavy fuel oil costing US$ 500 
per tonne. Even greater savings are possible if the 
engine is derated to a lower running speed (rpm) 
at the derated power to gain the benefits of a better 
propulsion efficiency.
There are already a number of standard ship 
designs delivered and on order with RT-flex50-B or 
even the original RT-flex50 engine. So it would be 
perfectly feasible to install a derated RT-flex50-D 
in further newbuildings to the same ship designs 
and obtain the benefit of the substantial savings in 
operating costs. The overall dimensions of the D 
version are identical to those of the B and original 
versions of the RT-flex50. There would, however, be
a modest increase in cost of the D version for the 
higher-efficiency turbochargers used, but the extra 
cost would soon be repaid by the fuel cost savings.
Derating with flexibility to full rating
Although derating offers attractive economics, it 
can be frustrating to buy more ‘engine’ than seems 
necessary. Yet there is an interesting option to retain 
an ability to utilise the full available installed engine 
power, even up to the full R1 rating for future use to 
obtain higher ship service speeds.
The concept would be to set up the engine for 
the derated output at the chosen reduced service 
speed. Then for a later date, the engine could be 
re-adapted to the higher output. However, this needs 
corresponding provisions in the selection and design 
of the propeller, shafting and ancillary equipment to 
meet the requirements of the envisaged higher power. 
Furthermore the engine would need to be tested 
and approved by the Classification Society for both 
ratings with all the necessary emissions certification.
RT-flex technology as an important 
contribution to fuel saving
Wärtsilä RT-flex technology plays an important role 
in fuel saving. Wärtsilä RT-flex low-speed engines 
incorporate the latest electronically-controlled 
common-rail technology for fuel injection and valve 
actuation. The result is great flexibility in engine 
setting, bringing benefits in lower fuel consumption, 
lower minimum running speeds, smokeless operation 
Table 7: Options for the Wärtsilä RT-flex50 engine type
Alternative engines: 6RT-flex50 6RT-flex50-D
Cylinder bore, mm: 500 500
Piston stroke, mm: 2050 2050
S/B ratio: 4.1:1 4.1:1
MCR, kW / rpm: 9720/124 10,470/124
CMCR, kW / rpm: 9720/124 9720/124
BMEP at CMCR, bar: 19.5 19.5
CSR at 90% CMCR, kW / rpm: 8748/119.7 8748/119.7
BSFC at CMCR, g/kWh:
– 100% load: 171 165.7
– 90% load: 167.6 163.0
Daily fuel consumption, tonnes/day:
– ISO fuel, LCV 42.7 MJ/kg: 35.2 34.2
– LCV 40.5 MJ/kg: 37.1 36.2
– As percentage, %: 100 97.3 – 2.7%
Annual fuel costs, US$: 4,637,000 4,513,000
Fuel saving, US$: 0 – 124,000
 — 11 — © Wärtsilä Corporation, June 2008
at all running speeds, and better control of other 
exhaust emissions.
Not only do RT-flex engines have a lower part-
load fuel consumption than RTA engines but they 
can be adapted through Delta Tuning so that their 
part-load fuel consumtion is even lower. [1]
Owing to the interaction between fuel economy 
and NO
X
 emissions, there is always the possibility 
that fuel saving measures will have an impact on 
NO
X
 emissions. As with all new marine engines 
nowadays, Wärtsilä RTA and RT-flex engines are all 
fully compliant with the NO
X
 emission regulation of 
Annexe VI of the MARPOL 1973/78 convention. 
Moreover, the engines in the Wärtsilä portfolio will 
be adapted to meet the coming IMO NO
X
 reduction 
level Tier II.
Conclusion
The paper shows that there are techniques to achieve 
worthwhile reductions in the fuel consumption 
of Wärtsilä low-speed engines when designing 
newbuildings. The key approach is to use the 
flexibility offered by the full power/speed layout field 
to select a better layout point with a lower BSFC and 
also possibly a higher propeller efficiency.
It must also not be forgotten that any fuel savings 
achieved at the ship design stage will have benefits in 
also reducing exhaust emissions.
If you have a project for which you wish to 
explore the fuel-saving possibilities through derating 
as set out in this paper, then please contact your 
nearest Wärtsilä office. Our experts will be delighted 
to calculate various alternatives for your evaluation.
References
1. German Weisser, ‘Fuel saving with RT-flex’, 
Wärtsilä Switzerland Ltd, July 2004.
Published June 2008 by:
Wärtsilä Switzerland Ltd
PO Box 414
CH-8401 Winterthur
Tel: +41 52 262 49 22
Fax: +41 52 262 07 18
www.wartsila.com 
	Introdução
	Geometria do navio
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	Ensaios experimentais
	Selecção do hélice
	Variáveis de optimização
	Tipos de problema
	Interacção entre casco e hélice
	Ensaios de propulsão
	Potência e velocidade
	Extrapolação dos resultados do ensaio de propulsão
	Instalações Propulsoras
	Introdução
	Propulsão diesel-mecânicaAccionamento de auxiliares
	Engrenagens redutoras
	Configuração ''pai-e-filho'' 
	Propulsão diesel-eléctrica
	Propulsão por motor eléctrico
	Propulsores azimutais
	Selecção do motor
	Turbinas e motores eléctricos
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	Índice Remissivo
	Previsão Baseada nos Ensaios de Propulsão
	Provas de velocidade e Potência
	Condições das Provas de Velocidade e Potência
	Selecção de Motores Propulsores
	Derating