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Problem 6.66PP
Consider the heat exchanger of Example 2.16 with the open-loop transfer function
G ( i ) = -
-5 i
(10s + 1 )(605+ 1)‘
(a) Design a lead compensator that yields PM > A5“ and the maximum possible closed-loop
(b) Design a PI compensator that yields PM > 45° and the maximum possible closed-loop 
bandwidth.
Example 2.16
EXAMPLE 2 .1 6
EXAMPLE 2 .1 6
Equations fo r M odeling a Heat Exchanger 
A heat excfanger b shown in Rg. 2 J7 . Steam enters the chamber through 
Equations fo r M odeling a Heat Exchanger
Rgura 2J7 
Heat exchanger
A beat exchanger b shown in Hg. 2J37. Steam eaters the chamber tbiDugh 
the controUable valve at the top, and cooler steam leaves at the bottom. There 
b a coostant Bow of water th ro n g the pipe that winds U uou^ the middle 
of the chamber so that k picks up beat from the «team- Find the diffemnial 
equatioas that describe the dynamics o f the measured water outflow temper­
ature as a functioo o f the area Aj o f the steam-inlet cootrol valve when open. 
The sensor that measures the water outflow temperature, being downstream 
from the exit temperature in the pipe, lags the temperature by ts sec. 
Solntion. The temperature of the water in the pipe will v a y contuniously 
along the p ^ as the heat flows from the steam to the water. The temperature 
the steam vrill also reduce in the chamber as it passes over the mare of 
pipes. An accurate tbennal model of tbb process b therefore quite involved 
because the actual heal transfer from the steam to the water will be propor­
tional to the local temperatures o f each fluid. For many cmkrol appUcatkms 
it b not necessary to have great accuracy because the feedback will correct 
for a considerable amount of error in the modeL Therefore, it makes sense 
to combine the spatially varying temperatures into single temperatures Tg 
X.
and Tw for the outflow steam and water temperatures, respectively. We flwn 
MSMine that the heat transfer from steam to water b proportional to the dif­
ference in these temperatures, as given by Eq. (2.81). There b also a flow of 
beat into the chamber from the inlet steam that depends on the steam Bow 
rate and its temperature according to Eq. (2.84X
qu = wjCw(7W — Tt),
Wg = KgAg, m an flow rtee of the w—m,
A( = area of the steam inlet valve.
Kg = Bow coefhcieat o f the inlet valve, 
cw = specific beat the steam,
Tgg = temperature of the inflow steam,
Tg = temperature of the outflow steam.
The net beat flow into the chamber b the difference between the beat from 
the hot incoming steam and the heat flawing out to the water. This net 
flow determines the rate of temperature change of the steam according to
Eq.(2.82X
Cgtg=AgKgCgg(rgi-Tg)-^(Tg-T,gy, (2.85)
Cg = iHgCgg is the thermal capacity o f the steam in the chamber
R = the thermal resistance o f the heat flow averaged over the 
eikiie exchangee
Utew ise, the difierential equatioo describing the water temperature is
1 (2.86)
Wv = mass flow rate o f the water,
Cew = specific heal of the water,
7 ^ = temperteure erf' the inenming wale^
Tw = temperature of the outflowing water:
l b complete the dynamics, the time delay between the measurement and the 
exit flow is described by the relation
vdiere 7^ b the measured downstream temperature of the water and b 
the time delqr. There may also be a rlehy in the measurement of the steam 
temperature Tg, vdiich would be modeled in the same manner.
Equation (2.85) b nonlinear because the quantity Tg b multiplied 1^ the 
control input At. The equation can be about Tgo (a specific value