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CHE426: Problem set #9
1. A circulating chilled-water system is used to cool an oil stream from 90 to 70oF in a tubein-shell heat exchanger. The temperature of the chilled water entering the process heat
exchanger is maintained constant at 50oF by pumping the chilled water through a cooler
located upstream of the process heat exchanger.
The design chilled-water for normal conditions is 1000 gpm, with chilled water
leaving the process heat exchanger at 60oF. Chilled-water pressure drop through the process
heat exchanger is 15 psi at 1000 gpm. Chilled-water pressure drop through the refrigerated
cooler is 15 psi at 1000 gpm. The heat exchanger area of the process heat exchanger is 1143
ft2.
The temperature transmitter on the process oil stream leaving the heat exchanger has
a range of 50-150oF. The range of the orifice-differential pressure flow transmitter on the
chilled water is 0-1500 gpm. All instrumentation is electronic (4 to 20 mA). Assume the
chilled-water pump is centrifugal with a flat pump curve (total pressure drop across the
system is constant).
1) Assuming linear trim determine Cv for the chilled-water control valve that is 25
percent open at the 1000 gpm design rate and has a maximum flow of 1500 gpm.
2) Determine values of the signals from the temperature transmitter, temperature
controller, and chilled-water flow transmitter when the chilled-water flow is 1000
gpm.
3) Determine the pressure drop over the chilled-water valve when it is wide open.
4) Determine the pressure drop and fraction open of the chilled-water control valve
when the chilled-water flow rate is reduced to 500 gpm. Determine the chilled-water
flow transmitter output at this rate.
Elevation 20'
Tank at atmospheric
pressure
Elevation 15'
Circulating chilled water
Hot oil
90oF
Refrigerant
FT
o
Elevation 0'
Cooler
50 F
Cooled oil
o
70 F
Heat
exchanger
TT
TC
2. Consider the two-tank, interacting liquid-level system shown below and its block diagram.
The data for the system are: A1 = 1 ft2, A2 = 0.5 ft2, R1 = 0.5 ft/cfm, R2 = 2 ft/cfm, and R3 = 1
ft/cfm.
m2
m1
c1
c2
R1
A1
M1
R2
A2
R3
C1
+
G11(s)
+
G21(s)
G12(s)
M2
+
G22(s)
+
C2
The transfer functions G11, G21, G12, and G22 are given as:
G11 =
s5
( s  1)( s  7)
G12 =
4
( s  1)( s  7)
G21 =
4
( s  1)( s  7)
G22 =
2  s  3
( s  1)( s  7)
Use Simulink to obtain the response of c1 and c2 for (a) M1 = 1/s, M2 = 0, and (b) M1 = 0, M2
= 1/s.
3.1 A tank is heated by steam condensing inside a coil. A PID controller is used to control the
temperature in the tank by manipulating the steam valve position. Derive the complete block
diagram and the closed-loop transfer function from the following design data.
Process. The feed has a density  of 68.0 lb/ft3 and a heat capacity cp of 0.80 Btu/lboF. The
volume V of liquid in the reactor is maintained constant at 120 ft3. The coil consists of 205 ft
of 4-in. schedule 40 steel pipe that weighs 10.8 lb/ft and has a heat capacity of 0.12 Btu/lboF
and an outside diameter of 4.5 in. The overall heat transfer coefficient U, based on the
outside are of the coil, has been estimated as 2.1 Btu/minft2oF. The saturated steam is
available at 30 psia; it can be assumed that its latent heat of condensation  is constant at 966
Btu/lb. It can also be assumed that the inlet temperature Ti is constant.
Design Conditions. The feed flow F at design condition is 15 ft3/min, and its temperature Ti
is 100oF. The contents of the tank must be maintained at a temperature T of 150oF. Possible
disturbances are changes in feed rate and temperature.
Temperature Sensor and Transmitter. The temperature sensor has a calibrated range of
100 to 200oF and a time constant T of 0.75 min.
Control Valve. The control valve is to be designed for 100% overcapacity, and pressure drop
variation can be neglected. The valve is an equal percentage valve with a rangeability
parameter  of 50, f(x) = x-1 = 50x-1. The actuator has a time constant v of 0.20 min.
a) Determine the parameters listed in the following table
Parameters for stirred tank heater.
A
241.5 ft2
4.93 min

CM
265.7 Btu/oF
0.524
min
c
o
o
3
KF
Kw
1.905 F/(lb/min)
 2.06 F/(ft /min)
o o
Ks
0.383 F/ F
KT
1.0 %TO/ oF
Kv
1.652 (lb/min)/ %CO
0.20 min
v
0.75
min
T
b) Use simulink to simulate the temperature response to a step change in both set point
and flow rate with a PID controller for two reset rates of 0.5 min-1 and 1.0 min-1, Kc =
2, and D = 1 min.
4. A liquid stream enters tank 1 at a volumetric flow rate F in cfm and contains reactant A at
a concentration of C0 [mol A/ft3]. Reactant A decomposes in the tanks according to the
irreversible chemical reaction
AB
The reaction is first order with reaction rate constant k1 and k2 for tank 1 and tank 2
respectively. The reaction is to be carried out in a series of two continuous stirred-tank
reactors. The tanks are maintained at different temperatures with tank 2 at a higher
temperature. We will neglect any changes in physical properties due to chemical reaction.
The purpose of the control system is to maintain C2, the concentration of A leaving tank 2 at
some desired value in spite of variations in the inlet concentration C0. This will be
accomplished by adding a stream of pure A to tank 1 through a control valve using a
proportional controller. A portion of the liquid leaving tank 2 is continuously withdrawn
through a sample line at a rate of 0.1 cfm. The sample line has a length of 50 ft and the crosssectional area of the line is 0.001 ft2.
Controller
Pure A
F, C0
F+m/A
V, T1
C1, k1
V, T2
C2, k2
(Set point )
Composition
measuring
element
Sample stream
Product stream
Heating coil
Data: MwA = 100, A = 0.8 lbmol/ft3, C0s = 0.1 lbmol/ft3, F = 100 cfm, ms = 1.0 lbmol/min, k1
= 1/6 min-1, k2 = 2/3 min-1, V = 300 ft3,  = V/F = 3 min.
1
mA
Kv = cfm/psi, Km = 400
, KT = 0.75 psi/mA, d = 0.5 min
6
lbmol/ft 3
a) Use simulink to obtain the response to a unit step change in both the set point and the
inlet concentration C0 for Kc = 2.
b) Find the value of Kc at which the system will have a sustained oscillation.
5.2 For the control system shown, determine the ISE, ITAE, and IAE for a unit-step load
disturbance. Let the controller be a PI controller with Kc = 6, I = 4.
load

+
Set point
-
+
Controller
Gc
+
1/(10s2 + 11s + 1)
Controlled
variable
Error
Measured variable
exp(-s)
Solution -----------------------------------------------------------------------------------------
6.2 Determine the Z-N and the C-C settings for the PI control system shown below.
7.1 The formula for the flow of saturated steam through a control valve is
msteam (lb/h) = 2.1Cvf(x)
P12  P22
In this equation, P1 is upstream pressure (psia) and P2 is downstream pressure (psia). The
temperature of the steam-cooled reactor shown below is 285oF. The heat that must be
transferred from the reactor into the steam generation system is 25×106 Btu/h. The overall
heat transfer coefficient for the cooling coils is 300 Btu/hft2oF. The steam discharges into a
25-psia steam header. The enthalpy difference between saturated steam and liquid condensate
is 1000 Btu/lb. The vapor pressure of water can be approximated by
T(oF) = 195 + 1.8P (psia)
To steam
header
Reactor
Steam
drum
Cooling
coils
Feed
Condensate
Product
For questions (1) and (2) the drum pressure is 40 psia.
a) Calculate the area of the cooling coils.
b) Calculate the Cv for the steam valve, assuming that the valve is half open at the
design conditions: f(x) = 0.5.
c) If Cv = 700 lb/hpsia, and the temperature of the drum at the maximum heat removal
(Qmax) of the system is 257.1oF, determine Qmax in Btu/h
8. Consider the following control system with U(s) = 5/s, Kc = 4 and D = 2.
Determine C(t = 1.25) and the offset.
9. Consider the following control system
R(s) +
Kc
-
1/(8s + 1)2
+
1/(1 + s)
Calculate the ultimate gain.
C(s)
IV. Consider a tank in which a solution (w1, x1) containing 10 weight % salt is blended with a
concentrated solution (w2, x2) containing 42 weight % salt, to produce 100 lb/h of a solution
(w, x) containing 20 weight % salt at design conditions.
SP
m1
w1, x1
V
SP
AC
FC
AT
FT
x
w
m2
w2, x2
a) Determine the open loop steady state gain on the mass percent of salt in the product stream
for an increase of 2.0 lb/h in w2
b) Determine the closed loop steady state gain on the mass percent of salt in the product
stream for an increase of 2.0 lb/h in w2 (and w1 is reduced by 2.0 lb/h).
References
1. Process Modeling, Simulation, and Control for Chemical Engineers by Luyben
2. D.R. Coughanowr and S. LeBlanc, Process Systems Analysis and Control, McGrawHill, 3nd edition, 2008.