_______________________
Last Name, First
CHE426: Problem set #6 1
1. A step change of magnitude 4 is introduced into a system having the transfer function
= s
2
10

1.6
s

4
Determine (a) Percent overshoot, (b) Rise time, (c) Maximum value of Y ( t ), (d)
Ultimate value of Y ( t ), and (e) Period of oscillation. Use the following formulas:
Step response for

< 1.
Y ( t )/10 = 1

1
1
 
2
exp
 
 t sin





1
 
2
 t
 tan

1


1
 

2




Overshoot = exp

 

1
 
2


1



2
1 f =
T
=
1
2

2. The two-tank system shown in Figure
E-2 is operating at steady state. At time t
= 0, 10 ft
3
of water is quickly added to the first tank. Determine the maximum deviation in level (feet) in both tanks from the ultimate steady-state value and the time at which each maximum occurs.
Data: A
1
= A
2
= 10 ft
2
, R
1
= 0.1 ft/cfm, R
2
= 0.35 ft/cfm.
Note: Q
1
= h
1
/ R
1
, if y ( t ) =

(0) (unit impulse) then Y ( s ) = 1.
3
20 ft /min h
1
A
1
10 ft
R
3
1
Q
1 h
2
Figure E-2
3.
1
Determine y ( t = 0), y ( t = 0.6), and y ( t = ∞) if Y ( s ) =
1 s
25( s

1) s
2 
2 s

25
A
2
4.
1
Sketch the response y ( t ) if Y ( s ) = exp(

2 s )/[ s
2
+ 1.2
s + 1]. Determine y ( t ) for t = 0, 1, 5,
∞.
R
2
Q
2

5
2
The two tanks shown in Fig. E-6 are connected in an interacting fashion. The system is initially at steady state with q = 10 cfm. The following data apply to the tanks: A
1
= 1 ft
2
, A
2
= 1.25 ft
2
, R
1
= 1 ft/cfm, and R
2
= 0.8 ft/cfm.
(a) If the flow changes from 10 to 11 cfm according to a step change, determine
(b)
H
2
( s ),i.e., the Laplace transform of H
2
where is the deviation in h
2
.
Determine H
2
(1), H
2
(4), and H
2
(∞).
Q h
1 h
2
R
1
R
2
Q
2
A
1
A
2
Figure E-6
6.
1
The overhead vapor from a depropanizer distillation column is totally condensed in a water-cooled condenser at 120 o
F and 230 psig. The vapor is 98 mol % propane and 2 mol % isobutene. The vapor design flow rate is 40,000 lb/h and average latent heat of vaporization is 128 Btu/lb. Cooling water inlet and outlet temperatures are 75 and 100 o
F, respectively. The condenser heat transfer area is 1000 ft 2 . The cooling water pressure drop through the condenser at design rate is 50 psi. A linear-trim control valve (air-to-closed, when CO = 20 mA, PV = 15 psig) is installed in the cooling water line. The pressure drop over the valve is
25 psi at design with the valve half open. The process pressure is measured by an electronic (4-20 mA) pressure transmitter whose range is 150-400 psig. An analog electronic proportional controller with a gain of 2 is used to control process pressure by manipulating cooling water flow. The electronic signal from the controller (CO) is converted into a pneumatic signal in the I/P transducer.
Vapor
Control valve
Cooling water
Condenser
PV
I/P
Reflux drum
PT
PM
PC
CO
SP a)
Calculate the cooling water flow rate (gpm) at design conditions. Water density is 62.3 lb/ft 3 and 1 ft 3 = 7.48 gal.
b) If the cooling water flow rate is 250 gpm at design conditions, calculate the size coefficient ( C v
) of the control valve.

c) Calculate the value of the signal PM at design condition.
d) Calculate the value of the signal PV at design conditions e) Suppose the process pressure jumps 20 psi, determine value for CO.
7.
Express the function given the graph in the t -domain
_________
8. A thermometer having first-order dynamics with a time constant of 1 min is at 100 o
F. The thermometer is suddenly placed in a bath at 110 o
F at t = 0 and left there for 0.167 min, after which it is immediately returned to a bath at 100 o
F. Calculate the thermometer reading at t =
0.5 min.