Exam #1
ChE 360 Process Control
1. (30 pts.) Consider the following transfer function:
G(s) 
Y ( s)
0.5

U ( s) s  0.1
Using properties of transfer functions and Laplace transforms (show your work), answer the
following questions:
a) What is the steady-state gain and time constant? (5 pts.)
b) If U(s) = 2/s, what is the value of the output y(t) when t → ∞?(5 pts.)
c) For the same U(s), what is the value of the output when t =10? What is the output when
expressed as a fraction of the new steady-state value? (5 pts.)
d) If U(s) = (1 – e-s)/s, that is, the unit rectangular pulse, what is the output when t → ∞?
(5 pts.)
e) If u(t) = δ(t), that is, the unit impulse at t = 0, what is the output when t → ∞? (5 pts.)
f) If u(t) = 2 sin 3t, what is the value of the output when t → ∞ ? (5 pts.)
2. (10 pts.) For each of the following cases, determine what functions of time, e.g., sin 3t,
e-8t, will appear in y(t). (Note that you do not have to find y(t)!). Which y(t) are oscillatory? Why?
Which exhibit a constant value of y(t) for large values of t? Why?
a) Y ( s ) 
2
(5 pts.)
s( s  4s  3)
b) Y ( s ) 
2
(5 pts.)
s( s  4s  8)
2
2
3. (20 pts.) A heater for a semiconductor wafer has first-order dynamics. The transfer function
relating changes in the heater input power level Q to changes in temperature T is
T (s)
K

'
Q ( s)  s  1
where K has units [°C/kW] and τ has units [min].
The process is at steady state when an engineer changes the power input suddenly from 1 to 1.5
kW. She notes the following:



The process temperature initially is 80 °C.
Four minutes after changing the power input, the temperature is 230 °C.
Thirty minutes later the temperature is 280 °C (essentially steady state).
a) What are K and τ in the process transfer function (show calculations)? Explain physically
why the gain K should be positive. (10 pts.)
b) If at another time the engineer changes the power input linearly at a rate of 0.5 kW/min,
what can you say about the maximum rate of change of process temperature: When will it
occur? How large will it be? Show all calculations. (10 pts.)
4. (40 pts.) On the next page, sketch the level response for a tank with constant cross-sectional area
of 4 ft2 as a function of time undergoing the following sequence of events; assume an initial level
of 1.0 ft with the drain open, and that level and outflow rate are linearly related. The steady-state
inflow and outflow are initially equal to 2 ft3/ min. The graph should show key numerical values
of level vs. time. Show supporting calculations using steady-state and dynamic equations below.
Hint: You can do the calculations for each part as if starting from t=0. Then you can renumber
the time axis based on actual clock time.
a) The drain is suddenly closed, and the inflow remains constant for 3 min (0  t  3) .
(10 pts.)
b) The drain is opened for 30 min, keeping the inflow at 2 ft3/ min, where a steady state is
essentially reached (3  t  33) . (10 pts.)
c) The inflow rate is doubled to 4 ft3/min for 30 min (33  t  63) . (10 pts.)
d) The inflow rate is returned to its original value of 2 ft3/ min for 27 min (63  t  90).
(10 pts.)
Draw level response below: