## CHEMENG3031 PROCESS CONTROL AND INSTRUMENTATION Solutions to Questions

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Complete the following questions and submit your solutions online via My Uni. Each submission must be
attached with an assignment cover sheet completed with names and signatures of all contributing team
members.

1. Three second-order processes are described by each of the transfer functions Gi(s) presented
below: For each of the processes:

a) Derive the transfer function for the process output (y), in response to a unit-step change in
process input (x)

b) Determine the time-domain response for the process output [y(t)] in response to the unit-step
change in input [x(t)]
c) Repeat steps (a) and (b) for a ramped process input (x), which increases at the rate of unity per
unit of time.
c) Plot each system response for the ramped input from part (c) above.

2. The dynamic behavior of a physical process may be represented by the transfer function: a) Determine the dynamic response to a step-change in the process input of x(t) = 3, and further
quantify the new steady-state system output.
b) For physical reasons, the process output must remain below 12. Determine the magnitude of
the largest step-change in input which may be tolerated without exceeding the maximum process
output, and further provide a time after the step-change for which the maximum output occurs.

3. As a process engineer you are assigned the responsibility of overseeing the operation of an
exothermic CSTR reactor. In order to better understand the dynamics of the process, you decide to
make a step change in the temperature of the reactor coolant temperature from 10 to 15°C.
Prior to this change, the reactor was initially operating at steady-state. You obtain the following
plot for the temperature of the reactor output following the change in the coolant temperature. a) Determine the process gain
b) Determine the natural period of oscillation (t)
c) Determine the damping coefficient (z)
d) Determine the decay ratio
e) Write the 2nd order transfer function for the process

4. A step change of magnitude is made to the following process: Determine:
a) percent overshoot
b) rise time
c) maximum value of y(t) and when this value occurs
d) ultimate value of y(t)
e) period of oscillation (t)
f) damping coefficient (z)

5. A thermocouple exhibiting first-order dynamic behavior with a time constant of 0.1 min and a
process gain of unity is mounted in a thermowell of an industrial fermenter which itself displays
first order dynamic behavior with a time constant of 2 minutes and process gain of unity.
During start up, the fermentation broth is linearly heated from 15°C to 38°C at a rate of 0.5°C/min.

a) Derive a dynamic equation for the measured temperature of the fermentation broth during the
start-up period and plot the response.

b) Derive a dynamic equation for the measured temperature of the fermentation broth during the
start-up period if the system also includes 5 minutes of deadtime and plot the response.

c) During normal steady-state operation the fermenter temperature is set to 38°C.
If the fermenter experiences an abrupt disturbance whereby the broth temperature suddenly
drops to 30°C, plot the dynamic response of the measured temperature and determine how much
time will lapse until the low-level temperature alarm sounds if it is set to 32°C.

d) Determine the limiting value of the time-constant for the thermowell to ensure the measured
temperature response of the system remains damped (ie. non-oscillatory