# CH3050 Process Dynamics and Control Assignment 2 solution

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1. An exothermic reaction A −→ 2B, takes place adiabatically in a stirred-tank reactor. This liquid reaction
occurs at constant volume in a 1200-gallon reactor. The reaction is first order, irreversible with the rate
constant given by k = 2.4 × 1015e
−20000/T (min−1
) where T is in ◦R.

(a) Develop a first-principles model for dynamics of cA and reactor (exit) temperature T. State all
assumptions that you make.

(b) Set up the SIMULINK model using DEE. Determine the steady-state exit temperature using findop
in MATLAB.

(c) Derive a transfer function relating T and cA to the inlet concentration cAi using MATLAB (linearise,
ss, ss2tf). Verify your result with hand calculation.

(d) Compare the step response (to a 10% step in cAi) of the non-linear and linearized systems. What is
the extent of error in steady-state values?

(e) Which of the output variables is affected more to a unit step change in cAi?
cAi,ss = 0.8 mol/ft3
and Fss = 20 gallons/min
Physical property data for the mixture
Ti = 90◦F, C = 0.8 Btu/(lb ◦F), ρ = 52 lb / ft3
and 4HR = −500 kJ/mol

2. This is a MATLAB Grader problem. Visit the URL at Matlab Grader.
3. (a) For a system described by the TF G(s) = (s + 1)/(s

3 + 10s + 31s + 30), write an equivalent SS
description using two different methods (i) partial fraction expansion method (call this SS1) and
(ii) nested integral method (call this SS2). Compare SS1 and SS2 descriptions. Can you find a
transformation matrix that takes SS2 to SS1? Explain.

(b) Suppose, for a single-input two-output (SITO) system, y1(t) = G11u1(t) and y2(t) = G21u1(t), where
G11(s) = 4s + 1
(s + 1)(s + 3) and G21(s) = 10s
(s + 2)(s + 3). Arrive at a minimal order SS realization for
the SITO system.

4. For the signal flow graph in Figure 1, (i) draw the block diagram relating R(s) to Y (s) and (ii) find the
transfer function Y (s)/R(s).
3s
s +3 4
s2 +1
−1
s
1
s +2
1
s +1
1
s2
-3
3
1 1
R(s) Y(s)
-6
s
Figure 1: Signal flow graph for Q.4