## Description

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?

Steady-state conditions

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