Please show your work either in writting or MATLAB code.

Consider a constant volume continuous stirred tank reactor in which the following isomerization reaction occurs A → B. Let CA and T denote the reactant concentration and
temperature, respectively, in the reactor. The reaction rate per unit volume is:

r(CA, T) = k0 exp(−E/RT)CA

The dynamic model equations are:

dCA
dt =
q
V
(CAf − CA) − r(CA, T)

dT
dt =
q
V
(Tf − T) + (−∆H)
ρCp
r(CA, T) −
UA
V ρCp
(T − Tj )

where V is the reactor volume, q is the volumetric feed flow rate, CAf and Tf are the
concentration and temperature, respectively, of the feed stream and Tj
is the cooling jacket
temperature. The reaction and heat transfer parameters (k0, E, ∆H, U, A) and the
physical properties (ρ, Cp) are assumed to be constant. Consider the following parameter
values: k0 = 3.8×107 h
−1
, (−∆H) = 3700 kcal/kmol, E = 13250 kcal/kmol, ρCp = 280
kcal/m3/K, UA = 90 kcal/h/K, R = 1.975 kcal/kmol/K, V = 1.8 m3
, q = 0.7 m3/h, CAf
= 20 kmol/m3
, Tf = 298 K and Tj = 298 K.

1. First, find all the steady states for the reactor at the specified operating conditions assuming CAf and Tj are the two manipulated inputs and CA and T are the two controlled outputs. Use the lowest conversion steady state for the model to find the linear state-space model at this operating point.

2. Prove that the steady-state gain matrix can be directly calculated from the state-space model as follows: K = −CA−1B. Use the relative gain array (RGA) to determine the preferred control loop pairing. Use the Niederlinski index to determine if your proposed pairing will yield a stable closed-loop system. At the end, clearly mention the input-output pairing in terms of the physical variables chosen for the control.

3. Use the linear state-space model to find the transfer function Gij (s) relating the j-th
input and the i-th output. Show that each transfer function has the general form:

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