Page 256 - Numerical Methods for Chemical Engineering

Optimal control 245

and the temperature is constrained to lie within the interval
298 K ≤ T ≤ 360 K (5.138)

To compute the steady-state concentrations, a dynamic model of the CSTR in overflow
mode is integrated until steady state is reached,
dc A υ(c A0 − c A ) dc B υ(c B0 − c B )
= − r R1 − r R2 = − r R1
dt V R dt V R
(5.139)
dc C υc C dc S 1 υc S 1 dc S 2 υc S 2
=− + r R1 − r R3 =− + r R2 =− + r R3
dt V R dt V R dt V R
Initially, the concentrations equal the inlet values. Time integration ensures that we
design only for stable steady states. The constrained optimization is performed by opti-
mal design CSTR.m. Starting at the initial guess
υ (guess) = 1 c (guess) = 0.5 c (guess) = 0.5 T (guess) = 310 (5.140)
A0 B0
maximizing the production of C by minimizing (5.133) yields
initial F 1 =−0.182 final F 1 =−27.148
υ = 360 c A0 = 1 c B0 = 1T = 360
(5.141)
c A = 0.913 c B = 0.924 c C = 0.075
= 0.011 = 9.8 × 10 −4
c S 1 c S 2
Thus, the optimal design is found at the limits of the constraints. A high flow rate yields
a low residence time and thus a small concentration of C in the outlet stream (note that
we maximize the product of this concentration with the volumetric flow rate). The A and
B inlet concentrations are maximized to yield the fastest rate of the first reaction, and the
temperature is at the upper limit. The loss of C by the third reaction is slight, as the short
residence time and low C concentration make the rate of this reaction small compared to
the first one.
Next, we maximize the economic benefit, by minimizing (5.134), starting at the design
(5.141), to yield the optimal design

initial F 2 =−46.37 final F 2 =−48.36
υ = 360 c A0 = 0.832 c B0 = 1.168 T = 360
(5.142)
c A = 0.749 c B = 1.094 c C = 0.073
= 0.009 = 9.5 × 10 −4
c S 1 c S 2
Here, at the cost of producing slightly less C, we have increased the economic benefit by
reducing the side product formation and the consumption of A by enriching the feed stream
with B.

Optimal control

N
Let us consider a dynamic system, described by the state vector x(t) ∈ , and governed
by the ODE system
˙ x = f (t, x(t), u(t); θ) (5.143)