Page 86 - Numerical methods for chemical engineering

72 2 Nonlinear algebraic systems

In a CSTR, we assume that the reactor is so perfectly mixed that the concentration field of
each species is spatially uniform. That is, every point in the reactor has the same concen-
tration of each species, governed by the set of mass balances
d
(Vc A ) = υ(c A, in − c A ) + V (−k 1 c A c B )
dt
d
(Vc B ) = υ(c B, in − c B ) + V (−k 1 c A c B − k 2 c C c B )
dt
(2.46)
d
(Vc C ) = υ(c C, in − c C ) + V (k 1 c A c B − k 2 c C c B )
dt
d
(Vc D ) = υ(c D, in − c D ) + V (k 2 c C c B )
dt
υ is the volumetric flow rate of the feed and outlet streams, V is the fixed reactor volume,
c j is the concentration of species j in the reactor (and in the output stream), c j,in is the
concentration of species j in the inlet stream, and k 1 , k 2 are the rate constants of each
chemical reaction.
At steady state, the time derivatives on the left are zero, and the concentrations of each
species within the reactor satisfy a set of four nonlinear algebraic equations. To put these
equations in standard form, we define the unknowns

x 1 = c A x 2 = c B x 3 = c C x 4 = c D (2.47)

to obtain the algebraic system
υ(c A, in − x 1 ) + V (−k 1 x 1 x 2 ) = 0
υ(c B, in − x 2 ) + V (−k 1 x 1 x 2 − k 2 x 3 x 2 ) = 0 (2.48)
υ(c C, in − x 3 ) + V (k 1 x 1 x 2 − k 2 x 3 x 2 ) = 0
υ(c D, in − x 4 ) + V (k 2 x 3 x 2 ) = 0

We see that in addition to the four unknowns, there are a number of other model parameters
whose values we must know before attempting to solve the equations. We collect these
quantities into a parameter vector Θ:
Θ = [υ Vk 1 k 2 c A, in c B, in c C, in c D, in ] T (2.49)

and write the set of nonlinear algebraic equations as

f (x; Θ) = 0 (2.50)

Newton’s method for multiple nonlinear equations

Again we use a Taylor series expansion to obtain Newton’s method, representing the ith
function in the vicinity of the current estimate x [k] as

N

[k] ∂ f i [k]
f i (x) = f i x + x m − x m (2.51)
∂x m
m=1 x [k]

N
N
2
1 [k] ∂ f i [k]
+ x m − x m x n − x n +· · ·
2 ∂x m ∂x n
m=1 n=1 x [k]