Page 82 - Numerical Methods for Chemical Engineering

Systems of multiple nonlinear algebraic equations 71

tet
V reactr c
ve j

inet
A B C
c j, in
C B D
Figure 2.6 CSTR with two chemical reactions.

practice, we may wish to identify complex solutions to complex-valued functions. To do
so, we write x as

x = a + ib a = Re{x}∈ b = Im{x}∈ (2.40)

and compute a and b by solving the two coupled real-valued equations

Re{ f (a + ib)}= 0 Im{ f (a + ib)}= 0 (2.41)
Thus, techniques for treating multiple nonlinear algebraic equations are required in this
instance.

Systems of multiple nonlinear algebraic equations

We now extend Newton’s method to solve a set of N simultaneous nonlinear algebraic
equations for N unknowns

f 1 (x 1 , x 2 ,..., x N ) = 0
f 2 (x 1 , x 2 ,..., x N ) = 0
. (2.42)
. .
f N (x 1 , x 2 ,..., x N ) = 0
More compactly, we define the state vector of unknowns

x = [x 1 x 2 ... x N ] T (2.43)

and write the system of equations as
f (x) = 0 (2.44)

As an example, consider a continuous stirred-tank reactor (CSTR), operated isothermally,
with negligible volume change due to reaction, in overflow mode with a constant fluid
volume V, and with the two chemical reactions (assumed elementary) (Figure 2.6)

A + B → C r R1 = k 1 c A c B
C + B → D r R2 = k 2 c C c B (2.45)