Page 96 - Numerical Methods for Chemical Engineering

Example. 1-D laminar flow 85

We could also use @calc f ex1 as the first fsolve argument rather than ‘calc f ex1’. For
the CSTR example, with the two chemical reactions

A + B → C r R1 = k 1 c A c B
(2.89)
C + B → D r R2 = k 2 c C c B
the steady-state concentrations

x 1 = c A x 2 = c B x 3 = c C x 4 = c D (2.90)
are obtained by solving the nonlinear 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
υ(c C, in − x 3 ) + V (k 1 x 1 x 2 − k 2 x 3 x 2 ) = 0 (2.91)
υ(c D, in − x 4 ) + V (k 2 x 3 x 2 ) = 0

CSTR SS ex1.m prompts the user for the values of the parameters and computes the con-
centrations, using the inlet concentrations as the initial guesses. For the parameter values

υ = 1 V = 100 k 1 = 1 k 2 = 1
c A, in = 1 c B, in = 2 c C, in = 0 c D, in = 0 (2.92)
running this program yields the output

Steady state concentrations:
[A] = 0.056614
[B] = 0.16664
[C] = 0.053409
[D] = 0.88998
infinity norm of f = 2.0326e-009

Example. 1-D laminar flow of a shear-thinning polymer melt

Again, let us consider laminar flow between two parallel plates separated by a distance B,
but now for simplicity we assume both plates to be stationary. Previously in Chapter 1, we
employed finite differences to compute the velocity profile v x (y) for a Newtonian fluid. We
now consider the same problem, but for a nonNewtonian fluid whose viscosity decreases
with increasing shear-rate, common behavior for many polymer solutions and melts. In this
problem, the (y-dependent) shear-rate is

dv x
˙ γ (y) = (2.93)
dy
y
A common model of shear-thinning behavior is that of Carreau and Yasuda (Yasuda et al.,
1981), for which the shear-rate dependent viscosity η(˙γ )is
η(˙γ ) − η ∞ a (n−1)/a
= [1 + (λ ˙γ ) ] (2.94)
η 0 − η ∞