Page 219 - Numerical methods for chemical engineering

Parametric continuation 205

the governing equations (4.239) can be written in dimensionless form

dϕ A γ (θ − 1)
= 1 − ϕ A − (Da)exp ϕ A = 0
dτ θ

dϕ B (in) γ (θ − 1)
= ϕ B − ϕ B + (Da)exp ϕ A = 0 (4.243)
dτ θ
dθ γ (θ − 1)
= 1 − θ − β(Da)exp ϕ A − χ(θ − θ c ) = 0
dτ θ
This is a set of three nonlinear algebraic equations with the six dimensionless parameters
(in) (in)
ϕ , Da, β, χ, γ , θ c .Weset ϕ = 0, and note that ϕ B can be obtained directly from the
B B
values of ϕ A and θ,

γ (θ − 1)
ϕ B = (Da)exp ϕ A (4.244)
θ
Therefore, we can remove it from the list of unknowns and compute it whenever needed;
i.e., we make it an auxiliary variable. Moreover, we note that ϕ B does not appear in either
the equation for ϕ A or that for θ, and thus we do not need to compute its value until the
solution is found. We therefore reduce the system to two equations,

γ (θ − 1)

f 1 (ϕ A ,θ) = 1 − ϕ A − (Da)exp ϕ A = 0
θ
(4.245)

γ (θ − 1)
f 2 (ϕ A ,θ) = 1 − θ − β(Da)exp ϕ A − χ(θ − θ c ) = 0
θ
with the five dimensionless parameters Da, β, χ, γ , θ c .
We now use arc-length continuation to draw curves of the dependence of ϕ A and θ upon
−2
Da for fixed, β, χ, γ and θ c = 1. We start our calculations at Da = 10 , for which good
initial guesses are ϕ A = θ = 1. Defining the parameter vector as
Θ = [log(Da) βχ γ θ c ] T (4.246)

using the fixed values β, χ, γ, θ c = 1, we vary Da from Da 0 = 10 −2 at λ = 0to Da 1 =
2
10 at λ = 1. nonisothermal CSTR Da scan.m performs this calculation using nonisother-
mal CSTR calc f.m.
Wepresentresultswith β =−1(exothermicreaction), χ = 0(noheattransfertocoolant
jacket), and θ c = 1, for various values of γ , the dimensionless activation energy. When
γ = 1, the temperature increases smoothly from a low Da limit of 1 to an upper Da limit
of 2 (Figure 4.17). As we increase the activation energy to γ = 5, the upturn becomes
more pronounced (Figure 4.18). At γ = 8 the curve becomes vertical, and the change in
temperature is very sudden (Figure 4.19).
At higher activation energies, e.g. γ = 12, the system exhibits multiple steady states
(Figure 4.20). As we initially increase Da from a small value, we reach a turning point at
which the steady state becomes unstable. Further increase in Da at this point results in a
sudden jump to a different steady state with a higher temperature (ignition). If we were then