Page 218 - Numerical methods for chemical engineering

204 4 Initial value problems

Example. Multiple steady states in a nonisothermal CSTR

We demonstrate the use of arclength continuation.m for the study of multiple steady states
in a nonisothermal CSTR. Consider a perfectly-mixed stirred-tank reactor with A reacting
to form B
(4.235)
A → B r = k 1 (T )c A
The temperature dependence of the rate constant is

E a 1 1
k 1 (T ) = k 1 (T ref )exp − − (4.236)
R T T ref
For constant volume CSTR with no volume change due to reaction, the mole balances for
a volumetric feed rate υ are
d
{Vc A }= υ(c A,in − c A ) − Vk 1 (T )c A
dt
(4.237)
d
{Vc B }= υ(c B,in − c B ) + Vk 1 (T )c A
dt
ˆ
Assuming constant density ρ and specific heat capacity C p of the reaction medium, the
enthalpy balance on the reactor is
d
ˆ
ˆ
{V ρ C p T }= υρC p (T in − T ) − V ( H)k 1 (T )c A − UA(T − T c ) (4.238)
dt
H is the heat of reaction (negative for exothermic reactions), UA is the product of the heat
transfer coefficient and area for a coolant jacket, through which flows at high rate a coolant
of temperature T c .
Dividing by V and setting the time derivatives to zero, we have the following three
algebraic equations for the steady-state behavior

dc A υ E a 1 1
= (c A,in − c A ) − k 1 (T ref )exp − − c A = 0
dt V R T T ref
υ 1 1
dc B E a
= (c B,in − c B ) + k 1 (T ref )exp − − c A = 0 (4.239)
dt V R T T ref
dT υ UA
ˆ
= ρC p (T in − T ) − ( H)k 1 (T )c A − (T − T c ) = 0
dt V V
Defining the dimensionless time, concentrations, and temperature
tυ c A c B T
τ = ϕ A = ϕ B = θ = (4.240)
V c A,in c A,in T in
the dimensionless Damk¨ohler number
k 1 (T in )V
Da = (4.241)
υ
and the scaled heat of reaction, cooling efficiency, and activation energy
UA
( H)c A,in E a
β = χ = γ = (4.242)
ˆ ˆ
ρC p T in υρC p RT in