Page 107 - Numerical methods for chemical engineering

Example. Steady-state modeling 93

M W is the molecular weight of water, a is the interfacial mass transfer area per unit mass
in the reactor and k m is a mass transfer coefficient. The mole balance on water is
d (in) (in)
{M[W]}= 0 = F [W] − F[W] − (k m a)M[W] + Mr W (2.134)
dt
is the rate of generation of condensate due to reaction, yielding
r W =−r λ 0
2
−1
0 = F (in) [W] (in) − F[W] − (k m a)M[W] + Mk fc λ − Mk fc K eq [W](λ 1 − λ 0 ) (2.135)
0
We next define the dimensionless quantities
(in)
λ [W] F
(in) k λ k
µ = µ k = ω = φ = (2.136)
k (in)
λ 1 λ 1 λ 1 F
the dimensionless Damk¨ohler number
k fc Mλ 1
Da = (2.137)
F (in)
and the dimensionless strength of mass transfer
(k m a)M
γ = (2.138)
F (in)
such that the three dimensionless balances for {µ 0 ,µ 2 ,ω} are
(in) 2 −1
f 1 = µ − φµ 0 − (Da)µ + (Da)K ω(1 − µ 0 ) = 0
0 0 eq
(in) −1 1
f 2 = µ − φµ 2 + 2(Da) + (Da)K ω − µ 3 = 0 (2.139)
2 eq 3
(in) 2 −1
f 3 = ω − φω − γω + (Da)µ − (Da)K ω(1 − µ 0 ) = 0
0 eq
In addition to these three coupled nonlinear equations, we have two auxiliary equations from
the moment closure approximation and the overall mass balance on the reaction medium,
µ 2 (2µ 2 µ 0 − 1)
µ 3 ≈ φ = 1 − γζω (2.140)
µ 0
ζ is the molecular weight of a condensate relative to that of a monomer,

ζ = λ 1 M W (2.141)

Effect of Da and mass transfer upon polymer chain length

polycond CSTR.m plots the number-averaged chain length ¯ x n = µ 1 /µ 0 , the polydisper-
sity, Z = ¯ x w /¯ x n = µ 2 µ 0 , the dimensionless condensate concentration ω, and the rela-
tive outlet flow rate φ as functions of Da and γ , for input values of the fixed parame-
ters {ζ, K eq ,ω (in) ,µ (in) ,µ (in) }. The calculations are performed for a monomer-fed reactor
0 2
2
with ζ = 0.2, K eq = 10 ,ω (in) = 0, and µ (in) = µ (in) = 1.
0 2
For this system, convergence of Newton’s method can be difficult, especially at high Da.
Homotopy is used; for each γ the sequence of simulations is started with the lowest Da,
for which the inlet values are reasonable guesses. Convergence can still be difficult at high
Da, thus we use here an additional convergence trick: we simulate an approximate set of