Page 106 - Numerical methods for chemical engineering

92 2 Nonlinear algebraic systems

The first moment is unchanged by the reaction, as is expected since the total number of
monomer units is conserved. The equation for the second moment requires that we know
the value of the third moment. To obtain a closed set of equations, it is common to postulate
a mathematical form of the chain length distribution (see Chapter 7) to relate the unknown
λ 3 to the known λ 0 , λ 1 , λ 2 . This approach yields the closure approximation

2
λ 2 2λ 2 λ 0 − λ 1
λ 3 ≈ (2.127)
λ 1 λ 0

Steady-state model of a stirred-tank polycondensation reactor
We now use these results to model the steady-state behavior of a polycondensation CSTR.
The mole balance on m-mer is
d (in) (in)
{M[P m ]}= F [P m ] − F[P m ] + Mr P m (2.128)
dt
Here, we represent concentrations on a per-mass basis, due to volume changes during
reaction. M is the total mass of the reaction medium in the reactor. F (in) and F are the inlet
and outlet mass flow rates.
The first term on the right-hand side of (2.128) is the flux of m-mer into the reactor from
the inlet stream, the second term is the flux of m-mer out of the reactor, and the last term
is the rate of change of m-mer concentration due to chemical reaction. Multiplying this
k
equation by m and summing over all m yields
(
∞
∞
∞
∞
d k (in) k (in) k k
M m [P m ] = F m [P m ] − F m [P m ] + M m r P m (2.129)
dt
m=1 m=1 m=1 m=1
This is simply the balance for the kth moment, and at steady state yields
d (in) (in)
{Mλ k } = 0 = F λ k − Fλ k + Mr λ k (2.130)
dt
With λ 1 held constant (as it is unchanged by the reaction), we have
d (in) (in) 2 −1
{Mλ 0 } = 0 = F λ 0 − Fλ 0 − Mk fc λ + Mk fc K eq [W](λ 1 − λ 0 )
0
dt
d (in) (in) 2 −1 1
{Mλ 2 } = 0 = F λ 2 − Fλ 2 + 2Mk fc λ + Mk fc K eq [W] λ 1 − λ 3 (2.131)
3
1
dt
and for λ 3 use the auxiliary moment closure equation
2
λ 2 2λ 2 λ 0 − λ 1
λ 3 ≈ (2.132)
λ 1 λ 0
We contact the reaction medium with a purge gas stream to remove the water, such that the
total mass balance on the reaction medium is

F = F (in) − (k m a)M M W [W] (2.133)