Page 117 - Numerical methods for chemical engineering

Problems 103

(I) (II)
Clearly φ = φ is always a solution; however, for fixed x,as χ increases, there may arise
1 1
additional solutions for which φ (I) = φ (II) . If the free energy of this heterogeneous system,
1 1
heterogeneous (I) (I) (I) (II)
g = g + 1 − g (2.178)
mix mix mix
is less than the free energy g mix for a homogeneous system, phase separation occurs.
(I) (II)
Generate a phase diagram, plotting φ 1 and φ 1 on the x-axis and χ on the y-axis for chain
lengths of x = 1, 10, 100, 1000.
2.C.2. For the system of Problem 2.C.1, use the concept of a bifurcation point to directly
compute χ c , the critical value of χ, above which phase separation occurs, given an input
value of x. Plot χ c vs. x.
2.C.3. In the polycondensation reactor example above, we reduced the number of equations
by deriving moment equations. This required a closure approximation to estimate the value
of λ 3 given λ 0 , λ 1 , λ 2 . Test this approximation by solving the complete set of population
balance equations
0 = F (in) [P m ] (in) − F[P m ] + Mr P m (2.179)

where the net rate of generation of m-mer is the sum of (2.120) and (2.124). Solve this set
of nonlinear algebraic equations for m = 1, 2,..., M max , and increase M max until you no
longer see an effect upon the polydispersity. Note that the required values of M max might be
very large, as even a small number of moles of such very high-molecular-weight polymer
chains contribute greatly to higher-order moments. Generate the plots of Figure 2.18 for
this full set of population balances and note when the closure approximation (2.127) fails
to give adequate results.