Page 63 - The engineering of chemical reactions

Variable Density 47

dca d x d x d x
- + x - x2
kf CA - 4kb(CAo - cAj2 = ( x -x1)(x - x2) = x-x1
where xi and x2 are the roots of the polynomial k&A - kb(CA,, - CJ2. This giVeS
CA
t = dC.4 -ji,:,+T&
s kc.4 - 4kb(CAo - cAj2
CAo CAo CA”
= In CA --xl - in CA -x2
CAo - xl CAo - x2

It is obvious from this example why we quickly lose interest in solving mass-balance
equations when the kinetics become high order and reversible. However, for any single
reaction the mass-balance equation is always separable and soluble as

CA
dC.4
t=- -
s r(CA>
CAo

Since CA = CA,(~ - X), we can also write this in terms of the fractional conversion X,

Since r(C,J) or r(X) is usually a polynomial in CA or X, one can solve this analytically
for t (CA) or t(X) by factoring the denominator of the integrand and solving the resulting
partial fractions for CA(t) or X(t). This can always be done, but the solution is frequently
neither simple nor instructive.
One can always solve these problems numerically for particular values of CAM and
k, and we will do this for many situations. However, it is still frequently desirable to
have analytical expressions for CA(t) because we can then solve this expression by just
substituting particular parameters into the equation. In any numerical solution one has to
solve the differential equation again for each set of parameters.
Another reason for searching for analytical solutions is that we can only solve numer-
ically a problem that is well posed mathematically. We must program a valid mathematical
expression of the problem on the computer or the answers may be nonsense. The need for
proper descriptions of the equations, initial and boundary conditions, and stoichiometric
relations among the variables is the same whether one is interested in an analytical or a
numerical solution.

VARIABLE DENSITY
The previous examples are all simple problems to integrate for CA(~) or at least for t (CA).
We assumed that the density was constant (the volume in the equations). This would be