Page 108 - The engineering of chemical reactions

92 Single Reactions in Continuous Isothermal Reactors

We could continue to write all these mass-balance equations as X(r) instead of
CA(r), but the solutions are not particularly instructive unless the rate expressions become
so complicated that it is cumbersome to write an expression in terms of one concentration
CA. For simplicity, we will use CA(t) rather than X(t) wherever possible.

Reversible reactions
Next consider the reaction
AZ B, I = kfCA -k&B
Since from stoichiometry CB = CsO + CAM - CA, the rate can be written as

r = k&A - kb(CAo + CBo - CA)
and the solution with CsO = 0 is
CAo - CA CAo - CA
t=
r(CA) = kfCA - kb(CAo - CA)
or
1 +kt,t
CA(r) = CAo
1 + (kf + ki,)t
Note that as t + co, this gives the equilibrium composition
CB kf
-=-
CA kb
as required by thermodynamics. Also note that if kb = 0, the expression becomes that for
the irreversible first-order reaction as expected.
We can continue to consider more complex rate expressions for reversible reactions,
but these simply yield more complicated polynomials r(CA) that have to be solved for
CA(t). In many situations it is preferable to write t(CA) and to find CA by trial and error
or by using a computer program that finds roots of polynomials for known kf, kt,, and feed
composition.

THE PLUG-FLOW TUBULAR REACTOR
The CSTR is completely mixed. The other limit where the fluid flow is simple is the plug-
Ilow tubular reactor, where the fluid is completely unmixed and flows down the tube as a
plug. Here we picture a pipe through which tluid flows without dispersion and maintains
a constant velocity profile, although the actual geometry for the plug-flow approximation
may be much more complicated. Simple consideration shows that this situation can never
exist exactly because at low flow rates the flow profile will be laminar (parabolic), while
at high flow rates turbulence in the tube causes considerable axial mixing. Nevertheless,
this is the limiting case of no mixing, and the simplicity of solutions in the limit of perfect
plug-flow makes it a very useful model. In Chapter 8 we will consider the more complex
situations and show that the error in the plug-flow approximation is only a few percent.
We must develop a differential mass balance of composition versus position and then
solve the resulting differential equation for CA(Z) and CA(L) (Figure 3-2). We consider a
tube of length L with position z going from 0 to L. The molar flow rate of species j is F’o
at the inlet (z = 0), Fj(z) at position z, and Fj(L) at the exit L.