Page 110 - The engineering of chemical reactions

94 Single Reactions in Continuous Isothermal Reactors

4. Constant tube diameter, and
5. A single reaction.

This equation is not appropriate if all five of these conditions are not met. We can
relax the third and fourth restrictions for the PFTR by considering the differential element
of volume dV = At dz rather than the differential element of length dz. The mass-balance
equation at a position where the fluid has moved from volume V to volume V + d V then
becomes
F’(V) meF’(V + dV) + dV I+r = 0
and taking the limit d V -+ 0 and dividing by d V, we obtain the expression

dFj - Vjr
dV
This equation can also be used in situations where the density and tube cross section are
not constants. The equation
dFA
- = -IA
dV
is described as the “fundamental equation” for the mass balance in a PFTR in the texts
by Levenspiel and by Fogler (with VAT replaced by rA). However, this equation cannot
be simply modified to handle transients, nor can it be used to consider other than perfect
plug flow, because for all of these situations we need equations in which the position z
is the dependent variable. Since situations such as laminar flow and dispersion caused by
turbulence are very important in all real tubular reactors, we prefer to use the constant-
cross-section, constant-density version of this equation so that we can easily see how it
must be modified to handle these situations.

CONVERSION IN A CONSTANT-DENSITY PFTR

We now consider solutions to the preceding equation for simple kinetics. For reactant species
A(UA = - 1) the equation becomes

For r = kCA substitution yields
dC.4
1.4 - = -kCA
dz
and after separation we obtain the differential equation
dC.4
- = -k dz
CA u
This equation must be integrated between z = 0, where CA = CAM, to position z, where
CA = CA(Z), to position L, where CA = CA(L),