We plan to reduce the energy balance into a more usable form. To achieve this form for plug-flow reactors, we begin by applying the balance to a small differential volume,, in which there are no spatial variations (Figure R13.7-1). The number of moles of species i inis

Substituting for N_i and dividingby gives

Taking the limit asand noting that gives

Rearranging, we have

(R13.7-1)

Comparing Equations (R13.7-1) and (R13.7-2)

(R13.7-2)

we note that the term in parentheses is just r_i. The rate of reaction of species i is related to the rate of disappearance of species A through the stoichiometric coefficient, 

Then

Finally,

where a is the heat exchange area per unit volume. Differentiating yields

Recalling that

we substitute these equations to obtain

Neglecting shaft work and changes in pressure with respect to time, we obtain

Transient energy
balance on a PFR

(R13.7-3)

This equation must be coupled with the mole balances,

(R13.7-4)

Numerical solution
required for these
three coupled equations

and the rate law,

(R13.7-5)

and solved numerically. A variety of numerical techniques for solving equations of this type can be found in the book Applied Numerical Methods.¹

We note, however, one can approximate the partial differential equation (PDE) of the unsteady PFR energy balance by replacing the PFR with a number of CSTRs in series. One then simply solves the coupled ODEs using Polymath™ or MATLAB™.

For steady-state operation in which no work is done by the system, Equation (R13.7-3) reduces to

(R13.7-6)

Substitution for the molar flow rates F_i in terms of conversion gives Equation (R13.7-7).

Use this
equation for
steady-state
energy balance on a PFR
with heat transfer

(R13.7-7)

As stated previously, this equation is solved simultaneously with the mole balance.