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 R9.6-1). The number of moles of species i inis

Substituting for N_i in Equation (9-3) and dividingby gives

Taking the limit asand noting that gives

Rearranging, we have

(R9.6-1)

Comparing Equations (R9.6-1) and (9-26)

(R9.6-2)

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,

To include radial variations in temperature, see Problem P8-29

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

Recalling that

we substitute these equations into Equation (9-22) to obtain

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

Transient energy
balance on a PFR

(R9.6-3)

This equation must be coupled with the mole balances,

(R9.6-4)

Numerical solution
required for these
three coupled equations

and the rate law,

(R9.6-5)

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 (R9.6-3) reduces to

(8-60)

Substitution for the molar flow rates F_i in terms of conversion gives Equation (8-56).

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

(8-56)

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