Page 185 - Elements of Chemical Reaction Engineering 3rd Edition

Sec. ,4.4 Pressure Drop in Reactors 157

Equation (4-28) will be the one we use when multiple reactions are
occurring or when there is pressure drop in a membrane reactor. However, for
single reactions in packed-bed reactors it is more convenient to express the
Ergun equation in terms of the conversion X. Recalling Equation (3-42) for F7,
FTo i
= F,, 1 +
F, = F,, + F,$X 6X (3-42)

and the development leading to Equation (3-43),

FT
=
- 1+EX
Fro
where, as before,

Equation (4-28) can now be written as

Differential form
of Esgun equation - _- - _ -
CY T Po
dP -
for the pressure (1 + EX) (4-30)
o
drop in packed dW 2 T PIP,
beds 1
We note that when 6 is negative the pressure drop AP will be less (Le.,
higher pressure) than that for E = 0. When E is positive, the pressure drop AP
will be greater than when E = 0.
For isothermal operation, Equation (4-30) is only a function of conver-
sion <and pressure:

(4-31)

Recalling Equation (4-21),

Two coupled dX
=
equations to be - F, (X,P) (4-21)
solved numerically dW
we see that we have two coupled first-order differential equations, (4-31) and
(4-21), that must be solved simultaneously. A variety of software packages and
numerical integration schemes are availabIe for this purpose.

Analytical Solution. If E = 0, or if we can neglect ( EX) with respect io 1 .O
&e., 1 %-EX), we can obtain an analytical solution to Equation (4-30) for iso-
thermal operation (Le., T = To). For isothermal operation with E = 0, Equa-
tion (4-30) becomes