Page 324 - Introduction to chemical reaction engineering and kinetics

12.3 Design Equations for a Batch Reactor 305

Substituting for (-rA)V from the material balance in terms of fA, equation 2.2-4, we
obtain

(12.3-27)

Since the relationship between df,ldt and dTldt is implicit with respect to t, we may
write equation 12.3-27 as

(-AHdnA,dfA = n,CpdT (12.3-28)

Equation 12.3-28 may be integrated to give T as a function of fA:

T = To + nAo d fA (12.3-29)

where To and fAO refer to initial conditions. The simplest result from equation 12.3-29
is for constant (-AH&, C,, and n,; then

T = To + (-A%‘“*0 (fA - fAo) (12.3-30)
t P
The time t required to achieve fractional conversion fA is obtained by integration of
the material balance, equation 2.2-4:

(12.3-31)

where (- I~) is obtained from a rate law and V from an equation of state. This also
requires simultaneous solution of equation 12.3-29, since the integral in equation 12.3-
31 depends on T as well as on fA.
An algorithm to calculate t from equation 12.3-31 to achieve a specified fA, and also
t0 obtain fA (T) iS as fOllOWS:
(1) choose Value Of fA; fAO 5 fA 5 fA (specified);
(2) calculate T at fA from equation 12.3-29;
(3) calculate ( -rA) from rate hW given;
(4) calculate V from equation of state given, if required;
(5) repeat steps (1) to (4) for Several vaheS Of fA;
(6) calculate t from equation 12.3-31 by numerical or graphical integration.

Alternatively, the E-Z Solve software may be used to integrate simultaneously the
material- and energy-balance expressions and solve the equation of state.

A gas-phase decomposition, A + R + S, is to be conducted in a batch reactor, with initial
conditions of T, = 300 K, V, = 0.5 m3, and a (constant) total pressure of 500 kPa. The