Page 315 - Introduction to chemical reaction engineering and kinetics

296 Chapter 12: Batch Reactors (BR)

12.3 DESIGN EQUATIONS FOR A BATCH REACTOR

12.3.1 General Considerations

The process design of a batch reactor may involve determining the time (t) required to
achieve a specified fractional conversion (fA, for limiting reactant A, say) in a single
batch, or the volume (V) of reacting system required to achieve a specified rate of pro-
duction on a continual basis. The phrase “continual basis” means an ongoing operation,
that is, operation “around the clock” with successive batches. This includes allowance
for the down-time (td) during operation for loading, unloading, and cleaning. The oper-
ation may involve constant or varying density (p), and constant or varying temperature
(T). The former requires an equation of state to determine V, and latter requires the
energy balance to determine T. We consider various cases in the following sections.
For a single-phase system, V always refers to the volume of the reacting system, but
is not necessarily the volume of the reactor. For example, for a liquid-phase reaction in
a BR, allowance is made for additional “head-space” above the liquid, so that the actual
reactor volume is larger than the system volume. In any case, we use V conventionally
to refer to “reactor volume,” with this proviso.

12.3.1.1 Time of Reaction
Consider the reaction

A + . . . + z+c + . . . (12.3-1)

Interpretation of the mass balance, equation 12.2-1, leads to equation 2.2-4, which may
be rewritten, to focus on t , as

1
(12.3-2) 1

where t is the time required for reaction in an uninterrupted batch from fractional con-
version fAl to fA2, and nAo is the initial number of moles of A. Equation 12.3-2 is gen-
eral in the sense that it allows for variable density and temperature. For specified It&,,
fAi, and fA2, the unknown quantities are t, V, ( -rA), and T (on which (-Ye) and V
may depend). To determine all four quantities, equation 12.3-2 may have to be solved
simultaneously with the rate law,

(--A) = IA(fAjT)
the energy balance, which gives

T = T(~A, V) (12.3-4)

and an equation of state (incorporating the stoichiometry),

v = v(nA, T, P) (2.2-9)

in which case, we assume P is also specified.
In equation 12.3-2, the quantity tlnA, is interpreted graphically as the area under the
curve of a plot of l/(-rA)V against fA, as shown schematically in Figure 12.1.