Page 45 - Introduction to chemical reaction engineering and kinetics

2.2 Batch Reactor (BR) 27

2.2.2 Material Balance; Interpretation of ri

Consider a reaction represented by A + . . . -+ products taking place in a batch reactor,
and focus on reactant A. The general balance equation, 1.51, may then be written as a
material balance for A with reference to a specified control volume (in Figure 2.1, this
is the volume of the liquid).
For a batch reactor, the only possible input and output terms are by reaction, since
there is no flow in or out. For the reactant A in this case, there is output but not input.
Equation 1.5-1 then reduces to

rate offormation of A by reaction = rate of accumulation of A

or, in mol s-l, sayr,

(-rA)V = -dnAldt, (2.2-1)

where V is the volume of the reacting system (not necessarily constant), and nA is the
number of moles of A at time t. Hence the interpretation of r, for a batch reactor in
terms of amount nA is

(-rA) = -(l/V)(dnAldt) (2.2-2)

Equation 2.2-2 may appear in various forms, if nA is related to other quantities (by
normalization), as follows:

(1) If A is the limiting reactant, it may be convenient to normalize nA in terms of fA,
the fractional conversion of A, defined by

(2.2-3) j
fA = @A0 - nA)inAo WV

where n&, is the initial amount of A; fA may vary between 0 and 1. Then equation
2.2-2 becomes

cerA) = (nA,lV)(dfAldt) (2.2-4) ~

(2) Whether A is the limiting reactant or not, it may be convenient to normalize by
means of the extent of reaction, 5, defined for any species involved in the reac-
tion by

d[ = dnilvi; i = 1,2, . . . , N (2.2-5) 1

‘Note that the rate of formation of A is rA, as defined in section 1.4; for a reactant, this is a negative quantity. The
rate of disappearance of A is (-r.& a positive quantity. It is this quantity that is used subsequently in balance
equations and rate laws for a reactant. For a product, the rate of formation, a positive quantity, is used. The
symbol rA may be used generically in the text to stand for “rate of reaction of A” where the sign is irrelevant
and correspondingly for any other substance, whether reactant or product.