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Unit Process Principles 63

Observed rate of change 4.3.3.3 Materials Balance: Mathematics
of mass in reactor
The mathematical statement of Equation 4.8 is given for
two cases: (1) a complete mix fluidized reactor, and (2) a
fixed bed column reactor, or more commonly called a ‘‘plug-
flow’’ reactor, in which the homogeneous volume element is an
Mass flow in Mass flow out
Reaction infinitesimal ‘‘slice’’ of the column. The term fluidized
[rate of reaction] means that the reactants are being mixed by turbulence,
vis-à-vis fixed bed, in which case the porous media reactant
is fixed in place as in a bed of GAC or a rapid rate filter
media bed.
FIGURE 4.4 Materials balance for a homogeneous ‘‘complete
mix’’ volume element showing terms in words, as corresponding to 4.3.3.3.1 Complete Mixed Reactor: Finite Volume
Equation 4.8.
Fluidized reactors include complete mix activated sludge,
rapid mix coagulation, and any other basin that has a high
case, the volume element is the whole reactor (since the reactor
level of energy input. Most often, this is done by a turbine
is ‘‘complete mix’’ and therefore the reactor as a whole is
impeller. The complete mix reactor, sometimes called ‘‘com-
homogeneous). The ‘‘rate of reaction’’ is the kinetic term,
pletely stirred tank reactor’’ (CSTR), is homogeneous by
which depends upon the particular kind of reaction, that is,
definition and therefore the materials balance equation applies
chemical, biological, coagulation, etc. With this basic concept,
to the whole reactor. For the ‘‘complete mix’’ homogeneous
reactor modeling is applicable to a wide variety of unit pro-
volume element of Figure 4.4, Equation 4.8 may be expressed
cesses. Figure 4.4 is for a continuous flow reactor, as is evident
mathematically as Equation 4.9, that is,
by the advection terms. For a ‘‘batch’’ reactor, that is, with no
advection in or out, the advection terms would equal zero.
q(C V) q(C V)
(4:9)
¼ Q C in Q C
2 3 qt o qt
Observed
Rate of
4 5
reaction in which
mass rate ¼[Mass flow in] [Mass flow out]
of change
C is the concentration of species of interest in the reactor
(4:8) and leaving reactor (kg=m )
3
3
V is the volume of reactor (m )
in which
C in is the concentration of species in flow into the reactor
observed mass rate of change ¼ the rate of change of the 3
(kg=m )
concentration within a given reactant as observed by 3
Q is the flow into and out of the reactor (m =s)
measurement (kg reactant A=s)
t is the elapsed time (s)
mass flow in ¼ the product of flow, Q, into the reactor
subscript ‘‘o’’ indicating ‘‘observed’’ mass rate of change
times its concentration, C (kg of A=s)
subscript ‘‘r’’ indicating reaction rate (kinetics)
mass flow out ¼ the product of flow, Q, out of the reactor
times its concentration, C (kg of A=s)
Figure 4.5 illustrates the same reactor as shown in Figure 4.4,
rate of reaction ¼ the rate at which A is undergoing change,
but with the terms depicted in mathematical form (instead of
which is V dC=dt (kg of A=s)
words). The continuous flow, complete mix (and thereby
homogeneous) reactor is the general case that embodies the
4.3.3.2 Comments on Materials Balance
reactor concept. This is the starting point for grasping the
To illustrate the flexibility of Equation 4.8, we may impose
mathematical modeling of the other kinds of reactors.
different kinds of special conditions, for example:
1. Steady state operation: The left side of Equation 4.8
is zero.
∂(C·V)
2. Rate of reaction is zero: The observed rate of change
∂t
equals the difference between the mass flow in and obs
the mass flow out, such as would occur for a solution
of one kind, say saline, replacing another solution,
Reactor
say pure water, in the reactor. Q·C ∂(C·V) Q·C
3. Batch reactor: There is no mass flow across the in
∂t
boundaries of the reactor, and the observed rate of r
change is equal to the rate of the reaction.
FIGURE 4.5 Materials balance for a homogeneous ‘‘complete
These conditions are illustrated mathematically in the sections mix’’ volume element showing terms in mathematical symbols, as
that follow, for example, Section 4.3.4. corresponding to Equation 4.9.