Biomedical Engineering Reference
In-Depth Information
Since the total amount of mass cannot be created or destroyed during chemical and bio-
reactions, we obtain through mass balance over the entire reactor as shown in
Fig. 5.6
,
ðrVÞ
d
t
¼ r
d
V
d
t
þ V
d
t
d
d
r
0
Q
0
rQ ¼
(5.36)
For condensed matters, density is only a function of temperature. In general,
X
N
S
r ¼
C
j
M
j
(5.37)
j¼1
Equations
(5.33) through (5.37)
are equally applicable to batch reactors. Therefore, these
equations are more general mole balance equations for any well-mixed reactors. In
a general case, Eqn
(5.35)
needs to be written each for every component (species) except
one involved. Like the plug flow reactions, momentum balance equation is needed to
close the problem. Thus, we would have
N
s
differential equations to solve even for
an isothermal reactor. If temperature changes, the energy balance equation must be
solved simultaneously as well. In addition, we need to know how
Q
would vary
with time.
We next consider more simplified case, where steady-state condition has reached. In fact,
most times when we say CSTR or Chemostat, we mean CSTR at steady state. At steady state,
nothing changes with time. Therefore, Eqns
(5.33) and (5.35)
are reduced to
F
j0
F
j
þ r
j
V ¼ 0
(5.38)
or
Q
0
C
j0
QC
j
þ r
j
V ¼ 0
(5.39)
which are algebraic equations.
For a single reaction that is carried out in a steady CSTR, there is only one independent
concentration or reaction mixture content variable and all other concentrations can be related
through stoichiometry as shown in Chapter 3. Without loss of generosity, let us use compo-
nent A as the key component of consideration. Equation
(5.38)
applied to species A gives
F
A
0
F
A
þ r
A
V ¼ 0
(5.40)
or
Q
0
C
A
0
QC
A
þ r
A
V ¼ 0
(5.41)
These algebraic equations can be solved easily.
F
A
0
F
A
r
A
F
A
0
f
A
r
A
V ¼
¼
(5.42)
or
Q
0
C
A
0
QC
A
r
A
V ¼
(5.43)
Note that the rate of reaction is evaluated at the reactor outlet conditions.
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