372
Chapter 9
9.2 The Plug Flow Reactor
The basic model for the tubular reactor is the so-called plug flow reactor model in which no
concentration (or temperature) gradients in the radial coordinate are admitted. Hence there is only
one spatial dimension, the distance z along the reactor axis. A mass balance for reaction component
i
in a volume element
Adz
where
A
is the reactor's cross-sectional area yields
dc
dc
=
- vs -= 7
+ 9,-(c>
(9.66)
dt
dz
where v. is the linear velocity in the z direction.
The transient mass balance for the plug flow tubular reactor is a hyperbolic partial differential
equation which can be solved using the method of characteristics, as described in Aris and
Amundson (1973).
c , ( z , /
=
0
) as well as
ci(z = 0,t)
must be known in order to solve Eq. (9.66). If
ci(z=0,t)
is constant in time, a steady-state profile
ct(z)
will gradually develop. An inlet
disturbance, e.g., a substrate pulse, travels along the reactor axis as a pulse of diminishing
amplitude if
c,
is being consumed. With these few comments on the transient equation, Eq. (9.66),
we shall devote the remainder of this section to the steady-state solution.
If, as assumed in the plug flow model v. is constant in the cross-section, the steady-state model
becomes
~ - *l(c)
(9.67)
at
where
c(/'=0) -c°
and
t' = z
/ v;
(9.68)
Equation (9.67) is mathematically identical to Eq. (9.2) with
t\
the time it takes a liquid element to
travel a distance z along the reactor axis, replacing
t.
Physically speaking, the two reactor operations
are, of course, quite different. The batch reactor is inoculated at time
t
= 0, and the condition in the
ideally mixed tank changes as a function of time. The plug flow reactor is studied at steady state, it
operates in a continuous mode, and no mixing even between neighboring fluid elements is
admitted.
A continuous injection of cells at z = 0 is impracticable, and inoculation of the continuous plug
flow reactor is done by placing it downstream from a continuous stirred tank. By this arrangement
an efficient reactor system for conversion of substrate to a very low residual concentration is
obtained. Recirculation of cells from the reactor outlet to z = 0 is another way of operating a
continuous plug flow reactor.
The integrated form of Eq. (9.67) can be taken from the corresponding expressions for the batch
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