balance) a simple analytical solution cannot be provided. An
approximation, such as the Runge-Kutta method,12 can solve
systems of such equations by an iterative, numerical tech-
nique. An adaptive step fifth-order Runge-Kutta algorithm,13
part of the Mathcad 8 Professional software,14 was used to
solve the equations defining the model.
The variables that can be manipulated to control the
system response include the diameter and length of the PFR,
the volumetric flow rate of the reactant, and the temperature
of the heat-transfer medium. [For simplicity in this analysis,
a constant temperature of the heat-transfer medium will be
assumed. Depending on the system, (the flow rate and heat
capacity of the heat-transfer fluid), this temperature may vary
along the length of the reactor in co-current or counter-current
operation. An energy balance on the heat-transfer fluid can
be included in the model and solved simultaneously with
the continuity equations for the reaction.]. A critical param-
eter, the overall heat-transfer coefficient, cannot be inde-
pendently set as a control variable. The fluid properties of
the reaction mixture and the heat-transfer fluid, the thickness
and conductivity of the wall separating the two fluids, and
the incorporation of baffles and fins used to improve mixing
and heat-transfer influence this parameter. The overall heat-
transfer coefficient must be determined for a given reactor
system.
To simplify and generalize the analysis, dimensionless
variables convert the differential equations to the following:
dX
djz
1 - ht
ht
) -khX exp - ꢀ
[
(
)
]
dht
djz
1 - ht
) âX exp - ꢀ
-Uh(ht - tc)
[
(
)
]
ht
where
X ) unreacted fraction of 4 [C(4)/C0(4)]
C0(4) ) the initial concentration of 4
jz ) z/L ) ratio of the distance (z) in the reactor to
the reactor length (L)
An early consideration of running this reaction in a batch-
recycle manner highlights the risk of insufficient completion
of this reaction in the PFR. The reactant within a batch
reactor (storage vessel) achieves partial conversion passing
through a heated reactor. Prior to the return to the storage
vessel, the material passes through a heat exchanger thereby,
returning to the safe holding temperature. During the
operation, a potential exists of introducing heated material
into the hold vessel in the event of loss of cooling to the
heat exchanger or circumstances that might upset the heat
balance in the system. Until sufficient material is converted,
a risk remains that the reaction could proceed in an
uncontrolled manner in the recycle vessel. Therefore, if the
operation strategy for a given PFR dictates incomplete
conversion, the reactor should be operated to convert the
majority of material in a single pass. This conversion should
ensure that any conversion of the material in the hold tank
would limit the temperature rise well below the boiling point
of the mixture and the onset of decomposition. As a
precaution, additional heat removal capabilities should be
designed for the hold vessel.
kh ) k0L/u ) dimensionless reaction constant
k0 ) A0 exp(- ꢀ)
ht ) T/T0 )
ratio of the reaction temperature to the inlet temperature
ꢀ ) E/RT0
â ) -∆Hk0C0L/FCpT0u
Uh ) ULa/FCpu ) dimensionless heat-transfer coefficient
tc ) T/Tc )
ratio of the reaction temperature to the jacket temperature
subjected to the initial conditions:
jz ) 0,X ) 1,ht ) 1
Figures 6 and 7 provide temperature profiles and conver-
sions along a PFR for three different regimes of operation.
The most desirable situation would be the isothermal case
wherein the operating conditions are set so that heat is
removed as fast as it is generated, and the reaction mixture
is maintained at the temperature of the heating fluid. As
shown in Figure 7, for a properly designed reactor, the
reaction reaches completion at the end of the reactor. As
long as the temperature of the reaction mixture does not reach
the regime where thermal decomposition begins, a less
desirable, but operable, case occurs where the heat generated
exceeds the capability for heat removal. With careful
consideration of reactor performance and good temperature
control, this can be a means to improve reactor throughput.
As the corresponding conversion curve in Figure 7 shows,
this particular reactor is over-designed as the reaction reaches
completion within the first 25% of the length of the reactor.
Capacity can be increased by increasing the flow rate, or a
Discussion of Modeled Reactor Performance
Application of the model to the reaction system implies
turbulent flow or more specifically a flat velocity profile
across the reactor diameter. Turbulent flow indicates good
mixing and hence a constant temperature profile. For the
nature of this reaction system, laminar flow should be
avoided for this very reason. Static mixers incorporated in
the reactor improve the cross-sectional mixing and enhance
the heat transfer. This discussion demonstrates the use of
the model as a guide for selecting appropriate reaction
conditions (temperature, reactor length, reactor diameter, and
flow rate) to ensure adequate control of a thermally sensitive
reaction.
(11) Evans, J. M.; Stemp, G.; Tedder, J. M. PCT Int. Appl. 1991.
(12) Carnahan, B.; Luther, H. A.; Wilkes, J. O. Applied Numerical Methods,
John Wiley and Sons: New York, 1969.
(13) Press, H. E.; Flannery, S. A.; Teukolsky, B P.; Vetterling, W. T. Numerical
Recipes in C; Cambridge University Press, New York, 1992.
(14) Mathcad 8 Professional; MathSoft, Inc.: Cambridge, MA, 1998.
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