# Comparison of One- and Two-Dimensional Models of Plate Reactors

This Demonstration compares three mathematical models of a plate chemical reactor in which an adiabatic reaction takes place. The fluid containing the reactants flows in the channel between two parallel plates; the distance between the plates is much smaller than their width or length. The models assume a steady-state condition with constant material properties. The first model is one-dimensional; it considers only horizontal convective transport. The others are two-dimensional models that consider horizontal convection and vertical diffusion as well; they differ only in the horizontal fluid velocity profiles: one is laminar flow and the other is plug flow.
There is no significant difference in the results of the average (cup) temperature or conversion of the one-dimensional and the two-dimensional plug flow models. However, these models overestimate both the average conversion and average temperature change of reactants for both endothermic and exothermic reactions.
The models show significant differences in both predicted temperature and conversion across the vertical direction. The changes are larger in slower (higher activation energy) reactions, and more noticeable near the center where the velocity of the two-dimensional model with laminar flow approaches the maximum velocity. Here the conversion is lowest; therefore, the temperature of the laminar flow model is lower than the other models in exothermic reactions and higher in endothermic reactions, whereas the conversion in the laminar flow model is always lower in both types of reactions.

### DETAILS

The equations for the one-dimensional model are:
and
.
These are the equations for the two-dimensional plug flow model:
and
,
and the two-dimensional model with laminar flow has these equations [1]:
and
.
The average (cup) values of concentration and temperature for the models with the flat velocity profiles are:
and
,
and for the model with the laminar flow:
and
,
where
is the distance between the plates
is the fluid specific gravity
is the mean horizontal velocity
is the concentration of component
is the fluid temperature
and are the horizontal and vertical coordinates, respectively
is the pre-exponential factor
is the activation energy
is the heat of reaction
is the gas constant
is the diffusivity
is the thermal conductivity
Reference
[1] R. B. Bird, W. E. Stewart and E. N. Lightfoot, Transport Phenomena, 2nd ed., New York: John Wiley & Sons, Inc., 2002.

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