Pressure Drop in a Packed Bed Reactor (PBR) Using the Ergun Equation
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A first-order, irreversible, gas-phase reaction A → B takes place in two isothermal, packed-bed catalytic reactors; they have the same diameter but different lengths (14 m and 21 m). Only A is fed to the reactors. The inlet pressure is the same for each reactor; the outlet pressure changes when particle diameter is changed with the slider, but each reactor has the same outlet pressure. As a result, the inlet molar flow rate of A is lower for the longer reactor. Show plots of conversion, pressure, reactant molar flow rate, and volumetric flow rate as a function of distance down the reactor for both reactors with buttons. The Ergun equation is used to model the pressure drop.

Details:
The pressure decreases down the length of the reactor, and thus the volumetric flow rate increases. As a result, the concentration of the reactant decreases (in addition to the decrease due to conversion), which lowers the rate of reaction.

The Ergun equation for pressure drop in a packed bed reactor (PBR) is:
where P is pressure, z is PBR length, φ is void fraction, v is volumetric flow rate, μ is viscosity, DP is the diameter of catalyst particles, AC is cross sectional area of the PBR, m is mass flow rate, B is a laminar flow term, and G is the turbulent flow term.

Material balance on reactant for a first-order reaction:
where CA is the molar concentration of reactant A, FA is the molar flow rate of A, k is a rate constant, and FA,0 is the inlet molar flow rate of A.

Since the reaction is A → B, and only A is fed to the reactor, the total molar flow rate Ftotal = FA,0.

Volumetric flow rate in reactor (from the ideal gas law):

where R is the ideal gas constant, and T is absolute temperature.

Conversion of reactant A: