Design of Non-Ideal Heterogeneous Packed Catalytic Reactors with Interpellet Axial Dispersion

22-4 DESIGN OF NON-IDEAL HETEROGENEOUS PACKED CATALYTIC REACTORS WITH INTERPELLET AXIAL DISPERSION 

If reactant A participates in one second-order irreversible chemical reaction and its stoichiometric coefficient is —1 in that reaction, then the one-dimensional plug-flow mass balance in dimensionless form, given by (22-11), is described by 

Subscript j is unnecessary when there is only one chemical reaction. When the kinetics are second-order, irreversible, and only a function of Ca, the dimensionless pseudo-volumetric reaction rate, based on bulk gas-phase concentrations, is 

The heterogeneous rate law in (22-57) is dimensionalized with pseudo-volumetric nth-order kinetic rate constant k that has units of (volume/mol) per time. k is typically obtained from equation (22-9) via surface science studies on porous catalysts that are not necessarily packed in a reactor with void space given by interpellet. Obviously, when axial dispersion (i.e., diffusion) is included in the mass balance, one must solve a second-order ODE instead of a first-order differential equation. Second-order chemical kinetics are responsible for the fact that the mass balance is nonlinear. To complicate matters further from the viewpoint of obtaining a numerical solution, one must solve a second-order ODE with split boundary conditions. By definition at the inlet to the plug-flow reactor, I a = 1 at = 0 via equation (22-58). The second boundary condition is d I A/df 0 as 1. This is known classically as the Danckwerts boundary condition in the exit stream (Danckwerts, 1953). For a closed-closed tubular reactor with no axial dispersion or radial variations in molar density upstream and downstream from the packed section of catalytic pellets, Bischoff (1961) has proved rigorously that the Danckwerts boundary condition at the reactor inlet is 

The numerical solution of second-order nonlinear ODEs with split boundary conditions requires trial and error integration of two coupled first-order ODEs. If one defines d p./di = Axial Grad, then the one-dimensional plug-flow mass balance with axial dispersion, 

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