Headspace diffusion limitations on heterogeneous catalysis in unstirred batch reactors

Headspace gas diffusion can affect measurements of catalytic rate constants in unstirred batch reactors when the reaction occurs in a small pile of catalyst within the reactor. To quantify these effects, the governing equations for catalysis in a round-bottom flask were solved numerically in a toroidal coordinate system. We also introduce a simpler model reactor geometry that preserves the essential characteristics of a typical bench-scale flask-reactor. The preserved characteristics include the reactor volume, an averaged diffusion length scale, and the area of the catalyst pile. An eigenfunction expansion solution for the model reactor closely parallels the full numerical solutions in the round-bottom flask reactor, thus confirming the validity of the simplified model reactor. Solutions for the model reactor show that concentrations measured above the catalyst pile decay exponentially to equilibrium even when transport limitations are important. Therefore, exponential decay rates in these reactors should not be equated to first order (or pseudo-first order) reaction rate constants without first checking carefully for diffusion limitations. Two dimensionless parameters govern the reactor performance. Effectiveness factors are computed for unstirred catalytic batch reactors over a wide range of the two dimensionless parameters. Our findings show quantitatively when headspace stirring is and is not necessary. When stirring is inconvenient or impossible, the tabulated effectiveness factors can be used to design reactors that are small enough, with kinetics that are slow enough, and with the catalyst dispersed over a large enough area to avoid headspace diffusion limitations.