We maximize the conversion of a first order reaction A → B in a catalyst pellet with a given volume. The pellet is made of a nanoporous material, for instance a zeolite. We introduce macro or meso pores going from the external surface to the pellet interior. Diffusion is faster through these pores than through the nanoporous material, meaning that the access to the pellet interior is improved. But, the intrinsic reaction rate per volume decreases when catalytic material is removed. Obviously, there is an optimal pore network. We are interested in properties of the optimal network, like, e.g., porosity, tortuosity, connectivity and the pore size distribution. We restrict ourselves to situations where the optimal pore network is not significantly affected by the lack of symmetry in the pellet's macroscopic geometry, meaning that we consider spheres, long cylinders and other geometries with a high degree of symmetry. Our results also apply to pellet geometries like squares and cubes when the corners are not dominating the behavior, i.e., for large values of the Thiele modulus (fast reactions). By limiting the study in this way, optimizing the entire pore network reduces to the optimization of a single pore going from the external surface to the interior of the pellet.
When there is pure molecular diffusion in the macro-/mesopores, it is optimal to have a very large number of these pores going from the pellet's external surface straight into the pellet. Remarkably, the pores should be distributed uniformly over the external surface in the optimal geometry, they should not be branched or connected to each other, and the tortuosity should be as low as possible. The local effectiveness factor could then approach 1 for all "islands of catalytic material" inside the pellet. Porosity is the most important parameter for the optimal network. The optimal value ranges from 0 for low values of the Thiele modulus (slow reactions, few diffusion limitations) to approximately 0.5 for very high values of the Thiele modulus (very fast reactions, strong diffusion limitations). The porosity should never exceed 0.5 when the meso/macro pores are disconnected. Gradients in porosity, from the external surface to the center of the pellet, are of marginal importance. Whether the pore size distribution is bidisperse or bimodal is also not important. The optimal bimodal network is always better than the optimal bidisperse network, but the differences diminish when the number of pores starting at the pellet's external surface becomes very large.
When the diameters of the large pores are comparable to the mean free path under reaction conditions, Knudsen diffusion will play a role, and the situation is altogether different. The optimal pore network will then generally be more complex. For pellets with a diameter, or other characteristic length, that is much larger than the mean free path under reaction conditions, the optimal network is similar to that for pure molecular diffusion in the large pores. The exception is for very fast reactions. Also when the pellet's characteristic length is comparable to the mean free path, the situation is different. While the pores should still be distributed uniformly over the external surface and be disconnected with an as low as possible tortuosity, branched pores and a bimodal pore size distribution are sometimes beneficial in these cases.
The increase in conversion of the optimal catalyst pellet compared to the purely nanoporous catalyst pellet varies with the Thiele modulus and the ratio of the diffusion coefficient in the macro-/mesopores to the effective diffusion coefficient in the nanoporous material. The gain is most significant when the Thiele modulus is high (fast reaction). When the ratio of the diffusion coefficients is also high (1E3 or higher), the production of the optimal catalyst pellet is several orders of magnitude larger than the production of the equivalent nanoporous pellet. This shows that there is great potential for improvements by optimizing the properties of the pore network in a catalyst pellet.