A three parameter one-dimensional model to predict effectiveness factor for an arbitrary pellet shape with linear kinetics

CLARISA MOCCIARO; NÉSTOR JAVIER MARIANI; OSVALDO MIGUEL MARTÍNEZ; GUILLERMO FERNANDO BARRETO

Ixtapa-Zihuatanejo

Congreso; International-Mexican Congress on Chemical Reaction Engineering; 2010

Instituto Mexicano del Petróleo

Several aspects should be accounted for the analysis of the performance of catalytic fixed bed reactor units. Fairly often the intra-particle diffusion-reaction inside the catalytic pellet becomes the limiting step. Due to the shape of the real pellets generally this phenomena takes place along two (2D) or three (3D) spatial coordinates. Consequently, the conservation balances inside the particle should be solved numerically. The computational task is readily affordable when a single set of conditions is undertaken. However, it is clear that even for the simplest practical case (e.g. simulation of a catalytic reactor with a single reaction) such evaluation have to be performed thousands of times. In addition, for applications as reactor optimization, the number of evaluations will increase by orders of magnitude. Besides, when dealing with a set of reactions the amount of numerical calculations is further increased. Thus, it is of paramount importance to avoid the use of 2D or 3D computations. Many years ago Aris (1965), simultaneously with other researches, presented a very simple approach to reduce 2D or 3D problems into a 1D problem showing that at large values of the Thiele modulus, the effectiveness factor for a single reaction does not depend on the pellet shape, but just on the ratio of pellet volume to external surface area, l. To perform approximate evaluations at low and intermediate values of Thiele modulus, any geometry satisfying the actual value of l could be adopted, as a simple slab of half-width l. The expected precision with this approximation is of the order of 20% for relatively simple kinetics. A more convenient 1D model was proposed by Datta and Leung (1985). For this model, here called Generalized Cylinder (1D-GC) model, it is supposed that diffusion takes place along a distance L of a hypothetical body of variable cross section according to zs, being z the non-dimensional coordinate. Values of the two model parameters, L (effective diffusion length) and s (shape factor), can be obtained by matching the value l of the actual pellet and one additional property related with the shape of the actual pellet. Mariani et al. (2003, 2008) proposed two different criteria to estimate the shape factor s by stating that the 1D-GC model response should exactly match the values of actual pellet either at high or low effective reaction rates, respectively. It is worth noting that using the high reaction rates criteria, s can be straightforwardly obtained just from the geometry of the pellet. The expected errors of the 1D-GC model employing the high reaction rate criteria is less than 3% for a variety of commercial pellets with normal kinetic behavior (i.e., no maximum in the effectiveness factor arises for intermediate Thiele modulus). This level of precision is judged as appropriate for any practical application; thus, it was adopted as a target. Nonetheless, it was detected that the use of 1D-GC model for commercial pellets previously analyzed (Mariani et al., 2009), but changing some of the ratios between geometrical dimensions, can lead to considerably larger errors than that of the above mentioned target. For example, for a holed trilobe pellet a twofold increase of the hole diameter, while keeping the external pellet diameter, the precision of the 1D-CG is reduced by an order of magnitude. In addition, 1D-GC model predictions start to deviate significatively for certain commercial pellets if the reaction order is low (e.g., for a four holed pellet with zero order kinetics the 1D-GC error is around 9%). Therefore, it is highly desirable to propose an alternative 1D model to restore a level of precision within the stated target. In this context, the objective of this contribution is to present a novel one-dimensional model to estimate effective reaction rates, called variable-diffusivity model (1D-VD). It is supposed that diffusion takes place in a slab with variable diffusivity along the non-dimensional coordinate z. Depending on the form of the variable diffusivity function, different number of parameters arises. For the here proposed 1D-VD model, the three associated parameters are obtained by stating that the 1D-VD model should match the behavior of the actual pellet at high and low effective reaction rates, simultaneously. The 1D-VD model warranties a precision higher than 2% for a vast set of particles whose geometrical parameters are considerably varied from those of the pellets available commercially when analyzing isothermal lineal kinetics.