Kinetic Modeling of Solid-Gas Reactions At Reactor Scale: a General Approach

Solid-gas reactions are of great interest in many industrial fields such as nuclear, chemistry, metallurgy, CO₂ capture_, etc… Industrial reactors where these reactions take place are difficult to understand. Indeed the solid phase is a granular medium through which circulate gaseous reactants and products. The properties of such a medium are modified in space and time due to reactions occurring at a microscopic scale. The thermodynamic conditions are driven not only by the operating conditions but also by the heat and mass transfers in the reactor. Several models have been developed to account for the complexity of these transformations such as the grain model [1] and the pore model [2] and all their improved derivatives. However most of these models are based on the law of additive reaction times of Sohn [3] for which the order respective to the gas in the kinetic rate equation must be equal to 1. However such condition is scarcely encountered in many gas-solid reactions so that erroneous results may be obtained using this simplified approach.

In order to overcome such problems, we have developed a multi-physic approach based on the finite elements method which combines the resolution of the thermohydraulic equations with the kinetic laws describing the heterogeneous reactions at the scale of dense particles.

Indeed, rather than the usual equation  based on the Arrhenius dependence of the rate and the choice of a f(a) function among a dozen of kinetic models, we propose a less restrictive kinetic modeling at the microscopic scale allowing non-Arrhenius and/or non f(a) behavior, and taking into account the real influence of the partial pressures of the relevant gases. The kinetic assumptions of the proposed models not only include those of the literature, but also offer various other possibilities such as surface-nucleation and growth processes, anisotropic or isotropic growth, inward or outwards development of the new phase, etc … Considering the three usual symmetries (spheres, cylinders, planes) such an approach allows to obtain about forty-five kinetic models well suited for reactions in solid-gas systems [4].

At a macroscopic scale, heat and mass transfer terms entering in the balance equations depend on the kinetics evaluated at the microscopic scale. These equations give the temperature and partial pressures in the reactor, which in turn influence the microscopic kinetic behavior.

At microscopic scale, the reaction fractional conversion is followed for a representative population of grains. By finite difference it is possible to calculate the reaction rate da/dt. This rate allows to evaluate heat and mass sources produced by the reaction. And using these sources terms, microscopic reactions have an impact on the spatial and temporal evolution of the thermodynamic processes at the reactor scale. Inversely since it modifies the areic frequency of nucleation and the areic reactivity of growth, thermodynamic influences fractional conversion of the microscopic reaction.

At the macroscopic scale, and for each of the three partial differential equations (heat, mass transport and hydrodynamic), two formulations have been constructed:

- a 2D finite element formulation in cylindrical coordinates,

- a formulation in 3D Cartesian coordinates.

The finite element method is based on a discretization of the weak forms for the equations of transport-diffusion of heat, charge and concentration. The discretization of these integral forms is obtained by choosing the projection functions from a set of linearly independent functions constitute a basis {a_i} and looking for the solution in the functional space generated by this base (i.e. the Galerkin method). By this way one can obtain a linear system}. The construction of the matrix and the second member of this system require the calculation of the functions a_i, and the numerical integration of functions defined on the whole domain.

 

The reactor is represented by a computational domain which can be of various shapes in a 2D or 3D space, described by a system of Cartesian or cylindrical coordinates. The mesh is made ​​using gmsh [5], a free finite element mesh generator.

The equations used for the macroscopic scale also involve physical properties that are necessary to calculate at various temperatures and various pressures. Physical properties to consider are for example: the density, the specific heat capacity at constant pressure, the thermal conductivity, the dynamic viscosity, the intrinsic permeability, the molecular diffusivity and the porosity.

These properties must be known when it makes sense for the different solid phases, for each of the gases and their mixture.

In this presentation, we propose a description of the physical models, the coupling between the various scales of calculation (grains population, aggregates and whole reactor), and its validation based on experimental results obtained in thermobalance. The results obtained in the case of the kaolinite dehydration are presented in order to illustrate the method capabilities.

[1] J. Szekely, J.W. Evans, Chem. Eng. Sci., 25 (1970) 1091-1107.

[2] S. Bhatia, D. D. Perlmutter, AIChE J. 26 (1980) 379-386.

[3] H.Y. Sohn, Metall. Trans., 9B (1978) 89-96.

[4] M. Pijolat, L. Favergeon, M. Soustelle, submitted to Thermochimica Acta.

[5] C. Geuzaine, J.-F. Remacle, Inter. J. for Numerical Methods in Engineering, 79, 11 (2009) 1309-1331.