(393g) 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.