Exothermic solid-solid reactions lead to sharp reaction fronts that cannot be captured by coarse spatial mesh size numerical simulations that are often required for large-scale simulations. We present a coarse-scale formulation with high accuracy by using a Taylor series expansion of the reaction term. Results show that such expansion could adequately maintain the accuracy of fine-scale behavior of a constant pattern reaction front while using a smaller number of numerical grid cells. Results for a one-dimensional solid-solid reacting system reveal reasonable computational time saving. The presented formulation improves our capabilities for conducting fast and accurate numerical simulations of industrial-scale solid-solid reactions.

Modeling of reactive flow has diverse applications in engineering
and science. Applications include heavy oil recovery processes, combustion in
porous media, ground water flow, and transport and reaction processes in
biofilms. An important class of reactions is that of solid-solid reactions.
Modeling of solid-solid reactions is of significant interest in various
industrial operations, including oxidation of metallic and nonmetallic mixed
powders, cement industry [

Accurate numerical simulation of solid-solid reactions is a challenging task due to the multiscale nature of the physical phenomena. Physical processes involved in solid-solid reactive systems include diffusive (heat and mass) and reactive processes. Reactions in porous media intrinsically often take place at the small scale, causing development of subdiffusive-scale concentration and temperature gradients, while heat and mass diffusive processes have scales orders of magnitude larger than the reactions. Large-scale simulation of such coupled processes is computationally expensive due to limitations in computational resources. Therefore, a formulation that captures the subdiffusive scale improves our capabilities for conducting industrial-scale simulation of the involved processes with high accuracy. In this paper, we provide a coarse-scale formulation to capture the subgrid-scale phenomena appropriate for large-scale numerical simulations. The paper is organized as follows. First, the fine-scale mathematical model used in this study is presented. Next, the coarse-scale model is described. Then, application of the coarse-scale model is given for a constant pattern reaction front, followed by summary and conclusions.

The exothermic solid-solid reaction is assumed to take place in a
one-dimensional semi-infinite domain. The system, which consists of a mixture
of two solid materials, is initially at temperature

The initial and boundary conditions are then given by

Numerical simulation of reactive front propagation with a large
number of grid cells is computationally expensive and therefore we are interested in using a coarse model that
preserves the accuracy of fine grid behavior. The
coarse-scale variable can be defined by

We intend to represent the differential equations (

In order to find the proportionality constant

The proportionality constant is obtained by the minimization of the
numerical error given by (

Based on the scaling groups used in nondimensionalizing the
governing equations, the frontal behavior of a solid-solid reaction depends on
two parameters, namely,

Dimensionless parameters for solid-solid reacting systems used in this study.

Reaction | |||||
---|---|---|---|---|---|

1 | 0.1 | 0.2 | 0 | 0.2 | |

2 | 0.0645 | 0.14 | 0 | 0.218 |

Coarse-scale
parameter

Figure

Reaction rate for the two reactions as a function of dimensionless distance.

Concentration and temperature
distributions at

Concentration and temperature
distributions at

The calculated numerical errors for the reactions given in Table

Numerical error in temperature (left) and
concentration (right) distributions for reactions given in Table

Ratio of coarse to fine grid (reference)
CPU times versus dimensionless grid size for the two reactions given in Table

Propagation of solid-solid reaction fronts often results
in a thin reaction zone that is difficult to resolve numerically unless a large
number of numerical grid cells are used. Such numerical simulations are
computationally expensive to perform. In this study, a coarse-scale formulation
for numerical modeling of a one-dimensional solid-solid reacting system is
presented. The presented formulation is based on a Taylor
series expansion of the reaction term
and presents the modification of the reaction term in the coarse model that
would allow an accurate solution. A key parameter in this formulation is

Heat capacity,

Concentration, kg/m^{3}

Molecular diffusion
coefficient, m^{2}/s

Activation energy,

Heat of reaction, J/kg

Pre-exponential rate constant,

Gas constant,

Time, s

Temperature, K

Coordinate, m.

Coarse-scale parameter

Inverse of dimensionless activation energy

Inverse of dimensionless heat of reaction

Numerical error

Coarse-scale volume, m^{3}

Fine-scale variable can be temperature, concentration, or reaction rate

Reaction rate,

Effective thermal conductivity,

Density,

Dimensionless temperature.

Adiabatic

Critical

Dimensionless

Ignition

Initial value

Scale value.

Deviation from coarse scale

Average or coarse scale.

The authors would like to acknowledge constructive comments from Dr. Brian Wood and Dr. Christof Meile. Helpful discussion with Dr. Mohsen Sadeghi is also acknowledged. The financial support of the Alberta Ingenuity Centre for In Situ Energy (AICISE) is acknowledged. The authors would like to thank the reviewers for useful comments.