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## Brazilian Journal of Chemical Engineering

##
*Print version* ISSN 0104-6632*On-line version* ISSN 1678-4383

### Braz. J. Chem. Eng. vol.20 no.2 São Paulo Apr./June 2003

#### https://doi.org/10.1590/S0104-66322003000200013

**Mass transfer in porous media with heterogeneous chemical reaction**

**S.M.A.G.Ulson de Souza ^{I}; S.Whitaker^{II}**

^{I}Departamento de Engenharia Química e Engenharia de Alimentos, Universidade Federal de Santa Catarina, Phone (+55) (48) 331-9448 - R.216, Fax (+55) (48) 331-9687, Cx. P. 476, 88040-900 - Florianópolis - SC, Brazil. E-mail: selene@enq.ufsc.br

^{II}Department of Chemical Engineering and Material Science, University of California at Davis, Phone (+1) (916) 752-8775, Fax (+1) (916) 752-1031CA 95616, U.S.A. E-mail: swhitaker@ucdavis.edu

**ABSTRACT**

In this paper, the modeling of the mass transfer process in packed-bed reactors is presented and takes into account dispersion in the main fluid phase, internal diffusion of the reactant in the pores of the catalyst, and surface reaction inside the catalyst. The method of volume averaging is applied to obtain the governing equation for use on a small scale. The local mass equilibrium is assumed for obtaining the one-equation model for use on a large scale. The closure problems are developed subject to the length-scale constraints and the model of a spatially periodic porous medium. The expressions for effective diffusivity, hydrodynamic dispersion, total dispersion and the Darcy's law permeability tensors are presented. Solution of the set of final equations permits the variations of velocity and concentration of the chemical species along the packed-bed reactors to be obtained.

**Keywords: **mass transfer, porous media, modeling.

**INTRODUCTION**

A packed-bed reactor is illustrated in Figure 1, where the w region represents the porous catalyst pellets and the h region represents the main fluid phase. In order to develop spatially smoothed equations for the transport of chemical species in the porous medium, illustrated in Figure 1, it is necessary to use the large-scale averaging volume, V¥ .

Knowledge of the transport equations within the w and h regions is required. The w region contains two phases: the g phase (the fluid phase inside the w region) and the k phase (solid). To understand the transport process taking place within the w region, transport equation for the g phase must be developed and this requires the use of the small-scale averaging volume, V.

**SMALL-SCALE AVERAGING**

The boundary value problem for the concentration of a chemical species, A, in the g phase, illustrated in Figure 1, with a heterogeneous chemical reaction, can be expressed as

_{}

in which C_{Ag} is the molar concentration of chemical species under consideration, k is the heterogeneous reaction rate constant, and D_{g} is the g -phase molecular diffusivity of species A. Here variable Age is used to represent the entrances and exits of the g phase at the boundary of the w region. Variable A_{gk} is used to represent the entire interfacial area within that region.

When the boundary value problem given by equations (1) through (4) is solved, the C_{Ag} concentration can be determined as a function of position and time. For design purposes, it is sufficient to determine the averaged concentration associated with the averaging volume, V.

The spatial averaging theorem (Howes and Whitaker, 1985) for the g - k system can be expressed as

in which A_{gk} represents the area of g-k interface contained within V. Application of the spatial averaging theorem (Whitaker, 1999) in equations (1) through (4) results in

The integral that appears in equation (6) is called the spatial deviation filter, because it acts as a filter, which allows some information to pass from the original point equation, and boundary condition, to local volume-averaged transport equation written for the intrinsic averaged concentration.

To obtain a closed form of equation (6), a representation for the spatial deviation concentration must be developed.

Vector b_{g} and scalar s_{g} are known as the closure variables and y_{g} is an arbitrary function.

The closure variables can be determined by the following boundary value problems:

**Problem I:**

**Problem II:**

When b_{g and sg} are determined by equations (8) and (9), it can be shown that

**Problem III:**

Here l_{i} represents the three non-unique lattice vectors that are needed to describe a spatially periodic medium.

The solution of Problem III is y= constant. Since this constant will not pass through the filter, the value of y plays no role in the closed form of the volume-averaged diffusion equation.

The closed form of the volume-averaged diffusion equation can be given by

The effective diffusivity tensor is defined by

and the dimensionless vector associated with the chemical reaction is defined by

The use of these two definitions in equation (11) results in

The second term on the right side of equation (14) represents the convective transport term that is generated by the heterogeneous reaction. This term is negligible for the case of diffusion in porous catalysts. Ryan (1983) has demonstrated that this term is equal to zero for the symmetric unit cells. Under these circumstances, equation (14) simplifies to

**LARGE-SCALE AVERAGING**

The w and h regions associated with the averaging volume, V_{¥} , are shown in Figure 1. The length scales associated with the w and h regions are designed by l_{w} and l_{h} and the volume fractions of these regions are identified by j _{w} and j_{h}, respectively. The boundary value problem that forms the basis for large-scale averaging is given by

where C_{A}_{b} is the b-phase molar concentration of chemical species under consideration, D_{b} is the b-phase molecular diffusivity of species A, and **v**_{b} is the velocity vector of the b phase. The h region contains only the b phase, which is the same as the g phase.

The boundary conditions given by equations (17) and (18) are based on the idea that the local volume-averaged concentration and flux are continuous at the h-w boundary.

The volume-averaged form of equation (16) is given by

The integral that appears inside the diffusion term is referred to as a spatial deviation filter.

The volume-averaged form of equation (19) can be expressed as

**THE ONE-EQUATION MODEL**

The one-equation model is based on the assumption that the mass transport process can be characterized by a single concentration. This assumption is valid when the system is in a state of local mass equilibrium (Whitaker, 1986a-b, 1991). If the size of porous particles is very small or the effective diffusivity is large, the local mass equilibrium condition can occur (Quintard and Whitaker, 1993, 1994a-e).

The intrinsic spatial-averaged concentration, given by

can be the proper single concentration to characterize the mass transfer process.

Under the local mass equilibrium condition, the following equation can obtained:

where <e> is the average porosity given by

The effective diffusivity tensor for the w–h system,D_{eff} , can be expressed as

_{}

To obtain a closed form of this equation, it is necessary to represent the spatial deviation concentration in the w region and h region as functions of the dependent variable. As the two source terms in the closure problem are proportional to the gradient of the intrinsic spatial-averaged concentration, the following can be written:

in which b_{h} and b_{w} represent the closure variables to obtain

**Problem I:**

It can be proved that y = x = constant, as discussed by Whitaker (1999). These constant values will have no influence on the closed form of the macroscopic equation, since they will not pass through the filter.

Equations (28) and (29) can be introduced into equation (27) to obtain the following closed form:

Here the effective diffusivity tensor for the w–h system is defined by

The hydrodynamic dispersion tensor can be defined by

The total dispersion tensor (Eidsath et al., 1983; Han et al., 1985) can be defined according to

The use of equations (36) through (38) in equation (35) results in

**DARCY'S LAW**

The physical process under consideration is that of a single-phase, incompressible flow in a rigid porous medium, such as the w-h system illustrated in Figure 1.

The boundary value problem is given by

Here p_{b} represents the total pressure in the b phase, while m_{b} and r_{b} represent the viscosity and density of the b phase, respectively. represents the h region entrances and exits.

The regional average in the h region of Stokes's equation can be expressed as

The third term on the right side of equation (44) is known as the Brinkman correction. This term is unimportant for a flow in homogeneous porous medium as discussed by Whitaker (1986c). In order to develop the closed form of equation (44), it is necessary to solve the following boundary value problem for the closure problem:

As the single source appears in this boundary value problem, a solution for the spatial deviation velocity and pressure can be proposed as

Here B, b, y, and x are the closure variables.

**Problem I:**

It can be shown that y is zero and x is a constant. Thus the constant will not pass through the filter. Under these circumstances, it is possible to obtain

and

Here K is the permeability tensor.

Equation (57) can be rearranged to give

and this is known as Darcy's law with the Brinkman correction.

For a homogeneous porous medium, the Brinkman correction is negligible compared to the pressure term, and in this case,

Darcy discovered experimentally this result in a one-dimensional version in 1850 (Whitaker, 1999).

Whitaker (1996, 1997) has presented a theoretical development for the Forchheimer equation, given by

where F is the Forchheimer correction tensor. When the Reynolds number is smaller than one, the closure problem can be used to prove that F is a linear function of velocity and an order of magnitude analysis suggests that this linear dependence may persist for a wide range of Reynolds numbers.

**CONCLUSIONS**

In this work the modeling of the mass transfer process in catalytic packed-bed reactors is presented. The volume-averaged form of the governing equations is obtained for the small and large scales. The one-equation model is used to describe the concentration inside the reactor. This model is associated with the condition of local mass equilibrium. The closure problems are developed subject to the length-scale constraints and the model of a spatially periodic porous medium. The expressions for effective diffusivity, hydrodynamic dispersion, total dispersion and the Darcy's law permeability tensors are presented. Solution of the set of final equations permits the variations of velocity and concentration of the chemical species along the packed-bed reactors to be obtained. This information is essential for the design of packed-bed reactors.

**ACKNOWLEDGEMENTS**

This work was developed while Selene Maria de Arruda Guelli Ulson de Souza was doing her postdoctoral research in the Department of Chemical Engineering and Material Science at the University of California, Davis and was supported by CAPES – Fundação Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – Brazil. This source of support is greatly appreciated.

**NOMENCLATURE**

**REFERENCES**

Eidsath, A., Carbonell, R. G., Whitaker, S. and Herrmann, L. R., Dispersion in Pulsed Systems III: Comparison between Theory and Experiments for Packed Beds, Chemical Engineering Science, 38, 1803-1816 (1983). [ Links ]

Han, N-W, Bhakta, J. and Carbonell, R. G., Longitudinal and Lateral Dispersion in Packed Beds: Effects of Column Length and Particle Size Distribution, A.I.Ch.E.Journal, 31, 277-288 (1985). [ Links ]

Howes, F. A. and Whitaker, S., The Spatial Averaging Theorem Revisited, Chemical Engineering Science, 40, 1387-1392 (1985). [ Links ]

Quintard, M. and Whitaker, S., One and Two-equation Models for Transient Diffusion Processes in Two-phase Systems, in Advances in Heat Transfer, 23, 369-465, Academic Press, New York (1993). [ Links ]

Quintard, M. and Whitaker, S., Transport in Ordered and Disordered Porous Media I: The Cellular Average and the Use of Weighting Functions, Transport in Porous Media, 14, 163-177 (1994a). [ Links ]

Quintard, M. and Whitaker, S., Transport in Ordered and Disordered Porous Media II: Generalized Volume Averaging, Transport in Porous Media, 14, 179-206 (1994b). [ Links ]

Quintard, M. and Whitaker, S., Transport in Ordered and Disordered Porous Media III: Closure and Comparison between Theory and Experiment, Transport in Porous Media, 15, 31-49 (1994c). [ Links ]

Quintard, M. and Whitaker, S., Transport in Ordered and Disordered Porous Media IV: Computer Generated Porous Media, Transport in Porous Media, 15, 51-70 (1994d). [ Links ]

Quintard, M. and Whitaker, S., Transport in Ordered and Disordered Porous Media V: Geometrical Results for Two-dimensional Systems, Transport in Porous Media, 15, 183-196 (1994e). [ Links ]

Ryan, D., The Theoretical Determination of Effective Diffusivities for Reactive, Spatially Periodic Porous Media, M.S. Thesis, University of California at Davis (1983). [ Links ]

Whitaker, S., Transport Processes with Heterogeneous Reaction, in S. Whitaker and A. E. Cassano (eds.), Concepts and Design of Chemical Reactors, Ch.1, Gordon and Breach, New York (1986a). [ Links ]

Whitaker, S., Local Thermal Equilibrium: An Application to Packed Bed Catalytic Reactor Design, Chemical Engineering Science, 41, 2029-2039 (1986b). [ Links ]

Whitaker, S., Flow in Porous Media I: A Theoretical Derivation of Darcy´s Law, Transport in Porous Media, 1, 3-25 (1986c). [ Links ]

Whitaker, S., Improved Constraints for the Principle of Local Thermal Equilibrium, Ind. & Eng. Chem., 30, 983-997 (1991). [ Links ]

Whitaker, S., The Forchheimer Equation: A Theoretical Development, Transport in Porous Media, 25, 27-61 (1996). [ Links ]

Whitaker, S., Volume Averaging of Transport Equations, in J. P. Du Plessis (ed.), Fluid Transport in Porous Media, Ch.1, Computational Mechanics Publications, Southampton, United Kingdom (1997). [ Links ]

Whitaker, S., Theory and Application of Transport in Porous Media: The Method of Volume Averaging, London, Kluwer Academic, 219p (1999). [ Links ]

Received: October 5, 2001

Accepted: December 2, 2002