Mass transfer in porous media with heterogeneous chemical reaction

Mass transfer in porous media with heterogeneous chemical reaction

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

^IDepartamento 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: [e-mail]

^IIDepartment of Chemical Engineering and Material Science, University of California at Davis, Phone (+1) [phone], Fax (+1) [phone]CA 95616, U.S.A. E-mail: [e-mail]

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_Ab 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_band 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

Received: October 5, 2001

Accepted: December 2, 2002

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Publication Dates

• Publication in this collection
  25 June 2003
• Date of issue
  June 2003

History

• Accepted
  02 Dec 2002
• Received
  05 Oct 2001