7.6. DIFFUSION OF GASES IN POROUS SOLIDS AND CAPILLARIES

7.6A. Introduction

In Section 6.5C diffusion in porous solids that depends on structure was discussed for liquids and gases. For gases it was assumed that the pores were very large and Fickian-type diffusion occurred. However, the pores are often small in diameter and the mechanism of diffusion is basically changed.

Diffusion of gases in small pores frequently occurs in heterogeneous catalysis where gases diffuse through very small pores to react on the surface of the catalyst. In freeze-drying of foods such as turkey meat, gaseous H2O diffuses through very fine pores of the porous structure.

Since the pores or capillaries of porous solids are often small, the diffusion of gases may depend upon the diameter of the pores. We first define a mean free path λ, which is the average distance a gas molecule travels before it collides with another gas molecule:

Equation 7.6-1


where λ is in m, μ is viscosity in Pa · s, P is pressure in N/m2, T is temperature in K, M = molecular weight in kg/kg mol, and R = 8.3143 × 103 N · m/kg mol · K. Note that low pressures give large values of λ. For liquids, since λ is so small, diffusion follows Fick's law.

In the next sections we shall consider what happens to the basic mechanisms of diffusion in gases as the relative value of the mean free path compared to the pore diameter varies. The total pressure P in the system will be constant, but partial pressures of A and B may be different.

7.6B. Knudsen Diffusion of Gases

In Fig. 7.6-1a a gas molecule A at partial pressure pA1 at the entrance to a capillary is diffusing through the capillary having a diameter d m. The total pressure P is constant throughout. The mean free path λ is large compared to the diameter d. As a result, the molecule collides with the wall and molecule–wall collisions are important. This type of diffusion is called Knudsen diffusion.

Figure 7.6-1. Types of diffusion of gases in small capillary tubes: (a) Knudsen gas diffusion, (b) molecular or Fick's gas diffusion, (c) transition gas diffusion.


The Knudsen diffusivity is independent of pressure P and is calculated from

Equation 7.6-2


where DKA is diffusivity in m2/s, is average pore radius in m, and is the average molecular velocity for component A in m/s. Using the kinetic theory of gases to evaluate the final equation for DKA is

Equation 7.6-3


where MA is molecular weight of A in kg/kg mol and T is temperature in K.

EXAMPLE 7.6-1. Knudsen Diffusivity of Hydrogen

A H2(A)—C2H6(B) gas mixture is diffusing in a pore of a nickel catalyst used for hydrogenation at 1.01325 × 105 Pa pressure and 373 K. The pore radius is 60 Å (angstrom). Calculate the Knudsen diffusivity DKA of H2.

Solution: Substituting into Eq. (7.6-3) for , MA = 2.016, and T = 373 K,


The flux equation for Knudsen diffusion in a pore is

Equation 7.6-4



Integrating between z1 = 0, pA = pA1 and z2 = L, pA = pA2,

Equation 7.6-5


The diffusion of A for Knudsen diffusion is completely independent of B, since A collides with the walls of the pore and not with B. A similar equation can be written for component B.

When the Knudsen number NKn defined as

Equation 7.6-6


is ≥10/1, the diffusion is primarily Knudsen, and Eq. (7.6-5) predicts the flux to within about a 10% error. As NKn gets larger, this error decreases, since the diffusion approaches the Knudsen type.

7.6C. Molecular Diffusion of Gases

As shown in Fig. 7.6-1b, when the mean free path λ is small compared to the pore diameter d or where NKn ≤ 1/100, molecule–molecule collisions predominate and molecule–wall collisions are few. Ordinary molecular or Fickian diffusion holds and Fick's law predicts the diffusion to within about 10%. The error diminishes as NKn gets smaller since the diffusion approaches more closely the Fickian type.

The equation for molecular diffusion given in previous sections is

Equation 7.6-7


A flux ratio factor α can be defined as

Equation 7.6-8


Combining Eqs. (7.6-7) and (7.6-8) and integrating for a path length of L cm,

Equation 7.6-9


If the diffusion is equimolar, NA = −NB and Eq. (7.6-7) becomes Fick's law. The molecular diffusivity DAB is inversely proportional to the total pressure P.

7.6D. Transition-Region Diffusion of Gases

As shown in Fig. 7.6-1c, when the mean free path λ and pore diameter are intermediate in size between the two limits given for Knudsen and molecular diffusion, transition-type diffusion occurs, where molecule–molecule and molecule–wall collisions are important in diffusion.

The transition-region diffusion equation can be derived by adding the momentum loss due to molecule–wall collisions in Eq. (7.6-4) and that due to molecule–molecule collisions in Eq. (7.6-7) on a slice of capillary. No chemical reactions are occurring. The final differential equation is (G1)

Equation 7.6-10


where

Equation 7.6-11


This transition region diffusivity DNA depends slightly on concentration xA.

Integrating Eq. (7.6-10),

Equation 7.6-12


This equation has been shown experimentally to be valid over the entire transition region (R1). It reduces to the Knudsen equation at low pressures and to the molecular diffusion equation at high pressures. An equation similar to Eq. (7.6-12) can also be written for component B.

The term DAB/DKA is proportional to 1/P. Hence, as the total pressure P increases, the term DAB/DKA becomes very small and NA in Eq. (7.6-12) becomes independent of total pressure, since DABP is independent of P. At low total pressures Eq. (7.6-12) becomes the Knudsen diffusion equation (7.6-5), and the flux NA becomes directly proportional to P for constant xA1 and xA2.

This is illustrated, in Fig. 7.6-2, for a fixed capillary diameter where the flux increases as total pressure increases and then levels off at high pressure. The relative position of the curve depends, of course, on the capillary diameter and the molecular and Knudsen diffusivities. Using only a smaller diameter, DKA would be smaller, and the Knudsen flux line would be parallel to the existing line at low pressures. At high pressures the flux line would asymptotically approach the existing horizontal line, since molecular diffusion is independent of capillary diameter.

Figure 7.6-2. Effect of total pressure P on the diffusion flux NA in the transition region.


If A is diffusing in a catalytic pore and reacts at the surface at the end of the pore so that AB, then at steady state, equimolar counterdiffusion occurs, or NA = −NB. Then from Eq. (7.6-8), α = 1 − 1 = 0. The effective diffusivity DNA from Eq. (7.6-11) becomes

Equation 7.6-13


The diffusivity is then independent of concentration and is constant. Integration of (7.6-10) then gives

Equation 7.6-14


This simplified diffusivity is often used in diffusion in porous catalysts even when equimolar counterdiffusion is not occurring. It greatly simplifies the equations for diffusion and reaction to use this simplified diffusivity.

An alternative simplified diffusivity can be obtained by using an average value of xA in Eq. (7.6-11), to give

Equation 7.6-15


where xAav = (xA1 + xA2)/2. This diffusivity is more accurate than . Integration of Eq. (7.6-10) gives

Equation 7.6-16


7.6E. Flux Ratios for Diffusion of Gases in Capillaries

1. Diffusion in open system

If diffusion in porous solids or channels with no chemical reaction is occurring where the total pressure P remains constant, then for an open binary counterdiffusing system, the ratio of NA/NB is constant in all of the three diffusion regimes and is (G1)

Equation 7.6-17


Hence,

Equation 7.6-18


In this case, gas flows past the two open ends of the system. However, when chemical reaction occurs, stoichiometry determines the ratio NB/NA, and not Eq. (7.6-17).

2. Diffusion in closed system

When molecular diffusion is occurring in a closed system, shown in Fig. 6.2-1, at constant total pressure P, equimolar counterdiffusion occurs.

EXAMPLE 7.6-2. Transition-Region Diffusion of He and N2

A gas mixture at a total pressure of 0.10 atm abs and 298 K is composed of N2 (A) and He (B). The mixture is diffusing through an open capillary 0.010 m long having a diameter of 5 × 106 m. The mole fraction of N2 at one end is xA1 = 0.8 and at the other end is xA2 = 0.2. The molecular diffusivity DAB is 6.98 × 105 m2/s at 1 atm, which is an average value determined by several investigators.

  1. Calculate the flux NA at steady state.

  2. Use the approximate equations (7.6-14) and (7.6-16) for this case.

Solution: The given values are T = 273 + 25 = 298 K, = 5 × 106/2 = 2.5 × 106 m, L = 0.01 m, P = 0.1(1.01325 × 105) = 1.013 × 104 Pa, xA1 = 0.8, xA2 = 0.2, DAB = 6.98 × 105 m2/s at 1 atm. Other values needed are MA = 28.02 kg/kg mol, MB = 4.003.

The molecular diffusivity at 0.1 atm is DAB = 6.98 × 105/0.1 = 6.98 × 104 m2/s. Substituting into Eq. (7.6-3) for the Knudsen diffusivity,


From Eq. (7.6-17),


From Eq. (7.6-8),


Substituting into Eq. (7.6-12) for part (a),


For part (b), the approximate equation (7.6-13) is used:


Substituting into Eq. (7.6-14), the approximate flux is


Hence, the calculated flux is approximately 40% high when using the approximation of equimolar counterdiffusion (α = 0).

The more accurate approximate equation (7.6-15) is used next. The average concentration is xAav = (xA1 + xA2)/2 = (0.8 + 0.2)/2 = 0.50.


Substituting into Eq. (7.6-16),


In this case the flux is only −1.1% low.


7.6F. Diffusion of Gases in Porous Solids

In actual diffusion in porous solids, the pores are not straight and cylindrical but irregular. Hence, the equations for diffusion in pores must be modified somewhat for actual porous solids. The problem is further complicated by the fact that the pore diameters vary and the Knudsen diffusivity is a function of pore diameter.

As a result of these complications, investigators often measure effective diffusivities DA eff in porous media, where

Equation 7.6-19


If a tortuosity factor τ is used to correct the length L in Eq. (7.6-16), and the right-hand side is multiplied by the void fraction ε, Eq. (7.6-16) becomes

Equation 7.6-20


Comparing Eqs. (7.6-19) and (7.6-20),

Equation 7.6-21


In some cases investigators measure DA eff but use instead of the more accurate in Eq. (7.6-21).

Experimental data (G4, S2, S6) show that τ varies from about 1.5 to over 10. A reasonable range for many commercial porous solids is about 2–6 (S2). If the porous solid consists of a bidispersed system of micropores and macropores instead of a monodispersed pore system, the above approach should be modified (C4, S6).

Discussions and references for diffusion in porous inorganic-type solids, organic solids, and freeze-dried foods such as meat and fruit are given elsewhere (S2, S6).

Another type of diffusion that may occur is surface diffusion. When a molecular layer of adsorption occurs on the solid, the molecules can migrate on the surface. Details are given elsewhere (S2, S6).

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