# Gaseous mass transport in porous media through a stagnant gas

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72 Ind. Eng. Chem. Res. 1987, 26, 72-77

xIw, xZw = as above, in water phase XAp, Xsp = adsorbed mass of AN and ST, respectively, per

unit mass of polymer, kg/kg of polymer XAW, XsW = dissolved mass of AN and ST, respectively, per

unit mass of water, kg/kg of water YA = p a / ( p A + p s ) , relative mole fraction of AN in the vapor

phase, mol of AN/mol of monomers 2 = global mass-fractional conversion ZAN = limiting conversion of AN in polymerization runs

(Figure 4) Zr, = limit mass-fractional conversion Greek Symbols u, /3,6 = liquid-liquid distribution coefficients of AN, ST, and

u, p = adsorption equilibrium constants of AN and ST, mass

ylo, y20, yH2O0 = activity coefficients of AN, ST, and H20,

yIw, yZw, T~~~~ = activity coefficients of AN, ST, and H20, 7, t , u, = in Kelen-Tudos method, defined in the text Superscripts o = organic phase w = aqueous phase sat = saturation P = polymer Subscripts 1, A = acrylonitrile 2, S = styrene

H,O, respectively, mass-fraction basis, eq 5-7

basis, eq 21 and 22

respectively, in organic phase

respectively, in water phase

L = limii min, max = minimum and maximum values in Kelen-Tudos

method Registry No. (AN)(ST)(copolymer), 9003-54-7; ST, 100-42-5;

AN, 107-13-1.

Literature Cited Abbey, K. J. ACS Symp. Ser. 1981, 165, 345. Alonso, M.; Oliveres, M.; Puigjaner, L.; Recasens, F. Proceedings of

the 3rd Congress of International Information on Genie Chimique; Society of Chimie Industrielle: Paris, 1983; Vol. 1, p 47-1.

Ballard, M. J.; Napper, D. H.; Gilbert, R. G. J . Polym. Sci., Polym. Chem. Ed. 1981, 19, 934.

Comberbach, D. M.; Scharer, J. M.; Young, M.-Y. Chromatagr. Newsl. 1984, 12, 4.

Elias, H.-G. Macromolecules; Wiley: New York, 1977; Vol. 2. Fineman, M.; Ross, S. D. J . Polym. Sci. 1950, 5, 259. Gardon, J. L. Symp. Polym. React. Eng. 1972, 137. Guillot, J.; Rios, L. Makromol. Chem. 1982, 183, 1979. Guyot, A.; Guillot, J.; Pichot, C.; Rios, L. ACS Symp. Ser. 1981, 165,

Hanna, R. J. Ind. Eng. Chem. 1957, 49(2), 208. Haskell, V. C.; Settlage, P. H. Presented a t the AIChE Annual

Hatate, Y.; Nakashio, F.; Sakai, W. J . Chem. Eng. Jpn. 1971,4, 348. Hendy, B. N. Adv. Chem. Ser. 1975, 142, 115. Johnson, A. F.; Khaligh, B.; Ramsay, J. Presented at the National

Meeting of the American Chemical Society, New York, Aug 1981. Kelen, T.; Tudos, F. J . Macromol. Sci., Chem. 1975, A9, 1. Kikuta, T.; Omi, S.; Kubota, H. J . Chem. Eng. Jpn. 1976, 9, 64. Kiparissides, C.; McGregor, J. F.; Hamielec, A. E. Can. J . Chem.

Kolb, B. Applied Headspace Gas-Chromatography; Heyden: Lon-

Min, K. W.; Ray, W. H. J . Macromol. Sci., Chem. 1974, C I l , 177. Ray, W. H.; Gall, C. E. Macromolecules 1969, 2, 425. Redlich, 0.; Kister, A. T. Ind. Eng. Chem. 1948, 40, 345. Schork, F. J.; Ray, W. H. ACS Symp. Ser. 1981, 165, 505. Shaw, D. A.; Anderson, T. F. Ind. Eng. Chem. Fundam. 1983,22,79. Smith, W. V. J . Am. Chem. Sac. 1948, 70, 2177. Tirrell, M.; Gromley, K. Chem. Eng. Sci. 1981, 36, 367.

415.

Meeting Chicago, 1970.

Eng. 1980, 58, 48.

don, 1980.

Received for review April 22, 1985 Revised manuscript received January 21, 1986

Accepted March 13, 1986

Gaseous Mass Transport in Porous Media through a Stagnant Gas

Ali Una1 Department of Metallurgy and Materials Science, Imperial College of Science and Technology, London, SW7 2BP England

An analysis is presented for gas-phase mass transport in porous media through a stagnant gas. The natural flux ratio rule for isobaric conditions, which requires that the fluxes be inversely proportional to the square roots of the molecular weights, is not met in this system. Accordingly, a pressure difference develops and this superimposes viscous flow upon diffusion. Equations are derived for the flux and the pressure difference, and it is shown that even when the pressure difference is small, the contribution of viscous flow to the flux can be considerable. Flux and pressure difference measurements carried out on the transport of carbon dioxide through stagnant nitrogen in a septum of coarse pores (average pore size 2.5 pm) are in agreement, within experimental error, with those predicted by the equations and hence provide support for the theories available for describing combined transport in porous media.

Gas transport in porous media is an important factor in designing isotope separation equipment and nuclear reactors, in predicting reaction rates in catalysts and in gas and solid reactions, and in drying of porous solids. Accordingly, this field has received a considerable amount of attention over the past 2 decades, and the main prob- lems appear to have been sorted out. The dusty gas model developed by Mason and Malinauskas (1983) has been instrumental in the attainment of the present level of understanding.

In transport under isobaric conditions (is., diffusion only), the fluxes of the components in a mixture are not

0S88-5SS5/81/2626-0072~0~.50/0

independent; they are related by

CNi(Mi)1/2 = 0 (1)

Nl/N, = -t(M2/M1)12 (2)

which, for binary mixture, becomes

This relationship, first reported by Graham (1833), has been verified experimentally, and its applicability is not restricted to the Knudsen diffusion regime; it applies also in the transition and mutual diffusion regimes. Indeed, deviations from this ratio are frequently regarded as prime evidence of surface effects in mass transport.

0 1987 American Chemical Society

Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 73

The isobaric flux ratio of eq 1 and 2 is exact, provided that the mean free path is much larger than the diameter of the pores, for then Knudsen diffusion mechanism will be operating and the diffusivity is inversely proportional to the square root of the molecular weight. In the mutual diffusion regime, the ratio holds for a different reason: the total momentum transferred to the walls, which is pro- portional to Ni and the mean molecular momentum (re- lated to the square root of the molecular weight), by all the molecular collisions must be zero for isobaric condi- tions. The ratio is approximate in this case, however, but as discussed by Mason and Malinauskas (1983), this is a good approximation. Hence, the flux ratio equations are applicable to gaseous diffusion in porous media over all pressures and pore sizes. When the ratio of fluxes is fixed externally, as is the case in chemical reactions or in drying, the system itself creates pressure gradients to satisfy the requirement of zero net momentum transfer to the walls of the pores. Gas transport in such cases then involves both diffusion and viscous flow. Flux equations for de- scribing combined transport were derived by Mason and Malinauskas (1983) and Gunn and King (1969), and the total flux was shown to be simply the sum of viscous and diffusive fluxes.

The general flux equations have been used to account for the pressure difference observed in equimolar coun- tercurrent transport (Jackson, 1977) and have provided good agreement with measurements. McGreavy and Asaeda (1982) applied them to unsteady-state diffusion in monodisperse porous solids and showed that diffusion fluxes can induce total pressure gradients. These authors used a nonisobaric model to interpret their data and showed that the general flux equations applied well. Mason and Malinauskas (1983) have reviewed the various applications of the equations.

In the present work, the transport of one gas (1) through a stagnant second gas (2) was investigated. Equations were derived for the created pressure difference and for calcu- lating the flux. These equations were tested by experi- mental measurements of the flux and the pressure dif- ference for the transport of carbon dioxide through stag- nant nitrogen in a porous septum of monodisperse pores (average pore size 2.5 pm). Reasonable agreement with theory was obtained when the values of the Knudsen diffusivity and the viscous flow parameter, determined separately from permeability measurements, were used in the calculations. The present system is of relevance to the drying of porous solids and calcination reactions and has been used in some previous work for measuring diffusiv- ities (Turkdogan et al., 1971; Unal, 1986). It also provides an effective and simple steady-state test for the general flux equations.

Theory Pressure Gradient. The flux equation for combined

diffusion and flow of a binary gas mixture may be written as

By making use of the identities P1 = X l P P2 = x,P x1 + x2 = 1

and noting that N2 = 0 for the transport of species 1 through stagnant gas 2, we obtain the following equation between the gradients of composition and pressure when

1 ,.' / U L

Figure 1. Variation of the pressure difference created in diffusion through a stagnant gas with the mean pressure. T = 293 K, x,(O) = 0.2, xl(L) = 0, p = 15 X 10" Paes, Bol2 = 1.5 N d, and M = 28 kg kmol-I.

N, is eliminated between eq 3 and the corresponding equation for species 2,

where x1 x2 (Dm)-l = - + -

DK1 DK2

and p refers to the viscosity of the mixture. Assuming DM and p do not vary greatly across the porous septum and may be represented by suitable average values, eq 4 may be integrated to obtain

(6)

This equation predicts a negative pressure difference; that is, the pressure gradient (resulting from the composition gradient imposed) acts in such a way as to stop the transport of gas 2 that would be obtained under isobaric conditions.

Curves plotted from eq 6 for four porous media with pore sizes of 0.1, 0.5, 1, and 3 pm are shown in Figure 1. In these calculations, the composition dependencies of p and D K A were neglected (Le., DK1 - DKA). For viscosity a value of 15 X lo4 Pas and Bol2 = 1.5 N s-l for the mutual diffusion coefficient were taken, and it was as- sumed that the same value of the tortuosity factor, q, could be used to obtain the effective Knudsen diffusivity, the effective binary diffusivity, and the viscous-flow parameter (see the Discussion section). Porous media parameters were calculated from

74 Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987

It is seen that AP goes through a maximum. This occurs at

For large pores this maximum is at low pressures and, hence, is not of much industrial significance. But for small pores the maximum appears a t higher pressures and for the typical values used in the calculations, it is a t 1 bar for a pore radius of 0.134 pm. The maximum pressure difference is given by

-+ -+ - P m a x PDKZ 012

and in the case of very fine pores and large composition gradients it can be of the same order of magnitude as the average system pressure, as shown in Figure 1.

Flux Equation. To obtain an expression for the flux of species 1, the following equation is obtained by adding eq 3 and the corresponding equation for species 2 and noting that N2 = 0

Then, dPldz is eliminated between eq 4 and 12, and in- tegration gives N,RTL =

In the integrated flux, eq 13, and eq 6, p and DKA are average values obtained in the system and will have to be chosen carefully. When the variations of these terms across the septum are great, the differential form of eq 13 may be used and integration carried out by numerical methods. However, for gases of similar molecular weight and vis- cosity, this equation reduces to

where the term in the second brackets is the Bosanquet (1944) extrapolation expression for effective diffusivity in the intermediate diffusion regime. For !he limiting case of large pores or high pressures (i.e., B$/pDKA >> 1 and DK1 >> Dl2) the flux equation (13) simplifies to

NIRTL = DlZF In (15)

and for the other limiting case of fine pores or low pres- sures,

The limiting case flux equations (15) and (16) may now be compared with the isobaric flux equations for limiting cases, which may be obtained from eq 13 by setting dP/dz = 0 and integrating. These are

(17) NlRTL = -DK~P[x~(L) - ~1(0)1

5 0 1 LO L

01 02 U 06 08 10 20 LO 60 83 13 P o r e rodtus e r r

Figure 2. Pore structure of the sample.

for the Knudsen diffusion regime ( D K ~

Ind. Eng. Chem. Res. Vol. 26, No. 1, 1987 75

Table I. Permeability Coefficient ( K ) of the Sample at Various Total Pressures (P) (Gas: Ha, Paos, T = 293 K)

= 8.885 X 10

P, Pa x io5 K , m2 s-l x P, Pa x io5 K , m2 s-l x 0.060 3.05 0.426 6.25 0.127 3.55 0.640 8.45 0.20 4.10 0.855 10.65 0.333 5.65 1.0 11.65

I I ? ? . t t ? / X I I P

LO*+ N2 I

( 0 1 I bl

Figure 3. Diffusion cell (a) and composition and pressure gradients (b).

pressure head, AP, was maintained across the sample, and from the measurement of flux by means of a capillary flow meter, the permeability coefficient, K, was calculated using the expression

(20) AP NRT = K- L

For higher accuracy, the permeability tests were carried out with pure H2 gas at mean pressures from 0.06 to 1 X lo5 Pa. The results obtained are given in Table I.

Diffusion Experiments. The flux of carbon dioxide through the sample septum was measured in the presence of stagnant nitrogen. The driving force for transport was obtained by absorbing dioxide onto sofnolite (soda lime) after it diffused through the sample. The flux was de- termined from the weight gained due to absorption.

The sample was mounted at one end of a cylindrical diffusion cell made from brass tubing of 16-mm bore and 25-mm height, Figure 3a. The other end was sealed, and the cell was partly filled with sofnolite. During experi- ments, the sample was suspended from a load cell (accurate to f 1 mg) and was placed in a tube (50-mm bore) through which the desired mixture of C02 and N2 was passed after the system was thoroughly flushed with pure N2. The rate of gas flow was 2 L/min (NTP). C02 in the gas mixture diffused through the sample and was absorbed onto the sofnolite which caused an increase in the weight of the cell. This increase in weight was a linear function of time after the initial transient period and was used to calculate the flux of cop

To measure the pressure difference between the outside surface of the sample and the interior of the diffusion cell, a plastic tubing of 2-mm bore and 50-mm length was at- tached to the hole on the lid of the cell. Separate exper- iments had to be carried out for measuing the pressure difference for which a differential micromanometer (ac- curate to f 0.5 Pa) was used. The results obtained are shown in Table 11.

Examination of the sofnolite after the tests indicated that only the first layer of the particles had partly reacted, i.e., Pco,(L) 0, and no significant temperature rise oc- curred as a result of the exothermic nature of this fast absorption reaction.

Table 11. Flux and Pressure Difference across the Sample in the Transport of Carbon Dioxide through Stagnant Nitrogen (T = 293 K) xi(b) N,RTL. N 5-l X lo-* -AP. Pa 0.30 0.50 0.70 0.75 0.80 0.85

4.15 8.41

12.66 14.49 16.21 17.53

54.7 132.5 194.5 208.7 243.2 265.5

Discussion Effective Knudsen Diffusivities and Viscous Flow

Parameter. These can be calculated from the results of the permea...

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