Multicomponent Diffusion of Gases in Porous Solids. Models and Experiments Chester F. Feng,’ Vladimir V. Kostrov,2 and Warren
E. Stewart*
Department of Chemical Engineering, University of Wisconsin, Madison, Wis. 53706
Multicomponent diffusion experiments are reported for the system He-N2-CH4 in y-alumina catalyst pellets, over a wide range of pressures (1-70 atm), temperatures (300-390 K), compositions, and composition gradients. These results provide the first multicomponent test of several pore diffusion models. The data are fitted with standard errors of 2-15% and maximum errors of 12-50%, depending on the model used. Special forms of the models are given for prediction of effective diffusivities in reacting systems.
Introduction Multicomponent diffusion of gases in porous solids is a feature of many chemical processes. Diffusion models for these systems are available but largely untested in multicomponent mixtures. The present work gives an experimental test of several models to assess their reliability in multicomponent systems. After establishing this point, the use of the models to predict effective diffusivities will be described. Theoretical Models The models to be tested here are shown in Table I. All but the last have been compared for binary systems by Feng and Stewart (1973). Each model includes viscous flow, gaseous diffusion, and surface diffusion. For the present experiments the viscous flow term is zero and the gaseous diffusion term is dominant. Model 1 is obtainable by adding a surface diffusion term to the model of Mason and Evans (1969), or that of Gunn and King (1969). This approach, however, leaves the role of pore structure undefined. An alternate method is to apply the Mason-Evans model to each pore; then Model 1 follows for a well-connected network of pores of equal radius (Feng and Stewart, 1973). Models 2, 3, and 3A take the pore size distribution into account by summing the flows with respect to r. Model 2 uses a detailed integration, with a constant tortuosity factor K for all pore sizes as assumed by Johnson and Stewart (1965). Model 3 uses a two-point quadrature and allows for a pore-size-dependent tortuosity K(r). Model 3A is like Model 3 except that pore size data are used to determine two of the constants. Model 4 is based on an expression for G, suggested by Aris (1965). It involves two approximations beyond those of Model 1: (i) use of Wilke’s Method 1 for the bulk diffusivity (Wilke, 1950), thereby equating the velocities of all species but i, and (ii) neglect of the convective term of Wilke’s treatment. Removal of either approximation gives coupled equations for the G,; removal of both gives Model 1. Experimental Method The experimental technique used here is a multicomponent modification of the Wicke-Kallenbach (1941) method. A porous solid is mounted between two different gas streams so that diffusion occurs through the solid. From the measured flow rates and compositions of the streams, the mass flux of each component through the solid can be
’
Present address Chemical Engineering Department, National Taiwan University, Taipei, Taiwan Present address, Department of Inorganic Chemical Engineering, lvanovo Institute of Technology, Ivanovo, USSR
calculated. We used a gas chromatograph to analyze the effluents so that multicomponent measurements could be made. A schematic diagram of‘the apparatus is given in Figure 1. Details are given by Feng (1972). The system consists of a high-pressure section .(for the diffusion process) and an atmospheric pressure section (for flow measurement and gas sampling). The pressures in the diffusion section were controlled by a pressure regulator and needle valve on each entering and leaving stream. The pressure difference across the porous solid was measured with a Pace KP15 pressure transducer and adjusted to zero with the needle valves. The temperature of the diffusion system was controlled with an electrically heated circulating air bath. The flow rates were measured with a soap bubble meter. The two effluent streams were analyzed by gas chromatography, with argon as carrier and molecular sieve 13X (ground to 80-100 mesh) as the column packing. The diffusion cell is shown in Figure 2. The main parts are a cylinder, a flanged piston, and a Teflon disk. The Teflon disk, which holds six pairs of pellets, is clamped between the cylinder and the piston, in which the gas mixing chambers are machined. On each side of the Te-! lon disk, a brass disk with six radial grooves is mounted to direct the flow of gas across the ends of the pellets. To improve the gas mixing and reduce the void space, each chamber is filled with %-in. diameter stainless steel balls (not shown). Catalyst Pellets The catalyst pellets and pore size data were supplied by Dr. Marvin F. L. Johnson of the Atlantic Richfield Co. The pellets were cylindrical, in. in length and diameter, with a particle density of 1.36 g ~ m - The ~ . pore size distribution, shown in Figure 3, was determined from nitrogen adsorption up to r = 500 A and from mercury penetration thereafter as described by Johnson and Stewart (1965).
Test P a t t e r n The test pattern is shown in Figure 4. It is designed to allow significant comparisons of the models, over a wide range of pressure, temperature and terminal compositions. Nine different pairs of inlet gas compositions were used, indicated by the nine lines in the triangular diagram of Figure 5. Figure 3 and Model 2 were used to select pressures distributed over the transition region from free-molecule flow to bulk gaseous diffusion. The experiments may be classified into three types, listed in Table 11, with A, B, or C denoting a pure component and A B or A + C denoting an equimolar mixture. The fluxes were measured with about 2% precision, except
+
Ind. Eng. Chern., Fundam., Vol. 13,No. 1, 1974
5
600
-
D I
VENT
a
-
W
5
U
400-
In In W
Figure 1. Flow diagram of the diffusion apparatus: A, gas cylinders; B, back pressure regulators; C, diffusion cell; D, dryers; G , pressure gauges; H, preheating coils; M, manometer; N, needle valves; P, pressure regulators; Q , pressure transducer; R, rotameters; S, sampling tees; T, constant-temperature bath; V, vacuum pump; Y, three-way valves
-
P
200-
0-
INLET
UPPER CHAMBER
I
/
90
I65
240
TEMPERATURE,
OF
Figure 4. Test pattern. The lines in each triangle denote the inlet composition pairs used from the triangular diagram of Figure 5 Table I. Summary of Models ~
General form
+ [G21
[Ns] = - [ c ]-p dp P dz
(viscous flow) (gaseous diffusion)
(surface diffusion) ~
PINLET
Special formsa for [GI Model 1: [G,] = - W ~ [ F ( r l ) l -d' - I C ] dz
Figure 2. Section view of the diffusion cell
-W,[F(rd I-'&
-
," 3 -
e
"3
Adjustable constants in [GI
d
[cl
-
Model 3A: Like Model 3, but with r1 and r2 determined by e(r1) = 0.677 e ( m ) t(r2)= 0.977 e ( - )
-
Model 4: G,,
=
Wl
{
WbW,
+-
1 X DOIK(rd (1 - xd ~
W1,n
X
0-
I
I
l l l l I l 1
102
IO
I
,
l ~ l l l l l
I
I
I
1
1 1 , s
RADIUS,
h#i
1
104
103
PORE
i
Figure 3. Pore size distribution of the alumina pellets for N A in~ tests of type 111, which was less precise and was not used in fitting models.
The matrix F(r) has the following elements for an ncomponent gas mixture Fij(r)
= (6ij
- 1)
X'
Dij
+ a i j {-DiK1(r) + 21 >} h= a i h hfi
(i
Model Fitting Each model of Table I was fitted to 283 observations of In Niz, with and without the surface diffusion terms. A general-purpose least-squares program (Stewart and S#rensen, 1971) fitted the ten models of Table I n in a total of 2.5 min on a Univac 1108 computer. In the least-squares calculation, each model was solved by one-step numerical integration (Az = L ) at each test condition; multistep integrations agreed within 1% for the present systems (Feng, 1972). The diffusion coefficients 6
W1,n
!nd.
Eng. C h e m . , Fundam., Vol. 13, No. 1, 1974
=
1, . . . , n)
( j = 1, . . . , n )
Table 11. Types of Experiments
Type 1 I1 111
Feed to upper chamber A A + B A + B
Feed to No. of lower experichamber ments B C A + C
58 33 34
Observations fitted NA~ N, B ~ N B ~Nc* , N B ~Ncz ,
N.42,
Table 111. Comparisons of Models with Experiments P = no. of parameters fitted to diffusion data
Parameter estimates, 8 Modela
@I
h
@*
h
r1
r2
83.8
0.1132
...
0.1162 0.0228 0.1127 0.00212 ... 0.1161 0.1001 ...
h
K
...
...
3 4 . 7 319 75d 2100d .,. 79.6 97.3 .
.
I
.
.
I
...
0.1106 0.00352 6 4 . 7 2527 2100d 0.1046 0.00284 75d ... 82.2 ... 0.1130
0.2160
...
... ... ...
0.2029
... ... ...
104fj8,CBr lo4fj8,,,
... ...
...
... ...
1.13 0.99 0.83 1.00 0.49
0.82 0.70 0.59 0.72
...
2 1 4 2 2 4 3 6 4 3
...
...
Sum*of
...
squares of deviations
Stdb dev
1.655 1,431 0.441 0,988 6.389 0,776 0,717 0.128 0.204 6.239
0.077 0.071 0.040 0.059 0.151 0.053 0.051 0.021 0,027 0.149
Maxc % dev 16.7, -24.4 24.9, -13.1 11.0, -16.0 1 2 . 8 , -17.5 48.1, -26.4 17.5, -21.0 21.6, - 9 . 8 1 2 . 0 , -12.2 12.8, -14.0 45.5, - 2 8 . 1
a A suffix g denotes omission of surface diffusion, * The standard deviation and s u m of squares are those of In Na, computed with (283 - P) degrees of freedom. c % deviation = 100 X (predicted Ni, - observed N t B/)(observed N , J . Determined from e(r);see Table I.
aL,were
calculated from eq 8.2-44 and Table I-A of Hirschfelder, et al. (1964), with Lennard-Jones parameters based on low-temperature viscosity data. The composition dependence of the D,i was neglected, since it amounts to less than 2% over the experimental range of mean compositions (0.25 < xt,*" < 0.5).
Comparisons of Models with Experiment The accuracies of the models differ considerably, as shown in Table 111. Model 3 fits best, followed by 3A, 3g, 2, and 1; Models 4 and 4g fit poorly. Figures 5 and 6 illustrate the excellent fit of Model 3 to the data. The models are further compared below. Model l g is simple and fairly accurate, with a standard relative deviation of 0.077 and a maximum deviation of -24.4%. It is remarkable that a model with one lumped pore size does this well. Model 2g does better, with only one fitted parameter, but its form is inconvenient for computations. Addition of a second pore size to Model l g gives Model 3g. This extension is physically reasonable and is supported by the experiments (see Table IV). Addition of surface diffusion to any of the gas-phase models is physically reasonable, and is also supported by the data of Table IV. As a further test for surface diffusion, we may compare the data with the extended Graham relation (Evans, et a[., 1961; Mason and Kronstadt, 1967)
2 Np2m0 =
(1)
,=1
which Models lg-3Ag give when the pressure and molar density are uniform. The deviations from this equation are summarized in Table V; they are small, but increase with pressure as the surface diffusion terms of Models 1-3A predict. Model 3A is like Model 3 except that two of its constants are determined'from pore size data. This makes the model easier to fit, but also somewhat less accurate as shown in Table 111. The poor results with Model 4 are attributed mainly to its omission of a convective term from Gf,. This omission makes the model inaccurate except in the Knudsen diffusion region.
Alternate Forms Models 1-3A can be used directly for computer modeling of reaction and separation processes. Alternatively, they can be expressed in terms of effective diffusivities for use with existing theories.
dl
I
I
2
3
I
4
1
5
I 1 1 1 1
I
20
6 7 8 9 1 0 P. afm
I
I
30
40
I
I
50 60
I
m
I
Figure 5. Pressure dependence of helium fluxes at 90°F for various pairs of inlet compositions. Curves calculated from Model 3,
Table I11
E
-1
06L
0
01
1 I 10
I 0.2
1
I
04
06
I
08
,
IO
"n4
(-a)o"
Figure 6. Effects of composition and pressure on nitrogen fluxes at 240°F.Curves calculated from Model 3, Table I11
Effective diffusivities for fluid components in a porous medium are commonly defined as follows.
(i Ind.
= 1,
Eng. Chern., Fundarn., Vo!. 13, No. 1, 1974 7
Table IV. Significance Tests
Models compared
Comparison sum of squares, A S
Comparison degrees of freedom, v
Comparison variance s2 = A s / v
Variance ratio*, F = s2/se2
lg us. 3g 1us. 3 lg us. 1 2g us. 2 3g us. 3 3Ag us. 3A
1.655 - 0.441 0.776 - 0.128 1.655 - 0.776 1 . 4 3 1 - 0 717 0.441 - 0.128 0.988 - 0.204
2 2 2 2 2 2
0.607 0.324 0.440 0.357 0.156 0.392
1315c 7Olc 95lC 773c 339c 848"
Null hypothesis5 Wt
0 0 D,, = 0 D,,= 0 D,,= 0 D,,= 0 W2
= =
in in in in in in
Model Model Model Model Model Model
3g 3 1 2 3 4
a Here the true parameter values are the subjects of hypothesis. The observational variance estimate, seZ = 0.000462 with 277 degrees of freedom, is obtained from the deviations from Model 3. Denotes F value large enough for confident rejection of the null hypothesis. The rejection level for 1 % risk, estimated by local linearization of the models, is F(2,277,0.01) = 4.69. For further discussion, see Draper and Smith (1966), sections 10.3 and 10.4.
Models 3,3A
Table V. Deviations from Graham Relation, Eq 1, Relative to Its Largest Term
P, atm
tests
Rootmeansquare dev
1.0 7.8 21.4 41.8 69.0
230 23 29 22 26
0.0276 0.028 0.037 0.050 0.075
No. of
Max dev 0 . 051a 0,051 0,077 0.075 0.103
i +i
Model 4
hfr
aTwo tests at 1.0 atm, with deviations 0.114 and 0.139, are excluded here but included in Tables 111and IV.
Since the fluxes N,, are defined relative to the solid, any bulk flow is here included in a>,,ff. Insertion of eq 2 into the models in Table I will give the corresponding results for Bi,ff. Since the introduction of a>ceff usually complicates matters, only special results are given here. In the low-pressure limit, we get the following expressions foraieff. Models 1,4 D,'O' =
Model 2 DlcO'=
K
wlDlK6-L) + Drs
s,;,
+
DLK(r)dt(r) Bis
(3a)
in the absence of pressure gradient and surface diffusion. This result is consistent with Models 1 and 4, but gives higher fluxes than Models 2, 3, and 3A in the transition pressure region. The stoichiometry in eq 4 is appropriate for one-dimensional steady flow or for one-dimensional steady diffusion in a catalyst particle with a single chemical reaction. If additional reactions occur, then (4) is replaced by m
(3b)
N l z = CaklNkz
(7)
k -1
Models 3,3A Dj'O'
E
2
+
Wh a&K(rh) 3 , s
(k)
h-1
The simplicity of these results is due to the absence of gas-phase collisions in the limit' of low pressure and the absence of adsorbate interactions in the limit of low surface coverages. In the high-pressure or large-pore limit, the results are complicated. For simplicity, we neglect pressure gradient and surface diffusion. Then if all the fluxes are proportional, i.e. N,, = a J 1 r (4) the limiting forms for Dieffbecome the following. Model 1
J
*I
Model 2
8
Here Models 1-3A all predict a dependence of Qeff on the flux ratios aJ, consistent with the Stefan-Maxwell law (Bird, et al., 1960) of ideal gaseous bulk diffusion. Model 4 fails to predict this dependence, because of the approximations mentioned earlier. Combining ajico) and D iin the manner of Bosanquet (1944) gives the interpolation formula
Ind. Eng. Chem., Fundam., Vol. 13, No. 1 , 1974
where the (Yk, are stoichiometric coefficients for a set of independent reactions, k = 1, . . . , m (Aris, 1965; Petersen, 1965). Equation 2 is awkward for m > 1, since the calculated values XIceff then vary with the ratios of the unknown fluxes NI,, . . . , Nm,. To obtain constant effective diffusivities, we would have to replace eq 2 by a matrix definition. We do not pursue this idea here, since the models in Table I remain applicable without any new notation. Conclusion
The results in Table III enable the user to select a model with the desired combination of accuracy and convenience. Model 3 is the most accurate, and the simpler Model l g is still quite good. Model 2 is useful for extrapolation of limited data and for guidance in catalyst development. Fluxes predicted from this model can be used to assist in fitting Model 1 or 3 when the experimental data are otherwise insufficient. The commonly used eq 2 is suitable only for the Knudsen diffusion region or for systems with only one independent flux as in eq 4. Applications of Models 1 and 3 to catalysis are forthcoming.
Acknowledgment We wish to thank the National Science Foundation and Atlantic Richfield Company for their support of this work, Jan wrensen for his assistance with the computations, and NormaRDraper for his helpful comments. Nomenclature [c] = column vector with elements c1, , . . , c, ci = molar concentration of species i in gas phase, mole cm-3 Dlerf = effective diffusivity of species i, defined in eq 2, cm2 sec -I D , ( O ) , a>,(-) = asymptotes of aleif for small and large p r in ideal gas region, cm2 sec-1 a,,= binary diffusivity of pair I , J as gases a t given temperature and total molar density, cm2 sec-l a l K ( r ) = r d 3 2 R T / 9 r M 1 ,Knudsen diffusivity of gas i in a pore of radius r, cm2 sec-l [Os] effective surface diffusion coefficient matrix, cm2 sec [F(r)] = matrix with elements defined in Table I, sec cm-2 G,, = gaseous diffusion contribution to flux N,, [G,] = column vector with elements GI, . . . G,, rn = number of independent fluxes in eq 7 M , = molar mass of species i [N,] = column vector with elements Nl,, . . . , N,, N,, = z component of smoothed flux of species i in the porous medium, mole cm-2 sec-l n = number of species in the gas phase p = pressure, atm r = poreradius, 8, r1,rZ = characteristic pore radii for given model, A T = temperature, OK V ( r ) = volume in pores of radius less than r, cm3/g of solid V( m ) = total pore volume, cm3/g of solid Wl, W2 = porosity-tortuosity coefficients for given model, dimensionless
x i = mole fraction of species i in the gas phase z = rectilinear coordinate, cm
Greek Letters w j = fluxratioineq4 f f k j = stoichiometric coefficient of species j in basic reaction k , with convention CYkj = 6 k j for j = 1, . , . , rn p = permeability constant, g c m sec-2 atm-I 6i, = Kronecker symbol; unity when i = j and zero when
i z j
s(r) = cumulative void fraction, ppV(r) 6 = least-squares estimate of parameter whose true value is 0 K = tortuosity factor, dimensionless p = gas viscosity, g cm-l sec-l pp = pellet density, g ~ m - ~ Literature Cited Aris, R., "Introduction to the Analysis of Chemical Reactors," Chapter 6, Prentice-Hall. Englewood Cliffs, N. J., 1965. Bird, R. B., Stewart, W. E., Lightfoot, E. N., "Transport Phenomena," Wiley, New York. N. Y.. 1960. Bosanquet, C. H., British TA Report BR-507 (1944). Draper, N. R., Smith, H., "Applied Regression Analysis," Wiley, New York, N. Y . , 1966. Evans, R. B., 1 1 1 , Watson, G. M., Mason, E. A., J. Chem. f h y s . 35, 2076 (1961). Feng, C.F., Ph.D. Thesis, University of Wisconsin, Madison, Wis.. 1972. Feng, C. F., Stewart, W. E., Ind. Eng. Chem., Fundam. 12, 143 (1973). Gunn, R. D., King, C. J., A.l.Ch.E. J. 15,507 (1969). Hirschfelder, J. O.,Curtiss,, C. F.. Bird, R. B., "Molecular Theory of Gases and Liquids," Second Printing with Notes Added, Wiley, New York, N. Y., 1964. Johnson, M. F. L., Stewart, W. E., J. Catal. 4, 248 (1965). Mason, E.A., Evans, R. B..Ill, J . Chem. Educ. 46,358 (1969). Mason, E. A., Kronstadt, B., J. Chem. Educ. 44, 740 (1967). Petersen, E. E., "Chemical Reaction Analysis," Prentice-Hall, Englewood Cliffs, N. J., 1965. Stewart, W . E., Sdrensen, J. P., Program No. 44.2.003. Engineering Computing Laboratory, University of Wisconsin, Madison, Wis., 1971. Wicke, E., Kallenbach, R., KolloidZ. 97,135 (1941) Wilke, C. R.. Chem. Eng. frogr. 46,95 (1950).
Receiced f o r review M a y 1, 1972 Accepted July 31,1973
Rate Model and Mechanism of Liquid-Phase Oxidation of Propionaldehyde C. V. Gurumurthy' and V. M. H. Govindarao** Department of Chemical Engineering, lndian lnstitute of Science, Bangalore 12, lndia
A rate equation is developed for the liquid-phase oxidation of propionaldehyde with oxygen in the presence of manganese propionate catalyst in a sparged reactor. The equation takes into account diffusional limitations based on Brian's solution for mass transfer accompanied by a pseudo m-. nth-order reaction. Sauter-mean bubble diameter, gas holdup, interfacial area, and bubble rise velocity are measured, and rates of mass transfer within the gas phase and across the gas-liquid interface are computed. Statistically designed experiments show the adequacy of the equation. The oxidation reaction is zero order order with respect to oxygen concentration, 3/2 order with respect to aldehyde concentration, and with respect to catalyst concentration. The activation energy is 12.1 kcal/g mole.
Introduction
propionic acid according to the reaction
Oxidation of liquid propionaldehyde in the presence of metal salts such as manganese or cobalt propionate gives Present address, Department of Chemistry, University of Southampton, England Present address, lnstitut fur Systemdynamlk und Regelungstechnlk, der Universitat Stuttgart, West Germany
The oxidation is an important step in the production of propionic acid from monoxide and Or from n-propyl alcohol. The acid is mainly used as an inInd. Eng. Chem., Fundam., Vol. 13, No. 1, 1974
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