Numerical Study of the Three-Component Gaseous Diffusion Equations in the Transition Region between Knudsen and Molecular Diffusion R. R. Remick and C. J. Geankoplis The Ohio State University, Columbus, Ohio 43210
A numerical solution to the equations for steady-state diffusion in a three-component gas mixture of helium, neon, and argon was obtained by trial and error in the transition region between Knudsen and molecular diffusion in an open system. The shapes of plots of the fluxes of helium or neon having large concentration gradients vs. pressure are very similar to those of a binary mixture, increasing with pressure and approaching constant values at high pressures. Osmotic diffusion was found for argon and, unexpectedly, the shape of its curve of flux vs. pressure was somewhat similar to that of a binary mixture. Reasons were given for this behavior. Osmotic or reverse diffusion cannot occur in the Knudsen region. Under certain conditions the binary flux equations can be used to approximate the fluxes in a ternary mixture and reduce computational time. The existence of a maximum or minimum point in the plot of concentration vs. distance was indicated for argon. Such a point was shown for molecular diffusion in a closed ternary system. The flux ratios of helium to neon or helium to argon can change slightly or markedly with changes in pressure, depending on the concentration gradient of each component. This i s contrary to the cases of diffusion in binary systems or multicomponent molecular diffusion in a closed system where the flux ratios are independent of pressure.
MANY
CATALYTIC PROCESSES and processes for diffusion of gases through porous solids involve multicomponent and not binary gases and are for an open system where the two ends of the diffusion path are not closed. The diffusion is generally in the transition region between Knudsen and molecular or bulk diffusion (Henry et al., 1967). The theory for binary diffusion of gases in the Knudsen, transition, and molecular regions is well understood (Evans et al., 1961; Rothfeld, 1963; Scott and Dullien, 1962). The theory for multicomponent diffusion is less well understood in the transition region. Toor (1957) derived exact analytical equations for three components in the molecular region for a closed system and obtained numerical values for fluxes showing the various types of diffusion phenomena that can occur. Cunningham and Geankoplis (1968) derived analytical equations for three components in the transition region for an open and isobaric system, but numerical values have not been obtained. A number of experimental studies for binary gases in the transition region are available (Henry et al., 1967; Rothfeld, 1963; Scott and Dullien, 1962; Wakao and Smith, 1962). However, there are no similar experimental or numerical studies for multicomponent gases in this transition region. I n the present paper a three-component gaseous mixture diffusing in the transition region in a capillary with a n open system was selected for theoretical study to give a better physical picture of the phenomena occurring in this region. The use of three components should show interactions, if any, as compared to binary diffusion. A capillary was selected for the diffusion path, since the irregular geometry of a porous solid would introduce uncertainties. The transition region equations derived by Cunningham and Geankoplis (1968) were theoretically investigated in detail and numerical solutions obtained by trial and error
206
Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
using a computer. The resultant fluxes and concentration gradients obtained were analyzed and compared to molecular diffusion and to results for binary transition equations. Forced flow due to total pressure differences was not considered. literature Review of Theory
Binary Diffusion and Basic Theories. The basic equation for a n open system and a binary gas mixture of A and B for Knudsen diffusion is
I n the transition region both molecule-to-wall collisions or Knudsen diffusion and molecule-to-molecule collisions or molecular diffusion are important. The transition region equation derived by Rothfeld (1963) and Scott and Dullien (1962) is as follows for an open system.
NA =
+ +
1
DKAP 1 - ~ X A , DAB/DXA RTL In [l - ~ X A , DABIDKA ~
For molecular diffusion the equation for an open system is
(3) The fluxes are related a t constant pressure and temperature by the following for any number of components and all three regions (Dullien and Scott, 1962; Evans et al., 1961; Rothfeld, 1963).
c N,dK i
=
0
For a binary system of A and B, this reduces to
(4)
NA/NB = - ~ M B / M A
(5)
For a closed system of two vessels a t constant P connected b y a diffusion tube in the molecular region, Fick's law holds and (Toor, 1957)
ENE = 0 i Multicomponent Diffusion. M a n y analytical solutions are available for multicomponent molecular diffusion in closed systems (Hoopes, 1951; H s u and Bird, 1960; Keyes and Pigford, 1957; Sherwood, 1937; Toor, 1957). Rothfeld (1963) and Scott and Dullien (1962) used Equation 4 and a niomentum balance of molecules in a capillary in an open system in the transition region and derived the differential equations for a multicomponent system. Evans et al. (1961) derived similar equations. For a ternary system of A, B, and C the equations are
-PdXa l v ~ X B N A- X A N B RTdZ DKA + DAB
Equations and Physical System Used for Numerical Solution
Equations. To increase the computational accuracy in subtraction of large numbers close to each other, Equation 11 for X B was rederived and called XBO, so it is in the same form as XA of Equation 10. This involves substituting B for A and A for B in Equation 10 to obtain xB0=
CloemlZ
+ CzoernZz+ F/C
(14)
I n a similar manner a new equation can be derived to calculate XC more accurately. Details are given by Remick (1968). This can also be done for X B for the sine-cosine solution (Equation 13) and for XC. A solution to Equation 10 for a given N A and N B a t Z = L and X A = X A is~ obtained when the right-hand side of Equation 10 is equal to the left side or when = 0. (&),,
=
+
+
ClernlL C2emzL E / C - XA,
(15)
Similarly for Equation 12,
+
e - B L / 2 [A1cos OL
( ~ l ) ~ = i ~ - ~ ~ ~
+ A 2 sin OL] + EIC - X A , (16)
(7)
Also, a solution to Equation 14 is obtained when (22)exp0 = 0 in Equation 17. =
DBC
(8) X A
+
X B
+
XC
=
1.0
(9)
Cunningham and Geankoplis (1968) integrated the above equations and obtained two sets of equations. For ( B 2 - 4C) positive, the equations are of the exponential type.
X A = Clem'' X B = - Cl(ml
+ C2em2' + E / C
(10)
+ Kll)emlz - Cdm2 + K d e m Z Z
+
Kio
F,,C
+ CZ0ernzL+ F / C - X B ,
Cl0emlL
(17)
Similarly, ( 2 ~ can ) ~also~ be ~ defined from Equation 11. The preceding equations were investigated to determine if any computational difficulties would be encountered, so that the computer could be programmed to handle them (see Appendix for details). Cunningham and Geankoplis (1968) showed that maxima and minima could exist in the plot of concentration us. Z but did not derive the actual equations. However, it is not possible to tell beforehand-Le., before solving for N A and X B if this occurs in the real distance between Z = 0 and Z = L or a t some fictitious distance of Z < 0 or Z > L which has no physical significance. The equation for the maximum or minimum has been derived (Remick, 1968) and is the following if in the exponential region.
Kio (11)
For the case ( B 2 - 4C) negative, the following sine-cosine solutions are obtained.
XA
=
e-BZ'2[Al cos OZ
+ A2 sin 021 + E / C
(12)
A similar type of equation has been derived for the sine-cosine region. The equation for the maximum or minimum for Toor's case (1967) has been derived (Remick, 1968) :
Z = -1 l n [( T B y] CII B'
Cunningham and Geankoplis point out that it is not possible to tell beforehand if the term ( B 2 - 4C) is positive or negative and it contains the unknown fluxes ~L'Aand N E . Hence, both sets of solutions must be tried. They showed the possible existence of maximum or minimum points in an X , us. Z plot. They pointed out that diffusion barriers and osmotic and reverse diffusion could occur in a manner similar to that in molecular diffusion as discussed by Toor (1957). The physical significance of these phenomena are not readily apparent because of the complexity of the equations. Yo numerical solutions to these equations are available.
Physical System. T o solve t h e transition region equations for a ternary system, the following must be specified : type of components, capillary tube radius, total pressure, temperature, a n d concentration boundary conditions. The components selected should have widely different molecular weights, exhibit no surface flow phenomena, and be able to be used eventually in actual experiments to test the equations. Argon, neon, and helium gases were used at 25OC. The capillary radius should be of such a size that the D K , values are of the same order of magnitude as the Dij values a t 1 atm. This provides for approximately equal contributions from Kiiudsen and molecular mechanisms to diffusion Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
207
Physical Properties of System Used at 25°C. and 1 Atm.
Table 1. Component,
X B C
Boundary Concentrations
DKir
I
He Ne = rir = =
Mi
Cm.Z/Sec.
4.003 20.183 39.944
0.837207 0.372868 2.65033 Table II.
0.01 0.1 1.0 10 100
0.4186035 0.38607534 0,2273822 0.04457350 0.004932115
Dij,
z=o Xa, = 0.5 XBQ= 0 XC, = 0 . 5
Cm.2/Sec.
DAB = 1.126 DAC = 0.729 DBC = 0.322
z=1 XAL
=
XBL = XCL =
0 0.5 0.5
Final Flux Values at 25°C.
-0.186434 -0,1718715 -0.1010107 -0,0197540 -0.00218470
in the middle of the transition region. A capillary radius of 1000 \vas selected. Pressures selected for the computer study were 0.01, 0.1, 1.0, 10, and 100 atm. or a range of 10,000 to 1 to cover the regions from almost pure Knudsen to molecular diffusion. The boundary colicelitrations are given iii Table I. The boundary concentrations of X C were set constant a t 0.5 to allow for possible osmotic diffusion. The experimental molecular diffusivities were taken from Reid and Sherwood (1966) aiid corrected to 25°C. by using the fact that, the diffusivity is proportional to The Kiiudseii diffusivities were calculated from
Discussion and Results
Fluxes N A and NB. T h e final values of the fluxes were calculat'ed b y trial and error using Equations 15 t o 17 (Table 11). The exponential equations were valid in this case. Details are in the Appendix. The final flux d a t a are plotted in Figure 1 as s , P / ( P / R T L ) us. pressure P. Up to about 0.1 atm. the diffusion of aiid 13 is alniost pure Knudsen, siiice the slopes are 45". At about 50 atm. the lines level out aiid become almost flat, which indicates essentially molecular diffusion. I n the region between 0.1 and 50 atm. the diffusion is in the transition region. These results indicate that the shapes of the curves of flux us. pressure for components A aiid I3 with large driving forces in ternary mixtures are very similar to those for binary mixtures.
0.0 -0.4730479 x -0.1802719 X lod3 -0,6876481 x -0.8394712 x
2.2454 2.2463 2.2511 2.2564 2.2576
( > I x 104) 0.8188 X l o 4 0.1261 X l o 4 0,06478 x lo4 0.05875 X lo4
I t is useful to see if a simple niethod can be used to approximate or bracket the curve for N A in Figure 1. Csiiig the binary Equation 2 and assuming oiily a mixture of ; iin 13, flux . V A was calculated and plotted in Figure 2. This was repeated for A in C and plotted. The plots show that the two biliary equations bracket the true ternary equation. At low pressures the two binary curves and the one ternary curve converge as expected, since Equation 1 should hold. -kt high pressures the binary curves differ by a niaximuni of +30 and -70yo from the ternary curve. This method can be used for a preliminary or quick method to bracket' the true value of X A in a ternary and reduce the computation time required for a final solution. This should work when the component has a large driving force. For a low or zero driving force, interactions will be inore important aiid the approximate results can be inaccurate. This problem is often encountered in linearizing multicomponent diffusion equations. Similar results were obtained in bracketing the true 'VBcurve using the binary eqaations. The shapes of the flux curves for S Aand S Biii Figure 1 for different values of the capillary radius, 7, can be predicted. For a larger capillary the low pressure line would be a 45' line parallel to the present line for S . 4 in Figure 1 but displaced to the left. The line for a smaller radius would be displaced to the right. At high pressures all lines would converge to the same horizontal limiting value. Flux Nc. I n Figure 1 the plot' of the flux of C I" pressure shows osmotic diffusion occurring. Cunniiigham and Geankoplis (1968) predicted t h a t this could occur in
-
.x 1 ~ 4
- 0.1 0
(NCI(P/RTLl)P
2
(-Ng/lP/RTLl)P
0.005 0.01
0.1
I.o P,atrn.
Figure 1.
208
Ind. Eng. Chem. Fundam., Vol.
~
P/RTl
10.0
vs.
100.0 200.0
P
9,No. 2, 1970
P, ATM.
Figure 2.
Comparison of binary and ternary equations
t h e transition region a n d Toor (1957) gives a n actual example of this for molecular diffusion in a closed system. Approaching very low pressures pure Knudsen diffusion occurs where no interactions between molecules occur. Hence, t h e flux of C should drop to zero a t very low pressures for zero driving force. At high pressures the flux of C should approach a constant value for true molecular diffusion, because both S Aand N E approach constant values. The flux of C in Figure 1 is leveling out a t higher pressures. It is concluded that further study is needed to determine more definitely the trends in N c when osmotic diffusion occurs. Concentration Gradients. T h e concentrations of X A , X g , and XC are plotted us. 2 in Figure 3 for P = 100 atrn. The Saline is concave or bowed out, as viewed from the a. For lower pressures the curvature decreases and the line is csseiitially straight a t 0.1 atni. At the higher pressures the curvature is probably due to interactions between the different species, and a t the low pressures no interactions are present because of Kiiudsen diffusion. The trends for the XB line follow those for the XA line but the curvature is in the opposite direction. For all pressures the plot of XCus. 2 was a horizontal straight line within the accuracy of the calculations. Calculation of the maxiniuni or minimuni point for component C iii Equation 18 shows that only for P = 10 and 100 should such a point occur in the real range of 2 = 0 to 1.0L. This point is a t 0.4L for 10 atm. and 0.6L for 100 at'm. However, these points are sensitive to slight changes in values of ivc. Hence, the minimum, if present a t 10 or 100 atm., is probably within the computational accuracy. From calculations in this work it is shown that a maximum or minimum point does not always appear for a species undergoing osmotic diffusion and that this point, may be outside the region 0 to 1.OL. This occurred for the cases at, 0.01, 0.1, and 1.0-atm. pressure. The fluxes must be known before this point can be calculated. Further work is needed to show if such a minim u m or maximum can occur in the transition region. Using gases with more interactions, such as H P , CO,, and HzO used by Toor (1957), might' give a larger effect on the maximum or minimum. Equation 19 derived for a closed system in the molecular range was used for a ternary mixture and a minimum or maximum was predicted a t 0.45L. I n this case the boundary conditions are the same as in the present system. Using the equations of Toor (1957) relating the concentrations wit'h 2, the coilcentration gradients were calculated and plotted in Figure 4 for the system H?-H20-CO2. The plot shows a minimum in the coilcentration plot for COS. This is the first or one of the first times a minimum has been shown. Flux Ratios. F o r a n open system t,he fluxes are related b y Equation 4. F o r a binary system in the present work for A = H e and €3 = Xe,
I
0.6
I
I
I
I -
-
z=L
I Figure 3.
X i vs. Z for P = 100 atm.
the binary systems for diffusion in the molecular or transition regions or the systems for multicomponent molecular diffusion where the flux ratios are constant. For two components in a ternary system which have the same but appreciable concentration gradients, the binary equation is a useful approximation for the flux ratio. However, the fact that reverse and osmotic diffusion can occur means that this approximation would be incorrect for components with small concentration gradients. Even though reverse or osmotic diffusion can occur in the transition or molecular region, it cannot occur in the pure Knudsen region, since Equation 1 holds for each component. Conclusions
The exponential and not the sine-cosine solutions were valid in the present case. Uncertainties still exist as to whether the sine-cosine solutions can ever be valid.
'mo
0.8
PREDICTED MINIMUM
0.6
(5) I n Table I1 the flux ratios for X and B in the ternary system are given. This flux ratio increases slightly as pressure increases, but the maximum difference from the binary value of 2.2454 is only 0.5%. The flux ratios of A to C in Table I1 decrease markedly by factors of over 20 as the pressure increases. This change might be of use in separation processes. The interesting conclusion is that the flux ratios in this ternary system in the transition region can change with pressure. This is unlike
0.OOZ 0.252 0.50Z 0.752
1.002
z=L
Z Figure
4.
Concentration profile for H2-H20-C02
system
Toor'r case c = con B = H20 A = Hz Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
209
The shape of the plot of the flux os. pressure of an individual coniponent’ with a large concentration gradient in a ternary mixture is very similar to those for components in a binary system. Under certain conditions the binary flux equat,ions can be used to bracket the ternary values and reduce computational time. The plots of flux us. pressure for different values of the capillary radius should be a family of parallel curves a t low pressures which converge to one limiting value a t high pressures. Osmotic diffusion was found for the component’ argon (C) in the transition and molecular regions and reasons were given for the somewhat similar shape of the curve of flux us. pressure and that of the other components. Osmotic or reverse diffusion cannot occur in the Knudsen region. A maxiniuni or minimum in the concentration-us.-distance plots call occur in the real region of 0 to 1.OL or in imaginary regions outside. The existence of a maximum or minimum point for component C was indicated from the calculations but was not conclusive. Such a point was definitely shown for molecular diffusion in a closed ternary system. The flux ratios of X to I3 and/or X to C can change slightly or markedly with pressure in the transition region. This could possibly be the basis of a separation process. This is contrary to binary mixtures for diffusion in the transition or molecular regions or multicomponent mixtures in molecular diffusion where the flux ratios are constant. Acknowledgment
= = = =
F2 5 R T
=
x’,
= = = =
z 01
8
i - B - ;/B2 - 4Cj/2 absolute total pressure, atm. radius of capillary, cm. gas constant, (cc.) (atm.)/(g. mole) (OK.) temperature, O K . mole fraction of component i distance in direction of diffusion path, em. 1 ;VB/N~
+
d ( 4 C - B2)/4
SUBSCRIPTS A , B , C = component A, B, or C = indexes referring to component A, B, or C i ,a = end conditions of diffusion path (boundary condi0, L tions) SUPERSCRIPT = denotes substituting B for A and A for B in Equations 14 and 17
0
The Gnion Carbide Co. provided a fellowship for this study and work was done under the Atomic Energy Commission Contract KO. AT(11-1)-(1675), The financial assistance provided is gratefully acknowledged. Nomenclature
-4 1 A4 2
B B’ C CII
References
Cunningham, R. S., Geankoplis, C. J., IXD. ENG.CHEM.FUNDAMENTALS 7,429 (1968). Dullien, F. A. L., Scott, D. S., Chem. Eng. Sci. 17,771 (1962). Evans, R. B., Watson, G. M., Mason, E . A., J . Chem. Phys. 35, 2076 (1961). Henry, J. P., Jr., Cunningham, R. S., Geankoplis, C. J., Chem. Eng. Sci. 22, 11 (1967). Hoopes, J. W., Jr., Ph.D. thesis, Columbia University, New York. 1951. I
c 1
C2
D ii
=
DKi
=
E E‘
= =
(mz - mJC molecular diffusion coefficient for binary of i a n d j , sa. cm./sec. Knidsen diffusion Coefficient for Component i~ sq. cm./sec. KioKi2 - KgKi4 KioKi2’ - Kg’Ki,’
210 Ind. Eng. Chem. Fundam., Vol. 9, No. 2, 1970
Hsu,H. W., Bird, R. B., A.1.Ch.E. J . 6 , 516 (1960). Keyes, J. J., Jr., Pigford, R. L., Chem. Eng. Sci. 6,215 (1967). Reid, R. C., Sherwood. T. K., “ProDerties of Gases and Liquids,” 2nd ed., McGraw-Hill, New York, 1966. Remick, R. R., 3f.S. thesis, Ohio State University, Columbus, 1968. Rothfeld, L. B., A.Z.Ch.E. J . 9, 19 (1963). Scott, D. S., Dullien, F. A. L . , A . I . Ch. E . J . 8 , 113 (1962). Sherwood, T. K., “Absorption and Extraction,” 1st ed., hlcGrawHill, New York, 1937. Toor, H. L., A.I.Ch.E. J . 3 , 198 (1957). Wakao, N., Smith, J. M., Chem. Eng. Sci. 17,825 (1962). Wilke, C. R., Chem. Eng. Progr. 46,95 (1950).
RECEIVED for review September 18, 1968 ACCEPTED December 8, 1969 For supplementary material order NAPS Document 00766 from ASIS National Auxiliary Publications Service, c/o CChl Information Sciences, Inc., 22 West 34th St., h’ew York, N.Y., 10001, remitting $1.00 for microfiche or $3.00 for photocopies.
