84
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
Katayama, T., Sung, E. K., Lightfoot, E. N., A.l.Ch.E. J., 11, 924 (1965). Kesselman, W. P., Hollenbach, G. E., Myers, A. L., Humphrey, A. E., J. Chem. Eng. Data, 13,34 (1968). Kirschbaum, E , Gertsner, H., Verfahrenstechnik, 1, 10 (1910). Lundberg, G.W., J. Chem. Eng. Data, 9, 193 (1964). Mathieson, A. R., Thynne, J. C. J., J. Chem. SOC., 3708 (1956). Max, G.M., Azapova, Z. H., Zh. Prikl. Khlm., 19, (5-6),585 (1946). Meertjes, P. M., Chem. Proc. Eng., 41,385 (1960). Michishita, T., Arai, Y., Saito, S., Jpn. Chem. Eng., 35, l(1971). Murakami. S.,Lam, V. T., Benson, G. C.. J. Chem. Thermodyn., 1, 397
(1969). Myers, H. S., lnd. Eng. Chem., 48, 1104 (1956). Myers, H. S.,Pet. Refiner, 36 (3),175 (1957). Nagata, I.,Yamada, T., I d . Eng. Chem. Process Des. Dev., 11,574(1972). Nigam, K. K.. Singh, P. P., Trans. Faraday Soc., 65, 950 (1969). Null, H. R., "Phase Equilibrium in Process Design", Wiiey New York, N.Y.,
1970. Pitzer, K. S., Curl, R. F., J. Am. Chem. SOC., 79, 2369 (1957). Prausnitz, J. M., "Molecular Thermodynamics of Fluid Phase Equillbria", Prentice-Hail, Englewood Cliffs, N.J., 1969. Prengle, H. W., Palm, G. F., lnd. Eng. Cbem., 49, 1769 (1957). Prengle, H.W., Pike, M. A,, J. Chem. Eng. Data. 6, 400 (1961). Prengie, H. W., Woriey, F. L., Mauk, C. E., J. Chem. Eng. Data, 6, 395
Ramalho, R. S., Deimas, J., J. Chem. Eng. Data, 13, 161 (1968b). Rastogi, R. P., Nath, J., Misra, J.. J. phys. Chem., 71, 1277(1967]. Rivenq, F.,Bull. SOC.Chim. Fr., 9, 3034 (1969). Roilet, A. P., Elkaim, G., Toledano, P., Senez, M.. C. R. A&, Science,242,2560
(1956). Rosanoff, M. A., Bacon, C. W., Schulze, J. F. W., J. Am. Chem. Soc., 36, 1999
(1914). Rothe, R., Dissertation, Gattingen, West Germany, 1958. Schnaible, H. W., Thesis, Purdue University, Lafayette, Ind., 1955. Schuberth, H.,J. P r a k Chem,, 8, 129 (1958). Sieg, L., Chem. lng. Tech., 22,322 (1950). Smith, Buford, communications from Washington University, St. Louis, Mo.,
1973-1975. Sturtevant, J. M., Lyons, P. A., J. Chem. Thermodyn., 1, 201 (1969). Thornton, J. D., Garner, F. H., J. Appl. Chem., 1 (Suppl. I), 574(1951). Watson, A. E. P., McClure, I. A., Bennett, J. F., Benson, G. C., J. Phys. Chem., 69, 2753 (1965). Wilson, G. M., J. Am. Chem. SOC., 86, 127 (1964). Zharov, V. T., Morachevskii, A. G., Shapil, L. G., Buevich, T. A,, Zh. PTikl. Khim.,
41,2443 (1968).
Received for review June 23, 1977 Accepted November 18,1977
(1961). Ramaiho, R. S.,Delmas, J., Can. J. Chem. Eng., 46, 32 (1968a).
Solid-Solid Reactions. Diffusion and Reaction in Pellet-Pellet Systems S. S. Tamhankar and L. K. Doraiswamy' National Chemical Laboratory, Poona 4 11 008, lndia
Analysis of coupled diffusion and reaction in a solid-solid pellet system is attempted. A theoretical model is developed for a first-order reaction, which is verified using reported electron probe microanalytical data, and the importance of a reaction zone in the analysis of solid-solid systems is brought out. From the model developed, the reaction zone thickness (or region of solid solution) can be precisely determined knowing the kinetic and diffusion parameters (the Thiele modulus). The treatment is extended to a general nth order reaction.
Introduction Solid-solid reactions, although of considerable industrial importance, have not received any significant attention in the chemical engineering literature. Practically all the solid-solid reactions, such as those associated with the manufacture of ferrites, semiconductors, ceramics, dry cells and some classes of catalysts, have been studied only empirically from the chemical engineering standpoint, and hardly any equation is available which can provide a firm theoretical basis for the prediction of the reaction rates. The usual manufacturing process involves mixing of the component solid reactants (usually transition metal oxides) in fine powdered form and firing at high temperatures to obtain the desired products. The role of diffusion in solid-solid reactions is far more significant than in any other reacting system, such as gasliquid, gas-solid, etc. As the actual three-dimensional geometry of solid-solid systems is quite complex, no simple mathematical analysis seems possible. T o facilitate mathematical analysis, however, pellet-pellet experiments may be carried out. The basic advantage in these experiments is that, the contact surface area being constant, reaction rates can be determined by direct measurement of the product layer thickness. In such studies, it is almost always assumed that the rate is controlled by the diffusion of the reactant species in the product layer. The diffusion and reaction of one of the reactants in the other (resulting in a reaction zone) is often ne0019-7874/78/1017-0084$01.00/0
glected. In the present analysis, assuming first-order kinetics, a model will be developed in which the role of the reaction zone will be explicitly brought out, and using reported data (obtained by electron probe microanalysis), the proposed model will be verified. The treatment will be extended to an nth order reaction and a procedure indicated for obtaining the order and the concentration profiles in the reaction zone. Considering the assumptions made in the development, the model should only be regarded as providing a predictive equation for pellet-pellet reactions.
Theoretical Development The Nature of the Problem. Let us consider the general class of reactions between two metal oxides A 0 and B203 A0
+ B2O3
-
AB204
Among the industrially important reactions belonging to this class are reactions between an oxide of a transition metal (Cu, Co, Ni, Mg, Zn, etc.) and Fe2O3, A1203, and Cr203 to give respectively ferrites, aluminates, and chromites. The important mechanisms of mass transport in these reactions are: (i) oneway diffusion of either A2+ and 02-or 2B3+ and 302-, (ii) counterdiffusion of 3A2+ and 2B3+,and (iii) one-way diffusion of A2+ and 2e- and of oxygen as gas ( 0 2 ) . In cases (i) and (iii) product growth is restricted to one side of the original interface, whereas for case (ii) it occurs on both sides. In all the cases the slower moving species will be rate controlling, and 0 1978 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978 85
different expressions for diffusivities are involved, as explained in the general treatment of Wagner and the later developments of Schmalzried (1973). As in the case of gas-solid (catalytic) reactions, several modes of diffusion are possible in solid-solid systems as well. The more important among these are: (i) vacancy mechanism, (ii) interstitial mechanism, (iii) surface diffusion, (iv) grainboundary diffusion, and (v) vapor phase diffusion. In systems where polycrystalline materials are involved, all or some of the above mechanisms may be operative simultaneously, and separate diffusivities for each mechanism cannot be determined. Hence an overall effective diffusivity De may be used as in the case of gas-solid systems. If, however, single crystals are involved, a single mode of diffusion (viz., volume diffusion mainly by vacancy mechanism) should be adequate to define the transport process. In the gross approach using an effective diffusivity De, one can apply Fick's laws of diffusion, as in the case of solid catalysts. From the three important mechanisms of mass transport in such reactions, it can be readily seen that all involve moving boundaries. In cases (i) and (iii) there would be only one moving boundary, whereas in case (ii) both the boundaries of the product layer would be moving, since product growth occurs on both sides. In addition, in a few instances, a different mechanism may be observed. Thus in certain ferrite formation reactions Fe3+ is transported as Fe2+ and is again oxidized to Fe3+ which reacts to form the ferrite spinel; simultaneously oxygen from Fez03 is transported in the same direction, while A2+ from A 0 is transported in the opposite direction (Kooy, 1964; Krishnamurthy et al., 1974). T o generalize the treatment, the reference axis may be fixed on one of the boundaries and the relative movement of the other boundary followed. Thus in all three cases there will now be only one moving boundary. Fortunately, since in all inorganic systems ionic species are involved in the diffusion process, the individual diffusivities are interrelated due to the condition of electroneutrality to be maintained. Hence, even in the case of c_ounterdiffusionof A2+ and B3+,an interdiffusion coefficient D has been defined based on the mole fraction and activity of one of the compounds, usually of A 0 (Wagner, 1969). This simplifies the problem to a certain extent, since we may now follow the concentration profile of only one of the species. In the case of counterdiffusion, for example, 3A2+ diffuse and result in three molecules of ABz04, whereas at the same time for charge balance ZB3+diffuse and form one molecule of AB204 at the other boundary. Hence, if we follow only the concentration profile of say A2+,the actual rate of product layer growth will be 4/3 times higher, which will be automatically taken care of by the rate constant of the process, since the reference axis is fixed on one of the boundaries. Reaction Modeling. Using the concept of an effective diffusivity, we shall now develop a model for solid-solid reactions based essentially on the moving zone theory for gassolid reactions developed earlier by Tudoze (1970) and Mantri et al. (1976). The validity of the theory will be verified by using the experimental concentration profiles from electron probe microanalysis (EPMA) data for a few systems reported in the literature. In fact, Schmalzried (1973) has postulated an effective reaction zone for phase boundary controlled reactions in solid-solid systems. An analysis of such a zone has been presented by Greskovich and Stubican (1969) wherein they have considered only the diffusion in this zone. Mention should be made a t this stage of a model proposed by Arrowsmith and Smith (1966). This represents perhaps the first attempt to present a quantitative analysis of solid-solid reactions. Two semiinfinite solid rods are considered and boundary conditions a t infinity are used. In the zone model developed below, on the other hand, boundary conditions a t
L
x i 0
I--+
t>O
t-0
NO R E A C T I O N
REACTION WITH
Product zone
+
ZONE
FORMATION
NO PRODUCT
LAYER
Product
Reaction
m e
x+
ZONE
Reaction
4zone
x-
I >o
REACTION
L
ixo
x-
f>O
ALONG WITH
A PRODUCT LAYER WITH DIFFUSION RESISTANCE
No
REACTION
ZONE ALONG W I T H
A PRODUCT LAYER W I T H A F I N I T E DIFFUSION RESISTANCE
Figure 1. Stages in the reaction between the solids A 0 and Bz03 in the reaction zone model.
finite distances are used. This is particularly important when spherical geometry is considered, since the infinite boundary conditions lead to a source or a sink a t infinity. Further, we shall assume a first-order reaction as against a second-order reaction assumed by Arrowsmith and Smith. According to the moving zone concept, reaction in a gassolid system occurs in three stages: zone formation, zone travel, and zone collapse. Mantri et al. have formulated mathematical equations for the three zones and have also derived an expression for the zone thickness for a first-order reaction. Experimental studies on solid-solid reactions using the EPMA technique give the concentration history of one of the solid reactants in the second reactant as the reaction progresses. Typical concentration profiles have been reported for a few systems (Yamaguchi and Tokuda, 1967; Minford and Stubican, 1974; Whitney and Stubican, 1971). An inspection of these profiles reveals some interesting conclusions, which point to a similarity in behavior with the zone model for gassolid reactions mentioned above. Hence we shall first formulate a model using the concept of a moving reaction zone, and then present a quantitative verification of the model using reported EPMA concentration profiles. The system can be most appropriately described in terms of two zones: (i) the product zone and (ii) the reaction zone. At time t = 0, the two solids are brought together and kept in contact under isothermal conditions. The reaction is initiated at the phase boundary and progresses inwards, forming a reaction zone of finite width. On completion, this zone starts moving as a whole, leaving behind the product zone formed as a result of the reaction occurring in the reaction zone. Further, the product zone formed may offer a diffusional resistance, thereby decreasing the concentration available for reaction a t the interface between the two zones. This new concentration at the interface may then be maintained a t its value by unsteady-state diffusion or may continue to decrease depending on the system properties. A few typical stages in the model are illustrated in Figure 1.The zone designated here as the reaction zone is referred to as the solid solution region in the EPMA studies. The situation visualized above involves a moving boundary problem which is normally solved by assuming a pseudosteady state. This assumption is valid in the case of gas-solid (noncatalytic) reactions, since the rate of transport of gas to
86
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
the moving boundary is much faster (-1000 times) than the rate of movement of the boundary, which arises because of the large density difference between a gas and a solid. This is obviously not the case in solid-solid systems. Hence, in the following treatment, unsteady-state behavior has been considered. We attempt an analysis of the system a t an intermediate stage. The equations obtained can be adapted to any particular stage by applying the proper boundary conditions. The two zones, viz. the product zone and the reaction zone, are considered separately and the mass balance equations for the two are solved with the corresponding boundary conditions, but attention is mainly focused on the reaction zone. (1) Product Zone. As mentioned earlier, the diffusivity is assumed independent of concentration. The governing differential equation can be written as
a c - d2C _ at - D~~
t>O
t 1 0
(3)
The solution satisfying initial conditions 2a and boundary conditions 2b and 3 is given by Crank (1975) as
C
=
X
C* erfc
(4) 2(Dpt ) 1/2 This solution, derived for a semiinfinite system, is also valid for a region bounded by one or two x planes (Danckwerts, 1950), if concentrations are the same at the same points in the two cases. New variables are now defined to convert the expression given by eq 4 into a dimensionless form (5) With these new defined variables, eq 4 becomes
w = erfc
20112 The dimensionless concentration at the end of the product zone w p can be obtained as z
wp = erfc-= 1 -erf* (7) 20112 20112 (2) Reaction Zone. As a first approximation, we shall assume the reaction zone to be of constant thickness. This assumption will be verified later. Since the time dependence will be implicit in one of the boundary conditions, viz. the boundary condition at the interface between the two zones, the governing differential equation for the reaction zone (with D, as the diffusivity in this zone) can be written as
with initial conditions 2a and boundary conditions C=C,; x = x p
c=o; x
=XI
]
d2w --
(#J,2w=
dz with the boundary conditions
0
]
=zp t>O (12) w=o; z=zr The solution of eq 11is straightforward and is given by w=wp;
2
AZ = Z , - Z , (14) represents the reaction zone thickness. Subsitituting eq 7 in eq 13, we get
t = O
The profile is assumed to be extended to infinity, with the additional boundary condition
c=o; x = m ;
Using eq 5 and 10, eq 8 becomes
where
where D, is the diffusivity of the species under consideration in the product zone (the subscript e indicating effective values being dropped). The initial and boundary conditions are
c=co; x < o ] c=o; x > o c = c * ;x = o ] c=c,; x = x p
where the second term accounts for the first-order reaction rate. Here, in addition to the dimensionless variables defined by eq 5, a Thiele modulus is defined as
t>O
Recent experimental results point to the essential constancy of w,. For example, the EPMA results of Whitney and Stubican (1971) on the system MgO-Al203 at 1695 OC clearly show that the interface concentration is constant and invariant with time. Thus, in our present treatment we shall restrict the analysis to wp = constant. Although in theory w p can have a value of unity, it would appear from reported experimental studies (Yamaguchi and Tokuda, 1967; Minford and Stubican, 1974; Whitney and Stubican, 1971) that wp is practically always less than unity because of the interfacial resistance. However, no explanation seems possible at present for the observed constancy of w,. The most useful aspect of this analysis of solid-solid reactions is that we can obtain an expression for the reaction zone thickness Az in terms of the Thiele modulus c $ ~ from eq 13. Such an equation can be readily developed and is identical with that given by Mantri et al. for a first-order reaction 1
+
+
Az = -In [& (dr2 1)112] (16) 9, A comparison of this model with that for gas-solid reactions reveals certain interesting points: (i) There is no counterpart to the gas film resistance in solid-solid reactions. In fact, the product itself moves in a direction opposite to that of diffusion, even in the case of one-way diffusion. (ii) Solid-solid reactions are essentially lattice rearrangements, and will always occur in a finite zone thickness, since diffusion and reaction rates are comparable. Verification of Model The model developed above will now be tested using the reported EPMA results for three distinct systems (involving a ferrite, an aluminate and a chromite) at different temperatures. The steps involved in the procedure used for testing are as follows: (1)Obtain the precise value of the zone thickness for a given system from the reported EPMA data. (2) Using the value of the zone thickness thus obtained and assuming first-order kinetics, calculate the Thiele modulus from eq 16. (3) From eq 13 calculate the concentration a t different positions in the reaction zone, using the values of Az and 4, determined in steps (1)and (2) and employing the value of w, obtained from EPMA studies. (4) Prepare a plot of w vs. position in the reaction zone using the values calculated as above
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
I
10
,
(
,
I
I
I
1
1
1
cr ~n 0 9-
I
1
I
I
I
I
II
I
I I
I
as 5
- SOLID
SOLUTION
i
I
400
300
I 1,
1
1
SYSTEM
- Mp0-F0203
TEMP
-1250.C
TIME
-7Ohr
1
1
-**-CALCULATED FROM MODEL EXPERIMENTAL (Yamoguchi and Tokuda.1367
08-
i
i
I
i
87
-
0 7-
06
8
1
I
1
I
200 N
100
0
Figure 2. Electron microprobe scanning figure for the system MgO-CrzOs (Yamaguchi and Tokuda, 1967).
1
L
SYSTEM -MpO-CrzO, TEMP,
-t400'C
TIME
-20hr
- 0 - 0 CALCULATED
DIMENSIONLESS FROM MODEL
-E XPER I M E N T A L (Yarnaguchi and
DISTANCE, 2
Figure 4. Comparison of the theoretical profile calculated from the model with the experimental profile obtained form EPMA studies.
0,71
0.6
SYSTEMTEMP
N10 - A 1 2 0 3
--1437'C
-
TIME 20 hr - 0 . U - C A L C U L A T E D FROM MODEL EXPERIMENTAL and Stubican, 1974 )
DIMENSIONLESS
DISTANCE. z
Figure 3. Comparison of the theoretical profile calculated from the model with the experimental profile obtained from EPMA studies. and compare with the experimental concentration profile obtained from EPMA measurements. The profile obtained by EPMA measurements for the system MgO-Ci-203 at 1400 O C after 20 h is reproduced in Figure 2 (Yamaguchi and Tokuda, 1967). From this the following precise values were obtained. (i) Az the reaction zone thickness, as calculated by normalizing the actual value Ax (by an arbitrarily chosen value of 1000 a t which EPMA results indicate no reaction): Az = 0.1. (ii) upthe dimensionless concentration a t the start of the reaction zone: w p = 0.5. The normalizing value of concentration was taken as that a t the beginning of the product zone where this value was available, as for the present system, MgO-CrzO3. In cases where this value was not available (as for the system NiO-Al203) the value at the end of the product zone was considered so that w p at this plane becomes 1.This is brought out in Figures 5-7 for the system Ni0-A1203. (iii) The Az value obtained from the experimental profile was used to determine the Thiele modulus for the system using eq 16: & = 45.6.
O0
I
I
I
0,l
02 DIMENSIONLESS
DISTANCE, z
Figure 5. Comparison of the theoretical profile calculated from the model with the experimental profile obtained from EPMA studies. (iv) Using the values of Az, &, and upin eq 13, the dimensionless concentration w was calculated at different positions, and the resulting plot of w vs. z is shown in Figure 3. The corresponding experimental profile is also reproduced in the same figure. The comparison shows remarkably good match. Similar calculations were made for: (i) the system MgOFez03 at 1250 "C from the data of Yamaguchi and Tokuda (1967) (the results are reproduced in Figure 4) and (ii) the system Ni0-A1203 a t 1437,1540,and 1630 OC from the data of Minford and Stubican (1974) (the results are reproduced in Figures 5,6, and 7, respectively). In all the cases the profiles prepared based on the model practically coincide with the experimental profiles, the average deviations being: MgOCr203,5.6%, MgO-FezO3, 9.7%, and NiO-A1203,3.0% (for all the three temperatures).
88
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978 I
I
,
1
SYSTEM
- Ni0-Al20,
SYSTEM
- N i O -A1203
TEMP
-1540T
TEMP
-163O'C
-
TIME -15Yz hr -*+.CALCULATED FROM M O M L EXPERIMENTAL ( Minford and Stubicon. 1974 1
TIME 8 hr -+O-CALCULATED FROM MODEL EXPERIMENTAL ( Minford and Slubican, 4974 )
-
y 8
10
9 08 5 v)
:
06
1
0 4
02
00 DIMENSIONLESS
DISTANCE,
Figure 6. Comparison of the theoretical profile calculated from the model with the experimental profile obtained from EPMA studies.
It would appear that the concept of an order of reaction, not explicitly brought out so far, is meaningful even in solid-solid reactions and can perhaps replace the concept of an index of reaction (Taplin, 1974). However, it should be noted that the present model does not necessarily validate the first-order assumption in view of the other assumptions also involved. It should be regarded as providing a predictive equation which represents the reported data satisfactorily on the assumption of first-order kinetics. The General Order Case In case first-order kinetics is not obeyed in a particular system, a general n t h order reaction has to be considered. Extending the treatment now to an nth order chemical reaction, the continuity equation for the reaction zone can be written in dimensionless form as
where the Thiele modulus +r now becomes kfp(n-1) i/z +r=~[?]
The following solution to eq 17 can be obtained with boundary conditions 1 2
Rewriting this equation for the end of the reaction zone, Le., for w=o; z = z r
04
0 2
03
DIMENSIONLESS
I
(20)
we obtain
or
If now concentration is normalized with respect to the value at the beginning of the zone, then w p = 1and we have
0 4
0 5
DISTANCE,
06
-
I
Figure 7. Comparison of the theoretical profile calculated from the model with the experimental profile obtained from EPMA studies.
(23)
Plots of log 4, vs. log Az for different values of n can thus be prepared. An interesting feature of eq 23 is that it gives unrealistic values of & for n > 1. No conclusion can be drawn from this, however, since there is no apparent physical reason for invalidating higher orders. Clearly, the EPMA profile for at least one experimental condition is necessary for obtaining the values of n and &. Once the reaction order is known, the concentration profiles for other sets of conditions can be predicted from experimental diffusivities (to give 4,). Conclusions A mathematical analysis of coupled diffusion and reaction in solid-solid systems (in the form of pellets) is presented considering two zones: a product zone, and a reaction zone. Continuity equations for the two are solved to obtain the concentration profiles. Based on these a relation between reaction zone thickness and the Thiele modulus is obtained. Reported experimental EPMA results are used to verify the predictive equation provided by the model on the assumption, among others, of first-order kinetics. A comparison of the theoretical and experimental profiles shows good agreement. The treatment is then extended to the general nth order case. The chief assumptions made in the development (in addition to first-order kinetics) pertain basically to the absence of external porosity in solids pelleted and sintered at sufficiently high temperatures and the applicability of the concept of effective diffusivity. It would have been instructive to compare the values of the Thiele modulus (and possibly of the effective diffusivity) obtained from the model with those reported. Unfortunately, this is not possible since values of De and k from which dr can be calculated are not available for the systems tested. However, the value of & = 45.6, estimated earlier in the paper for the system MgO-Cr2Os at 1400 "C, is based on the experimentally observed zone width, and is clearly well within the range for the existence of a reaction zone in gas-solid reactions (Mantri et al., 1976; Tudose, 1970). This is also true for the other two systems tested. Thus, while the value of 4r is physically meaningful and the model does provide an acceptable
Ind. Eng. Chem. Fundam., Vol. 17, No. 2, 1978
predictive equation, experimental data on De and k for solid-solid reactions are desirable for a more rigorous test of the model.
Nomenclature C = concentration of the species under consideration, mol/ cm3 CO = concentration of the species at t = 0, x < 0, mol/cm3 C* = concentration of the species at t > 0, x = 0, mol/cm3 C, = concentration of the species at the end of the product zone, mol/cm3 Q = diffusion coefficient, cm2/s D = interdiffusion coefficient, cm2/s D, = diffusion coefficient of the species in the product zone, cm2h D, = diffusion coefficient of the species in the reaction zone, cmz/s k = first-order reaction rate constant, l/s L = length of the pellet, cm t = time, s w = dimensionless concentration x = distance along the length of the pellet, cm x , = width of the product zone, cm x, = total width: product zone + reaction zone, cm z = dimensionless distance x / L z, = dimensionless distance x p / L z , = dimensionless distance x ,/L
89
Greek Letters 0 = dimensionless time as defined by eq 5 4, = dimensionless Thiele modulus as defined by eq 10 and 18
Literature Cited Arrowsmith, R. J., Smith, J. M.. lnd. Eng. Chem. Fundam., 5, 327 (1966). Crank, J., "The Mathematics of Diffusion", 2nd ed, Clarendon Press, Oxford, 1975. Danckwerts, P. V., Trans. Faraday Soc., 46, 701 (1950). Greskovich, C., Stubican, V. S., J. Phys. Chem. Solids, 30, 909 (1969). Kwy, C., "Fifth International Symposium on Reactivity of Solids", Munich, 1964, p 21, Elsevier, Amsterdam, 1964. Krishnamurthy, K. R.,Gopalkrishnan, J., Aravamudan, G., Sastri, M. V. C., J. lnorg. Nucl. Chem., 36,569 (1974). Mantri, V. B., Gokarn, A. N., Doraiswamy, L. K., Chem. Eng. Sci., 31, 779 (1976). Minford, W. J., Stubican, V. S., J. Am. Ceram. SOC.,57, 363 (1974). Schmalzried, H., "Defects and Transport in Oxides", Battelle Institute Materials Science Colioquia, Ohio, 1973, p 83, Plenum Press, New York, N.Y.. 1973. Taplin, J. H., J. Am. Ceram. SOC., 57, 140 (1974). Tudose, R. Z.,Bull. lnst. Polytech. Din /AS/, 16 (20). 241 (1970). Wagner, C., Acta Metall., 17, 99 (1969). Whitney 11, W. P., Stubican, V. S., J. Phys. Chem. Solids, 32,305 (1971). Yamaguchi, G., Tokuda, T., Bull. Chem. SOC.Jpn., 40, 843 (1967).
Received for reuiew December 22,1976 Accepted December 1,1977
Prediction of Changes in Bubble Size Distribution Due to Interbubble Gas Diffusion in Foam Robert Lemlich Department of Chemical and Nuclear Engineering, University of Cincinnati, Cincinnati, Ohio 4522 1
A theory is described for the change in the distribution of bubble sizes that results from the diffusion of gas between the bubbles of liquid foam. The theory involves an earlier concept of gas transfer to and from an effective fictitious intermediate bubble of zero holdup. The result is an integro-differential equation which is converted to a finite difference equation that can be applied via digital computation to any given initial distribution to yield the corresponding successive distributions. Such distributions are shown in generalized dimensionless form for a general unimodal initial distribution.
Introduction The distribution of bubble sizes in a liquid foam can change as a result of at least two different spontaneous phenomena (Lemlich, 1973). One involves the rupture of the lamellae between bubbles. Another involves the transfer of gas between bubbles by diffusion. Some liquid foams are extremely resistant to the first phenomenon, that is, to rupture. However, no liquid foam is immune to the second phenomenon. It is the prediction of the latter phenomenon that forms the subject of the present paper. Interbubble gas diffusion occurs in the following way. The bubble sizes in a foam are never exactly uniform. Indeed, they often vary widely. Consequently, the slightly higher pressure in the smaller bubbles forces gas to diffuse through the lamellae into the larger bubbles. Thus the smaller bubbles shrink and the larger bubbles grow. This changes the distribution of bubble sizes. In fact, the smaller bubbles can shrink to the 0019-7874/78/1017 -0089$01.OO/O
point of disappearance, thus causing a change in the number of bubbles as well. The higher pressure in the smaller bubbles results from their smaller radii of curvature. According to the classical law of Laplace and Young, the pressure difference across a curved surface is given by
.=.(?+$)
(1)
where y is the surface tension and rl and r2 are the principal radii of curvature. Applying eq 1 to the pressure difference between a large submerged spherical bubble of radius rL and a small submerged spherical bubble of radius r s gives = 27
1 1 (6 - --)
(2)
The factor 2 appears in the foregoing equation because for a sphere r l = r2. 0 1978 American Chemical Society
