to insert an adjustable cancellation dc voltage to restore the circuit to proper operation. However, this additional zero or balance adjustment is an inconvenience. At times, measurements of salinity profiles or stratification require the use of multiple probes. Multiple probes wherein each has its own cagehhield or needle probes to a reference distant ground plate have been employed. Using the frequency control method, only the cage electrode configuration provided the proper isolation between probes. Cross-talk can be very serious in this measurement method where frequency lock-in effects can take place. The multiple unguarded probe technique with a common ground plate cannot be employed in this conductivity measurement method. Further use of this measurement method will provide the full evaluation as to limitations and potential.
Acknowledgment The efforts of Siva G. Thangam, of the Department of Mechanical, Industrial, and Aerospace Engineering, in evaluating this method are appreciated, as well as the support of NSF Grant ENG73-03545-A01. Literature Cited Beckman Instruments. Inc., Cedar Grove, N.J. 07009, Technical Literature. Botos, B., "FET Current Regulators-Circuits and Diodes," Motorola Application Note AN-462. Khang, S. J., Fitzgeraid, T. J., lnd. Eng. Chem., Fundam., 14, 208 (1975). Leeds and Northrup, North Wales, Pa. 19454, Technical Literature. Wittlinger, H. A., Application Note CAN-5641, RCA Solid State Division, Somerville, N.J. 08876.
Received for review March 7, 1977 Accepted July 11,1977
A Transient Technique for Measuring Diffusion Coefficients in Porous Solids. Diffusion in Carbonaceous Materials Ralph 1.Yang,' Rea-Tiing Liu, and M. Stelnberg Department of Applied Science, Brookhaven National Laboratory, Upton, New York 7 1973
A transient technique has been developed for measuring effective diffusion coefficients of gases in porous materials. The technique is appropriate for measuring De in the range of low2to cm2/s. Comparisons between this technique and other techniques are discussed. Diffusion of two binary gaseous systems in a nuclear graphite and in a bituminous coal has been studied with this technique.
Introduction Problems connected with diffusion of gases in porous solids occur in many technically important areas. The voluminous literature published on this subject has been recently reviewed (Satterfield, 1970; Smith, 1970; Aris, 1975). Most of the research, both theoretical and experimental, has been concentrated on diffusion in catalysts. The results on various catalysts have been summarized (Satterfield, 1970). A standard experimental technique adopted in this area has been of the steady-state type which was perhaps first used by Buckingham (1904) and referred by many as the Wicke-Kallenbach technique (Wicke and Kallenbach, 1941). This technique employs flows of gases of different composition over two sides of a cell separated by a porous solid. Composition changes between gases entering and leaving the cell are measured and are related to the mass flux and the diffusion coefficient by using the solution to the governing diffusion equation. A problem arises when the diffusion coefficient is small. In this case, the gas flow rates must be kept low to effect a measurable change in the concentration of the diffusing gas. However, low flow would cause serious errors in data analysis because of the boundary layer problems. Here one recalls that in solving the diffusion equation of the system, the concentrations at the two boundaries are assumed to be equal to those of the bulk streams. A diffusion equation for this system with other boundary conditions has not been solved. With low flow rate, it can be shown that an error of 10-20% in diffusion coefficient can result from the boundary layer problem (Yang et al., 1977). With this constraint, the Wicke-Kallenbach technique is suitable and is normally used for measuring diffusion coeffi486
Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
cients of greater than cm2/s. It is, therefore, desirable to develop a rather simple technique with which diffusion coefficients of smaller than cm2/s can be accurately measured. This communication summarizes a transient technique which is suitable for diffusion coefficients in the range of 10-2 to 10-8 cm2/s. Diffusion in some technically important materials in this category, such as coal and graphite, has been studied and the results are discussed. It should be noted that some techniques for measuring very small diffusion coefficients have been developed previously and have been employed successfully for zeolites (Ma and Lee, 1976; Sarma and Haynes, 1974). It should also be noted that a transient technique similar to this one was published by Gorring and deRossei (1964). A comparison between these two transient techniques will also be given in the Discussion. Experimental Section Apparatus a n d Procedures. As shown in Figure 1, the apparatus consisted of a hollow, spherical chamber. The spherical solid sample was placed in the central portion of the chamber and was supported by three small posts. The chamber had a diameter of 2.54 cm and the spherical samples were of the sizes of about 2 cm in diameter. Temperature in this work ranged from 23 to 75 "C. The sweeping gas, gas B, was preheated with copper coils immersed in a constanttemperature oil bath. The line leading to the diffusion chamber was heated with heating tapes whose temperature was controlled. The diffusion chamber was also heated with temperature-controlled heating tapes. The temperaturecontrolling thermocouples were placed at the inner surface
TO VACUUM
IJ
D,,crn2/sec
Figure 2. Range of applicability of the technique. H
G
Figure 1. Schematic diagram of the apparatus: A, flow meter; B, constant-temperature oil bath; C, heated line (B to D); D, diffusion chamber; E, pressure gauge; F, sphere of porous solid; G, supporting post; H, O-ring seal; I, thermocouples.
et al., 1972). Total voids fractions of the graphite and of the coal are 0.173 and 0.167, respectively. Gases used were of the following grades: CH4, CP grade with purity = 99%; Cop, Coleman instrument, purity = 99.99%; and Nz, prepurified, purity = 99.996%. Calculation of Diffusion Coefficient Diffusion of A in the sphere is governed by the equation
ac,=.e(!%+zac,) at
of the diffusion chamber. Blank runs were made to calibrate the temperature of the solid from the temperature of the controlling thermocouples. Thermocouples placed at the center of the sphere (through a well drilled to the center) read less than 1 "C lower than the controlling temperature. Temperatures reported in this work were the calibrated or the solid sample temperature. The experimental procedure involved the following five sequential steps: degassing, saturating with gas A, diffusion (sweeping with gas B), equilibrating in the chamber, and analysis of the gas. All steps were performed isothermally. The chamber containing the sample was first degassed to Torr with a diffusion pump for a period of about 20 h. Gas A was introduced to the system and the pressure was gradually adjusted to about 77 cmHg. The sample was soaked in gas A for about 20 h. The preheated gas B was then swept through the chamber at a rate of 300 cm3 (STP)/min. The pressure was controlled at 77 cmHg by a needle valve downstream of the chamber. The transient diffusion of A out of the porous solid into the sweep gas B took place during this step. After a certain period of time, the diffusion chamber was closed and the gas composition was equilibrated inside and outside of the porous solid for a period of about 20 h. Gas in the chamber was then analyzed with a gas chromatograph. It was with this composition data that the diffusion coefficient was computed. Materials. A nuclear graphite (H-451) and Illinois No. 6 bituminous coal supplied by Pennsylvania State University (PSOC-26) were the porous materials studied. Methane was gas A and Nz or COZwas gas B. The solid materials were fabricated into spheres. The graphite sphere was machined to a diameter of 1.905 cm and the coal sample was manually fabricated to a diameter of 2.06 f 0.1 cm. Both spheres were cleaned of dust ultrasonically before diffusion experiments. The total volume of the diffusion chamber (between the two stopcocks in Figure 1)was 11.998 cm3. Analyses of the samples are available elsewhere (Yang et al., 1977; Gan et al., 1972). Pore size distribution of the coal sample at pore diameters greater than about 200 A has been measured with a Micromeritics mercury penetration porosimeter and the results agreed with that reported by the Penn State University group (Gan
r ar
with the boundary conditions and (3) and the initial condition A solution of the above system is
The total amount of A remaining in the sphere is
To calculate De, using eq 6, At is obtained from the experimental results. Figure 2 shows the ranges of applicability of this technique. In this figure, the amount diffused out of a sphere of radius 1cm is plotted against De ranging from to 10-9 cm2/s, for three periods of diffusion time. A diffusion time of 3 min is good for De as high as lo+ and 3 h is good for low De values such as Without a priori knowledge of the order of magnitude of De, two or more runs may be required to determine the appropriate diffusion time for accurate measurements. In calculating De using eq 6, a few terms (e.g., 3 terms) in the converging series are adequate, provided that the value of Det/R2 is greater than Also, for values of DetlR2 smaller than l O - l , the following approximation for error function solution may be used more conveniently
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487
Table I. Diffusion Coefficients in the PSOC-26 Coal Temp, "C 24 36
51 70 25.6 38 52 69
De, cm2/s For the Pair CH4-CO2 9.9 x 10-7 5.6 X 5.8 x 2.4 x 10-5 For the Pair CHd-Nz 8.3 x lo-+ 1.8 X low5 2.1 x 10-5 4.9 x 10-5
Dgas,cmz/s 0.165 0.172 0.190 0.210 0.22 0.23 0.25 0.27
Equation 7 has been derived previously (Crank, 1975). The approximation in eq 8 gives less than 0.01% error for Det/R2 < 10-l. Equation 8 will be used in this work since the above condition holds.
Results a n d Discussion Two sets of results will be discussed here, one with the PSOC-26 coal and one with the H-451 graphite. Before the measurements of the diffusion coefficients, control experiments were performed to test the validity of the initial condition, i.e., eq 4. This was necessary because it has been found by the authors that significant amounts of methane can be adsorbed at 1atm and room temperature by some coals (Yang et al., 1977). The adsorbed methane will desorb as well as diffuse on the surface during the diffusion experiment, and eq 6 and 8 will not be applicable. The control experiments involved saturation with methane followed by rapid flushing (for about 20-30 s) of the diffusion chamber and measuring the total amount of methane left in the sphere. This amount was compared with the calculated amount left in the void fraction of the sample. Any excess was accounted as the adsorbed phase. For the two samples reported here, the amount adsorbed was negligible and the initial condition was judged valid. Here the initial concentration CAOis taken as the product of the void fraction and the gas phase concentration of A. The above rationale and discussion are valid for the case that we are only concerned with diffusion in the pores. However, when adsorption and surface diffusion take place, the technique is also useful to measure the overall diffusion coefficient. In this case, diffusion in the pores and on the pore wall surface both contribute to the overall diffusion rate. The equations derived here are also applicable for this case provided: (1)CAO, which is the sum of the gas phase and the adsorbed phase, is accurately measured, and (2) the adsorption isotherm is linear (Smith, 1970; Gorring and deRosset, 1964). We have chosen to stay away from this situation because CAO is difficult to measure accurately and the assumption of linearity of the isotherm is often not valid. When local multilayer adsorption takes place at small relative pressures, and it often does, the surface diffusion rate is further affected (Yang et al., 1973; 1974). Values of diffusion coefficient measured with this technique are summarized in Tables I and 11. The binary systems CH4-CO2 and CH4-Nz were chosen because they are relevant to problems in coal mining technology, coal mine safety, and coal conversion processes. No comparison can be made between the results shown in Table I and those in the literature. Work done in this area was primarily concerned with flow of gases in coal due to a pressure gradient, instead of a concentration gradient (Nandi and Walker, 1975; Karn et al., 1975). In this work, it has been attempted to calculate the diffusion coefficient using information on pore size and voids distributions, and to compare the calculated results with that measured. The theoretical models used were the cylindricalpore model of Johnson and Stewart (1965) and the random488
Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
Table 11. Diffusion Coefficients in Nuclear Graphite H-451 For the pair CH4-COz Temp, "C De, cm2/s 23 36
51 61
1.61 x 1.72 x 1.84 x 1.90 x
For the pair CH4-Nz D e , cm2/s
Temp, "C
10-3
26
10-3 10-3
38 52 75
10-3
1.69 x 1.83 x 1.95 x 2.00 x
10-3
10-3 10-3 10-3
pore model of Wakao and Smith (1962 and 1964). The values calculated were about two orders of magnitude greater than those measured. The geometrical constant K was taken as 0.3 for the H-451 graphite and 0.2 for the coal sample. The value for coal is less than 0.3 because it is less isotropic (Nandi and Walker, 1975;Karn et al., 1975), although the value of 0.2 was arbitrarily taken. The models are therefore not applicable for diffusion in coal. This is because the models were constructed on the basis of the diffusion processes which are caused from collision, such as Knudsen, gas-phase and transient-region diffusion. Flow of gases in coal, due to pressure gradients, was thought of as an activation process (Nandi and Walker, 1975; Karn et al., 1975) due to the great fraction of the ultrafine pores of diameters of a few Angstroms in coal. The great discrepancy between the models and the experimental results would support the view that diffusion in coal is dominated by the activation process. Arrhenius plots of the data shown in Table I give apparent activation energies of 16.1 kcal/mol for the CHI-CO2 pair and of 8.5 kcal/mol for the CH4-N2 pair. The results with the nuclear graphite H-451 are shown in Table 11. Data in the literature on other types of nuclear graphite (H-451 data are not available in the literature) gave the De/Dgasratio in the range of 0.007 to 0.013 for various gas pairs including the two being studied (Hewitt and Sharratt, 1963). The results here give a ratio of 0.0097 for the CH4-COz pair and 0.0078 for the CH4-N2 pair. The agreement thus seems reasonable. Due to the larger pores (compared to that in coal) in graphite, the models have been used by the authors quite successfully in calculating the De values from information on the pore structure. Such results will be published elsewhere. Finally, it would be interesting to compare this technique with the technique developed by Gorring and deRosset (1964). In their technique, they measured the exit gas concentration at various times, while in this method, we measure the integrated amount left in the porous sphere at a certain time. Obviously, their method provides higher accuracy for greater diffusion coefficients whereas this method is better for smaller diffusion coefficients. Now we proceed to make a quantitative comparison based on the same conditions for two extreme cases: D e = and cm2/s. First, we assume that there is no adsorption (this assumption is made to simplify the comparison and it does not affect the comparison), or the adsorption coefficient ( R ) is equal to zero. The following values are used: voids = e = 0.3; radius of sphere = 1 cm; CAO = 4.17 X g-mol/cm3; VN and VH (flow rates of N2 and Hz, respectively, in their case) = 1.33 and 1.67 cm3/s. Under these conditions, for De = cm2/s, the concentration of the exit gas in their case = 9.6 X lo3 ppm (at 20 s) and the amount left ppm (at 3 min); for De = cm2/s, in our case = 3.4 X their concentration = 53 ppm (at 20 s) and our concentration = 7.03 X lo4 ppm (at 3 min). It is clearly shown that the two methods are complimentary to each other. In other words, it is more desirable to combine the two methods and measure the instantaneous concentration when De is greater than approximately cm2/s. The flow rates used in their method were kept low to effect higher concentrations whereas the flow rate is not a limiting factor in this method. The low flow rate can cause serious errors due to the boundary layer built around
the sphere, as has already been discussed in this note. The flow rate used in this work is about 3 times higher than theirs. For De smaller than about cm2/s, it should be more desirable to use this method. In summary, the transient technique developed in this study is appropriate for measuring diffusion coefficients in porous materials in the range of 10-2 to 10-8 cm2/s. The inherent factors which influence the accuracy of the measurement in the Wicke-Kallenbach technique are also operative in this technique. The factors are: accuracies in the gas analyses, disturbance at the boundary of the porous sample, and temperature and pressure controls. This technique can be applied to studies with carbonaceous materials as has been demonstrated in this work. Acknowledgments We acknowledge the helpful discussions and suggestions of Dr. A. L. Berlad of The State University of New York at Stony Brook. We thank Dr. C. N. Satterfield of Massachusetts Institute of Technology for bringing to our attention the paper by Gorring and deRosset, and for other helpful suggestions. Robert Smol offered invaluable technical assistance in all phases of this work. This work was performed under the auspices of the Office of Molecular Sciences, Division of Physical Research, U S . Energy Research and Development Administration, Washington, D.C. Nomenclature A0 = total amount of A remaining in pores a t time zero, g-mol At = total amount of A remaining in pores a t time t , 5:-mol CA = concentration of A in pores, g-mol/cm3 CAO = initial concentration or CA a t time zero, g-mol/cm3
D e = effective diffusion coefficient in porous solid, cm2/s erf = error function erfc = complementary error function ierfc = integral of erfc, defined in Crank (1975) K = geometrical factor in the parallel-pore model r = radial distance from center of the sphere, cm R = radius of the sphere, cm t = time, s Literature Cited Aris, R., "The Mathematical Theory of Diffusion and Reaction in Permeable Catalysts", Vol. 1. Claredon Press, Oxford, 1975. Buckingham, E., US. Dept. Agriculture, Bu. Soils, Bull. No. 25 (1904). Crank, J., "The Mathematicsof Diffusion", p 91 2nd ed. Claredon Press, Oxford, 1975. Gan, H., Nandi, S. P., Walker, P. L., Jr., Fuel, 51,272 (1972). Gorring, R. L., deRosset, A. J., J. Catal., 3,341 (1964). Hewitt, G. F., Sharratt, E. W., Nature, 198, 952 (1963). Johnson, M. F. L., Stewart, W. E., J. Catal., 4,248 (1965). Karn, F. S.,Friedel, R. A., Sharkey, A. G., Fuel, 54,274 (1975). Ma, Y . H., Lee, Y. T., AlChEJ., 22, 147 (1976). Nandi, S.P., Walker, P. L., Jr., Fuel, 54,81 (1975). Sarma, P. N., Haynes, H. W.. Jr.. Adv. Chem. Ser., No. 133,205 (1974). Satterfield, C. N., "Mass Transfer in Heterogeneous Catalysis", MIT Press, Cambridge, Mass., 1970. Smith, J. M., "Chemical Engineering Kinetics", 2nd ed, McGraw-Hill, New York, N.Y., 1970. Wakao. N., Smith, J. M., Chem. Eng. Sci., 17, 825 (1962). Wako, N., Smith, J. M., lnd. Eng. Chem., Fundarn., 3, 123 (1964). Wicke, E., Kallenbach, R., Kollold-t, 97 (2), 135 (1941). Yang, R. T., Liu, R. L., unpublished results on boundary layer calculations, Brookhaven National Laboratory, Upton, N.Y., 1977. Yang, R. T., Fenn, J. B., Haller, G. L.. AlChEJ., 19, 1052 (1973). Yang, R. T., Fenn. J. B., Haller, G. L., AlChE J., 20, 735 (1974). Received for review M a r c h 25, 1977 Accepted June 29,1977 T h i s work was performed under t h e auspices o f t h e Office o f M o l e c ular Science, D i v i s i o n o f Physical Research, U S . Energy Research and Development Administration, Washington, D.C. 20545.
COMMUNICATIONS
Thermodynamic Distribution of Trace Elements by Minimization of Free Energy
A technique has been developed for the efficient computation of the thermodynamic equilibrium species distribution for a large number of trace elements. Illustration was made for the case of an ideal gas phase and several pure condensed phases. Comparison with available data for a coal gasification process shows semiquantitative agreement of observed and equilibrium overall element volatilization.
The assessment of the fate of trace elements often involves the thermodynamic equilibrium distribution among the available chemical species. This note will not be concerned with the criteria for applicability of equilibrium, but with the ready determination of the consequences of that assumption. The method outlined below is capable of being generalized, but consideration will be limited to the case of a single mixed phase and several pure phases. This model will be adequate for many purposes. At least it will often be sufficient for the basic thermodynamic data available. The direct minimization of the free energy of a chemical
system to determine its equilibrium state has been discussed by many authors (White, 1967; White et al., 1958; Shapiro, 1969; Dantzig and Dehaven, 1962; Zeleznik and Gordon, 1968; Stinnett et al., 1974). General algorithms and computer programs have been developed to handle many elements and species in multiphase systems. However, the inclusion of 20-30 trace elements and their chemical species along with 6-8 major elements renders the problem of unwieldly if not unmanageable size. Problems of accuracy also may arise with trace elements in the minimization process. It will be shown how the distribution of trace elements Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
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