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
489
among their chemical species can be determined one trace element a t a time. The equilibrium composition for the major chemical elements and species excluding the trace elements will be the starting point. Analogous procedures could be developed using the classical equilibrium constant formulation. However, just the setting up of independent reactions and their equilibrium constants for 20-30 trace elements and 300-500 species would be a formidable task. Since free energy minimization uses a simple tabulation of the species formulas and free energies, problem formulation and computer programming are simplified. Also, the ease of determining the presence or absence of pure condensed phases in free energy minimization is particularly appropriate in consideration of trace elements.
Free Energy Minimization a n d Element Potentials The free energy minimization equations will be briefly reprised in the form best for trace element consideration. The total free energy of the system divided by the gas constant, R , and absolute temperature, T , GIRT, is
eq 4 and 5 in eq 1,interchanging the i and j summations, and substituting eq 2a with the result G - = Crjbj
RT Thus, the rj’s are dimensionless, intensive quantities which can be interpreted as the specific contribution to the free energy from each element. An analogy can be drawn with the sum of chemical potentials times species concentrations to give total free energy. The rj’s can then be termed element potentials (Powell and Sarner, 1969). Element Potential a n d T r a c e Species Given the equilibrium parameters for the major elements, in particular the element potentials, the trace species distribution can be determined. It will usually be sufficient to consider only one trace element per species. The trace element potential, rt, for that element can be separated from the rest in eq 4 (7) = ( C a l j r j h a j o r + altrt i An equation like (4) must hold for each pure phase species, regardless of whether it is trace or major. Only intensive quantities are present and not any involving element amounts. A trace element is defined as one which can be neglected in determining the system composition, to the closure error of the iterative free energy minimization process. The major element potentials, rj, are thus essentially independent of the trace element and its species, and we can solve eq 7 for at CL =
where zi is the number of moles of species i in the mixed phase, and the summation extends over the n’ species in the mixed phase. 5 is the sum of mixed phase species, Y z i . The subscript 1 represents pure condensed phase species, summed to a total of p pure phases; n is the total species in all phases, equal to n’ p . The ci contain all species and system parameters other than the mole numbers zi, and are written, for example, as follows: pure phase, GlOIRT; ideal gas species, GiOIRT In P; solution species with mole fraction activity coefficient yi, Gio/RT In yi. The superscript zero indicates standard state free energies. P is the system pressure in atmospheres. The equilibrium species values zi are those which make the total free energy of eq 1a minimum, subject to the constraint of the conservation of all of the chemical elements. These limitations are expressed as
+
+
+
where aji is the number of atoms of element j in species i, and bj is the total number of gram-atoms of element j . The summation extends over all species in all phases. The element balance conditions are included in the minimization process by means of Lagrange undetermined multipliers, r,. These quantities are part of the unknowns determined in the iterative minimization process. They were originally mathematical constructions, but turn out to have a physical significance and utility beyond the original intent. An equation (2) for each element j is summed with the r, and added to eq 1 j=m j= 1
rj(bj -
i=n
rt = [cl - (Caljrj)rnajorlla~t
(8)
In the usual situation there will be condensed, pure phases to consider, so the process is started with that presumption. Also, an equation like ( 7 ) cannot hold for more than one pure trace phase, since the major element potentials are not changed by various trace species distributions. The most likely pure phase is found by determining that one which gives the lowest (most negative) r t ,as found from eq 8. The next step is to use the rt determined above to find the mixed phase trace species values, zi, using eq 5. The amount of the trace pure phase identified above, zl, is found from the trace element material conservation equation
If a positive value of zl results from eq 9, that pure phase is stable and the composition distribution for that element is complete. A negative value indicates that the pure phase selected and all others for that element are unstable. Equation 9 will then be for the mixed phase only and must be solved for rt after substituting eq 5
ajizi) = 0
i=l
where m is the number of chemical elements. Differentiation of the augmented eq 1with respect to the mixed phase zi and pure phase zl gives (3)
(4) Equation 3 provides an explicit relation between the mixed phase zi and rj
The physical meaning of the rj’s can be shown by substituting 490
C aljrj
Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
Equation 10 is nonlinear in the unknown rt and can be solved by a standard technique such as Newton-Raphson. The allmixed phase zi values are now found from eq 5, as in the other case. The distribution of a trace element among the species available to it can thus be readily found. If kinetic or other information indicates, some species can be omitted from the set available to the trace element. A particular advantage of the present approach is the capability of setting up the computations in a general manner, without a priori regard to which pure trace phase might be stable under the given conditions. The computational error in the approximation used is less
Table I. Trace Element Volatilization in the COED Process
Element
Coal wt, PPm
Important species, %
Hg Se As
0.27 1.7 9.6
96 74 65
100 100 100
Pb Cd Sb
5.9 0.78 0.15
63 62 33
100 100 100
30 24 18
0.015 100
V Ni Be Cr B co cu Ge Mn Mo P Sn U Zn Ba
33 12
0.92 15 102 9.6 15 6.9 49 7.5
0.0
Hg Se As Pb Cd Sb V Ni Be Cr B co
cu
Ge Mn Mo P
Sn
0.0 0.0
100 41.2 33.4 100 0.03 83.8 100 100 0.0 100 4.0
71
4.8 1.3 272
130
Table 11. Minor Trace Element Species Considered in Thermodynamic Evaluation of the COED Process Element
Percent volatilized EquilibObserved rium
SDecies
than the iteration closure error, for ppm concentration levels. Results with a single element considered as both trace and major were essentially identical. Elements present in amounts up to the tenths of a percent level could be computed with the method given here to a reasonable approximation. This avenue was not pursued.
Trace Element Volatilization in Coal Gasification As an example of the foregoing technique, the trace element distribution has been computed for the outlet of a coal gasifier. Comparisons with some data on overall element volatilization in the COED process (Kalfadelis and Magee, 1975) are shown in Table I, for a final gasifier temperature of 1273 K and 1.5 atm pressure. Trace element concentrations in the feed coal for the first ten elements are those given with the volatilization data. Other feed coal values are typical ones for representative coals. Thermodynamic data were taken largely from the standard sources listed, with some estimation of missing values. The species listed in Table I are in general those having more than 0.1% of the trace element. Other species considered are listed in Table 11. The agreement of experimental and equilibrium values is seen to be semiquantitative. Elements found to be vaporized greater than 60% (Hg, Se, As, Pb, and Cd) were calculated to vaporize completely. Those measured less than 30% vaporized were calculated as essentially all condensed except for Be. More comprehensive data will help to establish the utility and range of applicability of the present considerations in this and other areas. Acknowledgment We thank P. S. Lowell and K. Schwitzgebel for helpful discussions. L i t e r a t u r e Cited
U Zn Ba
Chase, M. W., et al., J. Phys. Chem. Ref. Data, 3 (2), 31 1-480 (1974). Chase, M. W., et al., J. fhys. Chem. Ref. Data, 4 ( l ) , 1-175 (1975). Dantzig, G. B., Dehaven, J. C., J. Chem. fhys., 38, 2620 (1962). Kalfadelis,C. D., Magee, E. M., "Evaluation of Pollution Control in Fossil Fuel Conversion Processes. Liquefaction: Section I. COED Process", EPA 65012-74-009-3. Contract No. 68-02-0629. Linden, N.J., Exxon Research and EngineeringCo., 1975.
Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
491
Kelley, K. K., U.S. Bur. Mines Bull. 584 (1960). Parker, V. B., Wagman, D. D., Evans, W. H., Nat. Bur. Stand. (U.S.) Tech. Note, 270-6 (1971). Powell, H. N., Sarner, S. F., "The Use of Element Potentials in Analysis of Chemical Equilibria", Final Report, Vol. 1, Cincinnati, Ohio, Combustion Applied Research, Flight Propulsion Lab. Dept., 1959. Shapiro, N. Z.,J. SOC.lnd. Appl. Math., 960 (1969). Stinnett, S.J., Harrison, D. P.,Pike, R . W., Environ. Sci. Techno/.,8 (5), 441 (1974). Stull, D. R., Prophet, H., "JANAF Thermochemical Tables", 2nd ed, NSRDS-NBS 37, GPO, Washington, D.C., 1971. Wagman, D. D., et al., Nat. Bur. Stand. (U.S.) Tech. Note, 270-4 (1969). Wagman, D. D., et al., Nat. Bur. Stand. (U.S.) Tech. Note, 270-3, (1968).
White, W. B., J. Chem.phys., 46, 4171 (1967). White, W. B.. Johnson, S. M., Dantzig, G. B., J. Chem. Phys., 28 (5), 751-55 (1958). Zeleznik, F. J., Gordon, S..lnd. Eng. Chem., 60 (6), 27-30 (June 1968).
Kenneth A. Wilde* Michael E. Halbrook
Radian Corporation Austin, Texas 78766
Received for review January 10, 1977 Accepted July 25, 1977
Application of the Hildebrand Fluidity Equation to Liquid Mixtures
-
The Hildebrand fluidity equation &)/001has been applieg to simple binary and ternary_systems, = B( using the following mixing rules: V = XiXi7,", VO= 2Xivooi,B v = (61Vl0)"1 (B2v2')"2. . . , in which Vi" is the molar volume of a pure liquid, Toiis the limiting molar volume of a pure liquid, and vi is a volume fraction. These mixing rules allow prediction of the viscosities of mixtures from the properties of the pure liquids, with a maximum deviation of 10% in the 11 binary and 2 ternary systems considered. These predictions are shown to be comparable to, or better than, the best predictions of four commonly used "zero parameter" equations for viscosities of mixtures.
Recently, Hildebrand (1971) extended the concept of Batschinski (1913) to develop a two-parameter equation for the fluidity of simple liquids @ = B(V
- Vo)/Vo
(1)
The fluidity is considered to result from two contributions, the expansion of the liquid relative to a volume (TO) a t which fluidity ceases, and a parameter ( B )representing the ability of the molecules to absorb the energy of intermolecular collisions. The simple physical concept of this relationship suggests applicability to mixtures. The logkal approximations for molar volume ( V )and the parameter VOare simple molar -
v = X l P l O + XZV20 + . . .
(2)
+...
(3)
-
vo = X l V O l +
x2Po2
averages of the properties of the pure liquids, (Vio)and (Tol). Approximation of the B parameter for mixtures is not quite so straightforward. No clue to this combination can be gained from the dimensions of B (cm2/dyn-s),though the lack of molar dependence suggests that multiplication of B by some molar property might allow combination of the properties of the pure components. To investigate the behavior of this parameter for liquids, values of B were calculated from the fluidity of mixtures using the approximations of eq 2 and 3. Various combinations of the properties of the pure components were tested as approximations for the B parameter for mixtures. Volumetric combinations were clearly the most generally applicable, though there was no clear differentiation between several combinatorial forms (u, is volume fraction) (B8)mixture
+ U * ( B ~ P+ ~. . ~. )
=ul(~lVlo)
(BP)mixture
= ( B l V 1 " ) " 1 (BZV20)Q.. .
(4a) (4b)
and similar forms using limiting molar volumes (Voi)in place of molar volumes. The use of molar volumes provides a simpler mathematical form, and was chosen on this basis. In comparing approximations (4a) and (4b) for a number of binary mixtures, the two were found to have comparable applicability with approximately the same overall standard deviation be492
Ind. Eng. Chem., Fundam., Vol. 16, No. 4, 1977
tween calculated and "observed" values of B. However, a rough relationship was noted between the deviations of eq 4b and the nonideality of the solution as represented by AVexcess and AHexCes, which was not observed for eq 4a, indicating that eq 4b is probably the best approximation for an ideal solution. On this basis, the preferred form of the Hildebrand fluidity equation for mixtures was chosen as 9=
+ x2Vo2 + . . - ( X l V , O + X2V20 + . .
( B l V l O ) U ' ( B 2 V 2 0 ) U 2 .*
.[(XITO1
.)-I .)-I]
(5)
Calculations A serious difficulty in application of the fluidity equation to mixtures lies in the fact that the fluidities of pure liquids are quite sensitive to the parameters of eq 1 and are normally measured to a much greater degree of precision than can be calculated with this equation. Determination of these parameters requires viscosity data over a broad range of temperatures, data which have not been provided in the past by those workers who have studied the viscosities of mixtures. Therefore, a scheme has been devised to modify the parameters for pureliquids determined for broad ranges of temperature (B*, Vo*) to provide exact calculations of the viscosity for the reported conditions. An intermediate parameter 2 was calculated from the values of the molar volume and reported viscosity and tabulated values of B* and VO*(Hildebrand and Lamoreaux, 1972) or similar values calculated from literature data Z = B*V/@Vo*
(6)
Modified values of the parameters were calculated for application to the pure components of the liquid mixtures for which viscosities are reported.
E = @(Z- 1)
(74
vo = V(Z - 1)/Z
(7b)
-
These modified parameters do not differ appreciably from tabulated values (maximum deviation less than 0.3%) for pure liquids which obey the Hildebrand equation, but can be ex-
