Ind. Eng. Chem. Fundam. 1982, 2 1 , 262-268
262
Siiberberg, H.; McKetta, J. J.; Kobe, K. A. J. Chem. f n g . Data 1959,4 , 323. Soave, G. Chem. Eng. Sci. 1972,27, 1197. Starling, K. E. "Fluid Thermodynamic Properties for Light Petroleum Systems"; Gulf Publishing Co.: Houston, Texas, 1973. Stewart, D. E.; Sage,B. H.; Lacey, W. N. Ind. Eng. Chem. 1954,46,2529. Streett, W. B.; Staveiey, L. A. K. Adv. Cryog. Eng. 1988, 13, 363. Stuart, E. 8 . ; Yu, K. T.; Coull, J. Chem. Eng. Prog. lQ50.46,31 1. Tickner. A. W.; Lossing, F. P. J. phvs. Coibid Chem. 1951,55, 733. Tsonopoulos, C. AIChE J. 1974,20,263. van der Waals, J. D. Doctoral Dissertation, LMen, Holland, 1873. Van Ilterbeek, A.; Verbeke, 0.; Staes, K. Physice lB89,29, 742. Vargaftk, N. B. "Tables on the Thermophyslcal Properties of LiquMs and Gases", 2nd ed.;Hemisphere Pub. Co.: Washington, DC, 1975. Vennix, A. J. Ph.D Thesis, Rice University, 1966. Vohra, S. P.; Kobe, K. A. J. Chem. Eng. Data 1959,4 , 329. Wang. J. S.;Van Daei, W.; Starling, K. E. Can. J. Chem. Eng. 1978,5 4 , 241
Waxman, M.; Davis, H.; Sengers, L.; Klein, M. "The Equation of State for Isobutane: An Interim Assessment"; Thermophysics Division, NBS, 1978. West, J. R. Chem. Eng. Prog. 1948,4 4 . 287. Wlliingham, C. B.; Taylor, W. J.; Pignocco, J. M.; Rossini, F. D. J. Res. Natl. Bur. Stand. 1945,35, 219. Wilson, G. M.; Johnston, R. H.;Hwang, S.C.; Tsonopouios, C. Ind. Eng. Chem. Process Des. Dev. 1981,20,94. Yesavage, V. F. Ph.D. Thesis, University of Michigan, 1968. Young, S. J. Chem. SOC. 1900, 77,1145. Zwolinski, B. J.; Wiiholt, R. C. "Vapor Pressure and Heats of Vaporization of Hydrocarbons and Related Compounds"; API 44-TRC 101, 1971.
Received f o r review March 23, 1981 Accepted February 24, 1982
Gaseous Diffusion in Porous Solids at Elevated Temperatures Ralph T. Yang' Department of Chemical Engineering, State University of New York at Buffalo, Amherst, New York 14260
Rea-Tllng Llu State University of New York at Stony Brook, Stony Brook, New York I1780
The Wicke-Kallenbach (WK) technique was applied to the measurement of the binary gaseous diffusivity in a porous carbon at temperatures up to 700 OC. The severe leaking problem associated with the high-temperatwe application of the technique was solved by electroplating of soft metal layers on the sample. The diffusivity was also independentlycalculated from the reaction rate at a very low conversion through the use of a model. Comparisons of the two values showed that the WK diffusivitiis were substantially lower (by about 60%) than those calculated from reaction conditions. The commonly used expression in modeling and design studies, De, = c2D,, was found to yield values by over an order of magnitude too high in the temperature range of practical interest, e.g., 500-1200 'C. A remedy to this problem is to replace the molecular diffushrity (13,) in the expression by a transition diffusivity (Dt). D,may be approximated by using a hypothetical single pore size.
Introduction Knowledge of the rate of gaseous diffusion in porous solids at elevated temperatures is essential in understanding the kinetics of heterogeneous reactions. For the gas-carbon reactions, information on the pore diffusion at elevated temperatures is desirable in developments of coal conversion processes and of the high-temperature gas-cooled nuclear reactors. In our previous work, the experimental effective diffusion coefficients in several carbonaceous materials were compared with the predicted values using two structural models familar to workers in the field of heterogeneous catalysis (Yang and Liu, 1979). The predicted values were consistently higher than the measured data by a factor ranging from 1.3 to 15.4. The discrepancy was attributed to the highly tortuous path as well as the dead-end pores. That work was limited to temperatures below 100 "C. Diffusion data for the elevated temperature range are scarce, if not nonexistent, in the literature. Roberts and Satterfield (1965) have estimated the effective diffusivities in a carbonaceous material under reaction conditions by using an assigned temperature coefficient and extrapolated from the room temperature data. The WickeKallenbach type (WK) diffusion apparatus has been used for diffusion measurement at elevated temperatures. However, a severe leakage problem exists for the high-temperature work. It is caused by the stronger 0 196-43 1318211021-0262$01.25/0
temperature dependence (near !P/2) of leak through the space between the sample holder and the sample plug, as compared to the temperature dependence of pore diffusion (near P'2).The first experimental demonstration of the severe leakage problem was done by Growcock and coworkers (1977),in which the leaking rate could be an order of magnitude higher than the pore diffusion rate in a graphite sample at 500-800 OC. Golovina reported a temperature coefficient of 1.34 for C02/N2 in a graphite membrane at temperatures ranging from 20 to 600 O C (Golovina, 1969). Nichols measured diffusivities for several binary systems in a graphite at temperatures up to 700 " C (Nichols, 1961). In both cases, the leaking rate was uncertain, and not identified. Makaiho and Takahashi (1963) developed a sealing between graphite and glass by using silver and silver chloride, and they obtained diffusivity data for temperatures up to 200 "C. It is also worth noting that Olsson and McKewan (1966) measured the effective diffusivities under reaction conditions using a technique not related to the WK cell, but the technique is applicable to only a few gas-solid reaction systems. In this work, we solved the leakage problem in the WK cell by using an electroplated metal layer around the graphite sample and obtained binary diffusivities in the temperature range of 27-700 "C. In a parallel but independent study, the effective diffusivity was calculated from the overall reaction rate under reaction conditions, by the 0 1982 American Chemical Society
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982 HINOMETER
T l m
GAS A
STAINLESS STEEL F L I N G E
STAINLESS STEEL SAMPLE HOLDER GAS TANK ELECTROPLATED SAMPLE SURFICE
VEN
GRAPHITE SAMPLE
G.C
TWO-WAY SWITCH VILVE
283
GAS TANK
B
~
G
~
~
~
~
~
R
GAS B
, ! E NEEDLE VALVE
FURNACE
Figure 1. Schematic of flow system for the diffusion cell.
aid of a model developed here. Under the conditions where the porous structure is uniform and the effective pore diffusivity is near a constant, it is possible to extract the value of the pore diffusivity from the overall rate data provided that the rate expression is known. It could be a profitable approach to obtain the porous diffusivity under reaction conditions in this manner. The necessary tool for this approach is a reliable model which correlates the pore diffusivity, the overall rate, and the intrinsic surface rate. This approach has been applied to the carbon gasification reaction with COP Cylinder shaped graphite samples were used as the carbon reactants. The overall rates were measured gravimetrically. The extracted diffusivities will be compared with the values measured with the WK apparatus. This comparison may be made because the rate of the carbon gasification was measured at approximately 0.1% to 0.5% burnoff of the carbon, and consequently, the porous structure of the graphite remained practically the same as the sample used in the direct diffusion measurements. Experimental Section The carbon sample used in the study was a nuclear graphite. The pore structure and the other characteristics of the graphite have been given elsewhere (Yang and Liu, 1979). The porosity was 17.7%with the following bimodal pore size distribution: micropores centered at 40 A diameter with a porosity of 4% and macropores at 2000 A diameter with a porosity of 13.7%. The BET surface area was 4.98 m2/g. The overall reaction rate of COP with graphite was measured with a Mettler 2000 TG system. The TG technique has become rather standard. The details of the experimental method may be found in Yang and Liu (1979). Here, we shall only discuss the technique for the direct diffusion measurement. The flow system is shown in Figure 1. Any pressure differences between the two sides of the sample were read with the manometer and eliminated with the back pressure regulators, or needle valves. The downstream gas composition on both sides were measured by a gas chromatograph. The temperature was controlled within 2 "C. The most important part of the apparatus was the diffusion cell, which is shown in Figure 2. The pressure was 1atm. The sample graphite cylinder was cleaned to rid it of dust ultrasonically. The sample was subsequently electroplated with three successive layers, copper, nickel, and platinum, from the inner layer. The total thickness of the electroplated coating was 0.025 in. The coating was on the entire sample, while leak testa were done prior to the diffusion measurements. The coatings on the two ends of the cylindrical sample were cut off for the diffusion flux measurements. Further sealing between the sample and the stainless steel holder was provided by two gold O-rings, which were press-fitted by two stainless steel flanges, as shown in Figure 2. The leak tests involved two experiments: helium displacement and blank runs. The helium displacement experiment involved a measurement of the volume of helium
-GOLD
O-RINQ
Figure 2. The diffusion cell.
displaced by the entirely coated sample after the system was evacuated at room temperature. The porosity of the graphite, as mentioned, was 17.7%. The helium displacement was identical with the apparent geometric volume. The coating was, therefore, impermeable to helium at room temperature. The blank runswere conducted at temperatures up to 700 "C with the entirely coated sample in the diffusion cell. No flux was detected through the sample with Ar and N2flowing on the two sides of the cell. The sealing was hence concluded satisfactory, and the coatings on the two ends were cut off for flux measurements. In a typical flux measurement, the system was first evacuated. Gas A was introduced into the system on both sides of the cell. Gas B was subsequently introduced on one side. The typical flow rate was 2-5 cm/s. The downstream compositions were measured intermittently, while the temperature was controlled to a constant value with the aid of thermocouples inserted near the graphite sample. The lighter gas, N2, was always on the top side of the cell. Steady-state diffusion was reached when there was no change with respect to time in the downstream gaseous compositions. The time to reach steady state was less than 30 min for all runs. Model Derivation Method of Calculation of the Diffusion Cell Data. The flux for A in B in the sample is
The boundary conditions are
Upon integration of (1)
or
The flux, N A , can be calculated as the product of Xu and the flow rate of B, divided by the cross-sectional area of the sample. It is not uncommon to find calculations in the literature, using the WK cell, based on the Fick's law of diffusion DAB,eff,F
=
NAL CT(XA1
- XA2)
This method of calculation is mistaken when equimolar wall effects, or more precisely, the momentum transfer between wall and molecules, play any role in the overall diffusion.
264
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
This problem has been studied, and the ratio of the fluxes has been shown to be equal to the square root of the inverse of the molecular weight ratio of the two diffusing gases (Hoogschagen, 1955; Dullien and Scott, 1962). Equimolar counter diffusion is valid only in large channels. To our knowledge, a precise relation between the flux ratio and the pore size (and other factors) has not been established. Model for Calculating D& from Reaction Rate. Our approach is to use the Stefan-Maxwell flux equations, in which all the binary molecular diffusivities (Di.mol)are to be replaced by the effective transition binary diffusivities (Dij). The values of Dij are what we are to extract from the overall reaction rate data. Comparing our approach with the dusty-gas-model approach, the relationship between the two aforementioned diffusivities is (Evans et al., 1961; Scott and Dullien, 1962)
(7) The left-hand side of the equation is from the dusty gas model. The Di. in the right-hand side will be used in the Stefan-Maxweh equations for our derivation. The original Stefan-Maxwell equations do not account for the molecule-wall momentum transfer. The D i .values, however, are the effective transition values, which are to be experimentally determined. The overall rate is to be expressed using the flux equations. The flux equations should involve the effective diffusivities, which are the values to be used in our denB, such rivation. For a binary reaction system, A derivations using the dusty gas equations have been made (Apecetche et al., 1973; Kehoe and Aris, 1973; Jackson, 1977). It is possible to make an analysis for the multicomponent system, which is our case, using the dusty gas equations. However, there are several disadvantages in doing so. First, if one starts with the dusty gas flux equations, by match with the rate data, one would obtain the effective Knudsen diffusivities. By using the Bosanquet equation (with the added CY factor), one may then calculate an effective transition diffusivity. But here the Knudsen diffusivity would be entirely meaningless with respect to the pore structure. Secondly, for bimodal pore distributions, which is our case, poor agreement between the dusty gas model and experimental data has been found (Omata and Brown, 1972). Thirdly, such an analysis for a multicomponent system would be chaos mathematically, as compared to the approach which follows. The StefanMaxwell equations for the multicomponent system are
-
n
For the specific case of the carbon gasification reaction, a diluent (Ar) was used to slow down the reaction to facilitate accurate measurements at a low burnoff level. Therefore, we treat the three-component system: COz,CO, and Ar. For such a system, eq 9 yields two independent equations, which are subject to the constraint: XA + XB + X c = 1. The subscripts A, B, and C pertain to the species C 0 2 , CO, and Ar, respectively. Also, since the reaction is slow and the gaseous fluxes are several orders of magnitude greater than the carbon burnoff rate, the system may be considered as in a quasi-steady state. According to the stoichiometry
~ N A-NB; N c = 0 For C02
Similarly, for CO
To determine Db and DBm,we may substitute eq 11 into eq 10, and
Db=
1+ 2 x A
The above approximation is made because Dm z DAC. Similarly
Substituting eq 14 and 15 into eq 12 and 13, respectively, we have
(17)
The following Langmuir-Hinshelwood rate equation has been well established for the C-C02 reaction.
i
where i # j . We define a binary diffusivity Dimfor the diffusion of species i in a mixture, then eq 8 can be rewritten as
Ni = -CTDi,VXi
+ X ij=1 CNj
Now, a mass balance may be made by using the above rate equation as the local rate, and write in the cylindrical coordinates, for COz
(9)
and for CO
where n
Ni - Xi Dim =
n 1
-(XjNi j=1Dij where i # j .
N,
j=l
(10)
- XiNj)
Multiplying eq 19 by 2, and adding to eq 20
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
Table I. Effective Diffusivity of Ar-N, in Graphite flux D A ~ Nx, Dy2-&X ratio, T,K lo4,cmz/s 10 , cm'/s -N,/Ar 7.96 1.15 29 5 7.43 423 493 668 793 978
11.31 15.59 20.54 25.79 39.55
10.99 13.81 19.01 22.76 34.66
'
0
7
6.57 8.74 11.69 14.29
7.75 10.47 12.66 15.61
ratio,
-N,/CO, . . 1.38 1.42 1.17 1.19
This equation may be integrated with the boundary conditions XA = XAS; XB = XBS = 0 at r = R (22) and assume that DAC r DBC,we have
'
"
17
1
1
SLOPE = I I73
flux
T, K 291 380 480 593
'
/
1.06 1.28 1.17 1.29 1.31
Table 11. Effective Diffusivity of C0,-N, in Graphite Dco,-N, X D N 60 x lo4,cmZ/s 10": cm%/s
'
265
I
I
c
L
IO1,
2
3
4
5 6 78910
TEMPERATURE, Kxld.'
Figure 3. Effective diffusivity vs. temperature for Ar-N2.
The above approximation was made because dilute CO, in Ar was used, and XAS10.1 in all cases. A maximum of 3% error may thus be incurred. Substituting eq 23 into eq 19
where K1 = kl
: The boundary conditions are XA=XAS;
-dXA - -0;
dr the overall reaction rate is Rd=--
DAC
dXAi
1 + XM dr
r=R r=O i 1 I 2 3 4 TEMPERATURE,
(2rRLC.T)
,
1
5
,
I
,
,
6 78910
Krld.'
Figure 4. Effective diffusivity vs. temperature for C02-N2.
r=R
The values of Rd, K1, K,, and K3 were measured experimentally. Equation 24 was solved numerically using the Runge-Kutta-Verner fifth-order method, and a trial technique was adopted to guess the missing initial condition. From the numerical solution, DAC was calculated from eq 29. Results and Discussion Diffusivity from the WK Measurements. The effective binary diffusivities measured with the leakproof WK technique at temperatures up to 700 "C are summarized in Tables I and 11. The upper temperature for the C02-N2 pair was limited to 320 "C in order to avoid appreciable reaction between COz and the graphite. In the same tables, the ratios of the measured fluxes are also shown. The differences in two fluxes for the binary system,
NA in B and NB in A, are quite significant. The diffusivities calculated from the flux measurements based on Fick's law have also been obtained, which showed errors averaging about 20% of the correct values. The temperature dependence of the macropore diffusion is nearly 10/2(based on the Lennard-Jones 6-12 potential and the ideal gas law, although experimentally the exponent is 1.5-2.2), whereas for the micropores with Knudsen diffusion, it is nearly PI2. The theoretical values for the counter flux ratio are: -N2/Ar = 1.19 and -Nz/C02 = 1.25 (Hoogschagen, 1955; Dullien and Scott, 1962). Comparing these values with the data shown in Tables I and 11,we see a random deviation of less than 12%. Clearly, the earlier theories may be applied to the high-temperature range. The temperature dependences of the binary diffusivities are shown in Fig-
Id.Eng. Chem. Fundam., Vol. 21, No. 3, 1982
288
0,70/250 MESP ,50/170 *100/150 .80/l00 r60/80 a42/65 n28/35
Temp 1143 'C I otm
v
8
I
,i" 32-
.
>
I1 1
we35
Z
2
I
"
101
" "
'u
"
09t
84
IC-
'ME
4l
5 3
mri
Figure 5. Effect of size on the conversion-time curve.
Sire 100- 150pm
7
Somple Nucleor grophite H-451 I
bulk €!as, diameter, %CO, in. in Ar
1- a~12+ D i ] , m o l / D ! a
- ayil + Da),mal/Dki )/In(-) (30)
Here, e is used to account for the void fraction, and 7 is used as nothing but a fudge factor, which is called "tortuosity factor." With the bimodal pore distribution, one may assume many pore structures and accordingly calculate as many DlUvalues. For each Dh value, from our experimental data one may calculate a "tortuosity fador." Such an exercise and a comparison of the 7 value would not be very meaningful. Discussion based on other pore diffusion models will be included in later sections. Diffusivity from Reaction Rate. To apply the model derived in the foregoing for extracting values of the effective diffusivity, one needs data on the overall rate, Rd, and the rate constants, kl,k2, and k3. Before presenting the experimental data, we shall consider the temperature effech of the endothermic reaction. Using the Prater-type heat balance equation and data for the thermal conductivity of the graphite, the maximum temperature deviation from being isothermal was about 0.74 "C at the center of the sample (Liu, 1980). To obtain data on the rate constants, k l , k2, and k3, a series of rate measurements was performed, and eq 18 was used to treat the data. For the rate measurement, the pore diffusion limitation must be totally eliminated. This was done by using very fine particles. The particle size was reduced until there was no more increase in the overall rate, and this size was used for the rate constant measurements. Such size effect is shown in Figure 5. Accordingly, the size fraction of 100-150 km was used for the rate constant experiments. Furthermore, since the rate was constant up to about 30% conversion, all the rate constant measurements were done below this conversion. In calculating the three rate constants, eq 18 may be rewritten as
1143OC
Press I otm
ures 3 and 4. The slopes were between 1.1 and 1.2. These values clearly indicated that diffusion in both pore groups was significant. One may also discuss the measured diffusivity data based on the dusty gas model. For doing so, one must first assume a pore structure from which a single pore size is assumed. Using the single pore size, a Knudsen diffusivity may be calculated. Then, by using eq 7, the effective pore diffusivity from the model is
DL, = -In( ~D11,mol
Temp
2k
3/16 5/16 7/16 5/16 5/16
10 io 10 20 30
I
R d , g/min 0.65 X 1.18x 10-3 1.71 X 10'' 1.88x 10-3 2.45 X
I
Deii, cm21s
7.01 X 7.50 x 10-3 7.28 X 7.32x 10-3 8.00X
(I Temperature: 1143 "C; pressure: 1 atm; sample: cylindrical graphite 1.5 in. long.
Rates were measured at 1143 "C,at three ratios of Pco/ Pcoz, and the data were plotted according to eq 31, as shown in Figure 6. The slope yielded the value of kl,and from the intercepts we calculated k2 and k3. Their values were: kl = 2.92 X min-' atm-', k2 = 18.85 atm-', and k3 = 5.12 atm-'. The next experiment was to measure the overall rate, Rd. The overall rate was very low, which made the determination of Rd at about 0.2% carbon conversion feasible. To test the model, we measured the values of Rd for three sizes of carbon and three bulk concentrations of C02 and compared the extracted Deffvalues from these data. These results are summarized on Table 111. The DeHfor COz in Ar calculated from the reaction rates was (7.5 f 0.5) X cm2/s. Now, we shall compare this value with the data measured with the WK apparatus. We used the measured temperature dependence of 1.17th power and the inverse square root of molecular weight dependence to extrapolate the WK data to D,ff of C 0 2 in Ar at 1143 "C. Such an extrapolation resulted in a value of 4.6 X cm2/s. The comparison of the two sets of data yielded an uncompromising discrepancy; the diffusivity calculated from reaction conditions was 60% higher than that derived from the WK direct measurements. Such a great discrepancy cannot be explained by the following factors: the assumptions made in the model, the scattering of D,H as shown in Table 111, and extrapolation of the WK data based on the temperature dependence. The above comparison, i.e., that the Deftfrom reaction rates is 60% higher than that in the absence of reaction, is an interesting exception from previous studies. In a
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
0 ADSORPTION 0 DESORPTION
30
-
I
zoc -
1
*"
IO
c
016 ,,, ,
I 1 1 01 0 2 0 3 0 4 0 5 0 6 07 0 8 0 9 l b P/ Po
Figure 7. Adsorption-desorption isotherms at 77 K.
review by Stoll and Brown (1974), it was stated that in only one study where the two values agreed, the other five systems all showed a lower Defffrom reaction data. (The system studied by Stoll and Brown made it six.) We propose that the discrepancy was due to the deadend pores in the carbon. In the WK or any other measurement for diffusion rate, the dead-end pores do not contribute to the total flux. They do, however, participate in the heterogeneous reaction. For materials which have a large number of dead-end pores, such as carbonaceous materials, the effective diffusion in a reaction system should be higher than that predicted. The existence of the dead-end pores in the carbon sample was further indicated by the hysteresis loop of the adsorption-desorption isotherms as shown in Figure 7, although the hysteresis phenomenon is not yet understood. Effects of Dead-End Pores. Following the above discussion, it is interesting to address the question of the effects of the dead-end pores on Dee in light of the familiar models. Butt and co-workers first explicitly excluded the deadend pores from the flux computation in their model (Foster and Butt, 1966) by discounting the micropore volume as the dead-end pores. Dead-end pores were not excluded from the parallel pore model (Johnson and Stewart, 1965; Satterfield and Cadle, 1968). In the model by Wakao and Smith (1962), the dead-end pores were actually considered. Accordingly, the probability of a certain gas molecule entering a dead-end pore under given conditions is 2tj(l - ti - E,) + 2t,(1 - t i - t,) = 2t(l - t) (32) here t = ti + E,, and subscripts i and a denote micropore and macropore, respectively. The meanings of the two terms are obvious. The factor of 2 is included so one may calculate the porosity of the dead-end pores as predicted in their model. According to their model, furthermore, 241 - t) 2 4 1 - t) dead-end porosity = t =(33) 2€2 241 - e)
+
and the open-end porosity = t2/(2 - E). For a numerical example, if the total porosity is 50%,the dead-end porosity according to the Wakao-Smith model would be 33.3%. This need not be so. However, it is expected that the Deff values predicted by the Wakao-Smith model would be substantially lower than those by the parallel pore model,
267
if a reasonable tortuosity factor is used in the latter (e.g., tortuosity = 3). A brief survey of the published data indeed confirms this expectation (Satterfield and Cadle, 1968; Brown et al., 1969). For the carbon sample used in this study, the DeBpredicted from the parallel pore model (with a tortuosity of 3) is about three times as high as that predicted by the Wakao-Smith model (Ymg and Liu, 1979). Also, our experimental data were about 50% lower than the values predicted by the latter model (Yang and Liu, 1979). By counting the micropores as the dead-end pores, the Deffpredicted by the Foster-Butt model were about 4-5 times less than the corresponding values from the Wakao-Smith model, for the case of CO-C02 in NiO/A1203 with a bimodal pore distribution, at 325-370 "C (Steisel and Butt, 1967). Such low values predicted by the Foster-Butt model, as compared with the Wakao-Smith model, may be attributed to the series nature of the diverging and converging pores assumed in their model. Tortuosity Factors. A tortuosity factor is used in several models, e.g., the dusty gas model and the J-S model, for predicting the diffusivity or flux in porous materials. This factor may be calculated based on the models and the experimental data. If the value of this factor is large, for example, greater than 5, the factor no longer has a physical meaning; it is then a mere fudge factor. The tortuosity calculated based on the J-S model for the nonreacting case ranged from 12 to 15. Here the "geometric factor" was replaced by the tortuosity factor to fit the data. To use the dusty gas model, one would first assume a pore structure such that a single pore size may be used to calculate the Knudsen diffusivity, from which one may calculate 7 from eq 7. Assuming that the micropores branch off the macropores and only the macropores limit the flux, the values for 7 ranged from 15 to 21. For a more realistic structure, the micropores are also involved in transporting the flux,the 7 values will be much smaller than 15, and hence more meaningful. Inasmuch as the high T values are not meaningful, no more discussion on its values will be made. In addition, the pore structure changes with temperature, and we believe the changes are quite significant. Work on this subject is progressing in our laboratory (R.T.Y.). Estimation of Deffat Elevated Temperatures. In modeling and reactor design studies for heterogeneous reactions, especially for reactions involving changes in porosity, it is desirable to estimate Deffas a simple function of the total porosity, t (Hartman and Coughlin, 1976; Dutta and Wen, 1977; Ramachandra and Smith, 1977; Srinivas and Amundson, 1980). This has been an improvement over the models where Deffwas assumed constant in many previous studies (Szekely et al., 1976). Walker et al. (1959) measured DeHfor C02-CO in a spectroscopic carbon at various burnoff levels (up to 65% total porosity) under room temperature and pressure conditions, and successfully obtained the following relation Deff Dmo1c2 (34) Relation 34 also correlated well the room temperature data on H2-air diffusion in various porous materials ranging from glass spheres to kaolin to mica (Currie, 1960). Comparing our data on De, at elevated temperatures with the D& predicted from eq 34, the latter is 15-20 times too high. In the room temperature range, however, the values predicted by this equation are close to the experimental data. The reason for this inconsistency is the relative importance of Knudsen diffusion at elevated temperatures, as will be delineated as follows.
268
Ind. Eng. Chem. Fundam., Vol. 21, No. 3, 1982
For a porous material with a single pore structure, it can be shown that the Wakao-Smith model is simplified to
This is so regardless of whether the single pore size is micropore or macropore in their nomenclature. The model in this case simply means that the porous material is represented as a precise repetition of two randomly stacked layers. The value of ayAis normally negligible, as compared with the other two terms in the denominator in the above equation. The difference between eq 35 and eq 34 is the replacement of the molecular diffusivity in eq 34 by the transition diffusivity. For the systems studied by Walker et al. (1959) and by Curie (1960), the two diffusivities are about equal, because Dk is greater than Dmol. However, at elevated temperatures, this is no longer the case. The value of Dk is now only a fraction of Dmol,and D, is substantially lower than Dmal.To evaluate the denominator, l + Dmol/Dk,the “single” pore size must be known. For porous materials with a bimodal pore distribution, the “single” pore size becomes an adjustable or empirical parameter. For the carbon sample used in this study, we found that the Deffdata can be fitted well by taking 2h/(Sp) as the mean “single” pore size, which was 410 A. However, caution should be taken in using this procedure, which has been tested only with graphite. The generality of this procedure for calculating the “single”pore size needs to be tested on other porous materials. The mean pore size used in evaluating the denominator in eq 35 does not change greatly during the reaction for solid gasification reactions. A good example is seen in the calculation made by Peterson (1957). The pore radius was about doubled at near completion of the reaction (C + C 0 2 at 1100 OC). One may reasonably assume that the denominator in eq 35 remains a constant during the gasification reaction. This assumption should also be applicable to reactions with an increase in the solid volume, such as the sulfation of lime. Furthermore, our modeling studies for the carbon gasification reaction up to about 70% burnoff, as may be seen in Lids thesis (1980),showed that using the 2 dependence on Des could predict the overall rates satisfactorily. In conclusion, based on our experimental data and the above analysis, we recommend the use of eq 35 for predicting Defffor modeling and design studies. Dt may be estimated by the following equation
D, =
Dmoi -t D m a l / D k
(36)
where Dk may be evaluated based on mean pore radius (as shown in the foregoing) of the unreacted sample. The value of Deffcalculated in this manner agrees with the experimental data both on the temperature dependence and the porosity dependence. The values predicted by using eq 34 agree with those by eq 35 only in the room temperature range. The error is magnified as the temperature is increased, which is about 15-20 times too high at about 1000 O C .
Acknowledgment A grant from Alcoa Foundation and a grant from the National Science Foundation (CPE-8012357) made the work possible. Nomenclature CT = total molar concentration of gas mixture, g-mol/cm3 Deff= effective diffusivity, cm2/s Dij = effective pore diffusivity of i in j , cm2/s Dim = effective pore diffusivity of i in mixture, cm2/s Dk = Knudsen diffusivity, cmz/s Dmol= molecular diffusivity, cmz/s D, = diffusivity in the transition regime, cm2/s ki = rate constants for the reaction C + C 0 2 2CO L = length of the cylindrical sample, cm Ni = diffusion flux in pores of i, g-mol/(cm2s) Pi= partial pressure of i, atm r = radial coordinate of sample, cm R = radius of sample, cm Rd = overall gasification reaction rate, g-mol of carbon/s S = surface area, cm2/g W , = initial weight of carbon, mg W , = weight of carbon at time t , mg X i= molar fraction of i in gas mixture y = mole fraction z = axial coordinate of sample, cm
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Greek Letters (Y = 1 + NB/NA t = porosity p = bulk density [= solid density x (1 - e)], g/cm3
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Received for review April 3, 1981 Accepted March 9, 1982
