I n spite of all these difficulties, the results obtained in this study can be of important help. Each of the complications rnrntioned can be approached on a more successful a n d rational basis if one assumes that Equation 4 correctly denotes the purelJconductive heat transfer for the heterogeneous array. If a sample exhibits a strongly different behavior, then it would be best to assume that ,some other compound or mechanism o r geometry makes a major contribution to the thermal transfer and to search for that contribution. I t is believed that this \-iew is entirely justified by the results obtained, and that inorc sagacious attack:s upon these other difficulties ill bc i‘ound from it. Conclusions
7.
‘Thc thermal conductivities of widely different types ou heterclgeneous mixtures can be correlated using Equation 4. For mixtures in which the conductivity oi‘ the discontinuous p h , u e is the larger by more than a factor of 100. Equation 5 should be used to obtain n, accounting for thc influence of particle shape. For other mixtures the value ol )i in Equation 4 may be taken as 3. Nomenclature
thermal conductivity of thr mixturc: l3.t.u. sq. f t . hr.-’ F. ft. A, = thermal conductivitv of the continuous phase. B.t.u. s q . ft.-hr.-’ F. ‘ f t . K.,= thermal conductivitv of discontinuous uhase. B.t.u. sq. ft.-hr.-’ F.;ft. ’ 1.1 = volume fraction o f continuous phase, dimensionless 1 .: = volume fraction of discontinuous phase. dimensionless .Y = “shape factor” for ellipsoids in Fricke‘s equation. dimensionless =
= empirical shape Iacior, dimensiuiiless = sphericity, dimensionless = temperature? F.
’
Literature Cited (1) Brophy, J. H., Sinnott, \V. J . , 41st Annual Convention, Am. SOC.Metals, Preprint No. 139, 1959. (2) Brown, G. G., “Unit Operations,” p. 77, Wiley, New York,
1950. (3) Brown, \V. F., Jr., J . Chem. Phys. 23, 1514-17 (1955). (4) Dow Corning Corp., Midland, Mich., Bull. 9-399, Octobn 1959. (.5) Forsythe, \V. E., “Smithsonian Physical Tables,” 9th ed., pp. 136-43, Smithsonian Institution, Washington, D. C.. 1954. (6) Francl, J., Kingery, \V. D., J . Am. Ceram. Soc. 37, 99-107 (1 0541. \ - - ,.
Fricke. H.. Phys. Reu. 24, 575-87 (1924) (8) Hamilton, R . L., Ph.D. thesis. Universitv of Oklahoma, 1960. (9) Johnson, F. A , , .it. Energy Res. Estab. (GI. Brit.) AERE RIR 2578, June 1958. (IO) Knudsen, J. G., \Vang, R. H., IND.EKG. CHEM.50, 1667 (1938). (11) Kunii, O., Smith, J. M., .4.I.Ch.E. .Journal 6, 71 (1960). (12) Lauhitz, J., Can. J . Phys. 37, 789 (19.59). (13) Marathe, M. N., Tendolkar, G. S . , Trans. Indian Inst. Chrtri. Engrs. 6, 90-104 (1953). (141 Maxwell. J. C.. “.4 Treatise on Electricitv and Mametism.” Vol. I. 3rd ed.. p. 435, Dover, New York, 1934. (15) Russell, H. I$*,.J . Am. Ceram. SOC.18, 1 (1935). (16) Sibhitt, IV. L., Jefferson, T . B., \l’itzell, 0. \V., IND.ENG. CHEM.50,1989-92 (1958). (17) Vrirs. D. A. de. “The Thermal Conductivity of Granular Materials,” Institut International du Froid, Paris,’ France. (18) \Voodside: Ft’illiam, Can. J . Phys. 36, 815 (1958). (’)
tx\ o-component
G
ti
,3
~
RECEIVED for review July 31, 1961 .4CCEPTED May 24, 1962 IXvision of Industrial and Enginrering Chemistry, 140th Meeting. .\CS, Chicago, Ill., September 1961. Based on Ph.D. thesis of R . L. Hamilton at Oklahoma University Research Institute. \Vork supported by Texaco, Inc., and Celanese Corp. of America.
MULTICOMPONENT VISCOSITIES OF GASEOUS MIXTURES A T HIGH TEMPERATURES S. C. SAXENA’ AND T.
K. S. NARAYANAN Chemical Engineering Division,
Atomic Energy Establishment, Trombay, Bombay, India
Binary diffusion and pure viscosity data are used to compute binary viscosities according to an expression derived from rigorous kinetic theory for monatomic gases. The rigorous expression is based on the assumption of central forces. An approximate but simpler formula proposed earlier b y Wilke is also considered. A scheme is proposed for computing multicomponent viscosities from the known pure and related binary mixture viscosities at two compositions, and is tested for several ternary mixtures. This procedure permits calculation of the multicomponent viscosities for mixtures permuting out of the nine binary systems considered here with much less computation than for the rigorous method. are tabulated.
IGH
temperature viscosity d a t a for gaseous mixtures are
H extremely useful to design engineers in problems involving
fluid flotz.. Unfortunately, there are very few experimental data of this nature (70. 73). T h e only alternative. therefore, has been to calculate the multicomponent viscosity from the Present addwss, Univrrsity of Rajasthan, Jaipur, India.
The constants required for such a calculation
theoretical expression in conjunction with information about the intermolecular forces derived from some other source. hccording to the rigorous kinetic theory of Chapman and Enskog, the coefficient of viscosity of a u-component mixture, v”,,~.is given by Equation 1 . T h e development is based on the assumption o f crntral forces. VOL
1
NO. 3
AUGUST
1962
191
(l)
A- f
i
7 1 ~ . Si? and M iare the coefficient of viscosity, mole fraction. and molecular weight of component i. respectively. D 1 3is the binary diffusion coefficient benseen components i and j. X is the molar gas constant, Tis the absolute temperature, and p is the pressure. a4*tj is the ratio of reduced Chapman-Cowling collision integrals, and its exact value a t a given temperature involves the knowledge of intermolecular potential. Thus. il only a detailed knowledge about the force law is available. Equation 1 permits the computation of )I,,,)~. Following this procedure. Amdur and Mason ( 7 ) calculated for He-Ar mixtures a t 1000" to 15.000°K., using intermolecular poten-
Table I.
s, = 0
.\
2358 2347 3901 3860 5174 5046 5941 5-40 201' 2017 3259 3267 4268 4239 4868 4790 1787 1697 3031 2948 3983 3757 4762 4461 1932 1883 2909 2812 4093 3915 4755 451 3 1594 1706 2346 2444 3273
2384 2360 3982 3923 5271 .5131 6024 5830
.\ .\
B .I B 300
.\
B 600
.\
900
B A B
100
x B
300
x B
600
.\
B 900
B 200
.I B
300
.\
B 500 800 000 300 500 800
.\
B .4 B A B .4 B .4 B .\
=
[1
7,
1-1
Y
+
GL
(2)
k = 1 i: # i
in \vtiich
According to Equation 2,
T,,!,,
values can be calculated if the
1, =
.io
0 27
B
Y
Viscosities of Binary Mixtures as a Function of Temperature"
&e/
B
tials obtaincd I'roin scattering experiincnts of neutral atoins. Similarly, qlrrixcan be calculated for other systems for which scattering experiments have been performed (79). However, if the binary diffusion coefficients and pure viscosities are available from direct measurements, qm,x can be straightaway calculated from Equation 1, if .4*12 is also known. Fortunately, 4 * 1 2 does not vary much among realistic intermolecular potentials and with temperature (70). Also. qmlx is fairly insensitive to the value of =i*12, and consequently a n approximate value can be assigned to it without introducing a n y appreciable error in the calculation of qm,x. I n t h i s fashion q,,,,, can be computed from q i and D t j values without assuming any definite law for molecular interactions. This procedure also has been used previously (7, 70. 33), and this principle ol cvaluating one property from other measured properties and their temperature derivatives has proved very successful for nonspherical polyatomic molecules (8. 24, 33,34). A much simpler expression for q,ngy is proposed by IYilke (3(i). ivhich involves only v i and Mi. I t can be irritten:
1945 1938 3158 3154 4119 4081 4670 4591 1724 1620 3018 2904 401 2 3738 4840 4472 1785 1730 2757 2644 3930 3720 4589 4303 1879 1951 2768 2816 3845
2330 2314 3905 3867 5142. 5060 5862 5746 1861 18i6 3034 3023 3937 391 3 4436 4392 1621 1553 2916 2850 3900 371 3 4725 4473 1638 1602 2574 2500 3690 3550 4310 4118 2010 2039 2942 2957 4067
\it
0 27
0
so
0 75
R
3364 379.5 3878 1357 1376 2058 2049 2882 2830 3358 3271 1904 1856 2'75 2709 3785 3681 4353 4218 1421395 2261 2046 3445 3233 4177 3971 2244 2194 3012 2957 3674 3591 4231 41 38 491 8 4865
3875 4442 4458 1598 1622 2408 2393 33'6 3299 3928 3805 1993 1928 2900 2810 3975 3834 4581 4396 1773 1511 2675 2377 3849 3568 4539 4273 2161 2102 2948 2883 3624 3524 41 88 4076 4880 481 5
4067 4684 4674 1836 1853 2726 271 3 3797 3735 4395 4298 2049 1999 2978 2910 4095 3987 471 6 4575 1998 1786 2930 2698 4078 3869 4725 4532 2060 2021 2857 2814 3528 1461 4090 4014 4804763
\
B 300
\
R 500
\
B XOO
\
R 01)O
\
B 300
\
B 500
.\
B 800
.\
B 000
.I €3
O:-H:O
300
500
\
n
.\
B 800
oon (K)?-.Iir
400
\
B
.z B .\
B 600
\
800
B 1 B
1000
\
B 1300
\
B
pure viscosity data are available. Wilke (36) found a very reasonable agreement between the calculated qmix values and the experimental reported data for a number of gaseous mixtures but mostly around room temperature. Amdur and Mason (7) found that the calculated values according to Equations l and 2 for a He-Ar mixture differ by less than 474 in the temperature range 1000 O to 15,000 OK. Table I presents qmia values for a number of binary mixtures calculated by use of Equation 1 and experimental pure component viscosity and binary diffusion data (semiexperimental values) and of qmix calculated by use of Equations 2 and 2a. Tabulations cover only binary mixtures, the extension to multicomponent mixtures in both cases is straightforward. \Ye also propose another scheme for calculating the multicomponent mixture viscosities, which reduces essentially to Equation 2, but the Gik’s in this case are treated as disposable parameters and determined from the va1i.m of two binarv minture viscosities. Calculation of Viscosities
Experimental viscosity data for pure gases-viz., CO,, He, Ar? 0 2 . CO. CHd, HP:H 2 0 , and air-at different temperatures have been reported by a number of workers (3, 4, 6, 77? 72. 74, 75. 20-22, 26, 30: 37, 37). For each gas we have plotted all the experimental viscosity data and graphically interpolated at the required temperatures. T h e diffusion coefficient values for Ar-He, XZ-He, COZ-NZ, COz-02. 02-H2. 02-CH,. 0 2 - C O , and 0 2 - H 2 0 systems as reported by LValker and \Vestenberg (32) and for the C O T a i r system as reported by Klibanova! Pomerantsev, and Frank-Kamenetskii ( 76) have been used in this work. Other diffusion data for temperatures near or slightly above room temperature (2, 70. 25. 3.5) are not considered in the calculations described here. T h e A*12‘s for each system were approximated from available cabulations (70). Values of 1.12 and 1.17 were assigned for Ar-He and He-Sp systems, respectively, and 1.10 for all the remaining systems. values are, however, insensitive to ‘4*12. .4change of 10% in A * 1 2 for the Ar-He system produces a change of only 1% in the qm,. value when Ar is 2SYGand less than 0.3y0when Ar is 75 70. T h e effect of this change in qmi, with temperature is not pronounced and slowly ciecreases as the temperature increases. Utilizing these values of A*12 and experimental diffusion and pure viscosity data, vmlr for nine binary mixture:: is computed according to Equation 1. as a function of temperature and composition, (Table I, set A . ) Most of the pure component viscosity data are consistent within 1 yc. However, in a few cases the maximum deviation goes u p to 2 or 3%. and for air it goes to 3 70. T h e latter may be partly due to use of air of different compositions by the various workers. A change of 2 in pure viscosity causes a maximum change of 1 5% in the mixture viscosity value for the systems listed in Table I, and much less where the two components differ widely in masses. Walker and Westenberg’s diffusion data (32) usually possess a precision to i 1 %, which produces a maximum uncertainty of i 0.5 Yo in the qmiX values of Table I. Thus we can make a modest estimate of about 2 to 3 % over-all maximum uncertainty in our reported vmlr values. T h e viscosity values calculated according t o Equation 2 of Wilke (36) are also reported in Table I, set B. T h e two sets of values agree moderaixly well. We feel that the values obtained using diffusion data should be preferable, as the expected accuracy of Equation 2 is reduced by virtue of the simplifying assumptions involved in its derivation. I n Table I1 the direct measurements of viscosities available at high temperatures are compared with the calculated values according to
Table II. Calculated and Experiment01 Viscosities of c02-N~ (Experimental values refer to mixture 19.82% COz, 0.64% 0 2 , 7 9 . 5 4 7 , Nz. Calculated values are based on 20.0% COS and
Temp., K. 291.2 403 496 631 81 1 932 1047 1146
80.07, N?) lo6 X q,,,,,, G./Cm.-S c. Expil. Semzexptl. 11’1IX~’se q . 170 175 167 217 226 21 5 249 264 252 290 312 299 348 370 352 382 404 384 419 434 41 2 437 458 434
Equations 1 and 2. Agreement of both sets of calculated values with experimental values is good. As the formula given by Equation 1 holds rigorously only for monatomic spherical molecules, its use for N2 and COz molecules is likely to cause some discrepancy. Also, the limitations of Wilke’s Equation 2 are of uncertain nature and the deviation of the computed viscosities from the observed values will differ in sign and magnitude from system to system. iYe feel that use of Equation 1 and experimental diffusion data is preferable, because the rigorous expression from which it is derived has been shown to yield good viscosity values even for nonspherical molecules. However, accurate diffusion data are not always available. Wilke’s Equation 2 can be applied more widely. as it requires only pure component viscosity data. Further, in the use of Equation 1 it is tacitly assumed that the theoretical first approximation formula for the diffusion coefficient is equal to its experimental value. This, however. is justified because the contribution of the higher approximations is small and usually of the order of experimental uncertainty involved in diffusion measurements. Semiempirical Method of Calculating Viscosities of Multicomponent Mixtures. I t has been shown (5, 77, 23), that Equation 1 can be expanded into an infinite series which to a first approximation can be cast in the form of Equation 2. G,, is now defined as (3)
If the pure viscosities and the binary viscosities at tMo compositions are known, GI2 and G21 can be directly calculated from Equation 2, being treated as disposable parameters. T h e equation for qm,x so obtained with these empirical values of G I ? and GZImay be used to predict the composition dependence of vmlx in the entire range. Further, if the binary viscosities are known for a number of systems, the multicomponent viscosities may be evaluated from Equation 2. This calculation does not require a prior knowledge of the intermolecular potential energy and is also far simpler for multicomponent mixtures than the riqorous computation based on Equation 1. A similar procedure has proved very successful for thermal conductivity (77, 78, 27). Let us check this particular scheme for the case of ternary mixtures of 1-e-Ar-He, for which the direct measurements are available (29). Experimental data are also available for all three related binary systems (28). Table I11 lists the empirical values of Gii, for Ne-Ar, Ar-He, and Se-He systems at two temperatures, 373Oand473’ K., alongwiththepureviscosity data used in this calculation. I n Table IV are reported the observed ternary mixture viscosities at these two temperatures. Computed values of viscosities according to Equation 2 in conjunction with the empirical Glk’s and 7% values of Table VOL.
1
NO. 3 A U G U S T 1 9 6 2
193
Table 111.
Data Used for Calculation of Viscosities of Inert Gas Mixtures“
Temp., 72 GI? G21 2688 2300 0.327 2.173 3215 2694 0.278 2.673 Ar-Ne 2688 3623 0 716 1 386 3215 4220 0 948 1 118 1 035 Ne-He 3623 2300 0.709 4220 2694 0.706 1 040 a Subscript 7 r e f u s to heavier component. Cnzts of q are g./cm.-sec.
Gas Pair Ar-He
Table IV.
707 X q1 707 X
OK.
373 473 373 473 373 473
Calculated and Experimental Viscosities of Ternary Gas Mixtures of Ne-Ar-He
He
G./Cm .-Sec . Semiempiri- Wilke’s Theor. Exptl. cal eq. (9)
1754 3594 1983 5429 1754 3594 1983 5429
3237 3044 2886 2957 3790 3574 3415 3470
707
Temp.,
Mole Fraction
K.
Ar
373
0 0 0 0 0 0 0 0
473
Table V.
C02-Nz
CO2-02
02-H2
02-CHn
x
0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0
0
K.
300 600 900 100 300 600 900 100 300 600 900 200 300 500 800 000 300 500 800 000 300 500 800 000 300 500 800 000
02-HzO
C02-Air
300 500 800 1000 400 600 800
1000 1300
194
5576 3193 2166 2189 5576 3193 2166 2189
3215 3024 2895 2938 3739 3497 3398 3387
3223 2966 2829 2867 3782 3610 3337 3547
3205 3025 2895 2938 3752 3551 3425 3449
Values of Gik and 7%for Nine Binary Gas Systems as a Function of Temperature
O
N2-He
2
2670 3213 5851 2382 2670 3213 5851 2382
(Units of 7 are g./cm.-sec.) Temp.,
Gas Pair Ar-He
0
iVe
vrnix ,
2260 3790 4960 5630 1786 2920 3768 4222 1495 2750 3685 4465 1495 2375 3400 3955 2071 3010 4140 4755 2071 3010 4140 4755 2071 3010 4140 4755 2071 3010 4140 4755 1950 2750 3400 3955 4710
I&EC FUNDAMENTALS
1987 3186 4158 4745 1987 3186 41 58 4745 1786 2920 3768 4434 2071 3010 4140 4755 896 1255 1730 2005 Ill6 1680 2324 2694 1785 2608 3529 4040 985 1705 2860 3620 2300 3038 3660 4200 4910
GI2 0.311 0.313 0.303 0.296 0.353 0.358 0.353 0.341 0.652 0.703 0.704 0,700 0.684 0.699 0.699 0.690 0.353 0,336 0,326 0.320 1.018 0.994 0,963 0,936 0.968 0.966 0,968 0.905 0.834 0.827 0.813 0,799 0.730 0.750 0.760 0.765 0,794
G21 2.011 1.913 1.842 1.811 2.183 2.165 2.117 2.121 1.182 1.082 1.039 1.004 1.308 1.216 1,146 1.114 1.566 1.446 1.402 1.395 0.955 0.871 0.835 0.832 0.898
0,907 0.885 0.944 0.537 0.663 0,828 0.911 1.226 1.208 1,163 1.157 1.178
111 are given in column 6 of Table 1V. Also reported in this table are the calculated values of ternary viscosities according to Equations 2 and 2a. I n the last column of Table I V are listed the calculated values according to Equation 1, using the Lennard-Jones (12-6) intermolecular potential. T h e agreement between theory and experiment is very good. Wilke’s equation is also successful, because of the validity- of the assumption involved in deriving Equation 2 regarding diffusion coefficients for these spherically symmetrical molecules. Because of the excellent success of the proposed scheme in predicting ternary viscosities, it will be useful to determine G,k for all the nine systems of Table I. set A. I n Table V are reported the values of Gzh for all these systems obtained from binary viscosity values corresponding to mixtures having X,equal to 0.25 and 0.75 of Table I. These Gs;, will readily permit the calculation of multicomponent viscosities involving any permutation of these nine binary systems. This is important, for in problems of actual interest multicomponent viscosities and not just binary viscosities are needed. The GZx values of Tables I11 and V reproduce the binary viscosities over the composition range. I n this fashion a vast amount of binary viscosity data can be readily used to predict several very useful multicomponent viscosities. These Glh. values do not follow the simple Equation 3. IVe hope to throw light on this point in a later publication. I n this article we use binary diffusion data to calculate viscosities. T h e reverse of this approach would be very interesting. if high temperature viscosity data were available. I n fact. this approach for determining diffusion coefficients would avoid several experimental difficulties associated with a conventional two-bulb apparatus and might provide a method for investigating the composition dependence of the diffusion coefficient. Ac knowledgmenl
We thank H. N. Sethnn and Dipak Gupta for their interest and encouragement.
Literature Cited
(1) Amdur, I., Mason, E. A,. Phys. Fluids 1, 370 (1958). (2) Amdur, I., Schatzki, T. F., J . Chem. Phys. 27, 1049 (1957). (3) Bonilla, C. F., Wang, S. J., Weiner, H., Trans. Am. So(. Mech. Engrs. 78, 1285 (1956). (4) Bremond, M. P., Compt. Rend. 196,1472 (1933). (5) Brokaw, R. S., J . Chem. Phys. 29, 391 (1958). (6) Eucken, A , , Forsch. Gebiete Zngenieurw. 11B,6 (1940). (7) Fox, J. I V . , Smith, A. C. H., J . Chem. Phys. 33, 623 (1960). (8) Heath, H. R . , Proc. Phys. Soc. B66, 362 (1953). (9) Hirschfelder, J. O., Bird, R. B., Spotz, E. L., Chem. Revs. 44, 205 (1949). (10) Hirschfelder, J. O., Curtiss, C. F., Bird, R. B.. “Molecular Theory of Gases and Liquids,” LViley, New York, 1954. (11) Johnston, H. L., Grilly, E. R., J . Phys. Chem. 46, 948 (1942). (12) Johnston, H. L., McCloskey, K . E., Zbid., 44, 1038 (1940). (13) Kenney, M. J., Sarjant, R. J., Thring, M. IV., Brit. J . Appl. Phys. 7, 324 (1956). (14) Kestin, J.; Physica 25, 537 (1959). (15) Kestin, J.: IVang, H. E.: Zbid., 26, 575 (1960). (16) Klibanova, T. M., Pomerantsev, V. V.: Frank-KamenetskiI, D. S.. J . Tech. Phys. U.S.S.R. 12, 14 (1942). (17) Mason, E. A., Saxena. S. C., Phys. Fluids 1, 361 (1958). (18) Mason, E. A,, Von Ubisch, H., Zbid., 3, 355 (1960). (19) Mason, E. A , , Vanderslice, J. T., ”.L\tomic and Molecular Processes,” D. R. Bates, ed., Chap. 17. Academic Press, New York, in press. (20) Rammler, E., Breitling, K., W i r m e 60, 620 (1937). (21) Raw. C. J. G..Ellis. C. P.,J. Chem. Phys. 28,1198 (1958) (22) Zbtd.. 30, 574 (1959). (23) Saxena, S. C., J . Chem. Phys. 25, 360 (1956); Indian J . Phys. 31, 597 (1957). (24) Saxena, S . C., Zbid., 34, 449 (1960). (25) Saxena, S. C., Mason, E. PI., Mol. Phys. 2, 379 (1959). ,
I
(26) Shilling, I V . G., Laxton, A. E., Phil. Mag. 10,721 (1930). (27) Srivastava, €3. N., Saxena, S. P m . Phys. 5 0 6 . B70, 369 (1957) ; J . Chem. Phys. 27, 583 (1957). (28) Trautz, M., Binkele, J. E., Ann. Physik 5 , 561 (1930).
c.,
(29) Trautz, M., Kipphan, K. F., Zbid., 2, 746 (1929). (30) Trautz, M., others, Zbid., 2, 733 (1929); 5, 561 (1930); 7, 427 (1930); 20, 118, 121 (1934). (311 Vasilesco. V.. Ann. bhvs. 20. 292 (1945). (32) Walker, R. E., \V;,s;enberg, A.'A., 3. Chem. Phys. 29, 1139, 1147 (1958); 31, 519 (1959); 32, 436 (1960).
(33) Weissman, S.: Mason, E. A,, Physica 26, 531 (1960). (34) Weissman: S., Saxena, S. C., Mason, E. A,, Phys. Fluids 3, 510 (1960). (35) Ihid., 4, 643 (1961). (36) Wilke, C. R., J . Chem. Phys. 18, 517 (1950). (37) Wobser, R., Muller, F., Kolloid Beih. 52, 165 (1941). RECEIVED for review July 3, 1961 ACCEPTEDMay 7, 1962
RATE EQUATIONS FOR CONSECUTIVE HETEROGENEOUS PROCESSES KENNETH 6 . BISCHOFF1 AND GILBERT F. FROMENT Lahoratorium voor Organische Technische Chemie, Rijksuniversiteit te Gent, Gent, Belgium
A general discussion of the forms of rate equations for consecutive, heterogeneous processes is given.
The complexity oil the derivation of these rate equations depends on whether or not the process has parallel
steps combined with the consecutive ones. The procedures that can b e used for cases with only consecutive steps are outlined. The particular case of a heterogeneous catalytic reaction with more than one ratecontrolling step is covered in detail. It may be extremely difficult to determine if more than one step is controlling. However, kinetic constants derived from experimental data on the basis of only one step controlling may be greatly in error.
N hiAw
heterogeneous chemical reaction systems of engi-
I neering interest, the chemical reaction is only one of several
consecutive steps. If the rate coefficients of all of the other steps are extremely large lvith respect to that of the reaction itself (in other words, if their resistances are very small), the rate of the over-all process may be obtained by inserting the over-all potential difference in the rate equation for the chemical reaction. I n general? however, because of the influence of the other steps, the potential difference to be used in the chemical rate equation is some unknown fraction of the over-all potential difference, the only one that can be measured. However, since at steady state the rates of each of the steps must be equal, a set of equations may be written between which the intermediaize potentials can be eliminated by algebraic manipulation. This yields an expression that contains only the over-all potential difference and a coefficient that combines the rate coefficients or resistances of the different steps. T o chemical engineers. the most familiar applications of this method of eliminating intermediate, unknown potentials are the treatment of simple heat conduction through a series of la)-ers with different th-rmal conductivities and the t\ro-film theory of mass transfer. These are examples where the relations between rates and potential differences are linear. T h e method has been less intensively applied to chemical reaction systems, probably because chemical reaction rates are often dependent in a nonlinear way on the driving force, although this should not be considered as a restriction. I n addition, however, heterogeneous reaction systems are often not truly consecutive, and also contain parallel steps, which greatly complicate the treatment. This is the case when the phase in instance, in ivhich the reaction takes place is permeable-for fluid-porous solid systems or in fluid-fluid systems. I n nonpermeable reaction phases on the other hand-for instance, fluid-nonporous solid systems with reaction a t the solid surface
-only consecutive processes occur. This paper is concerned only with the nonpermeable reaction phase case-i.e., with truly consecutive processes.
Method of Combination of Resistances Linear Processes. T h e first case to be considered is that of a first-order reaction between one component of a gaseous mixture and a solid. T h e reaction at the solid surface will be in series with mass transfer through the fluid. T h e rate of mass transfer is expressed by kD(C
ID
- C,)
(1 1
while the chemical reaction rate is given by
(2)
r e = k&t
At steady state the two rates are equal, so that Y
= ~ D ( C- ci) =
kec,
from which c, may be found to be:
M'hen Equation 3 is substituted into Equation 1 or Equation 2> - c =
I
- A kD '
1
k,c
(4)
where
k, is the "over-all" rate coefficient, taking both reaction and mass transfer into account. Equation 4 is well known (3-5) and is of exactly the same form as the equations used in the two-film theory. When k , + m , the chemical reaction is said to be rate-controlling., and Equation 4 becomes
Present address, Univtrsity of Texas, Austin. Ter.
(6)
r = k,c VOL.
1
NO. 3
AUGUST 1962
195
