ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Most of the cited data were obtained on air and water systems a t ordinary room temperatures. I n addition, only a few of the annuli had diameter ratios greater than 2:1, so the scatter of points was of the same order as the differences attributable to geometrical effects. Consequently, the compilation cannot be viewed as an adequate test of the analogy equations, Equations 19 and 20. It does, however, indicate that curve c of Figure 4 represents the data for low viscosity systems within experimental accuracy in the high Reynolds number range. The present investigation has shown the stress patterns on which heat and mass transfer in annuli are superimposed. This affords a rational basis for describing and perhaps correlating the changes caused by particular transfer processes. More precise data are necessary before the relationships among heat, mass, and momentum transfer at the individual surfaces can be determined accurately. I n the meantime, caution should be used in extending experimental results t o other fluids and other conditions of transfer. It seems possible that rather large changes in the transfer coefficients can be caused by shifts in the shear stress distribution even when ordinary analogy relationships are valid. Nomenclature
D1= outside diameter of annulus core, f t . DZ = inside diameter of outer tube, f t . F4,
F8, Flz
= fluid drag on cores of 4-, 8-, and 12-ft. lengths,
respectively, (1b.-force) force on feelers, 1b.-force FI. F I I ,FIII = net fluid drag on first, middle and last 4-ft. sections of core, respectively, 1b.-force f l , fz = Fanning-type friction factors defined by Equations 12 and 5, respectively, dimensionless 90 = conversion factor = 32.2 (lb.-mass)(ft.)/(sec.2)(1b.-force) jD,, j ~ = , Chilton and Colburn j-factors for diffusion a t inner and outer boundaries, respectively, dimensionless j p , , j p 2 = Chilton and Colburn j-factors for friction at inner and outer boundaries, respectively, dimensionless j ~ ,jx, , = Chilton and Colburn j-factors for heat transfer at inner and outer boundaries, respectively, dimensionless N R ~ =~Reynolds number defined b y Equation 6, dimensionless
Fr
=
r r1, r2
rm S
B 2:
r
fi p T
T I ,T Z
$ $
radius from geometrical center of annulus to a point in fluid, f t . = radii of inner and outer boundaries, respectively, f t . = radius of maximum local fluid velocity, ft. = surface area of core, sq. ft.; subscripts correspond to subscripts on force terms, F = bulk average fluid velocity over entire cross section, f t . /sec. = axial distance from entrance of conduit, ft. = actual value of quantity (rg - r;), sq. f t . = fluid velocity, lb./(sec.)(ft.) = fluid density, lb./cu. f t . = local shearing stress a t a point in fluid r ft. from geometrical center of annulus, 1b.-force/sq. ft. = skin frictions a t inner and outer boundaries, respectively, 1b.-force/sq. ft. = function defined by Equation 17, dimensionless = apparent value of quantity (rz - r;) based on r m from Equation 3 =
literature cifed
Barnet, W. I., and Kobe, K. A . , IND.ENG. CHEM.,33, 436 (1941).
Carpenter, F. G., Colburn, -4.P., Schoenborn, E. M., and Wurster, A., Trans. Am. I n s t . Chem. Engrs., 42, 165 (1946). Chambers, F. s., and Sherwood, T. K., IND.ENG.CHEM.,Zg, 1415 (1937); Trans. Am. Inst. Chem. Engrs., 33, 579 (1937). Chilton, T. H., and Colburn, A. P., IND.ENG.CHEM.,26, 1183 (1934).
Gilliland, E. R., and Sherwood, T. K., Ibid., p, 516. Knudsen, J. G., and Katz, D. L., Proc. Midwestern Conf. on Fluid Dynamics, 1st Conf., No. 2, 175 (1950). McMillen, E. L., and Larson, R. E., Trans. Am. Inst. Chem. Engrs., 40, 177 (1944). Monrad, C. C., and Pelton, J. F., Ibid., 38, 593 (1942). Prengle, R. S., Ph.D. dissertation in chemical engineering, Carnegie Institute of Technology, 1953. Rothfus, R. R., M.S. thesis in chemical engineering, Carnegie Institute of Technology, 1942. Rothfus, R. R., Monrad, C. C., and Senecal, V. E., IND.ENO. CHEM.,42, 2511 (1950). Wiegand, J. H., and Baker, E. M., Trans. Am. I n s t . Chem. Engrs., 38,
569 (1942).
Zerban, A. H., Ph.D. thesis, University of Michigan, 1940. [Data are reported in ( I d . ] RECEIVED for review September 7, 1954.
ACCEPTED October 5, 1954.
Relations in Material Transport J. B. OPFELL
AND
California lnrfitufe o f
M
B. H. SAGE
Technology,
Pasadena, Calif.
ANY of the operations encountered in industry involve situa-
tions that depart materially from thermodynamic equilibrium. The molecular transport of components under such conditions is a n important part of absorption, extraction, and distillation processes and many of the processes involved in the production of petroleum and natural gas. Much that has beeu written about diffusion or molecular transport in gases was based on the kinetic theory and is subject to uncertainty when applied at elevated pressures. This article sets forth a few of the relationships that have been found to be useful in describing molecular transport in homogeneous systems. Particular emphasis has been placed upon the interrelation of the several diffusion coefficients in common use. The application of the more general expressions to specific situations is part of the objective. The analysis is presented in terms of the force-length-time system of dimensions (7). The basic equation of material transport rests upon the early work of Fick ( I I ) , Maxwell (19),and Stefan (s7-30). Maxwell 918
suggested a hypothesis based on simple kinetic theory which has been extended markedly in its accuracy of description of actual transport in real gases by the more elaborate treatment of Chapman and Cowling ( 8 ) ,which was based in part upon the analysis of Enskog ( I O ) for gas a t elevated pressures. Sherwood and Pigford ( 2 6 ) developed some of the simpler relationships for molecular transport and Jost ( I S , 14) treated molecular migrations in some detail. The general problem of transport was ably reviewed by Kirkwood and Crawford (15), who presented an excellent basis for the engineering evaluation of transport phenomena. This article presents a number of relationships that have been found by the authors to be of utility in predicting material transport in fluids. Chapman and Cowling (8) proposed the following relationship to describe diffusion in a binary system:
- $Dc.j.k 31. L72
uj-u.k=
INDUSTRIAL AND ENGINEERING CHEMISTRY
. .
(1) Vol. 47, No. 5
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT I n the study of the self-diffusion of carbon dioxide ( 2 4 , d l ) it was found that the experimental work deviated from the predictions of Chapman and Cowling only by 20% a t a pressure of 4500 pounds per square inch. Such accuracy is a n indication of the effectiveness with which it is possible to predict the influence of environment upon transport in gases. If the binary system is considered a n ideal solution ( 1 7 ) ,it follows from Equation 1 for isobaric, isothermal conditions that
Fick diffusion coefficient i s used to describe basic relations in transport phenomena
The equation of continuity is a n essential part of the analysis of transport processes. This expression, which is only a statement of the principle of the conservation of matter, can be expressed in a number of ways. I n the case of a system in which there are only changes in the z direction, the properties of the phase are functions of z alone and are independent of the coordinates y and z. Under these circumstances the equation of continuity may b e written as
(3) Maxwell (19)presented the following relationships which follow directly from the preceding expressions for the special case of perfect gases:
Equation 8 is expressed in terms of specific weight rather than density because of the use of the force-length-time system of dimensions. The extension of the equation of continuity to three dimensions results in
(4) (9) (5)
Fick proposed that the molecular transport be related to a coefficient, D F , k , and the concentration gradient as indicated in the following equation:
Four diffusion coefficients are in common use-the Chapman and Cowling diffusion coefficient defined in Equation 1, the Fick diffusion coefficient as defined by Equation 6 , the Maxwell diffusion coefficient ( 1 9 )related to Equations 2 and 3, and the volumetric coefficient (26). These are related by the following approximate expression:
For the purposes of simplicity all of the following analytical development is limited to one dimension, although the treatment may be extended to three dimensions without change in point of view. I n situations in which no chemical reactions are involved, there can be no change in the quantity of a component a t any point in the system except as a result of its transport to or away from the point by the processes under consideration. I n the case of component k , the equation of continuity as expressed by Equation 8 may be rewritten as
The component velocity,
uk,
may be defined in the following way: mkmX
=
Equations 1through 7 have as their basis the simple kinetic theory and they form a useful means of approximating the transport behavior in gases. It is found that the coefficients of resistance, C k and C,, of Equations 2 and 3 are nearly constant throughout a wide range of compositions for conditions under which the simple kinetic theory is applicable. I n the case of transport in liquids and in solids the prediction of the quantitative aspects of the phenomena is somewhat more difficult t o realize. Jost ( I S , 14) prepared a rather complete summary of the correlations of diffusion coefficients in liquids and in amorphous and crystalline solids. Little is known about the effect of composition, pressure, and temperature on the diffusion coeffi;ients in many of these systems. Until the kinetic theory has advanced materially, it probably will be necessary t o rely upon experiment for the evaluation of the diffusion coefficients encountered in solid, liquid, and gas phases a t elevated pressures. Onsager (80-22) was perhaps the first t o consider quantitatively the influence of one transport process (such as molecular diffusion, thermal transport, etc. ), on another. Such matters were treated in some detail by de Groot (12) and in a more limited fashion b y Denbigh (9). No consideration of these reciprocity relationships or the principles of irreversible thermodynamics will be given in this discussion. However, in any material transport where there exist temperature or gravitational gradients, consideration must be given to the interrelation of the coefficients. Such effects also appear to be of importance in turbulent transport (18). However, the interaction of the coefficients is not well understood for turbulent transport or for the transient behavior in molecular transport. May 1955
The component flux, A b , represents the total transport of component k per unit time per unit area in a specified direction. Throughout this discussion the force-length-time system of dimensions is employed. On this basis the material flux per unit area is expressed in pounds per (second) (square foot) of crosssectional area. All gradients in the phase are limited to the 2 direction in the interests of simplicity. For this reason the appropriate subscripts indicating this fact have been eliminated. The material transport or component flux as defined by Equation 11 is made u p of two parts, the gross transport associated with the hydrodynamic velocity and the molecular transport resulting from a concentration gradient. As used in this discussion, the hydrodynamic velocity may be related to momentum per unit volume of fluid in the following way:
rn u=U
Care must be exercised in order to distinguish clearly among the hydrodynamic velocity, the component velocity, and the diffusion velocity. These measures of transport are related in the following simple fashion: uk = u $.
Ud.k
(13)
As indicated in Equation 13, the hydrodynamic velocity is the same for all components and is the velocity to be employed in all the relationships commonly used in fluid mechanics (3, 16). The diffusion velocity for component k may be related to the concentration gradient as follows:
I N D U S T R I A L A N D E N G I N E E R IN G C H E M I S T R Y
919
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT The coefficient, D F . k , as defined by Equation 14, is the Fick diffusion coefficient and is a function of the local state of the system a t the point in question. I n addition, it may well be a function of the first derivative, with respect to distance, of the numerous potentials that may apply a t the point in question. For example, the Fick diffusion coefficient may be a function of a concentration gradient, the temperature gradient, and even a pressure gradient resulting from a gravitational or centrifugal field. The magnitude of these effects is not usually known with certainty except as established by the Onsager reciprocity relationships (60-22) which apply only to a steady state. D e Groot (12) and Denbigh (9) have considered these matters under steady conditions in some detail. For steady-state transport a combination of Equations 11, 13, and 14 results in
& = UkUk
= UfJk
uk - D F , k aax
(15)
It is again emphasized that Equation 15 applies only to transport in the z: direction and that a somewhat more elaborate expression results for the three-dimensional case. From Equation 15 and the conservation principle the following useful relationship (15)is obtained for one-dimensional transport:
I n this case the conservation principle may be expressed as mb
= m =
k = l
UkUk
=
uu
k = l
Equation 16 indicates that the total of the molecular transport for all components a t any point is zero. If Equation 16 is extended to molecular transport in three dimensions, there is obtained
reasonably successful in describing the behavior of gases a t low pressure. The more complete analysis of Chapman and Cowling (8) and Enskog (10)offered a number of distinct improvements but still left something to be desired in describing the transport characteristics of gases a t elevated pressures. The present discussion is limited to the more elementary concepts of Maxwell (19), using fugacity as a potential. The use of fugacity or chemical potential as the potential field was proposed by Onsager and Fuoss ( 2 2 ) and Jost (13, 14) and was considered by Denbigh (9) in the treatment of steady-state processes. For a binary system the component velocities may be related t o the fugacity:
Equation 20 indicates that the fugacity gradient is directly related t o the component velocity, a resistance term C, and' the product of the concentrations of all of the components present. The component velocities are defined by Equation 15 written in terms of molar quantities. If it is desired t o extend Equation 20 to a three-component system there is obtained
Equation 21 may be extended to any number of components. However, the treatment of the sets of equations that result when applied t o specific boundary conditions is somewhat involved. For this reason the Maxwell hypothesis is seldom extended to special cases involving more than three components (25). It is worth while t o consider, in the light of the Maxwell hypothesis, conditions commonly encountered in material transport. An example involves the transport of one gas through a second stagnant component. For the present, component k will be treated as in transport and component j as stagnant. The characteristics of the stagnant component are defined from Equations 11 and 13:
Aj
= ujui =
UiU
+
UiUd.i
=0
(22)
Equation 22 may be rewritten as u =
If it is assumed that the Fick diffusion coefficient is the same in value in all directions, which may not always be the case, Equation 18 may be simplified to
I n the application of Equation 19 the assumption of the isotropic nature of the Fick diffusion coefficient should be considered. If other potential fields are not involved, it follows from the Onsager reciprocity relationships that the Fick diffusion coefficient will not be the same in all directions. It is possible that a t high concentration gradients the Fick diffusion coefficient will be found to depend upon this gradient. These quantitative relationships are not considered in this discussion. The foregoing expressions set forth the greater part of the basic relations of interest in transport phenomena described in terms of the Fick diffusion coefficient. Similar expressions may be derived for cylindrical or spherical coordinates. The primary limitations in application arise from variations in the Fick diffusion coefficient with the pressure and :omposition of the phase. Since this coefficient was defined by Equation 14, it is not surprising that the properties of the phase exert a pronounced influence upon it. Studies of transport in the gas phase are reviewed
(23)
The diffusional velocity of the stagnant component is just equal and opposite to the hydrodynamic velocity associated with this transport process. If Equation 20 is rewritten for the diffusing component, k, there is obtained
Equation:2O, applied to the stagnant component, results in '
(25)
From a combination of Equations 24 and 25, it follows that:
Equation 26 indicates the interrelation of the fugacity gradients and the resistance coefficients for the diffusion of one gaseous component through a second stagnant component. Equations 24 and 25 may be integrated over a specific length of path under conditions of constant temperature and pressure. If the gas phase is assumed t o be an ideal solution (17), the fugacity may be related to the composition and the fugacity in the pure state in the following way: fh = ? k f t
The molecular transport characteristicfi of gases have been the subject of many investigations. Maxwell's early work (19) was
920
- ud,j
fi =
(1
- ?k).f?
INDUSTRIAL AND ENGINEERING CHEMISTRY
(27) (28)
Vol. 41, No. 5
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT Differentiation of Equations 27 and 28 with respect to the direction of transport results in
A combination of Equations 29 and 30 with Equation 26 results in fkoek
=
fPci
(31)
If each component is a perfect gas, its fugacity in the pure state is equal to the pressure, and it follows that
.
ci
=
Ck
(32)
Equations 31 and 32 indicate a relationship between the two resistance coefficients presented in Equations 24 and 25. A combination of Equations 24 and 29 and Equations 25 and 30 results in (33) (34)
(19), as set forth in Equations 24 and 25, is applicable to the transport process and that the phase is an ideal solution ( 1 7 ) . It is possible to derive expressions analogous to Equation 43 for phases following any specific equation of state such as that proposed by Benedict and coworkers (4-6). It appears unlikely that the deviation of the equilibrium behavior from ideal solutions will be greater than the deviation of the transport procese from the Maxwell hypothesis. If the gas phase is a perfect gas and a n ideal solution, the fugacity of a component is identical in the mixture to its partial pressure. p k
=
paTk = f k =
fh
(44)
Under these circumstances Equation 43 reduces to the conventional form,
(45) Equation 45 describes the transport of one component through a phase that may be treated as perfect gas. Equation 43 is possibly a more accurate representation of the characteristics of a transport process since it does not assume a particular equation of state. This equation is based on the assumption that the effect of changes in composition can be described by the relations applying to an ideal solution ( 1 7 ) .
From the definition of a molal concentration it follows that:
.
Uk
uj =
.
n k
= -
v
y - (1 V
(35)
Interrelation and deviations among diffusion coefficients may be significant
(37)
The widespread use of both the Fick and the Maxwell diffusion coefficients has resulted in confusion in their application. It appears to be desirable to delineate the basic relationships between these coefficients and to set forth certain other expressions describing the dependence of a number of auxiliary coefficients upon them. The Fick diffusion coefficient does not take into account the hydrodynamic velocity resulting from gross motion of the fluid, whereas the Maxwell diffusion coefficient pertains to the total transport. A combination of the Maxwell hypothesis, as expressed by Equations 2 and 3, and the definition of the Maxwell diffusion coefficient, given by Equation 41, results in the following:
nh)
V
Equations 33, 34, 35, and 36 may be combined:
(38)
Equation 37 may be solved for the component velocity t o yield (39)
Equation 39 may be rewritten in the following form to describe the flux of the component in transport:
Equation 15 may be written in a form involving two-component velocities.
It is conventional to define the Maxwell diffusion coefficient for component k as
(48)
A combination of Equations 48 and 16 results in ui = u
A combination of Equations 40 and 41 results in
DF +-uj
kbUk
(49)
ax
The substitution of the component velocities from Equations 47 and 49 in Equation 46 yields
If it is assumed that the Maxwell diffusion coefficient is independent of composition, Equation 42 may be integrated t o yield Equation 50 may be rewritten as (43)
[Mipi + Mkpk] - Xk - R2T2Dp.k -ax - O ax DM.k.i Mi a?k
Equation 4 3 permits the molal flux to be established from a knowledge of the differences in mole fraction and the properties of the phase. It involves the assumptions that the Maxwell hypothesis May 1955
(51)
The substitution of Equations 27, 33, and 34 in Equation 51 results in
INDUSTRIAL AND ENGINEERING CHEMISTRY
921
ENGINEERING, DESIGN, AND PROCESS DEVELOPMENT
For a binary system it follows from Equation 58 that Equation 52 may be simplified as follows:
n
(59) k = l
Further rearrangement of Equation 53 results in
(54)
If the concentration of component k is small or the two components have nearly the same volumetric behavior the denominator reduces to unity. Equation 54 is a general relationship between the Fick diffusion coefficient and the Maxwell coefficient in a phase which is an ideal solution. I n case the phase may be treated as a perfect gas, Equation 54 reduces to
Equation 60 is based on the premise that the pressure is constant throughout the transport system. The equation of state for a perfect gas may be described as
From the definition of equal molar diffusion, ukuk = -u,uj
=
ink
= -mi
(62)
The moles per unit volume of the component may be established from the equation of state,
(55)
Pi,
I n situations for which the concentration of one of the diffusing components is small, the ratio of mole and weight fraction of the other component may be considered as unity. Under such circumstances, Equation 55 reduces to the conventional expression,
Uk
Vi
.
=
RT
=
Pi RT
A combination of Equations 63 and 64 with Equation 62 results in Pjui
+
PkUk
0
(65)
Equation 65 may be rewritten Equation 56 has often been employed in situations in which the concentration of the diffusing component is significant, but under these conditions Equation 55 is preferred. The extent to which the Maxwell hypothesis utilizing fugacity as a potential yields a reasonable description of the transport behavior in gases a t elevated pressures remains to be established. From such information the utility of the relationship presented in Equation 54 can be evaluated. Expressions based on the basic expressions of Chapman and Cowling (8) may be derived to replace Equation 54, but until experimental transport data are available in greater abundance such an approach does not seem worth while. There exists another type of diffusion coefficient that is often employed in the engineering literature. Sherwood and Pigford (26) utilized this so-called volumetric diffusion coefficient in describing many transport processes. The relationship between the hlaxwell and volumetric diffusion coefficients is given by the following defining expressions :
The Fick diffusion coefficient is identical with the volumetric diffusion coefficient for a perfect gas under circumstances in which the concentration of the component in transport is small. Significant deviations among these diffusion coefficients exist under many of the conditions encountered in practice. Part of the difficulty in the treatment of transport problems results from confusion regarding the relationships between these three coefficients. Prediction of equal molar transport makes use of fugacity or chemical potential
A diffusion operation that is often discussed is the equal molar transport of perfect gases. If it is assumed that the gases are perfect, it is convenient to employ partial pressure as the potential since it is numerically identical with the fugacity for an ideal solution of perfect gases. Under these circumstances the partial pressure may be defined as 922
I n accordance with Maxwell's hypothesis, as defined by Equations 4 and 5 for two perfect gases diffusing under isobaric, isothermal conditions, the resistance coefficient, C, is identical for each of the components. A combination of Equations 4 and 5, written for components k andj, with Equation 66 results in
A combination of Equations 63 and 64 with Equations 67 and 68 results in
If Equations 69 and 70 are reduced to specific rather than molal quantities and combined with Equation 10,
Equations 71 and 72 correspond to expressions involving the Fick diffusion coefficient under such conditions that the hydrodynamic velocity is zero. As indicated by Equation 57, the volumetric and the Fick diffusion coefficients are equal only in the case of a perfect gas with components of equal molecular weight, or of an infinitely dilute solution. For this reason the discussions of Babbitt ( 1 , 2 ) concerning equal molal transport are limited to these conditions. It is only under these circumstances that the Maxwell and Fick diffusion coefficients yield comparable ex-
INDUSTRIAL A N D ENGINEERING CHEMISTRY
Vol. 47, No. 5
ENGINEERING. DESIGN, AND PROCESS DEVELOPMENT pressions, as is indicated in Equations 71 and 72. I n this case the hydrodynamic velocity would be zero and the Maxwell diffusion coefficients would be the same for both components. Such a state is typified by the phenomenon of self-diffusion which has been studied in some detail (24, SI). Under such circumstances there would be no change in the center of gravity of the system as a result of the transfer process. Any change in the center of gravity of s system of fixed total weight must be associated with a hydrodynamic velocity. I n the application of the Fick diffusion coefficient it is essential to take into account the hydrodynamic velocity. This necessitates a careful analysis of the boundary conditions to differentiate properly between gross transport of the phase, the hydrodynamic velocity associated with diffusion, and the molecular tran-port. An analysis of the evaluation of the Fick diffusion coefficient from transient, isothermal measurements involving a heterogeneous binary system in an isochoric vessel is available (25). Ac knowledgrnent
The interest and assistance of W. G. Schlinger in developing a number of the expressions reported in this article are gratefully acknowledged. \T. N. Lacey reviewed the manuscript and Evelyn Anderson assisted in its preparation. Nomenclature
Maxwell resistance coefficient of component k , (lb.) (sec.) (sq. ft.)/(lb.-mole): Ck.n = Maxwell resistance coefficient of component k , with respect to component n, (lb.) (see.) (sq. ft.)/(lb.mole)2 Dc,k = kinetic theory diffusion coefficient, sq. ft./sec. D F , ~= Fick diffusion coefficient of component k , sq. ft./sec. DE/I.= ~ Maxwell diffusion coefficient of component k , lb./sec. Dv,J; = volumetric diffusion coefficient of Component k , D K ~ / P; D F , k ( n k / n k ) , Sq. ft./SeC. = fugacity of component IC, lb./sq. ft. fk = fugacity of Component IC in pure state, lb./sq. ft. f’: l? = natural logarithm = weight rate of transport, lb./(sq. ft.) (sec.) r? 7 ~ 1 ~= transport rate of component k , lb./(sq. ft.) (sec.) = molal transport rate of component k , 1b.-mole/(sq. ft.) mi, (sec.) = molecular weight of component k , lb./lb.-mole Mk = weight fraction of component k nk = mole fraction of component k = pressure, lb./sq. ft., abs. P = partial pressure of component k , ( P n k ) ,lb./sq. ft. pk = universal gas constant, (ft.) (lk.)/(lb.-mole) ( ” R.) R T = thermodynamic temperature, R. U = local velocity, ft./sec. u d = diffusional velocity, ft./sec. = transport velocity of component k , (lb.)/(sec.) (sq. ft.) u k = molal volume, cu. ft./lb.-mole V &,y, z = coordinate axes, ft. Z = compressibility factor, P V / R T = specific weight, lb./cu. ft. cr = reciprocal molal volume, 1b.-mole/cu. ft. CT = concentration of component k , lb./cu. ft. Q,: = concentration of component k , 1b.-mole/cu. ft). ph z = summation of n terms = time, sec. = partial differential operator C k
=
nk
$
May 1955
Subscripts j refers to component in stream k refers to diffusing component 12 refers to component 12 2, y, z refer to 2, y, z directions 1 refers to greatest concentration 2 refers to least concentration Superscript O refers to pure state Literature cited
(1) Babbitt, J. D., Can. J . Phys., 29, 427 (1951). (2) Babbitt, J. D., C a n J. Research, A28, 449 (1950). (3) Bakhmeteff, B. A., “Mechanics of Turbulent Flow,” Princeton University Press, Princeton, 1941. (4) Benedict, M., private communication to B. H. Sage, 1946. (5) Benedict, M., Webb, G. B., and Rubin, L. C . , J . Chem. PhyS., 8, 334 (1940). (6) Ibid., 10, 747 (1942). (7) Bridgman, P. W., “Logic of Modern Physics,” Maemillan, New York, 1938. (8) Chapman, S., and Cowling, T . G., “hhthematical Theory of
Nonuniform Gases,” Cambridge University Press, Cambridge, 1939. (9) Denbigh, K. G., “Thermodyanamics of the Steady State,” Methuen & Go., London, 1951. (10) Enskog, D., Kgl. Svenslca Tfetenskapsakad. Handl., 63, No. 4
(1921). (11) Fick, A , Ann. Phys. u. Chem., 94, [2], 59 (1855). (12) Groat, S. R. de, “Thermodynamics of Irreversible Processes,” North Holland Publishing C o . , Amsterdam, 1952. (13) Jost, W., “Diffusion und Chemische Reaktion in festen Stoffen,” Theodor Steinkopff, Dresden und Leipaig, 1937. (14) Jost, W., “Diffusion in Solids, Liquids, Gases,” Academic Press, New York, 1952. (15) Kirkwood, J. G., and Crawford, B., Jr., J . Phys. Chem., 56, 1048 (1952). (16) Lamb, H., “Hydrodynamics,” Cambridge University Press, Cambridge, 1932. (17) Lewis, G. N., J. Am. Chem. Soc.. 30, 668 (1908). (18) Lynn, S.,Ph. D. thesis, California Institute of Technology, Part I, 1953. (19) hIaxwell, J. C., Sci. Papers, Cambridge University Press, Cambridge, 2, p. 625 (1890). (20) Onsager, L., Phys. Rev., 37, 405 (1931). (21) Ibid., 38, 2265 (1931). (22) Onsager, L., and Fuoss, R. M., J. P h y s . Chem., 36, 2689 (1932). (23) Reamer, H. H., Opfell, J. B., and Sage, B. €I., “Diffusion Coefficients in Hydrocarbon Systems-Methane in Liquid Phase of Methane-Decane System,” IXD. ENC.CHEM.,to be pub-
lished. (24) Robb, W. L., and Drickamer, H. G., J. Chem. Phys., 19, 1504 (1951). (25) Sherwood, T. K., “Absorption and Extraction,” LMcGraw-Hill Book Co., Sew York, 1937. (26) Sherwood, T. K., and Pigford, R. L., “Absorption and Extraction,” McGraw-Hill Book Co., New York, 1952. (27) Stefan, J., Ann. P h y s i k , 41, 725 (1890). (28) Stefan, J., Sitzber. A k a d . Wiss. Wien., Math.-naturw. Kl., 63, Abt. IIa, 63 (1871). (29) Ibid., 65, Abt. 11, 323 (1872). (30) Ibid., 83, Abt. 11, 943 (1881). (31) Timmerhaus, IC. D., and Drickamer, H. G., J . Chem. Phys., 19, 1242 (1951).
RECEIVED for review July
19, 1954,
INDUSTRIAL AND ENGINEERING CHEMISTRY
ACCEPTED December 13, 1954.
923
