A significant contribution in
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Thermal Diffusion Column Theory for Liquids T H m h i A L diffusion in a convectionless cell was first observed a century ago. T h e thermal diffusion column, which uses convection currents to enhance the separation. is a comparatively recent innovation, introduced in 1938 by Clusius and Dickel (2). I t is the more practical instance of thermal diffusion, and is the item of interest here. T h e column has been studied intermittently since its introduction. Earlv workers include Bardeen ( 7 ) . Debye ( 3 ) , Furry, Jones, and Onsager (7). van der Grinten ( 8 ) , de Groot ( 9 ) . Jones and Furry (73), Krasnv-Ergen ( 7 4 , Nier (75), Niini (76), Taylor and Glockler (27), and Waldmann ( 2 3 ) . Since the column was first used on a large scale by the Atomic Energy Commission during World War I1 to aid in the separation of uranium isotopes ( 6 ) ,it has received the attention of Drickamer and his coworkers (4,5), Powers and Wilke (78), Powers (77), Prigogine and conorkers (79, 22), and others. Novel modifications have been studied by Sullivan, Ruppel, and Willingham (20). Commercial application has been pioneered by Jones ( 7& 72). Much of the early work involved gaseous systems. Some investigators explored the application of the theory oi the column to liquid systems. I n handling viscosity and diffusivity, the equations used for liquid systems are somewhat inadequate, so that it is difficult to deal intelligently with the known instances in \I hich theory and experiment deviate. This article improves the theory of the thermal diffusion column for liquids by calculating the effect of variable viscosity and diffusivity on the operating equations of the thermal diffusion column. T h e theory is known to be lacking in other respects, such aq the effect of concentration on the properties, especially density, but these are not considered here.
Basic Theory Thermal diffusion in a convectionless cell involves a molecular process, as does ordinary diffusion. T h e theory of operation of the column, on the other hand, involves both these molecular processes and the smooth vertical convection brought about by the horizontal density gradient. T h e first complete presentation of the
A.
H.
EMERY, Jr.
School of Chemical and Metallurgical Engineering, Purdue University, Lafayette, Ind.
Up until now, it has been assumed that liquid properties are constant in obtaining the operating equations usually used for thermal diffusion columns.
Here are the results which
show that, for some systems, the effect of temperature on the properties i s important
column theory is that of Furry, Jones, and Onsager (7), who assumed that smooth, laminar convection currents pass upward along the hot wall and downward along the cold wall, and applied a viscous flow equation to determine the horizontal velocity distribution in the column. They combined this with the phenomenological equation for the flux of a species in a mixture, assumed that diffusion in the vertical direction is negligible and that the system properties are independent of concentration, made simplifications which correspond to a steady state in the batch column, and arrived a t the following differential equation:
Here h is the thermal conductivity, 7 is the viscosity, p is the density, D is the diffusivity, g is the acceleration of gravity, and G( T ) is a function obtained from the solution of Equation 1, from which one can calculate all the important characteristics of the steady-state batch column. These characteristics, the results of the calculation, are the variation of vertical velocity, horizontal mass flux rate, and concentration with horizontal position, and the variation of concentration with vertical position. T h e last result, the most important from the standpoint of column design, is obtained from the equation for the rate of vertical transport of species 1, rl,which is
Here c 1 and (‘2 are the concentrations of the constituents of a binary mixture, and z is vertical distance. The transport equation, Equation 2, occupies a n intermediate position in 1 he theoretical development, in that it is the relatively simple end result of the steps in which horizontal distance is eliminated by integration, and is the starting point of the development of operating equations which give concentrations a t the top and bottom of a column as a function of time in the batch column and of rate of throughput in the continuous c 01umn. I t is the end result of the calculation here; the effect of the variation of viscosity and diffusivity appears in constants H and K. T h e operating equations are unchanged by these effects, except that the values of H and K are different than heretofore supposed. H i s defined by
and K is defined by
K = K,+Kd
in which a is the thermal diffusion constant, calculated from convectionless cell data; 2w is the distance between the hot and cold walls; T is absolute temperature; TI and Tt are the temperatures of the cold and hot walls, respectively; A T is the difference between the wall VOL. 51, NO: 5
MAY 1959
651
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temperatures; and B is the horizontal dimension along the plates or cylinders which form the column. Constant K d represents diffusion in the vertical direction. This term is easy to handle in the calculation below, but it is unimportant in practically attainable columns using liquids, and is here ignored. As an example of an operating equation obtained from the transport equation, the steady-state separation in the batch column, which is the same as the limiting separation in the continuous column as throughput approaches zero, is
Same cases as in Figure 1
Cold wall on
Application to liquids Equations 1 through 4 are general and apply to liquids and gases alike. To evaluate H and K,however, some knowlege or assumption about the temperature dependence of the physical properties is required. One possibility is to assume that all the physical properties are independent of temperature. Jones and Furry (73) performed this calculation and obtained HO =
g ~ ( 2 W ) 3 ( ~ ~ ) 2 0 i ~ p
6!F7
(7)
HL K
lnp = -
where L is the height of the column and
Strictly speaking, the transport equation is valid only for the steady-state batch separation, because of the nature of the assumptions. However, this equation with H a n d K given by Equations 3 and 4 has been used as the starting point for equations which appear to be valid to a good first approximation for unsteady batch operation and, with a slight modification, for continuous flow. Bardeen (7) and Debye ( 3 ) derived equations for unsteady batch operation with dilute solutions, Powers (77) treated the case of unsteady batch operation in the middle concentration range, and Jones and Furry (73) presented the results for continuous flow.
652
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= ~-
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Here = - a p / d T and T is the arithmetic mean of the absolute wall temperatures. Equations 7 and 8 and the solutions of the transport equation which follow, such as Equation 9, may be applied as a first approximation to liquids and gases. The worst aspect of this application to liquids is the fact that the viscosity and diffusivity of liquids are sharp functions of temperature. For example, the viscosity a t the cold wall in a typical situation involving aqueous solutions may be four or more times that a t the hot wall. Thus a point in need of improvement in the theory for liquids is the effect of temperature on viscosity and diffusivity. T h e other physical properties do not change
INDUSTRIAL AND ENGINEERING CHEMISTRY
Slit Width
Figure 2. Horizontal mass flux rate is a function of horizontal position
Slit Width
Three different values of the exponent in Equation 10. left, hot wall on righi
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with temperature nearly as much as these two. O n e instance of a calculation of this sort appears in the literature. Niini (76) dealt with viscosity only, and concluded that the effect of varying viscosity on the steady-state batch separation will rarely be over 30%. This is undoubtedly true. However, the rate of approach is more severely affected than the steady-state separation, and the variability in the diffusivity is of greater importance. The effect of temperature on the viscosity of a liquid was here taken as 7 1 a T - n
(10)
where a and n are constants characteristic of a given liquid. This equation is almost as good as the more customary Arrhenius-type equation, and is easier to use here. T h e larger the value of n the more the change of viscosity in the column, and the worse the approximation afforded by Equations 7 and 8. About the low-est value of n exhibited by liquids is that of 1.5 for Freon 22. IVater has a value of about 6, and one of the highest values is about 20 for glycerol. The viscosity given by Equation 10 \$as substituted in Equation 1 and this equation was solved for the product DG( T ) , assuming that all other properties were independent of temperature. This yielded
in which the k's are functions of the wall temperatures of the column and the value of n for the system.
LlOUlD T H E R M A L DIFFUSION Three results may be derived a t this point, without having encountered the variation of diffusivity with temperature. T h e most important of these is the constant, H , which is obtained by substituting Equation 11 into Equation 3 and integrating. T h e other two results are the variation of vertical velocity, u , with horizontal position, which is proportional to the derivative of DG, and the variation of horizontal mass flux rate, J , with horizontal position, which is proportional to DG. Representative curves of these quantities are shown in Figures 1 and 2. T h e most regular curve in each case is for n = 0, indicating no effect of temperature on viscosity. T h e next curve is for n = 6 (water), and the last is for n = 20 (glycerol). All the properties in the ordinate groups. including viscositv, are to be evaluated a t T. T h e ordinates in Figures 1 and 2 are such that the curve for zero n applies to any system and operating condition. For nonzero n. the curves depend on the ratio of absolute Tvall temperatures. A typical ratio of 1.3 \vas used in the examples shoiz-n. These curves depict the effect of changing n, other things being constant. Thus the maximum velocity encountered in the column in the case of n = 20 is about 1.7 times the maximum in the case of zero n. For the calculation of K , it is necessary to include the variation of diffusivity. T h e effect of temperature on the diffusivity of a liquid is related to the effect of temperature on viscosity ( 2 4 ) . and consequently is here given by (12)
D = b p f 1
J\ here n is the same constant appearing in Equation 10. Equation 4 was solved using Equations 11 and 12. The resulting expressions for H a n d R may be written as
H
=
H ~ f ( nT, J T 1 )
(13)
K = Kag (n, T2lT1) (14) Here Ho and KO are the results for n = zero, given by Equations 7 and 8, in which all the properties including viscosity and diffusivity are to be evaluated a t T . Terms f and g thus measure the extent of the deviation from the simpler theory. They are functions which depend only on n and 7’z’T1, and are shown graphically in Figures 3 and 4 for some values of these variables. Both generally decrease with increasing temperature ratio. g somewhat faster than f. T h e ratio f,g,shown in Figure 5, is always greater than unity. because g is alM ays lower than f. One final result that may be obtained is the variation of concentration c \vith horizontal distance, shown at the steady state in Figure 6 for several types of calculation. T h e ordinate in Figure 6 is such that the curve for zero n applies to any system and operating condition at steady state. For nonzero n, the curves vary with Tz TI; a typical ratio of 1 3 was used here. The product c1c2 is fairly constant across the slit in most cases, so that the ordinate is essentially a constant times (c, - c ~ o ) where , 610 is the concentration a t the cold wall. T h e curve for n = zero is regular, as usual. T h e effect of the irregular velocity profile on concentration may be iso-
lated by taking a nonzero value of n for the viscosity, while holding the diffusivity constant. This results in the curve labeled “constant D” for n = 20. Finally, taking account of the variation of both viscosity and diffusivity with temperature results in the last curve. “variable D,”for n = 20. T h e variable diffusivity also has a pronounced effect on the values of g obtained. For example, consider the three cases of Figure 6. When n is zero, g is by definition unity. When n is 20 but the diffusivity is constant, g is 0.85; if diffusivity also is al!owed to vary, g is 0.54. One f curve in Figure 3, that for n = 20, is different from the others. The transport equation, Equation 2, is the integral with respect to horizontal distance of the product of kelocity and concentration. T h e first term, with which f is associated, has the significance that it is the rate of vertical transport of species 1 that would prevail if the Concentration varied with horizontal distance exactly as it does in the convectionless cellnamely, linearly. This linear concentration profile does not depend on either the viscosity or diffusivity of the system, and consequently f is different from unity only because the velocity profile is skewed by the variable viscosity. This has two effects: First, the rate at which the fluid moves u p and down the column is different from that predicted by the simpler theory, and second, the average concentrations (based, of course, on the hypothetical cell concentration profile) of the two streams are closer together than the simple prediction. For all the
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Temp erat ure Ratio Figure 3. Term f of Equation 13 i s a function o f exponent n and the ratio o f absolute wall temperatures
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Figure 4. Term g of Equation 14 i s a function of exponent n and the ratio of absolute wall temperatures VOL. 51, NO. 5
M A Y 1959
653
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curves in Figure 3 except that for n = 20, the rate a t which the fluid flows u p and down the column is close to or a little less than the simple prediction. Thus over most conditions it is the second effect which predominates. However, for n = 20 the rate of flow becomes appreciably greater than the simple prediction as the temperature ratio is increased. At a ratio of 1.3, the flow rate is 1% less than the simple prediction. At a ratio of 1.7 it is 2370 greater, and a t a ratio of 2 it is 8OY0 greater. T h e increased rate speeds u p the rate of vertical transport, and this is why the curve for n = 20 in Figure 3 goes through a minim u m and increases as the temperature ratio is increased. Significance of Results
I n the case of the batch column, the steady-state separation is determined by the ratio of H to K , as shown i n Equation 5, whereas the rate of approach to this limiting separation is dependent only on K. Thus the effect of variable viscosity and diffusivity o n the steady state separation is measured by the ratioflg, whereas the extent of this effect on the rate is given by g alone. Because flg is always greater than unity, the ultimate separation is actually increased over that given by the simpler theory; but a t the same time, g is always less than unity, and so longer times are required to reach a given fraction of the ultimate separation. Similar comments may be made about continuous operation. T h e limiting value of the separation as the throughput approaches zero is dependent on the ratio H / K , whereas the effect of throughput depends chiefly on K , so that the limiting separation is greater than that given by the simpler theory,
654
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0 Figure 5.
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Figure 6. Concentration may b e calculated as a function of horizontal position in three ways n = 0. Result if viscosity and diffusivity are assumed to be constant Constant D. Result if viscosity is assumed to vary, with n = 20, while diffusivity i s assumed to remain constant Variable D. The result if both are assumed to vary with temperature, with n = 20
but the throughput has a more serious effect than heretofore predicted. For many common liquids in common operating conditions, the effect is not large. Thus for aqueous systems and a temperature ratio of 1.3, f is 0.96 and g is 0.92. However, for a system with the characteristics of glycerol a t these same temperatures, f is 0.85 and g is 0.54, which means that the steady-state separation is 57% greater and the rate constant 47% smaller than that predicted by the simpler theory. Electrical heating and the higher temperature ratios which result have a similar effect. Acknowledgment
T h e author is indebted to the Kational Science Foundation for partial support during the work, and to Royce Stroud for assistance in calculating the values shown in Figures 3 and 4. literature Cited
(1) Bardeen, J., Phys. Reu. 57, 35-41 (1940); 58, 94-5 (1940). (2) Clusius, K., Dickel, G., Nuturwissenschaften 26, 546 (1938). (3) Debye, P., Ann. Physik 3 6 , 284-94 (1939). (4) Drickamer, H. G., Mellow, E. W., Tune. L. H., J . Chem. Phys. 18, 945-9 (1956). (5) Drickamer, H. G., O'Brien, V., Bresee, J. C., Ockert, C. E., Zbid., 1 6 , 122-8 (1948). (6) Fox, M. C., Chem. C3 Met. Eng. 52, 102-3 (December 1945).
INDUSTRIAL AND ENGINEERING CHEMISTRY
(7) Furry, W. H., Jones, R. C., Onsager, L., Phys. Rev. 55, 1083-95 (1939). (8) Grinten, W. van der, Nuturwissenschaften, 27, 317 (1939). (9) Groot, S. R. de, Physica 10, 81-9 (1943). (10) Jones, A. L., IND. ENG. CHEY.47, 212-15 (1955). (1 1) Jones, A. L., Petrol. Processing 6 , 132-5 (February 1951). (12) Jones, A. L., Foreman, R . W., IND. END. CHEM.44, 2249-53 (1952). (13) Jones, R. C., Furry, W. H., Reus. Modern Phys. 18, 151-224 (1946). (14) Krasny-Ergen, W., hrature 145, 742-3 (1940). (15) Nier, A. O., Phys. Reo. 57, 30-4 (1940). (16) Niini, R., Suomen Kemistilehti 2 0 , No. 9, 49-52 (1947). (17) Powers, J. E., Proceedings of Joint Conference on Thermodynamic and Transport Properties of Fluids, July 1957, p. 198, Institution of Mechanical Engineers, London. (18) Powers, J. E., Wilke, C. R., A.2.Ch.E. Journal 3 , 213-22 (1957). (19) Prigogine, I., de Brouckere, L., Amand, R., Physica 1 6 , 577-98, 851-60 (1 950). (20) Sullivan, L. J., Ruppel, T. C., Willingham, C. B., IND.ENG. CHEM. 49, 110-3 (1957). (21) Taylor, T. I., Glockler, G., J. Chem. PAYS. 8, 843-4 (1940). (22) Thomaes, G., Physica 1 7 , 885-98 (1951). (23) Waldmann, L., iVaturwissenschaften 27, 230-1 (1939). (24) Wilke, C. R., Chem. Eng. Progr. 45, 218-24 (1949). RECEIVED for review September 26, 1958 ACCEPTED January 21, 1959 Division of Industrial and Engineering Chemistry, 134th Meeting, ACS, Chicago, Ill., September 1958.
