= distance from bottom of diffusion column, cm. = critical compressibility factor, dimensionless
x t,
GREEKLETTERS Y
parameter in Equation 8, dimensionless = function defined in Equation 4 = T,ll2A,function defined in Equation 5 viscosity, g. cm.-’ set.-' = = 3.1415. . . = density, g./cc. = p / p , , reduced density = zero-point density (Riedel, 1954, 1957), g./cc. = Lennard-Jones collision diameter, A. = Pitzer acentric factor (1955), dimensionless = coordination parameter, dimensionless =
A 6 I . (
a P PT
PO u
w
€
Literature Ciled
Amagat, E. H., in “International Critical Tables,” Vol. 111, p. 38, McGraw-Hill, New York, 1926. Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” !%ley, New ’York, 1960. Cohen, M. H., Turnbull, ID., J . Chem. Phys. 31,1164 (1959). Crank, J., “The Mathenlatics of Diffusion,” Oxford University Press, London, 1956. Doolittle, A. K., J . Appl.Php. 22, 1471 (1951). Douglas, D. C., McCall, D. W., Anderson, E. LV., J . Chem. Phys. 34, 152 (1961). Eyring, H., Henderson, ID., Ree, T., “Progress in International Research on Equilibrium and Transport Properties,” Masi and Tsai, eds., Academic Press, New York, 1962. Eyring, H., Ree, T., Proc. .Vat/. Acad. Sei.47, 526 (1961).
Glasstone, S., Laidler, K. J., Eyring, H., “The Theory of Rate Processes,” McGraw-Hill, New York, 1941. Guttman. I.. Meeter. D. A.. DeDt. of Statistics. Universitv, of Wisconsin,’Madison, Rept. 37 (1564). Hirschfelder, J. O., Curtiss, C. F., Bird, R. B., “Molecular Theory of Gases and Liouids.” Wilev. New York, 1954. Kuether, G. F., Ph.D. thesis,’ University of Wisconsin, Madison, 1965. Li, J. C . M., Chang, P., J . Chem. Phys. 23, 518 (1955). Michels, A., Botzen, A., Schuurman, W., Physzca 23, 95 (1957). Michels, A., Michels, C., Wouters, H., Proc. Roy. Sac. A153, 214 (1935). Naghizadeh, J., Rice, S. A , , J . Chem. Phys. 36, 2710 (1962). O’Hern, H. A,, Jr., Martin, J. J., Ind. Eng. Chem. 47, 2081 (1955). Pitzer, K. S., Lippman, D. Z . , Curl, R. F., Huggins, C. M., Petersen, D. E., J . A m . Chem. Soc. 77, 3433 (1955). Reamer, H. H., Sage, B. H., Lacey, W. N., Ind. Eng. Chem. 41, 482 (1949). Ree, T. S., Ree, T., Eyring, H., J . Phys. Chem. 68,3262 (1964). Riedel, L., Chem.-Ing.-Tech. 26, 259 (1954). Riedel, L., Kaltetechnzk, Band 9, Heft 5, S.127 (1957). Robb, W. L., Drickamer, H. G., J . Chem. Phys. 19, 1504 (1951). Robinson, R. C., Ph.D. thesis, University of !$’isconsin, Madison, Wis., 1965. Smith, A. S., Brown, G. G., Ind. Eng. Chem. 35,705 (1943). Stakelbeck, V. H., Z . Ges. Kalte-Znd. 40, 33 (1933). Sutherland, !$‘illiam, Phtl. Mag. 9, 781 [ 1905). Tee, L. S., Robinson, R. C., Kuether, G. F., Stewart, LV. E., American Petroleum Institute, Refining Division, Houston, May 1966. Timmerhaus, K. D., Drickamer, H. G., J . Chem. Phys. 19, 1242 (1951). RECEIVED for review February 10, 1967 ACCEPTED July 24, 1967
VARIATION OF NONISOTHERMAL D I FFUS I C) N COEFFICI ENTS S E R G I O D I C A V E ’ A N D A L D E N H . E M E R Y , JR.2 School of Chemical Engineering, Purdue UniuersiQ, Lafaq’ette, Ind.
The diffusion coefficient measured in thermal diffusion is typically 20 to 40% larger than that measured in isothermal diffusion, and increases with temperature gradient. for the magnitude of the effect.
N THE
development of the phenomenological equations for
I thermal diffusion (Enskog, 191l ) , people naturally assumed
the minimum complexity needed to handle the data then available. One of the assumptions always made is that the diffusion coefficient measured in isothermal experiments, Di, is meaningful also in the nonisothermal conditions involved in thermal diffusion. Here attention is’ called to the data of Agar and Turner (1960a, 1960b) and data are presented that indicate that the diffusion coefficient measured in thermal diffusion, D,,is usually higher than Di. Previous Work
T h e effect was first called to our attention by I. Prigogine, who noted that Jeener and Thomaes (1954) had observed a D , 1
Present address, University of Rome, Rome, Italy. address, Cniversity of Leeds, Lceds, England.
* Present
Mass-transfer theory does not account
in their open cell which was significantly larger than D,in the same cell. Several pieces of research in this laboratory (Childress, 1957; Maniotes, 1959; Tope, 1960; Wilsbacher, 1964) followed. Results indicated that there may be a significant effect, but the precision of the work was not high enough for us to be sure. .411 of these researches were done in a two-compartment cell with a sintered-glass membrane of the type used by Drickamer (Saxton et al., 1954), with the hot side on top. This type of cell yields only one point per run, and the cell must be cleaned and reloaded for another point, which gives poor precision in the resulting diffusion coefficient. Cells that record continuously during a run, such as the one of Jeener and Thomaes which used an optical analysis, give much better precision. Agar and Turner (1960a, 1960b) used such a technique in work on dilute aqueous solutions in a n open cell, in which they fo1lov;ed the course of the diffusion by a conductometric
V O L 7 NO. 1
FEBRUARY 1968
95
method. A certain function of the measured resistances plotted against time gave exponential curves, and they obtained constants from these curves by two methods. I n one, they fitted the curves with the best values of D and the Soret coefficient, s. They made the important observation about the results that “in a majority of cases a value of D greater than the true value is required.” By “true value,” they meant D,, I n other words, using our terms, D , was usually greater than D,. They felt that this was probably due to convection, which would appear in the later stages of the process and thus would appear to increase D and decrease s. I n the other method of analysis, they obtained a value of s (call it )s, from the steady-state separation, and the product sD from the initial slope of the separation curve. From this product and the isothermal value, Di, they calculated another value of s (call it si). These two differed, and they preferred si because it was less likely to have been influenced by the convection they felt was present. The suggestion is made here that D , is not the same as D,, and that one does not need the hypothesis of convection to explain these discrepancies. I t is suggested that SJ and si are the same and that it is actually D which differs, being higher than they thought. Let 7 = D,/D,. Then r can be calculated from the second method by the ratio si/sf, and from the first method by the ratio D , / D i . I n their Table I, Agar and Turner present enough information to calculate 7 from both analyses for four systems, shown here in Table I. In two
Table 1.
Values of r from Data of Agar and Turner (1960b) +stem haCI RbCI CaCl CdSO4 7 From curve-fit D 1.42 1.05 0.99 0.97 From initial s and steadystate s 1.40 1.17 1.07 0.99
Figure 1 . 1. 2. 3.
4. 5. 6.
96
systems, 7 is different from unity, and about the same 7 results from Method 1 as from Method 2. There seems to be no reason to expect convection to affect D and s to the same extent, and this is taken to support our view. Agar has presented a n interesting argument why, under special circumstances, this might happen. Agar and Turner also present, in their Tables 3, 4, and 5, information from which r can be calculated by sJs, for 35 other systems. I n all but a few of these, Y is in the range 1.1 to 2.7, with an average a t about 1.4. Their data, by our interpretation, show that in dilute aqueous solutions of electrolytes, D is almost always larger under nonisothermal conditions. Additional evidence from their work comes from their measured effect of inclining the cell. They plotted concentration separation (their function of resistance) against time for several inclinations from zero to 45 minutes of arc, and used this to show that all of the inclinations gave the same initial slope. This was true, but a plot of separation us. inclination a t a given time suggests rather that separation leveled out nicely as zero inclination was approached, and that there may well have been no convection a t zero. In a later work, Snowdon and Turner (1960a, 1960b) presented results from a cell they felt was more carefully constructed to reduce convection. T h e values of s they obtained fell between the two sets obtained by Agar and Turner for the same systems, but since their analysis involved the central portion of the separation curves and not the initial slope, no conclusion can be drawn. Apparatus
Our cell, shown in Figure 1, consists of two hollow 1.7-ml’ cylinders separated by a sintered-glass membrane. T h e membrane was vertical, and several 0.06-ml. samples were drawn out of each side and analyzed on a Bausch & Lomb precision refractometer. T h e size of the samples was held constant, so that equal liquid levels were maintained on the two sides of the cell.
Diffusion cell
Brass body Hot water jacket Cold water jacket Sintered-glass membrane over bottom port of juncture of cylinders Teflon plate over top part of juncture Magnetic agitator
l&EC FUNDAMENTALS
7. Baffle to enhance mixing 0. Thermocouple 9. Teflon gasket 10. Sample port 1 1 . CUO-HIPO~ cement
Procedure
Six different values or A T were used, including A T = 0. At each value of A T , four runs were made, and in each run five pairs of samples were withdrawn a t times systematically varied to give a suitable spread over the range of times of interest. T h e system used was 1,1,2,2-tetrachloroethaneand 1,1,2,2-tetrabromoethanr: a t an average mole fraction of 0.5. I n the thermal diffusion runs, both sides of the cell were filled with the same fluid, and zero time was taken when the heating and cooling fluids were started through the jackets. I n the isothermal diffusion runs, on the other hand, the beginning was more arbitrary. First one side and the void volume in the sintered-glass membrane were filled; then the other side was filled. After a few minutes to assure that a linear concentration gradient had been established in the membrane, both sides were sampled; this time was arbitrarily called "zero." Working Equations
For each period between samples, the volumes of liquid in the two compartments are constant. T h e first part of the development below applies to one such period of constant volume; then these are combined to give the final equation. T h e flux of component 1 is given by
[
J1= Dp -
ax
-+
- x)
ax(1
bz
T
"1
(5) or
Concentration % A is eliminated from Equation 6 by means of the mass balance,
(7) which gives % A a t any time during the current constant-volume period in terms of the volumes of liquid during that period and the concentrations a t the beginning of that particular period, %A' and X B ' . These equations are combined to yield the ordinary differential equation
in which
bz
in which This equation is integrated from the beginning to the end of the period in question, yielding
= mole fraction of component 1 t = distance through. sintered-glass membrane
x
p
=
molar density of entire mixture
In
T = temperature a = thermal diffusion coefficient D = diffusion coefficilent
XB
XB'
- - %A' - Am Am
=--
DkzA
t
V B
in which t is the time increment of that period. We define a fictitious time increment,
Mass balances on the two sides yield
6 =
(12)
vBokzt/VBkzo
in which subscript zero indicates values a t the beginning of the run, and rewrite the equation in which
In
= volume of hot and cold compartments t = time
VB,VA
A
=
T h e temperature ancl concentration profiles through the sintered glass are assumed linear and unchanged by the transient nature of the process. This is a reasonable assumption because the void volume of the glass is a small fraction of the total volume. Thus, the concentration gradient in Equation 1 may be expressed in terms of the concentrations, x B on the hot side and on the cold. T h e thermal diffusion contribution to flux is ,assumed to be constant in time and space,
TB - x ) In = constant = Am TA
- % A - Am - %A' - Am
=
8 -T
(1 3)
Finally, by successive substitution, we can refer the concentrations to zero time,
area normal to the diffusion flux
ax(1
XB Xg'
A = xB
- %A
= Am[l
- exp(-O*/~)]
(14)
in which O* is the fictitious time since the start of the run. Thus the curve described is a n ordinary exponential when A is plotted against O*. T h e relaxation time is constant throughout the process, and is given in terms of initial volumes, T
=
L V B o / D k z d
(1 5 )
For the isothermal runs, by a similar development, A = A,, exp(-O*/r)
(16)
(3) Results
in which A m = xB - % A a t infinite time, as found from the solution of Equation 1 for J I = 0. J I is assumed independent of z for the particular time in question, though it varies slowly with time. Equation 1 can be rewritten (4)
T h e five sets of concentrations of each run were analyzed by regression to obtain the best values of the constants Am and r in Equation 14, or A. and r in Equation 16. Standard deviations of the constants were determined by the scatter around the regression curve. Averages of the four runs for each A T were then obtained from equations of the type
and, if D is assumed independent of z (and thus of T ) , and T linear in z, this may be integrated to give VOL 7
NO. 1
FEBRUARY 1968
97
0.018
An example of the thermal diffusion results is shown in Figure 2, in which A is plotted against O* for all four runs a t the highest AT. T h e solid line is Equation 14 for the average values of A , and r. T h e agreement between the runs is good. I n the isothermal runs, the value of T should be the same for all runs, but the value of A0 should not, because of the arbitrary time a t which each run was started. This is of no consequence for our work, since only T is wanted. However, to show all four runs on one plot, three runs have been adjusted to the A0 of the other run (Figure 3). T h e ratio r = D,/D, = T , / r t is plotted in Figure 4 as a function of AT; bars indicate 26 limits. T h e most important single conclusion to be drawn from these results is that all the values of r are significantly larger than unity-that is, the diffusion coefficient obtained from thermal diffusion runs is Significantly greater than the diffusion coefficient from isothermal diffusion. A second question is whether or not r is a function of AT. An F test shows that we can be 95% sure that the five values of r are not from the same population. From the figure, it appears that r is increasing with AT, but it is difficult to tell the nature of the function. T h e least complex function is a straight line not through the origin. T h e slope of this line is (4.8 4.4) X in which the second figure in the parenthesis is 2u (calculated from the values of u of the five points, not from the scatter around the line). Thus we can be 95% sure that r is an increasing function of AT. This best straight line, however, does not pass through the point (0, l ) , which we require of a reasonable function. Another two-constant curve which does this is
.016
-* .014 N "
r
"" .o I 2 C
.-0 t .010 0 Lc L
u ';5
.008
E u
,006
'ID
,004
,002 0
*
+ alAT f al(AT)2
0
12
8
20
16
e*,
B*.
Figure 3.
24
hours
Isothermal diffusion
Regression line Fictitious time, defined b y Equation 12, which causes curve to be exponential
(19)
1.6
Using the values of u shown by the bars, however, we find that a2 is not significantly different from zero, and thus the data
1.5
r = 1
4
I .4
0.018 .O 16
L
-* .014
1.3 1.2
ON
X N "
1.1
.012
C
0
I .o
'5. ,010 0
0
uL
=E rn
IO
20
30
40
50
60
70
AT, O F
,008 Figure 4.
Ratio of diffusion coefficients,
r
= DI/Di
Bars show 20. limits
,006 X
I %
.004 .002 0
0
4
8
12
16
20
24
hours
Figure 2.
Sample results of thermal diffusion
- AT
Regression line. There were similar curves for four other values of B*. Fictitious time, defined b y Equation 12, which allows curve to be a simple exponentiol
98
l&EC FUNDAMENTALS
make it possible to draw a curve no more complex than a straight line through (0, 1) (Figure 4). T h e value of the thermal diffusion constant, a, is plotted in Figure 5 ; bars show 2u limits. a appears to vary with AT, and the same questions can be asked as with r . First, a n F test shows we can be 95y0confident that the five values of a are not all from the same population. Second, the slope of the best straight line through the points is (-1.8 + 1.3) X and we can be 95% sure that a is a decreasing function of AT. T h e amount of variation with AT, however, is much smaller than that of 7. r varies 13% over the range studied, and CY varies 5%.
0
IO
20
30
40
50
60
70
AT, O F
Figure 5.
Thermal diffusion constant as a function
Of AT
Barr show 2v limits
This cell, with its vertical sintered-glass membrane, is subject to a n error from convection currents through the membrane generated by the difference in density on the two sides. This causes a to change with AT, whereas it should be constant. I t also contributes slightly to r, but not enough to alter the major conclusions. (The authors are indebted to H. L. Toor for suggesting the manner of treating this.) T h e primary effect of the convection is to speed up the remixing process, which ordinarily invohes only diffusion. This reduces the steady-state concentration difference, and increases the rate of approach to steady state. However, a secondary effect of the convection is that it changes the variation of concentration through the membrane from a linear profile to a n exponential one, which reduces the amount of diffusive remixing, and the net effect is slight. T h e broken line in Figure 5 was calculated from these considerations, from a membrane permeability chosen to fit these data best. (The permeability so chosen was 2.6 X 10-9 sq. cm., which is reasonable.) T h e effect on r is just the inverse of the effect on a ; the broken line in Figure 4 shows this result. Clearly. convection contributes in only a small way to r . Physical constants involved in the calculations were: a = 0.89 (Childress, 1957), q = 2.62 cp., D = 3.6 X 10" sq. cm./ g./(cc.)(" C ) ; all except a are from second, p = 1.97 X unpublished data taken in this laboratory. T h e cell constant obtained from the isothermal data is L / A = 0.263 cm.-l Potential Explanations
Several possible theoretical explanations of these experimental results were explored. These included modifying the analysis to allow for a variation of the diffusion coefficient with
the temperature, letting the thermal conductivity vary with the temperature, leading to a nonlinear variation of temperature with position through the diffusion plug, and allowing for the heats of transport in establishing the temperature profile. None of these offered a n explanation for our results. T h e thermodynamics of irreversible processes contains a number of hypotheses and limiting assumptions which might be relaxed or overlooked in a n effort to make the phenomenology more flexible. One might wish to include terms containing driving forces squared, or products of two drive forces, for instance. We would hope to be able to preserve those portions of the linear theory which have proved useful in the past, while modifying it to include this new effect. Unfortunately, there appears to be no way of doing both. By including either of the above modifications, or temporarily ignoring the Onsager hypothesis, we cannot derive equations which both include the new effect and preserve the useful notion that the thermal diffusion coefficient is given by a difference in net heats of transport. Not only is it not possible to produce equations with the proper functionality, affecting only D,but including these extra complications would destroy the developments which lead to the classic forms of the driving forces. Nothing short of a full-scale redevelopment of the entire theory would seem to do. Conclusions
Experimental evidence from three types of cells, and the authors' data from a fourth, indicate that D obtained in nonisothermal conditions is larger than D in isothermal conditions. This cannot be explained by current mass-transfer theory. T h e nonisothermal D increases with temperature gradient, but the data do not allow anything more sophisticated than a linear dependence to be derived. literature Cited
Agar, J. N., Turner, J. C. R., J . Phys. Chem. 64, 1000 (1960a). Agar, J. N., Turner, J. C. R., Proc. Roy. Soc. (London) 255A, 307 (1960b). Childress. D. E.. M.S. thesis. Purdue Universitv. 1957. Enskog, D., Phystk. Z. 12, 561533 (1911). Jeener, J., Thomaes, G., J . Chem. Phys. 22, 566L (1954). Maniotes, J., M.S. thesis, Purdue University, 1959. Saxton, R., Dougherty, E., Drickamer, H., J . Chern. Phys. 22, 1166 (1954). Snowdon, P. N., Turner, J. C. R., Trans. Faraday Soc. 56, 1409 (1960a). Snowdon, P. N., Turner, J. C. R., Trans. Faraday SOC.56, 1812 (1960b). Tope, W. G., M.S. thesis, Purdue Gniversity, 1960. Wilsbacher, C. L., M.S. thesis, Purdue University, 1964. / ,
RECEIVED for review November 7, 1966 ACCEPTEDOctober 17, 1967
VOL. 7
NO.
1
FEBRUARY 1 9 6 8
99
