e
=
angle of flow surface to horizontal, radians
p
= viscosity, g/cm sec
T
= = =
p
u
surface pressure, g/sec2 density, g/cm3 surface tension, g/sec2
Literature Cited
Bird, R. B., Stewart, W. E., Lightfoot, E. N., “Transport Phenomena,” p 41, Wiley, New York, N. Y., 1960. Burnett, T. C., Ph.11. Thesis, Queen’s University at Kingston, Kingston, Ont., Canada, 1965. Cook, R. A., Ph.D. Thesis, Queen’s University at Kingston, Kingston, Ont., Canada, 1969. Cook, R. A,, Clark, R. H., Can. J . Chem. Eng. 49, 412 (1971). Cook, R. A., Clark, R. H., Bacon, D: W,., Watts, D. G., “Analysis of the stagnant band on falling liquid films: a Bayesian approach,” Technical Report 72-3, Department of Chemical Engineering, Queen’s University, 1971. Cullen, E. J., Davidson, J. F., Trans. Faraday Soc. 53, 113 (1957).
Fulford, G. D., Advan. Chem. Eng. 5, 151 (1964). Kittler, G., 3 f . S ~Thesis, . Queen’s University at Kingston, Kingston, Ont., Canada, 1962. Lamb, H., “Hydrodynamics,” 6th ed, p 468, Dover Publications, New York, N. Y., 1945. Lynn, S., Straatemeir, J. R., Kramers, H., Chem. Eng. Sci. 4, 49 (1955). Matsuvama. T.. Mem. Fac. Ena. Kuoto Univ. 15. 142 (1953). ’ Rlerso;, R. L., Quinn, J. A., A:Z.Ck.E. J . 11, 391 (1965). Nusselt, W., V D I (Ver. Deut. Ing.) Z., 60, 541 (1916). Nysing, R. A. T. O., Kramers, H., Chem. Eng. Sci. 8,81 (1958). Ratcliff. G. A.. Reid, K. J.. Trans. Inst. Chem. Ena. ” 39,. 423 (1961). Roberts, D., Ph.D. Thesis, Imperial College of Science and Technology, University of London, London, England, 1961. Stewart, G., Ph.D. Thesis, University of Edinburgh, Edinburgh, Scotland, 1962. RECEIVED for review August 31, 1970 RESUBMITTED July 10, 1972 ACCEPTEDOctober 4, 1972
Studies of the Loschmidt Diffusion Experiment. I. A Perturbation Analysis of the Diffusion Cell Warren E. Stewart,* Sukehiro Gotoh, and Jan P. S9rensen Chemical Engineering Department, Vniversity of Wisconsin, Madison, Wis. 53706
A detailed mathematical analysis is given for the Loschmidt diffusion experiment. Variable properties and absorption and desorption at the cell midplane and ends have been considered. The profiles of composition, pressure, and molar density are evaluated by a perfurbation method, starting from the conventional solution with constant physical properties and no absorption. The results are useful for accurate determinations of binary diffusivities.
T h e Loschmidt diffusion experiment can give very precise diffusion coefficients. However, systematic errors can arise unless the experiment is analyzed with unusual care. In this paper r e analyze the influence of three normally neglected phenomena: variations in the diffusivity DAB,variations in the molar density c , and absorption or desorption of each component a t the seals of the cell. The three phenomena interact in a complicated way. In an isothermal experiment, the diffusivity can vary with composition and molar density. The molar density varies with composition and with pressure. The pressure may change because of volume effects of mixing and also because cf absorption or desorption. The combination of these effects has not been analyzed before, though some special cases have been treated (Gavalas, et al., 1968; Snider and Curtiss, 1954). hbsorption and desorption can be expected a t the midplane of most Loschmidt cells (see Figure 1) since the sliding surfaces there are generally sealed either by a lubricant (Boyd, et al., 1951; Ljunggren, 1965) or by a gasket (Berry and Koeller, 1960; Ivakin and Suetin, 1964). Additional gaskets a t joints, windows, or instrument leads may also absorb or release the diffusing species. Since absorption and desorption have been detected in some of our experiments, we include them here to a one-diinerisional approximatioil. 114 Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
Problem Statement
The system to be analyzed is showii schematically in Figure 1. It consists of the enclosed gases (phase I) and two uiiwanted absorbing layers (phases I1 and 111). Absorption a t the bottom can be added by superposition as indicated later. These absorbing layers correspond, in our experiments, to the window gaskets and the lubricant on the sliding plates of the cell. Each absorbing layer is considered deep relative to the penetration depth of each solute i over the test period. Our main interest is in the profiles for phase I, from which DABfor that phase is to be determined. The test begins a t time t = 0 with the joining of the two half-cells, a t temperature T and pressure PO. The initial mole fractions are ZA- in the lower half-cell and Z A + in the upper half-cell. Phase 11, the absorbing layer a t the top of the cell, is initially in equilibrium with the upper half-cell. Phase 111, the lubricant, is mixed by the closiiig of the cell; its initial composition is considered uniform, but unknown. The diffusion in the gas phase is analyzed in terms of the following dimensionless variables:
p
=
z/L
(1)
c = c/co
(3)
CD = CDAB/(CPAB)O
V,=
(5)
f xA-)/2
(6)
- XAO) e = (XA - Z A O ) / ~ n = (P - Po)/Po
(7)
= &A+
e = (%A+
CD
+ + 1 + Asl(0e) +
1
YOl(04
Ya2(0€)2
PHASE I
(8) (9)
Here 2r,* is the molar average velocity. The subscripts 0 indicate quantities evaluated a t the reference state (PO,T, ZAO). This state is taken a t the arithmetic mean initial composition in order that the initial conditions be exactly satisfied in the perturbation analysis. The gas-phase property ratios C and C D can be represented by truncated series expansions
c=
n
(4)
u,+~/(aAB)o
%A0
PHASE
+ YlOrI + . + A d I + ... , ,
A02(0e)~
(10)
t co
t
LO
Figure 1 . The Loschmidt diffusion experiment, in the presence of absorbing media (phases'll and 111)
The continuity equations for phase I11 are similar to eq 18. We omit them here, since for this phase a known solution is appropriate (see eq 30 and 31). The initial conditions for phases I and I1 are (at 7 = 0)
(11)
which reduce to unity a t the reference state. The coefficients y t j and A i , defined here can be evaluated from an equation of state and kinetic theory, or from experiments at neighboring conditions. The experiment is regarded as onedimensional and isothermal, so that the following continuity equations apply in phase I
The pressure gradient in the cell will be very small, and we neglect it. The equation of motion then will not be needed. Separate continuity equations are necessary for phase 11. We use a pseudobinary diffusivity Dj,,,I1 for each solute, and define the following dimensionless quantities
(14)
The last equation states that phase I1 is initially a t equilibrium with the gas in the upper half-cell. At the bottom of the cell (t = - 1) we have the boundary conditions
v, = 0
(24)
- =
(25)
0
which express the impermeability of that wall to both gaseous species. At the top ({ = 1) we have the conditions xj
(26)
= (3
cv, = Q
(.+ A ) - CD ar -Q
xjo
Xjf
a0
-
(27)
-k,(CD,)"
which express the equilibrium and conservation of each species a t the interface. An additional boundary condition for each species Here the k , are solubility constants for the following equilibria a t f = 1
z311= klzj
( j = A,B)
(17)
The continuity equations for each absorbed species then take the form
expresses the approximation of a semi-infinite absorbing layer. We treat phase I11 as a onedimensional source or sink for each species a t = 0. Thus phase I is divided into two regions, and to connect them we write the following boundary conditions for 7 > 0 :
r
when constant CII and DjmI1are assumed and convective transport in phase I1 is neglected. These assumptions are adequate here since our treatment is intended only for small solubilities in the absorbing phases. The total flux 9 a t ( = 1 can be expressed as follows, by use of Fick's first law for each solute species B
(z?+- Zjo)kj(CDj)11
Q = j =A
as
(19) f=l
The terms K17-I/1 and K27-l/s are based on the known sohtion for absorption in a semiinfinite medium after a step Ind. Eng. Chem. Fundom., Vol. 12, No. 1, 1973
1 15
1.0
e
.6
IFia+%I 4
.2
0.0
.e
.2
00
0.0
1.0
.e
.2
1.0
Figure 2. Zero-order gas composition profiles for the LoSchmidt experiment
Figure 3. First-order effect of yol on the gas composition profiles
change of interfacial concentration. The medium here is phase 111, which extends outward from the cell a t < = 0. The constants Kl and K z depend on the solubilities and diffusivities of h and U in the sealant, and on the area of sealant exposed a t the wall of the aligned diffusion chamber. Integration of eq 12 with respect to and use of eq 24 and 27, gives
Chemical Society, 1155 Sixteenth St., N.W., Washington, D. C. 20036. Remit check or money order for $4.00 for photocopy or $2.00 form icrofiche, referring to code number FUND73-114. Solutions of Order eo. These are the leading terms of ea 36, 37, and 39; they are the dominant terms for small c and small solubilities. To this order, the pressure in the cell remains constant, L e .
r,
Po(7) = 0 with a = 1for positive ( and a of this result into eq 31 gives
= -
1 for negative r. Insertion
(40)
This may be verified by combining eq 10, 35, 37, and 38, and equating the terms of order 8 . The composition profiles to this order are
(34) Integration of this equation from time zero to eq 10 and 20 to 22, owes 1
S-
C((,T)d{ 1
=
2
[+ 1
t2y$z
+
K17'/'
and use of
c (-I)% m
Xj({,7) =
Ir
-1 2
7,
1
- 2n=O
x
@(o)dw]
(35) This result is used with eq 10 to determine the pressure in the cell as a function of time. Equations 33 and 35 dispose of eq 12,20,24,27,and 31. Perturbation Expansions
To linearize the problem, the following perturbation expansions are introduced:
(42) with Pn =
(n
+ '/d*
(43)
Equation 41 is plotted in Figure 2. This is the solution customarily used to evaluate DABfor this experiment. We now summarize the corrections to this solution. Solutions of Order E. The pressure in the cell is constant t o this order also: Pl(7)
= 0
(44)
This may be verified by using eq 10, 37, and 41 to obtain the terms of order e in eq 35. The gas composition function of order e consists of two terms F1((,7) = (701
+ Aol)Fls +
(YJI
-
Aol)Flo
(45)
in which
Fis By inserting these expansions into the problem statement, and collecting the terms of order to, e, e 2 , K1, etc., we obtain sets of linear equations, which can be solved sequentially for the perturbation functions. The results are summarized here. Numerical tabulations of the functions will appear following these pages in the microfilm edition of this volume of the journal. Single copies may be obtained from the Business Operations Office, Books and Journals Division, American 1 16 Ind. Eng. Chem. Fundam., Vol. 12, No. 1, 1973
and
=
(1 - Fo2)/2
(46)
0 21
I
I
I
I
1
-Old
ao
0.2
0.4
5
o.6
I
I
Figure 5. First-order effect of absorption parameter K1 on the gas composition profiles
'.O
Figure 4. First-order effect of Aol on the gas composition profiles
'The separate contributions of variable c and variable %AB to Fi are plotted in Figures 3 and 4.Both are even functions of t. Solutions of Order 9. At this order we find a small change in pressure
which has been neglected in most published measurements. Gavalas, et al. (1968), have shown that this effect can be used to measure DABa t high pressures and have given a solution which agrees clohely n it11 eq 48 for T > 0.1. Solutions of Order K1 and K 2 . These solutions describe the effect of phase 111 to first order in the solubilities. The pressure terms are
1
&(T)
= - TI" 710
S(T)
=
0
as may be shown from eq 35. The gas composition function G consists of two terms
where
(49) (50)
Figure 6. First-order effect of absorption parameter K Z on the gas composition profiles
and
GR --
4
"
- T " '
3
n=O
pfle-@*"sin (On{)
(53)
The function I , introduced here is defined by
I,(cu)
=
1
e"-OL(w)-1/2du
(54)
The GR term in eq 51 can normally be neglected, since A d ylois very small a t ordinary densities. The gas composition function H is given by
2 (erfc 5 + erfc 2n + r)} -=
n=l
(55)
ad7
This is an even function of p, whereas G is odd (aee Figures 5 and 6). n'ote also that G has a discontinuous second derivative a t p = 0, as required by eq 31, and H has a discontinuous first derivative as required by eq 32. Ind. Eng. Chem. Fundam., Vol. 12,
No. 1 , 1973
1 17
t
I
I
00
5
IO
fi
I
I
I
I
I
l5
2o
I
- 1.0
25
-. 50
Figure 8. First-order effects of absorption parameters EkA and ekB on the gas composition profiles
Figure 7. First-order solution for flux into phase It
Solutions of Order ek5. These functions account for absorption and desorption a t the top of the cell. The flus p5 of eq 38 is obtained by use of eq 19 and 42
in which
(57) and
The coefficients Bjrno(7)have been tabulated numerically up to T = 2.0 f o r m = 0, 1, 2. . . 16. For BrnR(7), a tabulation up t o m = 9sufficed. The functions fA&, T ) and fso(P, T ) are shown in Figure 8 for a n experiment starting with pure A in the upper chamber and pure B in the lower. Absorption a t the bottom instead of the top can be handled by changing ( to - { and xj+ to zf-in eq 62 and 63. Absorption a t both ends can be handled by superimposing the two cases just described. Conclusions
m
4 = 2
C
(58)
( - I ) ' P ~ T " ~ I ~ ( P ~ ~ T )
n=O
Our results for the gas composition and pressure during a Loschmidt diffusion experiment are
The function 9 is initially zero, because phase I1 is equilibrated with the gas in the upper half-cell before the test. After the start of the test, increases rapidly t o a masimum (see Figure 7 ) and then decreases again. The pressure corrections are obtained by inserting eq 37, 38, and 56 into eq 35. This gives
+
(59) in which i
Pr
The functioiisf,(t, T ) of eq 36 have been obtained numerically, bj- application of Galerkin's method (Xmes, 1965; Snyder, et al., 1964) to a trial functionfj with time-dependent coefficients. The resulting solution can be written as F,(b,7)
= -*&,
[
130
+ Y-
(61)
A1o.fR] 10
in which the following trial functions are used to satisfy eq 25 and 28 exactly:
Bjmo(s)
m=O
'Os
{y
arid
1 18 Ind. Eng. Chem. Fundorn., Vol. 12, No. 1, 1973
-k
}I{
(62)
The relative molar density C is obtainable by using these results in eq 10. These results are useful for accurate determinations of DAB)^ from Loschmidt experiments, as will be shown in part I1 of this work. literature Cited
Ames, W. F., "Nonlinear Partial Differential Equations in Engineering," Chapter 5, Academic Press, New York, N. Y., 1965. Berry, V. J., Koeller, R. C., A.Z.Ch.E. J . 6 , 274 (1960). Boyd, C. A., Stein, N., Steingrimsson, V., Rumpel, W. F., J . Chem. Phys. 19,548 (19.51). Gavalas, G. R., Reamer, H. H., Sage, B. H., IND. ENG.CHEM., FCNDAM. 7,306 (1968). Ivakin, B. A., Suetin, P. E., Sov. Phys.-Tech. Phys. 8,748 (1964). Ljunggren, S., Arkiv Kemi Mineral. Geol. 24, 1 (1965). Snider, R. F., Curtiss, C. F., Universitv of Wisconsin Naval Research Laboratorv ReDort wIS-08R-9, Madison, Wis. z ; L. J., Spriggs, T.W:, Stewart, W. E.; A.1.Ch.E; J. 10, 535 (1964). RECEIVED for review October 29, 1971 ACCEPTED September 25, 1972 We are grateful to the National Science Foundation (Grants GK-678 and GK-17860) for support of this work.
S
