tion for the 110 forms, accurate estimates of MgSO4-7Hz0 crystal growth rate under a variety of conditions can be made.
=
y
constant in the Stranski-Kossel mononuclear twodimensional nucleation theory
literature Cited Nomenclature
A
=
D, D,
= =
AE
=
G K
= =
k~ ks
= =
N B =~ Nsc = R = S
T y
= = =
surface area of a crystal face diffusivity particle diameter energy of activation for inclusion of solute molecule into crystal face, cal/mole growth rate of crystal face, lb-moles/hr overall growth coefficient, lb-moles/hr ft2 mole fraction mass transfer coefficient, lb-moles/hr ft2mole fraction surface reaction coefficient, lb-moles/hr f t 2 mole fraction Reynolds number Schmidt number gas constant, 1.99 cal/mole O K supersaturation, ( y ~ ys)/ys absolute temperature, OK mole fraction solute, moles of solute/total number of moles; for system investigated, moles of MgSO47HzO/total moles
GREEKLETTERS = constant in the polynuclear two-dimensional nucleap tion theory
Amelinch, S., J. Chim. Phys. 47, 213 (1950). Berthoud, A., J. Chim. Phys. 10, 624 (1912). Botsaris, G. D., Denk, E. G., Jr., IND. ENG.CHEM.,FUNDAM. 9, 276 (1970).
Brian, P. L. T., Hales, H. B., A.I.Ch.E. J. 15, 419 (1969). Brian, P. L. T., Hales, H. B., Sherwood, T. K., A.I.Ch.E. J. 15, 727 (1969).
Cartier, R. M., Ph.D. Thesis, Polytechnic Institute of Brooklyn, Brooklvn. N. Y.. 1957. Gayner, i.G., Suckling, R. D., A.Z.Ch.E. J. 4, 114 (1958). Hixson A. W., Knox, K. L., Ind. Eng. Chem. 43, 2144 (1951). King, 6. J., A.Z.Ch.E. J. 10, 671 (1964). McCabe, W. L., Stevens, R. P., Chem. Eng. Progr. 47 (4), 168 /.nr, \
(lYOl).
Schlichting, H., “Boundary Layer Theory,” 4th ed, McGrawHill New York, N. Y., 1960. Sukanjnajtee, K., M.S. Thesis, North Carolina State University at Raleigh, Raleigh, N. C., 1968. RECEIVED for review July 2, 1971 ACCEPTEDApril 17, 1972 Presented at the 69th National Meeting of the American Institute of Chemical Engineers, Cincinnati, Ohio, May 1971. The support of the National Science Foundation under Grant GK1767 is gratefully acknowledged.
Diffusion of Gases in Liquids A. Akgerman* and John 1. Gainer Department of Chemical Engineering, University of Virginia, Charlottesville, V a . ,%2901
An equation for the prediction of diffusion coefficients of gases in liquids is presented. It is based on a modified absolute rate theory approach and contains no adjustable parameters. In addition, the quantities necessary for evaluation of the diffusivities are the physical properties of the species involved. A previous equation, that of Wilke and Chang, is used for comparison purposes since it appears to be the best previous equation. Although the Wilke-Chang equation is adequate for many systems, the equation presented here is better for the following cases: ( 1 ) systems involving a small solute gas molecule, such as Hz or He, (2) systems involving solvents having viscosities greater than 3-5 cP, and (3) for comparison of relative diffusion rates.
T h e available expressions for predicting diffusion rates in liquids have been successful in a variety of systems (Gainer and Metzner, 1965; Himmelblau, 1964; Johnson and Babb, 1956; Wilke and Chang, 1955). However, certain limitations still exist. For example none of the expressions is capable of predicting diffusivities for all systems involving a liquid solvent. This may be due to the fact that variations in intermolecular force fields were not taken into consideration. The widely used Wilke-Chang equation (Wilke and Chang, 1955) is not accurate when the solvent is a viscous liquid (Gainer and Metzner, 1965) or when the solute is a small molecule Akgerman, 1971). While the Gainer-Metzner equation (Gainer and Metaner, 1965) appears to work for cases in
* Present address, Department of Chemical Engineering, Bosphorus University, Istanbul, Turkey.
which the solvent is viscous, it is not applicable when the diffusing species is a gas. The object of this investigation, therefore, was to develop an equation for diffusion of gases in liquids based on molecular interactions and geometrical considerations. Dissolved gases appear to form a special class of molecules of small size and low enthalpy of vaporization in the broad category of solutes diffusing in liquids. Previous diffusion equations have been developed using four general theoretical approaches: hydrodynamical theories, kinetic theory of liquids, semiempirical expressions based on the preceding two theories, and absolute rate theory. For a detailed discussion of these, the reader is referred to the literature (Akgerman, 1971; Gainer and Metzner, 1965; Himmelblau, 1964; Johnson and Babb, 1956). We have chosen to use Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
373
Figure 1. A hole composed of four B molecules and the relative size of an A molecule fitting in
absolute rate theory since it appears to be able to account for intermolecular forces as well as being more tractable than the very rigorous kinetic theory. Theory
The development of an expression for diffusion coefficients of gases in liquids will begin using the following equation. This equation, from absolute rate theory, was developed by Eyring and coworkers (Ree, et al., 1958) and has since been modified (Gainer and Metzner, 1965; Olander, 1963).
k T XIB FBBFAB* - EDAB DAB= --exP(EpB R T €APB h2BX3B FBB*FAB ~
where DABis the diffusion coefficient of A in B, T is the temperature in OK, EA is a geometrical parameter describing the number of nearest neighbors around a central molecule, f i ~ is the viscosity of the solvent, X,B, X Z ~ X3B , are the distances between molecules in the 1, 2, and 3 directions, respectively, FBBand FBB* are the partition functions for viscosity in the equilibrium and the activated states, respectively, FAB and FAB* are similar partition functions for diffusion, and E,,B and EDAB are the activation energies for viscosity and diffusion, respectively. As has been done before, we will assume that X 1 = ~ XSB = x ~ B= (VB/N)'Ia. Olander (1963) assumed that the ratio of the partition functions was unity and other investigators have also done the same (Gainer and Metzner 1965). However, let us examine this assumption in more detail. The ratios of the partition functions can be given by the following equation, assuming the rotational and vibrational contributions cancel (Akgerman, 1971)
The free volume of the solution, (Glasstone, et al., 1941) VfAB'"
= XAVfAA'/a
VfAB,
+
can be expressed as
XBVfBB1/a
(3)
where X A and XB are the mole fractions of the constituents indicated by the subscripts. Since a number of gases are only slightly soluble in many solvents, one can assume XA = 0 and XB = 1.0. Therefore VfAB'la
=
VfBB'''
Hence, instead of unity, the product of the ratios becomes (4)
Figure 2. A hole composed of six B molecules and the relative size of an A molecule fitting in
two shortcomings. First it implies that the number of nearest neighbors around a central molecule depends on the size of the central molecule only. By intuition, one might expect that the number of nearest neighbors around a different type of molecule should actually depend on the relative sizes of the molecules involved. Secondly the selfdiff usivity expression is not applicable to gaseous solutes which are in a different phase in their bulk state. Therefore, a different, purely geometrical approach is employed in this development. First, assume that a small molecule like helium or hydrogen diffusing through a medium of large molecules like carbon tetrachloride has a different number of nearest neighbors than when diffusing through a medium composed of small molecules like water. Similarly, a large molecule like butane should result in a different €A value whether diffusing in water or in carbon tetrachloride. Let us also assume, as has been done many times before, that liquids contain holes in their structure (Eyring, et al., 1969; Glasstone, et al., 1941; Ree, et al., 1958, 1964) and solute molecules move into these holes in the process of diffusion. Now consider a hole composed of four molecules and for simplicity assume that the molecules or their intermolecular force fields are touching each other. As seen in Figure 1, each side of the square formed by four B in length and the diagonal is f i ( 2 R B ) molecules is ~ R units units. Therefore the radius of the largest molecule which can fit in the hole formed by four B molecules is given by the expression
Thus it can be concluded that if R A 6 0.414R~,[A = 4. Similarly consider a hole formed by six molecules. If the molecules are assumed to touch each other, they would form B and the a hexagon with each side being equal to ~ R units diagonal 2(2& cos 60') ~ R units B in length. (See Figure 2.) Thus the size of the largest molecule which could fit in the hole is given by
+
RA =
( 2 ( 2 R COS ~ 60') f ~ R B-) ~ R B 2 = RB
(6)
Thus if 0.414R~6 R A 6 R B , (A = 6. Similarly, figures for holes formed by 5, 7, 8, 9, 10, etc., molecules can be drawn, the size of the largest molecule that can fit in the mole can be calculated, and the ratio of R A / R B can be found. Finally, if a curve is fitted through the resulting values of [A vs. R A / R Bone obtains
&)
'/2
Next let us consider the geometrical parameter. [A, by definition, is the geometric configuration of the diffusing molecule and its nearest neighbors or, simply, it is the number of solvent molecules on a plane surrounding the solute molecule. Gainer and Metzner (1965) proposed to calculate [A from self-diffusion data. However, this method appears to have 374 Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
€A =
since R A
0:
VA'/~and R B
6
a
[A =
6
VB'/~,by substitution
(7)
Table 1. Activation Energies for Jumping, EBB’ EBBJ, cal/mole
Liquid
Temp range, OC
0-70 0-50 10-55 0-50 15-40 10-55 10-70 10-40 10-50 0-60 0-45 0-60 0-70 0-70
4,300 3,550 2,200 2,760 2 ,700 2,000 3 ,900 5,800 5,900 2 ,600 2,400 2,800 15,500 7,100
Water Ethanol n-Heptane Uarbon tetrachloride Benzene n-Hexane Cyclohexane Amyl alcohol Isobutyl alcohol n-Nonane n-Octane n-Decane Glycerol Ethylene glycol
A. VISCOSITY B. DIFFUSION Figure 3. Cell models for viscosity and diffusion w
lob
I
I
I
I
I
1
15 1
I I I I I
I
I
I
I
I 1 1 1 1 1
I
I l l 1 4
I
L
‘ a I
It
Thus eq 1 becomes DAB =
E EAPB
1
1
110
WEIGHT
20
OF
50
SOLUTE
100
MA
where {A is given by eq 8. It should be noted here that for self-diffusion the equation reduces to the form given previously (Gainer and Metzner, 1965; Olander, 1963)
since E,,B = EDAB,M B = M A , and [A = 6 as proposed by Glasstone, et al. (1941), Ree, et al. (1958, 1964), Olander (1963), and Gainer and Metzner (1965). This leaves only the activation energy difference in eq 1 to be estimated. Gainer and Metzner (1965) employed viscosity data to evaluate this difference. However, when the solute is a gas in its bulk state this method cannot be used. Therefore, another approach is followed here. I n a manner similar to that used by Gainer and Metzner (1965), let us also assume that the activation energy for viscosity is composed of two parts, the energy required to form a hole and the energy required for a molecule to move into that hole. Similarly for diffusion there is an energy required to form a hole and energy required for a solute molecule to move into the prepared hole. Hence E,,B
1
Figure 4. Activation energy of jumping for the solute vs. the molecular weight of the solute
(E,B R -TEDAB
(h7)’”a(*)1’2
2I
MOLECULAR
- EDAB = EBB’- EAB’
(11)
where the superscript j refers to jumping energy. The holeforming portions of the activation energy cancel each other (Akgerman, 1971). EBB’,the activation energy per mole for molecules to move into the prepared holes, is given by Glasstone, et al. (1941), as
EBB'(^-
&)=Rln(;)+
R :ln(2)
(12)
interesting to note that the values are independent of temperature in the range 0-70°C. The viscosity data for this table are taken from Reid and Sherwood (1966), “Handbook of Chemistry and Physics,” (1970), and Lange (1961). To calculate E A B ~consider , the cell models frequently used to represent viscosity and diffusion processes (Gainer and Metzner, 1965). In the model for viscosity there is a hole formed adjacent to the cell and the B molecule in the center jumps into the hole. (See Figure 3.) Since there are an equal number of molecules around any B molecule, the probability of a B molecule jumping, provided there is enough energy, is one. In the model for diffusion, however, there are f~ B molecules around the central solute A molecule. The probability that the A molecule will jump into the formed hole, then, is
and the probability that the jumping molecule will be B is
Thus to calculate E A Blet ~ , us assume that EABj
=
(EBB~)’B(EAA’)’A
(16)
since for this type of interaction the geometric mean seems to correlate effectively the mixture properties from pure component properties (Olander, 1963). Hence EABj = ( E B Bj ) E A / ( E A + l ) ( E A A j ) l / ( E A + l ) (17) and combining eq 11 and 17
where TL and Tz are any two temperatures in O K and and $2 are the fluidities a t these temperatures. By definition fluidity is the reciprocal of viscosity. Therefore
-
Ti
-Tz
E B B ~ ’calculated S this way are tabulated in Table I. It is
E A A for ~ various gases were determined by Akgerman (1971) from experimental diffusion data. The calculated values were independent of temperature as expected. Subsequent attempts were made to correlate these values with physical properties such as molecular weight, enthalpy of vaporization, and molar volume, and it was found that molecular weight correlated the data best. Figure 4 is a plot of EAA’ vs. moInd. Eng. Chem. Fundam., Vol. l l , No. 3, 1972
375
I
I
. I
I
30 -
0
6t
v
,
-
W
I
-
00.
/
0 ,
1
? 4-
N
i v
-
Lo
P
x 2m
-PROPOSED
04
EPN.
-PROPOSED
EQN.
---W-C I
I 0
IO
20
30
50
40
TEMPERATURE,
60
0
70
I
I
I
I
I
I
IO
20
30
40
50
60
C '
TEMPERATURE,
Figure 5. Diffusion coefficients of oxygen in water: 0, Davidson and Cullen (1 957); A, Duda and Vrentas (1 968); 0,Ferrell and Himmelblau (1 967); 0, Himmelblau (1 964); o, lbrahim and Kuloor (1 960); 9, Krieger, et. a/. (1 967); X, Vivian and King (1964); 0,Wise and Houghton (1 966)
EQN.
EPN.
----W-C
70
'C
Figure 8. Diffusion coefficients of helium in water: D, Baird and Davidson (1962); 0, Davies, et a / . (1964); X, Vivian and King (1 964); 0, Wise and Houghton (1 966)
% t I
4l
,
3
N\
5
/-
C 02- H20 -PROPOSED
EQN.
EQN.
-W-C
I
I
I
I
I
I
I
0
IO
20
30
40
50
60
TEMPERATURE,
CH4- H20
/
I
-
/
-PROPOSED
//
EQN.
/
EQN.
----W-C I1 0
I
I
20
10
I
30
I
I
50
40
TEMPERATURE,
1
70
OC
Figure 6. Diffusion coefficients of carbon dioxide in water: 0 , Davidson and Cullen (1957); A, Duda and Vrentas (1 968); 0,Ferrell and Himmelblau (1 967); 0,Himmelblau (1964); v, Reddy and Doraiswamy (1 967); D, Unver and Himmelblau (1 964)
I
60
1
70
'C
Figure 9. Diffusion coefficients of methane in water: 0 , Wise and Houghton (1 966) 6
I
I
I
I
I
5
I
0
2oL 15
w w m N\
5
10-
-
I-
8-
-
6ro
Q
I
I
EQN. I
I
-
C '
I
10
EQN.
----W-C
/ I
20
1
30
I
40
TEMPERATURE,
I
50
Thus, once E A A and ~ EBB^ are determined from eq 19 and 13, respectively, DAB can easily be calculated from eq 9.
1
60
70
C '
Figure 7. Diffusion coefficients of hydrogen in water: Baird and Davidson (1962); 0 , Davidson and Cullen (1 957); 0, Davies, ef a / . , ( 1 964); 0, Hirnmelblau (1 964); x, Vivian and King (1 964); 0,Wise and Houghton (1 966)
lecular weight showing this correlation. Each vertical line represents the range of E A A ~ for a particular substance of that specific molecular weight and the mean values are shown by the circles. When a curve was fitted to these data using a least-squares technique it was found that
E A A= ~ 5875.3M~-O.'ge 376
---W-C I
Figure 10. Diffusion coefficients of ethane in water: 0 , Baird and Davidson (1 962); 0,Wise and Houghton (1 966)
4-
2 0
/'
I
TEMPERATURE,
-
:
a4
/
Ind. Eng. Cham. Fundam., Vol. 1 1, No. 3, 1972
(19)
Results and Discussion
The proposed equation has been applied (Akgerman, 1971) to 44 different systems and over 300 data points and compared with the Wilke-Chang equation. Figures 5-10 and Table I1 summarize some of the results and the reader is referred to the original reference for the rest. The equation presented here has been compared to the Wilke-Chang equation since it works reasonably well for some gas-liquid systems. The association number in the Wilke-Chang equation is 2.6 for water, 1.9 for methanol, 1.5 for ethanol, and 1.0 for
Table II. Calculated and Measured Diffusivities Diffurivity X 1 O6 Solvent
HzO HzO
Temp,
Exptl
Eq 9
W-C eq
1.77 1.10 1.50 1.91 2.20 2.64 5.31 6.28 9.82 2.44 3.42 6.00 5.60 6.70 2.70 2.50 2.40 3.71 0.954 16.36 7.08 0.75
1.744 1.110 1.555 1.566 1.739 2.333 3.855 5.510 9.374 2.168 3.531 5.857 5.020 5.670 3.349 2.386 2.030 3.495 1.153 17.14 8.306 1.201
1.618 0.946 1.340 0.733 0.787 2.653 3.468 3.822 6.166 2.394 3.861 6.493 5.527 6.199 5.276 3.526 2.870 4.373 0.0034 13.31 4.500 0.2712
O C
16 0 10
25 25 29.6 29.6 0 25 0 25 30 30 40 20 20 20 25.4 25.4 25.4 25.4 25.4
unassociated liquids. However, it is not reported for amyl alcohol, isobutyl alcohol, ethylene glycol, and glycerol. Since the association number is highly dependent on the degree of hydrogen bonding it cannot be assumed to be 1.0 for these solvents. It is interesting to note that the association number can be approximated using XB =
(AH,,
of hydrogen-bonded substance)O.l AHvap of homolog
(20)
The homolog is defined as proposed by Bondi and Simkin (1957), Le., by substituting a “-CHa” group for the “-OH” group. For example, the homolog of water is methane, that of methanol is ethane, that of isobutyl alcohol is 2-methylbutane, etc. For systems where the association number is not reported eq 20 was used to estimate it. In that way it was possible to calculate somewhat better values for diffusion coefficients than calculated by setting XB = 1. As can be seen from Figures 5-10 and Table I1 the WilkeChang equation works quite well for some cases. However, it is not accurate when the solute is a small molecule such as H:, or He, or when the solvent has a viscosity higher than 3-5 cP. Although this behavior at higher viscosities has previously been reported for liquid-liquid systems (Gainer and Merzner, 1965) this investigation has shown that the WilkeChang equation is also in error when the solute is a small gas molecule whereas eq 9 predicts correct values. For all the systems studied the proposed equation was as good as or better than the Wilke-Chang equation, although the WilkeChang equation is easier to use and relatively accurate for many cases. It should also be noted that the predictions of the WilkeChang equation are highly dependent on the use of an empirically chosen correction factor called the association number. However, the use of such correction factors should be unnecessary with an adequate theoretical approach. The equation proposed has no such factors. Another advantage of the proposed equation is that it works equally well with associated and nonassociated systems. In addition, the ap-
Ref
Himmelblau (1964) Tammann and Jessen (1929) Tammann and Jessen (1929) Davies, et al. (1967) Davies, et al. (1967) Krieger, et al. (1967) Krieger, et al. (1967) Ross and Hildebrand (1964) Ross and Hildebrand (1964) Ross and Hildebrand (1964) Bennett, et al. (1968) Malik and Hayduk (1965) Malik and Hayduk (1965) Malik and Hayduk (1965) Groothuis and Kramers (1955) Groothuis and Kramers (1955) Groothuis and Kramers (1955) Akgerman (1971) Akgerman (1971) Akgerman (1971) Akgerman (1971) Akgerman (1971)
proach presented here results in an equation involving only the physical properties of the constituents involved. Another useful comparison of the proposed equation with the Wilke-Chang equation is possible. If diffusivities of different solutes in the same solvent are considered and the ratio DABIDCB,calculated, the Wilke-Chang equation results in
However, eq 9 gives
where the variable
+ is given by
Thus the Wilke-Chang equation suggests that the ratio of the diffusivities of two solutes in the same solvent is dependent only on the molar volume of the solutes. No factor is involved which accounts for the different types of interactions which may be present. However, the proposed equation results in an expression involving both the molar volume and the molecular weight ratios multiplied by a factor which is dependent on the activation energies of jumping. Values were calculated for two solutes diffusing through the same solvent and it was found that if M A = MC or VA = Vc then $ = 1.0 f 0.05 and 1.0 Z!Z 0.15, respectively. Thus the ratio DAB/DCBwas calculated setting $ = 1.0 for five pairs of systems in which the molecular weights were nearly the same and for six pairs in which the molar volumes of the solutes were nearly the same values. The ratios of diffusivities using experimental data were then compared with the calculated values and are listed in Table 111. As can be seen, the ratio DAB/DCBis correlated better with the proposed equation. It is not possible to generalize $ for solutes whose molecInd. Eng. Chem. Fundam., Vol. 11, No. 3, 1 9 7 2
377
Table 111. Comparison of DAB/DCB DABIDCB Solute A
Solvent B
Solute C
coz
CaHs C2H6 C2H6
Nz
02
so2
Clz
Ar Ar Ar Ne CHI Ar C2H6
c02 0 2
02 H2 02 N2 Cl2
MA
Mc
VA
vc
Exptl
W-C eq
Eq 22
44 28 32 71 44 32 32 2 32 28 71
44 30 30 64 40 40 40 20 16 40 30
34 31.2 25.6 48.4 34 25.6 25.6 14.3 25.6 31.2 48.4
74 51.7 51.7 44.8 28 28 28 16.14 29.6 28 51.7
0.955 1.111 1.333 0.798 0.946 1.188 1.022 2.305 0.857 0.937 1.811
1.595 1.353 1.525 0.955 0.890 1.055 1.055 1.075 0.091 0.936 1.040
1.138 1.126 1.088 0.937 0.923 1.134 1.134 3.226 0.725 1.173 0.657
Table IV. Diff usivitier for liquid-liquid Systems Solvent
r, oc
Exptl
Eq 9
Glycerin Ethylene glycol Methanol Hexane Propyl alcohol Amyl alcohol Hexane
Water Water Water Benzene Benzene Ethylene glycol Ethylene glycol
20 20 15 15 15 30 30
0.83 1.04 1.28 1.78 1.60 0.21 0.82
1.10 1.36 1.65 1.63 2.02 0.34 0.47
ular weights and molar volumes differ by more than 10%. Yet by using eq 22 with = 1.0, one can obtain a quick estimate of the diffusion coefficient of A in B provided the diffusion coefficient of another solute C, having about the same molecular weight or molar volume as A, in B is known. In summary, one can expect a relatively accurate estimation of the diffusion coefficients for gases in liquids using the equation developed here. Although the Wilke-Chang equation may result in predictions just as accurate for some systems, the equation presented here is superior for: (1) systems involving a small solute gas, such as H2 or He; (2) systems involving solvents having viscosities higher than 3-5 cP; and (3) comparison of relative diffusion rates. Any new equation presented for the calculation of diffusion coefficients in liquids should be applicable to all classes of solutes, whether they be gases, liquids, or solids in the bulk state. Most previous equations deal with only one class of solutes, and we are recommending the equations presented here for use in cases where the solutes are normally gases in the pure state. However, let us examine the equation for the case of liquid-liquid diffusion. The important quantity to be calculated when using the equation is, obviously, E.&. We have assumed that this can be found from values of E A A ~which , can, in turn, be determined from properties of the solute in the bulk state. The equation is founded on molecular behavior and should, theoretically, apply to all systems. However, a gas molecule when dissolved in a liquid is not the same as a gas molecule in the normal, pure state. The properties of the gas such as negligible intermolecular forces and binary collisions will not apply to dissolved gases because, in that state, they are really more similar to liquid molecules. Thus the equation ~ solutes which we have proposed for the calculation of E A A for that are normally gases, eq 19, should be different for the case of liquid solutes.
+
378
DAB X lo6, cm2/sec
Solute
Ind. Eng. Chem. Fundam.,.Vol. 1 1 , No. 3, 1972
Wilke-Chang
0.88 1.03 1.36 1.42 1.95 0.10 0.10
Metzner-Gainer
0.89 1.26 1.07 1.95 1.78 0.22 0.86
However, the general form of eq 19 should be correct for all types of solutes, with, perhaps, different constants. We have applied eq 19 in its present form to test this and have calculated liquid-liquid diffusion coefficients using eq 9. These values are compared in Table IV to those calculated using the Wilke-Chang and Gainer-Metzner equations. As can be seen, eq 9 and 19 are not too bad for these cases, and work is currently underway to develop an equation similar to eq 19 to use for liquid solutes. However, this is only to show the general applicability of the equations presented here. Wilke-Chang and Gainer-Metzner equations can, obviously, be used for the case of liquid-liquid systems with good accuracy. The equations presented here are, however, superior for the case of gas-liquid diffusion. literature Cited
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19,
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RECEIVED for review July 21, 1971 ACCEPTED April 10, 1972
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Kinetic Theory and Rheology of Dilute Suspensions of Finitely Extendible Dumbbells by Harold R. Warner, Jr.l Rheology Research Center and Chemical Engineering Department, University of Wisconsin, Madison, Wis. 65706
A kinetic theory is developed for a dilute suspension of dumbbells which are joined by finitely extendible ) R < Ro. Hydrononlinear elastic (FENE) connectors obeying the force-extension law F(c) = Ho/R( 1 - ( R / R O ) ~for dynamic. interaction is neglected or considered in “equilibrium-averaged” form. The equilibrium distribution function is calculated and steady-state shearing flow and small-amplitude oscillatory shearing flow are analyzed using this theory. The results for the various material functions computed vary between the Hookean spring dumbbell limit and the bead-string limit.
B e c a u s e of the success of the rigid dumbbell kinetic theory (Bird, et al., 1971) in describing qualitatively a wide range of shear-rate dependent viscoelastic phenomena, the development of a dumbbell model with a finitely extendible nonlinear elastic connector (hereafter referred to as FENE dumbbells) is a useful method of understanding the transition in behavior between the rigid dumbbell and the Hookean spring dumbbell. The finitely extendible connector is closer to the behavior of a macromolecule since a long chain molecule is neither rigid as is a rigid dumbbell nor infinitely extendible as is a Hookean spring dumbbell. Since little is known about the force-extension law for a macromolecule, the force-extension law chosen for the following calculations is one which expedites the mathematical manipulations required to obtain a perturbation solution. Although admittedly crude, the dumbbell model is considered since a dumbbell model seems to contain the essential physical ideas which are contained in models with a greater number of beads and connectors. However, the dumbbell model avoids the coupling of the equations which occurs in the multi-bead systems and which generally presents overwhelming difficulties in the mathematics. This paper is divided into four parts: development of the basic equations, calculation of the equilibrium distribution function, application of this theory to steady-state shearing flow, and analysis of small-amplitude oscillatory shearing flow using the FENE dumbbell model. The two flows which are studied most frequently are considered here. Other flows l Present address, Research and Development Department, Alantic-Richfield Co., Box 2819, Dallas, Tex. 75221.
can be analyzed using this theory; however, the math manipulations required to obtain even a perturbation solution to the equations for these flows would be very complicated. 1. Kinetic Theory-Basic
Equations
A. Basic Equations. The basic equations for the kinetic theory of a dilute suspension of dumbbells with an unspecified connector are well known (Bird, et al., 1971; Giesekus, 1966). Below are given the two equations, in the form derived by Bird, et al. (1971), which have to be solved to obtain the material functions for he various flows to be considered. 2kT b R]$ - --$ 1‘ 3 R
- 2- P@)$})
r
(1)
’5-’5
Equation 1 is called the “diffusion equation” and is a combination of the equation of continuity for the distribution function $(R) and the equations of motion for the two beads of thedumbbell. The distribution function $(R) is such that $(R) d R represents the probability of finding a dumbbell within the orientation range dR. The other symbols in this equation are R, the orientation vector for a dumbbell; K, the tensor specifying the homogeneous flow field; k, the Boltzmann Ind. Eng. Chem. Fundam., Vol. 1 1 , No. 3, 1972
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