CORRELATING DIFFUSION COEFFICIENTS IN BINARY GAS SYSTEMS Use of Vzscosities in a New Equation and Nomogram DONALD F.OTHMER AND HUNG TSUNG CHEN
Polytechnic Institute of Brooklyn, Brooklyn, N . Y .
Diffusion coefficients and viscosities of gases are closely related physically, and theoretical derivations show this relationship. Since viscosities are easily determined and well reported for many gases, the reference substance method has been used to correlate diffusion coefficients in binary gas systems with viscosities in a new generalized equation, using air as the standard substance. The theoretically derived correlation equalion, when tested with 59 binary gas systems, compared very favorably with the results of other methods which are usually more complicated to use. Straight lines of constant slope were obtained on logarithmic paper, from which a nomogram was developed to give directly the diffusion coefficient of any binary gas system over a temperature range from below 200" to over 1400" K. Only the criiical molal volumes and the molecular weights of the two gases are needed to use either the iheoretical equation or the nomogram over this wide range of temperatures.
in binary gas systems and their variawith temperature are most important for design engineers and research scientists. I n view of the scarcity of reliable experimental data. methods of predicting such information assume a useful role. T h e diffusion of the molecules of one gas through those of another is a phenomenon dependent on a concentration or partial pressure gradient. T h e diffusion rate. expressed by the coefficient: is dependent on the ability of the molecules of one species to pass betiveen the molecules of the other species. T h e viscosity of a gas composed of a single species is also dependent on a similar ability of molecules to flow past each other. This is usually expressed as dependence upon the momentum resulting from a velocity gradient of the individual molecule. Thus, it may be expected that because of the physical similarity of diffusion coefficients and viscosities. the latter property may be used to simplify the complex expressions heretofore used for diffusional phenomena. It has been shoirn by Othmer (8) that the method of plotting the property of one substance against the corresponding property of another under similar conditions-the so-called reference substance plot-tends to compensate the mutual irregularities or deviations from ideality. For the sake of this considerable advantage. a simple method for correlating diffusion coefficients is presented here. This includes a nomogram which is immediately usable and correct within the experimental limits of most data reported on diffusion. IFFUSION COEFFICIENTS
D tion
10
TEMPERATURE O K 400 600 800 1000
I
' / ' I
Development of Logarithmic Plot for Diffusion Coefficients
T h e diffusion coefficient for an ideal gas mixture is inversely proportional to pressure and increases \rith absolute temperature T as T b( 6 ) . Thus: DI?P= aTb Taking logarithms and differentiating:
(1)
Figure 1. Logarithmic plot of diffusion coefficients of various systems vs. temperature scale derived from viscosity of air 7. 5.
COz-CeHB 2. Ethane-propane Air-water vapor 6. Hz-CCla
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Similarly, the corresponding equation of a straight line for the variation of viscosity of gases with temperature, used previously for the viscosity correlation (3,7),can be represented as: dp
; =
dT q 7
where d, the distance between centers of unlike molecules on impact, may be replaced by the Lennard-Jones force constant, u, while u may be estimated by the simple empirical relation (2,5)
(3)
If Equation 2 is divided by Equation 3, always a t the same temperature:
u = 0.5894 V C " ~
(8)
Thus : d =
~ 1 = 2
7 + = 0.2947( Vclo.'
+ Vc10.4)
19)
(4) Combining Equations 7 and 9 : O n integration: b log Di2P = - log q
p
+C
(5)
Equation 5 states that if the log of D12P of any q s t e m is plotted against the log p of any substance (gas), a straight line results. This line is straight if the ratio b / q is constant over the temperature range in question, which is reasonably exact for most systems over wide ranges and more exact than the assumption that b or q alone is constant. These conclusions are borne out in Figure 1. in which air is the reference substance. A temperature scale was constructed on the X axis by indicating temperature values corresponding to viscosity values of air. Fortunately, most of the lines are straight with nearly identical slope of 2.74. Equation 5, then, can be rewritten as: log D n P
=
2.74 log u
+C
(6)
Evaluation of Constant C. The next step is to evaluate the constant C in Equation 6. At constant temperature, the hfaxwell equation ( 9 ) connecting the diffusion coefficients with the molecular properties of two diffusing gases can be replaced as:
+
r
Using a value of 0.0182 cp. for the viscosity of air at 298' K., Equation 6 becomes: log (DizP)2980K. = 2.74 log 0.0182
0.80
+C
(12)
Substitution of Equation 11 into Equation 1 2 gives:
Equation 6 and 13 are then combined to form:
-
r
1.00
+
Figure 2 shows the plot of (1 ' M I 1 .Zrl!)O 5 ( Vc10.4 Vc2.4)g us. Dl2P at a constant temperature of 298' K. for more than 50 different binary systems. Assuming a linear relationship, the following equation is obtained:
-
Note that p is the viscosity of air at diffusing temperature in centipoises. In the case of helium systems (such as heliumargon), the results calculated from Equation 14 have to be multiplied by 1.39, because of their deviation from the proposed line in Figure 2. Theoretical Development
The above derivation of Equation 5 is based on the reference substance plot. Such a plot may be made for diffusion coefficients of one system against those for the reference system, always a t the same temperature. Holyever, there is a close relationship between diffusion coefficients and viscosity, because both are based o n the same dynamic molecular phenomena-i.e.. collision integrals and molecular force constants. Hence, it was not unexpected that diffusion coefficients of any system may be plotted us. the viscosities of gas at the same temperature to give lines having a constant slope, as in Figure 1. However, the exact interrelation of gaseous diffusion coefficient and viscosity may be shown theoretically. The equation of Hirschfelder, Bird. and Spotz (5) is: D12P
=
+
0.001858 T 3 / 2 [ ( M 1 M21ilVl.Lfz]1i?
where uI2 is the Lennard-Jones force constant and Figure 2. Logarithmic plot of diffusion coefficients a t 298' K. vs. function of molecular weights and critical molal volumes of components 1 and 2 250
l&EC PROCESS DESIGN A N D DEVELOPMENT
collision integral.
(15)
Ui:nD
The last is the function of
n,
is a
kT
7; e is the Len-
nard-Jones force constant, k is Boltzmann's constant: and T is
I
I
I
I
I
1
1
r
I
1
r
-
c:
I
l
l
1
1
VISCOSITY
7
0.6
4
I
Figure
absolute temperature,
By plotting
us.
3.
kT
- correspond
)
,
I
I
I
20
I
I
I
I
I
I
60 80
40
Temperature dependence of collision integral
kT
on logarithmic
paper, all the consistent data of a, are well correlated by a unique curve. The generalized correlation is given in Figure 3. T h e curve consists of t\co definite regions. The higher values of
,
40 60 80 1.0
20
to the higher temperature region and.
therefore, higher energ! values for any particular system, and kT the lower y values correspond to lower temperatures and
Similarly the corresponding equation for correlating gas viscosity was developed by Hirschfelder, Bird, and Spotz (5) as: 0.0026693 ( M T )'
/1=
(24)
UTl
\\.here a,is the ..collision integral" for viscosity and u is the Lennard-Jones force constant. By using the method previousl) mentioned, the relationship between the collision integral, Q,,
and the function
kT
- has been shown, as in Figure 3. and the
lower energy values. Then collision integral Q,, as correlated in Figure 3. can be tentati\rel\ represented by:
corresponding correlations are as follows:
Substituting Equations 16 and 17 into 15 :
By substituting, Equations 25 and 26 can be rewritten as:
For
kT
< 2.5,
D1,P
=
€
0.001858[(2M1f M 2 ) / M 1 M 2 ] ' / ' T 2O0 k -0.50 uli X 1.47
(;)
(18)
where 0 is a constant for any particular compound. Taking logarithms of Equations 27 and 28 and combining \cith Equations 22 and 23>the following are obtained: For a binary system, nI2 is taken as the arithmetic mean of u1 and 0 2 and E as the geometric mean of € 1 and € 2 . ,411 are constants for any particular gas system. Equations 18 and 19 may be written as:
kT For e
> 2.5.
D12P=
where cy is a constant for any particular system. venience, Equations 20 and 21 may be written as : For
kT -
kT
For--
( 2 11
a"T167
For con-
< 2.5,
log D 1 2 P = 2.00 log T
+ log a'
(22)
> 2.5,
log D12P = 1.67 log T
+ log a''
(23)
For For
kT
; < 2.5,
kT
> 2.5.
log D12P
=
2.00 10% p
log D 1 2 P = 2.50 log
+
c
+c
(29) (30)
In Equation 29, the coefficient 2.00 is the slope of a line at an angle of 63@30', while 2.50 from Equation 30 represents an angle of 68"lO'. However, this slight error in slope is less than can be indicated by the accuracy of experimental data. Actually. in Figure 1, the slopes of the lines are all 2.74, corresponding to an angle of 7 0 @ . This slight variation from the theoretical represents the deviations, experimental and physical, from the theory of Hirschfelder, Bird, and Spotz (5). The conclusion is that use of the reference substance plot would have the same temperature range and utility as the diffusion coefficient equation of Hirschfelder, Bird, and Spotz. VOL. 1
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L
101
Rj'
5001
--IRi
40 450
350
*r 2004
2004
SI-
+
400
z
3
0
0
I; 150
I50
r
r
