Calculation of Gas Mixture Viscosities J. W. BUDDENBERG'
AND
C. R. WILKE
University of California, Berkeley, Calif.
An
empirical correlation has been developed to predict the constants i n the Sutherland-Thiesen equation for binary gas mixtures leading to the general equation: U1
U"
Reproduction of available data at moderate pressures over a temperature range from 11" to 225" C. is obtained with a n average deviation of 3.79'0 between calculated and observed values for 116 mixtures. The following general form to include multicomponent systems is proposed:
A and B are constants which are determined experimentally for each gas pair. Viscosities are assumed to be essentially independent of pressure in this development. Schudel (10) has effected a simplification of the SutherlandThiesen equation showing t h a t
leading t o a general equation containing only one constant @n
=
1
@'
+c9F
+ 1 + @*c 3 @ zz
51 P1
(3)
P2
Through Equation 3, therefore, i t is possible t o predict the entire viscosity curve for a binary mixture from a single experimental determination a t one mixture composition, 140 1
j#i
I
Data for three ternary mixtures and two quaternary mixtures are reproduced by the general equation with a n average deviation of 2.56%.
I
K MANY chemical engineering problems, particularly in the
interpretationand design of heat andinass transferprocesses, it is necessary t o know the viscosity of gas mixtures. Although viscosities for pure gases are fairly readily available, data for mixtures are scarce. It is desirable, therefore, to have a means of calculating necessary data for mixtures from the properties of the pure components. Uyehara and Watson (21) have developed a general method for estimating gas mixture viscosities based on a correlation of reduced viscosities for the single components and the pseudocrit,ical concept of K a y (8). T h e present method is proposed as an alternate means of calculation. Unfortunately, gas mixture viscosit,ies or fluidities are not additive on a simple mole fraction or other concentration basis. A plot of viscosity against mole fraction for binary systems niay show marked curvature with the viscosity of the mixture lying above or below t h a t indicated for a linear relationship, and in some cases rising to a maximum viscosity considerably greater than that for either pure component,. A number of general equations expressing binary mixture viscosit,ies as a function of concentration and pure component viscosities have been proposed (4,6, 9). These equations are generally complex and unwieldy and involve a number of arbitrary constants which must be determined from experimental data. A rather simple equation which has been found to fit a wide range of experimental data was first derived by Sutherland (11), and later derived independently by Thiesen ( l a ) .
1
Present address, Union Oil Company of California, Oleum, Calif.
0 Exp
%
H2
Figure 1. Fit of Hydrogen-Freon 12 Data by Schudel's Equation
T h e authors have further verified the application of Equation 3 in a study of five new binary mixtures ( 3 ) . Figure 1 shows the reproduction of the hydrogen-Freon 12 curve a t 25" C., and 1 atmosphere by this equation. DEVELOPMENT OF GENERAL EQUATION
The constant C in Equation 2 was calculated for available binary mixture viscosity data and compared with various properties of the systems. It was found t h a t C could be expressed as a function of the diffusion coefficient of the gas pair as shown in Figure 2. The results are summarized in Table I. The most satisfactory agreement was obtained with values of C based upon a n experimental point in the central region of the viscosity-mole fraction curve for each gas pair, and this procedure was followed in calculation of the d a t a plotted in Figure 2. Diffusion coefficients not available in the literature v e r e estimated by the method of Arnold (1). The line through the d a t a in Figure 2 is based on a least squares treatment of d a t a for 21 gas pairs at room temperature and atmospheric pressure. The data at higher temperatures were excluded from the least squares treatment because of uncertainty with respect t o the effect of temperature on the diffusion coefficient. However, the data a t higher temperatures
1345
I N D U S T R I A L A N D E N G I N E E R I N G CHEMISTRY
1346
TABLE 1.
SCHUDEII'S
Gas Pairb Hz-He (14, 17) Nz-SO (16)
COz-CaHs (18)
CONSTliNTS AND DIFFVSIOK COEFFICIEXU'TSa Temperature,
Din
C.
Rq. Cm.'/sec.
20 20 26.9 20 20
1.331A 0.2001A 0.0802.4 0.6831A 0.6998E 0.7413.1 0.7153E 0.676GA 0.4748.4 0.1433.4 0.1359.1. 0 . A13CE 0.0638A
0
C ~ e c . / ~ dCm. . 0.9266 6.4646 15.915 1.8382 1,9749 2.0313 2.0572 2,0328 3.3262 9.0547
Hz-SHI ( I # ) Hz-CHa ($0) 19 Hz-Nz (23) 19 Hn-CO ( I S ) 20 IIe-Ar (14, 17) 26.9 Hd-CaHa (18) 12-13 CzHeAir ( 7 ) 8.097 12-13 NHa-CzHa ( 7 ) 12-13 2.6112 Hz-CnH4 (7) COz-Freon 12 ( 3 ) 25 20.876 0.45179 25 3.6002 Hn-Freon 12 ( 3 ) 0 . 6158E 25 2.4504 Ha-C0z (3') 25 O.0965A 14.205 Nz-Freon (3; 25 1.. 093A Hz-Ne ( 3 ) 1.3743 11.1 0.7326A 1.9518 Hs-NO (7) 17 0,5308E 2,7614 Hz-SO2 ( 7 ) 0.2067E 26.8 0.6281 Nn-Oz ( 1 9 ) 0.7704E 20 1.9117 Hz-On ( 1 9 ) 99.8 0.2022.1 7.72 Air-CnHa ( 7 ) 1.705 99,2 0.800E Hz-COz ( 7 ) 1,21Y 199 Hz-802 ( 7 ) 1.10E 1, 9 15.4 0.6552 100 Hz-He ( 7 ) 22.5 4.2825 0.252.k CHa-CxHa (% 225 117.2 0.00926.ACHi-CrHs a A = diffusion coefficients calculated h) Ainald'8 method; E = diffusion coefficients based on experimental data. D h-umbers in parentheses indicate d a t a source. e A t 27.2 atmospheres' pressure.
fit the correlation satisfactorily using diffusion coeffieicnts calculated 011 t,he basis of the usual assumption that D is proportional t,o Tap.
7201. 41, N o ?
( k i t s alone, so t h a t better agreement rnight, be expectled with more accurate diffusion data. To illustrate further the utilit,? of the general eyua1,ion in the reproduction of viscosity curves having a wide variety of shapes seven binary systems were selreted as shown in Table IT. The average deviation betwvprn ca81culatetland observed values for the 35pointslistedis 2.15%.
Awucmiox TO MULHCOWPONE\ r
SY wmis
T h e form of the Sutherland-Thiesen type of equation for mulucomponent systems (9) suggeit- t h e following general extension of Equation 6.
u?
where U,,,, U,,, et(:., are the avemge diffusion coefficients of aomponents 1, 2, etc., with respect to the total ga.s mixture. .4n expression for the diffusion coefficient of one gas into a mixture of two or more other gases has been derived by Wilke (29) on the basis of hIaxv-ell's equations for diffusion. This relat,ion may he expressed as follows:
The equation for the !inc in Figure 2 g i v e the relation
Uirnerisioiial conbidel ations, howwer, require tlial tlie Sutherland-Thiesen coristanls be dimensionless. so that an exponent of unity is probably corrcct in I3quation 4 lrading to the relation
3101e
Gas Pair COz-Freon
whaie 1.385 IS n diniensioiiless constant
!\-itti nnplovenlent in qunntity and accuracy of diffusivitr data it is poqsible that this constant mar be sonieTrhat chang a d . By combining Equations 3 and 5 a single geneial rc.latiorr niav br written for thc T - i w m t v of biniiri ~ H mixtiirw' S
where ,ul and I.(.: = viscosities of pure compolit~iits1 and 2 at thcl temperature of the mixture p i and p2 = densities of pure components 1 and 2 a t the temperature and total pressure of the mixture. DL2= diffusion coefficient a t t,hc temperature and total pressure of t h r inixture rl and x2 = mole fractions of conipoiicnts 1 and 2 in the mixture Any consistent selection of units may be made so t h a t thc denominators of Equation 6 are dimensionless. For ideal gases the denominators are nearly indepcndcnt of temperature or pressure. For components a t temperatures below their boiling points hypothetical vapor densities may be estimated on the basis of the gas laws. Some theoretical basis for the relation between viscosity and the diffusion cocfficient is offered by the treatment of Chapman and Cowling ( 5 ) . Attempts t o develop Equation G from the kinetic theory %-ere not successful, howevkr.
0 7500 1.. 0000
xi. xi, ~~
~.'
H2-Fre011
He-A
142.0
133.6:
140.0
-1.4 1.5 -1.4
147.8
, . .
. , .
-"-I.
0.1864 0.2408 0,5893 0.5920 0.7822 1 ,0000
200.8 199.5 189.4 189.3 184.3 178.1
0.0000 0.2500 0.5000 0.7500 1.0000
124.0 128.1 131.9 135.1 88.4
on;
7
81.7 8 3 . ii 87.4 92.4 98.5 U8.7
97.0 89.1
11k:3
1b1:4
197, 2 184. 4 185 9 180.6
...
111:0
135.7 143.1
...
...
85.3 89 8 96.3 103.1 104.1 102.4
...
0~0000 0,1927 0.4096 0,4755 0,4755 0.6947 1 . 0000
174.5 171.7 165.1 161 . o 161 0 144.9
166.3 163 9 163.9 149. G
0.0000 0.8405
221.1 227.8
211:0
0.3660 0,3820 0.4906 0.5966 0.7565 1.0000
e
Calculated Viacosiw, Micropoises
"-";
0,9225 1.0000
Hz-C'C)
T.:~peri:uentaI Vixosity, Iliorogoises 121.0 130. 1 185.3
n nnnn
0~0000 0.1821 0 3704 0,5818 0.7882 0,8750
APPLIC9TIOS TO BPA 4 R Y SYSTEMS
To test the accuracy of Equation 6 viscosities weie calculated and compared with experimental data for the systems listed in Table I. The average deviation from experiment v a s 3.02y0 for 88 binary mixtures a t approximately room temperature. For 23 other mixtures at various temperatures ranging from 99 to 225" C. the average deviation was 6.9s0. These deviations are of the order of magnitude which might be expected to result from uncertainty in the estimated and experimental diffusion ooeffi-
Fraction I'irsl Gas 0.0000 0 2900 0 6000
87.4
Deviation,
% ..I
-
-1.5
-2.1 --2.6
-1.6 - 2.c .
.
I
-0.73 f2.B -i6 .0
+2.'!
f2.d
+4.2 $4.7
-4.7 -5.4
511'9
228.6 229.1 229.6 230.4 227.0 197.3
227.4 227.7 229.2 230.0 227.8
176.8
...
...
h l l d a t a obtained at 25' C. a n d atinospheric p r e m n e .
"
1.
-0.34 -0.52 -0.61 -0. sa
-0.17 $0.35 .
.
,
L .
M y 1949
INDUSTRIAL AND ENGINEERING CHEMISTRY
1347
DISCUSSION
Figure 2.
Correlation of C with Diffusion Coefficient for Binary Mixtures
with similar equations for the other components. relations such as 8 with Equation 7 lum
=
M1
__
-
~
Combining
.
.) + +
PZ
1.385~2 P.3
(9) Equation 9 may be written in more general form for ponents:
rb
P% 2=1
22
(10)
j=1 j#i
Equation 10 was tested by coinparison of calculated with experimental data obtained by the authors ( 3 ) for three ternary mixtures and two quaternary mixtures. These data are given in Table 111. The average deviation between the calculated and observed data for these mixtures is 2.57y0 which supports both the validity of Equation 10 for the gas viscosities and the validity of Equation 8 for diffusion coefficients in gas mixtures.
T A B L111. ~ COMPARISON OF CALCULATED WITH OBSERVED DATAFOR COMPLEX MIXTURES~ Experimental Visoosity,
Calculated Visoosity, Mipropoises 195.3 151.0
ProporMiproGasea tions poises Ne-Hz-COz (9) l/aeach 185.7 At-Nz-Freon 12 (8) 1/3each 146.9 Nl-COe-Freon 12 ( 5 ) 1/seach 145.8 144.6 Ne-H%-COeFreon 12 (3) I/aeach 168.1 162.3 He-Nz-COe-Freon 12 (3) 1/4each 147.1 148.2 All data obtained a t 25” C. and atmospheric pressure.
d arid B = roristants in thp Hutherland-Thieseri equation, dimensionless C = constant in Schudel’s modification of SutherlandThiesen equation, sec./sq em. I) = diffusion coefficient, sq. cm./sec. x = mole fraction P = viscosity, gram/cm.-sec. or micropoises p = density, gram/ml. Subscripts I,2,3, etc. = referring t o component 1, 2, 3, etc. = referring to i and j components 2, 3 m = referring t o mixture = referring t o n component LITERATURE CITED
1+1385u2 X*Pa
NOMEN
