Appl. sci. Res. Section A, Vol. 15

VISCOSITY OF MULTICOMPONENT GAS MIXTURES OF POLAR GASES

b y S. M A T H U R and S. C. S A X E N A

Physics Department, Rajasthan University, Jaipur, hldia

Summary

The rigorous binary viscosity expression ~mix as transformed to the form originally suggested by S u t h e r l a n d is studied for mixtures involving polar gases. Any attempt to simplify the ~o' of the Sutherland viscosity expression turns out to be only approximately successfui. A relation for ~~j/~ji is however derived, and the procedure suggested for computing ~mix on this basis appears to be very successful. The ~ij to a large extent are temperature and composition independent and it has been shown that this fact can be utilised with success for predicting *Tmix values at high temperatures.

§ 1. I n t r o d u c t i o n . At the first sight the formula t ion of t r anspor t coefficients for polar gases seems to be a ve ry compl ica ted job be- cause of the angular dependence of the in termolecular potent ia l . M a s o n , V a n d e r s l i c e and Y o s 1) made a r emarkab le contri- but ion in this direction b y showing tha t the results of the formal kinetic theory 2) still remain uneffected except the collision inte- grals are now eva lua ted differently and these now become the averages over all the possible potentials . M o n c h i c k and M a s o n 3) have recent ly even calculated such collision integrals for a Stock- m a y e r type poten t ia l 4) viz.,

q~is(r) : 4 e i j [ ( a i / r ) 12 - - ( a i / r ) 6 + ½Õ~J(aij/r) 3 ~ij], (1)

where the subscr ipts i and 7" denote the molecular species of the in te rac t ing pair, 6ij =/~i#/2(eija~3), # is the dipole m o m e n t and the o ther te rms are as defined b y thema) . The eij and ais are de te rmined fair ly accura te ly f rom the corresponding quant i t ies for the pure gases b y employing the geometr ic and a r i thmet ic mean rules re-

spect ively.

- - 203 - -

204 S. MATHUR AND S.C. SAXENA

§ 2. Formulae. The first approximation to the viscosity of a b inary mixture ~mix is s) 5)

X12~]1 4- f112XlX s _~_ 2 flSX2~2 ~]mix = X~ 4- b12XIX 2 @ fi2X 2 » (2)

where

and

B12 = (1 4- 2T~I @ ~]1~]21)/fl, (3)

Bs--(~1/~72)(«/fl), (4)

b12 = (~~1 _ V s ~ l ) / f l + ~~1, (5)

2RT 5M1 + 3A*12M2 Π- (6)

5pDls MI(M1 -/Ms) '

2RT 5M2 + 3A~~M1 f l - - , ( 7 )

5pDl2 M2(Mi 4- M2)

2RT 5 -- 3A12 (8) Y-- 5pD12 M1 q- M2

Here ~ is in gm cm -1 sec -I, p in dyne cm-% R the gas constant in erg mole -1 deg -1, Dl2 the diffusion coefficient in cm 2 sec -1, x the mole fraction, M the molecular weight, T the temperature in °K, and A12 is the ratio of two collision integrals. The values of A~~ as a funetion of d12 and T;2 = kT/elS are given by M o n c h i c k and M a s o n 3 ) .

The expression for flmix as given by (2) can now be easily trans- formed into the form given by S u t h e r l a n d 6) viz.,

where

and

~]1 7]2 7]mix : -~- (9)

1 -1- ~I2(X2/X1) 1 -~ qO21(Z1/X2) '

B12 _ BXl(X2 2_ B21Xl)-1 ~ l ~ = , ( l O )

1 4- Bx2(%2 -~ B21Xl) -1

B12 @ B~i 1 = bi2, (1 1)

B12B~I 1 = f12, (12)

B ~--- B21fi12(~]l 4- 7]2) - 1 - - 1. (13)

~021 can be generated from c~12, (l 0), by interchanging the subscripts.

VlSCOSITY OF GAS MIXTURES 205

We now look into the possibility of further simplifying the ex- pression for ~~j. following G a m b h i r and Saxena5) . For systems where M1 ~ M2 (we use subscript 1 throughout to characterise the heavier component) the factors

Bxl(x2 4- B21xl) -1 = X , and Bx2(x2 4- B21x1) -1 = Y

are usually smaller as compared to B12 and 1 respective]y, and we can write

B12 ~ B12 --- 1(512 ± Vb22 - - 482). (14)

Further as b~2 -- 482 is invariably small

q~12 ~--- ½b12. (15a)

If b~2 -- 482 ~___ 0 is a reasonable assumption, we get

Similarly,

B21 ~ B21 = b12 ± ~/b~2 - 482

282 __~" b12/282

_~ 1 /~ /~

Dividing (15b) by (16b) we get

B12 (5M1 4- 3A~~]~jr2) - - ~ 8 2 =

B21 (5M2 4- 3A12M1)

Based on relation (10), we can also write

~12 82(Q + x~) B21 (Q 4- 82 X2) '

where

~]1M2 ~~2M1

B21 x~ 4- (1 @ B12B21) xlxz 4- B12x~

(1sb)

(16)

(16a)

(16b)

(i7)

(18)

Q = B{Bel -1 (19) - - Æ12Æ2 (~1 -/- ~2) -1}

Equation (18) is not inconsistent with (17), because usually both Xl 2 and 82x22 are small compared wJth Q and in such cases both the expressions are identical. This interesting result establishes the equivalence of two different sets of approximations. We now pro- ceed to test the appropriateness of different expressions by per- forming numerical calculations on a few specific gas pairs.

2 0 6 S. MATHUR AND S. C. SAXENA

§ 5. Calculations and discussions. In t h e previous section we have succeeded in writing the ~mix expression as given by (2) in the form of S u t h e r l a n d ' s expression, (9), with three different sets of coupled values for 9*J. These may be ealled: (a) rigorous set i s one in whieh no approximation has been made and ~12 is given by (10) and 921 by a similar equation in which the subscripts referring to the molecular species are interchanged; (b) singIy approximated set as given by (14) and (16); and lastly (c) doubly approximated set (15b) and (16b). We do not suggest the set of ~ij values given by (i5a) and (16a) for computing ~inix, for this requires the knowledge of force field between the molecules and also the computational simplification is rather trivial as compared to the set (t 4) and (16). For these very two reasons the doubly approximated set should be most preferable if the calculations could establish its accuracy. We may now compute ~mix Oll the basis of (9) using these three sets of 9*J values with a view to assess their relative aecuracy. The nature of assumptions made in simplifying ~i~ dictate orte general requirement for choosing the gas pair viz., it should satisfy the condition M1 >~ M2. However, we consider in this article the specific binary gas combinations of NH3 with H2, N2 and CH4. It should be noted that for all these systems the approximation M1 >~ M2 is rather poor but for the same reason these systems offer a crucial test to the simplifying proeedure and the resulting formulae.

To know the degree to which the various speeific approximations hold for these systems we record in table I the computed values of the constants X and Y at two limiting compositions, and at the two extreme temperatures where experimental data are available. In eertain cases two sets of B12 values are obtained and for all such cases two values of X and Y are recorded for the same value of Xl. In such cases we have underlined the particular set for which the various approximations hold bet te t and therefore should be preferred. Similar calculations were performed for Æ21 and the final values only are given in the table for the sake of brevity. It should be noted that contrary to out expectations the two as- sumptions viz., X ~ B12 and Y ~ 1, are least valid for the system H2-NH3. For N~-NHs and CH4-NHs systems these approximations hold reasonably well and thus these calculations reveal that any assessment just on the basis of molecular weights may not be cor-

T A B L E I

System

Computed values of the singly approximated 9t3 and a few other faetors

Temp. o K

H 2 - N H a

N2-NH3

C H 4 - N H a

X l

293.16

523.16

293.16

523.16

287.66

0.1

0.9

0.1

0.9

0.1

0.9 0.1

0.9

0. I

0.9

VISCOSITY OF GAS MIXTURES 207

B12 "~ ~o12 X (14)

0.493 0.046

0.371 0.014

0.493 0.190

0.371 0.071

0.478 0.023

0.114

1.081 0.005

0.052 1.054 0.011

0.963 0.001

1.054 0.093

0.963 0.011

0.920 0.004

0.033

Y

0.410

0.125

0.021

0,008

0.204

0.013

0.048

0.006

0.096

0.011

0.010

0.001

0.035

0.004

! B 2 1 "~ 9 2 1

l (16) 2.698

2.029

2.698

2.029

2.093

I I 0 .899

1.038 0.949

1.038 0.949

1.064

rect contrary to our experlence with the mixtures of nonpolar gases. Finally, to judge the appropriateness of these singly ap- proximated 9ij we compute ~mix and these are listed in table IV column 5. The agreement with the experimental values, recorded in column 4, is not as good as obtained with the rigorous theory and indeed is unsatisfactory for H2-NH3. For the remaining two systems the agreement may be regarded as satisfactory. The aver- age absolute deviation being 2 . 8 ° , while the maximum and mini- mum deviations are 7.0 and 0.1 percent respectively. These numbers refer to all the mixtures considered in table IV.

T A B L E II

Computed values of the doubly approximated ~ij and the faetor (b]2 ~ -- 4~72)

Temp. ~1~ ~ ~/B2 ~~1 --~ (1/~//~Œ) System °K (bi22 4/~2) (15b) (16b)

H 2 - N H 3 293.16 + 0.015 0.427 2.339

523,16 0.000 0.478 2,093

N2-NH~ 293.16 -- 0.133 1.097 0 9 1 2

523.16 + 0.008 1.007 0,993

. C H 4-N H ~ 287.66 -- 0.073 0.930 1.076

In table Il, we record the values of the factor b~~ -- 4fi2 for all the three systems at the two extreme temperatures. In almost all cases it can be regarded as negligibly small. Thus, off-hand the doubly approximated 9~~ values are likely to be right for these eases. Computed values of such ~ij are given in this table while

208 S. MATHUR AND S. C. SAXENA

the /]mix values obtained on the basis of (9) using these ~i;" are recorded in table IV column 6. The agreement with the experi- mental values ig now slightly inferior than obtained on the basis of the calculations using singly approximated ~0i;" values. The differ- ence though is not appreciable in most cases. The average absolute deviation is 3.5%, while the maximum and minimum deviations are 10.2 and 1.0 percent respectively.

T A B L E III

Computed values of the rigorous q0«j

I ~~~ System Temp. ~12 modified

° K x l (I0) (I0)

H2-NH•

N 2 - N H s

C H 4 - N H a

293.16

523.16

293,16

523.16

287.66

0.1 0.9 0.1

0.9 0.1 0.9 0.1

0.9 0.1

0.9

0 .317

0.297 0.378

0.359 1.027

1.024 0 .952 0.951

0.885 0.884

1.653

1.755 1.591 1.669

0.851 0 .854 0 .937 0 .938 1.022 1.024

In table III we list the rigorous qq2 values for these systems as a function of composition and temperature obtained according to (10). ~021 values are also given in all cases and are based on the modified equation obtained from (10) by interchanging the sub- scripts 1 and 2. The calculated ~mix values using these rigorous ~ij values are also recorded in table IV column 7. The agreement with

T A B L E IV

Experimental and calculated ~mix values obtained according to different p~j

~mix X 10 v

System

H 2 - N H a

N~-NHa

C H 4 - N H a

Temp. o K

293.16

523.16

293.16

523. I6

287.66

xl Calc. (9), (14), (16)

E x p t l .

0.1 1004

0.9 1004 0.1 1505 0.9 1825 0.1 1081 0.9 1696 0.1 1927 0.9 2584 0.1 1099 0.9 1008

945 I

989 1399 1788 1057 1878 1925 2665

1065 984

Calc. (9) , (15b) , (16b)

902 978

1399 1788 1054 1673 I907 2647 1062

983

Calc. (% (10)

999 1003 1519 1826 1068 1691

1929 2670 1072 991

VISCOSITY OF GASMIXTURES 209

the experimental values, as expected, is much improved and the deviations with the experimental values are not very disappointing. The average absolute deviation obtained is 1.1%, while the maxi- mum and minimum deviations are 3.3 and 0.1 percent respectively.

From a critical s tudy of the tables I, II and II I certain im- portant features regarding ~0i~ are obvious. The rigorous 9~~ are seen to be temperature dependent though for most of the systems it is not likely to be pronounced. The composition dependence is negligible in all cases except for H2-NHs system, which is also somewhat unique as regards the composition dependence of ~mi» The singly approximated 9iJ values are dependent on temperature though the nature of approximation make it composition inde- pendent. Similar remarks also hold for doubly approximated 91j values.

As already stated the three sets of computed ~mix values ob- tained using the rigorous, singly and doubly approximated ~~~ values are recorded in table IV at a few representative tempera- tures and compositions along with the experimental values. A few conclusions are straightforward. The rigorous values are in excel- lent agreement with the experimental values in almost all cases. The discrepancies are pronounced in both the other sets of com- puted values and the difference is larger than the uncertainty associated with the experimental data. The interesting point to note is that leaving the H2-NHs system at the lowest temperature both the singly and doubly approximated ~ij values lead to very much the same values. This conclusion is of great practical utility for it suggests that fairly reliable values of ~mix are obtained using the doubly approximated ~~j values. No doubt rigorous ~i3" lead to better ~mix values but the associated handicap of tedious lengthy calculations and the knowledge of the interatomic potential is a seri6us limitation. Eren the singly ap- proximated ~vij values suffer appreciably for the same reasons. Thus, if we could employ these doubly approximated ~0ij values in such a way that the accuracy of calculated ~mix is increased we will have a great practical success of an important and useful nature. We explore this possibility in the ratio ~0~j/%'i obtained using these doubly approximated ~ij expressions.

Two relations for the ratio of ~12 and ~~1 have been derived in the previous section viz., (17) and (18). At first sight these look

210 S. M A T H U R A N D S. C. S A X E N A

quite different. The calculations given in table V, however, tend to clarify the ineIfectiveness of the Iactor (Q + x~) × (Q +/%x~)- i and thus the marked and pleasant equivalence of the relations of (17) and (18). To a reasonable degree the approximations that x~ and Ô2x~ should be small as compared to Q hold for these three systems. Consequently, the factor distinguishing the two expressions

T A B L E V

C a l c u l a t e d v a l u e s o f c e r t a i n f a c t o r s o c c u r i n g in (18)

S y s t e m T e m p . xl 2 oi~ 0 32xP (Q + x12)(Q +/hx22) -1

H2-N[H~ 293.16

N2-NH8

CH4-NI-ta

523,16

293.16

523.16

287.66

0.01

0.25

0.81

0.Ol

0.25

0.81

0.01

0.25

0.81

0.01

0.25

0.8I

0,01 0.25

0,81

- -2 .895

- -6 .282

-- 10.923

- - 4.446

-- 8.643

- -14 .222 --272.571

- -240.346

- -235.153

-- 793.844

- -789.450

- - 784.561

- -370.080

- -392.674

- - 415.922

0.148

0.046

0.002

0.185

0.057

0.002

0.974

0.301

0.012

O. 822

0.254

0.010

0.700

0.216

0.009

1.050

0.967

0.926

1.041

0.977

0.943

1.003

1.000

0.997

1.001

1.000

0.999 1.002

1 . 0 0 0

0,998

only inappreciably departs from unity. Its average absolute devi- ation for all the systems given in table V is 1.9% only. Thus, the simplified version of (18) as given by (17) may be regarded as fairly accurate and reliable. One further simplification is possible in (17) without impairing the aecuracy. This is because of the circumstance that the quantity A~2 which is the ratio of the two collision integrals is very insensitive to the nature of molecular interactions and the temperature. This factor has been tabulated by M o n c h i e k and Mason a) and we suggest a value of 1.10 for A~2 in conformity with its characteristic value even for nonpolar gases. The form of (17) therefore becomes as

~ 1 ~ _ filM2 (5M1 + 3.30M2) (20) B21 ~2M1 (5M2 + 3.30M1)

Thus, if pure viscosities are known the ratio ~v12/~21 can be accu- ra~ely calculated from the above relation. We further suggest to

V I S C O S I T Y O F G A S M I X T U R E S 211

determine the abso]ute 9# values occurring in (9) on the basis of the above relation between g0ij and ~3"i. and one ~mix value.

Before we discuss the detailed calculations of ~mix aceording to this procedure let us examine the temperature and composition dependence of the ratio (~12/g021). One expects this factor to be only weakly dependent on temperature because of the feeble de- pendences of (pD12/~T) and (~1/~2) on temperature. These are relatively better approximations for nonpolar gases but are not too unreasonable even for polar gases. To actually demonstrate the limitations of these approximations and their effectiveness we record in table VI the values of ~12/~21 for these three systems for

TABLE Vl

Computed values of ~12/~0fll and its likely composition dependence

System

H 2 - N H 3

N 2 - N H 8

C H 4 - N H a

Temp. o K

293.16

523.16

293.16

523. I6

287.66

I X1 X 0.1

0.5 0.9 0.1

0.5 0.9 0.1

0.5 0.9 Õ. 1

I 0.5 0.9 0.1

0.5 0.9

(14), (16) 0.183

0.228

1.202

1.015

0.865

I (15b),(16b) 0.183

0.228

1.203

1.014

0.864

] (18) I 0 .192

0 .177

0 .169 0 .238 0.223

0 .215 1.207 1.203

1.199 1.016

1.015 1.014

0.866 0 .864 0. 863

all the three cases i.e. when one uses singly approximated, doubly approximated and rigorous 9,j values. For the first two approxi- mate cases this is independent of composition but does depend on temperature, while in the rigorous case it is dependent both on temperature and composition. In the latter case, if we take the average of the values at the two extreme values of xl, we get very approximately the same value whieh is obtained at the middle composition and further this value is in excellent agreement with the first two sets of approximate values which can also be regarded as identical to a very great degree of accuracy. The sum total result is that in the rigorous method if we try to ascfibe a sort of weighted

212 S. M A T H U R AND S. C. SAXENA

T AB L E VII

Comparison of the calculated and experimental ~/mix values

[ Gas Gas pair

NH~-H2

NHa-N~

T~o~p 293.16

373.16

473,16

523.16

293.16

373.16

473.16

523.16

Xpolar [ I Exptl I

0,9005 1004 0.7087 1047 0.5177 1080 0.2975 1087 0.2239 1072 0. I082 1011

0.9005 1299 0,7087 1333 0.5177 I354 0.2975 1329 0.2239 1299 0.1082 1204

0.9005 1660 0.7087 "1680 0.5177 1676 0.2975 1610 0.2239 1560 0.1082 1432

0.9005 1825 0.7087 1837 0.5177 1823 0.2975 1737 0.2239 1678

0,8883 1092 0.7147 1254 0.8638 I383 0.2920 1585 0.111I 1690

0.8883 1398 0.7147 1569 0.5638 1710 0.2920 1920 0.1111 2031

0.8883 1768 0.7147 1946 0.5638 2085 0.2920 2296

0.1111 2408

0.8883 1939 0.7147 2112

0.5638 2250 0.2920 2460 0. i l l l 2572

10 v X ~mix (gin/cm-see)

pair

NHa-02

N H 3-Air

NHa-CHa

Temp. 10 7 X ~}mix

(gm/cm-sec)

Cale.

HCI-H2

°K Exptl . I

lOO5(+o.1) lO47(o.o) lO8O(O.O) lO87(o.o) lO72(o.o) lO14(+o.3)

13o2(+o.2) 1341(q 0.6) 1367(+1.o) 135o(+ 1.6) 132o(+1.6) 1227(+1.9)

1668(+o.5) 17o3(+1.4) 1718(+2.5) 1670(+3.7) 1621(+3.9) 1483(+3.6)

1836(+0.6) 1869(+ 1.7) 1879(+3.1) 1818(+4.7) 176I(+4.9)

1093(+0.1) 1254(0.0) 1383(0.0) 1586(+0.1) 1699(+0.5)

1406(+0.6) 1591(+1.4) 1739(+1.7) 1963(+2.2) 2084(+2.6)

1787(+1.1) 1989(+2.2) 2149(+3.1) 2383(+3.8) 2503(+3.9)

1960(+ 1.1) 2168(+2.7) 2332(+3.6) 2568( + 4.4) 2686(+4.4)

Xpplar

293.16 0.8758 0.7079 0.4786 0.2986 0.1351

373.16 0.8755 0.7079 0.4786 0.2986 0.1351

473.16 0.8755 0.7079 0.4786 0.2986 0.1351

!288.66 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0. I00

287.66 0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

294.16 0.8220 0.7179 0.5042

I. 0.2031

327.16 0.8220 0.7179 0.5042

i 0.2031

372.16 0.8220 0.7179 0.5042 0.2031

1143 1350 1604 1783 1924

1459 1689 1972 2170 2326

1840 2085 2390 2604 2773

1100 1203 1306 1403 1492 1576 1618 1713 1764

1008 1069 I061 1077 1091 1099 1105 1105 1099

1461 1469 1471 1342

1626 1632 1625 1472

1848 1855 1831 1629

Caic.

1143(0.0) 1350(0.0) 1604(0.0) 1784(+0.1) 1927( + 0.2)

1472(+0.9) 1718(+ 1.7) 2019(+2.4) 2228(+2.7) 2394( + 2.9)

1864( + 1.3) 2141(+2.7) 2473( + 3.5) 2700(+3.7) 2878(+3.7)

1093(--0.6) 1205(+0.2) 1308(+0.2) 1404(+0.1) 1492(0.0) I571(--0.3) 1642(+ 1.5) 1704(--0.5) 1755(--0.5)

1006(--0.2) 1037( -- 0.2) 1059(--0.2) 1077(0.0) 1 0 9 1 ( 0 . 0 )

I099(0.0) 1102( - 0.3) 1098(--0.6) 1089(--0.9)

1457(--0.3) 1467(--0.1) 147l(0.0) 1345( + 0.2)

1621(--0.3) 1630(--0.1) 1628 ( + 0.2) 1474(+0.1)

1849(+0.0) 1856(+0.o) 1845(+0.8) 1653(+ 1.5)

VISCOSITY OF GAS MIXTURES 213

TABLE VII (continued)

Comparison of the calculated and experimental ~/mi~ values

Gas pair

HC1-Air

1HC1-CO2

S O 2 - H 2

Temp. o K

427.16

473.16

523.16

298.66

291.16

290.16

318.16

343.16

Xpolar

0.8417 0.6989 0.5092 0.2409

0.8417 0.6989 0.5092 0.2409

0.7947 0.6312 0.5178 0.2991

0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

0.900 0.800 0.700 0.600 0.500 0.400 0.300 0.200 0.100

107 X ~]mix

(gin/cm-sec)

] ExptI. i Calc.

2099 2115(+0,8) 2104 2121(+0,8)

i 2053 2101(+2.3) 1866 1921(+2.9)

2311 2315(+0.2) 2304 2319(+0.7) 2261 2292(+1.4) 2024 2087(+3.1)

2527 2541(+0.6) 2507 2533(+1.0) 2454 2508(+2.2) 2281 2355(+3.2)

I489 1502(+0.9) 1545 1548(+0.2) 1592 1596(+0.3) 1638 1637(--0.1) 1678 1678(0.0) 1715 1718(+0.2) 1749 1755(+0.3) 1778 1788(+0.6) 1800 1819(+1,0)

1459 1444(-- 1.0) 1472 1466(--0.4) 1483 1483(0.0) 1492 1494(+0.1) 1499 1499(0.0) 1502 1499(--0.2) 1503 !1494(--0.6) 15oo 1485(- 1.o) 1495 1470(--1.7)

1293 1272(--1.6) 1350 1350(0.0) 1370 1376(+0.4) 1344 1363(+1.4) 1304 1330(+2.0)

1425 1416(--0.6) 1475 1489(+0.9) 1494 1508(+0,9) 1453 1490(+2.5) 1410 1449(+2.8)

! 1535 1527(--0.5) I 1557 1558(+o.1) I 1574 1579(+0.3)

Gas pair

SO2-CO2

Temp. °K

365.16

397.16

432.16

472.16

288.96

290.36

Xpolar

l 0 T X 'r/mix

(gin/cm-sec)

1587 I 1609(+1.4) 1596 1619(+1.4) 1551 1598(+3.0) 1500 i I550(+3.3) 1505 1547(+2.8)

1633 1627(--0.4) 1648 1654(+0.4) 1675 1675(0.0) 1682 1705(+1.4) 1640 1687(+2.9) 1573 1634(+3.9) 1577 I632(+3.5)

1797 1805(+0.4) 1814 1851(+2.0) 1801 1856(+3.1) 1685 1755(+4.2)

1942 1955(+0.7) 1960 2003(+2.2) 1942 2004(+3.2) 1803 1894(+5.0) 1748 1867(+6.8)

2118 2125(+0.3) 2121 2166(+2.1) 2098 2165(q 3.2) 1953 2007(+2.8)

1288 1262(--2,0)

1316 1296(-- 1.5) 1338 1329{--0.7) 1363 1358(--0.4) 1384 1384(0.0) 1407 ! 1406(--0.1) 1429 1424(--0.3) 1447 i 1436(--0.8) 1464 1442(-- 1.5)

1331 1330(--0.1) 1403 1403(0.0) 1469 1474(+0.3) 1535 1541(+0.4) 1603 1603(0.0) 1655 1660(+0.3) 1709 1710(+0.1) 1755 1752(--0.2) 1795 i 1785(--0.6)

0.8215 0.5075 0.2963 0.2286 0.1676

0.8028 0.5075 0.2963 0.2286 0.1676

0.8028 0.6999 0.6175

H2S-Air

0.4823 0.2963 0.2306 0.1676 0.1657

0.8028 0.6999 0.6175 0.4823 0.2306 0.1676 0.1657

0.6760 0.4698 0.3265 0.1636

0.6760 0.4698 0.3265 0.1676 0.1512

0.6760 O.4905 0.3265 0.1512

0.900 0.800

0.700 0.600 0.500 0.400 0.300 0.200 0. I00

0.900

0.800

0.700 0.600 0.500 0.400 0.300 0.200 0.100

Exptl. ] Calc.

214 S. MATHUR AND S. C. SAXENA

value for the ratio in reference to composition it will invariably be the same as that obtained from the two approximate procedures. This further emphasises that the ratio as given by (20) is to be associated with a high degree of accuracy. Of course the tempera- ture dependence of this ratio is not negligible and any procedure based on this assumption will have to counterbalance this inherent deficiency if reliable and accurate ~mix values are to be evolved.

Computed values of ~]mix for a large number of systems, where experimental data are available, as a function of composition ac- cording to the procedure which exploits (9) and (20) in conjunction with one experimental ~mix value are recorded in table VII. These caleulations refer to seventy-nine mixtures at the lowest tempera- ture for all the systems and the agreement between theory and experiment is excellent. The average absolute deviation is only 0.4%. This compares very favourably with the rigorous method where the corresponding deviation is 1.7°/o . The various ~v, 3. values along with the pure viscosity data are recorded in table VIII.

T A B L E V I I I

Da ta used for calculation of b inary viscosity

Temp. Gas pair °K ~1 x 107 ~]g x 107 912 9)21

NH~-He NH3-N2 N H s - 0 2 NH~-Air NH~-CH4 HC1-H2 HC1-Air HC1-COe SO2-H2 SO2-CO2 H~S-Air

293.16 293.16 293.16 288.66 287.66 294.16 298.66 291,16 290.16 288.96 290.36

982 ~) ~) k) 1758 a) 2031 d) 1797 a) 973a) g) k)

1433 e) h) 1455 c) ~1) 1451 a) 1230f) i) j) 1225f) i) j) 1252 b) e)

881 a) 982 a) 982 a) 975 a)

1073 a) 884 a)

1845 a) 1415 e) 876 a)

144! a) 1807 a)

g) k)

0.305 0.973 0.972 0.916 0.811 0.237 0.764 0.857 0.142 0.721 0.676

1.670 0.809 0.779 0.760 0.938 1.829

• 1.162 0.971 2.185 1.144 1,110

The subscr ipt 1 refers to the heavier component .

Though, as shown earlier the ratio ~12/~v21 is still dependent on temperature if we use this value at a higher temperature in con- junction with a ~mix value at that temperature, it is likely that the absolute ~v~j values so obtained may successfully explain the viscosity results. This approach demands for its success the cancel- lation of the consequence of assuming the ~0ij. ratio to be tempera- ture independent in the use of ~]mix at that temperature while

VISCOSITY OF GAS MIXTURES 2t5

fixing the absolute values of ~%.. This optimism is likely to be successful for it has been known that there is an inherent large flexibility in the ~%. values in (9) to repersent ~mix. As a check we calculate the viscosity for all these systems at higher temperatures as a Iunction of composition where experimental data are available. In table VII, we report for such ninety-five mixtures and the absolute deviation on the average is only 2.0 percent. This, how- ever, is not bad when one recalls the simplicity of the calculation procedure.

An interesting point which still remains to be investigated is the degree of the success that can be achieved if the whole procedure is directly performed at the higher temperature i.e. calculation of the ratio ~12/~v2» Indirectly this will check the assumption of the invariance of the ratio ~v12/~21 with temperature. We choose the systems H2-NH3 and N2-NHa for this purpose. Our results are, for the H2-NH3 system at 523.16°K, ~012 = 0.371, ~021 = 1.625, the maximum and average absolute deviations became Irom 4.9 and 3 to 0.2 and 0.08 percent respectively. In the same way for N2- NHa system at 523.16°K, ~12 = 0.977, ~21 : 0.963, the maximum and average absolute deviations became from 4.4 and 3.2 to 3.0 and 1.0 percent respectively. In general, by adopting this pro- cedure an improvement is expected though its degree seems to depend on the particular system in a somewhat unique fashion.

Received 15th January, 1956. REFERENCES

1) Mason, E. A., J. T. V a n d e r s l i c e and J. M. Yos, Phys. Fluids "~ (1959) 688. 2) C h a p m a n , S. and T. G. Cowl ing , The Mathematieal Theory of Non-Uniform

Gases, Second Edition, Cambridge University Press, New York, 1952. 3) Monch ick , L. and E. A. Mason, J. Chem. Phys. 85 (1961) 1676. 4) S t o e k m a y e r , W. H., J. Chem. Phys, 9 (1941) 863. 5) G a m b h i r , R. S. and S. C. S a x e n a , Trans. Faraday Soc. 60 (1964) 38. 6) S u t h e r l a n d , W,, Phil. Mag. 40 (1895) 421. a) B r a u n e , H. and R. LiIlke, Z. Phys. Chem. A 1 4 8 (1930) 195. b) G r a h a m , Phil. Trans., (1846) 573. c) Har l e , H., Proc. Roy. Soc. A 100 (1922) 429. d) H i r s e h f e l d e r , J. O., R. B. B i rd and E. L. Spo tz , J. Chem. Phys. 16 (1948) 968. e) R a n k i n e , A. O. and C. J. S m i t h , Phih Mag. 42 (1921) 615. I) S t e w a r t , W. W. and O. Maass , Canadian J. Res. A G (1932) 453. g) T r a u t z , M. and R. H e b e r l i n g , Ann. Physik 10 (1931) 155. h) T r a u t z , M. and A. N a r a t h , Anrl. Physik 79 (1926) 637. i) T r a u t z , M. and W. Weizel , Ann. Physik 78 (1925) 305. j) T r a u t z , M. and R. Zink, Arm. Phys ik7 (1930) 427. k) V a n e l e a v e , A. B. and O. ~ a a s s , Calladian J. Res. 18 (1935) 140.