Indian Journal of Pure & Applied Physics Vol. 41 , February 2003, pp. 121 - 127

Viscosity and thermal conductivity of gas mixtures Kuldip Singh & N K Sood

Department of Physics , Guru anak Dev Uni versity, Am ritsar 143 005

Received 29 April 2002: accepted 26 August 2002

A method is proposed for estimating viscosity and thermal conductivity of gas mixtures fro m the pure compo nent Jata. Dev iat ions of the resu lts predicted by the approx imate relationships developed are or the sa me order, as the uncertainty in the experimental measurements.

1 Introduction

There has been continued inte rest in the study of transport properties of gas mixtu res 1

•15

• Because of the limited range (compositi on as we ll as temperature), of experimenta l data on gas mjxtures, it has been of in terest to estimate the transport coeffi c ients from the approx imate expres­s i o n s~· ll.lfi·i~, based on f irst-order kinetic theory. On ly the uni versal functiona l approach 1 7 - 1 ~ (the extended law o f corresponding states), has been e ff icient in reproducing data on viscos ity and the rmal conductivity within experimenta l uncerta inty bu t, it makes use of a number of paramete rs. General express ions u.r..lo.l w have a lso been used to work out

higher-order contributions to the transport coeffi c ien ts. For a given inte rmolecular potent ia l, the computation of these contribut ions is always a cu mbersome job l. 16

'11

• In the present work, attempt has been made to deve lop approx imate re lat ionships for viscosity and thermal conducti vity , which invo lve onl )' the molecular weights and transport coeffi cients of the pure components for the given compos it ion of a gas mixture.

2 Present Approximate Expressions

(a) Viscosity - Start ing with the first-order kineti c theory expressions given by H irschfeldar et aP ., the viscosity of a bir~ary mixture can be written as:

1Jmix == _17.:..:,1 __ + 112

X, XI I+AI " - ~ I+A,I -- xl - x2

... ( I )

where 11; and X, are the viscos ity und concentration of the i 111 component , respecti vley. A11 is the

M 2 coeffi c ient depending upon the mass-rati o and

Ml

inte r-mo lecular potentia l parameters.

0 ~

c 0.5 , 0 0 I :,:; ' X X

X

e

1(a)

Q Q ~ g

.~ I o ~ ~ o 1--hor---,-----r---"---. '0 210

~ c ~ 500 75~ 1000 Ill ' u -0.5 j '- I

~ I

-1 ·

0 j . !l. c

!l

0

Temperature(K)

1(b)

g 0.5 J 0 0 0.2677 l'il I 6 t;

') ' ~ 0 0 0 0 0.5316

8 X 0

Ill I 0 & ~ 0 ~ ___ .:,_o -~~~~+-.. ___ 6 0.8687 -

~ 2~0 500 750 1000 c -0 5 j Q) • I

u ' ~ I !l. -1 ~

0 0

Temperalure(K)

XH,

0 0.22

o0.62

6 0.43

X 0.75

0 0.90

Fig. I- Percentage deviation (PO) versus temperature for viscosity or (a) He-Ar and (h) He-Xe mixlllrcs

( . F . ·1 I . M , t) or srmr ar mass va ucs 1.e. --- =I Ml

122 INDIAN J PURE & APPL PHYS, VOL 41 , FEBRUARY 2003

Table l - Percentage deviation (PD) of experimental viscosity llcxp from linear relationship Tlmi.< =[ XI 771+ x 2 Th] for various mixtures at different temperatures

Mix- Mass Tem- XI ll cxp Tlmix PD ture ratio pera- (JlP) (pP)

ture (K)

298

Kr-Xc 1.57

773

298

Or Ar 1.25

667

298

667

298

He-Ar 9.98

773

298

1-l e-Xe 32.80

778

0.2727 236.86 237. 10 -0.1 0

0.4627 241.26 24 1.49 -0.09

0.2727 54! . 13 542. 16 -0. I 9

0.4627 545.23 544.77 0.08

0. 1762 223 .00 222.76 0.11

0.3948 218.90 218.38 0.24

0.1762

03948

0.6216

0 .8093

0.62 16

0.8093

0.2 18<+ 0.62 18 0.8953

0.2236 0.6270 0.90 I 0

0.2677 0.53 16 0.8687

0.2677 0.53 16 0.8687

4 13.70

404.10

188.80

183.50

339.30

328.80

230.74 231.22 218.85

467.01 459. 19 420.99

241.78 254.43 255 .03

41 3.39

403 .83

183.7 1

183 .40

340.4i

329.25

220.25 2 19.09 201.53

448.43 416.40 394.65

222.20 213.70 202.86

552. 13 500.78 558.48 460.50 51 3.54 409.05

0.07

0.07

0.05

0.06

-0.33

-0. 14

4.76 12.20 8.60

4.14 10.26 6.67

8.81 19.06 25 .72

10.25 2 1.28 25 .54

Eq . ( I) reduces to the I in ear relationship :

.. . (2)

This linear relationship reproduces viscosity of

b; nacy m; xtmes such as Kc-X e l ~: = I .57 ). 0 ,-

Ac( ~: = 1.25) and N,-0{ ~: = 1.14) w;th;n the

uncertainties of experimental measurement (Table I ).

0.5 l

~ I XN, 00.2153

_Q 0.25 I ro I oo.434o ·:;; i l>0.6195

2(a)

X

1:1

~ 02~-r--~->:0.7747 750 ~---0000 10 ) tj <> X

t . 0 ~ -0.25 l a ID I ~

()_ : -0.5 ;

Temperature(K)

2(b) 0.8 '

j

~ ~-: ~ A 0

o o.s / ~ -~ 0.4 I Ill o

0 ¢07785

i ~· ~ ~ X ~~ ;:~ ~ 0.1 j ~ 0 o X0.5866

Q_ 0 ------8--------r· ·-- ------~-------- --0. 12~0 ° 500 750

Temperature(K)

1000

Fig. 2- Percentage deviation (PD) versus temperature for viscosi ty of (a) Nc-N2 and NT Kr mixtures

( ii) For different mass values, experimental results deviate much from the linear relationship g iven in Eq . (2) e.g. mixtures containing He (Table I). The coeffic ient !(' A,i has to be modified (Appendix A) and is given by:

... (3)

SINGH & SOOD: VISCOSITY AND THERMAL CONDUCTIVITY 123

Tab le 2- Percentage deviation PO or the predicted vi,cosity Table 2 .. . Cmlld from the experi mental value for the various binary mixtures

Mixture Tempe- XI 'l oxp ll mix. PO Mixture Tempe- XI Tt exp Tt n:ix PO ratu re (f.lP) (f.lP)

rature (f.lP) (f.lP) (K ) (K ) 0.2 165 259.95 260.05 -0.0-l

0.4376 265 .99 265.80 0.07 0.1997 266.93 266.8 1 0.04 298 0.5953 268 .32 267.77 0.21 0.3945 280.88 280.54 0.1 2 0.7041 266.77 266.03 0.28

298 0.5329 292.71 290.75 0.67 0.8721 249.69 248.8 1 0.35 0.6582 300.79 299.97 0.27 He- Kr 0.8300 3 I 1.88 311.37 0.16 0.23 01 606.64 60H.35 -0.28

e- Kr 0.4512 604.91 605.67 -0.1 3 0. 1997 6 17.0H 6 17.45 -0.06 873 0.6089 596.12 595 .80 0.05 0.3945 629.26 629.00 0.04 0.7177 582 .01 5HO. I7 0.32

873 0.5329 639.56 637 .03 0.40 0.8957 5 17 .46 5 15.69 0.3-1 0.6582 646.29 643.71 0.40 0.8300 652.31 650.64 0.26

0.2677 24 1.78 24 1.39 0.16 298 0.53 16 254.43 253.72 0.28

298 0.4800 273.99 273.66 0. 12 0.8687 255.03 254.2 1 0.32 0.7882 306.82 306.74 0.02 l-le-Xe

Ne-Xe 0.2677 552.13 550.26 0.34 773 0.4800 580.47 580.49 0.00 778 0.53 16 558.4H 555.56 0.53

0.7882 604.63 604.58 0.01 0.8687 513.54 5 11.34 0.4.i

298 0.5774 236.24 237.01 -0.32 0.2 153 198.89 199. 15 -0. 13 0.8093 233.47 234.10 -0.27 298 0.4340 224.25 224.51 -0.12

Ar-Xe 0.6195 249 .57 2-19.91 -0.1-1 773 0.5774 515 .28 51 2.26 0.59 0.7747 274.36 274.62 -0.1 0

0.8093 493.55 490.79 0.56 Ne-N2

973 0.4340 503.84 503. 15 0.14

0.1988 307.43 306.92 0.17 0.6 195 556.78 556. 14 0. 12 298 0.3699 295.49 294.2R 0.41 0.7747 61 1.1 I 608.77 0. 38

0.5934 273.1 2 271.3 1 0.67 0.7663 245.80 246.55 -0.30 0.2363 243.80 243.6 1 0.08

I-l e-Ne . 298 0.4041 234.20 233.% 0. 10

0. 1988 683.31 678.42 0.72 0.5866 22 1.1 4 220.86 0. J:l 973 0.3690 658.66 653.86 0.73 0.7786 203.67 203.5 1 0.08

0.5934 609.06 607.77 0.21 N2-Kr 0.7663 559.80 557.09 0.49 0.2363 517.76 515.96 0.35

767 0.4041 489.36 486.3 1 0.63 0.2 184 230.74 230.74 0.00 0.5866 453.25 450.06 0.71

0.4300 234. 16 234.00 0.07 0.7786 409.26 406.79 0.61 298 0.62 18 234.60 234. I-I 0.20

0.7506 231.22 230.56 0.29 298 0. 1990 2..J9.10 249.35 -0.10 0.8953 2 18.85 218.49 0.16 0.407 3 242.50 242.5') -0.0-1

He-Ar OT Kr 0.2312 507 .7X 507.63 0.03 667 0.1990 482.60 -1811 () 0.31 0.4428 507.98 506.60 0.27 0.4073 -16'1..90 460.60 0.50

873 0.6346 500.36 49!U.\9 0.29 0.7634 487.99 485 .99 0.41 0.9086 457.81 455.82 0.4-1 I { r ... Con:d 4M M . -1

where L= I I and K is a function of (M;+M)

1

temperature and mass-ratio. The exp li cit form of K is given in Appendix A .

124 INDIAN J PURE & APPL PHYS, VOL 4 1, FEBRUARY 2003

Table 3 - Percentage deviation PD of the ca lcul ated mixture viscosity from the experi mental value at diffe rent lemperalu res

Mixture Tem- Com Tl cxp ll mix PD pera- posi- (JlP) (JlP) ture lion (K)

258.03 257.43 0.23 298 II 258.97 259. 16 -0. 19

Ill 265 .21 264.86 0. 13 IV 278.63 278.13 0. 18

Ne-Ar- Kr 609.98 60716 0.46

973 II 635.72 63 1.08 0.73 Ill 632.47 627.54 0 .78 IV 654.9 1 652.02 0.44

373 v 334.22 332.59 0 .49 He- 1e-Kr 483 v 399.2 1 399.49 -0.07

673 v 498 .95 499.39 -0.09

373 VI 300.45 299 .58 0 .29 He-Ar-Kr 483 VI 363 .33 36 1.80 0.42

673 VI 459 .0 1 458 .99 0 .0 1

Composit ion x, x~ x~ I 0.2796 0.5036 0 .2 168 II 0 .1 703 0.2886 0.54 11 III 0. 3075 0. 189 1 0.3436 IV 0.4407 0. 189 1 0 .3702 v 0.5435 0.3282 0 .1 283 VI 0 .5022 0. 3050 0.1928

Extending the re lati onship given in Eq . (3),to N­

component gas mixture, 1imix is given by:

l l- 1

N N X 11m;, = I 11; I + I Au _ _}_

•= I J=l X; J"''

.. . (4)

(b) Thermal conductivity- Following a similar procedure, thermal conducti vity A,nix o f a gas mi xture is:

l ]- !

N N , X j A111;x = I A; I + I Au -

1= l f = l X . . · ' l '"'.!

.. . (5)

where ?c, and X; are thermal conduct ivity and concentrat ion of the i'" component, respecti vely. The

explicit form of A ~ i • as a function of thermal

conducti vity A; and mass-rati o M 2

Ml Appendix B.

. . . ts g tven m

0 a. c 0

·~ ., ~ 0 ~

"' ro c ~ 0

I> a.

1.1 1

0 9 XN, XAI 0 . ' a.. I o .2100 .sros Q 07 1 0 .1703 .2886

-~ 0 5 I 6 3075 .3489 ~ . 1 0 X .4407 .1891

8, 03 1· 0

~ 0.1 I, ~ ~ 0 . 0

0

0

X 6

3(a)

0

~ -0. 12F-o--2~--~-----~---

' ·0.3 .

i -05 ;

1 -,

I

0.5 '

:

i

0

0

0 .l--.

I

0

0

0

X 0

Temperalure(K)

0

0

3(b)

x,k x,, "' .5435 .3282

XH, XAI rr. .5022 .3050

-<>----r--- - · -·· .. ... 0

1000

2 0 500 750 1000

-0.5 Temperature(K)

Fig. 3 -- Percentage dev iati on (PD) versus temperature for viscosity of (a) Ne-Ar- Kr and (b) He-Ne-K r & He-Ar- Kr mixtu res

T he express ion g iven in Eq . (5) is not valid fo r mi xtures containing diatomic and po ly-atomic gases, as it has been derived from the one that does not take into account ine lastic co ll is ion terms.

3 Results and Discussion

Us ing the approx imate express ions deve loped in the previous section, an atte mpt has been made to reproduce 0 .1-0.3% accurate viscos ity data of Kes tin et a l. 20

.2x fo r different compos itions of vari ous

binary and ternary inert gas mi xtures over a wide temperature range 300- 1000 K; binary mi x tures ~0-~ 1

of N2 with Ne, Ar, Kr, 0 2 and 0.1 % accura te thermal conducti vity data of Van Dae l & Cauwenberg~9 and 0 .2% accurate data of Clifford et a/.5 and Asseal et a f.'A. The results of such computati on for viscosity have been compiled in Tabl e 2 for Ne-Kr, Ne-Xc,

SINGH & SOOD: VISCOSITY AND THERMAL CONDUCTIVITY 125

Ne-N2, N2-Kr, 0 2-Ar, He-Ar and He-Xe mixtures for the various compositions at two temperatures, in Table 3 for ternary mi xtures Ne-Ar-Kr, He-Ne-Kr and He-Ar-Kr; and in Table 4 for thermal conducti vity of binary Ne-Ar, Ar-Kr, He-Ar and ternary Ne-Ar-Kr mixture . The last column of all these Tables 2-4 gives the percentage deviation

(PO) [=(Tcxr-Tmix) x iOO/T"'ix] of the experimental values from the calculated ones. A detailed picture of the variation of PO with temperature has been presented in Figs 1-2. Finally, Table 5 presents average deviation, average absolute deviation and standard deviation fo r the above-mentioned mixtures.

Table 4 - Percentage deviation of the calcul ated values of Amix fro m the experimental ones Acxr for the various composi ti ons of the binary mixtures and one ternary mixture

Mixture Tern-pera-lure (K)

Ne-Ar (Ref.S ) 300.65

Ne-Ar 296 (Ref.29)

He-Ar (Ref.S) 300.65

Ar-Kr 308 (Ref.4)

Ne-Ar-Kr (RefS) 300.65

XI

0.2623 0.4724 0.6325

0. 11 26 0.2693 0.47 14 0.5948 0.6772 0.7864 0.8864 0. 1895

0.3597 0.6547 0.7762

0.4088 0.705l)

X1=0.4887 X2=0.2565 X3=0.2548

A cxp Am ix PO

(mWm.1K-1)

23.36 23. 19 -0.73 28.78 28.63 -0.52 33.81 23 .64 -0.50

19.91 19.86 -0.25 23.34 23.24 -0.43 28.50 28.40 -0.35 28.50 32. 11 -0.28 32.20 34.89 -0.40 35.03 38 .99 -0.36 39. 13 43.26 -0.46 43.46 28 .62 -0.63

41.04 41.33 0.70 74.95 74.54 -0.55 95.26 95.3 1 0.05

12.43 12.51 0.64 15.00 15.00 0.00

24.68 24.65 -0.12

It has been observed from Tables 2-5 and Figs 1-2 that :

( I ) The present approximate re lationships reproduce viscosity and thermal conducti vity data within the same order as the uncertai nty in experimental measurement. Compared with the Brokaw approximation'\ which poorly reproduces

data on viscosi ty, especially, for mixtures with large mass-ratios, e .g., He-Xe maximum dev iati on is 3.9%. Similarly, Mason-Saxena approx imation'~ is also inadequate in reproduci ng the thermal conductivity data, e.g., for He-Ar mi xture, the maximum deviation of the pred icted results from the experimental results be ing 4% .

Table 5 - The average deviation , average abso lu te deviation and standard deviation of calcu lated values from the experimental mixture viscosity for binary and ternary mixtures

Mixture

Ne-Ar Ne- Kr Ne-Xe Ar- Kr Ar-Xe He-Ne He-Ar He- Kr He-Xe Ne-N2

N2-Ar N2- Kr Or Kr Ne-Ar-Kr He-Ne- Kr He-Ar- Kr

Average deviati on

-0.03 0.05 0.07 0. 19 0.67 0.35 -0.09 0.03 0.1 1 0.00 0.33 0. 12 0.30 0.24 0.28 0.38

Average Standard abso lute devi ation devi ation

0.05 0. 11 0.08 O. IY 0.38 0.46 0. 19 0.44 0.73 0.60 0.35 0.53 0. 12 0.26 0.09 0.22 0. 19 0.28 0.05 0. 10 0.33 0.3 1 0. 13 0.22 0.31 0.26 0.30 0.30 0.34 0.32 0.38 0 .28

(2) Universal functional approach 17-19 is efficient

enough in reproducing the transport coefficient data but, it involves a large number of parameters, e .g .

for a given mixture these are a;i, Eii• quantum­mechanical deBoer parameter A *ii• the low and high

temperature scaling parameters c; , p~ and v.o· , l 11 I} I)

respecti ve ly. The present approx imate express ions make use of the parameters 7Ji, Mi for viscosity and It;, Mi for thermal conductivity (which arc four in number for a binary and six for a ternary mi xture for a given tran sport coeffici ent). These express ions reproduce the experimenta l data within the same order of uncertainty as the uni versal functiona l approach and make the computation much simpler.

References

Ferziger J H & Kaper H G. Math e111atical theorr '!f transport processes in gases (North-Holl and , Amsterdam), 1972.

2 Hirschfeldar J 0, Cu rtiss C F & Bird R B, Molecular theory of liquids and gases (John-Wiley, New York), 1964.

126 INDIAN J PURE & APPL PHYS, VOL 41 , FEBRUARY 2003

3 Assael M J, Wakeham W A & Kestin J, fnt J Th ermophys, I ( 1980) 7.

4 Assael M J, Dix M, Lucas A & Wakeham W A, 1 Chem Soc Faraday Tranv I, 77 ( 198 1) 439.

5 Clifford A A , Fleeter R & Kestin J, Physica, 98A ( 1979) 467.

6 Pavlov A V . Sov Phvs Tech Pitys, 26 ( 198 1) 80.

7 Amcling W & Lucas K , lnt J Thennophys, 7 ( 1986) I 007.

8 Ameling W & Lucas K , /111 .I Thermophys, 8 ( 1987) I .

9 Matika T , lnt J Thenno11hys, 10 ( 1989) 727.

I 0 Si ngh K, Dham A K & Gupta S C, Physica A , !59 ( 1989) 369 .

I I Singh K. Dham A K & Gupta S C, Indian 1 PHre & Appl Pln•s. 28 ( 1990) 135.

12 Xi ufeng L , Qifeng C & Xi L . Indian 1 Pure & Appl Pln·s, 34 ( I 996) 87 1.

13 Bczerra Jr A G, Reinecke S & Kremer G M. Contin Mech Thernwdvn , 6 ( 1994) 149.

14 Singh K, Dham A K & Gupta S C, 1 Phys 13: At Mo l Opt Ph,·s. 29 ( 1996) 11 43.

15 V csovic V , lnt .I Themwphvs, 22 (200 I ) 80 I.

16 Brokaw R S, .I Chem Pln•s , 42 ( 1965) I 140.

17 Najafi !3 , ~ ason E A & Kestin J, Physico A , 119 ( 1983) 389.

18 Kestin J, Knierim K, M ason E A Najafi B, Ro S T & Waldmann M, J Phrs Chem Re.fData, 13 ( 1984) 229.

19 !3zowski J, Kestin J, Mason E A & Uribe F J, .I Pln•s Chem Re( Data , 19 ( 1990) I.

20 Ke:ain J, Ro S T & Wakeham W A, J Citem Pln•s , 56 ( 1972) 4086.

21 Kalekar AS & Kestin J, .I Chem Phys . 52 ( I 970) 4248.

22 KL:stin J. Wakeham W A & Watanabe K, J Chem Phvs, 53 ( 1970) 3773 .

23 Kcs tin J. Ro S T & Wakeham W A, .I Chem Phys , 56 ( 1972) 5837.

24 Kestin J, Ro S T & Wakeham W A, J Chem Ph\'S, 5H ( 1972) 165.

25 Kestin J, Ro S T & Wakeham W A, 1 Chc111 Phys, 56 ( 1972) -W89.

26 Kestin J, Khalifa H E & Wakeham W A, Physico A , 90 (1978) 215 .

27 l-lcllemans J M , Kestin J & Ro ST. Physica , 71 (1974) I.

28 Kestin J, Khalifa II E & Wakeham W A . .I Chcm Phys , 9 ( I 977) 4254.

29 Van Dacl W & CauwenbL:rg H, Phys1ra , 40 ( I %8) 173.

30 Hellemans J M , Kestin J & Ro S T , Physica , 65 ( 1973) 362 .

3 1 Hellemans J M , Kestin J & Ro S T, J Chem Phys, 57 ( 1972) 4038.

Appendix A

In the expression [Eq . ( I 0) o f Re f. l6], for

mixture vi scos ity 7J111;x, first-order viscosities are

replaced by experimental va lues 77 1 and 7h The interaction viscosity 7] 12 is expressed in terms of 7] 1

and 7]2 by:

... (A.I )

where the parameter K is dependent on (i) mass­

ratio M 2 and (ii ) temperature. (Here, in writing Eq. M ;

(A.I ) use of Eqs (8.2-18) and (8 .2-20) of

Hirschfe lder et aU for viscosity [11 )1 and [77 1 ~ ) 1 has been made.)

( i) To bring out the dependence of K on the

mass-ratio M 2 , the values of 7J P have been Mt -

obta ined for the various inert gas mix tures at 298 K, us ing the first-order express ions of [ '1]111;, ] 1 [Eqs (8.2-22) of Hirsch felder et aP] and accurate data of Kestin et al'0

-24

. The val ues of K1 are then obtained from Eq. (A. l) and are plotted for the va ri ous mass­ratios. From the trend fo llowed, the best fit is given by:

K, ~ 1.0+0.013{ : : -1.4) for M ,>M,,,

~ 2.0~ exp{ -0.0001 { :: r} for M,~M,, .. (A2)

( ii ) Similarly, temperature dependence of K has been studied for different binary mixtures and the form suited has been worked out . Finally, comb ining both temperature as we ll as mass dependence, the expression forK is:

K=K, exp{ -0.0066 aiL }for M 1>M 1k

SINGH & SOOD: VISCOSITY AND THERMAL CONDUCTIVITY

where a=-- -- 1 , M? (T l M 1 T0

7;,=298 K.

Whe n M 2 = I , the linea r re la tion g iven in Eq. (2) M,

for 7J111;, automatically follows from Eqs ( I) & (3).

Appendix B

For the rma l conduct ivity, A ~~ (modified from Ref. 16, in a s imil ar way, to that in Appendix A) is given by :

w ith

K =~(1.01546L -0.0 1 403) L-

127

. . . (B.I )

... (B .3)