Barua, A K

Pal, Arun K

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EFFECT OF CLUSTER FORMATION ON THE VISCOSITY OF DENSE GASES ARUN K, PAL AND A. K. BARUAIndian Association fob the Ctn,TivA»ioN or Science, CAtCOTTA {R eceived  S ep tem ber 16, I960)ABSTRACT, The effect of dimerization on thoVisoosity of donee gases has been oaloula, ted for the rigid sphere molecules by utilising an ^tension of the Enskog theory for dense gases. This treatment as distinguished from tho earlier ones, takes into consideration the effect of dimerization on the oollisional transfer ter*i. Sample calculations made for Argon show that the contribution of dimerization to the viscosity is negative at low pressure and it becomes positive with tho increase of pressure,I N T R O D U C T I O NFor the calculation of the transport properties of dense gases it is neces­sary to consider the collision transfer (Chapman and Cowling, 1952) in addition to the transfer by molecular interaction. Enskog (Chapman and Cowling, 1952) long ago, first formulated a theory for the dense gases by considering the mole­cules to be rigid spheres. The advantage of this assumption is tha t tho probabi­lity of multiple encounters become negligible. I t  has subsequently been found (Hirschfelder, Curtiss and Bird, 1954) that the Enskog theory can represent the transport properties of dense gases only approximately. Several attempts have been made to improve the Enskog.theory by introducing scmi-emprirical correc­tions. Recently, Snider and Curtiss (1958, 1960) and Snider and McCourt (1963) have made a significant advance in the theory of dense gases. They have modified tho Boltzmaim equation which is an approximation to the series development of Bogolubov (1946) and Hollinger and Curtiss (1960). Curtiss, McElroy and Hoffman (1964) have used numerical methods to consider tho effect of three body collisions on the first density term by a generalization of the Enskog rigid sphere expression.Another factor which is of importance in the dense gases is tho formation of clusters or bound molecules. As has already been pointed out by Das Gupta and Barua (1965) a t sufficiently high pressures the gas should be treated as a mixture of monomers and j-mers. Stogryn and Hirschfelder (1941) have considered the formation of dimers in non-polar gases on realistic potential models. The effect of dimerization on the transfer by interaction has been considered by them. This treatment has recently been improved by Kim and Boss (1965) who considered the presence of quasi-dimers. These treatments are valid only for the initial pressure dependence. For higher densities it is necessary to consider the change37323324 Arun K. Pal and A. K, Baruain the colliaional transfer term (hio to cluster formation. At not too high pressures it is sufficient to consider dimerization only. In this paper a start has been made in this direction. We have made a sample calculation of the effect of dimeri­zation on the viscosity of argon on the rigid sphere model. The results will show if the contribution of dimerization is of significant magnitude.M E T H O D  OF C A L C U L A T I O NThe co-efficient of viscosity of a d( r^iso gas Tjp on the Enskog theory may be expressed as (Chapman and Cowling, 1952)Vp =  Vobp [ A -  - I- g- +0.10Ubpx ], ... (1)where t/q is the zero-pressure viscosity, p the density, bp =  |  cr being thecollision diameter. The quantity x  given by,a; -  l+#p6+0.28G9p2ft2_|,...  ^ (2)The Enskog theory for a pure dense gas has been extended to the case of a binary mixture (Chapman and Cowling, 1952) and the viscosity for such a mix­ture may be written as,[rip\mix =  g [b -i  -iW i2  ( l  +  lh p JX ^  \~ ^ b '^ p ^ X ti)^— 2 6 i _ i  ( l  +  s  ■f'T ^ i ^  tP iX n )+  {»h‘WllYlV +  ^ (2'«0- l^-'l^2)*»»l«2A’x2O’l2‘‘+»W2*W2*AW}]. ••• (3)In this formula,fi - 7T ■ ... (4)X l  =  1 +  J2 '^ %0’l®+j[J^2{0'l®+16orjij3_12(ri2Vi) +  .,.Xl2 — I +  J 2 3cTj/frj2)-)-j  ^M2(T2*(8 — 3o’2/eTj2)4-... ... (6)with a corresponding equation for x^- The subscripts 1, 2 stand for the compo­nents, 1, 2 respectively. We have also the following relations viz :-'ll f'’ii+nj^2Xi^\JXia,^ - 1 - 1  —  ^ - 1 - 1  +  ^ 2 1  x j ^ " ~ l ~ l l X l Z >= - 5 k m - A ) I E ,  b'li =  5kTi^+M ^AIM ,)IE,b \ i  =  5kTI2[‘>ii]i,The constant A  depends on the ratio of the collision integrals.(6)(7 )(8)(9)(10)The percentage of dimers at any particular temperature and pressure may be obtained from th(  ^ equilibrium constant for dimerization,K2(T) =  — y ... (11)=  n^Vjn^^ ... (12)Bb{T) and Bm{T) being the constribution of the bound double molecules and metastably bound molecules which have been calculated on the Lennard-Jones (J2:6) model by Stogryn and Hirschfelder (1959). are respectively thenumber densities of monomers and dimer^.In order to obtain cTg it is necessary to ojulculate the viscosity of dimers. On the Lennard-Jones (12 :6) model it has been shown by Stogryn and Hirschfelder(1959),Effect of Cluster Formation on the Viscosity, etc. 325Lia =  1.03; 1^2 ^ 1.32 ... (13)We have also the combination rules'12 i  (o-i+o a^) ; cii =  (e^  . eg)* (14)Thus by knowing o*j, 1^2 possible to calculate 0*2 and from Eqs.(13) and (14). With theso force constants for the dimers on the Leimard-Jones (12:6) model the viscosity values were calculated from the expression^XlO’ ^- 266.93 V M T ... (15)wlicre jg ^ reduced collision integral which has been tabulated as afunction of 5T* =  kTje, From the values of viscosity at any temperature as obtained from Eq. (15) cr^  was calculatc'd on the rigid sphere model from the expres­sion,7 X 1 0 7 - 266.93 (16)For sample calculation of the effect of dimerization we have chosen the viscosity data of argon at 25'^ C (Michels, Botzen and Schuurman, 1954) The results obtained are shown in the Tabic I. I t  may be seen from the table that at lower pressures the effect of dimerization is to lower the viscosity which is in agreement with the results obtained by Barua and Das Gupta (1963) for steam and ammonia. How­ever, at higher pressures the dimerization makes a positive contribution to visco­sity. This may be due to the effect of dimers on the collisional transfer which overshadows the effect on the transfer by interaction, which is evident from the columns 4-6 of Table I. We have not made the calculations at higher pressures which will need the consideration of larger clusters. In fact a t the highest pres­326 Arun K. Pal and A. K. Baruasures considered by us, the positive contribution becomes less which is probably due to the importance of larger clusters at that pressure. As our main purpose in this paper is to find the effect of cluster formation on the rigid sphere model, wo have not compared with the experimental viscosity data. In  a subsequent paper wo propose to make more realistic calculations.Pressure(Atm.)Density(g./co.) %of Dimersmixcalcmlatod from eqn. (1.3) PoiseTJ^XlO*calculated tjcoi)X iO*calculatedfrom eqn. in %Poise65.9 0.1115 6.41 2.327 2,380 -2 .2 3105.4 0.1806 9.65 2.640 2.636 0.72175.2 0.3016 14.50 3.055 2.985 2.30351.2 0.5558 21.92 6.171 4.955 4.44465.5 0.6728 24.90 6.754 6.505 3.80The authors arc grateful to Prof. B. N. Srivastava for his kind interest and encouragement.R E F E R E N C E  SBarua, A. K. and Dasgupta, A., 196S, T r a n s ,  F a r a d a y  S o c ., 59, 2243.Bogolubov, N. N., 1946. J . F h y s .  ( U .S ,S ,R . ) ,  10, 205.Chapman, S., and Cowling, T. G., 1952, T h e  M a th e m a t ic a l  T h e o r y  o f  N o n - U n i fo r m  Qaaes^ Oxford University Press.Curtiss, C. F., McElroy, M. R., and Hoffman, D. K., 1964, T G  1-46, May*Dasgupta, A., and Bama, A. K., 1905, J. C h e m . P h y s , ,  42, 2849.IIirschfolder, J. O., Curtiss, C. F. and Bird, R, B., 1951, M o le c u la r  T h e o r y  o f  G o a ts  a n d  L iq u id s ,  John Wiley & Sons, N. Y.,Ilollinger, H. B., and Curtiss, C. F., I960, J . O h e m . P h y s . ,  88, 1386.Kira, S. K., and Ross, J., 1965, J . Chem. Phys, 42, 263.Michels, A., Botaons, A., and Sohuurman, W., 1954, P h y s io a ,  20, 1141.Snider, R. F,, and Curtiss, C. F., 1958, P h y s ,  F lu id s ,  1, 122.-----------, 1960, P h y s  F lu id s ,  8 , 903.Snider, R. F., and McCourt, F. R., 1963, P h y s .  F l u i d s ,  6 . 1029.Stogryn, JD. K., and Hirschfelder, J . O., 1959, J. G h c m , P h y s . ,  81, 1531.

Effect of Cluster Formation on the Viscosity of Dense Gases

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Effect of Cluster Formation on the Viscosity of Dense Gases

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