Thermoosmosis of Mixtures of Oxygen and Ethylene
The Journal of Physical ChemWy, Vol. 82, No. 21, 1978 2341
Thermoosmosis of Mixtures of Oxygen and Ethylene R. P. Rastogi* and Byas Mishra Chemistry Department, Gorakhpur University, Gorakhpw (U.P.),India (Received March 3, 1978)
The thermoosmotic pressure difference and the thermoosmotic concentration difference of binary gas mixtures of oxygen and ethylene through an unglazed porcelain membrane have been measured for various temperature differences. The heats of transport of mixtures along with the thermal diffusion factor, a12(eff),and the individual heats of transport, Q1* and Q2*, of the component gases in the mixture have been estimated from the observed data. The values of a12(eff)have been found to be greater than the values obtained from diffusion-thermoeffect measurements. Furthermore, a12(eff)is found to linearly depend on the reciprocal of the mean temperature. Attempts have been made to correlate the thermodynamic theory of thermoosmosis with the kinetic theory of thermal transpiration through porous medium based on the “dusty gas” model. The dependence of Q1* and Q2*on the mean temperature has been studied. It is found that both Q1* and Q2* depend linearly on the mean temperature.
Introduction The phenomenon associated with the development of a steady state pressure difference and a steady concentration difference across a porous barrier, the pores of which are comparable to the mean free path of the permeant species, is called thermoosmosis. Studies on thermoosmosis, which is a classical nonequilibrium phenomenon, have been made by several workers1-l1 in recent years. The thermodynamic theory of irreversible processes has been applied successfully to interpret the properties of the thermoosmotic steady state. The existence of a thermoosmotic concentration difference due to a finite temperature difference was theoretically anticipated much earlier12but the effect has been experimentally demonstrated quite r e ~ e n t l y . ~In ? ~the present investigation we report data on the measurement of the thermoosmotic pressure difference AP and the thermoosmotic concentration difference Acl corresponding to different mean temperatures for binary gas mixtures of O2 + C2H4and the data have been utilized to estimate Q1* and Q2*, the heats of transport of oxygen (1) and ethylene (2), respectively, in the binary mixture. The quantities Q1* and Q2* have been estimated a t different temperatures from the experimental data in order to study their dependence on the mean temperature since this has not been done previously. However, the thermodynamics of irreversible processes by itself cannot give an insight into the mechanism of transport through the membrane and hence it is desirable to examine the phenomenon from the viewpoint of molecular kinetic theory in order to have a satisfactory physicochemical picture of flow processes. Hence, in the present communication, an effort has also been made to correlate the thermodynamic theory of thermoosmosis with the kinetic theory of thermal transpiration based on the dusty gas approach14J5of Mason et al. and the observed data have been analyzed accordingly. Thermodynamic Interpretation of the Heat of Transport It can be shown by the thermodynamics of irreversible p r o c e s s e ~ , ~that ~ J ~the heat of transport of pure component i, [Qi*]O, i.e., the heat transported per unit of mass transport when the temperature difference on the two sides of the barrier is zero, obeys the relation: ( f l / A T ) j Z = o = [Qi*lo/uLT
(1)
where Ji is the mass flow, uiis the specific volume, and T 0022-365417812082-2341$01 .OO/O
is the mean temperature. In the above equation, AP denotes the thermoosmotic pressure difference corresponding to the temperature difference AT. For a twocomponent system and assuming ideal gas behavior, we have
1
and
Aci - cic2[Q2*/M1 - Qi*/MzI M _ AT ci/Mi + c2/M2 RTZ
P
(3)
where Qm*is the heat of transport of the mixture defined by Qm* = clQ1* c2Q2*. It should be noted that Q1* and Qz* are the individual heat of transport of component gases 1and 2, respectively, in the binary gas mixture given by the following relations:
+
and
where
M = ClMZ + czM1
(6)
and = -Ac~ (7) Here, P is the mean pressure and R denotes the universal gas constant, and c1 and c2 are the mean mass fractions of components 1 and 2, respectively. We suppose that in chamber I, the pressure is P1,the temperature is T1, and the mass fractions of components 1and 2 are clI and c?, respectively. Similarly, in chamber 11, the pressure is Pz, the temperature is T2,and the mass fractions of components 1and 2 are clD and Q, respectively. Now, we define AP = P1- P2 A T = Ti - T2 AC1
AC= ~
~ 1’ ~ 1 ”
0 1978 American Chemical Society
2342
R. P. Rastogi and B. Mishra
The Journal of Physical Chemistry, Vol. 82, No. 21, 1978
A c ~=
~ 2 '- ~ 2 "
The subscripts 1 and 2 denote the components oxygen and ethylene, respectively. Using nonequilibrium thermodynamics,I2 it can be shown that Qi* = (LiqLzz - L2qL12)/(LiiL22 - Li2L21) (8) and
Q2* = (LzqLii - LipL2i)/(LiiLz - Li2L21)
(9)
where Lik (i, k = 1,2, q ) are phenomenological coefficients. Also, for pure components, we have
[Qi*1° = Lip/Lii
(10)
[Qz*lo= L,/L22
(11)
and
account the shape, geometry, and interlinking of channels of the membrane.14J5 (iv) In estimating total number density, mass density, and the pressure which appear in multicomponent diffusion and viscous flow equations, the stationary dust particles are included in the counting. (v) Since the dust is stationary, it does not contribute to the viscosity coefficient and thermal conductivity unless a temperature gradient exists. (vi) Explicit expressions can be written in terms of gas and dust properties for (Did),the diffusion coefficient of component i with respect to dusty gas d and f f i d , the thermal diffusion factor with respect to i and d. Thermal Transpiration in a Single Gas. For a Lorentzian gas, which is considered to be a binary mixture of gas and dust in the framework of the "dusty gas" model, the thermal transpiration flux, J , is given byi5si6
It should be noted that Lik in eq 8 and 9 need not be the same as in eq 10 and 11. Now, on substituting eq 10 and 11, as a first approximation, into eq 8 and 9, we obtain
and
However, since cross phenomenological coefficients have a smaller magnitude compared to straight coefficients, Le., Ll1LZ2>> L12L21, we can write
Qi* = [Qi*lo-
L12
[Q2*lo
where nd is the number density of the dusty gas, n'is the number density of molecules including dusty gas molecules, agd'is the thermal diffusion factor of the gas with respect to the dusty gas, k is Boltzmann's constant, and q is the viscosity of the gas. is the Chapman-Enskog second approximation to the Knudsen diffusion coefficient and Bo = l / R o n dis a characteristic constant of the porous medium. Ro is a geometric constant. Now on comparing eq 19 with the corresponding thermodynamic flux equationi2
J = L11(-uAP/7')+ L i q ( - A T / p )
and
L2l Q2* = [Q2*lo- g [ Q i * 1 °
(15)
Now, if only Lik > B$ and %?[D2K12>> B$ This is possible, when P 0. Experimental results show that Q1* and Q2* depend markedly on temperature. Plots of Q1* and Q2* against the mean temperature are shown in Figures 2 and 3, respectively. Using least-squares analysis, the results are found to be best fitted by the following equations: Q1* (cal/g) = -0.067‘ + 12.0 (51)
-
and Q2*
(cal/g) = 0.387’ - 81.0
(52)
The temperature dependence of Q1* and Q2* is intimately related to the temperature dependence of a12(eff). The values of a12(eff)are given in Table V for various temperatures. On plotting the values of a12(eff)against the reciprocal of the absolute temperature, considerable scatter is obtained, as shown in Figure 4. However, the best fit is given by a12(eff)= 6.14 - 1304.16/T (53) When eq 53 is substituted into eq 44 and 45 one obtains a linear dependence of Qi* (i = 1, 2) on temperature as suggested by eq 51 and 52. The values of alz(eff)for an equimolar mixture of oxygen and ethylene has been found as 0.001 at 372 K by Waldman using diffusion thermoeffect measurement^.'^ This value is much less than the corresponding values of a12(eff)found by the thermal transpiration data recorded in Table V. In order to examine this point further, we shall use the kinetic theory of the dusty gas model. Equation 34 shows that alz(eff) is made up of a term a12/due to separation caused by thermal diffusion in the normal region and another term due to separation by the porous barrier. However, for the normal thermal diffusion phenomenon the contribution of the second term would be negligibly small. From eq 34, it follows that cu12(eff)>> a12/if (Yld’pV2 >> (YZd’Pfl1, since x d is always much greater than (xl’ + x i ) . The values of P71 and Pq2 are of the same order of magnitude because [D1K]2 and [&]2 would be of the same order for the experimental gases. The magnitude of aid' and q d ’ would depend on the extent of nonelastic collisions. Since the sorption coefficientls of component 1 (oxygen) is 2.67 f 0.23, and that of component 2 (ethylene) is 5.2 f 0.45 atm-’, hence the probability of nonelastic collisions of the second component with the membrane
T,,K T , K
AT,
323.0 326.0 330.5 343.5
350.0 54.0 0.068 357.8 63.5 0.076 367.3 73.5 0.085 -0.41 398.5 110.0 0.107
372.0 402.0 425.0 438.5
320.0 327.0 332.5 335.5
346.0 52.0 0.062 364.5 75.0 0.081 378.8 92.5 0.092 -0.38 387.0 103.0 0.099
T o t a l pressure =
K
AP,
T,, K 377.0 389.5 404.0 453.5
c m H g [Q,*]o,cal/g
k
0.02
t
0.02
26.8 c m H g .
material would be much larger as compared to that for the first component. Hence21 aid' > (Y2d’. Mason and co-workers20have shown that a12(eff)can assume a very high value for a particular Knudsen number. It is very likely that, in the present case, we may be near such a maximum. However, it is necessary that experiments be performed at different pressures in order to test this point.
Acknowledgment. Thanks are due to the Department of Atomic Energy, India, for supporting this investigation. The authors are grateful to Professor E. A. Mason, Brown University (U.S.A.) for helpful correspondence. Supplementary Material Available: Tables I1 and V containing the thermoosmotic concentration differences and thermal diffusion factors for binary gas mixtures of O2 and C2H4(2 pages). Ordering information is available on any current masthead page.
References and Notes R. P. Rastogi, K. Singh, and H. P. Singh, J. Phys. Chem., 73, 2798 (1969). R. P. Rastogi and H. P. Singh, J . Phys. Chem., 74, 1946 (1970). R. P. Rastogi and A. P. Rai, J . Phys. Chem., 70, 2693 (1974). R. J. Bearman and M. Y. Bearman, J. phys. Chem.,70,3010 (1966). H. J. M. Hanley and W. A. Steeie, Trans. Faraday Soc., 61, 2661 (1965). R. P. Rastogi and K. Singh, Tram. Faraday SOC.,62, 1754 (1966). R. P. Rastogi, A. P. Rai, and M. L. Yadava, Ind. J . Chem., 12, 1273 (1974). R. P. Rastogi and A. P. Rai, J . Membr. Sci., in press. R. P. Rastogi and B. Mishra, J. Membr. Sci., in press. E. A. Mason and A. P. Maiinauskas, Trans. Faraday Soc., 87, 2243-2250 (1971). R. E. Jenkins and E. A. Mason, Phys. Fluid, 13, 2478 (1970). S. R. DeGroot, “Thermodynamics of Irreversible Processes”, North Hoiiand Publishing Co., Amsterdam, 1952. R. Haase, “Thermodynamics of Irreversible Processes”, Addison Wesley, Reading, Mass., 1968. R. B. Evans, 111, G. M. Watson, and E. A. Mason, J. Chem. Phys., 35, 2076 (1961). E. A. Mason, R. B. Evans, and 0. M. Watson, J. Chem. Phys., 38, 1808 (1963). E. A. Mason, A. P. Maiinauskas, and R. B. Evans, J. Chem. Phys., 48, 3199 (1967). J. R. Partington, “Advanced Treatise on Physical Chemistry”, Vol. I V , Longman, Green and Co., London, 1953. R. P. Rastogi, A. P. Rai, and B. Mishra, Id.J. Chem., 14A, 831-835 (1976). W. Jost, “Diffusion in Solids, Liquids and Gases”, Longman. Green and Co., London, 1960. E. A. Mason and A. P. Malinauskas, J. Chem. phys., 41,3815-3919 (1964). E. A. Mason in “Kinetic Processes in Gases and plasmas”, Hochstein, Academic Press, New York, N.Y., 1969, p 88.
