Indian Journal of Chemistry Vol. 32A, May 1993, pp, 414-417 Thermal diffusion factor in gas mixtures and Dufour effect A Maghari* Department of Chemistry, University of Tehran, Tehran, Iran and A Bordbar & A Boushehri Department of Chemistry, Shiraz University, Shiraz, Iran Received 31 March 1992; revised and accepted 21 December 1992 A new formula for thermal diffusion factor, ay-, is pre- sented on the basis of thermodynamics of irreversible processes. By using the virial equation of state, the noni- deality effects are considered, and the pressure depend- ence of diffusion thermoeffect is interpreted from the theoretical results. Furthermore, by a modified appara- tus, a study of the Dufour effect is made for three differ- ent gas mixtures: H 2 /C 2 H 4 , H 2 /C 2 H 2 and H 2 /NO for which the Dufour effect have not been measured earlier. The value of the thermal diffusion factor are evaluated for the above mixtures according to the obtained formula. In our earlier investigation, we have derived a general formula for thermal diffusion factor on the basis of thermodynamics of irreversible processes. In the present work, we report the modifications of our ear- lier calculations of ar using perturbation technique. Also by applying the virial equation of state, the noni- deality of the gases have been considered. As a result, the interpretation of the pressure dependence of ther- mal diffusion factor has been made possible. Further- more, in this work the Dufour coefficients of three dif- ferent gas mixtures have also been investigated, General formulation In view of the local approximation, when the diffu- sion and heat conduction take place simultaneously, it may be shown that', pCp(oTlot)= V' {AVT+ PI (0ftl1 OClh,.p" DI2 aT VCd ... (1) where p, Cp,ftl' CI,A, D12 and aT are total mass density, heat capacity, chemical potential, mass frac- tion of component 1,thermal conductivity, diffusivity and thermal diffusion factor, respectively. At the maximum temperature, the illS of Eq. (1) is zero and we must have, AVT max + PI (Oftl 1 oC l h,.p,Pl2 aT VCI = K(t) ... (2) Notes where K (I) is a function of time only, of course.inde- pendent of coordinates. Rastogi and Madan", calculated aT from Eq. (2), while they were assuming K is zero, and also Boush- ehri! considered K as a constant dependent on the geometry of the diffusion apparatus. In this work we have evaluated from Eq. (2) the thermal diffusion fac- tor. Equation (1)can be rewritten as, oTI 01= (Ai pCp)V 2T + [C1 (0ftl I oClh".p,P12 aTI Cp]lV 2 C 1 • ... (3) where the terms involving VA and VaT are neglected for the present work. Now, suppose that two gases labelled with sub- scripts 1 and 2, initially at the same temperature and pressure are placed in the two regions separated by a diaphragm, as shown in Fig. 1. The temperatures are monitored at points X I and X 2 placed symmetrically about the origin, X 2 =- XI = L12. Wecanassumethatthethermaldif- fusion may be neglected according to the diffusion equation and we can write, oCllot=D1202ClloX2 ... {4} The appropriate solution of the above equation can be written as, C 1 (x; t) = ~~ Eifc [x12 (D12 1)112] . .. (5) where Erjcis a complementary error function, and C? is the initial mass fraction of component 1 in region 1. 2 o " '" Fig.l-SchematicviewofDufoureffect
Transcript
Indian Journal of ChemistryVol. 32A, May 1993, pp, 414-417
Thermal diffusion factor in gas mixturesand Dufour effect
A Maghari*
Department of Chemistry, University of Tehran, Tehran, Iranand
A Bordbar & A BoushehriDepartment of Chemistry, Shiraz University, Shiraz, Iran
Received 31 March 1992; revised and accepted 21 December1992
A new formula for thermal diffusion factor, ay-, is pre-sented on the basis of thermodynamics of irreversibleprocesses. By using the virial equation of state, the noni-deality effects are considered, and the pressure depend-ence of diffusion thermoeffect is interpreted from thetheoretical results. Furthermore, by a modified appara-tus, a study of the Dufour effect is made for three differ-ent gas mixtures: H2/C2H4, H2/C2H2 and H2/NO forwhich the Dufour effect have not been measured earlier.The value of the thermal diffusion factor are evaluatedfor the above mixtures according to the obtained formula.
In our earlier investigation, we have derived a generalformula for thermal diffusion factor on the basis ofthermodynamics of irreversible processes. In thepresent work, we report the modifications of our ear-lier calculations of ar using perturbation technique.Also by applying the virial equation of state, the noni-deality of the gases have been considered. As a result,the interpretation of the pressure dependence of ther-mal diffusion factor has been made possible. Further-more, in this work the Dufour coefficients of three dif-ferent gas mixtures have also been investigated,
General formulationIn view of the local approximation, when the diffu-
sion and heat conduction take place simultaneously,it may be shown that',pCp(oTlot)=V' {AVT+ PI (0ftl1 OClh,.p" DI2 aT VCd ... (1)where p, Cp,ftl' CI,A, D12 and aT are total massdensity, heat capacity, chemical potential, mass frac-tion of component 1, thermal conductivity, diffusivityand thermal diffusion factor, respectively.
At the maximum temperature, the illS of Eq. (1) iszero and we must have,AVTmax + PI (Oftl 1 oClh,.p,Pl2 aT VCI = K(t)
... (2)
Notes
where K (I) is a function of time only, of course.inde-pendent of coordinates.
Rastogi and Madan", calculated aT from Eq. (2),while they were assuming K is zero, and also Boush-ehri! considered K as a constant dependent on thegeometry of the diffusion apparatus. In this work wehave evaluated from Eq. (2) the thermal diffusion fac-tor.
Equation (1 )can be rewritten as,oTI 01= (Ai pCp)V 2T
+ [C1 (0ftl I oClh".p,P12 aTI Cp]lV2C1•
... (3)where the terms involving VA and VaT are neglectedfor the present work.
Now, suppose that two gases labelled with sub-scripts 1 and 2, initially at the same temperature andpressure are placed in the two regions separated by adiaphragm, as shown in Fig. 1.
The temperatures are monitored at points X I andX2 placed symmetrically about the origin,X2 = - XI = L12. Wecanassumethatthethermaldif-fusion may be neglected according to the diffusionequation and we can write,oCllot=D1202ClloX2 ... {4}
The appropriate solution of the above equation canbe written as,
C1 (x; t) = ~~ Eifc [x12 (D12 1)112] . .. (5)
where Erjcis a complementary error function, and C?is the initial mass fraction of component 1 in region 1.
2
o"'"
Fig.l-SchematicviewofDufoureffect
Using the virial expansion of compressibility, Z = 1
BP ith . ft h h .+- + ...Wl truncatron a ertwoterms, t ec emt-RT
cal potential of component 1 in a binary mixture ofreal gases can be written as,
where MI , Ml are the molar masses (we assume thatMl> MI), and !l.12= 2BIl - Bll - 822, The coeffi-cient B II refers to a pure component, and 812 is calleda cross-coefficient.
Furthermore, the variation of the concentrationdistribution of component 1 is defined as,oCI (x;t) = C~ - CI (x;t) ... (7)
Substituting the above relation into Eq. (6), usingthe perturbation parameter, and neglecting the qua-dratic and higher order terms in oC1, we can obtain,
Putting Eqs (8 )-(10) into (3) and applying the per-turbation parameter k, we obtain,aT/ {}t= KT 02 T / ax2 +; e'c, / ax2 +A(x;t)
... (11)in one dimension, where K T is the thermal diffusivity.The first two terms in Eq. (11) correspond to the un-perturbed expression, and the third term is the firstorder correction which will be treated as a perturba-tion. The perturbation term A (x; t) is,
A(x; t) = k<P e'c, / a2x2
+ D~ aT (Q'_ Q)(acl/ax/In Eqs (11) and r12)KT=)./pCp
RTo aT Dll n12 (n12 - 1)<P = -~-'----"-"-'-:'="':'---'-=-"":'MICp
_ 2MI M2 Po aTD12!l.12Co (Co _ CO _ 3 n Co)Cp(MI C~ + M2 C~)3 I 2 I 12 2
... (15)
Modification and derivation of aTIn this section, we have evaluated an explicit form
of the thermal diffusion factor by exact solution ofEq.(11).By using the Fourier transformation technique,Eq. (11) transforms to,af( w,t)/ at+ K T W2 f( w; t) = x( w,t) + A( w; t)
... (16)where f( w, r), Al (w, t) and x( w; t) are the Fouriertransforms of T( x; t), A I (x; t) and x( x; t) respectivelyso that,
The complementary error function can be written as,00
Erfc(X /2 Y) = 2 n-112 Y exp( ~ X2 14 y2) f dUo
x exp( - y2U2_ XU) ... (22)
Hence by inserting Eq. (21) into (19) and applying(22), the second term ofEq. (19) becomes,
2(K:~~12) [Erf [2(D~ t)1I2]- Erf [2(K~t)1I2]1
+ G(x;t) ... (23)where G( x; t) is a function of t and x. It can be shownthat G(x; t) remains invariant under the inversiontransformation of coordinates, that is,G(x,t) = G( - x,t) ... (24)
Consequently, the temperature distribution in thesystem is,
... (2S)
The measured temperature difference is defined as,
oT= T(X2;t)- T(XI ,t)= T (k, t) - T (-t, t)... (26)
Substituting Eq. (2S) into (26), and using (24), themaximum temperature difference, c5Tmax , would be,
y=KTIDI2=A/pCpDI2 ... (30)Therefore, estimation of the thermal diffusion factorfrom Eq. (28) is possible.
ExperimentalThe apparatus employed was similar to that used
by Boushehri ' with an appropriate modification byputting the bulb with the lighter gas at the top of thebulb with the heavier gas (Fig. 1),so that the error dueto convection is avoided. The apparatus consists oftwo double-walled glass bulbs (1,2) of capacity SOO-rnl each. A fivejunction thermocouple and a leeds andNorthrup potentiometer ( ± 1,uV)wasusedbywhichc5T could be estimated to ± 0.004°C. The distancebetween two thermocouples was equal to 36.S ern. Allruns were made at 3SoC under 700 torr. The maxi-mum temperature difference c5Tmaxwas obtained as afunction of time after the stopcock connecting the halfcells was opened and interdiffusion was allowed tooccur.
Table I-Thermal diffusion factor for gas mixtures
Systems dTmax aT' from aT fromEq. (28) thermal diffu-
Results and discussionThe values of maximum temperature difference,
oTm3• observed in the present work and their values ofa T obtained from Eq. (28) are given in Table 1.Thevalue of aT for H2 IC2H4 obtained from thermal dif-fusion measurement is shown for comparison pur-pose. The data for H2/C2H2, and H2/NO are notavailable. The difference between the found value ofH2/C2H4 from the present calculation and the ther-mal diffusion measurement may be due to the use ofpure component values of l, PI and Cpin Eq. (28).
Finally,Eq. (28)whichcorrelates the thermaldiffu-sion factor, can be explained by the pressure depend-ence of a T from experiments; it can be shown that the
" RT( 1+ BPI RT )correlation quantity y = K T I D12 = -----'--------'-Cp
NOTES 417
(A.!pDul appearing in Eq. (30 lis nearly constant withincreasing pressure. As a result, the effect of increas-ing pressure on the thermal diffusion factor nearlyparallels the pressure dependence of oTmax which isobtained from these experiments. This conclusion isin agreement with Mason's treatment".
