Calculation method of electrical conductivity, thermal conductivity and viscosity of a partially ionized gas. Ilona Lázniková

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1 www. energyspectru.net Cacuaton ethod of eectrca conductvty thera conductvty and vscosty of a partay onzed gas Iona Láznková Brno Unversty of echnoogy Facuty of Eectrca engneerng and Councaton Departent of Eectrca Power Engneerng echncka 66 Brno Czech Repubc te: 5 9 ea: azncka@feec.vutbr.cz ABSRAC he cacuaton ethod of the transport propertes (eectrca conductvty thera conductvty and vscosty of a partay onzed gas s descrbed n ths paper. he cacuaton ethod s based on the Chapan-Enskog ethod and on the knowedge of the coson ntegras for each par of speces of a partay onzed gas. he generay ethods of the coson ntegra cacuaton are ntroduced. Keywords: Eectrca conductvty thera conductvty vscosty partay onzed gas Chapan-Enskog ethod IRODUCIO he knowedge of the transport propertes of the gas xture s very portant because the operatng propertes of soe technca devces are deterned by the transport processes n the gas xture. he thera pasa created by the eectrca arc gves the portant advantages for technoogca appcatons the pasa hgh teperatures n the nterva of 5-5 kk the hgh densty and the hgh degree of the energy transport the hgh speed of the checa reactons n the pasa. hese propertes are essenta for soe technca appcatons. he chef characterstcs of these appcatons are the usng of the dfferent gas xtures of the dfferent transport propertes. CHAPMA-ESKOG MEHOD AD COLLISIO IEGRAL he knetc gas theory s the base for the deternaton of the gas transport propertes ] ] and t s based on the knowedge of the dstrbuton functon. hs equbru dstrbuton functon s deterned for every partce n the gas on ters of the sae teperature for a partces n the gas and of the sae speed of the a partces. he equbru dstrbuton functon foruates the equbru of the every speces and the equbru between speces n the gas syste too. hen the gas syste s not n the equbru the dstrbuton functon agrees wth the Bozann ntegradfferenta equaton. he Chapan-Enskog ethod ] of the souton of the Botzann equaton s the theoretca base for the transport propertes cacuaton of the partay onzed gases. hs ethod was extended for the utcoponent gas syste by Hrschfeder Curts a Brd ]. For the every speces n the gas syste s foruated one Botzann equaton and therefore the utcoponent gas syste s descrbed by the syste of the ntegrodfferenta equatons. he Chapan- Enskog ethod akes over these equatons to the near equaton syste and gves severa approxatons of the dstrbuton functon. If the utcoponent gas syste contans speces and t s used -th approxaton the near equaton syste for varabes s soved. Chapan and Enskog ntroduce the new varabe caed the coson ntegra Ω and foruated for a par of partces. he deternaton of these coson ntegras s the portant part of the cacuaton of the transport propertes. he coson ntegra s defned by the equaton (s k ( exp(- s ( Q ( g ( Q ( g ( cos γ ( d b db ( s the tota transport cross secton of two codng partces and. In the equatons ( and ( k s the Botzann constant s the teperature s the reduced ass (/ / / and s the reduced nta reatve speed of th and th speces g k ( g s the nta reatve speed s the ange of defecton between th and th speces ( g b b r dr r ( r b ϕ r g (

2 b s the pact paraeter and s are the paraeters defnng the order of the coson ntegra and s. he coson ntegras are cacuated not ony for a par of codng partces but aso for the gven cobnaton of the paraeters and s. hese cobnatons resut fro the used degree of approxaton. he deternaton of the coson ntegras s possbe by usng three souton ways. he tabes of coson ntegras are pubshed for soe pars of codng partces. In the ost cases the coson ntegras are cacuated fro the potenta energy between two oecues (r and by usng ( (. he ntegraton of experenta vaues of the oentu transfer cross secton and the usng of ( ( can be cacuated for soe nteracton ncudng cosons between eectron and neutras. he spest way of the coson ntegra cacuaton s the ode of rgd spheres ] and t can be used for a types of cosons except the Couob nteracton and t s equa to ( ] ( σ σ Q rs ( (5 ( s k ] ( s! ( Q ] rs µ rs σ σ are the daeters of the partces and. he ethod of the Lennard-Jones potenta can be used for a par of neutra partces but the Lennard-Jones paraeters of both partces ust be known. he spe approxaton ethod of the Lennard-Jones potenta s descrbed n ] and t s based on the approxaton Q ( ( g ( Qap ( ( ( ( ( g A g B g C D g (6 and on the expresson of the coson ntegra n the for ( ( A ap ( s ( B Γ( s ( ( C D ( s! Γ( s 5 ( s! he vaues of ( ( ( ( A B C D are gven n the ab. ]. ab. he coeffcent vaues of poyno ( ( Q g A ( B ( C ( D ( For the cosons of neutra-on the poarsabty ethod can be used ] ( s A ( ( s Z e ξ π ε k (! s ( ( (7 ( e s the eectron charge ε s the pertvty of the ( vacuu A ( are the nuerca constants A ( ( 6566 A ( ( 5. he ethod s based on the knowedge of the poarsabty ξ of the neutra. For the soe cosons the experenta data of the oentu transfer cross secton can be used. hese data are n the terature n the for of the graphs or n the tabes. he experenta data can be approxated by the functon n the for Q A ε Bε Cε ε Dε (9 and the coson ntegra s cacuated fro the reaton ( s Ω Cε k A ε Γ( s 5 Bε ( s! πµ ( ε Γ ( s 7 Dε ( s ]! In the equatons (9 ( ε k/e. CALCULAIO MEHOD OF ELECRICAL CODUCIVIY he reaton for the eectrca conductvty cacuaton of a partay onzed gas s n the for 5] e n ρz σ ( pρ Z n D nk Z k k the charge of a partce s arked ez (Z e - and f the condton n s satsfed the equaton ( s odfed k k Z k e n Z Z n D pρ σ ( In the gven equaton ( n or s the nuber densty or the ass of the th speces n s the tota nuber densty p s the pressure s the ass densty. In the sus ony the charge partces are ncuded because Z s not zero ony for these partces. D s the dffuson coeffcent defned by the equaton c s one souton of near equaton syste ( ( δ δ h ζ D ρ n k ( c ( n k δ k ( n ( ( δ δ c n hk ( In ths equaton δ δ δ δ are Cronecker deta h k ζ -. For the eectrca conductvty cacuaton of a gas syste thrd approxaton was chosen

3 (ζ therefore the syste of equatons ust be soved and can have vaues or. Ony for the eft sde of equaton ( can be dfferent fro zero. Reatons for cacuaton of coeffcents ( are functons of the coson ntegras ( s and for the thrd approxaton they are gven n the reatons (5 5]. ( ( 9 ( n ( ( 5 5 ( ( 5 ( n n ( ( ( ( n n ( n n ( δ ( n ( δ δ n ( δ ] ( 5 ( ( δ δ 5 ( 7 ( ( δ ( ( 5 ( ( δ δ ( 5 ( 5 ( ( ( ( ( ( ( δ δ 7 ]} 6 ( 5 ( 7 ( δ δ ( 6 5 ( 5 7 ( ( ( ( δ δ ( 7 ( nn 5 6 ( 7 ( (5 ( 6 ( ( ( δ δ 5 ( δ δ ( n n ( ( 6 n n ( ( ( 6 nn ( 5 ( ( ( ( δ δ 7 ( δ δ ( δ δ ( ( 5 ( ( ( 5 9 (5 ( 77 7 ( ( ( ] ( δ δ. CALCULAIO MEHOD OF HERMAL CODUCIVIY hera conductvty can be cacuated as the su of three coponents 5] λ λ λ λ (6 tr nt r λ tr s the transaton thera conductvty λ nt s the nterna thera conductvty and λ r s the reacton thera conductvty. he frst coponent λ tr of the thera conductvty s the transaton thera conductvty due to the heavy partces and t can be wrtten n the for

4 λ 5 tr k n nk the coeffcent equaton syste k ( a (7 ( a s one souton of the foowng λ r R A A ' A A A A A A ( 5 n k δ ζ ( n n ( δ ( oδ o a ( (. he coeffcents ( are the functons of the coson ntegras (5. In the equaton (7 the notaton the product of three atrxes fored fro the bnary dffuson coeffcents and fro the bnary thera dffuson coeffcents n D n ρ D n D s gven n the equaton ( and by the equaton D s (9 s cacuated n k ( D a ( a s the one souton of equaton syste (. ( he second coponent of thera conductvty λ nt s the nterna thera conductvty soetes s tered as the Eucken thera conductvty λ Euck. It s gven by the equaton 5 k λ nt 6 π A ( c 5R p x σ x M M M M ( In ths equaton A s Avogadro constant R s the perfect gas constant x or x respectvey M or M s the oar fracton respectvey the oar ass of th or th speces c p s the specfc heat at the constant pressure of th speces and s the radus of partces and ( /. he thrd coponent of the thera conductvty λ r s the reacton thera conductvty arsng ke a consequence of checa reactons n gas syste and t s defned by the expresson 6] A k k A A R k p ν k ν k x x ν k H k. k ν ν x ( In the equatons ( ( s the nuber of the ndependent checa reactons H s the heat of the ndependent checa reacton k k are the stochoetrc coeffcents for speces k ( n reacton (. 5 CALCULAIO MEHOD OF VISCOSIY he vscosty can be cacuated by the usng of the reaton 5] k n b ( η ( ( b s one souton of the equaton syste ζ ( ( 5nδ o b ( ( he vaues of the coeffcents are cacuated ony for and and the reatons are n (5. 6 COCLUSIO In ths paper the cacuaton ethod of the eectrca conductvty the thera conductvty and vscosty s descrbed. he portant part of cacuaton of these propertes s the knowedge of the coposton of the gas syste. In the paper the generay cacuaton ways of the coson ntegras are showed. For soe pars of partces whch have donant concentraton n the gas syste t s better to use the experenta vaues of oentu transfer cross secton (for eectron-neutra or to use the tabe vaues.

5 ACKOLEDGEMES he present paper contans the resuts of research work funded by Proect /6/7 of the Grant Agency of the Czech Repubc. REFERECES ] Chapan F. Cowng. G. Matheatca theory of non-unfor gases (n Russan. Izdatestvo nostranno teratury Moskva 96. ] Hrschfeder J. O. Curts Ch. F. Brd R. B. Moecuar heory of Gases and Lquds. John ey ew York 95. ] Kenek P. Contrbuton to the Cacuaton of Coson Integras for the Lennard-Jones Interacton Mode. Acta echnca SAV. 6(99 no ] Chervy B. Cacu des propretes de transport et etude du pouvor de coupure des eanges hexafuorure de soufre (SF6 fuorure de carbone (CF ou CF6 et hexafuorure de soufre vapeurs de cuvre. hese no. D Ordre: 997. Unversté Pau Sabater France ] Kenek P. nka V. Eectrca Conductvty hera Conductvty and Vscosty of SF6 at the teperatures 5 K n the pressure range.. MPa. Acta echnca SAV. (9 no ] Brokaw R. S. hera Conductvty of Gas Mxtures n Checa Equbru. II. he Journa of Checa Physcs. (