The isentropic exponent in plasmas

Transcription

1 Te isentroic exonent in lasmas Burm K.T.A.L.; Goedeer W.J.; cram D.C. Publised in: Pysics of Plasmas DOI: / Publised: 01/01/1999 Document Version Publiser s PDF also known as Version of Record (includes final age issue and volume numbers) Please ceck te document version of tis ublication: A submitted manuscrit is te autor's version of te article uon submission and before eer-review. Tere can be imortant differences between te submitted version and te official ublised version of record. Peole interested in te researc are advised to contact te autor for te final version of te ublication or visit te DOI to te ubliser's website. Te final autor version and te galley roof are versions of te ublication after eer review. Te final ublised version features te final layout of te aer including te volume issue and age numbers. Link to ublication General rigts Coyrigt and moral rigts for te ublications made accessible in te ublic ortal are retained by te autors and/or oter coyrigt owners and it is a condition of accessing ublications tat users recognise and abide by te legal requirements associated wit tese rigts. Users may download and rint one coy of any ublication from te ublic ortal for te urose of rivate study or researc. You may not furter distribute te material or use it for any rofit-making activity or commercial gain You may freely distribute te URL identifying te ublication in te ublic ortal? Take down olicy If you believe tat tis document breaces coyrigt lease contact us roviding details and we will remove access to te work immediately and investigate your claim. Download date: 18. Aug. 018

2 PHYIC OF PLAMA VOLUME 6 NUMBER 6 JUNE 1999 Te isentroic exonent in lasmas K. T. A. L. Burm Deartment of Alied Pysics Eindoven University of Tecnology P.O. Box MB Eindoven Te Neterlands W. J. Goedeer F.O.M. Institute for Plasma Pysics Rijnuizen artner to te trilateral Euregio Cluster P.O. Box BE Nieuwegein Te Neterlands D. C. cram Deartment of Alied Pysics Eindoven University of Tecnology P.O. Box MB Eindoven Te Neterlands Received 15 December 1998; acceted 6 February 1999 Te isentroic exonent for gases is a ysical quantity tat can ease significantly te ydrodynamic modeling effort. In gas dynamics te isentroic exonent deends only on te number of degrees of freedom of te considered gas. Te isentroic exonent for a lasma is lower due to an extra degree of freedom caused by ionization. In tis aer it will be sown tat like for gases te isentroic exonent for atomic lasmas is also constant as long as te ionization degree is between 5% 80%. Only a very weak deendence on te electron temerature and te two nonequilibrium arameters remain. An argon lasma is used to demonstrate te beavior of te isentroic exonent on te lasma conditions and to make an estimation of te value of te isentroic exonent of a customary lasma. For atmoseric lasmas wic usually ave an electron temerature of about 1 ev a sufficiently accurate estimate for te isentroic exonent of lasmas is American Institute of Pysics X I. INTRODUCTION Te isentroic relations for a flowing gas system relating ressure density and temerature to eac oter are usually used in situations of atmoseric ressure and of local termal equilibrium LTE. Local termal equilibrium means tat te electron temerature may not deviate from te eavy article i.e. atoms and ions temerature te lasma is socalled equitermal and tat ionization may not differ from aa s ionization equilibrium LE. Te isentroic relations can be used instead of more comlex energy equations wen viscosity and eating lay no significant role. Te Debye lengt must be small and te Reynolds number sufficiently ig tyically in te order of 100 or even larger. Te isentroic exonent as used in tese relations is a constant making te isentroic relations a convenient tool to analyze many systems wit gas flow. In tis aer we try to extend te isentroic relations to atmoseric lasmas. Te main difference between gases and lasmas is tat in lasmas ionization needs to occur to sustain te lasma. Plasmas are not in te LTE wic imlies tat we need to distinguis a temerature for te electrons and one for te eavy articles and tat te actual neutral density does not equal te aa neutral density. An imortant question is ow muc te isentroic exonent deviates from its LTE value due to ionization as well as due to equitermal disequilibrium. In order to stress te influence of ionization and te disequilibria we will use an alternative descrition in wic te lasma ressure te ionization degree and two nonequilibrium arameters b 1 for deviations from aa s relation and for equitermal disequilibrium are used instead of te electron temerature te electron density te eavy article temerature and te eavy article density. Te isentroic exonent is usually investigated as a function of te temerature. However in tis aer we will sow ow te isentroic exonent beaves as a function of te ionization degree ratio wic is te arameter tat sows best te difference between gases and lasmas. It sould be stressed tat te ionization degree ratio and te electron density in lasmas vary over many orders of magnitude wereas te electron temerature varies over a relative small range e.g. for atmoseric lasmas te electron temerature is always in te range of 1 ev. Next we will consider tree lasmas in articular; a monoatomic LTE lasma a monoatomic non-lte lasma and a diatomic non-lte lasma. II. THE IENTROPIC RELATION In gas dynamic teory it is common to use te isentroic relation relating ressure and density wic reduces te comlexity of te ydrodynamic descrition. Altoug te isentroic condition is not always valid it is often used as a first aroximation of comlex gas dynamical systems. In tis aer we discuss tis isentroic relation for lasmas in order to ave te same tool to describe lasmas and gases. Te isentroic relation relating te ressure or te temerature T and te mass density reads T C C X/99/6(6)/6/6/$ American Institute of Pysics Downloaded 1 Mar 005 to Redistribution subject to AIP license or coyrigt see tt://o.ai.org/o/coyrigt.js

3 Pys. Plasmas Vol. 6 No. 6 June 1999 Burm Goedeer and cram 63 were C and C) is a constant and is te so-called isentroic exonent wic is only a function of ow energy is distributed internally in te considered fluid i.e. of te degrees of freedom. Note tat by determining exression 1 te relation for ressure vs temerature is also determined. In termodynamics exression 1 is called Poisson s relation. Te isentroic exonent equals 5/3 for monoatomic gases wic ave tree kinetic degrees of freedom e.g. an argon gas and equals 7/5 for diatomic gases wic can also rotate and vibrate e.g. nitrogen and ydrogen gases. A lasma as extra degrees of freedom because of te occurrence of ionization. In ordinary lasmas eat is used to ionize and create te lasma wic is an extra freedom in te energy distribution inside a lasma. Te isentroic exonent of a lasma will terefore differ from tat of a gas. To calculate te isentroic exonent we will use te definition in gas dynamics; te isentroic exonent equals te ratio of te eat caacity at constant ressure c and te eat caacity at constant volume c V. For a full derivation of te isentroic exonent of monoatomic and diatomic gases we refer to Matsuzaki. 1 Furter we will consider only singly ionized ions and make use of quasineutrality i.e. te ion density equals te electron density n i n e. Te eat caacity at constant ressure at constant volume and te isentroic exonent are defined as c H T T s T c V T e s c 1 3 T T T ln ln 4 s s wit H te secific entaly e te internal energy and s te entroy. At constant entroy c T s T Ts c V TT s Ts wic we use to derive c c V 1. 5 T For a lasma c c v and are functions of te ionization degree. Te ionization degree is defined as n i 6 n in wic n te eavy articles number density is equal to te sum of te atom density n a te ion number density n i and if not equal to zero te molecule number density n m. Note tat we use number densities wic equals te number of articles er molecule times te density of tat molecule. III. THE IENTROPIC EXPONENT A. Te monoatomic LTE lasma case For lasmas we need to focus on te contribution of ionization in te secific entaly and te internal energy. We will first discuss te secific entaly and te internal energy before we derive ow te temerature influences te ionization degree. We need tis latter result in te derivation of te eat caacity at constant ressure at constant volume and te isentroic exonent see formulas 3 and 5. We will see tat due to ionization decreases towards 1.16 for a lasma in comarison to 5/3 or 7/5 in case of an ideal monoatomic resectively diatomic gas. Te following set of lasma conditions is considered first: 1 monoatomic lasma like for e.g. an argon lasma; te electron temerature may not deviate from te eavy article i.e. atoms and ions temerature i.e. local equitermal equilibrium LEE; and 3 ionization may not differ from te aa equilibrium i.e. local aa equilibrium LE. Later on we will discuss non-lte and diatomic lasmas. Te basic equations governing te termodynamic roerties of te lasma are te secific entaly and te internal energy including ionization. Te secific entaly can be obtained by using te metods of statistical mecanics. For most of te monoatomic lasmas like an argon lasma tis includes only te translational energy and ionization energy since te excitational energy is negligible. 3 Te secific entaly H can be written as H 5 ion RT1 7 m a wit R te mass secific gas constant ion te ionization energy and m a te atom mass. Te internal energy e is related to te secific entaly by eh HRT1. Exressions 7 and 8 are similar to te exressions for gases wen te ionization degree is set equal to zero. Next we will derive te exression for te isentroic exonent in a lasma starting from te aa balance 4 and Dalton s law. Te aa equation gives te relation between te electron density n e te ion density n i and te density of te ground level neutrals n s n in e g i n s g a m e 3/ex ion 9 were m e equals te electron mass te Planck constant T e te electron temerature and g i and g a are te statistical weigts of te ion and atomic ground state. Notice tat T e 3 ion 10 T e and in LTE TT e and n a n s. Dalton s law in LTE reads 8 Downloaded 1 Mar 005 to Redistribution subject to AIP license or coyrigt see tt://o.ai.org/o/coyrigt.js

4 64 Pys. Plasmas Vol. 6 No. 6 June 1999 Burm Goedeer and cram n a n i n e ktn kt1. 11 Dalton s law and quasineutrality togeter wit te aa balance 9 gives n e e n kt. After solving tis quadratic equation in n e weget 1 kt 1 or kt Taking te derivative of te ionization degree wit resect to te temerature T leaving ressure constant 1/ /kt/t /kt 11//kT T 3 kt 1 T 1 T. 15 Tis yields conform Refs. 1 and 5 using exressions 10 and 14 T 11 5 ion 16 kt. T We need exression 16 in te derivation of te eat caacities wit resect to te temerature at constant ressure resectively at constant volume wic can be written as c 5 R1RT 5 ion resectively kt T 17 c V 3 R1RT 3 ion kt. 18 T Note tat te derivative of te ionization degree wit resect to te temerature at constant density can be found in a similar way as exression 15 from 13 wit an extra term exressing te derivative of ressure wit resect to te temerature at constant density. Tis yields T 1 T 1T 19 T Combining exressions 16 and 19 yields 1 3 ion T kt. T T. 0 Hence te eat caacity at constant ressure becomes for a monoatomic LTE lasma kt c 5 11 R1R 5 ion and te eat caacity at constant volume 1 c V 3 1 R1R 3 ion. kt Te isentroic exonent from exression 5 can now be analyzed furter. By making use of T RT1T 1 RT1 1 T 1 and from exressions 13 and T we can write for te isentroic exonent of a monoatomic LTE lasma c c V 1 c 1 3 c V 111/ as in agreement wit Matsuzaki 1 using termodynamic metods and Owczarek 6 using te metod of caracteristics. Notice tat te exressions for te eat caacities and te isentroic exonent are similar to te exressions for gases wen te ionization degree is ut equal to zero. We can calculate te isentroic exonent from exression 3 wit use of te above exressions for c and c V. B. Te monoatomic lasma in non-lte In tis section we allow te lasma to deviate from te two tyes of equilibria we considered. We ave te following set of lasma conditions: 1 monoatomic lasma like an argon lasma; te electron temerature T e may deviate from te eavy article i.e. atoms and ions temerature T i.e. non-lee; and 3 ionization may differ from te aa equilibrium i.e. non-le. For ig density lasmas wic are not too far from local termodynamic equilibrium LTE te commonly used termodynamic lasma variables are te electron temerature electron density eavy article temerature and eavy article density or ressure. In tis section we use an alternative descrition in wic te lasma ressure te ionization degree or electron density and two nonequilibrium arameters b 1 and conform cram et al. 7 b 1 n a n s and T T e are used. Note tat in local termodynamic equilibrium b 1 and are bot equal to one. Tis alternative descrition will turn out to be more convenient as we will see wen we discuss te results in te next section. In tis aer b 1 and are considered as indeendent variables. We start again wit te exressions for te secific entaly and te internal energy wic are Downloaded 1 Mar 005 to Redistribution subject to AIP license or coyrigt see tt://o.ai.org/o/coyrigt.js

5 Pys. Plasmas Vol. 6 No. 6 June 1999 Burm Goedeer and cram 65 H 5 RT T e ion 4 m a resectively ehrt T e. 5 Dalton s law in non-lte reads n a n i kt n e n kt T e. 6 Dalton s Law and quasineutrality togeter wit te aa balance 9 now gives b 1 n e 1n e 0. olving tis quadratic equation in n e we get b 1 or b Taking te derivative of te ionization degree wit resect to te electron temerature T e leaving ressure constant as we did before in exression 15 yields T e T e ion. 30 To find exression 30 we used exressions 10 and 9. Note tat te nonequilibrium arameters b 1 disaears in exression 30 since we made use of exression 9. In a similar way we find T e T e ion. 31 Hence te eat caacity at constant ressure becomes in non- LTE c H T 5 R R resectively te eat caacity at constant volume c V e T 3 R R 3 ion 1 ion Note tat te nonequilibrium arameters b 1 remains to be disaeared. For analyzing te isentroic exonent in non-lte we use T e RT e T e 1 RT e 1 T e and from exression T e Te isentroic exonent exression 5 for monoatomic non-lte lasma becomes c 1 c V Notice tat te exressions for te eat caacities and te isentroic exonent are identical to te corresonding exressions in LTE wen is set equal to one. Like before we can calculate te isentroic exonent from exression 34 wit use of te above exressions for c and c v. C. Te diatomic lasma in non-lte In tis section we consider a diatomic lasma e.g. ydrogen wic deviates from te two tyes of equilibria i.e. deviations from LEE and LE. Te exressions for te secific entaly and te internal energy exressions 4 and 5 do not differ from te monoatomic case after rotation and vibration are added H 5 RT T e RT ion 35 m a resectively ehrt T e. 36 Dalton s law in te diatomic molecular case differs in te number density n a n i n m kt n e. 37 Dalton s Law and quasineutrality togeter wit te aa balance gives an exression similar as exression 8 before for te ionization degree b 1 Also exression 9 is similar b 1 1 n kt m e. 38 n m Taking te derivative of te ionization degree wit resect to te electron temerature T e leaving ressure constant resectively density yields again similar exressions as in te monoatomic case. Exression 30 resectively 31 becomes Downloaded 1 Mar 005 to Redistribution subject to AIP license or coyrigt see tt://o.ai.org/o/coyrigt.js

6 66 Pys. Plasmas Vol. 6 No. 6 June 1999 Burm Goedeer and cram T e T e T e T e kt n m ion 11 3 ion kt n m T. T Hence te eat caacity at constant ressure becomes in non- LTE c 5 R R R 5 ion 5 ion 1 1 kt n m T resectively te eat caacity at constant volume c V 3 R R R 3 ion kt n m T. ion 4 43 Te isentroic exonent exression 5 for a diatomic non- LTE lasma equals te one for a monoatomic lasma c 1 c V Again we can calculate te isentroic exonent from exression 44 wit use of te above exressions for c and c v. FIG. 1. Electron density (m 3 ) as function of te electron temerature (K) (b and 1.0. In te simulations te relation between te electron temerature and te electron density must be calculated first by solving te aa balance and te ressure balance simultaneously. To solve tese equations in wic te ressure is assumed to be given te bisection metod is used. Te result is deicted in Fig. 1. Te results for te isentroic exonent from te comuter calculations are deicted below in Figs. 3 and 4. We conclude from Figs. 4 tat te isentroic exonent of an argon lasma is almost always equal or close to Under rater extreme lasma conditions like very low (5% and very ig (80% ionization degrees te isen- IV. REULT From te exressions for te eat caacities and te isentroic exonent in te revious section it is noted tat tese variables are only functions of te ionization degree te ionization energy te electron temerature te ratio of te eavy article temerature and te electron temerature and te mass via R but not of te ressure. In tis aer we used an alternative descrition in wic te lasma ressure te ionization degree or electron density and two nonequilibrium arameters b 1 and conform cram et al. 7 are used. In te alternative descrition we make use of te lack of ressure influence on te eat caacities and on te isentroic exonent and of te relatively weak deendence of te isentroic exonent on te electron temerature togeter wit te small range of relevant electron temeratures over wic a lasma can exist. In comuter calculations considering ig density argon lasmas we found tat te isentroic exonent ardly deends on te ressure and te nonequilibrium arameter b 1. However some deendence on te ionization degree and on te nonequilibrium arameter were found. FIG.. Te isentroic exonent as function of te ionization degree to or ionization ratio i.e. n i /n s ) bottom and te ressure Pa (b and 1.0. Downloaded 1 Mar 005 to Redistribution subject to AIP license or coyrigt see tt://o.ai.org/o/coyrigt.js

7 Pys. Plasmas Vol. 6 No. 6 June 1999 Burm Goedeer and cram 67 FIG. 3. Te isentroic exonent as function of te ionization degree to or ionization ratio i.e. n i /n s ) bottom and te nonequilibrium arameter ressure Pa b FIG. 4. Te isentroic exonent as function of te ionization degree to or ionization ratio i.e. n i /n s ) bottom and te non-equilibrium arameter b 1 ressure Pa 1.0. troic exonent may increase or decrease somewat but in general te 1.16 value is a very good first estimate. Wen we consider oter lasmas tan te argon lasma only te ionization energy and statistical weigts differ in te exression for te isentroic exonent subject to situations in wic 3 ion kt n m T 3 ion 45. Exression 45 will not old in lasmas tat are dominated by wall association or dissociation. It olds only for lasmas wit a relative small molecule density. ubject to exression 45 all lasmas give similar result as exressed in Figs. 4 for te argon lasma. Only te value of 1.16 as an estimation of te isentroic exonent in all lasma conditions must be adjusted but can be exected to be close to 1.16 in most cases. Te ionization energy ranges tyically from 13.6 to ev for ydrogen oxygen nitrogen and argon atoms resectively. Terefore an isentroic exonent of 1.16 is a reasonable value for common lasmas used in daily ractice. V. CONCLUION Te isentroic exonent for a lasma as been derived and its beavior as a function of te ionization degree te ressure and te nonequilibrium arameters b 1 i.e. wit resect to te eavy article density and i.e. wit resect to te temerature ave been investigated. Te isentroic exonent of a lasma is lower tan tat of a gas wic is due to ionization. For ionization degrees between 5% 80% common lasmas of about 1 ev ave an isentroic exonent of R. Matsuzaki Jn. J. Al. Pys. Part K. C. Hsu and E. Pfender Plasma Cem. Plasma Process H. N. Olsen in Pysical Cemical Diagnostics of Plasmas edited by T. P. Anderson et al. Nortwestern University Press Evanston M. Mitcner and C. H. Kruger Partially Ionized Gases Wiley New York J. C. M. de Haas Non-equilibrium in flowing atmoseric lasmas tesis Eindoven University of Tecnology J. A. Owczarek Fundamentals of Gas Dynamics International Textbook Comany cranton D. C. cram J. C. M. de Haas J. A. M. van der Mullen and M. C. M. van de anden Plasma Cem. Plasma Process s Downloaded 1 Mar 005 to Redistribution subject to AIP license or coyrigt see tt://o.ai.org/o/coyrigt.js