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Analysis of Electron Transport in the Plasma of Thermionic Converters

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Analysis of Electron Transport in the Plasma of Thermionic Converters [••iw'-'^-Tr-' "'"^ •*"$J?,«*WS' •^••"w**" Analysis of Electron Transport in the Plasma of Thermionic Converters Maria L. Stoenescu and Peter H. Heinicke Plasma Physics Laboratory, Princeton University Princeton, New Jersey 085<f<* ABSTRACT Electron transport coefficients of a gaseous ensemble are expressed analy­ tically as function of density, and are expressed analytically as function of tempera- & ture urp to an unknown function 6 _ which has to be evaluated for each specific e,eo electron - neutral atom cross section. In order to complete the analytical tempera- ture dependence one may introduce a polynomial expansion of the function 6 _Qt or, one may derive the temperature dependence of a set of coefficients, numbering thirteen for a third approximation transport evaluation, which completely determine the transport coefficients. The latter approach is used for determining the electron transport coefficients of a cesium plasma for any ion neutral composition and any temperature between 500 K and 3500°K. The relation betweea the transport coefficients of a fully and partly ionized gas is readily available and shows that, in the classical formalism, electron-ion and electron-neutral resistivities are not additive. The present form of the transport coefficients makes possible an accurate numerical integration of transport equations eliminating ten0.hy computations which are frequently inaccessible. It thus provides a detailed knowledge of spatial distribution of particle and energy transport and makes possible the determination of one of the three internal voltage drops, surface barrier, siieath and plasma, which are linked together experimentally by current density versus voltage characteristics of thermionic converters. 2 I. INTRODUCTION The advantage of introducing cesium vapor in the thermionic converter which improves the thermionic emission by lowering the emitter work function is counter­ acted by the increase of plasma resistance due to the presence of neutral particles. Optimization studies of current density at given interelectrode electric potential difference based on variation of geometrical factors, additional electrodes, additional ion sources, etc., require a complete and flexible analytical model which can produce adequate information for theory experiment correlation analysis. The current analytical description ai the thermionic converter behavior is based on the numerical solution of the transport equations in the plasma region, on simple model extrapolation of the plasma parameters through the sheath and on Poisson's equation solution for the electrostatic potential in the sheath. The solution of the transport equations is determined by the boundary conditions at the plasrna-sheath interface and by the transport coefficients, which reflect local processes. These transport coefficients are calculated solving Boltzmann equation for a specific form of the velocity distribution function assumed to be a perturbed Maxwellian where the perturbation is linear in the gradients of the macroscopic variables, as introduced by 12 1 Enskog and Chapman and Cowling and presented in detail by Hirschfelder. In the following two paragraphs a parametric representation of the transport coefficients versus density and temperature, for a given electron-cesium atom elastic cross section is provided. A similar representation, for a hard sphere cross section, was offered by Wilkins and GyftopouLos. The present formulation makes possible a numerical integration of the transport equations, for realistic cross sections, updating the transport coefficients in space and time with local values of the density and temperature. In paragraph k an analysis of the electron transport in the plasma of a thermionic converter and of the plasma resistance as a result of electron neutral atom and electron-ion interactions, for typical converter conditions, is presented. 3 The last paragraph summarizes the conclusions and indicates trie areas in which additional theoretical and experimental work is considered necessary. [J. EXPLICIT DEPENDENCE OF TRANSPORT COEFFICIENTS ON DENSITY AND TEMPERATURE The Enskog-Chapman-Cowling ' solution of the lincar.aed Boltzmann equation is appropriate for constructing analytical expressions of the tranport coefficients versui dtnsity and temperature, if the interaction forces are known. Using the additivity property of the linearized Boltzmann equation with respect to the perturbing causes which act against the equilibration of the velocity distribution function, expanding in a polynomial series the unknown functions of velocity of the first order approximation to the equilibrium solution, taking moments of the Boltzmann equation by multiplying it with terms of these expansions and integrating over the velocity space, leads to relations of the following form, as results from the h formalism presented by Wilkins and Gyftopoulos: /^Ve^e = n¥o # ("t ^) /<f"° (V"^) dve (1) where i identifies the species, and z n represent the coefficients of the Sonine polynomial expansion of the unknown velocity functions Z (v* ) of the linearized Boltzmann equation solution which are used to calculate transport coefficients: Vl^-ent^M^K) r=.,b (2, Here ue is the velocity normalized to the electron thermal velocity ^eTM2kVme)1/2- By definition; V=(V2kTe)"27e k Sn M = h (q+j) l<n-j)! The matrix elements: ^,n)syG(m,n) (v^,^)^ have the dimension: ( m n) 1 [6 e ' ] = LV and the le'.t side elements in (1) have zero dimension according '.o their definition: 2 for m = 0 2 -C-/ffM*.-ir;/* KK'"'.- 0 for m/ 0 -15/* form = 1 y> K* (\) *^ir W Kl K - i)S<V 0 formM i» thus the dimension of the coefficients z is: [*] • L-3T The analytical dependence of the matrix elements ° ,n on density and temperature results from the following considerations. Labeling with ee, ei, eo the electron-electron, electron-ion and electron-neutral atom contributions to the matrix element 6* ' , one may writes ,,{m,n) ,(m,n) ,(m,n) _,_ J.m,ri °e,ee H ae,ei + °e,eo W The term s ' is identified in Chapman-Cowling with a sum of two terms: 5 2 6(rn,n) 1 La £o /«. -. , -_ s -' . s -jj' \ s ^ <Td fld v'd'v' e,ee 2/e \ne n ne n / m e° e e ^ I m e n e J e = Jim I S„ u\, Sm ut | + lim [Sn"iT, S_ u" | (5) ._e I n e' m i|ei ._e | n e m e I ^ Where: Z [Sn u*,e' S m uTi J1 . =Jn —n- / /f°fe ?i I(s n ue* - Sn "ue' /^ Sm utg<Tdi& 5id vtdI 7e 1 ei e i J y 2 [ S n ITe ,' Sm u*e 1J . = —n^—j /f°f/ e A°i JSn ue* - JnS ief Jm5 eu*g«Td6 nd iv* d ev* (6) ^nd: — 3/2 -» s u u p = m,n s = e,i P s = V N s According to (9.33,1), (9.33,2) in Chapman-Cowling": m+- n+- ,<i) lSn"l>Sm"ll = 6t>A; 2 M. 2 £ A™nr ° ii (r) (?) I n e m i J . t e *^'- mnr ei * ' Where: a*," <r> • n** y.-e V*? (l - oo.^/ g b d b dT (8) tf*> = 211/(1-003^) bd bs2n/(l-cos£x)<r. (g,X)sinXdX (9) memi ei ~ m + n>j 6 1/2 •Or) and where unlike in (2): 1/2 u V v. e = i 2kT / e «. ^ (50 i the parameter T being the same for all species. The A . are numerical coefficients evaluated in [2]. For electrostatic forces the differential cross section is: Zie 1 (10) € m .s i o ej° (i - cos xr and thus: cos** *<f (g) = 211 ' L sin XdX 2kTeo& It cos xr V Z e r /2 /I i 2n (11) 2 2kTeog where: 2 2kT g0bg" v = (12) 2 o V and where vo l. denotes the upper limit of v o . The finite practical limit assigned to v . in order to eliminate the divergence of the integral (11) is based on the observation that particles farther apart than the Debye length D are less numerous than the closer encounters and do not undergo \ 7 ~Z, binary interactions. 5etting b = D in relation (12) and substituting g by its mean value equal to 2 leads to the following exnression of the cutoff limit of the integral in (ID: ftDkT (13) h<2 The Debye length in a binary mixture is approximately equal to: 1/2 kT D s (if) triZje ne Substituting (11) into (8) and taking into account (13) and (H) yields an expression for £2* (r) of the form: 1/2 Z e °(i)(r) i ei (%r) 2kTe„g' vol(VT) 1 rj^-f' / vo dv o d e6 So** 3/2 = T" F(ne,T) (15) where the truncation of the integral in (11) introduces a supplementary dependence F n ,T on temperature and electron density. The function F varies slowly with temperature and density as may be seen from expression (10) for i = 2 , which is the only value of t, needed in the present evaluation: 8 1/3 J2) _ J2) i ^-"(^M?) AnT + in s"!2 R) »k3T3 z e i nz?.«ne = ^n( Jin 1 (16) ,2KT €og< nZ?e6n„ In a similar way, according to (9.4,15) in Chapman-Cowling one may evaluate the second bracket integral in expression (6): S S 8 n <r) (17) [L nV m"e] . = Z \w* ef ei r,* Combining (5), (7) and (17) yields the electron-electron contribution to the <5e * matrix element; ..(nn,n) ^ m+n+1 fi A + A (18) e,ee = 2- ,(I) ' mnri mnri "tf « r,x. L which for m,n = 0,1,2 as given in (9.6,7) - (9.6,10) of [2] is: S S [ oV mV]e = ° [*ivsi«.] - «42)<2> 2) ( 2) [s^e.SjS;] = 7 ^ (Z)-2 n e (3) [Ve'^e] = T nf(2)- 'n^O)* nl2>W (19) ,(2). where fi (r) is provided by (S) and (16) substituting T with Tg and mei with me/2.
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