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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 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 ^) / 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 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 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 ^ 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 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+- , a*," (8) tf*> = 211/(1-003^) bd bs2n/(l-cos£x) 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** * 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 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. For a mixture of species thennalized around their individual temperatures the electron-ion term jn (3) is evaluated by Wilkins and Gyftopoulos using the following approximative expression: 6(m,n(m,n)) = JJ__ [/f ofo u2u2 s s d - (20) Sfel " n2 J e eeeim n e where the 'momentum collision frequency" vgi in the approximation vg » v; and m «. m. is: e i 2 v 2i. == n.vn.v e JI (1-cou -cos X)x; ao\,e. . (v(v„,xe,X) ad "S (21} Following Trubnikov in eliminating the divergence by choosing the minimum value of the deflection angle to be the ratio of the impact parameter for X = 3 and the Debye length: 3 .1/1 2 T T y b(n/2) i /I ec + i• _^ec \I ,__* inin D " n 2 fm n\ ; T'3/2 „l/2 t ,,, f -u >>"> » 2 ,f J, T3/2 in A / e e u S S Ai (23) ' ne e m ' k3'2 e J e m n e o e A= 1 - cos X. <»> min An electron-neutral atom term may be evaluated as in [*] based on the same approximations used in the electron-ion term: 10 m»ni - A. /"f°,,2 ,eo " 2 / Je -e e ve o Sm 5n d ve 1/2 2 "0 , /2"P \ /" -u s * V ^ Hf) 7 e "J S-SnQeo (ve)due «» where the total cross section is: s2n (1 cos X) o-eg (ve,x) sinX dx (26) (M / - The temperature dependence of expression (25) is determined by the presence of a Qeo(v ) function in the integrant summed over a u ~T~ ' v variabJe, as results from approximating Q with a sum of segments of constant slope v s 3Qeo/ove and observing that 3Q /^ e > independent of temperature contrary to aQeo/3ue, and that Sdv = a? du: /"dQ Qeo(ue) = Qeo[Ue(Ve)] = ^ + j ~*f dVe 1/2 2kT (27) - £ m . m) , (-,) s = o where: /v = u Q°sQ„ \ e e s„ - 1 are the number of fracture points of Q (uj along the ug direction and u M may be finite or infinite. In this case (25) beccmes: II 1/2 1 / 2kTe \ (m,n) ^ % -372 (-Sj—1 e,eo n n \ e / / \ s+1 e"u u5S S du 5 v S S+1 m n s = o ' . J if 1/2 S+1 2 <- e_u u6 S S du (28) m n v For a constant slope ^Qeo/^ e along the practical interval, expression (2S) becomes: fm,n) no 1 /2kTe\'/2 />2 5 < , opM%i n du (29) The values of Q and 3Q/ 5v for various values v of the variable v, respectively, u of the variable u are independent of temperature. The two integrals in (28) depend on temperature through their limits according to the transformation I ™ \1/2 U =(2kT ) V _ 2 , and may be integrated analytically leading to products of e" and polynomials in u . For the first two Sonine polynomials 5 , S, , the integrals in (28) are: u 5 u 5 je" u SQSQdu = /e" u ciu /e-Vs^d^/e-^f-u2). du f 2 C 2 j e"u u^du-J e"u u5 (| - u2) du 12 2 u2 2 4 8 Je"" u& du = - e' (l2 + 12u • 6u + 2u° • \ u ) u2 2 4 6 /e^Vdu = -e" (3 + 3u +|u tiu ) u2 5 u2 2 /e- u du = -e- (l+u + iu*) u2 6. /e-VsoSodU = /e- u du du 2 f^u's^to =/e-u2u6(f - u2) du -u2 10. g!! / -u2 -u2g!! / 2 3 22 5 23 7 2* g\ / e u du=£_ye du.e Ji_^u + _u +_u +_u +^|USj u2 8 7!! 2 u2 7 5 3 7 5 3 '/„- e .,uw d„u = —j - L-"/e ^,d, u - =-e /I "—v'—,. u + "—s -.. u +7 -= • 5u +1 ? 7u\ 1 / 2* J V 2* 2J 2^ J / I -u2 6. 5.'! / -u2 -u2 /5-3 ,53 1 5\ (30) ye udu= -5 je du-e |-j u.-j u + j u j They can be used for evaluating the second order approximation of the mobility p ryi as shown in the next paragraph. In conclusion the density and temperature dependence of the 5e ' matrix element, as resiilts from expressions [(18), (8), (16)] , [(23), (24), (22)] (25) and (2S) may be summarized as follows: 13 (m,n) _ (m,n) , "j M,m,n) , T t + — #m,n) /T ) (31) d T + d + lJ ^e * e,ee (V eJ ne e,ei (V e''i) ne °e,eo ('«/ " (m,n) -3/2W,. , ni -3/2 &m,n) , . "o -1/2 fcm.n), . .„, 6e =Te 6e,ee (ne'Te)+F;Te 6e,ei (ne'Te'Ti)+ r£ Te fie,eo (Te) (32) A A A where the 6 , 6 . and 6„„„ functions are defined to be weakly e.ee e.ei e.eo ' dependent of densities and the <5 , 6 . and 6 functions are defined to be r e,ee e,ei e,eo veakJy dependent of temperature; in particular <5 is independent of temperature for a hard sphere cross section. According to their definition: * ,.-3/2 « (33) e.ee "" e,ee e e,ee 3/2 6 = Hi £ s .Hi r * (34) e,ei n e,ei n e ue,ei 6 T 6 (35) e,eo ne e,eo n e e,eo r£ 'e l e,eo * e e,eoj 6 T e,ee (V e) = £ ^ S |(?) Amnr2 + A^ X 3 (36) J e 2 T17C3/m"'k 2 e e t m 3 1 2 o e z8e /yr/ nn v ' 1 - cos kTe I -TeTi k / u2uS u2 2 x /e- J ) Sn(u )du (37) 1* 1/2 1/2 u frm.n) ,„ ,_ 2 k f " e 5 5 / 2\c 2 „ . J 6e,eo (Te) = 73721172-J e ue Sm(ue IVe Qeo(ve) due f38) n m, Sfl l/2 A(rr,n)I,T ,_ 1 /2k\ V e u T 5 du (39) °e,eo e ' 572 nT 2- J mSn 1 e (#1S,S+ L n \ / s-o v. stl &m,n)rt,_ . _ 2k /SQ\ /* -u2 . { 'du (40) 00 /&Q „.\ _ 1 /2k \I/2 ^o /* -u2 : o , ct 5 S du (4 0 ,,«> bv = ) = 372 (mj ^ ye u m nn * e,eo(ff = Ct) = 372 n7 ("if) .„ J e u 5rAdu M) TI ' e ' 'v =o 11 e o where: 1 -5- f - r «; 6e,eo = ° &Te *e,eo ' ' & i-2 A The functions 6_* and 6^ . are weakly dependent on electron density and e^ee e^ei temperature for practical variations of these two parameters due to the logarithmic functionality. It results that in a first approximation in which the logarithmic functions are averaged over the efectron density and temperature rsnge of hterest, Eq. (30) becomes: .(mjn)„ ^3/2 U(m,n) _^i §(mfn) + V2 | /lM e e e,ee n e,ei n e e,eo \ el B(m,n)CS = --3/2 (£ where the averages are effectuated with respect to density and temperature, T1 n cs results from (38) - (' Substituting (31) and (32) in (1) the density and temperature dependence of the unknown functions z when the indices m and n are truncated to n,. has the e,r ivi following explicit form: For a general cross section: e l=o ,,, AL Xrk In e / TJ/2 J£=SL V e ' ( ) = W ; W A j where Y. are the coefficients of (n-Au) in the expansion of the principal determinant 6 ™'n of order n.. + 1 , of the system of Eqs. (1) and Y. are the i HI 1 n n T in the coefficients of ( 0/ e)( e ) expansion of the same determinant after factoring out T" from each element; X |™ are the coefficients of In In \k in the expansion of the determinant of order, n,, + 1 , conjugate with the unknown z , in in the the system of Eqs. (1) and X ? are the coefficients of (V"e) (Tjj J372 expansion of the same determinant after factoting out f~ '* from each element which is of the form (31). As previously with the 6 , the single hatted X and Y suggest that they are S A A weak functions of densities, while the double hatted X and Y are functions of temperature primarily through 6_ if one neglects the very weak temperature ft A ' dependence of 6 and 6 e,eo e,ei 16 For a cross section independent of v (hard shpere): e > A(n)Hs/no _2\k rk »HS & >e M e.r n +1 +1 [ M ] "M * s J j=o *r (H n M £(n)CS,l/no _2\ &n)CS,ll/n© _5,'2\" X T X T (n)CS 3/£ 2 E rk \K~e e/ * rk ^ e / z T* k=o <«5) e,r [n l] e M+ n., + 1 ^+ [*CS4 /% 2\ *CS,n fc> 5/2V jto" L» \"e e/ * \ne e /. vn)HS $HS *(n)C1 S #SC where the coefficients X ''"", Y"~ , X '" ', Y""" are independent of temperature. The transport coefficients have thus the following density and temperature dependence taking into account their definition and expression (M); The electron mobility: Z._ d v* 3m n I e e J 1 n=o k=o \ e / H7^ nM+1 i=o "M L MM n n,k=o, y . j l/^V e MA /n\' e k=o \ e / 7 n +I IF ^: M («> j=0 17 The thermal diffusion coeffic.ant: T f k s -L—?£ Zebdve e "e 3mS fe°«e x E E S.xW/jSj 1 3e n n tl "e W^ni e u E A /"oV j=o 1 Se X k=o \ e/ "e VPV /n oVv ' r. Y, r3/2 E ub k k=o (^ *)' (47) *« M?75™. nM + 1 ?-M The thermal conduction coefficient: k2n T K = // e e e e E E Knx<"k> U\ 8k' n=o k=o \ e/ - ^:f| kT)nT TIT e e \2 + 8k' T £M5) - A kl (f +k )nT m' 'Pfcl c e \2 e/ e e 2 bl< n e 8k T5/2 k*o \ e / e n,. + l , j " " Ve"(i+keJneTe(*8> where: 2 :n,/e- ^sn(u) du (Cn = 0 for n / 0 u2 S <„* /e- u Sn(u ) du ( Kn = 0 for n = 2 j (*9) (n) L = fCc XX rk" ^ n rk r = a, b r = b rk ~ r. rr k L i = E C X1"' r = a,b rk *-> n rk n 0 r =b (50) Mi k. =T* - Kn X r?k' M A, 5 2 rk • k ' ) *S M A - V'2) Y. (5 0 19 III. ELECTRON TRANSPORT COEFFICIENTS OF A CESIUM PLASMA Electron transport coefficients for a hard sphere cross section are derived by Wilkins and Gyftopoulos. In the present paragraph the electron transport coeffici ents are evaluated for a cesium plasma for an experimantal (Nighan) and a theoretical (Karuli) electron-neutral cesium elastic cross section. The evaluation is effectuated to the third order approximation thus n = 0, 1, 2; m = 0, 1, 2; n., = 2 in (30) - (51). The two functions z^fji ar>d 2^31 corresponding to the general expression (1*3) are: . f MM 2 >)/no\' (h) k=0 ^ ^ , . . mij=o where the temperature dependent coefficients are: X(°j = (/?ee) + (/Jei) + (/Sie) + (/3ii) X^ s(j3eo) + (|3io) + (0oe) + (/3oi) X^^oo) A(il ; xyo = (ege) + (e/Ji) + (i/3e) + (1/3 i) A(0 Xal = (e/?o) + (i/3o) + (o;Se) + (o/Ji) X^ = (oflo) A(2) Xao' = (ee3) + (ei/3) + (ie/3) + (i.i/3) X'al' = (eoj3) + (io/3) + (oe/3) + (oi0) X The coefficients Xj^ ,.... XJ^' have similar expressions with X^ , .... X^2j with the exception that replaces A YQ s (eee) + (eei) + (eie) + (eii) + (iei) + (iie) + (iii) 4 (iee) A Yj ^- (oee) + (oei) + (oie) + (oii) + (eeo) + (eio) + (ieo) + (iio) + (eoe) + (eoi) + (ioe) + (ioi) A Y = (eoo) + (ioo) + (oeo) + (oio) 4. (ooe) + (ooi) Y3 = (000) (55) The following notation is used: AAA AAA AAA llrtf) = dfo AAA AAA AAA" ^20^11^02 _ ^00^21^12 _ ^10^01(5^22 (J6) together with: *e,e„ 9*P ,y A Expressions for pm , Tm =>re given in (3) and are adimensional. The resulting transport coefficients defined in (46) - (48) and in the additional relations (49 - (51), are evaluated as follows: ? * "O A K\2 , Lao + Lai n~ + La2 iT J "*] • iJk^ : i-' '•» BU e &•*$' 21 L + L L DboO bDl l iT* Db2(in g T 1 Se "\ e/ (59) k *[3]-*e[ e 3^172 e e ^bo+ "«bl n k T ^) e e 1 Se Ke[3] = "e -T- jT —172, 7T n m A A n^ A /n \ A /n \ !(| • kj) (SO) *:(' where according to (50) and U9): ALrk = Co^rk (Cn = ° f°r n *)° *»rk = Ko *K + Kr *rk (K2 = °) They are completely determined by the set of 13 independent coefficients Lao '"" Y3 wnicn are Siven on FiS- ' and Table 1 as function of temperature for the best guess and theoretical cross sections indicated on Fig. 2. As indicated in the preceding paragraph, and as resuJts from expressions !*9) - (50), for a hard sphere cross section the 13 coefficients in (58) (60) are by definition independent of density and depend on temperature as follows; t» = (T-3/2)V)k £™ «tHK*T = 0 2 k L™ = (r^2) (T2, £HS d%m/dl = o MS= (T^)2^jk fl™ S 3 2 2 Y» = (r ' ) (T )" ?«s ^H/aT = 0 22 It is sometimes interesting to find out what influence the electron-neutral elastic cross section has on the mobility. As even a second order approximation expansion of the functions 2_ r in (2) gives satisfactory results for the mobility, it is e,r sufficient for this purpose to evaluate z ryi and z r-T from (I) which yields: li z° e°6 e,a[2j 6°° 6U - 610 o01 10 1 0°6 z n - e,a[2] - 6°° 611 - 510 601 and to use the poligonal approximation of the cross section expressed in (27) and the integrals expressed in (30) in order to evaluate 6 from (28), (35), (39), (40) and thus Z from (2) and the mobility from {lid). The resulting mobility is an explicit expression of densities, temperature and slopes of the cross section. The transport coefficients expressed in (58) - (60) depend primarily on the three independent parameters n , n , T . They are represented on Fig. 3 versus electron temperature 500 K £ T £ 3500 K for fixed values of the electron and neutral / 15 -3 13 -i\ particle density In = &*10 cm ,n =8x10 cm j. In particular the electron mobility in (58) varies as follows: "ene = f fe • '•) -.2 thus at each T on Fieb . 3 and for n n = 10 the mobility scales as 1 n . e o e ' e The actual distribution of densities and temperatures in a real converter results from conservation equations solution and clearly, due to the spatial inhomogeneities and lack of interspecies thermalization maintained by the boundary fluxes, their values cannot be predicted by equilibrium equations as perfect gas and Saha. Figure 4 shows how the set of variables chosen as input for the results presented on Fig. 3 differ from the perfect gas and Saha predictions. The solution of 23 the transport coefficients versus distance from emitter in a thermionic converter is presented on Fig. 5, based on an example of integrated plasma parameters provided by Dr. C. C. Wang and shown on Fig. 6. IV. ELECTRON TRANSPORT AND PLASMA RESISTANCE IN TYPICAL THERMIONIC CONVERTER CONDITIONS The electron particle and heat flux result directly from their definitions, once the transport coefficients are known: ^ = e / v"efed v^ = - /Je (grad p& + enjf + k^ ngk grad TeJ (62) -» mV 3 5 *U T 5 % - -J Vefed V^| kTe + -I S kTe - Ke grad Te (63) They depend on transport coefficients and on gradients of the electron pressure, temperature and electrostatic potential and are illustrated on Fig. 7 as function of the distance frorrrthc emitter for the conditions shown on Fig. 6. As seen on Fig. 7 there are two main contributions to the electron current density, one attributed to the pressure gradient, the second attributed to the electric field. The plasma resistivity is related to the conductivity as follows: Pe = V^e^e (M) and is represented on Fig. 8 for the same conditions described on Fig. 6. Its spatial variation is determined by the spatial variation of the product electron mobility times electron number density which results from expression (58). The negative effect of the neutral cesium atom to the electron passage through the plasma may be isolated by evaluating the mobility in the presence of neutrals, nQ 4 0 , and in their absence, n = 0 , in expression (58). The resistivity in the absence of neutrals is due to the electron-electron and electron-ion long range electrostatic interactions, and the values shown on Fig. 8 coincide within interpola- tion errors with the values obtained by Spitzer. 2* Av Te -- 3000°K (Pe )n 4 a/(fie\ = 3.667 / no = ° andatTe = 2500°K (^) yfe) = 2.0 The resistivity of a fully ionized gas is given in order to serve as reference to the resistivity of a partly ionized gas, and it does not mean that in converter conditions a fully ionized gas can exist at the considered temperatures. V. CONCLUSIONS The electron transport coefficients in a cesium plasma ar<_ expressed analytic ally as function of density and temperature dependent coefficients which are numerically tabulated as function of temperature for an experimental and a theore tical •electron-neutral elastic cross section. They are used to evaluate the pjasma resistance to the electron transport across it. According to (62) and (6*), for giver gradient of electron pressure, electron density and electric field, the resistivit> determines the electron current density and also influences the current indirectly through its effect on pressure and electric field. It reilects the interaction of the electrons with various species (ions or neutrals) and evidences their individual influence on the electron current density. Setting n = 0 in Eqs. (58) - (60) leads to the transport rc-iiVjents in the absence of neutrals and shows how much penalty one has to pay in electron current resulting from the cesium presence in the inter- elec trode distance, for the advantage of lowering the cathodes work function through cesium adsorbtion and on the surface. For given boundary conditions at the electrodes, interelectrode spacing and pressure, the solution of the transport equations and Poisson equation which yields the spatial distribution of the density, electric current, temperature and electric field, is unique. The optimum thermionic converter is the one which has the lowest 25 internal voltage drop defined as the work done by a unit charge to cross the iiiterefectrode spacing, which is equal to the line integral of the electric field. This optimum is therefore reached by varying the interelectrode spacing, pressure and physical boundary surface conditions. Improving the converter performance beyond the optimum obtained as described above, rests therefore on the capability of altering physical conations which are reflected in terms of the conservation equations and Poisson equation, as, for instance additional ionization rate, or as additional boundary conditions generated by supplementary electrodes. In any case an accurate solution of transport equations is needed in order to analyze the physical model and seek beneficial solutions. The transport coefficients treated in the present work m^ke possible a more accurate integration of transport eqiations at a given level of computational complexity. ''"nere are two main difficulties encountered in the mathematical modeling of a converter. One is related to the fact that one cannot discriminate experimentally the electrode work function from the internal voltage drop, as the work function in converter operation conditions which produce cesium adsorbed layers on the electrodes, is inaccessible to experiments. The second is a result of the invalidity of the transport equations in the sheaths regions neighboring the electrodes due to the assumption of close to equilibrium (Maxwellian) velocity distribution functions, which is implicit in the expressions of the transport coefficients. The present effort helps to analyze the contribution of the plasma region described properly by transport equations, to the converter performance. A complementarv effort is felt necessary to analyze the sheaths and the electrode work function in converter operating conditions. 26 ACKNOWLEDGMENTS The authors gratefully acknowledge the contribution of Timothy M. Smith and Stephen R. Channon to the Coulomb collisions evaluation. 27 REFERENCES 1. D, Enskog, Kinetische Theorie der Vorgange in ma^ig verddnnten Gasen, Tuaugurai Dissertation, Uppsala (1917). 2. S. Chapman and T. G. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press (1939). 3. J. O. Hirschf elder, C. F. Curtiss, R. B. Bird, Molecular Theory of Gases and Liquids, 3ohn Wiley & Son , (195«. ^. D. R. Wilkins and F P. Gyftopoulos, Transport Phenomena in Low Energy Plasmas, 3. Appl. Phys., 37» 3533 - 35*0 (1966). 5. B. A. Trubnikov in Reviews of Plasma Physics, Consultants Bureau Enterprises, Inc., 1965. 6. W. L. Nighan, Physics of Fluids, ]£, 1085 (1967). 7. E. Karule, "The Spin Polarization and Differential Cross Sections in the Elastic Scattering of Electrons by Alkali Metal Atoms," 3. Phys. B; Atom. Molec. Phys., 5, 2051-2060 (1972). 8. L. Spitzer, Physics of Fully Ionized Gases, Interscience Publishers, 3ohn Wiley & Sons (1962). 28 8-0l* WW m * to ( _ 1 ill 1 in =3 c\i 1- < UoJ: Q_ 2. 0 TE M z S / /g 1 o a: if in i— 1/ A^T/- —: o o o O 1 SS^ D SSI S \s^\s /—LxSr / J^T^T W Rl - _oixqi'Dn Figure 1a Temperature dependent coefficients of electron transport coefficients. 29 Figure lb Temperature dependent coefficients of electron transport coefficients. 30 If) —1— ~ '\ '\ / ' to \ / ~° \\ \ / ~ -^ \ / 0 - \ 1 »< X tf> / X o CD t 1 m >- >^" 1 >- (B G < \ >- PU IJ O J \ ^ UJ BG ) m •—• // 1 O •*"" II I O X. \ - X o v / t^ X I CM X 1 s / 2 \ 1— \ //— >- io l / "^ L >- J /r J/ o**• ' X ' X \ m —" 1 1 1 O O ID 9 o o c\l in Figure '-c Temperature dependent coefficients of electron transport coefficients. 31 - to ^r a LU o LU O. t/> ITS 2 8 O cc in \- <\i UoJ o _) UJ in o o ir> { Z"J3) N0I103S SSOHO Figure 2 Electron - cesium atom elastic cross section i 32 — r~ 1 | 1 1 - 0. 8 to t-_ to to - 'E o - o O X m - a> CO _ oCO II it W O IO. 1 -I 1 1 1 1 J. 1 A ocncor^ u) m Figure 3a Electron temperature coefficients versus ekectron temperature Tj =T = 2000 °K >(torr) 0 1.69 i.70 1.71 —i— "~i Tj =T = 1000 "k p(torr) 0 085 0.8S tt87 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3 I0 ELECTRON TEMPERATURE Te(°k) Tj = T0 = 2000°k 1.70 0.87 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 3.2 3.4 3 I0 ELECTRON TEMPERATURE Te(°k) PRESSURE = 1.42 Torr 2.0 1 • i ' ~r~ 12.0 '*= • data from Fig.6 1.6 n0Ti = IOOO°K 1.2 OT •"eTi = 2000»K ^^3 0.8 o o h __^n0T| = 2000 K cr i- 0.4 UJ neon Fig.3 — — . _i LU — 2.5 3.0 o I03 ELECTRON TEMPERATURE (°k) m o o o 2 4 6 18 20 22 24 I0"3DISTANCE FROM EMITTER (cm 8 20 22 24 10 DISTANCE FROM EMITTER (cm) l.Br 1 1 1 ( 1 1- 1 1 1 1 i 2 (c) O 1.6 h- " s^* """^""-^ CJ a 1.4 z o o 1.2 \^LB - ,—-—" '——• N 1.0 -^^^^=^. \ ce ~ UJ a: 0.8 ~r/^ ^^^^V 0.6 . o a: * ^^ ---\8G\S 0.4 ^\.^>S 0.2 i i i i i i i i i > 0 10 12 14 16 18 20 22 24 I0"DISTAI\ICE FROM EMITTER (cm) 2.9- 2.8- E27' ^2.6 5 2.5 2.4 2.3 0 2 4 6 8 10 12 14 16 18 20 22 24 -3 10 DISTANCE FROM EMITTER (cm) Figure 6 Example of integrated thermionic converter plasma parameters versus distance from emitter J844iura WOJJ. aouE+sip snsjaA A+isuep 4_uejjno UOJ+33|3 L SJnBiJ ELECTRON CURRENT DENSITY Je (CGS) Ofr 2 4 10 12 14 18 20 22 24 I0"3 DISTANCE FROM EMITTER (cm