Energy Conversion. Vol. 8, pp. 91-95. Pergamon Press, 1968. Printed in Great Britain

Plasma Thermocouple Figure of Merit Based on a Semi-Conductor Thermocouple Analogy

JOHN RANDALLt and RICHARD A, KENYONT

(Received 20 January 1968)

Plasma thermocouples, thermionic diode energy con- verters with positively charged plasma between their electrodes, can be examined from several points of view. The purpose of the plasma is to neutralize the negative space charge barrier that exists between the electrodes and thus enhance the efficiency of the converter while preserving its advantage of having no moving parts.

The efficiency of the high vacuum thermionic con- verter has been examined extensively and the reader is referred to Hatsopoulos [1] for the derivation of this efficiency. This analysis suggests that the simplest way to estimate the efficiency of the plasma thermocouple would be to neglect all space charge terms in the thermionic analysis.

However the very presence of the plasma forces one to conclude that this approach is oversimplified because, by regarding the plasma as a continuous medium between the electrodes, one must also consider, in addition to space charge neutralization, the possibility of thermal and electrical conduction in the plasma. Since there is both thermal and electrical conduction through the plasma, caused by the temperature gradient between the elec- trodes, the Seebeck effect should be considered. The Thompson effect will be neglected throughout the paper because of slight variation of the Seebeck coefficient with temperature.

A partial approach to this problem has already been made by Kenyon [2] whose analysis considers the thermal conduction and Seebeck effects in the plasma but does not consider electrical resistance.

Both analyses mentioned have the further short- coming that they apply only to converters operating over a very small temperature difference so that even their reversible efficiencies are quite small. The purpose of this simplification is to linearize the Stefan-Boltzmann equation for thermal radiation so that the resulting equations for heat and electric current flows may appear in the Onsager reciprocal form, thus allowing application of the figure of merit concept.

The present analysis will attempt to consider the effects of electrical resistance in the plasma and will not necessarily assume that the Richardson-Dushman equa- tion is descriptive of the electric current flux in the plasma. This will be done by treating the plasma thermo- couple in the same fashion as one would treat a semi- conductor thermocouple. Throughout the analysis which

t Comell University, Ithaca, New York. .+ Clarkson College of Technology, Potsdam, New York.

91

, r

I Tcold

/ / Qout LOAD Qout

~VVVVv~

, Tho t

Fig. la. Semiconductor thermocouple.

CONDUCTOR

CA THODE

(+) ~) l (_1 (+)

PLASMA (+) (-)

(+) (-)

ANODE

/ Qout

Fig. lb. Plasma thermocouple.

- - T o l d

follows space charge sheaths which occur in the vicinity of the electrodes when the plasma is present will be neglected. This is somewhat analogous to neglecting junction resistances in a semiconductor thermocouple analysis.

Comparing Figs. l(a) and l(b), the plasma portion of the plasma thermocouple will be assumed to be analo- gous to the P-leg of the semiconductor thermocouple, while the conducting leads of the plasma device will be assumed to be analogous to the N-leg.

The procedure is to optimize the efficiency of the plasma thermocouple with respect to its useful output voltage and its geometry. Both heat and electric current flows must be considered as well as the coupling between: them. This can be taken care of by proper application of Onsager's reciprocal relations, which can be used to relate fluxes of the proper extensive properties (entropy

92 J O H N RANDALL a n d RICHARD A. KENYON

and electric charge) to the gradients of the proper intensive properties (the reciprocal of temperature and the electrochemical potential divided by temperature) in the general form:

J1 : L l l X 1 -4- L12X2

J2 = L21X'I -'}- L22X2 (1)

L21 = L12.

The coefficients L~j are the proportionality factors between the fluxes & and the gradients, or driving forces as they are sometimes called, X~. L21 (or L12) accounts for the coupling between J1 and J2.

Following the procedure of Callen [3], Equations (1) are usually rearranged so that heat flow, temperature, and voltage are considered in preference to entropy flow, the reciprocal of temperature, and electrochemical potential, respectively. Adapting Callen's procedure to the present analysis gives:

F7 ] JQ = -- TS* + K' 7T, (2) P

J 1 = 1 7 v S* - - - -- VT. p P

For the fairly simple geometry under consideration the temperature and voltage gradients may be found by assuming linear variations of temperature and voltage along the flow directions in the system, remembering that the temperature gradient is negative while the voltage drop between the electrodes is positive. This would lead to:

V T = AT AV _ - - - a , v v = - - u, (3) L L

where L is some convenient reference length and fi is a unit vector.

A V, the output voltage of the system, can be multiplied by Jz to obtain the output power of the system, and dividing this product by the heat flowing into the system, Jq, will give its efficiency 7.

gzA V (4) ~/ - - - j o •

Differentiating Equation (4) with respect to the output voltage A V and setting the derivative d~/dA V equal to zero gives an optimizing value of the output voltage of:

AVopt = P-K'-AT [zT + 1 -- v / (ZT + 1)1, (5) S* T

where" Z = S*~ (6)

pK"

Substitution of Equation (5) into Equation (4) gives:

[ x / ( Z T + 1 ) - -~ ] AT (7)

The term Z is called the figure of merit and accounts for the efficiency's dependence upon system properties, as Equation (6) indicates. At this point the efficiency of the plasma thermocouple has been optimized with respect to its output voltage.

To optimize the efficiency term with respect to the geometry of the system under consideration, the figure of merit Z will be optimized, with respect to system geometry, since the efficiency will increase with increasing figure of merit.

Each term in Equation (6) has to apply to the whole system, which consists essentially of two parts, the conducting leads and the plasma region. The net Seebeck coefficient, due to the net Seebeck effect within the system, is:

S* S* * = P L - S c . (8)

The overall electrical resistivity p can be evaluated by noting that the entire system is in series electrically so that a total resistance based on plasma geometry can be found from:

LPL LPL Lc (9) p ~ --- pPL ~LPL ~- PC A~6"

where pet is evaluated considering multiple angle Coulomb scattering and collisions between neutral atoms and electrons. The reader is referred to Lewis and Reitz [8]. The overall thermal conductivity K' can be evaluated by noting that the entire system is in parallel thermally so that a total conductance based on plasma geometry can be found from:

K' Apt __ • Apt + Kc Ac (10) LPL PL LPL L-g"

Combining Equations (9) and (10) gives:

o K ' = ( p p L + pCr)(RPL + ~ ) , (11)

where:

LcApL (12) r - - - -

L p L A c

That value of r which minimizes pK' yields the value of the figure of merit which is optimized with respect to system geometry

Z = (S*L -- S*)2 (13) [(RpLpPL) 1/2 + (Kept)l~2] 2"

At this point attention should be given to RpL, the equivalent thermal conductivity of the plasma, since it must account for thermal conduction and radiation through the plasma. Thermal reradiation by the plasma is neglected because the operating temperatures of the thermionic device are relatively low and reradiation is not considered significant. If the temperature difference across the plasma is small the total thermal energy transport per unit time through the plasma can be written as:

ApL ]~pLAT ApL = K p L A T + 4¢rFAcT3ATApL, LPL

where KpL is evaluated considering neutral particle collisions and a Wiedemann-Franz electronic effect in the plasma according to procedures outlined by Lewis and Reitz [8].

Plasma Thermocouple Figure of Merit 93

T 1 T T 2 T 3

%

in in+l

n

T C

Fig. 2. Staging the plasma thermocouple.

Now geL becomes:

J~PL = KpL .3f_ 4LpLF.4coT z (14)

and Equation (13) becomes:

is;* - s*]~ (15) Z = {[pPL(KPL ~-4oFAcLPLT3)] 1/2 + (pcKc)l/g) 2'

where S~* is found by use of an equation given by Lewis and Reitz [8]. Equation (15) is the complete expression for the figure of merit for a plasma thermo- couple, based on the semiconductor thermocouple analog. As previously claimed it accounts completely for the efficiency's dependence upon system properties. Since it has been optimized with respect to system geometry, insertion of Equation (15) into Equation (7) will give an efficiency optimized with respect to both output voltage and system geometry.

The previously discussed developments by Hatsopoulos Ill and Kenyon [2] each arrived at efficiency equations identical to Equation (7) but their figures of merit are not the same as Equation (15) because of the more limited scope of their analyses. Adapting the Hatsopoulos figure of merit to the present situation only requires the neglecting of the space charge barrier, yielding:

Z ----- (16) [(2k 2T q- 4(rFAcT4 j~,) I/2 + (pcKc)U2] 9:

The first two terms in the numerator of Equation (16) can be regarded as a vacuum Seebeck coefficient, often called the thermionic power. As expected there are no terms in the equation which account for electrical or thermal conduction in the plasma. The Kenyon figure of merit, however, does consider thermal conduction, but still not electrical resistance, and is given by the following equation:

S* z : [ ~ - s ~ ] 2

APL~ 1/2 (pcKc)l/Z] z" [ Ok 4oFAcT4 + TKpI, ~pL] + J

(17)

Equations (15), (16), and (17) may all be used in Equation (7) to evaluate the plasma thermocouple efficiency, provided the thermocouple operates over a very small temperature difference. Since most plasma devices do not operate over very small temperature differences, a way must be found to adapt Equation (7) to a large temperature difference situation. This can be

done by staging the device with respect to temperature by dividing it into several stages operating over small temperature differences. Staging the device will over- rate its efficiency but if it is broken into an infinite number of stages an accurate accounting can be made of the temperature dependence of the properties of the system. This procedure, called infinite staging, is presented by Sherman, Heikes, and Ure [4], and will be outlined briefly here.

The plasma diode is broken into several stages as shown in Fig. 2. For each stage one can write,

qi ----- P~ + q~+l ----- 7/~q~. -k qi+l. (18)

Beginning with stage 1 and going through stage n by repeated use of Equation (18) gives:

qn+l ---- q~ [I (1 -- ~). (19) 1

If there are enough stages so that the temperature drop over each stage is small, ~, can be estimated from Equation (7) as:

T , - T,+I [~/(ZT, + I ) - 11] '~ ' = - T , L ~ / ( 2 ~ r ~ - ( : l ) - "

As n approaches infinity T ~ - T~+I becomes dT, T~ becomes T, and qn+l becomes the rejected heat of the plasma diode while qt is its added heat. Now Equation (19) would become:

2'H

(20) ' / 'C

and noting that the overall efficiency ~ is

~ 1 -- QO~/T (21) QI~

there results:

n = 1 -- exp

TH (22) H / ( z r + l) + 1I , j

TC

At this point comparisons among the analyses re- suiting in Equations (15), (16), and (17) could be made using Equation (22) but this would be incomplete because there would be no common basis of com- parison to use as a standard. To overcome this problem, a computer-simulated engineering model has been proposed by Kenyon [2] which crudely estimates the plasma thermocouple efficiency without using the figure of merit concept. The model accounts for the

94 J O H N R A N D A L L and R I C H A R D A. K E N Y O N

where:

useful output voltage of the system by subtracting the 24 voltage drop through the leads from the difference in Fermi levels of the electropes. The heat supplied to the device is due to electron cooling, thermal con- duction and radiation through the plasma, and thermal conduction through the conducting leads. The resulting efficiency expression is: g

"cLc~] [ c(VF + ~ + 2kT,) ---- VF -- JApc Ac ] J [

- JA(VF + ¢ + 2krc) + ,,F~,c(r~ -- r~)

+ KPL(TR -- To) q_ , (23) LPL APLLC "~ S I

( + +], Jc = AT~ exp ) ~ - /

- ¢ J A = AT~exp (~-T-c).

The efficiency of the plasma thermocouple as given in Equation (23) is optimized with respect to the load voltage V given by:

V -~- VF - - J A p L p o l e (24) Ac

by incrementing VF so that a plot of ~7 vs V is obtained. From this can be taken the maximum value of 7. Geo- metric optimization is accomplished by use of the following equation due to Randall [5].

A p L L c _ ( pPLKC ~ 1/2

AcLpL .p,(KpL -f- aFAv(T~ + T~)(rn + T~)/ (25)

The only geometric dimension which needs to be speci- fied is the electrode spacing LPL. All electrical resistivifies and thermal conductivities were estimated at the arith- metic mean temperature of the device, leading to some inaccuracy in the use of Equation (23).

Figure 3 shows how the overall efficiency as evaluated by the four methods discussed varies with electrode spacing. Tantalum electrodes were used with a cathode temperature of 3200°K and an anode temperature of 2000°K, with cesium plasma undergoing resonance ionization between the electrodes.

As Fig. 3 quite clearly shows, none of the figure of merit analyses follows the engineering model over the entire range of spacings considered. The Kenyon. figure of merit, accounting for heating losses in the plasma due only to thermal conduction and radiation, increases with electrode spacing because temperature gradients across the plasma become less severe with larger spacing. The figure of merit in Equation (15) decreases with electrode spacing primarily because of its inclusion of electrical resistance effects. Since thermal and electrical conduction follow similar laws, the figure of merit in Equation (15) would be independent of spacing were it not for the consideration of thermal radiation. The Hatsopoulos figure of merit is spacing independent (when space charge is ignored) because it does not consider any plasma transport phenomena.

I I

,~ Randall (Eq. 15)

(Eq. 16)

Kenyon (Eq. 17)

Eng:tneering Model (Eq. 23)

I I : I • 002 .02 .2 , ~.0

Electrode Separation (Centimeters)

Fig. 3. Thermal efficiency variation for several figures of merit at varying electrode spacing.

Using the engineering model as a measure of trends and not absolute values, the use of Equation (17)in Equation (22) yields similar efficiency trends for small spacings while use of Equation (15) seems to giye better correlation for larger spacings. Use of Equation (16) is, of course, an oversimplified approach to the problem. Either Equation (15) or (17) appears to be but a half- extension of the figure of merit concept to plasma thermocouples, at least for the range of spacings con- sidered. At extremely small electrode spacings, where the spacing approaches the mean free path in the plasma, the entire discussion of transport parameters becomes somewhat meaningless.

In the preceding paragraphs the authors have de- veloped the figure of merit concept for a plasma diode operating over a differential temperature gradient and have indicated that bY means of infinite staging it may be extended to use over finite temperature ranges. Considerable work yet remains to verify the quantitative value of this analysis.

I QIN

QovT J~

xi Jo J1 T P S* VV VT

Nomenclature

electric current rate of heat addition rate of heat rejection flux of an extensive property Onsager phenomenological coefficient gradient of an intensive property heat flUX electric current flUX absolute temperature (degrees Kelvin) electrical resistivity (dverall) (D-cm) Seebeck'coefficient (overall) (V/°K) voltage gradient (a vector) temperature gradi~fit (a vector)

Plasma Thermocoaple Figure of Merit 95

g I

A T A V Z sL

P P L

Pc LpL

A P L

Lc

Ac Rm~ Kc~

F

KpL cf

FAC

k J, q~ qi~l P~

T~ Tl~t

thermal conductivity (overall) (W/cm-°K) temperature drop between cathode and anode voltage drop across load figure of merit plasma Seebeck coefficient conducting lead Seebeck coefficient (neglected in calculations) plasma electrical resistivity conducting lead electrical resistivity electrode separation electrode surface area length of conducting leads cross-sectional area of conducting leads overall thermal conductivity of plasma thermal conductivity of conducting leads dimensionless geometric parameter plasma thermal conductivity Stefan-Boltzmann radiation constant gray body angle factor between anode and cathode anode work function (4.19 V for Ta) Boltzmann's constant saturation emission at temperature T (A/cm z) heat delivered to stage i heat rejected by stage i power delivered from stage i efficiency of stage i upper temperature of stage i lower temperature of stage i

n total number of stages Tc lower temperature of plasma diode (anode

temperature) Tu upper temperature of plasma diode (cathode

temperature) Jc saturation emission at cathode temperature JA saturation emission at anode temperature J net flux of electric current through system

(Jc -- JA) VF difference between Fermi levels of cathode and

anode thermal efficiency

V useful output voltage

References [I] G. N. Hatsopoulos, Direct Conversion of Heat to Electricity.

Chapter 3. Massachusetts Institute of Technology (1960). [2] R. A. Kenyon, Optimized Figure of Merit for a Plasma Thermo-

couple. Ph.D. Dissertation, Syracuse University (1965). [3] H. B. Callen, Thermodynamics. Wiley, New York (1963). [4] B. Sherman, R. R. Heikes, and R. W. Ure, Jr., J. appl. Phys.

31, 1 (1960). [5] J. D. Randall, Optimized Figure of Merit for a Plasma Thermo-

couple Operating over a Finite Temperature Range. M.S. Thesis, Clarkson College of Technology, Potsdam, N.Y. (1967).

[6] W. B. Nottingham, Direct Conversion of Heat to Electricity, Chapter 8. Massachusetts Institute of Technology (1960).

[7] L. B. Robinson and R. C. Beaty, J. appl. Phys. 36, 142 (1965). [8] H. W. Lewis and J. R. Reitz, J. appl. Phys. 30, 1838 (1959). [9] W. H. McAdams, Heat Transmission. McGraw-Hill, New

York (1954). [10] G. N. Hatsopoulos, J. Kaye, and E. Lanberg, Direct Con-

version of Heat to Electricity, Chapter 4. Massachusetts Institute of Technology (1960).