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N 7 2 - 11207 


NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 


Technical Report 32-1545 

DEXTER —A One-Dimensional Code for Calculating 
Thermionic Performance of Long Converters 

C. D. Sawyer 



JET PROPULSION LABORATORY 

CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 


November 1 5, 1 971 



NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 


Technical Report 32-1545 

DEXTER— A One-Dimensional Code for Calculating 
Thermionic Performance of Long Converters 

C. D. Sawyer 


JET PROPULSION LABORATORY 

CALIFORNIA INSTITUTE OF TECHNOLOGY 
PASADENA, CALIFORNIA 


November 1 5, 1 971 



Preface 


The work described in this report was performed by the Propulsion Division of 
the Jet Propulsion Laboratory. 



JPL TECHNICAL REPORT 32-1545 


iii 


Contents 


L Introduction 1 

II. Problem Definition 1 

A. Fuel Element Description 1 

B. Physical Model 1 

C. Electrical Model 3 

III. Program Description 5 

A. Method of Solution 5 

B. Capabilities and Restrictions 6 

IV. Examples 7 

A. Variation of Lead Geometry 7 

B. Variation of Lead Arrangement 7 

C. Collector Temperature Effects 8 

D. Central Short 8 

V. User's Manual 10 

VI. Error Messages 10 

Nomenclature 12 

References 13 

Tables 

1. Data table available in DEXTER 7 

2. Net performance of shorted converter 10 

3. Input data for DEXTER 10 

4. Error messages in DEXTER 11 

Figures 

1. Typical external-fueled converter design 2 

2. Analytic converter model 2 

3. Converter electrical model for normal operation 3 

4. Converter electric short model 4 

5. Converter electric model for internal short 4 

6. Converter electric model for emitter-emitter short 4 


JPL TECHNICAL REPORT 32-7545 


v 


Contents (contd) 


Figures (contd) 

7. Converter electrical model for collector-collector short 5 

8. Temperature, current and voltage profiles 7 

9. Effect of lead geometry on converter output 7 

10. Performance of 12.7-cm parallel electron flow converter 8 

11. Voltage and temperature profile for liquid metal cooled converter 8 

12. Coolant and collector temperature profile for liquid metal 

cooled converter 8 

13. Effect of central short on converter temperature profile 9 

14. Effect of central short on converter voltage profile 9 

15. Effect of varying short conditions on converter temperature profile 9 

16. Effect of varying short conditions on converter voltage profile 9 


JPL TECHNICAL REPORT 32-1545 


Abstract 


This report describes a versatile code for computing the coupled thermionic 
electric-thermal performance of long thermionic converters in which the tem- 
perature and voltage variations cannot be neglected. The code is capable of ac- 
counting for a variety of external electrical connection schemes, coolant flow paths 
and converter failures by partial shorting. Example problem solutions are included 
along with a users manual. 


JPL TECHNICAL REPORT 32 -1545 


vii 


DEXTER — A One-Dimensional Code for Calculating 
Thermionic Performance of Long Converters 


I. Introduction 

An early version of a one-dimensional thermionic per- 
formance code has been previously described (Ref. 1). 
This early version was, however, limited in scope to a 
specific converter geometry operating under nominal full 
power conditions and with the converter collector tem- 
perature fixed. A new code, DEXTER, replaces the earlier 
version and is much more flexible in scope. DEXTER 
allows a variety of boundary conditions to account for 

(1) Counter current or parallel electron flow. 

(2) Collector temperature variations. 

(3) Converter failures by partial shorting. 

In addition, the physical properties are allowed a linear 
temperature dependence. Since the running time is ap- 
proximately 30 s per case on a UNIVAC 1108 for a 13-cm- 
long converter, restart capability is built in to allow 
recovery from computer error or to observe the progress 
of the convergence. A sample problem is built into the 
code, so that the user, with FORTRAN NAMELIST input 
need only enter a minimal amount of data to execute a 
problem. 


II. Problem Definition 

A. Fuel Element Description 

Due to the one-dimensional nature of the model in this 
code, it is possible to study a variety of converter geome- 
tries, including both internal- and external-fueled. In 
describing the code, however, attention has been focused 
on the external-fueled converter. 

Current designs of external-fueled thermionic convert- 
ers are typified by Fig. 1. Nuclear fuel supplies the heat 
to drive the emitter. Electron emission aids in cooling the 
emitter, and electrons are collected at the collector. Waste 
heat is removed by means of liquid metal or heat pipe 
cooling of the collector. At one or both ends of the con- 
verter there is an emitter lead to carry electrons from the 
external circuit back into the emitter. 

B. Physical Model 

The converter is approximated one-dimensionally by an 
annular emitter-collector arrangement as shown in Fig. 2. 
The fuel-emitter body is replaced by specifying the emit- 
ter heat input as a function of distance. In the fuel ele- 
ment described above, the emitter fins and outer fuel 


JPL TECHNICAL REPORT 32-7545 


1 



Fig. 1. Typical external-fueled converter design 


= C e (Q ec (i, T e , T c , T cs ) + Q rad (T„ T e ) 

+ Q cs (T e , T c , T cs ) - jv (*)) + it (x) 
-C c h[T c (x)-T co (x)] (3) 

dTo (x) _ Qcjx) 

dx A c k c (T c ) { 1 


dTco (x) 

dx 


C'h 

wC„ 


[Te(*)-T„(*)] 


(5) 


Q in (x) 



The radiation term is determined from a simplified but 
quite accurate formula (Ref. 2): 


Qrad ~ V (CeT* — € c T* c ) 


( 6 ) 




1 

(T e ) 


+ 


e c (\[T7r c ) 


(7) 


1 _ 1 , 1 , (B) 

e e (VT7r"c) + *c(T c ) w 

The cesium conduction term is given by the Kitrilakis- 
Meeker formula (Ref. 3) 


Fig. 2. Analytic converter model 


cladding also can conduct heat and electricity to the ends 
of the converter. Provision is made for this in the analy- 
tic model by providing different effective emitter cross- 
sectional areas for heat and electron conduction. Axial 
fuel heat conduction is neglected, a good approximation 
for most ceramic fuels such as U0 2 . 


Q cs = 1.1 X 10- 


(Tl * ~ T‘-») 


d + 1.15 X 10- 5 


(T e + T e ) 


0) 


„ 2.45X10“ 7 8910\ 

p ' = -^r e n-— ) (10> 

The collector-coolant heat transfer coefficient is taken 
from a simplified correlation (Ref. 4) 


The equations describing the heat transfer processes in 
the converter can thus be written as (refer to nomen- 
clature for definitions): 


h = 


7 k c 

Dca 


(11) 


= C e (Qin (x) ~ Q ec (j, Te, T c , T cs ) - Q rad (T„ Te) 



- Q cs (T e ,T c ,T ct )) + il(x)^^ 

(i) 

dT e (x) _ 

Qe (X) 

(2) 

dx 

A e2 k e (T e ) 


Experience" has - shown" that even - for ^collectors ap- 
proaching 0.3-cm thickness, it is a good approximation to 
neglect axial conduction in the collector. This simplifies 
Eqs. (3-5) to 

^fx ~ = ^ [ Ce (<?ec + Qrai + Qc * ~ iv) + il f] 

(12) 


2 


JPL TECHNICAL REPORT 32 -1545 



Tc — T co + [c. ( Qec + Qrad + Qcs ~~ fo) + ?c 

(13) 

An additional simplification in the case of heat pipe cool- 
ing can be made, namely, 

T c — constant (14) 


EMITTER 


rr 

7 i 

r 

t 

LL 

JJ 

L 

J 


| COLLECTOR 
MP 

a. COUNTERFLOW 


>-■ 1 -* 


EMITTER 


ZZZZ 




COLLECTOR 
PARALLEL FLOW 


Fig. 3. Converter electrical model for normal operation 


The left boundary conditions to be applied to Eqs. (1, 2) 
and (12, 13) are 


C. Electrical Model 


Qe( 0 ) = 0 \ 

T e (0)=T eo ( (15) 

T co ( 0 ) ~ T co o I 

The emitter lead design forms an important part of the 
overall performance optimization of the converter. This 
is because conduction heat losses down the lead are in- 
versely proportional to the lead length-to-area ratio, while 
lead electrical losses are directly proportional to the same 
quantity. In many designs the problem of optimization is 
made complex by the additional ability of the lead to 
radiate to the collector. Lead conduction losses are given 
by 


Q e (l) = h 


[T e (l)-T c (l)] 

\ 



(16) 


I. Normal operation . Two possible converter electron 
flow patterns are considered within DEXTER: counter- 
flow and parallel flow, indicated schematically in Fig. 3. 
In the counterflow case the total current appears at the 
emitter and collector leads on the same end of the con- 
verter. At every position within the converter the emitter 
and collector currents are identical and of opposite sign, 
decreasing to zero current flow at the converter center. 
The voltage and current relations are given by 

™ 

= C e j(v, T e , T c , T cs ) (19) 

i c ( x ) = -i e ( x ) (20) 

The left boundary condition is 


In an attempt to compensate for radiative losses, within 
the spirit of the one-dimensional approach, the second 
term of Eq. (16) is not included in the model used for 
DEXTER. In addition, one can show that if lead conduc- 
tivity k L varies linearly with temperature, the appropriate 
temperature for evaluating this property is the mean lead 
temperature. Therefore, Eq. (16) becomes 


<«( 0)=0 ( 21 ) 

The net power and voltage obtained at the end of the 
emitter lead is 

Pnet = U ( l ) V net (22) 

Vnet =V(1)- i 2 e (l) f> L A (23) 


\ — k L (T L ) 


[T e (l)~T c (l)] 

Qe(l) 


In the parallel flow case, the emitter and collector leads 
(17) 

are on opposite ends of the converter. The appropriate 
relations are 


This equation forms the normal right boundary condition. 


For certain cases involving converter shorts, described 
in more detail below, the short is characterized by a ther- 
mal conductivity. In these cases Eq. (17)‘can be solved 
for k or k/X to determine the appropriate right boundary 
condition. 


dv (sc) 
dx 


Pe(T e y 

A C1 


i e (x)-f c (T c )i c (x) 


(24) 


^■ = C e j(v,T e ,T c> T cs ) (25) 

ic (x) = I - i e ( x ) (26) 


JPL TECHNICAL REPORT 32-1545 


3 


with the same left boundary, Eq. (21), and the additional 
restriction that 

i e (l) = I (27) 

2 . Abnormal operation— converter shorts . A generalized 
circuit diagram involving two series-connected fuel ele- 
ments with the possibility of internal shorting and emitter- 
emitter and collector-collector shorting is shown in Fig. 4 
(only one-half of each converter is shown for a two-ended 
counterflow converter, which is assumed for the base 
case). Here the shorts are all represented by means of 
electrical resistances located at arbitrary positions along 
the converter. The interconnection between converters is 
assigned a lead resistance Rj. 

For an internal short resistance R s , only one complete 
converter need be examined, provided that the change 
in effective load resistance seen by the converter can be 
specified from a data map obtained for normal perform- 
ance converters. The electron flow pattern is indicated 
schematically on Fig. 5. There are, in general, four regions 
of interest in the analysis. The two outermost regions, 
a and d , can be treated as counterflow converters, with 
length as a parameter to be determined. Depending on 



Fig. 5. Converter electric model for internal short 


the overall converter connection pattern, l L and I R or the 
voltages v L and v R may be known, or a combination of the 
two such as 

— h Rl (28) 

may be known. Regions b and c are also counterflow con- 
verters of length to be determined with the right bound- 
ary being 

J*(J) + !„(!)=/, (29) 

v b (l)=v c (l) = l s R s (30) 

The thermal equations also require that T e (0) and Q e (0) 
be parameterized. 1 

For an emitter-emitter short resistance R e «, two com- 
plete converters must, in general, be studied. Figure 6 
shows the breakdown of the problem into six regions, of 
which a, d , e , and f are counterflow and b and c are 
parallel flow. The necessary boundary conditions may be 
derived with the aid of the figure. Again, region length 
T e (0) and Q e (0) must be parameterized in effecting a 
solution. Note that regions e and f are just the counterflow 
converter with excess current flowing in the emitter. 

Finally, the analytical model for a collector-collector 
short is shown in Fig. 7. For an external-fueled converter, 
this short cannot take place except through the coolant 
channels or leads external to the converter, but the case 
is shown in generality for other possible converter con- 
figurations in which such a short is possible. 

1 An equivalent solution method is to define the length of each sec- 
tion based on the location of the temperature maximum and have a 
non-zero current at that location. 



Fig. 6. Converter electric model for emitter— emitter short 


4 


JPL TECHNICAL REPORT 32-7545 




Fig. 7. Converter electrical model for 
collector-collector short 


III. Program Description 

A. Method of Solution 

I. Counterflow converter with constant collector tem- 
perature. To show the nature of the solution process, 
attention will be first given to the counterflow converter 
with constant collector temperature. The equations to be 
solved are 


dQe 

dx 


Ce ( Qin — Qec ~ Qrad ~ Qcs) + ( U + le»Y ~T~ 

Si el 


( 31 ) 


In order to solve these equations it is necessary to be 
able to determine Q ec and j at each location. A data map 
of Q €C vs j and v vs / for a variety of emitter temperatures, 
collector parameters and cesium reservoir temperatures is 
available from the SIMCON code (Ref. 5). The v-j data 
are single-valued if ; is the independent variable, but not 
always single-valued if v is the independent variable, 
which is the case above. Thus the data were modified at 
the expense of accuracy in the lower-temperature ex- 
tinguished mode regime so as to force a single-valued 
j(v). Also the data were extrapolated to T e = T c in order 
to study converter shorting. The extrapolations were 
rough in some cases, but fortunately there is very little 
electron cooling at these low temperatures so that the 
effect of errors in the extrapolation will be minor. Because 
it is far simpler to determine a uniform grid if / is the 
independent variable, an iterative process is required at 
each location. Thus, knowing T e> T c and T cs , a table 
lookup can be performed for a given / to find v. A modi- 
fied Newton-Raphson iteration process is used to find that 
j(x) which corresponds to a known v (x). Then QecW is 
also known. 

In solving Eqs. (31-35) one of the boundary conditions 
is known only at the right boundary, so a shooting tech- 
nique is required. An initial value of v (0) is estimated and 
then varied until the desired right boundary condition is 
satisfied. A good estimator for the first guess, v° (0), can 
be obtained by assuming that 


dT c _ ^ Q e 
dx A e2 k e 


(32) 


Qin (0) = Qec (0) + Qraa (0) + Qcs (0) (36) 


dv / pe , pc \ . 

~ \A^ + A c ) te 


Thus / (0) is iterated until Eq. (36) is satisfied. A fourth- 
order Runge-Kutta method with automated step-size 
selection was used to perform the integration. 



(34) 


where I es is excess (short) emitter current, if any, and the 
dependence of the variables is not shown. Boundary con- 
ditions are 


Qe( 0) = 0 
Te( 0) = T eo 

ie( 0) = 0 


Te(l)~T c 

L Qe(l) 


= A 


(35) 


It turns out that the trajectories obtained [particularly 
T e (x)] are quite sensitive to the choice of v (0), requiring 
a knowledge of the order of five or six decimal places to 
get a meaningful solution. In order to handle this prob- 
lem, a logarithmic algorithm is first used to get values of 
v (0) in the correct decade. In this algorithm, upper and 
lower bounds are estimated for v (0) and modified accord- 
ing to whether the temperature trajectory goes to oo or 
— oo. After modification of the appropriate limit [reduc- 
ing o(0) causes the T e temperature trajectory to head 
toward +oo], the new estimate of v (0) is obtained from 


t>(0) i+1 


K + V L 
2 


(37) 


JPL TECHNICAL REPORT 32-7545 


5 



In this manner it is usually possible to obtain meaningful 
T e ( l ) values after about 10 iterations. 

It is possible to continue iterating with Eq. (37) until 
the required right boundary condition is satisfied. How- 
ever, the lead length-to-area ratio is usually desired as a 
parameter in the output in order to obtain an optimum 
fuel element design. By trial and error, it was determined 
that once three meaningful trajectories are obtained by 
using Eq. (37) it is possible to fit the results by 

o(0,\) = o + — z— (38) 

A C 


and use relation (38) to fill out a parametric performance 
map for varying values of A. 

2. Parallel flow converter with constant collector tem- 
perature . The parallel flow case replaces Eq. (33) with 


dv 

dx 



(39) 


ic — ie {l) le 


(40) 


It is possible to solve this new set of equations exactly 
as in the counterflow case. However, since i e (l) is un- 
known, several solutions are required with i e ( l ) treated 
as a parameter until the estimated value of i e ( l ) equals 
the value obtained by integration. At this time, this addi- 
tional iteration is not automated in DEXTER, so the user 
must provide the iterations with overlay cases, each time 
specifying a new value of i e ( l ) as input. Because of the 
running time of DEXTER, this was done so as to allow 
maximum ingenuity on the part of the user in obtaining 
a minimum cost solution. 


3. Counterflow converter with varying collector tem- 
perature. For this case, additional relations are added to 
Eqs. (31-35): 


Qrej — C e (Qec ~b Qrad ~b Qcs jv) "b (i e ~b I C s Y ^ 

(41)- 


dT co _ Q rej 
dx wC p 


(42) 


T c — T co + 


Qrej 

C c h 


(43) 


T C o (0) = T C0( 


(44) 


where I C8 is excess (short) current flowing through the 
collector, if any. Since the thermionic properties Q ec and 
/ also depend on T Ci it would appear that additional itera- 
tion is required at each location in order to determine T c . 
However, since the thermionic properties are not strongly 
dependent on T 0 , it was decided to update T c after each 
integration step to avoid the iteration. T c (0) is deter- 
mined from one iteration of the equations at x = 0 to 
determine Q re j (0). 


The solution procedure is described here for a double- 
ended converter, since in this case the solution is no 
longer symmetric about the converter midplane. It was 
found that a convenient way to obtain a solution involves 
solving two counterflow converter problems, one with 
positive and one with negative flow rate. In matching the 
solutions it was found that there is a nonzero i e at the 
location of the emitter temperature maximum; i.e., the 
voltage and emitter temperature do not peak at the same 
place. Thus i e (0) is given to one converter, — i e (0) to the 
other, and the level is adjusted so as to match v (0) and 
i e (0) at the point of joining the two “half” converters. 
In the case of a double-ended converter it is desirable to 
have identical electrical voltage from either end in order 
to provide a good voltage match at the reactor bus. There- 
fore, it is additionally necessary to treat the boundary 
between the two “half” converters as a parameter; i.e., 
the length of each side must vary while satisfying the 
constraint that the total length is a constant. Thus para- 
metric information must be obtained both for i e (0) and 
the length of each section. Once again, this procedure has 
not been automated into the present version of DEXTER 
so as to allow human ingenuity to minimize the invest- 
ment in machine time for a complete^solution. 


4. Abnormally operating converters . The techniques 
described above provide sufficient flexibility to cover any 
of the shorted converter situations. No automated solu- 
tion is programmed into the present version of DEXTER. 
An example of the solution of a converter central emitter- 
collector short is given in Section IV-D. 


B. Capabilities and'Restrictions 

Once the set of equations and boundary conditions -is 
fully understood by the user, his ingenuity may discover 
ways to apply DEXTER to situations not explicitly con- 
sidered in the original formulation of the problem. For 
instance, the author has found that DEXTER can be used 
in conjunction with a fuel redistribution analysis code to 
solve the problem of fuel movement to form an isothermal 


6 


JPL TECHNICAL REPORT 32-1545 



internal cavity. In this problem Q in (x) is not known and 
must be determined by iteration. 

The present version uses a thermionic data table for- 
mulated by the SIMCON code for a specific diode per- 
formance characteristic. The user may desire to provide 
his own data tables. If so, it is recommended that the user 
also replace subroutines TABLE and BOUNDS, which 
were written for the interpolation of a specific data ar- 
rangement. The table lookup is up to four-dimensional 
but has built into it reductions to lower dimensions in the 
event that the collector temperature or cesium reservoir 
temperature exactly matches one of the tabulated sets in 
the data map. The tabulated data are arranged as shown 
in Table 1 for each of four collector temperatures: 850, 
1000, 1150 and 1300 K. For each T c , T e and T C8) there are 
15 current density values: 0, 0.5, 1.0, 2.0, 3.0, 4.0, 6.0, 8.0, 
10.0, 12.0, 15.0, 20.0, 25.0, 30.0, and 35.0 A/cm 2 . For initial 
performance scoping it is highly recommended that the 
tabulated values of T c and T c8 be selected so as to reduce 
table lookup time. Any attempt to request data outside 
the table, e.g., T C8 = 650 K for T e — 2000 K, will be re- 
jected by the code. 


temperature, current density and voltage profiles along 
the converter are shown in Fig. 8 for a typical near- 
optimum lead geometry. These profile shapes are repre- 
sentative of the types of shapes to be expected in long 
converters. Note that very little electric power is gener- 
ated in the last 2 cm of the converter (near the lead). 
Figure 9 shows how the net electric output and net volt- 
age vary with lead geometry. 



DISTANCE FROM CONVERTER MIDPLANE, cm 


CN 

E 


< 


LT) 


z 

LU 


o 

x 

> 


o * 



3 O 
> 


Fig. 8. Temperature, current and voltage profiles 


Table 1. Data table available in DEXTER 
(d = 0.0254 cm) 


r fl ,K 

Tc, K 

1400 

540-620 steps of 10 K 

1500 

540—620 


1600 

540-620 


1700 

540-620 


1800 

540-630 


1900 

540-640 


2000 

560-640 


2100 

580-640 


2200 

590—640 


2300 

600-640 


2400 

600-640 \ 



o 

o 

O' 

h- 

o 


LU 

t— 

LU 



LU 

< 


O 

> 


LU 


Fig. 9. Effect of lead geometry on converter output 


IV. Examples 

Physical property and geometric data used on all ex- 
ample cases to be reported here are the built-in values 
for the code (see Section V). Variable names correspond 
to those listed in Section V. 

A. Variation of Lead Geometry 

For this example the collector temperature is fixed at 
1000 K. The converter is assumed to be symmetric about 
the midplane and electron flow is countercurrent. Emitter 


B. Variation of Lead Arrangement 

In this example the performance of a converter of 
12.7-cm length with emitter and collector leads at oppo- 
site leads of the converter (parallel electron flow) is com- 
puted. The solution of the parallel electron flow case in- 
volves an iteration on total current flow (see Eq. 27). 
Thus performance data were run assuming i c ( 0 ) = 400, 
450 and 500 A (Fig. 10). The solution fine is the locus of 
i e (12.7 cm) = i c (0). This line then was used to interpolate 
the solution for net power, also shown in Fig. 10. 


JPL TECHNICAL REPORT 32-1545 


7 







LEAD GEOMETRY FACTOR x, cm" 1 


350 

340 

330 

320 

310 

300 


Fig. 10. Performance of 12.7-cm parallel electron 
flow converter 



Fig. 1 1. Voltage and temperature profile for 
liquid metal cooled converter 


C. Collector Temperature Effects 

In this example, the converter studied was 25.4 cm long 
with electric leads at both ends (countercurrent electron 
flow). The coolant flow rate was set at 30 g/s, and an 
objective of 1000 K coolant outlet temperature was set. 
This problem is not symmetric about the converter mid- 
plane; however, it is desirable to match the voltage out- 
put from each end of the converter. As a first guess, the 
length of each converter section studied was set at 12.7 cm, 
the coolant temperature at midplane was set at 925 K, 
and current flow through the midplane was varied until 
the center voltages from the solution of each converter 
section matched. This solution provided insight into the 
selection of a second estimate of the boundary between 
left-side and right-side converters in order to match the 
output voltages of each side and into the second estimate 
of coolant temperature at the boundary plane. It was 
found that the voltage peaked some 0.6 cm to the outlet 
coolant temperature side of midplane, so the second esti- 
mate placed the boundary at 12.1 cm from the inlet end 
(i.e., converters of 12.1 and 13.3 cm in countercurrent 
electron flow were studied). The results of this iteration 
were quite close to a good final solution, as can be seen 
in Figs. 11 and 12^The^ net_pqw^from_each,side_was— 
Closely matched and equal to 282 W. 

D. Central Short 

In this example, a central short in a 25.4-cm converter 
with leads at both ends was studied. To simplify matters 
the collector temperature was fixed at 1000 K. The sym- 
metry of this problem allows the study of a 12.7-cm con- 



Fig. 12. Coolant and collector temperature profile 
for liquid metal cooled converter 


verter. The normally operating converter characteristics 
are given in Section IV-A, where the output voltage and 
current correspond to a load resistance of 1.45 mQ for the 
half-converter when A = 4 cm 1 . 


The first case studied was that of short resistance equal 
to 0.725 mn (1.45 mfi for the half-converter studied in 
symmetry). For a first estimate, the boundary between the 
two converters that must be studied to achieve a solution 
(central short creates symmetry, see Fig. 5) was set at 
6.35 cm, i.e., one quarter of the 25.4-cm converter. The 
right-most converter, “d,” (Fig. 5) is required to have a 
load resistance of 1.45 mf2 and a lead length-to-area ratio 


8 


JPL TECHNICAL REPORT 32-1545 





VOLTAGE, V EMITTER TEMPERATURE 


A of 4 cm -1 . The left converter, “c,” has a short resistance 
of 1.45 mfi computed from 


R. 


v(!l 

ieQ ) 


and a short thermal conductance given by 


(45) 


(*) = m (46) 

UA (T e (l)-T c (l)) (4b; 

Solution was obtained by varying T e (0) until the required 
right boundary conditions for converter “d” were obtained 
(Figs. 13 and 14). The short conductance obtained was 
0.3 W/K. In order to vary the short conductance to any 
other desired value, the boundary between converters 
V and “d” would have to be varied. 


To study an even lower short resistance, the next case 
arbitrarily moved the boundary between converters V' 
and "d” to 9 cm from converter midplane; i.e., converter 
V was 9 cm long and converter “d” was 3.7 cm long. In 
this case, in addition to varying T e (0), it was also neces- 
sary to vary the current flow through the boundary u (0) 
to achieve a voltage match between converters “c” and 
“d.” Fortunately the matchup of the two converters is not 
sensitive to the choice of right boundary condition on 
converter V* (the X boundary condition in DEXTER) as 
can be seen in Figs. 15 and 16. This allowed a third plot 
in Figs. 13 and 14 at a value of ( k/X ), nearly the same as 
for the first short case to show the effect of varying short 
resistance only. For the short cases studied, Table 2 
shows the net power and voltage obtainable from the con- 
verter as a function of short resistance. 



DISTANCE FROM CONVERTER MIDPLANE, cm DISTANCE FROM CONVERTER MIDPLANE, cm 


Fig. 13. Effect of central short on converter 
temperature profile 


Fig. 15. Effect of varying short conditions on 
converter temperature profile 


1.0 
0.8 
0.6 
0.4 
0.2 
0 

DISTANCE FROM CONVERTER MIDPLANE, cm DISTANCE FROM CONVERTER MIDPLANE, cm 




Fig. 14. Effect of central short on converter 
voltage profile 


Fig. 16. Effect of varying short conditions on 
converter voltage profile 


JPL TECHNICAL REPORT 32-7545 


9 







Table 2. Net performance of shorted converter 


R t , mS2 

(k/\)„ W/K 

Pnet, W 

Vnet, V 

00 

00 


0.62 

1.45 

0.3 


0.39 

0.067 

0.4 


0.18 


V. User's Manual 

Every attempt has been made to simplify the use of 
DEXTER. The input NAMELIST format and built-in 


values for each variable are included so that only desired 
changes need be made to execute the code. Table 3 lists 
the input data required by variable name and FORTRAN 
name and shows the built-in values. 


VI. Error Messages 

Table 4 lists the current error messages and action 
taken. 


Table 3. Input data for DEXTER 


Variable 

FORTRAN 

Definition 

Built-in value 

name 

name 


■SEB^ 

Coefficients in p e = RE1 + RE2*T 

r— 1.203 X lO^n-cm 

P* 



L 3.445 X 10 -8 fl-cm/K 

L 

1 

Coefficients in k e = TKE1 + TKE2*T 

1" 1.33 W/cm-K 




L — 1.5 X 1 0'* W/cm-K’ 


rci"| 

Coefficients in p c — RC1 + RC2*T 

r 6.0 X 10'°n-cm 

po 

RC 2 J 

L 3.77 X lO _8 a-cm/K 

L 

TKCll 

Coefficients in lc c = TKC1 + TKC2*T 

T0.472 W/cm-K 

Kc 

TKC2J 


L 1.1 27 X KT 4 W/cm-K 


RCOL1 "1 

Coefficients in p co — RCOL1 4- RCOL2*T 

p 3.8 X 10'* J2-cm 

pco 

RCOl2 J 


L8.28 X 10~ 8 fi-cm/K 

L 

TKCOL1 “1 

Coefficients in k co = TKCOL1 + TKCOL2*T 

T0.2154 W/cm-K 

Kco 

TKC012 J 


L.7.667 X 10- 5 W/cm-K’ 

J 

DCOOL1 "1 

Coefficients in d co = DCOOL1 + DCOOL2*T 

T 0.943 g/cc 

Oco 

DCOOL2 J 


L“ 2.467 X 10“ 4 g/cc-K 


CPCOL1 "1 

Coefficients in C p = CPCOL1 + CPCOL2*T 

fO.801 J/g K 

v-P 

CPCOL2 J 


L 8.5 X 10' 5 J/g-K 2 

kL 

TKLD1 "I 

Coefficients in k L = TKLD1 + TKLD2*T 

r 1.33 W/cm-K 

TKLD2 J 


L-1.5 X 10'’ W/cm-K’ 

ph 

RL1 ”1 

Coefficients in p L — RL 1 + RL2*T 

r- 1.203 X 10-*n-cm 

R 12 J 


L 3.445 X l0‘ s ft-cm/K 


EME1"1 

Coefficients in e, = EME1 + EME2*T 

r0.026 

€ e 

EME2 J 


L1.168 X lO'VK 


EMC IT 

Coefficients in e c = EMC1 + EMC2*T 

Too 

€ c 

EMC2J 


Li. 11 x io Vk 

A el 

AE1 

See Nomenclature 

2.29 cm 2 

A e 2 

AE2 

See Nomenclature 

2.29 cm 2 

Ce 

CE 

See Nomenclature 

4.39 cm 

Ac 

AC 

See Nomenclature 

0.9673 cm 2 

Cc 

CC 

See Nomenclature 

2.394 cm 

A CO 

ACOL 

See Nomenclature^ 

, __ Q - 

\ 

~ TEND 

Desired value of X for right boundary condition 

4.0 cm* 1 

Deo 

DEQ 

See Nomenclature 

0.762 cm 

l 

ELDIOD 

See Nomenclature 

12.7 cm 

d 

DGAP 

See Nomenclature 

0.254 cm 

Tco 0 

TCOLIN 

See Nomenclature 

1000 K 

w 

TCOLOT 

See Nomenclature 

30 g/s 

Tee 

TCS 

See Nomenclature 

620 K 

Teo 

1 

TEMAX 

See Nomenclature 

2000 K 


10 


JPL TECHNICAL REPORT 32-1545 















Table 3 Icontd) 


Variable 

name 

FORTRAN 

name 

Definition 

Built-in value 

Qin 

PTH 

Average value 

40.0 W/cm 2 


[■P{10) 

Unnormalized distribution function for Q< n 

1.0 


LpX(10) 

Points at which function is specified 

0.0 cm 


ANPX 

Number of entries in P-PX tables 

1.0 


XPLT (50) 

Specified points for data printing 

0.0, 1.0, 2.0 12.0, 12.7 cm 


ANP 

Number of entries in XPLT table 

14.0 

ie (0); 

CURREN 

Left boundary current 

0.0 A 

ie (0) for // 
flow 

ASAVE 

Switch — 0. No effect 

0.0 

*(0) 

VOLT 

= 1. VOLT, VU, VL input 
Left boundary voltage guess 

0.0 V 

Yu 

VU 

See Nomenclature 

0.0 V 

YL 

VL 

See Nomenclature 

0.0 V 


DDV 

If ASAVE = 0., VU = computed v fO) (by 

0.1 V 


COL 

Eq. 36) + DDV, VL = computed v (0) — DDV 
Switch = 0. Heat pipe cooled collector 

0.0 


AIT 

= 2. Liquid metal cooled converter 
Switch = 0. Want a data map vs X 

0.0 


ACHT 

= 1. Iterate to satisfy input TEND 
Switch — 0. Switch not used 

0.0 


AFLO 

Switch = 0. Countercurrent electron flow 

0.0 

f«. 

CURE 

= 1. Parallel electron flow 
See Nomenclature 

0.0 A 

let 


See Nomenclature 

0.0 A 


Table 4. Error messages in DEXTER 


Message 

Definition 

SEAR 2 

Unable to converge on desired right boundary 
condition (X). T„ (/), VOLT, VU, VL printed and 
case is terminated 

SEARB 

Unable to find a value of j which corresponds to 
desired v, T e , T C/ and T e $. Te, T Ct j , T c * , v, /», ji, 
printed and case is terminated 

STEP SIZE 
TOO SMALL 

Runge-Kutta failure in satisfying integration error 
criterion. Case is terminated 

AVLO or 
ENTPL 

Power distribution function not specified at x == 0. 
Problem terminated 

AVHI or 
ENTPH 

Power distribution function not specified at x = 1. 
Problem terminated 


JPL TECHNICAL REPORT 32-1545 


11 










Nomenclature 


A c Collector cross-sectional area, cm 2 

A el Emitter cross-sectional area for electron flow, 
cm 2 

A e2 Emitter cross-sectional area for heat flow, cm 2 

C c Collector surface area per unit length, cm 

C e Emitter surface area per unit length, cm 

C p Coolant heat capacity, J/g-K 

D co Equivalent flow diameter for coolant, cm 

d Emitter-collector spacing, cm 

h Collector-coolant heat transfer coefficient, 
W/cm 2 -K 

I cs Excess (short) current flowing through 
collector, A 

l es Excess (short) current flowing through 
emitter, A 

i c Collector current, A 

i e Emitter current, A 

j Thermionic current density, A/cm 2 

k c Collector thermal conductivity, W/cm-K 

k co Coolant thermal conductivity, W/cm-K 

k e Emitter thermal conductivity, W/cm-K 

k L Lead thermal conductivity, W/cm-K 

l Converter length from plane of zero heat flow, 
cm 

Q c Collector axial heat conduction, W 
Qca Cesium heat conduction, W/cm 2 
Q e Emitter axial heat conduction, W 


Q ec Electron cooling, W/cm 2 
Q in Input heat, W/cm 2 
Qrad Radiation heat transfer, W/cm 2 
R C8 Resistance of collector-collector short, Q 
R es Resistance of emitter-emitter short, Cl 
R/ Resistance of interconverter lead, Cl 
R l Load resistance, Cl 
R s Emitter-collector short resistance, Cl 
T c Collector temperature, K 
T co Coolant temperature, K 
T ca Cesium temperature, K 
T e Emitter temperature, K 
T l Lead mean temperature, K 
v Emitter-collector voltage, V 
v L Lower limit to v (0), V 
v u Upper limit to t> (0), V 
to Coolant flow rate, g/s 
x Distance along converter, cm 
e c Collector emissivity 
€ e Emitter emissivity 
X Lead length-to-area ratio, cm -1 
p c Collector resistivity, Q-cm 
Pco Coolant resistivity, a-cm 
p e Emitter resistivity, fi-cm 
p L Lead resistivity, Cl- cm 
(j Stefan-Roltzmann constant, W/cm 2 -K 4 


12 


JPL TECHNICAL REPORT 32-1545 


References 


1. Davis, J. P., et al., “One-Dimensional Diode Heat Transfer Program Coupled 
to Thermionic Performance,” Space Programs Summary 37-59, Vol. Ill, pp. 211- 
214. Jet Propulsion Laboratory, Pasadena, Calif., Oct. 31, 1969. 

2. McCandless, R. J., and Hill, P. R., “Radiation Heat Transfer Calculations,” in 
1967 Thermionic Specialist Conversion Conference , Palo Alto, Calif., Oct. 1967. 

3. Kitrilakis, S., and Meeker, M., J. Adv. Energy Conv ., Vol. 3, pp. 59-68, 1963. 

4. Nuclear Engineering Handbook , First Edition. Edited by H. H. Etherington, 
McGraw-Hill, New York, 1958. 

5. Wilkins, D. R., S1MCON , GESR2109, General Electric Company, Pleasanton, 
Calif., 1968. 


JPL TECHNICAL REPORT 32 -1545 

NASA - JPL - Coml., L.A., Cafif. 


13 


tin-in^ 

TECHNICAL REPORT STANDARD TITLE PAGE 


1. Report No. 32-15U5 

2. Government Accession No. 

3. Recipient's Catalog No. 

4. Title and Subtitle 

DEXTER — A ONE- DIMENSIONAL CODE FOR CALCULATING 
THERMIONIC PERFORMANCE OF LONG CONVERTERS 

5. Report Date November l£, 1971 

6. Performing Organization Code 

7. Author (s) 

C. D. Sawyer 

8. Performing Organization Report No. 

9. Performing Organization Name and Address 

JET PROPULSION LABORATORY 
California Institute of Technology 
4800 Oak Grove Drive 
Pasadena, California 91103 

10. Work Unit No. 

11. Contract or Grant No. 

NAS 7-100 

13. Type of Report and Period Covered 
Technical Report 

12. Sponsoring Agency Name and Address 

NATIONAL AERONAUTICS AND SPACE ADMINISTRATION 
Washington, D.C. 20546 

14. Sponsoring Agency Code 

15. Supplementary Notes 

16. Abstract 

This report describes a versatile code for computing the coupled thermionic 


elec trie -thermal performance of long thermionic converters in which the tem- 
perature and voltage variations cannot be neglected. The code is capable of 
accounting for a variety of external electrical connection schemes, coolant 
flow paths and converter failures by partial shorting. Example problem 
solutions are included along with a user's manual. 



17. Key Words (Selected by Author(s)) 

Computer Programs 
Power Sources 
Thermionics 

18. Distribution Statement 

Unclassified -- Unlimited 

19. Security Classif. (of this report) 
Unclassified 

20. Security C 
Unclas 

lossif . (of this page) 
ssified 

21 . No. of Pages 

13 

22. Price 






















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