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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. 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