THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COTLEGE OF ENGINEERING PRELIMINARY STUDIES OF A HIGH TEMPERATURE, GAS-COOLED NUCLEAR,RAACTOR Wan-Yong Chpn Richard E. S i ahl Harold A. Ohlgren September, 1958 IP- 321

LI.mg, 11i- 5 ~

ACKNOWLEDGMENT The authors wish to express their sincere appreciation to Dr. Richard K. Osborn of the University of Michigan, who has been a constant source of technical advice and suggestions throughout the investigation. Further acknowledgment is extended to Dr. Raymond L. Murray of North Carolina State College of Agriculture and Engineering, to whose competent publications many references have been made, especially in nuclear calculations. ii

TABLE OF CONTENTS Page ACKNOWLEDGMENT.................................................................. ii SUMIARY................................................... INTRODUCTION................................................. 2 MATERIALS FOR HIGH TEMPERATURE REACTOR....................... 4 1. General............................................ 4 2. Uranium Carbide.................................... 5 3. Zirconium Carbide................. 7 PRELIMINARY CONSIDERATIONS.......................................10 SLIGHTLY MODIFIED TWO-GROUP THEORY......................... 14 1. Fast Fission Factor....... 14 2. Modified Two-Group Theory................... 16 SEVERAL APPROACHES ADOPTED IN THE TREATMENT OF HIGH TEMPERATURE CONDITION..................................................... 18 GROUP CONSTANTS........................................... 22 1. General Approach........................... 22 2. Calculation........................................ 22 3. Results......................................... 26 TREATMENT OF COMPLETELY REFLECTED CYLINDER CORE.................. 27 TWO-GROUP CALCULATION.............................................29 1. General Approach................................... 29 2. Murray's Arrangement of Two-Group Calculation...... 29 5. Results....... e.................................. 30 FLUX DISTRIBUTION................................5..........33 EVALUATION OF TEMPERATURE COEFFICIENTS........................... 35 Xe- AND Sa-POISONING............................................ 36 REACTIVITY SPECIFICATIONS FOR CONTROL DESIGN..................... 37 iii

Page EVALUATION OF BASIC THERMAL CONSTANTS............................ 41 1. Conductivity, Viscosity and Specific Heat.......... 41 2. Mass Flow Rate..................................... 42 3. Heat Transfer Coefficient.................... 42 GENERAL RELATIONSHIPS.......................................... 45 1. Local Gas Temperatures................... 45 2. Relation Between the Total Thermal Power and the Maximum Power Density................... 48 3. Fuel Surface Temperature and Inlet Gas Temperature. 49 MAXIMUM CORE TEMPERATURE.................................. 52 1. Fend's Work................................... 52 2. Derivation of Maximum Core Temperature for the Reactor System.................................... 5 3. Channel Diameter and Power Output.............. 57 4. Channel Diameter and Reflector Inlet Temperature... 57 PRESSURE AND TEMPERATURE DISTRIBUTION IN THE REACTOR............. 60 1. Power Generation and Coolant Temperature Drop in Thermal Shield................................... 60 2. Power Generation in Reflector...................... 61 3. Local Coolant Temperatures in Reflector and Thermal Shield........................ 62 4. Pressure Drops Through the Cooling Channel and Number of Channels.........6....................... 62 5, Pressure Drops Due to Entrance and Exit Pressure 2osses in the Reflector........................... 67 SUMMARY OF HEAT TRANSFER DATA.................................... 69 CONCLUSION.................................. *..... 70 BIBLIOGRAPHY..................................................... 71 APPENDIX.............................................. 72 iv

LIST OF FIGURES Figure Page 1 The Constitutional Diagram of the System Uranium Carbon...............~ * ~ *................. 6 2 Tentative Constitutional Diagram of the System Zirconium Carbon.................................... 8 3 Geometric Buckling as a Function of Fuel-to-Moderator Volume Ratio with Enrichment as Parameter (One group One region).*........................................ 4 Core Radius as a Function of Enrichment............. 12 5 Conversion of Completely Reflected Cylinder Core into Bare Slab System.......2,...............X.. 27 6 Determination of Critical Radius............... 31 7 Flux Distribution................................... 34 8 Helium Film Coefficients as a Function of Mass Flow Rate with Diameter of Channel as Parameter (1200~F, 1000 psia)..........................................44 9 Cooling Channel and Neighborhood.................... 52 10 Geometry of Cooling Channel and Vicinity............ 53 11 Coolant Flow Pattern (Helium at 1000 psia)......... 63 v

SUMMARY Parametric studies on a helium-cooled, hetero-homogeneous composite-type power reactor employing highly elevated temperature and gas pressure were carried out. The fuel element is uranium carbide solid solution in carbon where an enrichment of 20% and a fuel-to-moderator volume ratio of 5 x 103 and up were used for the preliminary design. Maximum fuel element temperature of 2700~F and core pressure of 1000 psi were adopted, allowing improved characteristics of the reactor, although a series of unique approaches for both the nuclear and thermal designs were inevitably needed. High power density and thermal efficiency can be incorporated with a fair degree of inherent safety of operation in this reactor, although some ambiguities still remain due to the lack of nuclear data at highly elevated temperatures. This system, coupled with a closed-cycle gas turbine plant may be of use for application wherein extreme compactness and lightweight are necessary. Also such a system may be of interest for a rocket heat source, although in this case even higher temperatures may be required.

INTRODUCTION Although the limitations imposed on the materials used in nuclear reactors restrict the maximum attainable temperature in the core and thus the thermodynamic efficiency in the associated power loop, attempts have been made to overcome the difficult task of obtaining new reactor core configurations as well as materials which can withstand highly elevated temperature and pressure. The formidable problem of cutting the nuclear power cost to the commercially competitive level may be solved partially by the enhanced thermal efficiency thus attained. As found in other high temperature applications, metals in carbide form are among the possible materials on which attempts along this line can be based and thus uranium carbide or any fissionable or fertile material in the form of carbide compounds seems a very natural choice. Considering the fact that some of these fissionable materials, especially uranium, have phase-transients at medium-high temperature regions, any genuinely heterogeneous set-up of a fuel element should be avoided since it will most likely cause severe structural problems. One configuration, in which uranium carbides are dispersed homogeneously in graphite, can be considered suitable as a possible core material. Apart from these material considerations, nuclear and heat transfer problems encountered under the specified conditions are -2 -

-3 - unique. All nuclear data obtained at room temperature or comparatively low temperature regions should how be modified for the new high temperature and certainly this can not be achieved without using several rather crucial assumptions. The temperature to be used in this study is far higher than those which have been employed by many investigators, as it will appear in the basic design parameters below. Average reactor pressure 1000 psi Maximum fuel element temperature 2700~F Fuel element UC solid solution in carbon Fuel element cladding ZrC (0.3 mills thick) Moderator High density three-dimensional graphite Reflector Same as moderator Thermal shield 5" of iron outside of reflector Once the materials for use are specified, the nuclear and heat transfer calculations can be carried out as usual. One may not have any particular difficulties.in following the conventional approches in heat transfer studies except that the figures used are very high. On the other hand, some bold assumptions must be made in nuclear calculations, thus any unnecessarily elaborate multigroup theory calculations will not be justified in the preliminary design studies. Rather, the use of a slightly modified two-group theory is thus recommended in the first effort to tackle these enhanced conditions, considering the subsequent ease and satisfactory accuracy in the calculation.

MATERIALS FOR HIGH TEMPERATURE REACTOR General In general, the materials to be used for nuclear reactors should fulfill certain physical and chemical requirements. In high temperature nuclear reactors, these requirements will be much more severe, since the enhanced conditions will not be sustained safely for any long periods of operation without extremely superior mechanical and chemical stabilities. The nuclear requirements, however, will not be very different from those for any conventional medium-temperature reactor. At highly elevated temperatures, good thermal stabilities and chemical inertness are especially required. The answer of presentday technology to this problem may be the metal -carbides, due to the fact that many carbides fulfill the above mentioned properties and, at the same time, carbon is a widely used moderator and reflector for thermal and intermediate type nuclear reactors. Possible carbides for each reactor component are summarized below. Fuel Thorium Carbides ThC ThC2 Uranium Carbides UC U2C3 UC3 -4 -

-5 - Cladding Material Titanium Carbide TiC Zirconium Carbide ZrC Uranium Carbides As observed on the constitutional diagram of the UraniumCarbon system in Figure 1, the composition for solid solution of uranium carbon ranges from 50 - 66.7 carbon atom percent or 4.8 - 9.2 weight percent. One can easily notice that the melting point of the solid solution is very high (2400~C in the vicinity of) throughout the range and the system can stand the proposed reactor core temperature. Any allotropic transformations such as those that appear in pure uranium at 650 - 770~C do not show up for the present composition and consequently the system has thermal stability. The homogeneous dispersion of the powdered uranium carbide into the reactor grade graphite may be easily incorporated during the formation process of the graphite blocks. Considering the fact that the volumetric ratio of uranium carbide to total graphite in the core should be very small, the overall physical properties of the core material will be those of pure graphite and thus the composition seems perfectly ideal for medium or low enrichment fuel systems.

-6 - CARBON, WT-% 1 2 3 4 5 6 7 8 9 10 11 12 13 14 eoUt. 260C ) ) ) I I i I I I I I I I I 2 Liquid - Liquid + solid so/ution 240C 1 L / Liquid / graphite UC, f graphite 220C 0 o - 2000 i3 < 1800 I 1 Sw a. 1600 Liquid +UC I, e~~~~~~~~Z: I Solid so/ut/on, \ ' UC UC, \\ \ I — I. mLp I 1400 1200 I I I' I I -4800 4400 4000 IL 3600 so 3200 a. LU -2800 - 2400 2000 1500 1100 To/ I I I L,- Gamme uranium 800'- UC Beta uranium + UC 600 A, Apha uranium 0 UC i I I I I I I I I I I I I I 29 50 66.7 CARBON, ATOMIC-% *Figure 1. The Constitutional Diagram Uranium-Carbon. of the System * By permission from The Reactor Handbook, Vol. 3, Section 1, General Properties and Materials, USAECD 3647, by H. A. Saller, McGraw-Hill Book Company, Inc. I

-7 - Zirconium Carbide For elevated temperature application, titanium carbide is outstanding for its thermal shock resistance and high tensile strength, but the absorption cross section for thermal neutrons is rather high. Zirconium and its alloys have somewhat better high-temperature strength but certainly their upper limit should be under 1000~F at the most. Zirconium carbide, on the other hand, has a very high, melting point with a composition of carbon of around 50. In its optimum composition, temperatures of 1800 - 1900~C can be reached in solid phase. In case this material is to be used as a cladding material, however, the process of forming zirconium carbide at the surface of the fuel element will cause some technical problems. Although when graphite is used as the fuel-bearing core block material, the formation process will not be a very difficult one. Considering the fact that several different types of metal carbides have been successfully formed on the surface of graphite, it can be expected that similar experimental trials have been done for zirconium carbide by some investigator, though no further detailed descriptions are attempted here. In a high-temperature reactor, the diffusion of fission products through the cladding material will be much more accelerated and the requirement must be more stringent. In these design studies, the thickness of zirconium carbide was tentatively set as 0.3 mills, which was believed to give enough thickness to withhold the fission products in the fuel element.

-8 - CARBON, ATOMIC-% 0 10 20 30 40 50 o3UU 3700 3500 3300 3100 2900 o 2700 o t 2500 D 2300: 2100 - 1900. 1700 1500 1300 1 100,900 700 snn I I I I L iquid 00000 00000 I I 7000 6600 6200 5800 5400 5000 4600 4200 3800 / -I Liquid + ZrC LL. 0 o sL D Iw 0. w F I Zrg-f Liquid r r Zrc s I 7,r.,7, LC 7rr. L - I T ^ I O B Zrn.C I -3400 -3000 2600 2200 1800 1400 - 100 13 IZr Z Zr r i I I 4- ZrC I I I I I I 6i 0 I 2 3 4 5 6 7 8 9 CARBON, WT-% 10 II 12 *Figure 2. Tentative Constitutional Diagram of the System Zirconium-Carbon. * By permission from The Reactor Handbook, Vol. 3, Section 1, General Properties and Materials, USAECD 3647, by R. W. Dayton, McGraw-Hill Book Company, Inc.

,9 - a'. I0 -2 10 N. 10 C =0.03 = rN.^^ Figure 3. Ou I IUU Bg Geometric Buckling as a Function of Fuel-to-Moderator Volume Ratio with Enrichment as Parameter. (1 group 1 region

PRELIMINARY CONSIDERATIONS To obtain a quick insight into, the overall optimum conditions, bare one region and one group calculations were first carried out for various fuel-to-moderator ratios and enrichments, while the temperature was taken as 270~K. Thus the inverse geometric buckling versus fuel-to-moderator volume ratio VF/VM were plotted, the enrichment being one of the parameters. The result, in spite of its crudeness caused by the over-simplified assumptions, gives some means of a quick check. The product of resonance escape probability and fast fission factor p ~ e was assumed to be unity which was later verified to be a fairly good assumption in the more elaborate one-group-tworegion and modified two-group-two-region calculations. As seen in Figure 3, as soon as the enrichment of 0.1 (10%) is attained, further increase in the enrichment does not affect the buckling considerably. On the other hand, the maximum p value for the system is around 0.8 and the decrease in p value will increase the fast fission factor c, thus changes the reactor into a faster one. The difficulty already encountered by the elevated temperature implies that the extremely fast reactor employing an elevated temperature, such as, 2700~F is not desirable in the design work as yet when the ambiguities still dominate several critical nuclear problems. Thus the enrichment of 20% and VF/VM ratio of 5 x 10-3 are found suitable for the core considered here. The trend observed for the cold -10 -

-11 - reactor should still be valid for a high-temperature reactor core in a primary evaluation, since the relative buckling will not change too much with changes in temperature. This can be clearly seen in Figure 4, which is the plot of the result of two-group calculations for the operating conditions at 27000F where the core height and reflector thickness are held fixed as indicated therein. Assumptions Made for Nuclear Calculations A highly elevated temperature employed in the reactor necessitates a somewhat unique approach in the reactor calculations. Some difficulties encountered here are the uncertainty of cross sections and ambiguity of resonance region characteristics including Doppler broadening effect at elevated temperatures. The whole flux distribution will shift and little information is available on this critical problem. Neither core nor reflector expansion can be neglected. Thus the difficult task of choosing group constants becomes more complex, the situation still being such that the calculations based on the cold reactor can not be logically used. Main assumptions as well as approximations which will be made are as follows: 1. Absorption cross sections follow - laws. The product of non - - factors and most probable-to-average neutron velocity correction factors for U-235 and U-238 were selected through the extrapolation of the data for considerably lower temperatures.

-12 - 14C 13C 12C 110 ir C) C) or 100 90: _ L Cylindricol core, UC Reflector graphite Temperature 2700~ F max. Void 30% H= /28cm Ts = 23 cm TE = 84 cm pr -l 80 I 70 60 i 50 40 1 0.03 005 0.1 0.2 ENRICHMENT, C Figure 4. Core Radius as a Function of Enrichment

2. The neutron cross section does not shift considerably at the epithermal energy region, thus the cross section data can be taken as they appear in BNL-325 for the region. 3. Thermal expansion of fuel and moderator is independent of change of neutron average energy due to the high temperature. The product of these two factors contributes to the correction of reactor material constants. 4. The effect of Doppler broadening in the resonance region is slight for the present system. 5. Voids for the cooling channel through core and reflector are homogenized in the nuclear and thermal calculation.

SLIGHTLY MODIFIED TWO- GROUP THEORY The determination of minimum critical mass is the basic part of reactor analysis. Before going into the calculation, however, it would be beneficial to know the present reactor system. This is important in deciding the possible approaches. The fast fission factor was first examined from this viewpoint and from this, the approach was determined. Fast Fission Factor First, composite multiplication factor K was defined as K = K + KF where KT = K-thermal KF = K-fast Accordingly fast fission factor e is K KT+KF KT KT On the other hand, KF and KT are calculated as follows: 00 KF = v f dE' *E, ET + f'dE' PI (E',E) v E=ET E'=E 00g a KT f f'dE' PI' ( E E'=E d aT -14 -

-15 - where E' = Initial neutron energy ET = Thermal threshold energy f = Fast fission spectrum CZ = Macroscopic fission cross section Z = Macroscopic scattering cross section EL = Macroscopic absorption cross section J = Macroscopic total cross section PI = Probability of neutron slowing down from energy E' to ET. = Average logarithmic energy change per collision v = Average number of fast neutrons generated per fission P ana f can be further expressed by 2Pe (EE' i, + D"B 2 PI (E',E) = e(- DB.2) E S +DBg2) E" and f = 0.484 e sin h f2E where E is in The first term of KF, the virgin fast fission, was found to be rather small, i.e., 1.67 x 10-3 for the system, but the secondary fission part of KF is appreciably larger although the double manipulation of graphical integration gives some difficulty. The value is computed to be 5.22 x 10-1 and this, together with KT = 1.58, gives the composite multiplication factor of K = 2.10, thus 2.10 cE - = 1.33 1.58

-16 - The reactor is slightly fast. The logical theory to be used in the critical mass calculation would be multigroup theory where several epithermal energy regions are specifically handled. However, the fact that there is little information on the neutron flux shift in eqithermal regions due to the high temperature, makes the use of any multigroup handling somewhat awkward. Considering the saving of time and effort achieved by simple two-group calculation, a slightly modified two-group theory would be the proper compromise. Modified Two-Group Theory A slight modification can be made by using the fast fission factor with conventional neutron balance equation for a thermal reactor. The new equation will be Core: D~1 cPD1 + q2fE = 0 D2I2 P2 - hi + 0 = Reflector: Dlr - =T ~r D2r2 %P2r - P2r2 r rr 0 Tr where D1 = Fast diffusion coefficient of the core D2 = Thermal diffusion coefficient of the core CPi = Fast flux in the core P2 = Thermal flux in the core

-17 - T = Age of the core ^2 = Macroscopic thermal cross section f = Thermal utilization p = Resonance escape probability D = Fast diffusion coefficient of the reflector Ir D2r = Thermal diffusion coefficient of the reflector Tpr = Fast flux in the reflector c2r = Thermal flux in the reflector Tr = Age of the reflector 2r = Macroscopic thermal cross section of the reflector

SEVERAL APPROACHES ADOPTED IN THE TREATMENT OF A HIGH TEMPERATURE CONDITION As described in the introduction, all reactor constants are to be corrected for thermal effects. Absorption Cross Sections Temperature dependent absorption cross sections are, in a simplified form, aa)2 -= fTl ya) 2 aa)l T2 where a = Average neutron absorption cross section a T = Temperature ~K 1 = Reference to 293~K 2 = Reference to 1753~K aa)2, thus obtained, were further corrected for core void, i.e., homogeneously diluted core composition to obtain the macroscopic cross section. Thus Fa)2 = aa)2 [N x dilution factor] where N is Avogadro's Number. Scattering Cross Sections The scattering cross section for graphite is almost independent of temperature.

-19 - C = Graphite UC = Uranium carbide Age Temperature corrections for age were carried out using the relation E E 0o D dE 1 dE T = 5ZI, t th E tho Eth Eth s thus Eth T Y r / 1 i dE E fo 5 st E th Specifically for the present system, this can be further approximated as T = o- (53 jt) Tn o since scattering cross sections are regarded temperature-independent, thus energy independent. Diffusion Length Diffusion length is temperature sensitive and a decisive factor which affects the criticality. From the original definition, it follows that L2 = 1 = 1 1 3z s t 3 aotO~ 9 _(n+1/2) = L2 n+l/2 = 0

-20 - Thermal Expansion Contribution For most group constants, the corrections which account for the thermal expansion of core and reflector should not be neglected. Fast Diffusion Coefficient As usual, flux averaged value of fast fission coefficients were used. Thus r t I J D(u) cp (u,r) du D1 = 0 ut ut p (u,r) du 0 J D(E) q) (E) 00 J p (E) dE ET Et of 0.166 ev, which is the corrected value of termal threshold energy 0.025 for room temperature, at 27000F, was used. Thermal Diffusion Coefficient The relations = 1 [ Z, (c) + Z (UC)] core D2c = 3 tr tr D =1 [I " (c) ] reflector 2r 3 tr were used to obtain the thermal diffusion coefficients, where Z = Temperature-, void- and thermal expansion-corrected macroscopic transport cross section of core 2tr = Temperature-, and thermal expansion corrected macroscopic transport cross section of reflector

-21 - where G = T/To Lo = Diffusion length at 293~K n = 0 for graphite. Resonance Escape Probability Lawrence-Dresner's data on the effective resonance integral gives the value of p = 0.820. On the other hand, Murray's formula gives approximately 0.781 which was considered safe from the viewpoint of criticality achievement and therefore adopted. Very little reliable data are available on the type of hetero-homogeneous core configuration similar to the one described later in the heat transfer studies. Doppler Effect Estimation of Doppler effect at an elevated temperature such as 2700~F for any intermediate type reactor is almost impossible at the present time. In this design calculation, the effect was simply neglected. The essential homogeneity of the present core may possibly allow this approximation. Nevertheless, a clearer understanding of this effect would certainly be helpful in the design of high-temperature reactors in the future.

GROUP CONSTANTS General Approach Group constants are computed according to the previously described approaches. The process is as follows: First a modified one-group calculation based on age theory is used to obtain the approximate critical mass of cold reactor, upon which the thermal distribution for core as well as reflector is determined. The maximum core temperature is set to be 2700~F, from which the side and end reflector temperatures are to be determined. The group constants are recalculated again at this point for the corresponding temperature in both core and reflector region. Ideally, an infinite number of repetitions of this process would give the best result but one or two trials are considered to be sufficient for practical purposes. Calculation Since the evaluation of group constants and general reactor constants are main problems in any high temperature reactor analysis, somewhat detailed computations are presented here for further reference. Microscopic Cross Sections Choosing 2700~F as the reference temperature of the core, T 9 = T = 5.99 ~ 1/2 = 2.44, 9-1/2 = o,409 -22 -

-23 - For U-235: non 1 factor, fl(2700~F) 1.10 most probable-to-average, f2 = 4n/2 using these, one obtains aa = 274 b, af = 231 b ac = a - af = 43 b. For U-238: aa = 1.00 b For Graphite: aa = 1.16 x 10-3 b. As can be seen, the figures are greatly reduced at the elevated temperature. Thermal Volume Expansion Using volume expansion coefficient a = 75 x 10-6 for U-235 and U-238 and a = 13 x 10-6 for graphite, the number of nuclei per unit volume is calculated. The results are shown in Table I. TABLE I THERMAL CROSS SECTIONS OF PURE MATERIALS AT 2700 ~F Pure Substances at 2700~F u-235 U-238 C N 4.76 x 1022 4.69 x 1022 1.01 x 1023 a 231 274 - 1.00 - 1.16 x lo-3 a s 10.0 4.8 11.0 11.0 a 13.03 0.0469 _4 1.17 xlO s O.469 0.485

-24 - Void Correction As indicated previously, cooling channel voids were homogenized in the calculation. The ratio of moderator volume ratio of 5 x 10-3 is used to obtain the final cross sections corrected for temperature and void as shown in Table II. Material U-235 U-258 C Total TABLE II MACROSCOPIC CROSS SECTIONS OF REACTOR MATERIALS, TEMPERATURE-, VOID CORRECTED C, Ca 4 7.63 x 10-3 9.03 x 10-3 _- 1.32 x 10-4 1.32 x 10-3 8.19 x 10-5 0.539 7.63 x 10-3 9.24 x 10-3 0.340 it 9.03 x 10-3 1.45 x 10-3 0.339 o.350 Thermal Diffusion Coefficient for Core, D2 D2c - 30oS) but o = 0.0553 Thus D2c = 1.01 cm Since Z or Et does not vary too much with temperature change, D2c is not very sensitive to temperature. Thermal Diffusion Length for Core L2c L2 = Thus L = 10.45 cm L2c is related to fZ and thus very sensitively temperature dependent. Thermal Utilization f and rl Thermal utilization is temperature independent. Thus P = n Q77 for this system at all temperatures. v varies slightly

-25 - with internuclear relative velocities but the change is regarded negligible at the temperature range considered here. Calculation gives ' = 2.08. Fast Diffusion Coefficient for Core and Reflector Dc and Dr. These are almost temperature independent, but the void and thermal expansion corrections should be made. Age T Age T is slightly temperature dependent, and T decreases somewhat at elevated temperatures, but void correction gives increased T value. Cross Sections for Reflectors A separate heat transfer calculation which will be discussed in later chapters gives rough estimations of reflector temperatures. Portion Average Temperature Top reflector 1650~F Bottom reflector 1050~F Side reflector 1300 ~F For convenience of calculations, it was assumed that the temperatures of all three reflectors are equal and 1300~F. Temperature and void corrections give the values shown in Table III. TABLE III THERMAL CROSS SECTIONS OF DIFFERENT PORTIOI OF REACTOR _ Z Cbot End Reflector 0.344 1.12 x 10-4 0.344 (a = 0.2965) Side Reflector 0.460 1.49 x 10- 0.460 (a = 0.06) where a is void fraction.

Group Constants for Reflectors Using the above calculated figures, D1, D2, T and L of side and end reflectors are obtained. Results Results of calculations are shown in Table IV. TABLE IV SUMMARY OF GROUP CONSTANTS CORE 2700~F a = 0.2965 I END REFLECTOR I SIDE I 1300~F a = 0.2965 1300~F REFLECTOR a = 0.06 Constants Fast hThermal Fast Thermal Fast Thermal T D L a S.s f 706 1.62 726 1.01 1.60 10.45 9.24 x 10-3 0.340 231 b 407 1.19 1.03 95.9 0.767 71.7 1.49 x 10-4 o.46o 1.12 x 10-4 0.344 f p K 00 2.08 0.977 0.781 1.33 2.10

TREATMENT OF COMPLETELY REFLECTED CYLINDER CORE At the present time, a completely reflected cylinder can not be analytically handled. Therefore as an alternative approach, the core was first considered to be a slab of thickness H', where H' includes the extrapolation distance with the top and bottom considered to be reflected. REFLECrOR CORE 2 ' 'H FiguREFLECTOR e Conv on of Coy Rd2 Figure 5. Conversion of Completely Reflected Cylinder Core into Bare Slab System. This is an approximation but it roughly provides the height of an equivalent bare reactor that can be used in solving the jacketed part of the cylinder by conventional two group calculation. The height which would have to be added to the reflected height H' to give the equivalent bare height is found by Ho' H' 2 2- -27 -

-28 - Ho' = New thickness of bare core H' = Initial thickness of a reflected core 6 = Reflector saving The reflector saving 5 was calculated from the expression for a slab reactor. 1 D B 6 Bc [tan-1 ( — c tan h KcTE) ] Bc Dp r^ tr rr 1 where Be = Geometrical buckling Dc = Diffusion coefficient for core Dr = Diffusion coefficient for reflector Kr Inverse diffusion length in the reflector TE = Reflector thickness For the arbitrary value of TE = 84 cm, 6 is computed as 6 = 54 cm. The values of all material constants such as Dc, Dr and Kr are based on the appropriate temperatures as previously calculated.

TWO-GROUP CALCULATION General Approach In obtaining the critical mass of the reactor, efforts have been made to make the final reactor shape the proper one under the high-temperature and high-pressure conditions. The usual approach of attempting to give it an approximately cubic shape is questionable under these conditions. Rather, a slim reactor which will facilitate the manufacture of the high-pressure vessel outside the reactor and which will cut down the pressure vessel cost accordingly was considered preferable. Considering the auxiliary reactor units to be attached, the overall core height-to-diameter ratio of around 3/2 was chosen. Thus after several trial calculations, the basic dimensions were assigned as given below: Core height, H = 128 cm Side reflector Ts = 25 cm thickness End reflector T = 84 cm E thickness Radius of Core R = To be determined. Murray's Arrangement of Two-Group Calculation Among the several computational set-ups for two-group calculations, Murray's arrangement seemed most convenient and was therefore adopted in this work. As described in the foregoing chapter, equivalent barecore height Ho' was computed and corresponding buckling was found to be -29 -

-30 - 1.78 x 10. Material constants 12 and v2, which are the solutions of modified two group critical equation discussed previously, were calculated. These material constants are further corrected for geometric buckling B2. Coupling coefficients S1, S2 and S3 are calculated. Murray's handling of critical condition is interesting. A quantity a' in the critical equation varies slowly with reactor radius. On the other hand another quantity a, which is the counterpart of a' in the critical equation changes quickly with reactor dimensions. Both quantities are computed for different sets of reactor sizes and the critical radius is found. The result for the system is as shown in Figure 6. Results The result of computations gives the following basic figures. Core height 128 cm Core radius 40 cm Side reflector thickness 25 cm End reflector thickness 84 cm Actual core volume 4.5 x 105 cc (less void) Apparent core volume 6.5 x 10 cc (with void) Total graphite in the 920 kg or core 0.92 ton (wt.) Total uranium 42.1 kg Total U-235 8.42 kg Average thermal flux 2.54 x 1014 The reactor is rather compact and the power density is extremely high as will be shown in heat transfer considerations in a later discussion.

-31 - -2 x 10io -2 6x 10 I I II a! 5x 10 -a,-a' 4 x 10'2 3 x 10'2 2x 10'2 I x 10-2 I I II m _I I I I I 1 40 45 50 55 60 R (CM) Figure 6. Determination of Critical Radius

-32 -Without proper core materials such as were proposed previously, the above features may not be easily attainable.

FLUX DISTRIBUTION Flux distribution was evaluated and plotted in Figure 7. Fluxes were interpreted on the basis of reactor thermal power output of 50 megawatts. The thermal flux distribution is surprisingly flat as seen from the boundary values shown below. Central fast flux cp1 1.52 x 1015 Central thermal flux P2c 2.69 x 1014 Average fast flux 7l 1.15 x 1015 Average thermal flux '2 2.54 x 1014 Core surface fast flux cp 7.88 x 1014 Core surface thermal flux 2.70 x 1014 Ratio, central thermal to aver. thermal CP2c/P2 1.06 Ratio, core surface thermal to aver. thermal cp2s/2 1.06 The flat thermal distribution offers a great advantage in obtaining flat power density in the core. However, it should be noted that the actual reactor flux under excess loading conditions is different from the above unless a special effort is made in the actual fuel loading. The presence of control-, safety-, and shim rods will further distort the flux to a minor degree. -33 -

-I 2x10 x -J LLa 10-I 0 0 10 20 30 40 50 R(CM) 60 65 Figure 7. Flux Distribution

EVALUATION OF TEMPERATURE COEFFICIENTS Temperature coefficients were found to be -1.85 x 10', using a simple migration area model. The coefficient is for the operating temperature, i.e., 1480~C or 2700~F, whereas a rough calculation shows the overall temperature coefficient for 20~C to be around -5.9 6 6.0 x 10', which indicates that the negativity of temperature coefficient is offset as the reactor temperature rises. Thus, the high temperature reactor is not as safe as low-temperature reactors. -35 -

Xe AND Sa POISONING Xe and Sa poisoning at operating reactor flux were calculated from the equilibrium concentration of each nuclei. The diffusion of fission fragments through the zirconium carbide coating was tentatively assumed negligible, thus poison is not removed during the operation period. The assumption is obviously not correct, although the result thus calculated will.give safe-side figures in the estimation of reactor reactivity due to the poisoning. TABLE V EQUILIBRIUM POISONING OF Xe AND Sa Equilibrium Nuclei ~a Concentration Ya Xe135 2.7 x 106 b 8.18 x 1019 221 Sa149 5.3 x 104 b 8.71 x 1020 46.1

REACTIVITY SPECIFICATIONS FOR CONTROL DESIGN Reactivity change due to fuel depletion and isotope production during the long operation should be taken care of by the enclosed shim rods. However, the control rods should be able to handle the considerable amount of excess reactivity above the level which the shim rods may override at any moment of the fuel cycle. Approaches If one defines zL/J- where Zu = Macroscopic cross Zm = Macroscopic cross Zp = Macroscopic cross P = Poison then reactivity becomes P P +Z 1+Z and P - u U section of uranium section of moderator section of poison Respective reactivities for different origins will be briefly reviewed here. Equilibrium Reactivity Due to Xe Poisoning Equilibrium poisoning, Po can be expressed as p =aX (YI.-. E \ *..A + 7X) q + axo

where X = Xenon I = Iodine a = Microscopic cross section X = Decay constant 7 = Yield fraction. For the system, Po = 4.74 x 10-2 Z = 1.02 x 102 Hence p = PZ = - 4.7 x 10-2 1+Z Equilibrium Reactivity Due to Sa Poisoning This is practically independent of thermal flux and -0.012. Xenon Poisoning After Shut-Down and Reactivity Equivalent Amount of poison P(t) after the reactor shut-down is ex pressed by; (t) x(t) a P(t) = X(t) u where X(t) = I Io (e XIt -eXt) + X0 e-xt AX - XI At around eleven hours after reactor shut-down, poison reaches the maximum value, and P max o. 6o Hence Pmax 0.60

Sammarium Poisoning After Shut-Down and Reactivity Equivalent After a conventional consideration, the maximum poisoning PO is expressed by PO = 1.5 x 1-16 cp + 0.012 where q) is the thermal flux. The asymptotic value for the present system is P = 5.01 x 102 00 Hence P = 5.01 x 102 Reactivity Equivalent Due to Temperature Rise Since Ptotal' 20 ~C and Ptotal, 1480~C Average value will be paver. - Hence, reactivity change = -5 = -6.0 x 10-5 = -1.85 x lo0 -)2 x 10-5 but AT = 146ooC 5.72 x 10-2. Reference Dead Time After the Shut-Down At 30 minutes after shut-down, the reactivity will be decreased by ebout 0.050. This 30-minute basis seems a reasonable dead time between the reactor shut-down and reactor re-startup in

-40 - case any nuclear power plant has mechanical difficulties. Hence, the evaluation of shut-down override is based on 30 minutes after the shut-down event. Summary The above results are summarized below. AK/K requirements for Due-To safety and control rods I Temperature + 0.057 qgu&ilbrium - 0.059 Xe and Sm hut-Down - 0.050 override Total - 0.052..______________________________________________.._ _._ _.._ _._ _.__-,_ _.__._ _._ _. It should be noted that the excess reactivity due to the temperature change (from operating temperature to room temperature for shut-down) takes a reverse sign and thus lessens the burden of control rod and safety rod excess reactivity requirements. Thus, if one assumes 100% safety margin for rods, overall reactivity requirement will be around 0.10 or 10%.

EVALUATION OF BASIC THERMAL CONSTANTS Since elevated conditions are used in the present system, the task of determining physical constants related to heat transfer calculations is one of the major problems encountered here. The extrapolation of physical properties at any known temperature and pressure region seems to be the possible approach in man cases. Conductivity, Viscosity and Specific Heat At one atmosphere pressure, helium has the properties listed below. Density #/ft3 Specific Btu/#~F Viscosity #/(hr)(ft) heat e aF p F OF 0.01191 0 1.240 all 4.58 x 10-2 45 temp. 0.0087 212 5.81 405 7.63 765 9.26 1125 10.76 1485 Thermal conductivity Btu/hr-ft2~F/ft K ~F 8.03 x 10-2 45 10.1 242 11.7 441 12.5 548 18.2 800 16.7 1000 -41 -

-42 - Among these, the variation of Cp is very slight and accordingly was set at 1.250. With reference to several previous investigations, one can further assume negligible effect of pressure on viscosity and thermal conductivity. Extrapolations of already existing data give for helium at 1200~F, 1000 psia, K = 0.1870 Btu/(hr)(ft2) (F/ft) p = 0.224 lb/ft3 = 0.0926 lb/(hr)(ft) Mass Flow Rate If one sets the inlet temperature to core to be 9000F, outlet temperature 1500 F, AT = 600~F but Cp = 1.25 and Q = wCpAT where w = Helium flow rate Q = Rate of heat removal Then w = C T 4= 4.55 x 105 P lb/hr where P = Power in MW. For P = 50 MW, Q = 170.65 x 106 Btu/hr and w = 227.50 x 103 lb/hr. Heat Transfer Coefficient For turbulent flow in the tube, the Nusselt number is expressed by the correlation

hD 23 ( 0.8 0.4 K: = 0.023 (Re) (Pr) where Re = and Pr = DG Dpv K K where v = velocity, ft/hr. Hence, K0.6 h = 0.023 0.2 Go-8 )'4 G where CT = wp Using the values found in h = 0.0252 or h = 0.0239 the previous section, one obtains K0.6 G0..8 K-6 G. Btu/hr-ft2-_ F 0.4 D0.2 G0.8 D0.2 Btu/hr-ft2- OF. The plot of h values as a function of G and D are shown in Figure 8.

-44 - 10 104 10l G(#/HR-FT ) Figure 8. Helium Film Coefficients as a Function of Mass Flow Rate with Diameter of Channel as Parameter. (1200~F, 1000 psia).

GENERAL RELATIONSHIPS Before proceeding further into any detailed specific discussions, more general relationships between the various reactor hea transfer parameters will be reviewed. Using the expressed for total t - On the other hand, qT = If one approximates then Local Gas Temperatures nomenclature shown in Table 6, basic relationships are heat accumulated in distance traversed, qT, as t = qT and w = G(a2) wCp z AcQdz o Q by sine functions, i.e., Q = Qo sin JT, Z z z qT = AC Q dz = AC % sin z dz qT )Qdz =Ac %n QO sin -- dz dz o 0 QoAcZ [ 1- cos Z ] I( z Therefore, QAc Z (1-cos -) tf = tI + G X a2Cp But (1) tf = t2 at z = Z, -45 -

TABLE VI NOMENCLATURE FOR HEAT TRANSFER ANALYSIS t = Initial mixed mean fluid temperature, ~F tf = Local mixed mean fluid temperature, ~F ts = Cladding temperature. ~F t = Fuel temperature at any point, ~F T = Maximum fuel temperature at any given z, ~F TM = The maximum fuel temperature, ~F a = Radius of channel, ft. 2b = Distance between channel centers (equilateral pitch), ft. r = Distance from center of channel, ft. G = Angle between axis and vector r, ft. z = Axial distance measured from entrance of channel Z = Total axial distance of free position, ft. Q = Local power density Btu/hr-ft3 D = Diameter of channel, ft. D,, = Diameter of channel, in. h = Film coefficient, Btu/(hr)(ft2)(~F) n = Number of channels = Void fraction in core L- = Length of core, ft. R = Radius of core, ft. QA = Average power density

-47 -TABLE VI (CONT'D) Qo = Maximum power density A = Heat transfer area of free position G = Mass flow rate lb/hr-ft2 t2 = Final mixed mean fluid temperature, ~F.

Thus 2QoAc Z t2 = t + 2a2Cp G On the other hand, nta2 = aiR2. n a2 = aR2 (2) But A = (l-c) R2 c n (l-a) i a2 Hence t2 t = 2(l-a) oZ (3) 2 i Cp G Relationship Between the Total Thermal Power and the Maximum Power Density If one lets P = Total thermal power of reactor, Btu/hr V = Volume of core (ft3) Then P = V QA z Q A Qdz where Q A Z Substituting Q = Qo sin -, one obtains Z 2 Q Z QA = Qosin dz o 2Z Q0 Thus QA = 2 Qo (4)

-49 - Hence Q = Q 2 0 A 2 V 2 But, V = (1-a) n R2Z and P = P(MW) x 3.143 x 106 Btu/hr Q = 17.065 x 10 P (MW) Btu/hr (5) 0 (l-c) R2Z Fuel Surface Temperature and Inlet Gas Temperature The following basic relationship holds through the channel wall, dq = h dA AT where AT = ts - tf Furthermore, dq = QdV = Q Adz and dA = 2r a dz Then QAcdz = h (ts - tf) 2j( a dz or QAc = Qo sin Iz (1-a),a = h (ts - tf) 2ra thais (1 -c) aQQ sin (6) 2h(ts - tf) = (6) 1 A a sin

-50 - At 1200~F and 1000 psi, o0.8 h = 0.0239 -G0 DO.2 On the other hand, G = w and q = 2.413 x 106 P = w Cp (t2 - tl) 3.413 x 106 P Cp (t2 - t1) 3.413 x 106 P or n x a2Cp (t2 - tl) Thus h = 0.0239 L11 3.413 x 106 P t a2Cp (t2 - tl) 3.413 x 106.... o ~ ~ 0'.8 1' or 1 (2a)0-2 p0.2 (! 1" h = 0.0239 LaX -o.8 1I R2Cp (t2 - tl). Let us call 0.0239 r.413 x 106 ]0.8 L0 9 t R2Cp (t2 - t 1) 0.2 ( = K Then pO. h = K --- Substituting this into equation (6) to obtain KP0. 2K\aO.2, (ts - tf d-a) a (t5 -tf) = a s a~~~~~c (%Q sin z)

-51 - Hence, or (i-c) a t = tf + 2K t$ = tl + ZOAc ts = G2a2Cp G (QO sin,z) ) (1- cos z) Z + (i-c) a 2aK (Qo n z ) 0( Z from equation (1).

MAXIMUM CORE TEMPERATURE Considering the fact that the temperature employed in the reactor system is very high, a safe estimation of the maximum temperature in the core is essential in order to prevent any possible fuel burn-up. Accordingly, a full chapter was devoted to the discussion of this problem. Fend's Work In a large solid reactor generating heat Q uniformly, with coolant channels placed on an equilateral pitch, the differential equation is r2T 1 aT 1 a2Tl k [ + + + =0 a- Cr 30 0k~~ X Ta Figure 9. Cooling Channel and Neighborhood Since the region has 30~ symmetry, only the shaded area bounded by the adiabatic planes q = 0, cp = 30~ and x = C need to be considered. The -52 -

-53 - temperature at the inner wall is assumed to be constant, Ta. No exact solution can be found because all the boundary conditions can not be satisfied in both circular and rectangular coordinates. An approximate solution in circular harmonics has been obtained by Fend et al. The result is _[T T bn (E(r)(r2+ (a l Q b 2/K 4 I a b b - 0.o1484 - cos 6 p - 0.00021 cos 12 Derivation of Maximum Core Temperature for the Present Reactor System Figure 10. Geometry of Cooling Channel and Vicinity As indicated previously the system uses equilateral pitch for cooling channels, that is, each tube is enclosed in a hexagon of diagonal 2b. The total area of the hexagon is 6 x A. But A = bh/2 and h = 3~f 2 Thus the area of hexagon = 2-b = 3 bl2 _ a2 2 3: 3 b2 1 a2 2 a

-54 - In our new nomenclature, Fend's result can be written as (t - ts)K Qb 2 - 1 [ 3 In 4, IC (,) a - ) 2 + b b i - 0.01484 (b)6 cos 6 b p - 0.00021 (-) b cos 12 qp The maximum fuel temperature at any given axial position will occur at r = b, cp = 30~. Then, (to - ts)K Qb2 1 r5 an 4 L 2g (b+ (a)2 a b 1] + 0.01496 - 0.00028 Substituting the relationship b2 2 a2 3 ~3cx into the above equation, one obtains 30ac3 (to - ts) K 2 ia2 Q - 4 2.n ('3 2 ) 3a's 3 + 2~3 -1] 2ic + 0.01468 Hence t - ts a2Q 2ir K K =12 K 1 12ct $5 K i 32$ 2it en (2 —) 7Q v3 + 23Q 3 2it - o.94128} or to = ts + a2 Q K1 In a short form, the above relationship can be expressed by K1 = 1- In (1.21/cz) + a -1.141 K1 4 uK c

But as seen previously, Q = 17.065 x 105 P. i z (1-ac) R2 Z z Thus, after some rearrangements, to = t + a2 Q k = [740 D2P {In (1.2 1/0) + a- 1.14}] sin z + t ra (i-a) R2Z K or to = t + K1 sin Z where K - 74o D2P {fn (1.21/a) + a - 1.14) 1 a (1-cX) R2Z K~ On the other hand, we previously had ts = tf + 2i (Qo sin Z ) t + D1 3 2 C (t2-1)0 = tf + G4.4D C (t2-tl)}0 f ~~~~?~ Z R0. 4.. ()0.2] sin - or t = tf + K2 sin z where K2 = [above Combining these expressions, to = tf + K2 sin LZ + K1 sin tZ where,i = - z

-56 - Using the previously obtained value for tf, Ac z to = t1 + QOAZ (1 -0 a2CpG p os ) + ( + K cos Z) + (K1 + K2) sin p4 Z = tl + K (1 - cos - ) + K4 sin v Z where QoAc Z K3X = aC 3 it EL2C G P - and K4 = K1 + K2 Now, to obtain the maximum core temperature, the condition is imposed so that dTo = 0 dz.'. K3 sin Pz + CJK4 cos kz.'. z = tan-1 (- K ) Thus T takes the form ofK.Thus Tm takes the form of = 0 Tm = t + K - K3 cos + K4 sin [tan'l ( [tan-1 ( K4 K 3 = tl + K3+ K2 + K42 - tI 2 tl + 3 t2 But K3 = 4 t2 + tl 2

-57 - one arrives at 18.85 x 10-4 D2PC + 0.529 D1 2 Pc = 1 The trial and error method gives D and Pc values satisfying the equation as follows. D P D P I D P D 5/8 190.0 17/16 14.5 1 1/2 2.0 11/16 127.0 1 1/8 10.5 3/4 85.o 19/16 7.7 13/16 59.0 1 1/4 5.8 7/8 4.0 21/16 15/16 28.0 1 3/8 3.4 1 20.0 23/16 Thus, there exist distinctive maximum power limits for any cooling channel diameter of core under each set of reactor conditions. In this particular case, D should be around 13/16" and less to keep the maximum core temperature 2700~F and below. Actually 3/4" size was chosen for the present design. Channel Diameter and Reflector Inlet Temperature Setting Tm, a, R, Z, K and P in the equation (8), one obtains (12.97 x 106 - 7200 T)1/2 = 280 D2 + 20.4 D12 T0.8 where T = t2 - t1

-58 - Thus finally, t2 + tl m 2 +/ 14 (t tl2+740 D2P {.n (1.21/c) + c- 1.14} [ 4 C2t a (1-a).Z K + 14.3 D12 [CAt 08 pO.2 2 Z Ro.4 oO.2 Channel Diameter and Power Output of Core Substituting Cp = 1.20 into equation (7), one obtains t -t T = t2 -tl Tm 2 + [ (t2 tl)2 + [740 D2Pc {n ( ) + a - 1.14} c (1-a) R 2 K 17.1 D1 2 (t2 - tl)0- Pc02 2 -1/2 + Z.R4.0a.2.. J (8) Setting the temperature and critical conditions as previously observed, tl = 900~F t2 = 1500~F Z = 128 cm = 4.20 ft R = 40 cm = 1.31 ft Qc = 0.2965 Tm - 2700~F Kc = 100

-59 - The result of calculation shows t2 D t2 D t2 D 1400 1.05 1900 0.55 2400 0.27 1500 o.88 2000 0.47 2500 0.22 1600 0.77 2100 0.42 2600 0.16 1700 0.68 2200 0.37 1800 o.60 2300 0.52 For convenience in further calculations, the values of cooling channel film coefficient h for a different set of n and D, were calculated. The total heat transfer area A is then calculated from the relationship of A = nItDL. The products hA as a function of channel number n are then plotted for different reactor portions, the channel diameter D being a parameter. One set of such plots which is for thermal shield is shown as an example in the Appendix. Similar plots were used for core and side reflectors, though they are not shown.

PRESSURE AND TEMPERATURE DISTRIBUTION IN THE REACTOR In the analysis of a high-pressure and high-temperature reactor, the distribution of temperature and pressure patterns is especially important in connection with the selection of structural materials and thermal and pressure shock evaluations. A general approach will be observed here. Power Generation and Coolant Temperature Drop in Thermal Shield. If one lets Pc = Power released in the core = Total power released in the core reflector and shield, and assume Pc = 0.85 PT' then Pc = 0.85 PT = 50 MW Pc = 58.8 MW 2 x 108 Btu/Hr If one further assumes 5% heat generation of the total in the thermal shield (abbrev. T. S.). QTS = 107 Btt/Hr. hA(T ) = 107 Btu/Hr Assume again the average temperature of thermal shield to be 11000F. Furthermore, from 2Cp (T2 - T1) 2 107 Btu/Hr, one can evaluate T - T - 10 = 2Cp = 35.2~F. -60 -

-61 - This is the coolant temperature drop in the thermal shield under the conditions assumed. Power Generation in Reflector Let us set Inlet temperature to side reflector = T2 Outlet temperature from side reflector = T3 Inlet temperature to bottom reflector = T3 Outlet temperature from top reflector = T4 Heat generated in side reflector = Hs Heat generated in end reflector = HE Due to the additional gamma heating of reflector and thermal shields, it is not easy to assign any definite power generation to the different portion of the reactor core through purely analytical means. As has been the practice, a rough assumption is made on the basis of estimated power flux and relative gamma emission effect. Thus it is assumed that (Hs + HE )/PT = 0.10 Further, Hs/HE = Side area of core/End area of core = Z/(l-a)R =2.16 Then, Hs + HE =.10 PT = 3.16 HE ' Hs = 13.68 x 106 Btu/Hr HE = 6.32 x 106 Btu/Hr

-62 - Local Coolant Temperatures in Reflector and Thermal Shield From the relationship u Cp (T3 - T2) = 13.68 x 106 Btu/Hr T3 - T2 = 48.1~F and, T4 -T3 = 622.20F T4 -T = 704~F T + T3 = 2400~F From these relationships, local temperatures are found to be T4 = 1511.1~F T = 888.9~F T8 = 840.8~F T1 = 805.6~F Furthermore, the average temperature for thermal shield is T + T2 Tav)TS = 825.2F. The coolant flow pattern for the reactor core.is shown in Figure 11, together with the local coolant temperatures evaluated thus far. Pressure Drops Through the Cooling Channel: Number of Channels For flow of fluids in channel, the pressure drop gradient can be expressed by d_ = 2fGu _ 2fG2 dx Dgc Dgcf which is the accepted Fanning's equation, where the notations are con ventional. But DG -0. 2 f = o.o46 (-)

-65 - Figure 11. Coolant Flow Pattern (Helium at 1000 psia)

Thus 0.2 o.8 p o 0.092.G dx P gcD -14 0.2 1.8 9.27 x 10 o G L AP 1.12 psi (19.7)2 p D 1. where o = In cp G = In lb-mass/ft2-Hr L = In feet Dtt = In inch p = In lb-mass/ft3 But for the.system L = 9.85 ft = 22.75 x 104 lb/Hr..AP 1.85 x 104 p0.Psi pn.8Du,4.8. =o and p are temperature dependent and thus the pressure drop is a function of temperature, number of channels and channel diameters. TABLE VII CHANNEL PRESSURE DROP AS A FUNCTION OF TEMPERATURE, CHANNEL NUMBER AND DIAMETER Temperature A P (~F) (psi) 825 3.29 x 104/.nl D 4.8 870 3.39 x lo4/nl8Dt 4.8 go0 3.46 x 104/nl'. 4.8 1200 4.22 x 104/n1 D.8 1500 5.05 x 104/nl'8D,4'8

-65 - There can be two approaches in finding the optimum cooling channel diameter and numbers as follows. i) Specify n and D abitrarily and find pressure drop through the channel. ii) Specify pressure drops and find n and D. From economic and operational viewpoints, pressure drop is the controlling factor, hence the latter approach was adopted in the work. Let us require that Ap through the thermal shield and side reflector/ are both 2 psi. Thermal Shield From Ap = 5.29 x 104/n18D 4.8 psi Since Tav)He = 8250F, ff n = 220 for D =1 As previously seen, hA(AT ) = 107 Btu/Hr But from the plot hA = 60 x 103 for n = 220, D = 1" Thus TS = 990~ This is the thermal shield channel -surface temperature. Side Reflector From Ap = 5.39 x 10 /nl D,, for Tav)He = 870~F, one obtains n = 76 D = 1 1/2" Tsr = 1265 F

-66 -Summary of Reactor Component Temperatures and Powers It would be interesting to observe the overall temperature distributions for different reactor components as calculated previously. Results are shown in Table VIII. TABLE VIII SUMMARY OF REACTOR LOCAL TEMPERATURES AND POWERS Reactor: Component Thermal Shield Side Reflector Bottom End Reflector Top End Reflector Core Average Temp. ~F 990 1265 1026 1637 Average Helium Temp. oF 523 865 894 1506 1200 Helium, Local: Inlet to Thermal Shield Inlet to Side Reflector Inlet to Bottom Reflector Inlet to Core Reflector Inlet to Top Reflector Outlet from Reactor System Local Power: Core Thermal Shield Side Reflector Bottom Reflector Total 806~F 841 ~F 889 F 900~F 1500~F 1511~F 50 MW 2.94 MW 4.00 MW 0.95 58.80 MW

-67 - Pressure Drops Through Core and End Reflectors 53t After setting the channel diameter D =, the number of channels through the core, the end reflectors were determined to be n = 521. Using the pressure drop equations as previously done for side reflectors, pressure drops were then calculated. Results are AP = 0.93 psi core APTR = 0.74 psi AP BR 0.51 psi AP = 2.18 psi total Pressure Drops Due to Entrance and Exit Losses in The Reactor Through the coolant path, enlargements, contractions and 180~ bends are experienced by helium as shown below. Enlargements AP1 = Entrance to plenum chamber AP3 = Exit from thermal shield AP5 = Exit from side reflector AP7 = Exit from top reflector Contraction AP2 = Entrance to thermal shield AP4 = Entrance to side reflector AP6 = Entrance to bottom reflector AP8 = Exit from top plenum chamber Bend (180~) AP = 180~ bend at top of thermal shield and side reflector APO = 180~ bend at bottom of side reflector and bottom reflector

-68 - Using the conventional approach, pressure drops and gains were calculated. Results are shown in Table IX. PRESSURE Portions AP2 AP3 A?4 AP5 AP 7 AP8 SP9 oP10 4Total TABLE IX LOSS OF COOLANT FLOW DUE TO ENTRANCE AND EXIT Pressure Drops (psi) - 0.17 1.47 - 0.22 2.59 - 0.19.o4 - 0.02 1.8 0.05 0.02 5.57

SUMMARY OF HEAT TRANSFER DATA The contribution of radiant heat transfer to the overall reactor heat transfer was separately investigated but found to be negligible for the present system. Some additional thermal and coolant flow data for the reactor core are summarized as shown below. Total Eeat Transfer Area Average Heat Flux Maximum Heat Flux Average Power Density Specific Power Film Coefficient Helium Flow Rate Total Reactor Pressure Drop 429 ft2 5.98 x 105 Btu/Hr-ft2 6.25 x 105 1.51 x 105Pc(Btu/Hr) Btu/Hr-ft3 34.15 Pc Btu/Hr-ft3 1238 Btu/Hr-ft2- ~F 4 22.75 x 104 lb/Hr 11.55 psi -69 -

CONCLUSION As has been observed in the foregoing discussions, the application of high-temperature nuclear reactors to power generation seems quite feasible with some safety margins in material, nuclear and thermal characteristics. Crude reactor design and analysis with accuracy of first approximation can be performed using presently available technical information and techniques. Thermal analysis can be as conventional and straightforward as it has been for the usual low- or medium-temperature reactors designed, although a rather careful treatment will be essential in the estimation of the maximum core temperature to secure the ultimate reactor safety. The problem of handling nuclear properties at the highly elevated temperatures is not an easy one at the present time however, and several basic assumptions are inevidably needed. Further studies on high-temperature reactor materials as well as nuclear properties at elevated conditions will certainly be helpful in a more elaborate analysis of reactors of this type. -70 -

BIBLIOGRAPHY 1. Chernick, J.. The Dependence of Reactor Kinetics on Temperature, BNL 173, 1951. 2. Fend, F. A., et al., An Approximate Calculation of the Temperature Distribution Surrounding Coolant Holes in A Heat Generating Solids BMI-T-42 USAEC. 3. Glasstone and Edlund, Elements of Nuclear Reactor Theory, D..Van Nostrand, 1952. 4. Hughes, D. J. et al., Neutron Cross Sections, BNL-325 United States Government Printing Office) 1955. 5. Kennard, E. H., Kinetic Theory of Gases, McGraw Hill, New York, 1938 6. Keyes, F. G.,"Heat Conductivity, Viscosity, Specific Heat and Prandtle Numbers for 13 Gasestt Technical Report 37, Project Squid M.I.T., April 1, 1952. 7. McAdams, W. H., Heat Transmission, McGraw Hill, New York, 1954. 8. Murray, R. L., Nuclear Reactor Physics, Prentice Hall, Inc., -1957. 9. Ohlgren, H. A., Hammit, F. G., Component Optimizations for NuclearPowered Closed-Cycle Gas Turbine Power Plants, IP-245, Univeisity of Michigan, September 1957. 10. Osborn, R. K., Personal Communications (1957 - 58). 11. Peaceful Uses of Atomic Energy: Proceedings of the International Conference in Geneva, August 1955. Volume 2, Physics, Volume 3, Power Reactors. 12. Roe, G. M., The Absorption of Neutrons in Doppler Broadened Resonances, KAPL 1241, 1954. 13. Schneider, P. J., Conduction Heat Transfer, Addison Wesley, 1955. 14. U. S. Atomic Energy Commission Reactor Handbook (Materials) McGraw Hill, New York, 1955. -71 -

APPENDIX -72 -

1300 1~200W 1 ( Helium film coefficient as function of number of channels i\ ~~~ ~~\ ~with diameter of channel as parameter. Il00 \ h =Bfu/hr/ftZ/~F Based on reactor power level of 58.8 mev heat to 1000 \ shield 55% of reactor power. ~900-~ V~ \ //e/Helium flow rate =w -22.75 x/04#/hr 800 700 h 600 500 - ' = 1/2" 400 300 200 I - D 1 1 1/2 | 00 0 20 40 60 80 100 120 140 160 180 200 220 240 260 280 300 320 340 n THERMAL SHIELD HEAT REMOVAL I — A wo

A I I 0 20 40 60 80 100 120 140 160 180 200 220 240 n 260 280 300 320 340 360 THERMAL SHIELD HEAT REMOVAL

130 120 1 10 Product of helium film coefficient and heat transfer as function of number of channels with diameter of channel as parameter Helium flow rate =w -= 22.75 x I04#/ based on reactor leve of 58.8 mf heat to shield - /x /0 7Btu/hr h A Btu/hr/~F to 0 x.c - %f< D =3/4" D=/" \ I D = -/2" 20 40 60 80 100 1200 200 220 --- 80 00 20 140 160 180 200 220 n 240 260 280 300 320 340 360 THERMAL SHIELD HEAT REMOVAL

UNIVtHWIIIY I MIU MIUIAN 3 9015 02651 0720