- o . 4 - L ! Y Pl 2 . ] &oean g T W @iy M UNCLASSIFIED MIT-5000 REACTORS ~' RESEARCH AND POWER NUCLEAR PROBLEMS OF NON-AQUEOUS FLUID-FUEL REACTORS October 1 1952 Authors Clark Goodman John L. Greenstadt Robert M. Kiehn Abraham Klein Mark M. Mills Nunzio Trallil Consultants -Harvey Brooks Henry W. Newson shington 25, D. C. NUCLEAR ENGINEERING PROJECT MASSACHUSETTS INSTITUTE OF TECHNOLOGY " Manson Benedict, Director CLASSIFICATION CANCELLED DATE FEB 28 1957 U.S. ATOMIC ENERGY COMMISSION -Hr 2 ) NEW_YORK OPERATIONS OFFICE N h N\ QA lE Chief, Declassification Branch LEGAL NOTICE This report was prepared as an account of Government sponsored work, Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, express or implied, with respect to the ac- curacy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, methed, or process disclosed in this report may not in- fringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report. As used in the above, ""person acting on behalf of the Commission” includes any em- ployee or contractor of the Commission to the extent that such employee or contractor prepares, handles or distributes, or provides access to, any information pursuant to his em- ployment or contract with the Commission. UNCLASSIFIED . [ ] e & e ' e £y TABLE OF CONTENTS Chapter I General Considerations sseecceessccsnces 1.1 INtroduction eesecocecrocsassesce 1.2 Tast ReaCltors cevececcccvossacces 1.3 Thermal Converters cecesceesccceces Glossary of Symbols Used in Chapter I .. Chapter II Fast Reactors ® S0 5000 H P SO S 0SS OSSO B l. Nuclear Constants O 0O O B0 O 0P OO R P OO SO EE NSRS E O PDE 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2. Calculation 2.1 2.2 2.3 2.4 Total Cross-SectionS.cccecccccses Transport Cross=-SectionSeeecseees Inelastic Cross~-Sectlons eceeseess Fission Cross-Sections eseeesecoses Captule Cross-Sections eeecececse Scattering Cross-Sections eceeees Neutron Yields .eeeccecscscccccnas Neutron Spectra .ecseceesceescsscce Recommendations eeeececccccecenens MethOdS ® O & 0 0 &8 0 0 050 2P L BO e ST S SLSDS Bare Reactor Multigroup Method .. Multigroup Estimates for Blanket * & 00 0" P 5" OO S Oe s 08N P Two-Group Two-Region Method ..... One-Group C310Ulat10ns o¢ o8 e 3. Results of Fast Reactor Calculations seeesceee 3.1 Introduction .eeeceececcecccccccccsce 3.2 Bare Reacltor ececececccccsscccnces 3.3 Two-Group Two-Region Calculations 3.4 One-Group Three Region ceeecvvens 3.5 Special Multigroup Calculations . . N3 5, ¥ £, ) - Page No. 11 11 12 20 24 25 25 25 25 27 30 32 36 38 38 39 48 48 50 56 59 66 66 66 88 95 97 .‘,rvr Y ! - n" v 3. Results of Fast Reactor Calculations (contd.) 3.6 General TrendS cceceescscscscaves 3.7 Final Design of Fused Salt ReaCtor o0 & 0 0 00 O 6P BSO OO0 s ODS Y}, TFast Reactor PolsSoning sececescssccccssssocces 4.1 Introduction eeeececeeeescrcsccas L.,2 Fission Products eceeccoeveccocses 4.3 Higher ISotODES ccereececcsenvcoas 4.4 Engineering Considerations ...... 5. ContrOl MethOdS 5 O % B 0 080 %08 SO B S S e LN ON SN POSRDS 5.1 General Considerations .eceeeecees 5.2 Control Calculations sececcesccsce 5.3 Control of the Fused Salt Reactor o & 5 5 & & ¢ 488 8 S5 80 S B2Ee T e S 0SS Glossary of Symbols Used in Chapter II . Acknowledgments cvesescecccncscsosnscncena Chapter III Thermal Reaclors seceeececsscsocsoseccosse 1. Introduction ...cccevvvvvrenecccercnscescannns 2. Bare Homogeneous ReactOr ceeeccccsrecscccsccsss 2.1 Definitions and Basic Constants . 2.2 Analytical Expressions ..cecceceee 2.3 The Conversion Ratio, CiRe ceeose 2.4 Results of Calculations sceceecees 3. Poisoning Effects S 0 8 08 800 8800 00 5e BSOS e s seR 3.1 Uranium 236 S 9 9S00 H S PSSO N PR S SSDR 3.2 Tast Neutron Reactions of Beryllium ..000‘00.0.0....0..... 3.3 PFission Products eceeeecevecceescne 104 111 127 127 127 128 130 131 131 132 140 145 150 151 151 153 153 159 164 165 166 166 173 174 4., Heterogeneous ReacCtorS eeeeccecccsscsssccsesss 4,1 The "Immoderate" Blanket ccececes 4.2 The Bffect of Be Lumping eecveeee 5. Final Design CalculationsS .eecececvecscccssssne 5.1 Reactor Constituents and PrOPerties ® S 00 & 0P S P O OE PO eY S8 b 5.2 Calculation Procedure c.ecessseesee 5.3 Results .!.......‘............... 5.4 Critique of ReSultS cesececcccces 6. Recommendations ciceccescccscessccssscscancans 6.1 Design Studies cececeveccccsccces 6.2 Nuclear Data Studies cecececscces Glossary of Symbols Used in Chapter III Chapter IV Comparison of Fast and Thermal Conver- TerS ceeessctscccvcsscssonsscsnncsnnne 1. Core STruCture eceevvsvecsesesesvsesenssessscnce 2. Reflector Structure ..eeeceeevesvcocscsccsnscsns 3. Blanket Structure ceceecccecceccessecscsssossssscs %, Critical Mass and INventory .eeeeeecceecessess 5. Conversion Ratio stesescececcsconssoscscnnncnscs 6. Parameters and Processing of POlSONS seeeecces 7¢ CONtrol ceeeeececccesvessacssccscascsosscncsse Glossary of Symbols Used in Chapter IV Appendix A Procedure for Homogeneous, Fast, Bare Reacltorl teeesvncscccscccscosccccncsncncne Appendix B Summary of Calculations on Bare Homo- geneous Thermal Reacltors .sseececcssses Appendix C Secular Equations eeeeesesceocssccescsss Appendix D Effect of Fast Neutron Reactions of Be on the Conversion Ratio ceeceececencens Appendix E Two-Group, Two-Region Reactor Equations. References ® 0 % O BB OO OO B USRS PRNETOSESNEPEES OO EER e Acmow:—edgments ® 08 009 00 02T PO OE OO RO NSO S OO IYTEOIEOSORTEYOPRETS r ¥ i * s 9 0 e & * - L J - . . Page No. 177 178 181 187 187 188 192 197 202 202 202 205 208 208 209 209 209 210 212 21k 215 216 231 238 247 257 261 264 TABLE II-l.2-1 11-103-1 II-1.3-2 IT-1.4-1 II-lOS-l II"lo 5-2 II"lo 6“"1 II"lo 9-1 II"3 02-1 II"3 . 2"'2 IT-3.2=3 II-3.2-4% 11-3 * 2-5 II"3 . 3"'1 II-3.5-1 II"B . 5"2 II-3.5-3 II-3.7-1 II"'307"‘2 ;:.u : PTERT L L LIST OF TABLES = Transport Cross=Sections eeeecececes Inelastic Cross=Sections .cceeceses Assumed Spectral Distribution of Inelastically Scattered Neutrons. FiSSiOH CrOSS-SGCtionS S o0 P OO BSBLIEOGOEOETRTS Capture CTOSS-SGCtiODS 2209 98 e0s 0000 Effect of Changing <(Bi) on C.R. in U-Bi SYStems S e P e B IPIPOERTEOIRNSEY Degradation: Product of Scattering Cross=-Section and Mean Loga- rithmic Energy Decrement (ficé)... Resume of Nuclear Data Needed .ce.. System Constituents for Bare Re- actor Multigroup Calculations ... Breeders - Results of Multigroup Bare Reactor Calculations ..cceee. Converters - Results of Multigroup Bare Reactor Calculations ,...e¢e. Breeders - Neutron Balance ...cccee Converters - Neutron Balance .eesee Comparison of Special Multigroup Calculations with Two-Group Calculations * & 8 & 0 0 & & 09 PO Ve S e e 0o System Constituents for Special Multigroup Calculations ececeeeee Results of Speclal Multigroup Calculations cesccececcevcsccscnee Special Systems - Neutron Balance . System 24: System Constituents and Results es e s e s e s s sOBREREEGEE Systemp 24 - Neutron Balance escecese e 2724 +05 : THNRE Page No. 26 28 29 31 34 35 37 k7 71 72 73 7 75 89 99 100 101 112 113 - e W x g | Page No. TABLE II-3.7-3 System Constituents of Fused Salt | Re&CtOP ooooo & & ® ¢ & O " O S S8 B O B g ¢ e s 118 II-3.7-4 Results of Fused Salt Multi- region Calculation .¢ccccseeccee 119 II1-3.7=-5 Neutron Balance - Fused Salt Reactorj Multigroup Calculation . 120 II-5.4=1 Delayed Neutrons ......ecceeeeeees 137 I1I-2.1-1 Temperature Corrections for Nuclear Datl ececeocosssessscessss 15U ITI-2.1=2 Thermal Neutron Properties of Thermal Reactor Constituents .... 155 111-2.1-3 Properties of Isotopic Uranium Mixtures ® 8 5 % 0 80 S PO GBS HSE BB s 157 III-2.1-4 Resonance Escape Probability ...... 159 III“2.2—1 Age tO Therm&l ® 0 0 9 08 0606006 BB s e et oS 161 I1I-2.2=2 Age and Density of Bi-Be Mixtures . 163 III-3.1-1 Effect of U-236 on Conversion Ratio in Thermal Reactors «...... 171 III-4,2-1A Results of Cell Calculations ...... 183 III-4.2-1B Required Data for Evaluation of p . 184 ITI-5.1-1A Reactor Constituents and Constants. 188 I1I-5.1-1B Reactor Constituents and Constants. 189 1II-5.2-1 Summary of Reactor Properties ..... 191 III-5.3-1 Results Qf Calculation ecscecesessces 192 III-5.3=2 Neutron BalanCe .....ceovesessescees 193 I11-5.3=3 Delayed Neutrons ...ccececessccsnecs 196 III-5.4-1 Homogeneous Reactor Results ...... 198 III-5.4=2 Gain in Production Ratio by Use of Blanket S 2 5 8 65 86 00Ot O SO B OSSN SS 201 S | EY ;-;Q %x‘ TABLE B-l B=-2 B-3 Bl B-5 B-6 B-7 B-8 B-9 B-10 B-11 C-1 C=2 C-3 D-1 D=2 w Page Summary of Calculations on Bare Homogeneous Thermal Reactors .... " . & 88 " ® o e ® & & & n L 2 N " n .‘.Q. Numerical Values Used in Computa=- tion *© 5 0 0 8 06 0SS S BRIV B0 SN OO G LS U-236 Concentration and Bulldup Time [ I N B BN B BN O BN NN BN R BN N NN NN CEE I S BN N B N BN B AR AN Summary of Secular Results ....... Data for Fast Neutron Reactions of Be.'.... ..... * o 0 0 5 0P OSSO B0 PSS e L16 Concentration Factors .....c.. No. 232 232 233 233 23% 234 235 235 236 236 237 ol 245 246 252 253 FIGURE II-2.4-1 IT-3.2-1 IT-3.2=-2 IT-3.2=-3 II-3.2-k4 IT-3.2-5 IT-3.2-6 II-3.2=7 I1-3.2-8 II-302-9 IT-3.2=-10 II-3.2-11 IT-3.2-12 II-3.3-1 II-3.3=2 II-3.3=3 II-3.3-4 LIST OF FIGURES Three Media-One Velocity Systems .. Pu013-U01h Breeders S e s s 0ess s res e PU**U-238——Bi Breeders "o 0000w U-235'-U-238--Bi COnverterS ses s esee UClh‘N&Cl Converters sss0sevesses s System 1l4: Spectrum of Fissions and Fraction of Fissions above u System 14: Flux SpectITUlm seveeess. System 15: Spectrum of Fissions and Fraction of Fissions above u System 15: Flux SpecCtrull ceeecseese System 16: Spectrum of Fissions and Fractlion of Fissions above u System 16: Flux Spectrum ....ccc.. System 17: Spectrum of Flssions and Fraction of Fissions above u System 17: Flux SpecCctrul .eeseecees Reflector Size vs. Core Size sccese Reflector Size vs. Core Size eceecee X.C.R. X.C.R. (Bare Reactor) versus __Core Size Bare Core Size X.C.R. X.C.R.(Bare Reactor) versus Core Size Bare Core Size =i » ¢ & ¢ 0 &0 %" s OB P S E NS e * 00 080000 P eSO PS e Page No. 76 77 78 79 80 81 82 83 8l 85 86 87 91 92 93 9l FIGURE II-3.4=-1 II-305‘1 II-3.5-2 II"3 . 7-1 II-307'2 II"‘3 . 7-3 I1I-3.7-l II-3.7-5 II-3.7-6 I11-2.1-1 II1-3.1-1 ITI-3.1=-2 III-‘-!-.2-1 III"L"Q 2"'2 IT1I-5.4=1 Reflector Thickness and X.Co X.C.o R BN N W NN N R I N A versus Core Slize System 52: Comparison of the Spectrum of Fissions of a Bare Reactor and a Blanketed ReaCtor "TEEEEREEEEERE XN NN B I B B BN N I S B N N Comparison of the Flux Spectrum of a Bare Reactor and a Blanketed Reactor System 24: Spectrum of Fissions and Fraction of Fissions above u .. ¢ 600 F O S0 s o080 System 24: TFlux Spectrum ..ceveceees Fused Salt Reactor - Flux Spectrum .. Fission Power Distribution in the Fllsed Salt Reactor ® $ 6 0 6% &0 S OO OO e S . Total Neutron Flux per Unlt Radial Distance vs. Radial Distance ecc¢eeee Normalized Flux Spectrum at Various Radial Distances ® 5 5 €9 5O 0 e 8 s s 0 0 Experimental Resonance Absorption Integral, A, versus "¢ /U" ........ Reaction Equations ...cceevveccscccosse Higher Isotope Chains ....cecoeoeseos f and p versus Radius of Be Lump for Various N(Be)/N(BL) ....ccveeee Product pf vs. Radius of Be Lump ,... Critical Mass versus Production Ratio for One-Zone and Two-Zone U-Bl-Be Reactors & ¢ & &6 & ¢ 8 058P0 0L eSS OSSN _9- Page No, 96 102 103 114 115 121 122 123 124 158 167 168 185 186 199 -10- Page No. FIGURE A-1 System 21: Spectrum of Fissions | and Fraction of Fissions above u 224 A-2 System 21: Flux Spectrum ...ccceee 225 u D-1 q)(u) =J:,J(f (u') du' as a Function of ENergy .cccsccecoscns 254 D-2 3% = 7 {HEY . Number of Be Atoms Undergoing (n,a) Reaction per Unit Source Neutron vs. X. ... 255 D-3 g = Average Fraction of Final L16 Concentration as a Function of M(U-235) ® & 5 & 90 @ s SO O PSSP LS 0O PSSP PSS 256 -11- % I. GENERAL CONSIDERATIONS 1.1 INTRODUCTION &l . .. . The primary purpose of the Nuclear?Efigfifiéé}ffifii?roject at M.I.T. during the summer of 1952 was to investigate the problems of reactors using non-aqueous fluid fuels for the production of plutonium and to recommend a program of research and development to supply the information needed to provide a sound basis for the engineering of this important type of reactor. Aqueous fluld-fuels are receiving attention at 0Oak Ridge and elsewhere. The results of this Project are being described in three companion reports., l. "Engineering Analysis of Non-Aqueous Fluid-Fuel Reactors" - (MIT-5002), 2., Y"Chemical Problems of Non-Aqueous Fluid-Fuel Reactors" - (MIT-5001), and 3. This report. 5 The first of these reports describes the objectives of the Project, the lines of investigation pursued, and the main conclusions drawn. It describes in detail the engineering studies carried out by the Project and the bases for them. It summarizes all recommendations for future research and de- velopment. The second of these réports describes the chemical studies conducted by the Project and gives details of the program of chemical and chemical engineering research re- commended by it. The present report treats the nuclear studies conducted by the Project. The basic nuclear data and design methods are described and the results of the nuclear studies are given in detail. A research program on nuclear properties of importance to non-aqueous fluid-fuel reactors is recommended. == ¢ ?,l =12~ Chapter I of this report outlines the considerations which led to the choice of two reactors for detailed study by the Project: (1) A fast converter using as fuel a solution of UCl,, | in fused chlorides | (2) A thermal converter using as fuel a liquid alloy of U in Bi, and lists the main characteristics of each reactor. Nuclear studies on the fast and thermal reactors are described in Chapters II and III, respectively. Chapter IV compares the two reactors. The Appendices contain details of calculation methods, and the results of nuclear studies not directly related to the reactor processes given engineering study. 1.2 FAST REACTORS The two main guestions regarding fast reactors asked at the beginning of the Project were: (1) Should the fast reactor to be investigated be a converter or a breeder, and (2) What fuel system should be chosen? BREEDERS VS. CONVERTERS. = As shown in Chapter II of thls report, fast, fluid-fuel breeders may yield a breeding gain of the order of 0.6, whereas a fast converter using the same type of fuel except for the interchange of U-235 for Pu-239 will give a conversion ratio of around 1.15. Cost analyses described in more detail in the engineering analysis report show that plutonium can be produced more economically in a fast converter than in the corresponding breeder, under the cost bases adopted for this project. The cost advantage of the fast fluid-fuel converter compared with a feasible fast breeder arises from two main causes: 1). The high unit cost of Pu compared with that of U-2353 i.e., U=-235 is cheaper to burn or store than Pu-239 on a gram for gram basis, because the -13- projected cost per gram of Pu-239 is still larger than the present cost per gram of U-235, 2). The inventory charges on Pu-239 are based upon the total critical mass of the breeder, but only on the equilibrium concentration of Pu-239 in the converter, which may be quite small in comparison. Hence for the low specific powers considered for the fused salt reactor (~ 300 watts/gm), an equivalent breeder (also with a. specific power of 300 watts/gm) would have such a largé inventory charge assessed against it, that it could not compete with a converter in the economic production of Pu-239. Higher specific powers, if possible engineeringwise, would decrease the disparate cost estimates for Pu-producing breeders and con- verters. Even if no charge was made for inventory, however, some cost advantage would remain with the converter under the cost bases adopted by this project. SELECTION OF IFUEL. - Fast reactors must have high con- centrations of uranium and low concentrations of absorbing diluents which should not moderate excessively. An initial survey of possible fluid core materials showed that few, if any, fluid alloys with uranium have sufficiently satisfactory nuclear and engineering properties to be practical for fast reactors. Fused salts are an alternative to the fluid alloys. The fluorides were eliminated on the basis of thelr excessive moderation which would increase the overall a of the melt. The bromides and iodides were eliminated on the basis of their un- favorable capture cross-sections (associated with high atomic weight). These decisions were based on the following useful rule-of-thumb for estimating the upper limit for the micro- scopic capture cross-section of diluents, <,(D), in fast con- verters: -14- BN < (D) < N-%z-?- x 0.2% barns (1.2-1) c N D . o at 0.2 Mev mean neutron energy. This relation assumes as a practical basis that the decrease in the conversion ratio resulting from capture in diluents should be less than one-tenth the conversion ratio for a non- poisoned reactor. For example, in a hypothetical mixture: U235Brh, which is certainly the lowest possible concentration of Br in a bromide fused salt, S, (D) = 0.24/% = 0.06 barn ‘ max - at 0.2 Mev., However, Br has an estimated ©_ = 0.1 barn at this energy and<3;(iodine) is even greater.c Bromine and iodine were ruled out for this reason, leaving chlorine as the only possible halogen, T,(C1l) = 0.003% barn at 0.2 Mev. The fused salt, uranium chloride, was chosen on the basis of metallurgical, engineering and nuclear considerations to be the subject of the fast reactor investigations by this project. The corrosion problem remains as its most detri- mental feature. To lower operating temperatures in the fused salt, Pb Cl, and NaCl were added to the UCly. The optimum ratio of U-238/U~-235 was then determined by minimizing the estimated production cost of plutonium in this system, in- cluding the inventory charges on external holdup. Internal or external cooling is a major consideration in the design of a fast reactor. Externally cooled systems have the advantages of safety and replaceablility of heat exchangers, and absence of parasitic loss of neutrons to cooling flulds and internal structural materials. Internally cooled systems have the important advantage of lower critical mass due to absence of non-reacting inventory in external heat exchangers. ' For the fused salt system selected for detailed investigation, an engineering study suggested that the externally cooled system would be preferable. For a liquid-metal reactor, it is probable that an internally cooled system would be preferred. -15- For fast reactors the highest possible Conversion Ratios (and Breeding Ratios) occur for U-238/U-235 ratios of ~%—§% equal to or less than 2.5. Essentially, the internal conversion ratio depends only upon the ratio ”%_?% implying that the ratio‘ggég%% is a constant over the reglon between 0.1 and 1 Mev., The upper limit of g 38 is about 125 higher ratios will lead to either in- finite critical mass or intermediate to thermal spectra, and the systems are no longer considered to be fast re- actors. For externally cooled reactors, it turns out that slightly higher ratios of N 28 y 1loe. 3 to 5, will give the minimum overall cost, even though some conversion ratio has been sacrificed, because the higher dilution with U-238 decreases the holdup mass of U=235, | Internally cooled systems will contain structural material (Fe, say) and coolant (Na, say) to remove the heat. Admittedly, external holdup is decreased, but with a sacrifice in C.R. due to parasitic absorption in the above materials. The high densities of liquid metals imply small critical masses of fissionable material, but most known uranium alloys involve high operating temperatures when used in the fluid form. | Captures in fuel material and reactor poisons reduce . the conversion ratio to about 1.15 in a practical fast con- verter. The relatively low capture cross-sections of the fission products and the feaslbility of removing them by processing make the poisoning effects of fission products negligible in fluid-fuel fast reactors. . In fact, large fractional burnup - and hence large buildup of fission products - may be tolerated in fast reactor designs. This can be seen quite readily from Eq. (1.2-1). . When -16~- N(F.P.) = N(25), i.e. 33 per cent burnup, qE(F.P.) should be less than 0.24% barn if the decrease in C.R. is to be less than 10%, whereas, in fact, the estimated average value UE(F'P') is 0.2 barn at 0.2 Mev. U-236, formed by U-235 parasitic capture, is a trouble- some poison in fast converters. Unfortunately, aé(U-236) is not known in the fast region. It was assumed to be the same as U-238 in all NEP calculations, i.e. g (U-236) = 0.22 barn at 0.2 Mev. On this basis, U-236 is about equivalent to the average F.P., atom for atom. However, because it 1s isotopic with the primary fuel, U-235, the removal of accumulated U-236 and its subsequent separation, presumably by gaseous diffusion, would be quite expensive. Thus re- processing costs dictate the upper limit, and the loss in conversion ratio by parasitic capture in U-236 sets a lower limit on the rate of reprocessing the fuel for U-236, For Pu Eroducing reactors, one should note that the ratio %%%52% 1s smaller in the fast region than in the thermal. Therefore, Pu-240 contamination of the product Pu-239 is of less importance in a fast converter than in a thermal converter. The fast converter structure studied consists of a semi- spherical core, reflector, and blanket. The reflector is used to reduce the inventory of critical mass. In general, the re=- flector should be a dense material of high transport cross- section and low capture cross-section, such as lead. Iron could be used as a combination container-reflector provided it were not too thick. Reflectors should not be designed such that they more than replace fissionable material, for large thickness of iron (or, to a lesser extent, lead) seriously reduce the conversion ratio by decreasing the leakage flux reaching the outer blanket, <, -17- The blanket structure should be sufficiently thick that it does not allow an appreciable number of neutrons to leak out of it. Moreover, the blanket should completely cover the core so that no leakage gaps are present. This last requirement is difficult to achieve in practice, since cooling pipes and control mechanisms must be in- serted through the reactor. It is obvious that the blanket should be processed often enough to keep the inventory of Pu-239 to a minimum. It is well to keep in mind that the fast fission effect in U-238 approximately balances the parasitic captures in structural materials in a well de- signed blanket. Constructional problems may dictate that a suitable nuclear reflector is impractical. It should then be remembered that the blanket, the containing structural materials (Fe), and the reflector control rods (Pb) will act to some extent as an effective reflector if properly de- signed. As the generation time (<10~ reactor is much shorter than in a thermal reactor, fast reactors are inherently harder to control. Moreocever, in externally cooled fast reactors with their rapid fuel flow rates there can be up to a 60% loss of delayed neutrons in the heat exchanger, depending upon the relative fuel transit times. Flinid fuels will however have large negative temperature coefficients due to thermal expansion giving a distinct safety advantage. The use of absorbers for control mechanisms is not practical, as absorption cross- sections are low in the fast region and because the addition of absorbers will decrease the conversion ratio. Reflector control is the most efficient method of controlling fast 6 seconds) in a fast reactors, because its use will not significantly destroy 18- the conversion ratio. Fluld fuels have the additional feature that they are not as sensitive as solid fuels to radiation damage. Processing problems are also greatly reduced, enabling the concentrations of fission products and Pu to be kept small. In general, as diluent materials are added to the composition of the non-aqueous fluid-fuel reactor core, the critical mass at first rises quite slowly since the diluent merely replaces fissionable material (lower leakage loss and number density of U-235 compensate for fuel dilution). The neutron spectrum is rapidly degraded to a mean energy of about 200 Kev. As more diluents are added, the neutron loss by parasitic capture more than compensates for the improved transport cross-section (less leakage). The neutron spectrum falls into the 100 Kev region, or less, and the critical mass Increases rapldly - in many cases it becomes infinite. As further dilution of fissionable material 1s brought about, especially with moderating material, the critical mass of the reactor may again become finite as the neutron spectrum is degraded to thermal energies, for in this region the fission eross-section of U~-235 increases rapidly with decreasing energy. Further dilution with good moderators causes the critical mass to go through a minifium, until the uranium is so diluted that the loss of neutrons to capture and leakage is not balanced by the neutron production from U-235. The critical mass again goes to infinity even though the microscbpic fission cross-section 1s very largé in the thermal region. It is evident that only for particular combinations -19- of enriched fuel, moderator and other diluents may a reactor become critical in the thermal region. One notes that the critical size of a reactor depends upon the buckling and atom density of the fuel material, and that the critical mass is inversely proportional to the square of the U=-235 atom density. As the conversion ratio is decreased by parasitic capture, the best converters have the following general design specifications. 1). small parasitic capture 2). large macroscopic transport cross-section 3). large U-235 atom density 4). high specific power Items 1, 2 and 3 imply small critical mass (large buckling) and item 1 implies large conversion ratio. To bring in the last, important general design parameter for Pu-producing reactors, we note that the inventory charges on the final product are inversely proportional to specific power. To reduce the effect of inventory on the cost of Pu-239, we ~add item 4 to our list, Recommendations for future work involving nuclear data which are important in fast reactors are to be found in detail in Section II-1.9, The major deficlency in data is to be found in the fast inelastic and capture cross- sections. | Paralleling this lack of information are a number of engineering deficienciles which are to be found in detail in the engineering analysis report. The major reason why these deficiencies are important nuclearwise 1s that they set an upper limit on the specific power of a fast reactor. Until these deficlencles are removed by further basic research, we can not be certain of the feasibility of fast, non-aqueous, fluid-fuel reactors for the economical pro- duction of Pu-239. | - s TS . e, " - - 4“("‘_' =20~ M l.3 TIHERMAL CONVERTERS Preliminary analysis of the U-Bi system showed that this was the best liquid alloy fuel for thermal reactors and that a reactor using this fuel would have an acceptable con- version ratio and critical mass. Since a fused salt system had been chosen for the fast reactor, it was decided to study the U-Bi system for a thermal reactor, in order that l1iquid metals could also be investigated by the Project. A limited amount of study was also given to fused salts for thermal reactors, with the conclusion that solutions of UF), in fused fluorides were the only ones competitive with the U-Bi fuel chosen for detailed investigation. Fluoride fuels merit further study. Of other fused salt possibilities, ci, Se, Te, Bry, I, N and CN were eliminated because of un- favorable cross-sectionss carbides, oxldes, sulfides, silicides and arsenides because of high melting point; phosphates, sulfates and nitrates because of poor thermal and radiation stability. A more detailed report of the search for fused-salt mixtures for thermal reactors 1s des- cribed in the engineering analysls report. The particular U-Bi thermal reactor studied consists of a core in which U-Bi solution flows through holes in a Be matrix. Some of the special problems which must be con- sidered in choosing a particular design are conversion ratio, critical mass and inventory, processing rates, internal vs. external cooling, uniform or variable Bi/Be ratio, and control. The maximum possible C.R. in a thermal converter employing U-235 is 1.10. This maximum is reduced by para- sitic capture and leakage. A rough and ready criterion to test whether parasitic capture by the basic constituents of the reactor is excessive is N(s)ag(s) N(s)gz(s) N(25)0;(25) * N(DRoL(25) <.1, (1.3-1) e Z:: ————-_.. - € -21- where N(s)4, elastic scattering is the only process for de- grading the neutron energy. The inelastic cross-sections used are given in Table II-1l.3-1l. The yield of inelastically scattered neutrons in other groups due to inelastic scatter- ing in a given group is presented in the calculation sheet, Appendix A, More generally the assumed energy distributions of inelastically scattered neutrons as obtained from KAPL are summarized in Table II~l.3-2. & * .y - an ee s i» . . s 4% § 448 4 sus e - ¥ ’ & & o . * » * * o * » - . es » " . . a aw . . o . . . *ne . . % » s ¢ s & . | . e - «928- TABLE ITI-1.3~1l. TInelastic Cross-Sections These cross-sections (in barns) used in all NEP fast reactor calculations Group 1 2 3 L Pu-239 7 o7 | o5 U-238 2.5 2.5 2.1 | .85 U=-235 1.2 1.2 9 Bi .8 .7 L Pb .95 .88 oM .0 F.P. .4 78 31 Fe .8375 .7875 .3125 .0 * Negligible above 4th group The values for U-238 are based on a re-analysis of the Snell and similar experiments (see KAPL=-741). .9 * "RENS s . . - - . - a®ts . L] * " TN . . - - - . e - » ’ . - » *ee ‘ - » ‘ L R ] ] - ® . . - *e e » » * yee \\\\\\ ooooo ....... iiiii N 7 —— TABLE II-1.3-2. Assumed Spectral Distribution of Inelastically Scattered Neutrons The fractional yield(X inelastic s 1 . =¥ é£in each of the energy groups (2-6) is shown due to ttering in a given higher energy group. Group| Range U-238 and U-236|Pu-239 and U-235|Fission Products and Fe|Pb and Bi|Cl N ifl 123wl 2 | 3 1 2 3 1 12{3]1 2 1-2 A5 - - - }.10 - - L - - M| -] - 3 2-3 «351.35| =~ - 1.20 .20 - M .6 - 31.6] = 4 [3-3.75 [.2%].33].50 .30 | .33 .51 |. Rn 1.0 2.3 ]9 5 13.75-4.5].16|.22].3%|.80].20 22| W3 | - - - 1l.1.1 6 4,5-5.5 |.10{.10{.15|.20].20 25| .15 | - - - -] =] = These data, except for Cl, obtained from KAPL. There is considerable uncertainty in all values, since so few experimental measurements ofjxiflj have been made. -68— 30 %.;\ 1.4 TFISSION CROSS-SECTIONS The fission cross-sections ar used are listed in Table II-1l.4-1. These values are based on recent repeat - measurements at Los Alamos which indicate a reduction of about 10 per cent is necessary in the fast region. For ‘ example, @y (Pu-239) has been reduced from 1.9% to 1.75 . barns, which is of considerable practical importance, since it results in a significant reduction in estimated breeding ratios. Similarly, the reduction in the accepted value of 0} (U-235) lowers practically attainable conversion ratios in the production of plutonium. This revision in g7 (Pu-239) has not been made in BNL-170. Apparently the revision in g (U-235) has been made in this recent publication. It is uncertain whether the revision has been made in G} (U-238) and 0 (U-236). The values for 5. The capture cross-section of fissionable nueclei is generally expressed as the ratio a = 6:/ G';.. Unfortunately, the data on a in the fast region are so sparse and so un- certain that only rough estimates can be made of the varia- tion with energy. Unpublished values of a(Pu) obtained from KAPL are: En(kev): 0.15 l.2 3. 10. 122. a (Pu) 0.7%0.1 0.6%0.15 0.52%0.17 .43%.09 o.1%0.1 9 For want of any better data, a smooth curve was drawn through the mean values and extrapolated to a = .02 at 10 Mev. The values of 0‘5(Pu—239) for the 8 energy groups listed in Table I1-1.5-1 were obtained from this very approximate curve. Similarly the values of O';(U-235) were obtained from KAPL's unpublished curve of a(U-235) as a function of energy. tered neutrons with a broad energy spread of mean value 15 kev. -33- Roughly the same uncertainty exists in «(U-235) as in a (Pu-239). Recent information from BNL and KAPL suggests that the values of G‘é(Bi) given in Table II-1l.5-1 should be re- vised. Measurements with the pile oscillator at ANL and by activation using thermal neutrons have established that Bizlo formed by B1209(n,Y) decays both by B and a emlssion with a branching ratio of about 50 per cent. Hence the single experimental value of 03 (Bi) = .003% barn at 1 Mev (group 3) as measured by Hughes from induced § activity should be doubled. Since the values for groups 1 and 2 are based on Hughes value at 1 Mev, it might be concluded that these should also be doubled. However, some unpublished ex- periments at KAPL suggest that 07 (Bi), unlike its non-magic neighboring nuclei, is essentially constant over the eight energy groups. Transmission measurements at KAPL indicate no detectable variation.of‘O;(Pb) with neutron energy. Other experiments indicate no significant difference in this respect between Pb and Bi. If these conclusions are correct JZ(B1)2=i.OO7 barn for each of the 8 energy groups rather than increasing with decreasing neutron energy as assumed in Table II-1.5-1. Corrections of the fast reactor calculations for Bi systems can be accomplished easily by reference to the detailed balance sheets of Section II-3.2. However, an estimate has been made based on the one-group method which indicates that the correction in the conversion ratio is generally less than 104. Table II-1.5-2 presents the effects of changing the capture cross-section of Bi on the conversion ratio for System 12. ¥ ------ aaaaaa oooooo ------ lllll sssss ...... ------ ------ . >4 TABLE II-1.5-1. Capture Cross=-Sections These cross-sections (in barns) used in all NEP fast reactor calculations u .5-1 1-2 2=3 3=3.75 3.75-k.5 %.5-5.5 5.5-7 7-10 Group 1 2 3 4 5 6 7 8 Pu-239 .05 .07 .088 .1k .23 M1 L7 3.3 U-238 015 .05 .13 .16 .22 32 .52 1.0 U-235 .065 .065 .075 .12 . 207 375 .726 2.k% Bi .0025 .003 .003% .007 .009 015 .03 .07 Pb .0025 .0025 .0025 .0025 .0025 .0025 .0025 .0025 F.P. . 002 .006 .052 .125 .195 27 .38 . 576 Fe .001 .001 .0013 .0039 . 0062 . 0097 .01 .013 cl .0006 .0007 .001 .0018 .003k4 .0063 .0085 .0113 Al . 0004 . 000k . 000% . 000k% . 000k « 0004 LOoo0F ,00092 Na .00027 .00027 .00027 .00027 .00027 .00028 00049 ..02211 For Pu~-239 and U-235 see text. Values for U-238 from BNL-170, p. 35. Valves for Cl and Bi based on single measurements by Hughes at 1 Mev extrapolated to other energies with average slope obtained from measured values for nelghbor- ing elements. Estimates for F.P. based on yields and measured values when avallable. Considera- tion given to low values for magic numbered isotopes; however, energy variation assumed to be the same as for non-magic. All other cross-sections from XAPL. G (Pb) probably should rise with decreasing energy although magicity (2=82) for all isotopes and (N=146) for Pb~208 may cause 0~ to remain essentially constant as assumed by KAPL. -vg- TABLE II-1.5=-2 -35- Effect of Changing 67 (Bi) on C.R. in U-Bi Systemss One-Group Calculation, System 123 U-235:U-230sBiz1l:3s Assumed value of a_é(Bi) 1n fifth u group IQCOR. XOCOR. T.C.R. .009 barns «339 .683 1,022 .00’+ barns 0339 0786 1.125 -36- 1.6 SCATTERING CROSS-SECTIONS Since the scattering cross-sections 05 always enter the calculations in the form &7, where § = the mean logarithmic energy loss, this product, called the degradation, is given for each of the elements in each energy group in Table II-1.6-1. & is calculated from the atomic weight in the usual manner. The values of Eflg are conslidered quite reliable, except, possibly,., for that of Cl1 (which is the principal source of elastic degradation unfortunately in the fused salt reactor). ...... s e e “eszew ..... tttttt ------ TemeaN ...... TABLE II~1.6-1. Degradation: Product of Scattering Cross-Section and Mean Logarithmic Energy Decrement EEO‘Sl: | These values used in all NEP fast reactor calculations sarrerat — g — R—— M—— u .5-1 1-2 2-3 3-3.75 3.75-4.5 %.5-5.5 5e5=7 7-10 | Group 1 2 3 L 5 6 7 8 Pu-239 .037 .037 . 046 .061 .071 .078 .08k .089 U-238 - .037 .037 .Ol6 .061 .071 .078 . 084 .089 U-235 . 037 .037 .O46 .061 .071 .078 .084 .089 Bi .038 .039 . 045 .063 .085 <111 .120 .125 Pb .038 .039 045 .063 .085 <111 .120 .125 F.P. .055 . 081 .106 «120 .122 122 .121 .119 Fe .103 .096 .09 122 122 0137 <147 o cl »139 o 114 179 «197 .206 .217 234 L55 Al .176 .192 «256 .288 «303 .215 .082 .11 Na . 194 .209 347 .323 346 .351 .369 o5l Be .213 <320 . 64 .787 2955 1.103 1.16 1.211 These values obtained from KAPL except for the following: Pu-239 taken same as U-238 since 0t 's are in good agreement between 0.3 to 3.5 Mev (BNL-170). The absence of data on Cl necessitated an interpolation of the KAPL values for Na, Al and Fe. The values for F.P., obtained by plotting averaged £ o for known F,P. versus energy (welghted for yield), averaging and interpolating. S L w - t -38- 1.7 NEUTRON YIELDS One of the most important parameters in the calcula- tions of critical mass and conversion ratio 1s the number of neutrons per fission v. The experimental values used in NEP calculations were: 2.47 2.97 - v(U=235) v (Pu-239) Assumed values were: v(U=236) = v(U-238) = 2.50 The latter are more important in fast reactors than in thermal reactors, since the fast effect (in U-236 and U-238) contributes from about 8 to 15 per cent of the total number of fissions in the fast reactors considered. 1.8 NEUTRON SPECTRA . - , The fission spectrum for all but delayed neutrons (which are substantially lower in energy(a)) is given in the following table: R Group u = 1n(E_/E) . : 1 0.5 = 1.0 .13k 2 1 -2 522 3 2 -3 .295Y% L 3 - 3.75 .0807 5 3‘75 - L"‘S 00373 These are the KAPL ylelds of fission neutrons in each of the indicated groups. They have been used for all fissionable (a) No correction was made for the fact that the delayed .- neutrons have a lower mean energy than the prompt neutrons. | -39- v _ elements in all NEP calculations. While there 1is probably some variation in‘J( between different fissionable nuclei, this difference is probably not sufficient to affect seriously any of the NEP calculations. Since only limited attention was given to control problems by the NEP, accurate values for the half-lives and yield of delayed neutrons were not essential. The values quoted by Glasstone and Edlund were used, namely: Tfi in sec. Fraction By Energy in Mev 0.43 0.0008% 0.42 1.52 0.002% 0.62 4,51 0.0021 0.43 22,0 0.0017 0.56 55.6 0.00026 0.25 1.9 RECOMMENDATIONS In the course of the NEP studies, the need arose for new nuclear data and for improvements in existing measure- ments., This section outlines these needs specifically, and makes some suggestions regarding methods of satisfying them in the future. The fast reactors considered by NEP have neutron spectra peaked in the range of 100~-300 kev., This energy region is beyond the reach of velocity selectors and below the fission spectrum. However, it is conveniently covered by electro- static generators, for example, via the Li7(p,n) reaction. By means of such sources the fast cross-sections needed can probably be obtained most expeditiously. | Fewer gaps appear in the thermal data. These can be filled quite readily using pile sources (ANL, ORNL and BNL) provided qualified personnel can be attracted to these prob- lems. CAPTURE CROSS-SECTIONS. - In the thermal region capture cross-sections may be measured by transmission methods (large G:) and by diffusion methods (small G;fi in addition to being measurable in some cases (both large and small.G;) by activa- tion methods. However, in the region of primary interest (100 - 300 kev) for fast reactors, the capture cross~sections of all elements are so much less than the scattering cross- sections that transmission and diffusion methods cannot be used. Hence, nearly all of the measurements in this region have been made by bombardment of the elements with neutrons of more or less well-defined energy followed by measurements on the reaction productss either induced radiocactivity or isotopic abundance. These methods have a number of limitations. When radiocactive isotopes are produced in even atomic-numbered elements which often have several isotopes, there may be some uncertainty as to which isotope is activated. 1If this is not a serious rroblem, it may still happen that the abundance of the isotope activated may be small and non- representative of the element as a whole. The half-l1ife of the radioisotope formed must be convenient for activation and measurement. The radiations emitted must be sufficiently energetic to be measured and must be readily distinguished from radiations (if any) emitted by the target element. Many elements, especially those of promise in reactor design, have too low cross-sections to he measured except in large neutron fluxes. The last limitation is especially critical when an electrostatic generator is being used as the neutron source, Despite these difficulties, enough data are available to allow some fairly reliable general conclusions to be drawn. As indicated in Section II-1l.5, experiments have established that 0"; increases rapidly with decreasing energy below 1 Mev for most elements. It is also well established that the cross-section increases with increasing atomic weight and that the magic nuclei (atomic number and/or neutron number -41-~ 2, 8, 20, 50, 82, 126) have unusually low capture cross- sections. Some rough measurements at KAPL suggest for Pb and possibly for Bi that O‘; is not nearly as dependent on energy as for non-magic nuclei. This hypothesis is of practical value and should be investigated experimentally. If this proves to be true, certain of the fission products will be less absorptive than has heretofore been assumed. What is more important, the magic nuclei, Pb and Be, which have good engineering properties, will also be shown to have unusually good nuclear properties for fast reactors. Measurements of capture cross-sections in U-235 and Pu-239, obtained from the U-236 and Pu-240 formed, give only rough indications of their energy dependence. Danger co- efficient measurements are only of slightly better reliability. Improvements in the knowledge of these capture cross-sections are urgently needed. CHLORINE. - If fused salt reactors continue to be of practical interest, further information is needed on the cross-sections of chlorrne.cr'(Cl) has been measured only up to 280 ev. CT'(Cl37) = .74 mb has been obtained by Hughes et al for flSSlon neutrons, (mean energy about 1 Mev). Assum- ing this value applies to chlorine as a whole (25% C137), 0.74% mb was used as the single experimental point on a curve whose slope was obtained from a combination of theory and experiment (TMS-5). Average cross-sections for the other seven energy groups were obtained from this curve (linear on semi-log plot). The activation.cross-sectionuGE(Cl37) should be extended below 1 Mev, and the capture cross-section of Cl as a whole determined from 0.1 to 1 Mev if possible. IRON GROUP (Mn, Fe, Co, Ni). - These elements are of importance as structural materials. Only Fe has been con- sidered in the NEP calculations. As indicated above, the values of(Tg(Fe) in the fast region were obtained from KAPL. These probably were derived by analogy with neighboring 42~ nuclel since there appear to be no measured values for Fe in the literature. Activation measurements on 0059 and Ni made only at 1 Mev. Fairly complete data on Mn have been obtained from .03 to 3 Mev. It would be desirable to have similar data for Fe, Co and Ni. 64 have been Pb AND Bi. -~ These two magic nuclei are so important as possible constituents in fast reactors (coolants, re- flectors, controls and blanket components) that a special effort should be made to establish G;(Pb) and G;(Bi) over the energy range 0.1 to 1 Mev. U=236. - In order to keep down the reprocessing costs, it is necessary to allow substantial buildup of U-236 (0.2 atoms U=-236 to 1 atom U-235) to occur before sending the fluid fuel of the fast converter through the gaseous diffusion plant for separation of U-235 and U-236. TFor this reason accurate knowledge of(T;(U-236) is of importance. In the NZP fast reactor calculations, it was assumed (purely ad hoc) that U-236 has the same capture cross-section as U-238. This was necessary since no data on,U;(U-236) in the fast region appear to be available. Obviously this uncertainty should be removed by experimental measurements of QZ(U-236). U-235 AND Pu-239. - Better knowledge of the capture cross-sections of U-235 and Pu-239 in both the slow and fast regions is urgently needed. This lack results in serious un=- certainties in the design of the most economical reactors for the production of plutonium, | Not only are these measurements difficult to make, but the results are difficult to interpret unambiguously. Weisskopf has reviewed the situation in a preliminary report (NDA Memo-15B-1) received near the end of the summer (1952). As he points out, the easiest region to measure is at low ‘. -:*:.. M v -43- v energles where scattering is negligible compared to fission and capture. Under these conditions a is determined by measuring the peak values of 7% and Opt it = 0t ¢ = T )peak He states, however: "Unfortunately the resolving power of ‘the relevant measurements is not adequate, and the observed peak value corresponds to the actual one only for the first few ev. At higher energles, the observed peak value is much less than the actual one and in the case of the total cross- section it contains a good deal of the potentlial scattering taking place between rescnances". Reference 1s made to KAPL-377 and KAPL-394 for an analysis of the U-235 levels made at G.E. based upon very unrellable data. While theory 1s of value as a guide, what 1s really needed for the low energy region is an improvement of resolving power in the fission as well as the total cross-section measurements. Two types of measurements have been made at higher energles - both give only rough indications of the energy dependence because of the broad spectral distributions of the neutrons. In the first method (KAPL-183), samples of U-235 and Pu-239 are enclosed in shields of different thick- nesses and composition, and are irradiated at Hanford. o (0-235) is obtained from mass spectrographic measurements of the U-236 produced and ¢ (Pu-239) from measurement of the spontaneous fission in the Pu-240 produced. The second method consists in measurement of the danger coefficlent at different locations in several reactors. The energy spectra of the neutrons at these locations are known only approximately. a can be calculated from the danger coefficients. This method requires a correction for the thickness of sample irradiated - a procedure which 1s not very reliable. -44- Thus it appears that improvement in the knowledge of the a's in the fast region can be attained by these two methods only with considerable further experimental effort. This effort is amply Jjustified by the need for these data. A possible technique which apparently has not been tested is to determine a from the prompt gamma rays emitted during fission and capture. Through the use of scintillation counters (NaI-T1I) one might be able to distinguish between the fission and capture gammas since the latter should have a higher energy component. If this is possible, monoenergetic neutrons from the Li7(p,n) reaction could be used to irradiate a ccnical sample of U=-235 or Pu-239 surrounding the scintilla- tion detector which is shielded from the direct beam by a cone of boro-paraffin, | TRANSPORT CROSS-SECTIONS. - Measurements of O, have been made for only a few of the elements in the NEP calcula- tions. Fortunately, except for Cl, values of U; are available and these set upper limits for the values of OL,.. In addition the theory is fairly reliable and can be used for interpola- tion between measured elements. It would be of particular value to have G¢.(Cl) or alternatively (T;(Cl) in the fast region. This should not be difficult - C Clh might be used as scatterer. At preseni;G;(Cl) has been measured only to 400 ev. G;r(Bi) has been assumed the same as G:;I.(Pb) (some experimental data ) O"{r(F.P.) was obtained by interpolation and (T;r(Pu-239) by extrapolation of the limited experimental values for other elements. 'hile large differences are not expected, reliable experimental measurements on the elements of interest should be obtained as soon as possible. SCATTERING CROSS-SECTIONS. - Values of Qg can be - obtained with sufficient reliability from 0. Hence, the only urgent data needed are for chlorine (see above). -45- FISSION CROSS-SECTIONS. - Except for the 10 per cent downward revision, which appears to be in process of general accertance, and the unknown value of S¢(240), the fission cross-sections used by N.E.P., appear to be of adequate re- 1liability for such preliminary design calculations. For more refined calculations, more accurate values of S in the region of 100 to 300 kev wounld be regquired, INELASTIC CROSS~SECTIONS. - The need for improvement in inelastic crcss-sections is widely recognized. This has been glaringly evident for some time in shielding studies and is becoming of increasing urgency in reactor calculations. These measurements are unusually difficult to make with electronic detectors,and easy,but extremely, tedious with photographic emulsions as detectors of the knock-on protons. In the NEP calculations the KAPL values of g7 and :[i*j‘were taken over without critical review even though it was recognized from the outset that these data are probably quite uncertain. These cross-sections and the assumed energy distribution of inelastically scattered neutrons enter the calculations very significantly in that they determine to a great extent the average energy of the neutron spectrum, and therefore the mean effective a and the percentage of fast fissions. In addition to adding to the theory of the nucleus, the c¢ross-sections and energy distributions of inelastically scattered neutrons by reactor fuels, diluentsyand structural materials are urgently needed for accurate design calculations of fast reactors. NEUTRON YIELDS. - In all NEP calculations it was tacitly assumed that the v of each fissionable element 1s independent of energy. While this is very likely, experimental confirma- tion is desirable. | It was also assumed that v(U-238) = v(U-236) = 2.50 46~ and v(U=-235) = 2.47. This important parameter has only been measured reliably for the last of these three isotopes. The fact that in fast reactors, U-236 and U-238 can account for 15% of the total number of fissions and that the fast fission effect in U-238 in the blanket helps to offset the neutron loss due to parasitic capture in structural materials emphasizes the importance of accurately determining v of U-238. RESUME_OF NUCLEAR DATA NEEDED: J. Capture cross~sections. a. Ascertain whether maglc nuclel are less energy dependent than non-magic. | b. Improved resclving power 1s required in the fission and total cross-section measurements in order to obtain reliable values of the capture cross-sections of U-235 and Pu-239 up to several hundred electron ~ volts. | c. Obtain.dz(01) and d;(U-236) as functions of energy in the fast region. d. Obtain U”C(Fe), O"C"(Co), O'E(Ni), O'é(Bi) and G’C(Pb) as functions of energy in the fast region. 2. Transport cross-sections. a. @3, (or CT;) for chlorine particularly needed in fast region. b. G, (B1) Q. (Pu-239) and U of intermediate elements representative of the fission products should be obtained. 3. Inelastic cross-sections. a. Improvement should be made in the experimental values of the inelastic cross-sections of U-238, Pb, Bi and Fe. b. Measurements ofG';_’ (Pu-239), OE’(U-235) and O'i'_' (Cl) especially in the energy range 0.2 to 1 ~47- Meve. c. Additional information is needed on the energy distribution of inelastically scattered neutrons even though it is recognized that these measure- ments are difficult and tedious. | 4, Pission Cross-sections. a. af(Pu-ZhO) as a function of energy. The following table checks the most urgently needed nuclear data in the fast region (100 to 3000 kev) TABLE II-1.9-1. Resume of Nuclear Data Needed Element | v dc Op o4 Gy Jg U=-235 3* * U-236 * 3# 3 U-238 #* #* Bi * 3#* 3* 3# Pb * * 3 * Cl * # #* #* Fe * 3* Co 3# * Ni # * Na * Pu-239 * * % * Pu~-240 | & %* * * * 3# Pu-241 | i * #* * * #* anik A e SW AEL SIS SRS AU GE A GNS ORE VAN MW WEP SN OME EIG LS GSS AUl WEA WE TMD WN GID 4N GES WML GED I NS S SRR GUr awm A -48- m | 2. CALCULATION METHODS 2.1 BARE REACTOR MULTIGROUP METHOD At steady state the rate of removal of neutrons from each energy group a (leakage + absorption + inelastic scatter- ing + degradation by elastic scattering) equals the rate of addition (fission + elastic and inelastic scattering from higher energy groups). The bare reactor multigroup approxima- tion to the age equation for this steady state condition can be written: —(i—lfh)fl v‘@d + (z“)NQN + (z'i)u ix + (“f':'%)“ §°‘ = p = oty (2.1=-1) B=n ('flf)“ 1)‘.J:£F;.)fi @p +(_é_223_) §or -1 +£('x‘)p-£z5)p§9 These and all other symbols used are defined in the glossary which is appended at the end of this report. In these equa- tions it is implicitly understood that v, Z and the inelastic spectra are summed over the appropriate nuclear species. Since the flux distribution has the forms: §°ffl= §—ill?-l-c—-1: \ 28 I—N 26 Exfernal Breeding Ratio 1.2 29 \ 22 1.O \\ 20 / 18 18 \ 8 16 f16 , t-"‘ 14 / \ ,fi; 12 6 /“ l —12 .// "/’5 / /\ \ 10 \ —=110 Internal Breeding \ Ratio 4 Xle 06 6 2 ' - - -04 4 / Critical Mois | 02 A ) 2 - 1 0 o I 2 3 4 5 6 7 8 9 Ned) ot —— N(49) ¥ ~ "o - - - T | ENEORFE—= i 7y Bare Reactor Critical Pu Mass, metric tons ————> -7 = -78- Conversion Ratio ——» 73 .5/ T T T T 1 1 T 30 FIGURE II-3.2=3. U-235--U-238--Bi B CONVERTERS |, ¢ SYSTEMS 11 TO 13 26 'Totol Conversion .24 24 -\\\\ Ratio \\\/ .22 5 \ 20 20 \\ \\ 18|18 CExieran \ »T’ 6| 6 Conversion : Ratio / / 1 14! 14 (/\ T 12|, / = \ / | 10 4 / 10 Internal \ / 0 "Clgn\’r'ersion 8 atio \/ X 08 . Y / A N a | / \ fi 2 ' [ Critical \ Mass ] 0 l 2 3 4 5 6 7 8 N(28)_~A N(ZS)-F Bare Reactor—Critical U-235 Mass ; metric tons _—= -79- FIGURE II-3.2-4, UCl),-NaCl CONVERTERS SYSTEMS 1% TO 17 Conversion Ratio —= 30 28 Total Conversion Ratio 26 26 \ s | 24154 N 22 |,, w \ External Conversion 5 Rotio - | ] .20 o l / I8 E / —18 _ — & ol 1 116 g | 16 |, 8 / 14 a4 o / N\ / 3 12 x £ 2 5 | Internal /\ / 10 0 § Conversion / \ , 5 Ratio / / S N oglsg « o @ / / Nk / / \Critical N\ y / Mass. \ 02l 5 v N N o - p W o)) 7 8 9 10 -80- e, FIGURE II-3,2-5 SYSTEM 14: SPECTRUM OF FISSIONS AND FRACTION OF FISSIONS ABOVE u 30 128 26 /Spectrum of Fissions 24 | * 20 20 © ——\ > o 0 o 9 18 » \ et o 8 16 m \ » w 7 14 =z \ 550 Kev. o / - .6 1 112 o < c 5 10 4 08 3 06 2 04 N D2 % 2 8 9 10 FISSIONS/ UNIT u — e T E A T il i T -81-~ G =3.2-6 SYSTEM 1lk4: FLUX SPECTRUM 30 .28 26 24 22 —— Flux Spectrum @ UNIT y———— o X FLU — N 06 o4 -82- FRACTION FISSIONS above u—— FIGURE II=3.2=7 SYSTEM 15: SPECTRUM OF FISSIONS AND FRACTION OF FISSIONS ABOVE u 10 ¥ S;;ectrum of Fissions N L7 S0 26 24 .22 20 18 12 28 — FISSIONS AJNIT y ————— 26 24} 22 -83- FIGURE II-3.2-8 SYSTEM 15: FLUX SPECTRUM ® FLUX/UNIT U = . Flux Spectrum 7 Ty Rl g 3 - 1 10 23 -84- FRACTION FISSIONS obove u——= S @ é ) FIGURE II-3.2-9 SYSTEM 16s: SPECTRUM OF FISSIONS AND FRACTION OF FISSIONS ABOVE u .30 @ 0 / \ Spectrum of Fissions o = ~ ro 87 Kev o 106 02 —_ FlSSIONS/UNIT u FLU&/’UNH'U -85~ FIGURE IT-3,2-10 SYSTEM 16: FLUX SPECTRUM Flux Spectrum 3 4 S -86- FRACTION FISSIONS above u—— FIGURE II-3,2-11 SYSTEM 17: SPECTRUM OF FISSIONS AND FRACTION OF FISSIONS ABOVE u ~ 30 28 26 24 22 1.0 20 18 Spectrum of Fissions 8 4 16 7 14 12 6 61Kev i / o 4 ‘ \\ 08 X \ 06 2 04 N 02 A \ o | 2 3 4 5 6 7 8 9 10 ~ — U —a% C6 . ’1 SE T A RSRSEEE RN FISSIONS/ UNIT u ¥ FLUX/ UNIT u ity s -87~- SYSTEM 17: FIGURE II-3,2-12 FLUX SPECTRUM .30 28 26 24 22 20 A Flux Spectrum, e 14 12 10 .08 06 .04 .02 0 3 4 5 6 7 8 9 o) 8"{ = y———--= . SSECRET ‘.,,,q % + (B mrerariony 'l | -88- 3.3 IWO-GROUP TWO-REGION CALCULATIONS The several two-group calculations that were carried out can be divided into two distinct sets: (a) Computations were performed for a given core and infinite blanket to determine reflector savings and to com- pare breeding ratios with those obtained by means of special multigroup methods for the blanket. The results are given in Table II-3.3-1, which compares the result of a given two- group calculation with the multigroup calculation which was taken as the basis of cross-sectlion averaging in the given case, It is seen that the requirement of satisfying boundary conditions in the two=-group case leads to a small adjustment of breeding ratios away from the extreme values given by the special multigroup methods. However, this adjustment 1s not sufficient to warrant an extensive number of such calcula- tions for present purposes. It is further seen that the re- flector savings are not great compared to those obtainable with a good reflector because of the low density of the blanket. (b) Core size as a function of reflector size was in- vestigated for system No. 1l coupled to a Pb reflector. We have also obtained the X.C.R. as a function of the same para- meters. This was calculated as X.C.R. = Leakage from reflector *vett T Pission + Capture of U-235 in core ’? consistent with the assumption made in the bare pile multi- group calculations that all neutrons which leak out of the core are avallable for conversion. Three distinct calculations were performed. Two of these were two-group calculations, which differed in the choice of reflector cross-sections. The results for the critical size are compared in Figure 3.3-1, where we have plotted reflector size as a function of core size, both given in units of the bare core size, a = 56 cm. The overall TABLE I1-3.3-1, Comparison of Special Mult] tions with Two-Group Calcul -89- Lgroup Calcula- lations System Number Multigroup Two-Group 52: Strong Coupling 53¢ Driven Blanket Multigroup Two-Group T.B.R. 1.556 1.58% 1,728 1.717 X.B.R. 1.170 1.233 1.421 1.403 I.B.R. 0.386 0.351 0.307 0.314% Core Radius - 1.000 0.681 1.000 0.688 Bare Core Radius e rmnr oL ae*ee . ,‘1 ase ; - < ews . @ -* - * - * adee - 9 O - o~ — o M size of the reflected reactor relative to the bare reactor is then obtained by adding abscissa and ordinates. C.M. can then be directly obtained from the value given in the previous section for the system in question (see Table 3.3-1). The curve labeled "Fast Reflector Spectrum" was obtained on the extreme assumption that the reflector has the same spectrum as the bare core, whereas the one marked "Slow Reflector Spectrun" resulted from the assumption of a substantially slower energy distribution. In Figure 3.3-3 the same compar- ison is made for the X.C.R.'s., In all cases, the points actually computed are indicated. | In addition to the results found above, we have also performed a one-group calculation for the same system, using the fast spectrum to determine average cross-sections. The results of this computation are compared with the two-group fast spectrum result in Figures 3.3-2,k. As was to be expected, the slower reflector spectrum predicts smaller critical masses because it results in the assumption of larger transport cross-sections for the re- flector than in the fast spectrum case. The results for the conversion ratio quite likely underestimate the conversion loss, since there is no information contained in the calcula- tions made about the degradation of the leakage spectrum relative to bare core leakage. The actual rise of the X.C.R. as seen in Figure 3,3-3 may be understood on the basis of 'the remark that for small reflector sizes, the overall size of the reflected reactor is actually smaller than that of the bare pile. The most important result to be gleaned from the graphs, however, is that with little loss in conversion ratio, one can approach optimum reflector savings. ," L] _91_ 1.8 l ‘ Legend 16 A Fast Reflector Spectrum __ ' ® Slow Refiector Spectrum Bare Core Size=56cm. 14 L2 Q® ~N B £ FIGURE II-3,3-1 o0 — ;c:': Reflector Size vs. Core Size $—a)= !1?#_ 7 kd _ . - Core Size 1.0 -92- L6 e TR e CRET 2 Group Asymptotes % @e=——r_ /? Group 5 bl | Legend @ | Group @ 2 Group Bare Core Size= 56 cm. FIGURE II-3,3-2 Reflector Size vs. Core Size i\ X h FIGURE II-3.3- m ©w O = X.CAXC) bare [ 4 -93- X.C.R. vs Core Size X.C.R. (Bare Reactor) * Bare Core Size Legend A Fast Reflector Spectrum @ Slow Reflector Spectrum. o ) s H 5 .6 7 8 9 1.0 kd n. 83 v—-:______,.,_—-—- e R -94- s N 2 Gro | | G:ogg} Asymptotes FIGURE II-3.3-4 X.C.R. vs Core_Slze X.C.R. lfiEre Reactor) * Bare Core Slze ] Legend ® | Group A 2 Group " T ‘TL .9 2 2 8 O 25' 7 ~ O § > 5 4 - 2 Jd ) 4 5 6 7 8 9 K = -95- 3.4 ONE GROUP - THREE REGION The methods developed in Section II-2.4 were used to estimate the loss of conversion ratio and the reduced critical core mass which would be obtained for various thicknesses of a Pb or Fe reflector. Equation (II-2.4-3) was applied to the systems of interest. As an example, the results of applylng this method to a system # 15 core, Pb reflector and an infinite UCl, blanket are represented graphically in Figure 3.4l. The fifth energy group was used to determine cross-sections for all three regions. If the core buckling is low, loss of external con- version ratio does not permlt the utilization of the maximum possible reflector savings. Economic considerations will de- termine how much loss of X.C.R. can be tolerated, or traded for decreased inventory. The one velocity -~ three region techniques are applicable for comparing and optimizing similar reactor systems. For an actual reactor, however, these results indicate trends, and are not expected to give precise values as a detalled account of the neutron distribution as a func- tion of space and materlals was neglected. ~-96- | I Reflector Thickness and X.C. vs. Core Size __| 8 X.CCO 7 1.3 ] 6 \ 1.2 @ Il Q : \ Ig S e/l.0 S 9 § / \xc g 4_ \ XCO .8 ’_.f I XC. B \ X.Co x| 3 6 | \ _ 5 \ Reflector 2 \<£ Thickness 4 ‘\\\\\\‘ J | \ i \\.I D 6 kb . — Core Thickness — 97~ 3.5 SPECIAL MULTIGROUP CALCULATIONS Additional multigroup calculations were carried out to investigate specific problems encountered. For the bare re- actor in most cases, interpolations proved satisfgctory for intermediate values of parameters. System 18 examines the effects of Fission Product poisoning on conversion ratio and critical mass. System 51 examines the effects of mixtures of Pu-239 and U-235. Systems 52 and 53 are self-consistent calculations which were wndertaken to evaluate the assumptions made concerning the effect of a blanket on the spectrum and conversion ratios of the bare reactor, see Section II-2.2. System 54 examines the effect of cross-section varia- tions on critical mass and breeding ratio. TABLES AND GRAPHS Table 3.5-1: System Constituents for Special Multi- group Calculations Table 3.5-1 lists the atomic constituents and gives a code number for each of the special multigroup calculations. Table 3.5-2: Results of Special Multigroup Calcula- tions Table 3.5=2 summarizes the results of the multi- group calculations for each special reactor system considered. Table 3.5-3: Speclal Systems, Neutron Balance Table 3.5-3 gives a detailed neutron balance for each special system considered. The basis for these balances was 1 neutron absorbed in U-235 and/or Pu=-239 in the core. [ T Figures 3.5-1,2: Comparison of neutron spectrums by strong-coupling theory. Figures 3.5-1,2 compare the neutron spectrum of System 2 with the spectrum given by the strong- coupling theory. One notes that the mean fission energy is lower and a larger in the strong-coupling case. o o ...... lllllll ssssss ooooo ----- ...... '''''' ...... tttttt nAada®E TABLE IT-3.5-1. 77 M System Constituents for Special Multigroup Calculations System il U-235|U-238|Pu-239| Cl |Na Remarks 51. 1 4,5 .25 |28.75| 6 | Bare Reactor calculation (Compare with 15.) - Mixture of Pu=-239 and U=-235 52. core 3 1 15 Strong Coupling Blanket Calculation Blanket 1 L (Compare with #2.) 53. core 3 1 15 Driven Blanket Calculation Blanket 1 L (Compare with #2.) 5, 3 1 15 10% Revision of ¢, and ¢, for Pu, S R)=19%b. -66- e e : . ; R e . -001- ]7. ¢ ;gz TABLE II-3,5-2. Results of Special Multigroup Calculations System| Atom Density (metric) - ] ----- : 51. |N(25)=.852x10°% | .M. (25)=1.847 4,520.161/1.302 | 0.888 | 0.414 ¥°/3=375.4| Bare Reactor Calculation Ll - C.M. (49)=0.470 E§9=0.176 3 52, @ = .235 [1.556 |1.170 {0.386 | I = .11 |Strong Coupling Blanket : Calculationy Compare with No. 2. 53. 1.728 |1.%21 | 0.307 | A = 12.% | Driven Blanket Calcula- ...... tionj; Compare with No. 2. T.B.R. X.B.R. I.B'R. 5, N(u9)=1.687x1021 CeM. (49)=20.926|3=0.125 [1.778 |1.512 | 0.266 k2/3=2h2 <, and o, of Pu-239 increased by 10%. ...... ...... lllll flflflflfl ...... ------ oooooo rassaw sresan TABLE I1I-3,5-3, by Special Systems - Neutron Balance B System Number 51, 524 53 54, Core Blanket Neutron Production v(25)-Fissions U-235 1.6974 v(28)+Fissions U=-238 0.2276 0.1045 0.3167 0.1174 .272 v(49) Fissions Pu=239 0.5105 2.4892 2.639 ‘Leakage into blanket 1.4606 Total source 2.4355 2.9104 1.5780 2.911 Neutron Consumption Fissions in U-235 0.6872 Fissions in U=-238 0.09103 0.04180 00,1267 0.04697 .1088 Fissions in Pu-239 0.1719 0.8381 . 8886 Captures by U-235 0.1106 Captures by U-238 0.4141 0.3999 1.,2115 1.434] _. 2664 Captures by Pu-239 0.03032 0.1974 <1114 Captures by Cl 0,04121 0.05724% 0.07689 0.08806 . 0202 Captures by Na 0.001490 Leakage 0.8876 1.5116 Total 2.4355 2.9495 1.5691 2907 ! ok o ek 1 ~-102-~ :_:W FIGURE II-3.5-1 SYSTEM 52: COMPARISON OF THE SPECTRUM OF FISSIONS OF A BARE REACTOR AND A BLANKETED REACTOR Legend 30 — Bare Reactor ~ ——— Strong Coupling 2 26 1 .24 22 20 J 1 S e I"“'u_.fi - .8 : : E 16 ] ; 5 | | w o 14 1 ! L | | 2} | | al ] J j | I~ f| 10 1 | ] | og) —- ' ! | 0 — | O4 1 1T 1 | 0 | | o | 2 3 4 5 6 7 8 9 0 102 e NEUTRON FLUX/UNIT y—» =2 -103- vv—T Legend Bare Pile 30 ~——Strong Coupling 28 FIGURE II-3.5-2 *! SYSTEM 52: COMPARISON 26 ! OF THE FLUX ] I SPECTRUM OF 24 | A BARE RE- ‘ I ACTOR AND A | BLANKETED 22 | ~ REACTOR I I 20 | I8 I I 16 ; I | L. .41 —— 12 ! 10 4 I .08 % 0 ; | | 04 — I .02 T l I % 7 8 9 10 -104- 3.6 GENERAL TRENDS TRENDS OF BREEDING OR CONVERSION RATIO. - The breed- ing(a) characteristics of a reactor are determined by the average final fate of its neutrons. If we assume the blanket large enough to capture practically all the neutrons that leak into it from the core, then the breeding ratio will depend on the outcome of the competition between radiative capture by 28 on the one hand, and parasitic capture by 49 or by reactor contaminants and diluents on the other. While degradation by collision and leakage cannot be considered the "final fate" of a neutron, the extent to which they in- fluence the neutron spectrum will have an important effect on the absorption competition referred to, principally because of the tendency with slower spectra for capture cross-sections to increase relative to the fission cross- section of 49. The following discussion attempts to explain the general trends of breeding ratio and critical mass for chemically similar systems as variations are made in the relative atomic ratios of the reactor constituents. The results of such an investigation will appear as Figures 1 to 4 of Section II-3.2. An understanding of these results en- ables one to generalize fast reactor characteristics to such an extent that optimalizations may be made without lengthy calculations. a). Internal Breeding Ratio (I.B.R.) The internal breeding ratio is given by: IT.B.R = S Zs‘““?d“dv / .L,‘z“-.‘w?‘“w (3.6-1) Corg It happens that the ratio z““-”/z*qis not strongly energy dependent in the range of energy where all the in- "Conversion" and "25" may replace "Breeding" and "49% in the subsequent discussion. __04 A gt ~105~ vestigated neutron spectra are concentrated. Therefore, if an average value for é%%gg 1s taken outside the integrals, the remaining integrals cancel, and we have: B.R. = Z G = EEESLEEEED .6~ T Zi.. (4) N (Q4) 0z 49) (3.6-2) Thus we see that the I.B.R. is proportional to the atomic ratio of 28 to 49, and is roughly independent of the neutron spectrum. b). External Breeding Ratio (X.B.R.) In the case of internal breeding, we noted that the competition for neutrons between 28 and 49 takes place "locally"; i.e., in each small volume element of the core, the number of neutrons available to 28 and 49 is the same (for all energies), since the flux is the same for both. Hence, the I.B.R. does not depend on geometry, other con- stituents, parasitic absorption, etec. The situation in the case of the X.B.R. 1is entirely different. Here the competi- tion is between 49 in the core and 28 in the blanket. Between the time that a neutron is emitted in the core and leaks into the blanket to be captured by 28, it has to run a gamut of parasitic absorption by 49, diluents and contaminants and fast fission absorption by 28. The effect of this gamut on decimating the blanket-bound neutron population varies with core and blanket density, spectrum composition, and geometry. One can always wrlte for the core Leakage + Absorption = Production (3.6-3) The assumption underlying our blanket breedling estimates from bare reactor calculations is that the fast fission effect in 28 compensates for non-productive absorption in and possible leakage from the blanket. On this basis, all the neutrons that leak into the blanket are captured by 28 to form 49 | - S, "106" M—.—-. finally. Therefore, the rate of external breeding is Just equal to the leakage rate divided by the rate of absorption in (49). The production is given by P = 9(49)[2*(4«\¢dml\" + Dlzs)yz*wypduow (3.6-1) and the absorption by: A= ffl.;(«)cpcluow' + S\Z“(z?)tfdualv-{- gflc(p\cpduc“- where Zi (f) expresses the parasitic capture by diluents and contaminants. We rewrite the balance equation: L = N (44) { 7)(44)!@'(«) cpduow' + %'r_f(_?_:.z\z)(z'i) jq;(z&) P cl::j]g ) _ f oz @ ¢dudY - %%a% Yo; 29) @dudV - flN;((% S'GE ) ‘fdudv:} For the fused salt systems, the parameter of interest 1s the ratio N(28)/N(49); therefore, denoting § 23 by ¥ ~ and Sd'?mby ¢V , we have: | L = KOO FY | D) )~ 69 N .m0 _ M Tan —26)T e § In the range of energy where most of our spectra fell, the various cross-section averages were roughly constant, though not exactly so., Therefore, the changes in the neutron spectrum due to changing vy were reflected in relatively small changes in the mean cross-sections. Moreover, one notes that if %%E%% is a linear function of y, where N(Cl) is the number density of chlorine, L is a linear function of vy, with negative slope. We may write L = N(49)-(G - Hy) (3.6-7) - T . .06 ———-_...‘__ E -107- M___.___— and | X.B.R. I I/ZM9) V = Silos— = Aly,-y) (3.6-8) and we remember that A is a slowly varylng function of ¥y 1in virtue of the small changes in G caused by spectral shifts with increasing dilution. These small variations cause, in fact, a slight downward curvature in the plot of XBR vs. vy. ¢). Total breeding ratio (T.B.R.) The T.B.R., being the sum of the I.B.R. and X.B.R., will also be an approximately linear function of y. Moreover, the leakage does not play a direct part in the overall neutron economy, since we have assumed that whatever neutrons leak out of the core are captured by 28. Therefore, any changes in T.B.R. will be due to differential parasitic capture with changing y. This can occur because of the shift in spectrum that takes place with dilution. a increases, as well as the relative parasitic capture by diluents and contaminants. Therefore, the T.B.,R. falls off somewhat with greater dilution although not as strongly as the X.B.R. Moreover, for the dilutions at which the T.B.R. does begin to drop more sharply, the critical mass becomes infinitej for feasible masses, the whole variation of T.B.R. is about 15%, TRENDS OF SIZE AND CRITICAL MASS. - a). Trends of size Formula (3.6-7) gives the leakage L as an approximately linear function of the 28 to W49 ratio y. For a bare pile, the following formulae are true: La K = 3 Ztr (3’6"9) kb - T (306"10) 07 F=ETEEr -108~ e emea! T T ——— where k2 i1s the buckling and b is the radius of the bare plle. For chloride systems (PuCl3-UClh), the transport cross- section,‘Z:tr, is approximately independent of y since the Tip of 28 and 49 are approximately equal(a) and the total number density of heavy atoms(b) is practically independent of vy, being 20% less for y =z 10 than for vy = O, In fact, we can write: N(49) + N(28) = N = const.j therefore: No N49) §1+v§ = N5 N(49) = 5o (3.6-11) Combining formulas, we obtain for b2: ' 2 2 = 1ty 1 1ty bT = e——— . . = C (3.6=12) 304p N, Ay,-v) YooY Strictly speaking, the constants in this formula vary weakly with y. However, (3.6-12) still displays the main feature of the critical size: slow variation for small y, rising rapidly to infinity for y -+ Yoo One notes that the X.B.R. goes to zero as the critical mass goes to infinity. b). Trends of mass. The mass of 49 in the bare pile is given by: M = % 3« N(49) » m(49) (3.6-13) where m(49) 1s the mass of the 49 atom. Substituting for b gives: \ 3/2 N M = -31& mm(49) c3/2(%3_fi\,) . T (3.6-1%) T (y,-m)3/2 According to this formula, the critical mass varies relatively slowly for small y, then increases to infinity for Y Yoo (Yo 1s of the order of 10). Therefore, there is an extensive range of y wherein inventory 1s roughly the same (8) nis is even truer for 25 mixtures. (b) As well as that of Cl atoms <08 | o — e sl LTI L - Y a——— - 1 0 9 - . and therefore can be a secondary consideration in choosing ry. c). Temperature effects on density. The bare-reactor multigroup method of calc¢ula- tion shows that the spectrum is independent of density; there- fore, one need only consider density changes in analyzing critical size and mass changes with temperature. Since the proportions of a chloride mixture are independent of temperature, the temperature variation of N (number density) - for each component - will be the same. If we assume: p=p, £ (t) (t = centigrade (3.6-15) temperature) then we can write: N(49) = N (49) £ (t) (3.6-16) N(28) = N_(28) £ (t), ete. where £(0) = 1. Now we rewrite the formulas for b° and M, keeping track this time of all number densities. We have: L = N(#9) }G - H v} (3.6-17) 2 e L1 (3.6-18) 5552m'N Therefore, 2 p? g ——T (3.6-19) 3Ty, N-N(9) | G-Hy} 2 b2 - —EE—_————— . “ufl_;—_—_—_ . 1 :El_gg__fi 3T 36-H% Ny * Ny (%9) £(8)% £(t) ®ocsoece s (306‘20) 10 ——= 7 bd e N(49) b3 (3.6-21) w2« N (49)£(E) £(t) WIE WIF Mo Mz—7 | (3.6-22) £(t) Equation (3.6-22) shows that M is more strongly dependent on temperature than the critical radius. Estimates of f(%t), made by chemical engineers for the fused salt systems, gave: F(E) ™1 - .25 x 1073 « & (3.6-23) These formulas are of use in calculating the temperature co- efficient of reactivity due to density changes., TRENDS OF &. - The only effect of y on a is the change it makes on the neutron spectrum, viz., the spectrum will become slower as y increases. However, this effect will be less marked for larger vy it is found that the spectral shift becomes very slow for y's within several units of the point of infinite critical mass. Therefore, although a(%9) changes rapidly with energy, the median spectral energy of the reactor changes slowly for this range of y so that a(49), the average over the spectrum also changes slowly. For small values of y (0O to 3 or 4), a follows the relatively rapid shift in neutron spectrum. For interesting values of y (those which lead to reasonable critical mass and inventory} a is less than 0.2. N Lo -, .- “m" b 150 . . ol . -111- 3,7 FINAL DESIGN OF FUSED SALT REACTOR BARE REACTOR CALCULATION. - A bare reactor calcula- tion was made for the system described in Table II-3.7-1. The atomic constituents were chosen by optimalization of pro- cessing and production costs. One notes that the fission product concentration is quite low, and the U-236 content relatively high (see Section II-4). Table II-3.7=2 is a detalled neutron balance of system 24 based on 1 neutron absorbed in Pu-239 and U-235. Excess degradation is that small amount of neutrons which are de- graded below the 8th y group, and which are assumed to be lost by paraslitic capture. Figures II-3.7-1,2 are the fission and flux spectrums for the prescribed core. This calculation was used as a preliminary basis for the desigr and evaluation of the fused salt reactor discussed in the engineering analysis report. The one-velocity methods of Section II-2.4 were used to compute the reflected critical mass and the Ak assoclated with the molten Pb control rods. -112- System 2k System Constituents and Results System No. U25 U26 28 Pu239 Na Cl Pb 2L 1 | 0.2 |3.25 | 0.0048 |1.81 0.173 1.126 0. 744 0.382 (125' - T.C.R. N XoCoRo - I.C.R. Fraction Median Fission Energy = 1ll1 Kev Critical Radius = 129.39 cm Critical Mass (Bare)=z 2.918 N(25) = .825 x 10°! atoms U-235 p = 4.2 gm/ce 28.75 of Fast Fissions = 8.2% /ce 4. 57 0.017 metric tons of U=235 - IABLE IT-3.7-2, System 24 - Neutron Balance Basis: 1 neutron Neutron Production absorbed in U-235 and Pu-239 v(25)+*Fissions of U-23%5 v(#9)-Fissions of Pu-239 v(28)+Fissions of U-238 v(26)+Fissions of U-236 Fissions of Total Neutron Production Neutron Consumption U-235 Fissions of Pu-239 Fissions Fissions Captures Captures Captures Captures Captures Captures Captures Captures of of by by by by by by by by U-238 U-236 U-235 Pu-239 U-238 U=236 Pb Na Cl r.P. Excess Degradation Leakage =13 - Total Consumption lllll 2.094728 0.011942 0.173925 - 0.017257 2.297852° 0.848068 0.004021 0.069570 0.006903 0.147120 0.000790 0.382281 0.023525 0.005060 0.000735 0.052288 0.001480 0.011659 - 0.74%356 2.297856 -~113- -114- e ———— FIGURE II-3.7-1 . [ SYSTEM 24: SPECTRUM OF FISSIONS AND — 130 FRACTION OF FISSIONS ABOVE u 28 .26 124 | ’ 22 = ® 10 20 o £ z \ | D Total Fissions w “ .8 1 6 u > \ e .7 14 - \ < x 6 12 L. 1 Kev. .5 10 4 N 08 3 \ 06 /Fost Fissions \ 2 04 A 02 O —~1_ o 0 2 3 4 5 6 7 8 9 IO FISSIONS/ UNIT u —= . -115- FIGURE II-3.7-2 SYSTEM 24:; FLUX SPECTRUM 30 .28 26 .24 r N N O /Flu x Spectrum ® o B FLUX /UNIT u—e o o Q @ 06 04} 02 -116- MULTIREGION CALCULATION. - Through the cooperation of Dr. Ehrlich at KAPL, NEP obtained two machine calculations for the three region multigroup problem of the fused salt re- actor. The calculation was made to determine the correct re- flected critical mass, and to determine the decrease in con- version ratio brought about by absorptions in the reflector. Such a calculation by hand was not considered practical. The reactor was approximated by a spherical core en- closed by a reflector and a blanket. To make the calculation for the blanket and reflector easier, these regions were homogenized with their containing structural materials, i.e., the core container, the inside edge of the blanket contalner, the two-inch void, and the Pb control rods were homogenized into one region called the reflector; similarly, the rest of the blanket container and the blanket material (UCL,,) were homogenized into one region called the blanket. In the machine calculation, the reflector uranium was assumed to be pure U-238, and the blanket uranium was assumed to be depleted uranium containing 0.3% U=235. The neutron balance achieved by the machine computation was .984 produced neutrons for 1.000 source neutron, using v = 2.50 for U-235. Table II-3.7-3 summarizes the results of the homogeniza- tion and gives other facts concerning this reactor. The re- actor constituents were changed slightly from the bare reactor calculation above. | ~ Table II-3.7-5 gives the neutron balance based on the machine calculations, adjusted to v = 2.47, to be consistent with the rest of this report. Results of perturbation calculations to force the neutron balance, and to include the effects of small changes in Pu-239 and U-235 concentrations in the reflector and blanket are to be found in the engineering analysis réport. SR b -117- Table II-3.7-4 presents the results of the multigroup multiregion calculation, and by comparison with Table II-3.7-1 indicates that the assumptions made using the bare reactor calculations and one-group theory were satisfactory. The only significant change brought about by the more elaborate calculation was a 5 per cent reduction in reflected core radius. The blanket was designed to be four diffusion lengths thick, on the basis of one velocity theory, to reduce the leakage loss out of the blanket to a minimum. The multi- region calculation substantiates this decision. Figure II-3.7-3 compares the averaged flux spectrum in the three regiohs. By comparison with Figure II-3.7-2 one sees that the core spectrum is only weakly coupled to the ‘blanket. | Figure II-3.7-4 represents the powér distribution within the reactor as a function of the radius. Figures II-3.7-5,6 portray the integrated flux vs. radius, and the shift in spectral distribution of the neutron flux with radius for the fused salt reactor. The areas under each curve in Figure II-3.7-6 have been normalized to unity. Only one spectral curve has been drawn for the interior of the core, Figure 3.7-6, because the shape is practically invariable almost all the way out to the core-reflector interface. This effect, the truth of which was assumed in the bare-reactor multigroup calculation, demonstrates why the bare reactor calculations give good results. | The rather small shift in spectrum at the outer surface of the core shows that the’back-couplipg from the reflector to the core is weak, and hence that the "strong-coupling" method outlined in Section 2.2 would not be expected to give reliable results. On the contrary, the "driven-blanket" method, which assumes the blanket to be coupled to the leakage spectrum of the core and to exert no reciprocal influence on - the core, rests on reliable assumptions and is the calculation oot f— ) ------ ...... oooooo ------ [1& TABLE II-3.7-3. System Constituents of Fused Salt Reactor Region | U-235 | U-236|0-238 | pu-239 | Na | C1 Pp | Fe | F.E. N o atom/ce gm/ce Core 1.000 0.20 | 3.2% |.oou8 1.89| 29.20 | 4.77 0.016 |N(25)z.810x10°T 4.2 Reflector .035 0.141 | 0.271]1.000 N(Fe)=23.798x10°T - Blanket .003012 .996988 4,000 0.736 N(28) = 6.043x10°Y 4.05 -81i1- 117 w 3¢ Results of Fused Salt Multiregion Calculation - e m TABLE II-3 07-’+o ...... _ C.M. (25) Reflector | Blanket | Fission (Including U-235) ?:;: . T.C.R.{ X.C.R. | I.C.R. a25 netric bons Core Radius Thickness | Thickness | Power (1n blanket) g Core Blanket and o ' Reflector ;.:.:' (given) (given) ot 5 1.126 0.7%5 | 0.381 | 0.1381 1.168 95.9 cm 15.25 cm 150 cm 97.13% 2.874 * From Table II-3.7-5 i — — v o 027 s ¥ ...... ...... ------ aaaaaa nnnnn nnnnn 000000 000000 ¥ TABLE II-3.7-5. /120 w Neutron Balance - Fused Salt Reactor: Multiregion Calculation Basis: 1 neutron absorbed in U-235 and Pu-239 in the core Core Reflector . Blanket All Regions = Neutron Production ® *v(25)+Fissions of U-235 2.080600 0.033276 v(26)+Fissions of U-236 0.015642 v(28) «Fissions of U-238 0.157183 | 0.005895 0.028103 v(49)-Fissions of Pu-239 | 0.011740 Total Production 2.265165 0.005895 0.061379 2.332439 Neutron Consumption , Fissions: U-235 0.842348 0.013472 U-236 0.006257 ’ U-238 0.062873 0.002358 0.011241 Pu-239 0.003953 Captures: U-235 0.152868 0.003437 U-236 10.023531 U-238 0.381222 '0.040077 0.704953 Pu-239 0.000830 Na 0.000905 c1l 0.052761 0.002442 0.042803 Pb 0.005033 0.002192 Fe - 0.024205 0.010443 F.P. 0.001383 Net Leakage 0.795872 | =-0,071486 -0.71901k Total Consumption 2.329836 -0.000212 0.067335 2.396959 - - - ” - - - - - - - -l - - - - - - - - - - - - - - - - - L] — - - - - - - - - - - - - * v(25) = 2,47 was used in this table to be consistent with rest of report. v=2.50 was used in original machine calculation. 1 1 * 1 L Flux /Unit u -121- .30 Fused Salt Reactor - Flux Spectrum 28 1 26 . Reflector Spectrum. 24 P 22 =T =" 2 e==i— ' i J Blanket | .. ' Spectrum 18 16 —core Spectrum y I 14 i 12 ! — | I E Lo — 10 - — ] | I 08 1 06 04 i____ 02— _____l L.................. | L] | 2 3 4 5 6 7 8 9 10 I U — ~21 PTTI T J 22~ ¢z }6 -221- ...... oooooo cccccc ...... 6 - \ —— ~— Reflector ...... Relative Fission Power oooooo ...... +————— Core - — Blanket 30 60 20 120 150 180 2l0 240 r (cm)— FIGURE II-3.7-4., Fission Power Distribution in the Fused Salt Reactor /23 FIGURE II-3.7-5. Total Neutron Flux per Unit Radial Distance vs. Radial Distance (r2(P, arbitrary units) 0 AN ------- - \ Reflect 3 | / \ eflector lllll nnnnnn ...... ...... DDDDDD ,,"__ P —_— "'“P , Arbitrary units ——= p [ — / Core Blanket . // \\ L/ . \\ \ 20 40 60 80 00 120 140 160 180 200 220 240 260 r i (cm) €21~ 1 ek /24 0 ; S i }-2 o > 40 T D ...... h V o | - ...... o S ------ L. @ ...... ‘J ...... = o e e C . + = By i de — - ° e b IIE. Q : i3 5 2 ’ : L o S 5 15 b 2 ...... o | o~ I .05 0 9 c Y FIGURE II-3.7=6. Normalized Flux Spectrum at Various Radial Distances *» method of choice for computing the blanket spectrum. The spectrum in the blanket degrades slowly with in- creasing radius as the flux gradually shifts to the fundamental mode (the "equilibrium blanket"). However, it is evident that the blanket ends long before that stage. To evaluate the core radius for an exact neutron balance, an extrapolation procedure gives 95.9 cm. One will note that the neutron balance is slightly inconsistent due to round-off error and the slightly non-critical multiplica- tion constant assumed. The figures in Table II-3.7-5 are given to 6 figures only to allow intercomparisons to be made by the reader, and not as an indication of the accuracy of the calculation. Based on the preceding results, the following calcula- tion procedure appears appropriate in future work: a)., Surveys: Use the one-velocity techniques described in section II-2.4 to compute con- version ratios and critical size. b). Design parameters for particular systems: Utilize the multigroup technique for the bare reactor to compute conversion ratios and core bucklings next, use the core spectrum to choose appropriate average cross-sections to be used in a one-velocity calculation for the three-region problem giving reflected core radiussy then use the driven blanket technique to determine the characteristics of the blanket. ¢). Final design: Use multi-region, multigroup calculation methods. A second calculation for the fused salt reactor was undertaken to determine the effectiveness of the Pb reflector control rods. The second calculation substantiated (within 15%) the one-velocity estimates (Section II-5.2) for the effectiveness of the Pb control rods. The change of -126- : """‘"—-—-h.._ reactivity according to the multigroup multi-region calcula- tion is Ak =5 = 0.027 if all of the Pb is removed from the reflector. 126 = 4, FAST REACTOR POISONING 4.1 INTRODUCTION Fast reactor poisons are detrimental because they decrease the conversion ratio and increase the critical mass of fuel. The two principal types of fuel poisons are the fission products and the higher isotopes. The features of these two types are different in important respects. The concentrations of both types of polisons are, how= ever, governed by secular equations of standard form: The rate of change of element "a" (= —2) is equal to its pro- duction by fission (= Z'f?fiY(a)), by neutron absorption by element "3" (= N(3) c%(fi)i?) and by radiocactive decay of a predecessor a' (= A(a') N(a')), minus its loss by neutron absorption (= g% (a)$N(a)), by radicactive decay (= A(a) N(a)) and by processing (=« R(a) N(a)). (Here Zs is the macroscopic fission cross-section, @ is the neutron flux integrated over the core, Y(a) is the yield function for fission products, N(x) is the number of atoms of element "xV in the core, A is the reciprocal mean 1life and R(a) is the processing rate.) The balance equation (for each "“a") is then: igfltal = 2, P¥(a) +Po(8) N(&) + n(a') N(a') -§a%(a) N(a) - A(a) N(a) - R(a) N(a). (4,1-1) 4,2 FISSION PRODUCTS As the fission product capture cross-sectlions are small in the fast region, loss of fission product atoms by neutron capture 1s a relatively small item in comparison with the high chemical extraction rates possible in fluid fuel systems. In equation (h.l-l), processing is assumed to be done at a constant rate R. Moreover, the small capture cross-sections (relative * R RS RET— i e c vt 27 -128- to the fission cross-section of U-235) make it possible to allow large concentrations of fission products to build up without too adverse an effect on the conversion ratio. In fact, _ (PP . a (F.P.) = E%TE?TH- < .12 4,2=1) Accordingly, if a 10% loss of neutrons to fission products 1s tolerable, the fractional burnup allowed is 53%, for: fract. burnup v-1-0 (tolerable neutron loss) ceees (W2-2) o g - 1.27 5%. 15 X +1 = 53 The direct effect of fission product capture on conversion ratio is actually as follows: 2 (F.P.) Loss iI'l XOC.R. = Z__(-gr- (L"02-3) a If the radiocactive decay half-life is long, the ex- traction rate is much larger than the decay rate; on the other hand, if the decay half-life is short, one can consider the atom to decay immediately into its daughter isotope, (in effect, not to exist at all). Then, competing events (viz., burnup, processing) are negligibly probable by comparison with decay. Fortunately, the decay half-lives of the fission products are either much longer than or much shorter than the processing half-life, which leaves the processing rate the dominant quantity in determining fission product concentra- tions. 4.3 HIGHER ISOTOPES In contrast to fission products, higher isotopes have larger capture cross-sections and fission cross-sections W T AR gyt R A 0,\ L » — -129- as well. Moreover, the role of capture-decay chains is much more important in the formation of higher isotope poisons. (These chains are described in detail in Section III-3.1.) Therefore, the neutron capture and radioactive decay half-lives are of the same order of magnitude as economically feasible processing half-lives. This means that higher-isotope equilibrium concentrations depend more strongly on nuclear quantities than do fission product con- centrations. For converters, in particular, the formation of U-236 is of prime importance, because chemical processing techniques cannot be used to remove it from the reactor. Hence, its concentration will build up to a much greater degree than those of fission products, thus rendering its effect on the conversion ratio much more deleterious than fission products, in view also of its higher capture cross-section (on an atom-for-atom basis). A mitigating feature of higher isotopes is their non- vanishing fission cross-sections. However, this effect does not nullify the loss due to capture. If one denotes the fission cross-section of these elements averaged over the neutron spectrum as Zf(H.I.) and assumes v neutrons per fission, the loss in X.C.R. is: (H.I.) = (H.I.) Loss in X.C.R. = za Z‘ a(2‘5";2f (11'03"'1) Except for meager data in BNL-170, little is known about the higher uranium isotopes. U-236 was assumed to be similar to U-238 for all neutron processes except fission., In breeders, the principal higher isotope poison 1s Pu~-240, Experiments at L.A. indicate that gp(2k0) < . L"o - and one may claim on the basis of general nuclear considerations | L TR _ — P TR - N that (24+0) %—2—3-87 ~ 1 (%.3-3) a 4.4 ENGINEERING CONSIDERATIONS A more detailed discussion of the secular equations and their solution is contained in the engineering analysis report, It is evident from Eq. (4.1-1l), however, that the relations between concentrations under steady-state conditions will depend only on the ratios of cross-sections. In general, the ratios of cross-sections vary much less with energy than the cross-sections themselves. Therefore, it 1s possible to make estimates of steady-state concentrations using values of cross-sections taken at the mean neutron energy of the core. It remains only to choose processing rates. Because the processing cost is strongly dependent on the processing rate, economic factors, as well as design feasibility factors,come into play in the choice of pro- cessing rates. These must be balanced against the loss in conversion ratio and the increased inventory that attend high poison concentrations. The arguments above show that this balance will be well on the side of low processing cost for fission products, because of the ease of processing as well as the high allowable concentrations. ¥For higher isotopes, this balance is not so readily struck, because it is expensive to extract them and expensive to leave them in. Thus, the higher isotopes are much the more serious poisons for fast liquid-fuel reactors. 217¢ 130 S sSsw B oo .e * . . 23cC e * . e - . e = ~131- 5. CONTROL METHODS 5.1 GENERAL CONSIDERATIONS When considering the kinetics of fast reactors, one is always struck by the disastrously short periods which would result by even small Instantaneous increases in k. However, in nature nothing happens instantaneously, and, 1in fact, in a reactor the speed at which the multiplication constant of the reactor may change is to a considerable extent under the control of the designer. The possible methods of control consist in adding absorbers, removing fissionable material and changing leakage by moving parts of the reflector. As the absorption cross-section of most elements is less than 1 barn in the fast region, absorption control " implies moving large amounts of material in short periods of time. One must also consider that additions of absorbing material upset the neutron balance and lower the conversion ratio. Control by removing amounts of fuel is more effective in fast reactors, but one is then faced with the problems of removing large amounts of heat generated within the control element. U-238 control rods would not destroy the conversion ratio, but then the rods must be processed from time to time to recover the valuable product produced. For large fast reactors, reflector control implies moving a large mass of reflector in short periods of time, a difficult mechanical accomplishment. However, reflector control has the advantage that it does not destroy conversion ratio, as those neutrons which leak out are captured by the blanket to produce useful Pu. One type of fast reactor control which has not been examined thoroughly but has some merit, is moderation control. Some moderating material is added to the reactor, = -132" M slowing down the reactor spectrum, and decreasing the reactor reactivity by essentially raising the overall a. This method of control would also destroy conversion ratio, and in this light is not as advantageous as reflector control. 5.2 CONTROL CALCULATIONS REACTOR KINETICS. - We consider now the time behavior of a reactor., If n 1s the number of fissions per unit time occurring in a reactor, then the equations describing the secular behavior of n are(a)z éi - (5.2-1b) § i >J |ul. Q e + = = where B; is the number of delayed neutrons of 1 th type arising from each fission divided by the number of prompt neutrons so arising, €{ is the "effectiveness" of the ith type neutron, S is a neutron source term, L 1s the prompt generation time, and Cy is related to number of delayed neutron emitters of type 1. If one calculates the complementary function of (5.2-1) ot wt by assuming n = ne" ", ¢y = Cy, € 9 One obtains for w: hfiili L - w=A+F TR (5.2-2a) or, substituting for A: B1€12y Lo = k_ =1+ 24 ———— (5.2-2b) k@ 1 w + ki In general, if there are N delayed groups, there wlll be N+1 solutions of (5.2-2), of which only one can be positive. The reactor reproduction constant k, is obtained by setting w = 0 and requiring that k = 1 for this case (steady state). (a) Equations (5.2-1) and, (5.2-2) are based on: H. Hurwitz, Nucleonics, %, p. 61 (July 1949). AR EC R Pi— . i w o~ 29 see v see W . (X} j . s .. . e v s v @ o ® . "y ) e @ ae o . * . g - . e - e 2 s » . o @ » ; - . » - . . + . s e & * . e . e an3 s . . ¢ sea . e ates s g % -133- Then we have: 0 = Xk, "1‘*23-3%61 (5.2-3) and thus ky + 2By € =k, Kzl (5.2-4) Eliminating kp between (5.2-2b) and (5.2-4) yields: A w = k -1+ Zsiei[—-——w,rii - 1% (5.2-5) L Now, let us consider a reactor with a fixed k, and change the effectiveness €1 of the 1% delayed neutron group. (That this can happen will be shown in the next section). We wish to find the induced change in o resulting from changing €; . Therefore, we differentiate equation (5.2-5) with respect to €; to obtain: w A\; Bi€: \ ) 4 ——J—- — -# _&)_ ° - L ééej = fi"iwfl\j '} (;Z(wwli)z) IE; (5.2-6a) or, rearranging: {L + 7 P& f W = - Bl g, ( (w+A)* 963 a.J+AJ Now consider a rising period (k» 1l). The brace in (5.2-6b) is always positive as are all the factors on the right-hand side. Therefore, the derivative )w/aij 1s negative, so that if €) decreases, w will increase. A similar argument for the case when w< 0 (k<1) shows that g-lé";—'<0; thus, the absolute magnitude of w always increases when ej decreases, and vice versa., Now consider the steady state of the reactor for k<« 1. (For k » 1, there is no steady state). We set all time- TR S Ee— — —— —— . . - i 4 e . dee 4 -« - L & 9 Y @ s 3 s 8 [ (1] . e o » . . . e . . . @ . -y . s .. « s » @ . s 0 € . are & 2 re e e -134- i EEEE derivatives equal to zero in equation (5.2-1) and eliminate ci. We obtain: oo (7 or, after substitution: Pt'E,; S . ™ )h + T =0 (5.2-7a) n = g (5.2-7Db) This relation will be useful in calculating heat generation under subcritical conditions. We shall now derive some approximate consequences of (5.2-1) by dropping delayed neutron and spontaneous fission terms. This procedure is to the point when the reactor 1is prompt supercritical (an "accident"). Since Ajcy 1s positive, we have: n=An:t Z)‘i éi + %) An (5.2-8a) ¢ or * -1 Bsa- EP_T_ (5.2-8b) We integrate this equation with respect to time from © to T (maintaining the inequality), and obtain: T In n(T) - In n_ > %j‘(l&)-l) dt (5.2-9) © We assume now that: k,=1+Rt (5.2-10) i.e., kp is increasing linearly with time and at t - O, the reactor has just become prompt critical. Equation (5.2=-9) becomes, for this case: 134 | ST -135- 1n{nnT >-23—I: 2 (5.2-11) Now, when t = T, we assume that the reactor is well into the super-prompt critical condition and that delayed neutrons can be neglected. Then, from (5.2-8a), a | k -1 RT Lo (= _E_L - —L') (5.2=12) By definition, %E% where © 1s the reactor period. There- fores fclt'T‘S' = %2 (5.2-13) Combining this with (5.2-11) (by eliminating T), we obtain: mifi%fl 7%5 ’ (%{}2 = %"fi : 1—2' (5.2-14%a) T or y q:>[ L ] = (5.2-14b) °R In { i n\e Finally, assume that a control element which moves along a coordinate x has a linear coefficlent of reactivity %}1—: equal to s Assume also that the distance x covered in time t (from a standstill) under the influence of gravity will be + g t2. The reactor period will then vary (from (5.2=12)) as follows: k--l-Sm-w,i;gt2 1_ & D - = ENA(t) = T, (5.2-15a) Using (5.2-10), we have, finally, 0 1, k-1 me-3% 2 1, Rt-3Ppt? T L L = To L s 000 s (5'2-15b) ~-136- These formulae will be used in Section 5.3. 1,0SS OF DELAYED NEUTRONS. ~ In externally cooled fluild fuel reactors, the fuel circulates through the core and the heat exchanger. Accordingly, some of the delayed neutrons are produced in the core and some are produced in the heat exchanger. To estimate the control dollaf*hevalua- tion due to this loss of delayed neutrons, one may assume uniform progressive flow through the core and heat exchanger. The fuel spends a time T1 in the core and a time T2 in the _heat exchanger. If one assumes that the delayed neutron emitters are produced uniformly throughout the core, one may write the fractional number of delayed neutrons produced within the core during the continuous cycling process at constant power as:s (- Yi-&™) 6,‘_ = | 3, Z, |- e"(fo“'zz) (5.2-16) where Z, = -'.1_-5. and Z, = I.‘%— L L T, = mean life of delayed neutron emitter i. Table I1-5.4-1 summarizes the result of applying equation (5.2-16) to each delayed neutron emitter, assuming representative values of Ty = 3.36 seconds and T, = 5.96 seconds. One dollar =%;eifli’ For all delayed neutrons emitted in the core@eifii = 0.0073 in Table II-5.4-1 (data from ORNL-1099). =36 A .‘ ‘ e == -137- TABLE II-5.4-1, Delayed Neutrons Half Life (sec) Bi(no flow) €;B4 (flow) €; 0.43 0.0008% . 000687 .818 1,52 | 0.0024 .001236 515 4,51 0.0021 . 000808 .385 22,0 0.0017 . 000614 .361 55.6 0.00026 . 000094 .360 0.0073 .003439 One sees from the table that the delayed neutron dollar has been deflated to 47 per cent of its original (no flow) value. In view of the discussion in the section on kinetics, one notes that for a given k, the reactor period has become smaller, One may also compute the generation time from equation (5.2-5) when w&«X;, as E ’-‘-‘Z%é‘-:- [, = .035% secs. (5.2-17) for the fast reactor for small Ak changes. If no delayed neutrons were lost, the generation time would be L. = Z_ IG‘;+ |_, = 0942 secs. (5.2-18) One notes that for reactivity changes greater than one delayed neutron dollar, the pile period becomes the order of the mean lifetime of a fast neutron in the reactor. For a fast reactor | ‘ L.""viiz‘ 2 10'"6 secs. It is obvious that such reactivity changes could not be tolerated in a fast reactor. Furthermore, one notes that the e . s | e pp——pr— - ! - #4 ee 4 ess & ww . v« @ s P . . . . " » . . ow . - nee - s = @ . * n * - loss of delayed neutrons lowers the upper limit of controllable reactivity changes by about a factor of 2. ~-138- DENSITY CHANGES. = It is of some interest to estimate the amount of reactivity change due to thermal expansion which will be found in the fused salt reactor. It is ex- pected that such a reactor will have a large negative temperature coefficient. From the one-velocity relation (II-2.4=3) Oa ) o 0 o — N? ¥ 'b% ’ (5.2-19) If one assumes that the reactivity, é% is approximately<%3 ) then, approximately 8k ody o M2 % ng_@.Ii (5.2-20) kK~ v b N ~=T zlt-erf For the fused salt reactor NN (1~ .25x 1073T) (5.2-21) where T 1s in degrees centigrade. "Iz fi,dv Hence ~ 2 = 0,25 x 103 4T (5.2-22) and the negative temperature coefficient of the fused salt reactor 1is approximately 2.5 x 1o‘“/°c (5.2-23) For metallic systems the negative temperature coefficient due to thermal expansion is of the order of 10'5/°C; while for an aqueous reactor, the coefficient 1s of the order of 1073 /°c. It 1s evident that the fused salt reactor has a large inherent stablilizing feature due to its large and negative L — Z 3R e O A TV AT “_:—--_._;___. E | ~-139~ ———— density temperature coefficient. It is expected that the nuclear temperature coefficient will be much smaller. REFLECTOR CONTROL. -~ To estimate the effectiveness of reflector control, it is assumed that the removal of a slug of reflector may be represented by a homogeneous macroscopilc density change over the entire reflector. From the results of section II-3.4, it is obvious that the core radius depends upon the thickness of the reflector. Accordingly, the effectiveness of the reflector control also depends upon the thickness of the reflector. The reflected critical radius is first computed for a given core buckling (kg fixes the dimensions of the reactor). Using these same dimensions a new fictitious core buckling, k2, is computed assuming the reflector density has decreased by an amount equivalent to the removal of a control rod of reflector. The change in core buckling can then be con- veniently expressed as an equivalent change in v. The change of v so computed will give an approximate result for the effectiveness of the reflector control, as Ak ~2 Ay k v As an examplé, a lead reflector 2.6 cm thick on core system No. 15 will give a reactivity change of Ok o, Ov kfi-’- vN0.025 if it is completely removed from the reactor. (See Figure II-3.4-1). | e Ml ket -140- M 5.3 CONTROL OF THE FUSED-SALT REACTOR The fast fused-salt reactor, being composed of a homogeneous melt, can stand rather considerable overloads in temperature for short times. This is a distinet advantage inasmuch as it renders control more feasible and permits of the use of the quenching action of a temperature rise (via the negative temperature coefficlent of the reaction demonstrated in Section 5.2) as a safety mechanism. CONTROL DURING CHARGING. - Let us consider what sorts of accldents may occur in the operation of reactor and what thelr consequences will be. At some interval not known, it will be necessary to f11ll the core with the uranium chlorilde melt, and it will also be necessary to start up the reactor by inserting the liquld lead reflector which is to be installed to act as a control rod. The lead reflector occuples a space about 2 feet high and about 2 inches thick over the surface of the core. The total effect on k of this amount of lead will be about 3%(8). If we are therefore able to insert this lead in about 5 minutes, the time rate of change of k will be .01% sec'lg or R will be 10'” sec™T. (See eq. (5.2=10)). The rate of filling the core may be calculated as follows: approximately 1 liter of melt will be equivalent to the addition of a Sk of .0001. If, therefore, 1 liter of melt is added per second, the rate of change of k will be .01% sec™t and R will be 10" sec™! which is the same as when the lead reflector is moved. The total time for filling the core will then be 10,000 seconds, or roughly 3 hours. Using equation (5.2-14b) with the above value of R, and assuming L to be 10 -~ seconds and g g to be e3° (a rather pessimistic estimate), we obtain a runaway period of %5 SeC., l.e.y, 11 milliseconds. (a) See the analysis in Section 5.2. The effect derived there (2.5%)1s for a reflector thickness of 1 inch. A solid-angle factor has also been included here to take account of Incomplete covering of the core by the reflector. - A D s 3 -141- Under normal circumstances, the intensity of the reactor and the rate of change of intensity will be observed on suitable instruments, and the insertion of melt or lead will be stopped when the reactor is on a stable period of about 10 seconds. However, if by some maloperation the reactor should run away, we have the problem of stopping its rise before damage to 1ts structure can occur, when the period is 11 milliseconds. For this purpose valves will be opened which allow the lead to flow out of the reflector at the same rate as if it were falling free under the influence of gravity. To this situation we apply formula (5.2-15b). If the total movement of the lead (2 feet) i1s equivalent to 3% in k, then the value of S is 5 x 10~ cm'l. Equation (5.2-15b) shows that the power will rise to a maximum and then fall rapidly. The time tm at which it will reach the maximum may be obtained by setting the left side of equation (5.2-15b) equal to zero and solving for t.. This gives a time of < .0l seconds between the instant that the lead starts to fall and the instant that the reactor power starts to decrease, The fractional rise during this time may be calculated by integrating equation (5.2-15b) from t = 0 to t = tm (computed as above). This gives a factor of rise of about 2. However, one must consider the fact that after the instruments have detected a runaway period, there will still be some time delay while a solenoid valve is operated to drop the lead. This delay may be estimated as about 30 milliseconds, and during this time the reactor will rise by a factor of e |(~ 20). However, instrumentation is now well developed which will detect a runaway period at a power level which is a thousandth of full power. A short-period runaway (as calculated above) may be checked and the reac¢tor shut off by means of the moving ‘lead reflector in a much shorter time than that in which appreciable heating of the melt would occur. Consequently, reasonable times for inserting the lead or filling the core seem to be consistent with safety. ' S |9 *e s e - . * . . . LN ] - * - L I ) ;e . - w9 & > * & . . e 3 anm t“i e 2 a * . ¢ o . . « @ e s 9 a *‘,}a XX . « o @ - - - g - - . * . . . . - » qes - » * & ad . & ——— - It may at some time be necessary to fill the blanket while the core is already full, and since the blanket has a powerful reflector action, the possibility of overshooting criticality during the filling will exist. In order to fill the blanket safely under these conditions, it should be arranged that the blanket and the core cannot be filled simultaneously and that the times to fill the blanket be about the same. We have shown above that convenient operat- ing times are quite safe, and in fact, if it were desirable to be more conservative, considerably longer times for filling core, blanket, and reflector would probably be operatlionally tolerable. CONTROL DURING OPERATION. - We should now consider accidents which are not connected with the start-up or the filling of the reactor. A serious possibility of accident might be anticipated if the reactor were run with a partially filled core. The rapid rate of circulation of the core material and the large pump connected therewith might be ex- pected to shift this level appreciably and thereby cause a runaway accident at the operating power level. To avert this possibility, careful precautions should be taken to bleed the top point of the core so that no pockets of gas can exist within the core volume itself. The only other accident which seems at all likely would be a precipitation of 25 in the heat exchanger as, say, a scale, and then a rather sudden break-off of this scale, which, when swept into the core,would cause an increase of k. In order to change the quantity of U-235 within the core by an amount sufficient to exceed the delayed neutron fraction, about 5 liters of break-off scale would have to be trans- ferred from the heat exchanger to the core. If a sufficlent amount of this scale were to be swept into the core to drive the reactor over prompt critical, the time to do so would be roughly the circulation time of the core, or about one second. This would give a rate of change of k (i.e., R) of -143- 5 x 1073 sec™l. 1If we now substitute this into equation (5.2-14b) and take 1n(2$4) as only about 1 (since this accident would be most dangerous if it occurred at full power), we agaln get a runaway period of about .0l sec. This would cause a power surge to about 50 times full(power; but since its duration would only be .01 sec., the total surge would be equivalent to an operation at twice power for one second. This would raise the temperature of the melt by about 200°C. Such an increase of temperature of the melt would cause boiling. However, an increase of only a few degrees in the témperature of the melt would be sufficlent to pull the k of the reactor below prompt critical,and there-~ by to decrease the intensity and duration of this surge by a large factor. Under these assumed conditions, the pile would probably survive this accident without damage. This type of accident, however, is very difficult to characterize quanti- tatively, and an analysis in terms of equation (5.2-14b) must be regarded as very rough indeed. Nevertheless, it indicates that even a rather large amount of scale formation might be tolerable as far as this accident is concerned. Moreocever, even such an approximate analysis as this points clearly to the desirablility of éontaining materials which will not form corrosion products which precipitates clearly also, precau- tions must be taken to remove solid materials as soon as possible after they are formed, Naturally, before one attempted a more specific design for such a reactor, the accident possibilities and con- sequences would be analyzed in much greater detail than has been done above. However, a preliminary consideration such as this indicates that, with reasonable design points and operating conditions, the Operafion of this reactor would be quite comparable in safety to that of such thermal reactors as the Naval reactor or the MIR. In case of signs of maloperation, it is proposed that the melt from the core would be drained into a sump. Since ——— — O s S . SU it LT S ————_y by €3 et s oFs & tea s . M I .. ¢ s . e ® . @ .. . . » . * . . « o » * s ] « & B | . . -144- there is a considerable cooling problem in the sump, and since the reason for draining the core might well be mal- operation of the heat exchangers, it 1s important to estimate the times that it would take to stop the chain reaction. Any maloperation in pumping the core would of course scram the reactor and drop the lead reflector out in the manner des- cribed before. This would reduce k by 3% during the total time of fall of the lead, which would be about 1/3 second. The power due to all sources would decrease by about a factor of 10, since the power due to fission can drop to much less than that due to the decay of fission products, at least over short intervals of time. This can be shown as followss Equation (5.2-7b) gives the total rate of fissions in a sub- eritical reactor induced by a source of neutrons. For the purpose of this calculation, we can consider the delayed neutrons as originating from a constant source (even though this source really decays with the mean 1life of delayed neutrons). The reactor under this assumed circumstance was suberitical by about .3% when it was running steadily. When the lead is dropped out, the reactor is then subcritical by about 3%. The ratio of the fission rates in the two condi- tions (remembering that there is the same source for both) is, according to (5.2-7b), just equal to the reciprocal of the ratio of their "criticality defects", i.e.: N. (after shutdown) 1 -k = —-————-9-%-1 (5.3-1) VNb‘Tbefore shutdown) = 1 -k, = 3 = * This shows that the power will drop to about 1/10th its value (and subsequently decay exponentially). B.R. C.M. av ' F.P. ~145- GLOSSARY - CHAPTER II Neutron absorptions in the neutron balance equations A(t) 1is related to the reactivity of the reactor in the reactor kinetlec equations: 1 A(t) = ETE;:TT One=-velocity neutron flux coefficients as defined in Figure II-2.4-1 Sum of all neutron losses, except the degradation, in u group a Critical radlus of bare critical pile or reactor. Actually this is the extrapolated critical radius, no account being made for difference between extrapolated radius and radius of actual material to achieve bare critical Breeding Ratio (see T.B.R.) Variable related to the number of delayed neutron emitters, my3 i.e., €114 C; = i L vp where v_ is the number of prompt neutrons per fissionP Critical mass of fissionable material Differential volume element Fission Products Ratio of average slowing down density g in energy group a to slowing down density at bottom of energy interval qy: - P = Fy q'; '(—_FL)a§a Acceleration of gravity ot Lo P dwanee dhvae > . ot » - - -146- H.I. I.B.R., I.C.R. W & l‘(a) K%a) t R e e m(a) N(s) R(a) '!’; *é;-fihuun;-..' Higher l1lsotopes :£!K£l.§!;£2£§__- the eigenvalue in the "strong { ¢tr) av blanket ’ coupling" blanket calculatlons Internal Breeding Ratio = Gross atoms of Pu-239 produced in the core per neutron absorbed in Pu-239 by fission and capture in the core Internal Conversion Ratio = Gross atoms of Pu=239 produced in the core per neutron absorbed in U-235 by fission and capture in the core Effective reproduction constant Prompt reproduction constant Buckling, with atom number density factored out, for multiplying media in multigroup calculations One-velocity buckling for multiplying region a One-velocity buckling for non-multiplying reglon a Fictitious cross~section representing neutron leakage Neutron leakage in the neutron balance equations Prompt geheration time Generation time including delayed neutrons Mass of atom a Number of fissions per unit time in reactor Atom number density of material s, atoms per cm3 Neutron production in the neutron balance equations Time coefficient of prompt reproduction constant Processing rate, fraction of (a) inventory pro- cessed per unit time Source in reactor, i.e., spontaneous fission source . - . sy 38 ¢ & s .e . » * are *e . . . * F @ » * ® - L 2 a0 *soRPS o I X.B.R, X.C. X.C.R. X.C.o Y(a) Z a o s fi;E“ " -147- ——— A neutron source term in u group a Centigrade temperature Time (seconds) A definite value of time, t Time spent by circulating fuel in core Time spent by circulating fuel in pipes and heat exchangers Total Breeding Ratio = I.B.R. + X.B.R. Total Conversion Ratio = X.C.R. + I.C.R. In (E/E) where B, = 107 e.v. width of energy interval a 1n lethargy units Neutron velocity Coordinate x External Breeding Ratio = Gross atoms of Pu~239 produced in the blanket per neutron absorbed in Pu-239 by fission and capture in the core External Conversion Ratio with reflector in two and three region calculatlons External Conversion Ratio = Gross atoms of Pu-239 produced in the blanket per neutron absorbed in U-~235 by fission and capture in the core External Conversion Ratio without reflector in two and three region calculations Yield function: atoms of (a) formed per atom fissioned T/t5, a dimensionless parameter Ratio of capture to fission in fissionable nuclei, for example a(Pu) = o-. (Pu)/g%(Pu). From the context this is easil? distinguishable from the following second definition of «a. -148- | @ - a Index labeling the energy group. In the present calculations a runs from 1 to 8 (n = 8), the groups being selected on a logarithmic energy scale as follows: as 1 2 3 L 5 6 7 8 us 05"1 1"2g 2"333"3-75 3-75"‘"05,?"'- 5"50 535- 5'7,7‘10 : 4.87,2. 51,.92#,.367, 173, .076, .025,.0048 - _ _fi (), £ B (5 as) in maltigroup calculations , capture to fission ratio for U-235 =2 N W ! By Fraction of delayed neutrons of type 1 Y = N@8)/N(M9) €4 "effectiveness" of ith delayed neutron emitter on reactivity 3(2 Buckling, with atom number density factored out for non-multiplying media in multigroup calc tions Radiocactive decay constant (sec'l) Ratio of externally supglied neutrons to blanket (by leakage from core) per fission source neutron in the blanket for the driven blanket calculatlion v(a) Number of neutrons per fission of element a v Number of neutrons emitted per fission (must be summed over all fissionable materials). vo = calculated value of v. g - Mean logarithmic energy decrement per elastic collision = average change in lethargy q Microscopic cross-section (per atom)s subscripts same as for Z Macroscopic cross-section; the subscripts on X are defined as: tr = transporty a = absorption (fission + capture); s = elastic scatterings is= inelastic scatterings f =« fission Jr Summation sign (to avoid confusion withZ)) \“ o T E > -149- M— Reactor Period (sec); defined by Equation II-5.2-13 Mean life of neutron emitter 1 (sec) One-velocity neutron flux, if subscript a is missing Neutron flux integrated over the reactor;-—uSCPdV Neutron flux at point r integrated over the energy interval denoted by a Spatial part of neutron flux, § Fission yleld function Fractlion of fission neutrons emitted in energy range a, crall aXq =1 Probability that a neutron inelastlcally scattered in group g will be scattered into group a, ofa()ti)fi—*a =1 A root of the inhour equation -—_rt - TETe— -150- Pt s ACKNOWLEDGMENTS The physics group of the Nuclear Engineering Project is greatly indebted to Dr. H. Hurwitz, Dr. R. Ehrlich and the staff of the Knolls Atomic Power Laboratory for the assistance, advice, and data they gave for use in the fast reactor calculations. Through their generosity, the N.E.P. received invaluable aid in the form of 1). Nuclear Data, 2). Fast Reactor Calculation Methods, and 3). Machine Calculation Assistance g 15p e -151~ ITI. THERMAL REACTORS 1. INTRODUCTION In the earlier stages of the project, calculations were made for fast reactors using Uranium-Bismuth solutions. It soon became clear, however, that the concentrations of Uranium in Bismuth available at reasonable temperatures would not provide a sufficiently fast neutron spectrum. Only by going to slurries could such a fast spectrum be obtained. Because of the large number of uncertainties in the behavior of slurries under reactor operating conditions, in particular, in regard to their stability, it was concluded that further investigation of the nuclear properties of such systems would be of academic interest only. The available concentrations, as well as the low thermal capture cross-section of Bi made a U-Bl fuel mixture a strong contender when the decision to study a non-aqueous thermal converter was made. At the time no fused salt thermal system seemed more promising. It was therefore decided to in- vestigate the nuclear properties of a Uranium-Bismuth-Beryllium reactor. A large number of bare homogeneous thermal converters were studied. The variable parameters were the U to Bi and Be to Bi ratios as well as various U-235 atomlic enrichment fractions. The calculations are described in Section 2. While the results of these early calculations (See Appendix B) were crude and approximate, they described trends correctly so that a choice of the optimum reactor could be made on both physical and economic bases. ' The modification of the results of Section 2 due to the effects of polsons 1s investigated in Section 3. There we have studied higher isotope buildup, in particular that of U-236, the fast neutron reactions of Beryllium and the poisoning due to fission products. ~152- \ | Actually the conversion ratios of the bare homogeneous reactors with sufficiently low critical mass turn out too small for economically feasible operation. This is due to the loss of neutrons which leak from the reactor. Both the fast and slow leakage could be reclaimed by the use of a sub- stantially pure U blanket. It is generally conceded, how- ever, that a truly multiple region thermal reactor is out of the question because the increase in conversion ratio does not compensate for the increased inventory, additional chemical processing, and the added engineering problems. The fast leakage, however, can be reclaimed by the simple ex- pedient of reducing the Be to Bi ratio in a peripheral zone of specified thickness. The manner by which this is accomplished is described in Section 4%.1. In Section 4.2, the effect of Be lumping is briefly considered. Finally, in Section 5, we report in detall the results of calculations on the final reactor design. In connection with the latter, there are some problems, both nuclear and enginéering, left unfinished, which might alter the significance of our specific study. There are also some uncertainties in the nuclear data which might modify both our particular results as well as the general status of the type of reactor considered in this chapter. These are discussed and specific recommendations for further study made in Section 6. ! *y -153- 2. BARE HOMOGENEOUS REACTOR The computations for unreflected thermal reactors utilized simplified approximate expressions. The reactors were assumed to be spherical and free of poisons. The effect of poisons 1s considered in Section 3. The results of the calculations reliably show trends but absolute values (e.g. critical mass) are uncertain. The calculated values of con=- version ratios and thermal utilizations are considered re- liable. The computations are made for neutrons of 760°C. thermal energy (see Table III-2,1-1). 2.1 DEFINITIONS AND BASIC CONSTANTS The important nuclear data used in the calculations of thermal reactors are summarized in Table III-2.1-2. Thermal neutron properties at room temperature (2.2 km/s. neutron velocity) have been converted to the assumed operating temperature of 760°C by means of the conversion factors given in Table III-2.1-1. The factors which determine the value of the multi- plication constant, kbo’ are a). The number of neutrons produced per neutron absorbed in uranium, "M s defined by: — (25) e fl%‘ gfifi (2.1-1) where o= 2e@) L X es) =68+ 26s) ;. Zy(U)=265)+DuGs) Z 4 (5) By usling the values of the basic constants given in Table ITI-2.1-1 and the definition of the atomic enrichment fraction, namely, N§2§% the expression for V] at 760°C may be reduced to: 63R N = 2.10 3337 + ':1i.1?911-37 (2.1-3) *n se & F43 & o4& & . .8 » s s & T - . . . *e « - . 408 .0 ¢ e v 33 . v 0 L L ’ [ - * shn gh -154- TABLE I11-2.1-~1 Temperature Corrections for Nugléar;pata The computations were made for neutrons of 76000. thermal energy since a % per cent U in Bi solution was initially contemplated. In BNL-170 cross- sections are given for monoenergetic (not Maxwell) neutrons at 2.2 km./sec. These are thermal (Maxwell) for 20°C.,, if the cross-section is 1/v. BNL-170 gives factors for cross-sections not 1/v. Energy Comparison Ordinary Thermal in Thermal Bl Reactor Temperatures 20°c¢. 760°C. 293°K 1033°K 68°F. 1400°F 527°R 1859°R Neutron velocity, km./sec. 2,20 .13 Neutron energy, €.V. 0.0250 0.0881 1/v factor 0.5326 (1.04)(3) x 1/v factor 0.5539 -----‘--------_fl----‘—fi---—--fl- (a) Extr%polated from BNL-170 factor-temperature curve, for U“"23 - .54 - hhhhh """" ...... lllll lllll ...... oooooo ...... aaaaaa AN AR TABLE 111-2.1-2, Thermal Neutron Properties of Thermal Reactor Constituents, The cross-sections o are in barns,N in Values at 760°C calculated from thermal data by means of factors given in Table III-2.1-1. (atoms/cm3) x 10~2, Nuclear U-235 | U-238 Bi Be Parameters), 2 um/s. 760°C || 2.2 km/s. 760°C| 2.2 km/s.| 760°C | 2.2 km/s.| 760°C Op 549 303 o, 101 60 2.8 1.%91 || 0.016 0.00852 [ 0.009 0.00%473 op + 0, 650 363 2.8 1.%91 [ 0.016 0.00852 | 0.009 0. 00473 o 9.3 9.3 || 9.2 9.2 6.1 6.1 Oy 9.3 9.3 || 9.2 9.2 5.67 5.67 (Za,) 0.088 0.088 [1.23 1.23 a 0.183 0.197 v 12.51 2.51 n 2,12 2.10 p gm/ce 9.#6(a) 1.85(b) 1.798(°) N 0.0272 0.1199 App(cm) 3.997 [|1.3'P) | 1um xg (cm) 3.997 1.368 2 (em™ | 0,250 0.731 (a) (b) prom BNL-170 (c) Corrected density assuming 3.66 x 10'5/°C. volumetric expansion, Esbach. Interpolated, Table I, p. 31, "Liquid Metals Handbook." -GG1- . iy ;“‘ . ) . . ~156~ *\ * The average absorption cross-section per uranium atom is dz(U) = 363 R + 1.49 (1-R) (2.1=4) In Table III-2.1-3 1 and o are tabulated for various values of R. b). The thermal utilization, f, is defined ass . Z.W) (2.1-5) f - 2o o (4at) where: Zzgtot.) = ZJfU) + Z(structure, coolant, moder- ator). The cross-sections used ln the calculations ofvland f are quite sensitive to the thermal neutron temperature. In general the data for Bi are less reliable than those for Be and U. In particular the value of o, (Bi) = 0.016 barn, accepted during the early stages of the Project and used in most of the thermal calculations, is probably low by a factor of two. The higher and more reliable value Gb(Bi) = 0,032 | barn was obtalned later and was used only in the final design calculations of the thermal reactor. ¢). The resonance escape probability, p, was determined from the expression In (1/p) « BP0 (2.1-6) where A is the experimental resonance absorption integral: A =(fSa (238) 4) eff and EE?s represents the quantity &; N, (G‘s)1 summed over all the 1 atoms. The scattering cross-sections were evaluated near 7 ev, the largest, lowest lying U-238 resonance. Two curves of A vs. "o_/U", the total scattering cross- section per atom of U-238, are plotted in Figure III-2.1-1. The optimistic values of p thus derived and used in the calculations slightly underestimate the conversion ratlos. However, they more stroggly affe¢t the multiplication constant and the critical sizes &nd masses. e asm o 6 .. < . o‘ cow : . ® L I . s & @ .. . .W . . s 2 PA 8 i Nty "-W"Mn o .. ' il »t! TABLE IIT-2.1-3. R atom fract. 0.01 0.02 0.03 0.0k 0.05 0.06 0.07 0.08 0.09 0.10 1.00 (For Maxwell neutrons, 760°) Ta(U) barns 5.106 8.721 12.336 15.951 19.566 23.182 27.797 30.412 34,027 36.642 363 L I Ny 1.4929 1.7482 1.8538 1.9116 1.94%80 1.9730 1.9913 2.0053 2.,0163 2.0251 . . iiiiiii - Zio(U) /Z3(235) -157- Properties of Isotopic Uranium Mixtures (for conversion ratio) 1.4+066 1.2013 1.1328 1.0986 1.0780 1,064k 1.0546 1.0472 1.0415 1.0370 1.00 1 o vy ¥ * Teeaey /S¥ ~100 - | This optimistic curve was i used in the calculations. ’ £ b 3 - s L 3, I<| =10 T T 1T T 1T FIGURE III-2.1-1l. Experimental Resonance Absorption Integral, A, versus "a—s/U". The value A = 9.25 barns for U-238 metal is from CL-697. and the asymptotic value for infinite dilution, 240 For "rst" > 2000 barns, the curve was faired into the asymptote. The lower curve, based on"the more recent asymptotic value given in BNL-170, was used The straight line portion of this curve is from ci'-51-5-98. For "r’/U"> 2000 barns, this curve was also faired into the asymptote. also from CL-697. in the NEP calculations. The straight line portion barns, of the upper curve are -8g1- 1 iu- METAL, 9.25b 1 1.1 1 y o3 1111 1 L1 & 111 L1l 100 "0 /U" barns 10000 00000 e A 4y i, S T —— -159- N ——" TABLE I11-2,1-k4 Resonance Escape Probabillgz No/ig, = 0.01 0.02 0.03 0.0k NBe/NBi | - Resonance Escape Probability, p 1 0.541 0.410 0.329 0.273 2 .69 . 583 . 510 455 L .80 .722 .656 .612 6 . 849 . 776 . 721 677 8 .876 .802 . 764 «733 10 .893 .836 791 752 12 « 907 .852 .81k .780 14 .916 .867 .831 . 784 20 « 936 . 894 . 864 .838 In Table III-2.1-k are given the values of p correspond- ing to different values of the ratio NBe/NBi for NU/NB1 = = 0.01, 0.02, 0.03, 0.0%. These p values correspond to an enrichment fraction of 0.05. Since the enrichment has little effect on the values of p, the same values were used for R = 0.03, 0.05, 0.07. d). The fast fission factor,€ , was assumed to be equal to unity in the calculations. 2.2 ANALYTICAL EXPRESSTONS a). The thermal diffusion area, 12 1s given by the expression L = | 'E.' 3 Z:tr Z“’ - (2.2=1) in which Z,. and Za represent, respectively, the quantities Ni(d'{;r)i and Ni(aa)12summed over all the i atoms. The value of L= 1s dependent on the energy of the thermal neutrons. In the calculations thils energy corresponded o e e » *en * » * .s s & T a2 » [ L +« ¢ @ * . [ R -160~ --"-'--—-- g to 760°C. (T0 = 1033°K) or 0.0883 ev. Through L the critical radil R, and masses Mc will vary with the temperature T according to the relations R(D)~ R, (7)) {1 - + 4 123 (2.2-2) M (D) a2 M (1) {1 - 32 18P} (2.2-3) in which A L2 = L2(T0) - 1%(T). If it is assumed that the absorption cross-sections vary as 1/v, L(T) = L(To) {T/Tag%. Hence A L2 - L2(TO) {1- T/1.} (2.2-}4) b). The age to thermal, 1.=- In Table III-2.2-1 are given the values of « which were used in the calculations together with additional values, denoted by Tqs obtained by a detailed calculation. The values used in the calculations were obtained from the crude expression (Ezgtrzzs)Bi (EztrzsjBe in which t(Be), the age to thermal (760°C) in Be, was taken to be 86.5 cm2. In the detailed calculations Tq wWas determined by numerical integration of E S m d(1nE) 3L T, (2.2=5) T = T(Be) Ta Etn in which = " N MeV, Eth = 000883 eV, (NEOo)g, + (N E G )py = (NG, + (N ?Er)Bi’ -} r)Be . LIRS ! £y e -161- TABLE IIT-2.2=-1 Age to Thermal Be/NBi T T4 1 895cm2 779cm2 2 472 431 Y4 266 256 6 202 8 171 10 133 12 141 14 13 20 11 Check calculation: + td(Be) to indium resonance = 78.1 (= 80-2, BNL-170) In these calculations g3 for both Be and Bi was assumed equal to @ and the curves of o vs. E in SENP I, (33), pp.#02-403, h98-h99, were used. As a check on the calculations the value of ©t(Be) to the indium resonance was calculated. The value obtained agreed with the value 802 cm® given in BNL-170. Although the values of t used in the calculations are somewhat higher than those obtalned by the detailed calcula- tions, the discrepancy is appreciable only for low values of the ratio NBe/NBi’ Furthermore, any error in v will affect the critical radii and masses but will alter the conversion ratio only slightly. Better values of the critical radii and masses than those used in the calculations may be obtained from the relations Ryg=R, (1 - % At/M) (2.2-7) M, (1 - 3 Oc/M2) (2.2-8) where Rc and Mc are given in the tables included as Appendix B, and Mg = M2 = 12 + T 3 b=z -oT4 ‘162“ ARy - « ¢). Density.- Since in the reactor there is a mass of Be in Bi rather than a solution, the densities were calculated on the assumption that the volumes of Be and Bl were additive. It was further assumed that the uranium (1 - 4%4) dissolved in Bi without increasing the volume. The densities of Be and Bl given in Table III-2.2-2 are for a neutron temperature of 760°C. which corresponds to a solution of 4 atomic per cent U in Bi. For lower U to Bi ratios the densitlies will be somewhat highéf than those tabulated because of the lower eutectic temperature. d). Critical radii and masses. - The critical radiil R, given in Appendix B, Tables B-1 to B-ll were calculated from the relation R, =7 (22 + ¥opnh (2.2-9) The critical masses Mc were calculated using (2.2-9) and the appropriate density and composition. The expression for the c¢ritical radii is crude and optimistic. Better values of R, and M, than those used in the calculations may be obtained by using the results of two-group calculations. Denoting by Bc(a) and MC(Z), respectively, the critical radii and masses obtained from the two-group calculations, 1t is found that Rc(a) = B, §2 u™t ({170 - 1)} o (2.2-10) and , M ?) o n {2 v (VL - 1)} -3/2 (2.2-11) where u o= % (k1) B2 (1-gD) .52-_.1._1'_ = 2 L+t - M2 ’ o 3 — — —r—— .‘WM‘ ,,,,,, 4 ¥ mgtoms" = No. of atoms x 10~ 2k 153 ————— -163=- TABLE III-2.2-2. Age and Density of Bi-Be Mixtures (assume additive volume, 760°C.) Atom ratio Age Be Bi N(Be)/N(Bi) , (0 1 P - 35215537 ( - 1) represents the conversion ratio for an infinite, clean thermal reactor at steady-state critical. The last two terms of (2.3-3) represent, respectively, the slow leakage loss and the fast leakage loss. 2.4% RESULTS OF CALCULATIONS The results of the calculations on the poison-free reactors are summarized in Appendix B, Tables B-1 to B-1ll. The cases treated are itemized below. Table Nyy/Npy _JE_ B-1 0.0k 0.03 B-2 . O4 .05 B-3 Ol .07 B-4 0.03 0.03 B-5 .03 .05 B=6 .03 | .07 B-7 0.02 0.03 B-8 .02 .05 B-9 .02 .07 B-~10 0.01 0.03 B-11 .01 .07 ~-166- o o 3. POISONING EFFECTS 3.1 URANIUM-236 U~-236 1s generated in thermal reactors by non-fission -~ neutron capture in U-235. It has a half-life of 107 years., ‘In converters operating at 10 to 105 watts/gm. it reaches equilibrium in approximately 10 to 1 years, respectively (see Aprendix C). At equilibrium there are 10 to 15 atoms of U~-236 to 1 atom of U-235. This concentration of U-236, if it does not shut down the reactor, will reduce the conversion ratio substantially. . U-236 generates a chain leading to Pu(238) which has a spontaneous fission rate 6.8 times that of Pu(240). The re- action equations for this chain, together with those of the U-238 chain leading to Pu(240) are given in Figure III-3.1-1. The general chain scheme is outlined in Figure III-3.1-2. The secular equations for both chains are considered in Appendix C. The effect of the U-236 on the conversion ratio is two=- fold: It introduces a factor 1 - N(2§é?3E)N(238) in the contribution due to the resonance capture (Section 2.3), if we assume the same resonance structure for U-236 as for U-238, and adds the term.-%%éégggé to the contribution from thermal capture. The resu %ng expression for the conversion ratio is then z, (0 Kol v (1-—921:)&2 C.R. = CO - P -W —F = T+a l+'l:3&2 + S(C.R.) s o000 e (301-1) in which | Zg(230) 3. N(236 $er) = - Py - 125 wme) SNGTE (3012) TS R . e ‘ Y v . ¥ P . FIGURE IIT1-3,1-1 Reactlon Equations Lower chain Upper chain /U(236) U(235)-+11\‘ U0(238) + n—1U(239) fission fragments U(239) 23220, Ny (239) U(236) + n —~U(237) U(239) + n —U(240) u(237) 284, Np(237) B(2koy ¥ 1 oy (B) U(237) + n —U(238) (&) Np(237) + n —s Np(238) Np(239) 239, pu(239) Np(239) + n-Np(240) (¢) Np(238) —2:12,_ py(238) Np(238) + n—fission fragments Pu(239) + n ’216' PR APPENDIX A Procedure for Homogeneous, Fast, Bare Reactor The accompanying calculation sheet gives the computed values of the indicated quantities for each of the eight energy groups. In order to assist those unfamiliar with multigroup calculations of this type, a step by step description of the procedure is presented below. 1. The relative proportions of the reactor core materials are chosen primarily from engineering considerations. The example to be considered was system =I""21, one of the fused salt converters. It was assumed that the core material consists of a homogeneous mixture of 102351, + 3u238cy, + ‘+PbCl§ + 4NaCl which is equivalent to atomic proportions: 10235, 3U230, 4Pb, 4Na and 28Cl. 2. From experience with these calculation methods a judicious guess of k2/3 = 295 is made. This guess in- volves choosing the correct energy group in which the average of the neutron spectrum lies. For dense systems (1ittle 238 and/or other materials) the 3rd group 1s likely; for more dilute systems (3-6 atoms 238 to 15-30 diluent atoms) the 4th or 5th group is appropriate; for very dilute systems the 6th or 7th group should be selected. In the present example since there are 3U 36 diluent atoms per U235, the 5th energy group was chosen. TFrom accompanying data on cross-sections and 238 and other nuclear parameters we obtain: 2.47 N(25) = .8hh579x10-3x102u atoms/cm3 Z,. = 186.454 N(25) Zp(25) = 1.74N(25) L, = 2.71328 N(25) Z,(25) = .207N(25) < " -217- - One=-velocity diffusion theory yields the relation: K2/3 = Ty, (vE~Z)N25) = 295.6N(25) Since we normalize all cross-sections to N(25) = 1 for convenience, the value k2/3 = 295 was selected for use in the trial calculation. As shown in this example, a judicious choice of the energy group enables a balance between sources versus absorption and leakage to be attained in only one or two sets of calculations. Since each set represents several man-hours of work, there 1s a distinct advantage in reducing the number to a minimum. In the present case the balance achieved on the first calculation, after making the minor corrections dis- cussed below, was sufficliently accurate. The foregoing method of estimating k2/3 usually yields results which check the assumed source strength within 10%. The second value of k2/3 is then computed by means of the formula: 2 2 3 3 1l - F, - Cl where, 5, =z computed source strength for k2 = kl2 Fl = number of fissions using k2 = k12 Cl = number of captures using k2 = kl2 - This formula has been found to give 1 per cent accuracy or better in all cases. In thé above list of cross-sections many figures are included beyond the significant place. The same is true in the calculation sheet. It was found convenient in using hand calculating machines to standardize on six figures and not attempt to round off the last signifi- cant figure until the final step in the calculation. The fraction of fission neutrons emitted in each of the first five energy groups,Xf, was obtained from KAPL. A 3 w T SRy - T ki, _218_ M;M Above the 5th group, 1l.e. up>k4.5, and below the 1lst group, i.e.u 0.5,the contribution from the fission spectrum 1s considered negligible. The tabulated values of (xi)aéa given for groups 2-6 inclusive under the heading "fraction" were likewlse obtained from KAPL, except for chlorine (Cl) in which case our own estimates were made. All of these values are only approximate since reliable information on the energy distribution of inelastically scattered neutrons is not yet available. No appreciable inelastic scatter- ing is expected above the 6th group, since the first excited states of even the heavy nuclei appear to be greater than this energy. The U23 data are based on a re-analysis of the Snell and similar experiments by KAPL, reported in KAPL-741. The figures entered under “sources" opposite Group I are the contributions to each of the lower energy groups from energy group 1l resulting from inelastic scattering by each of the reactor constituents which contribute. Similarly those entries opposite Group II are contribu- tions from inelastic scattering in energy group 2, etc. Sodium does not contribute appreciably, chlorine contributes only from Group I and 238 is the only impor- tant inelastic scatterer from Group Iv. Another contribution to the total source of neutrons 1is from elastic scattering from group (a~l) to group «. These sources are entered opposite "previous gq+". The total source Xy in each energy group is obtained by simple addition. The first entries in the lower half of the calculation sheet, where the losses are summarized, are in the left- hand column of each energy group. These figures are the average macroscopic cross-sections normalized to one T3 O L atom of 235 for each energy group. The leakage L is re- presented by a fictitious cross-section k2/3 2 tp entered opposite L in the left hand column for each energy group. The degradation resulting from elastic scattering is ~ also represented by a fictitious cross-section EZS/FU. The product EZS for each energy group 1ls determined by standard methods. The width of each energy group, U, is determined by the arblitrary selection of the energy intervals. A modification, used by us, in the manner of ing F in the degradation term {Z,/FU saves time ‘choos~ as compared with the graphical method used by KAPL. By assuming that (linear approximation)s: + + q + q - a-1 a qa - 2 (A"'2) and using the definition of Fa (see Glossary) we can eliminate q)a from eq. 3 and obtain + (Lt 951 F, = T (A-3) 2 (%) - (Ay/By) a3 where (x1\¢ = right hand side of eq. 1I-2.1-3, sec. 1.1 2 AU. - (k /3z|tr a (Z ) (Zi)a B = 5 s a ( U )q This is an adeguate expression for Fa in all but the last group or possibly the last few groups where the choice of F is of least relative significance. Some= what better approximations, based on an 1mproved approximation for qa, have been tested. The results show a negligible correction for a representative system, The next step is the stepwise computation by groups -2, - kA & [E Loy ) ¥ -220- i ——— b —— 10. 11. 12, 13. R Ir s-ve - SRR beginning with group 1. Since there 1s no contribution to this group from higher energies, q:_l for the first group is zero. Hence from Egq. (4-3) above,Fa = 0.5 as entered at the bottom of the group 1 column. From this value of F the degradation (& ZS/FU) = 19.872 which is entered together with the other cross sections. The total cross section 1s obtained by addition of all cross sections, real and fictitious. The total source Yy is obtained by addition of the entries in the source column., In group 1 this is simply the fission source Xf. The relative flux in group 1, P = (XT/a-'T), is then obtained by division. Each of the cross-sections is multiplied by ¢ and these products, which are the fractional leakage and losses, are entered in the right hand column. The sum of these should equal X, e.g. 1.34340 = .13%4, the difference simply resulting from the rounding off of the figures. This completes the entries under group 1. The source terms in group 2 consist of x?, the group I inelastic sources,obtained from the product of the fraction for each component times the fractional in- elastic loss from group l)plus the loss by degradation from group 1l. The calculation of F for group 2 and subsequent groups is obtained through Eq. (A-3). (Aa/Ba) is computed separately. In some calculations F may be negative in groups 7 and 8. In which case it is assumed that the degradation out of the group 1s zero. The procedure for groups 2 - 8 follows the same steps as for group 1. The only source terms for groups 7 and 8 are by degradation from higher energy groups. The calculation must be internally consistent on two counts, The fraétional leakage and absorption terms summed over can -221- all groups should = 1 (see last column)., If this is not the case, within the rounding off error, there must be a numerical mistake in the calculation . Iikewise the total fractional fission term for each fissionable component is multiplied by its v, and the sum of these products, the fission source strength, should also = 1. If this is not the case the calculations do not represent the critical condition and must be repeated using a new k2/3 estimated from Eq. (A-l) employing the above fission source strength. ‘ After a consistent calculation has been completed, one obtain the following information for the bare reactor a). Internal conversion ratio b). External conversion ratio ¢). Mean effective @ d). Critical radius and critical mass e). Spectrum of fissions, i.e. percentage of fissions due to neutrons in a given energy range f). Spectrum of neutron flux g). Fraction of fast fissions h). Detailed neutron balance ) Sy - - LI a ¢ . - EE XK ] ~222- Typical Tabulation. - The following tables and graphs exemplify the procedure for system 21. 1s corrected by changing the fission cross-sections slightly. The resulting calculation error in conversion ratios is then less than the original error in the neutron balances in this case 0.6%. a). b). c)e. d). e). £). g)e ¥ § -~ I.C.R. = rfi%@%%%am = 0.355681 X.C.R. = 5o3oed 15062 = 0.780702 - 0. 06640 ayg - 5733333% = 0.176203 b = EEETE?T = 125,1 em C.M. « 4/3 wb3 = 2.693 tons U-235 assuming N(25) = .8ul46 x 10+2l atoms/cc. See Figure A-1l See Flgure A-2 Fraction of fast fissions by U-238 = .0276 - - 0-396865:0.007887 068369 = 6.8% ke 29w 5 088 9§ e L] . - 2 e 40 . * The neutron balance -—-_“—* _223_ Detalled Neutron Balance: basis - 1 source neutron Neutron Production Uncorrected Corrected v(25) ¢+ Fissions in U-235 0.936471 0.930857 v(28) + Fissions in U-238 0.069560 0.069143 Total Production 1.006031 1.000000 Neutron Consumption Fissions in U-235 0.379138 0.376865 Fissions in U-238 0.027824% 0.027657 Captures by U-235 0.066405 0.066405 Captures by U=-238 0.157663 0.157663 ~ Captures by Pb 0.001949 0.001949 Captures by Cl 0.0226%45 0.022645 ‘Captures by Na 0.00075% 0.00075% Leakage 0. 343494 0. 346062 Total Consumption 0.999872 1.000000 LT TR T L 02k -224- FRACTION FISSIONS above u ~——»— FIGURE A-1l SYSTEM 21: SPECTRUM OF FISSIONS AND FRACTION OF FISSIONS ABOVE u Spectrum of Fissions e 14 Fraction Fissions \ 12 .10 08 04 102 ——FISSIONS/ UNIT u—> -225- FIGURE A-2 SYSTEM 21: FLUX SPECTRUM Flux Spectrum N o FLUX/UNIT u— O ‘ : Q0 o & -2926- M DERIVATION OF "EXPONENTIAL"™ Fa' - A somewhat improved approximation for_F;, which is expected to be more accurate than A-3 when ¢ varies strongly within a group, may be derived as follows: We start with the age diffusion equation: & “ ' (-5‘;‘1‘& + 2w L) s 34 =7.=’+ LK(u,u')z,.pru (A-1) We shall denote the parenthesis on the left by Z,.and the right hand side br S (the total source). We replace (f by A CB and denote Zr by h. We now have 325 25 hg +%& = S (A-5) uWe can integrate (A-95) by multiplying by expg Su_ h(u')du'} and integrating over u. The final result is: q{u) = e 4 q(u-) + € U(, “:,u u,uuu _s_fldu - u_elduj-cs“'fia S‘Ju' @ ceeees (A=6) Now let u~ be the lower limit of a lethargy group and u be the upper limit. If we make the assumption that both h(u) and S(u) are constant throughout the lethargy interval [u', u+_] , then we have: ~hU at = e g7+ g — s (A=7) where U = u’ - u~. Now we define T =US and hU = X, and we have - . gt ceFqT e - ¥ (A-8) Returning now to (A-5), we integrate it another way by assuming h and S to be constant and we obtain the multigroup equation: -227- hgU+q -q =8 (A-9a) or | Ix +q -a" = 8 (A-9b) We now assume that: q = aq” + bq+ | (A-10a) q = Fq© | (A-10Db) Equation (A-10b) is, in a sense, superfluous, since it only fulfills the function of defining ¥. However, in the actual multigroup calculation procedure, F is the useful quantity. From (A-9b) and (A-10a), we eliminate g and solve for q+, with the result: + l=ax - 1 T = T5px T T Tox © B | (A-11) In the multigroup calculation procedure, q and S are known, as 1s x. If we wish formula (A-1l1l) to agree with formula (A-8), we must have: %i%% - e X (A-122a) -X I%Ek = 1'2 (R-12b) These equations can be solved for a and bs 1 e~* a= = - - (A-13a) X 1-e-% 1 1 b=-=4% - (A-13Db) X q_gX 3, . o | - -228- e Note that a + b = 1, rigorously. Now, we can eliminate q+ and ¢~ from (A-9b), (A-10a) and (A-10b) to obtain for F: b Xy + & q F - m (A-1k4) where J(Ta. 2+ q . We can write this formula for the o.th group as follows: b, (Xp), + 2, q, F_o- S a1 (A-15) a - (XT) - X, a, q -1 where we have replaced q_ by qZ_l. Going back now to the definition of x (z hU), and re- placing h by 2 /EE we haves £ s U X, = (Hg), < . (A-162a) We can now recognize x, : XC& = ACL / Ba (A-16Db) In an actual calculation, a(x) or b(x), given by (A-13), can be tabulated or plotted accurately and then read off for various X, If a is obtained, b is just l-a, and vice versa, This operation, together with formula (A-15), replaces step 6 in the procedural outline above. Examination of particular cases shows that this method of estimating ¥, is much better than the one which assumes a, = b, = %, for groups wherein q and q are very different. Moreover, Fa can never become negatlve wzth this procedure, which is a more realistic behavior than if it can be negative. On the other hand, calculations on a representative case gave the result that, while the spectrum of the bare pile was changed somewhat, the differences between the calculated rexr aw L e Y R —229' critical sizes, conversion ratios and other neutron balances were negligibvle. However, for a spectrum which varies much more sharply than the one tested, there might be a non- negligible difference. 1In any case, there is very little more work involved in using the method with variable a and b. 2 3¢ NUCLEAR ENGINEERING PROJECT Calculation Sheet System No.2I Atomic Proporfions u"“ U®*Pb Na CI -0€¢~- K/3 =295 (estimated from one velocity approximation) 3 4 4 28 u interval 5= | -2 2-3 3-375 |375-4.5 45-55 5.5 -7 7-10 Summa- Group | 2 3 4 5 6 7 8 tion Fraction|Sources |Fraction Sources| Fraction| Sowces |Fraction| Sources | Fraction|Sources |FractioSources | Fraction Sources|FractionSources Fission Source Xe | 1344 :-'tfaa .39 5% Logo? loa73 | ! 1 | | | l Gl’OUp I 235 l .l }-9001“‘ .2 |.0“483 3 }.00072" A :,009483 ‘6 }.“04’3 I ! NS 238 I s r o016 .35 |.o0suma] a4 l.oo361.o de ecadid | .to l.oo,s-as ; 1 EO Pb l -4 |r 00305y - 3 [+002293( .2 |.oo|fas .l [-eeoted ! — I [ | LU Cl | 2 |sonaez|-e lozanis| .2 renaenl — | — | I AR Group II 235 | l 2 |ooues1| .33 | o0T700| 122 |-e05133 | .28 | 005234 | [ SC 238 | |' a5 liosios] .33 lodwiaci.ze e3a034| .10 |.o14s3e | ] TE Pb | | o otoez| .3 :.ozoss'i I |.oobyds| — ] | I'S Group III 235 | f | 51 013179 34 |.oos136 | AE 100370 | I C 238 | | | 51 otaasy 34 |.obisos | oIS 023t | | Pb | * ' (4 LT[l Lests|—— | | Group IV 238 1 ! ! | 8 loviesz|.n loisam ! | Previous g+ =(¥/FU), 4., : | o3v9¢3 et 174 i-mm' : 365 303 I.‘il‘o!'.-. l.aflm o o | = . Total Source Xr i:sw | so¥qgs 1595 197 {.5‘[1230 l.s9¢ o |.494¢q2 - 3T4éN |0 TRAY | (Xr/e3) = N 1. poaol | 019945 loag 713 .039903 |.oa7193 .o'llffla 029779 00 5397 s LEAKAGE : L=k¥/3e, [5.39¥%tl. cosr3s potssag | osvaus 2,209 |, oudaré [Lir304s |.os4ars r.:qm.qo|.osqmz Layswet | o8 091237787 | 037014 |, ti6ads) 004778, ad 344y o . . Fissions 23512 l.eeatis [z loasasy |f,21 j.o35030| 143 }.oiriu 4 oHATIe | 2 .o88054 |33 l.owm L5y lowse].579)87 0 3x 238,42 l.oozsos|hits |-earyy| o513 |.ootqn3| — —_ 027324 crecen S Captures 235 ].065 lioooisi .05 Lootaey).oms |.ooaiss) iz :-»35'87 07 007649 375 IolS‘?.'ts' Py TR i.oub‘ia a4 lLeiss |. 0be dos S 3x238 |- ot5 |ovooso|.i5 ooy |30 lowif| 4E Lewszi.be |oatsN7| 'Iaqaasa wse hotéen | 3.0 lemen|isyiey g 4x Pb|.el | oec0ael. ol |.oaoHl| 0 |.oooal’? ol .000aq9| .0l |:000372| 0| |.oao'm 0l ‘I.Oooaqq 1ol i.uoo.fl WYILLL] 28 x Cl |.ot68 |oocosr|.0i% joeo0dPt| c02? .ooo¥e¥| .oJ0Y I.ools‘o‘? (0452 |.0036Y] | L1764 [,007347 | 38O 00 TIII| 316Y |i00186L], 02RENT 4 x Na |.o0i08 lLooceoz|: ool0¥ |.oe00l | .0010¥ |1000031 | . 00108 |.000032 00108 | :000041| .pon2 I.oaoc'#'] 00I% l.oooosq 103%%4 | 000521 |, 000T3Y Inelastics 235 1z ooy 3| ha J.o®33t|. 9 |.ea5¥4] — l l i 3x238(7ns Loisew3|hs |Msyag[é3 Lisosaz| A.ss |0%af3| T | | | | 4x Pb |3z |.oo7b4z 352 |.o0éryd| he l.oqm; — — 1 | | | 28x Cl [240 Lo56308|— | ' | - l | | | Pegradation$%/FU |i11.9% .o 39963(v. 341953 : et v.w'iml.nv:sr 1.R78507]. 365203 |K. 195757 416 032 |4.56706| 2746 1. 5762810 TTCHY —— | . 1 i 1 | ] 1 | — Totai Cross Section o3 6. Boe3 7, 1343 46, 175702 |.5°08 917 e tagéial. SIs180 e 3ss0at].suv 2 ISMI7a7] .S109 bE5¥600 1934690 HT8375 127689 ;a..ums}.ms;; .999%712 F . os %a3763 T5velo L1985 426533 7 269 k% ¥39 ©0 Mi | -231- M APPENDIX B SUMMARY OF CALCULATIONS ON BARE HOMOGENEQUS THERMAL REACTORS Tables B-1 to B-ll contain the results of calculations of a number of bare, spherical, critical, thermal reactors composed of mixtures of enriched uranium, beryllium and bismuth. Each of the eleven tables gives the results for a specified U/Bi ratio and an assumed enrichment for a range of Be/Bi ratios. These tables have been computed for a Bi capture cross~-section of 0.016 barns at 0.025 ev., the value accepted at the beginning of the Project. The final calcula-' tions, reported in Chapter III use a cross-section of 0.032 barns, the value now considered more reliable. Results of this Appendix may be expected to show trends correctly, but are inconsistent numerically with the final calculations, ...... ...... ...... ..... ooooo ...... ...... QQQQQQ seasey T & 132 TABLE Bl Ny/Ng; = O Ok R = 0.03 M = 1.853 0 0 ;B.Q , . N -~ L2 T M2 Rc Vc MBi MU M25 MBe | Bi (en?) [(ea®)| (en®) | (em) |(10%em| (108m) | (1050m) 110°cm) (mégm) 1 - - - - - - - - - - - - - 2 | 0.455 | 0.965|0.813| - - |- - - - - - - - 4 612 | .947|1.074| 0.903| 98.8| 266 | 365 | 221 | 44.95 |223. |10.2 | 306. 38.5 6 617 .930 1| 1.166 718 109.7| 202 | 312 136 | 10.6 42.2 1.933 | 58,0 10.9 8 733 | W91311.241| .605|121.2f 171 | 292 | 109 | 5.48 | 18.4 | 0.842| 25.3 | 6.36 TABLE B-2 Ny/Ngy = 0.04 R = 0.0 "= 1.94798 Ny L T M R, Ve Mps My Mos Mpe B | P LT T P (ad)| (@B ead) | (e |08y (10%m) | (108em) | (10%m)| (108em) 1 |0.273 |0.9833|0.5229| - - - | - - - ~ - - - R | W455 | J9774| .8663] - - - | - - - - - - - 4 | 612 | .9658|1.151 |0.801 63.54| 266 |329.5| 146.7| 13.22 | 65.57 {3.000 |150., |11.32 6 | 677 | .9546[1.259 | .632| 71.04| 202 |273. | 102. | 4.444 | 17.78 |0.8136 | 40.68 | 4.604 8 | J733 | J9435|1.347 | .500{ 78.97| 171 |250. | 84.32| 2.511 | 8.43 | .3861 | 19.31 | 2.913 ..... ------ ------- Y 253 TABLE B-3 Nyy/Ngg = 0.0k R = 0,07 N = 1.991 N, ¥y | YR Ve M3 Yo | M | Mg — P £ k C.R. "1 . (cw®)| (eu®) | (ca®) | (cm) | (20%en’)| (10°em)| (10%m) | (10%gm) | (1084m) 1 | 0.273 |0.988] 0.537 | - - - - - - - - _ - 2 455 | 984 .891 | - - N - - _ _ - _ 4 612 | J976|1.189 |0.751 |46.8 | 266 | 313 |128. | 8.75 | 43.4 |1.986 | 139. | 7.49 6 677 | .968|1.304 | .583 |52.6 | 202 | 255 | 90.9]| 3.15 | 12.6 |0.577 40.4 | 3.26 8 2733 | J960 | 1.400 | .452 |58.6 | 171 | 230 | 75.3| 1.786 | 6.00 | .275 19.2 | 2.07 TABLE B-l NU/NBi - 0003 R - 0.03 Q - 1085380 NB I..2 T M2 Rc vc MBi MU M25 MEe e f 1 k | CR, 2.0, 2 2 6 3.0, 6 6 3 6 Bi (cw®)| (em®) | (en®)| (em) |(10°cw’)|(10%gm)| (10°%m) (10°gm) (10%am) 1 }0.329/0.9653]0.5887| - - - - - - - - - - 2 0510 09534 09014 - - - - - - - - - - 4| 656 .9304{1.131 |0.792 | 129.4| 266 | 395.4| 172.6| 21.54 |106.8 | 3.649 |109.5 |18.44 6 | 721 .9085[1.214 | ".639 | 143.0| 202 | 345. | 126.1| 8.399| 33.60 | 1.148 | 34.44| 8.700 8| .764| .8877|1.257 | .548| 157.1| 171 | 328.1| 112.2] 5.915| 19.87 | 0.6790 | 20.37| 6.861 1 N o W ! ....... tttttt ------ ------ 00000 ...... ...... TABLE B- NU/NB:I. - 0.03 R = 0.05 "N = 1,94798 e 2| | R | Ve | M| M| s | W | 2] L5 T | (a@d | () ()| (em) |(108en®) (10%m) (20%m)| (10%gm) (20%m) 1 10.32910.9778|0.6266| - - - - - - - - - - 2| .510| .9701| .9637| - - - - - - - - - - 4 | 656 .9550|1.2204| 0.686| 83.75 | 266 | 349.8(125.1 | 8.202 | 40.68 [1.390 |69.50 | 7.024 6 | 721 .9403|1.3207| .531] 93.28 | 202 | 295.3| 95.30 3.625 | 14.50 [0.4955 | 24.78 | 3.755 8 | 764 .9261|1.3783| .438|103.34 | 171 | 274.3| 84.57) 2.534 8.514 | 2909 | 14.55 | 2,940 TABIE B-6 NU/NB:I. = 0.03 R = 0.07 Vt - 1.99133 12 T | R Ve Mpy My Mos Mo ':_:f P () () (@) (o) |0Pa?) (108 (10%m) | (10%am) | (20°em) 1 1 lo.329)0.983700.6445 - | - | - | - | - - - - - - | 2 | 510| .9780| 9932 - - - - - - - - - - L | 656 .9667|1.263 |0.637 | 61.91 266 |327.9(110.9 | 5.714 | 28.34 |0.9684 | 67.79 | 4.892 6 | J721| 9557 (1.372 | 482 | 69.24|202 |271.2 84.83| 2.557 | 10.23 3496 | 24.47 | 2.649 8 | 764 | 9449 |1.438 2386 | 77.00|171 | 248.0| 74.77| 1.751 5.883 | .2010 | 14.07 |2.023 -$EC- TABLE B- Ny/Ngg = 0402 R = 0.03 v = 1.8533 e p | £ | x | c.n e el Yo 1 | % M5 | MBe %1 (en®) | (ea®)| (en®)| (cm)|(20%’) (10%m) |(10%gm) | (10°em)| (10%gm) ...... 1l =~ - - = - - - - - .- | - ~ - ------ 2| 0.5630.932|1.007 } 1.003 | 175 | 472 | 647 | 983 |3980. |2590. |so2. | 17800. |2234. ,,,,,, 41 J722| .899{1.203 | 0,641 | 187 | 266 | 453 | 156 | 15.9 78.9| 1.81 54.3| 13.64 e 6 76| .869|1.249 | .531| 205 | 202 | 407 | 127 | 8.59| 34.4| 0.789| 23.7| 10.88 8| 802 8411250 | .486 | 223 | 1TL| 394 | 124 | 8.04| 27.0| .618| 185| 9.3% B TABLE B-8 s Ny /Mgy R = 0.05 M = 1.94798 “Be P £ | x |c.R L: T2 Mz R_c Z" 3 :Bi 35 H2° N1 (en®) [(em®)| (em®) | (em) [(10°en”)| (207gm) | ( (10°gm) | (10°gm) ool - |- | - - | -] - |- - - - - - 2| .583|0.9558)1.085(0.897| 113.38| 472 | 585.4 | 260.8 74,31 |483.8 | 11.07 |553.5 |41.76 4| o722| .9339|1.313| .533| 122.91| 266 | 388.9 | 110.7 5.684 | 28.19 | 0.6450| 32.25 | 4.868 ::l 6| 76| 9130 1.380| .418| 135.94| 202 | 337.9| 93.67| 3.443 | 13.77 | .3151) 15.76 | 3.566 'g 8| .202| .8932/1.395| .375| L49.54| 171 | 320.5 | 89.47| 3.000 | 10.08 | .2306| 11.53 | 3481 ...... ...... QQQQQ ----- ...... ...... mmmmmm 23 TABLE B-9 NU/NBi - 0002 R = 0.07 fl - 1.9907 2 . lde L T Mz Rc ¢ . MBi Ml] M25 MBe Nps (em™) | (cm™)| (cm™)| (em) |(107cm”)| (107 gm)| (10 gm) | (107gm) |(10 gm) ‘\A,} - 7 — 5 10 12 14 X = N(Be) /N(Bi) -256- L.OK I I 1 i FIGURE D-3 g = Average Fraction of Final 9 - 11® Concentration as a Function | of M(U-235) Reactor power = 500 MW 8 7 6 s\ o 4 \ 3 \\ 2 \‘\, R O 50 400 -257- APPENDIX E TWO-GROUP, TWO-REGION REACTOR EQUATIONS E.1 EQUATIONS FOR THE FLUXES Let ‘?1 = flux in the ith energy group (i1 = 1 repre- sents the thermal group and i = 2 the epithermal) A= 1 = transport mean free paths 17 (2,4 {2, — - " ' 22 = m = total "fast absorption' (with Ej = 2 Mev), Z& = total thermal (true) absorption Ay T=3x, =88° 2 M L™= a2 thermal diffusion area I35, Superscripts (0 for the core and 1 for the blankét) will be used to designate the region of the reactor to which any of the above quantities belongs. There are two equations of the form ,T V2¢2 - ¢2 +nt % ¢l =0 (E.1-1) 12 v2 ¢, - ¢+ p % $p =0 (E.1-2) for each region. The fast fission effect, € , has been ignored. ' On assuming a solution of the form SEEn , - Fal w oo -258- (E.1-3) and substituting in (E.l-1) and (E.1-2) there results the well-known secular equation 1+ 1°%%) @+ 1. The two roots of (E.1-4) are )(2 = }(02 2u 1§ \}1+u - 1} (E.1-5) where k -1 k -1 Ko —I?':: = :I% y U = (koo-l) L|-fi‘2(1-[32), 52 = ;2-'!’- . 2 Here, one root x‘ = /u"" is positive while the other z(:_' = -v2 is negative. b). Blanket. koo< l. Here € = Wov ‘1{ i-v - 1} (E.1-6) where xoz = %Q y V = (1'kco_) ""fia (1-32), 82 = ? . 2 In thls case both roots )(3 ,+ a"?\l 2 9 respectively, are negative. The equations for the fluxes are (o) ¢ = A Snsr, g ..S.i__nh.;.;?_.l: (E.1~7) " 4 s sinpr,cg slabvr (E.1-8) e et— T -259- N‘_—H o sinh 24 (r-b) sinh A, (r-b) ¢=F r + G T (E.1-9) al sinh )\, (r-b) sinh A, (r-b) ¢"°=F 5y = + G 8, — >0 0088 (Eol“lO) where b is the outer radius of the spherical reactor and the coupling coefficients Si'are given by 2 S - = Eol"ll) 1= W2 /4, P 1+I.-2-z.(12 ( The constants A, C, F and G are determined from the boundary conditions which state the continuity of flux and current at the boundary, r = a, between the core and blanket. Deriva- tion of convenient expressions for the critical determinant and for these constants 1s straight forward, though tedious. E.2 THE CONVERSION RATIO The quantities required for the derivation of the ex- pression for the C.R. are Total source(fast) =ylf(°)2{°)5¢l(°) d(core) + s Z{l)jfil) d(blanket) . s(@) 4+ (D | (E.2-1) ~ U-235 consumed = 28 (5(®) + (1)) | (E.2-2) : (o) d (o) . Fast leakage from core = I.f(o) = -1l-1ra2 A2 ¢2 l 3 _ ar lraa ) eosces e (E02-3) -260- M — ——————— A1) g4iL) (1) _ 222 9% Fast leakage from reactor = Ls s-Hrb 3 ar r=b ee s e e (E02""+) The method of derivation of the expression for the C.R. of the two-region reactor is the same as that employed for the bare reactor except that the contributions from both the core and blanket must now be considered. The contribu- tion from the resonance capture in the core is (S(o) - Lf(O)) (1 - p(o)) — (E.2-5) Lo (glo) 4 (D), The corresponding contribution from the blanket is D 41,0 _ 1 Dy - 1) t e i 2 (E.2-6) Te (5@, gD, The contribution due to U-238 thermal capture is 250 €0 2 o0 acore + 200 ¢ FHRfDnzman _1-_‘;';}, (S(O) + ST]')) — v 2a238) 4 Z (0 2T @ § 25 - L (E.2-7) This simple result is a consequence of the fact that L, (238)/7, (U) 1is independent of the region of the reactor. The total C.R. is the sum of (E.2-5), (E.2-6) and (E.2-7). After a simple rearrangement of terms, there results the expression c.R. =C, =P~ § + 51 -4, + 55 (4.1=2) of the text. 1. 9. 10. 11. 12. 130 | tememeetet—— -261- REFERENCES A-4315, R. Ehrlich, H. Hurwitz, and J. R. Stehn, "A Malti-Group Method for Computing Critical Masses of Intermediate Piles", May 9, 1947. AECU-2040, D. J. Hughes, et al, "Neutron Cross-Sections: A Compilation of the AEC Neutron Cross-Section Group", August 1952. AERE- T/R/108, H. Melvin-Melvin, "A Method of Cal=- culating the Critical Dimensions of Spherically Symmetrical Reactors-I", November 30, 1950. BNL-170, D. J. Hughes, "Neutron Cross-Sections - A Compilation of the A.E.C. Neutron Cross-Section Advisory Group", May 15, 1952. BNL~-Log No. C-5754 and BNL-170, D. J. Hughes, et al, "Classified Neutron Cross-Sections Compila%ion", January 25, 1952. (To be reissued as part of another report). Cr-2881, F. L. Friedman and A. T. Monk, "Macroscopic Theory of Breeders and Converters", March 26, 1945, CF-51-5-98, A. M. Weinberg, L. C. Noderer, "Theory of Neutron Chain Reactions - Volume I -~ Diffusion and Slowing Down of Neutrons", May 15, 1951. CL-697, A. Turkevich, H. H. Goldsmith, P. Morrison, Volume II, Chapter 4, "Neutron and Fission Physics", Figure 7. CpP-1121, G. Young, "Displacement of Delayed Neutrons in Homogeneous Piles", December 7, 1943, Cp-1456, G. Young, Murray and Castle, "Calculations for Some Pile Shapes of which the Boundaries are Partly Spherical®, February 25, 194k, | KAPLQI#, H. Hurwitz, "Pile Neutron Physics I", December L4, 1947, | KAPL-33%, H. Brooks, “Secular Equations for Breeding", April 28, 1950. | | KAPL-346, R. Ehrlich, "Multi~Group and Adjoint Calcula- tions for SAPL-5", May 10, 1950. -262- 14+. KAPL-611, "The Plutonium -Power. Breeder Reactor", 15. KAPL-§3%, "A Discussion of Possible Homogeneous Reactor Fuels". | 16. KAPL-741, R. Ehrlich and S. E. Russell, "Fast Neutron Cross-Sections of U-235 and U-238", May 19, 1952. 17. LA-140 and LA-140A, "Los Alamos Handbook of Nuclear Physics", September 30, 1944, 18. LA-994, W. Nyer, "Summary of Fast Fission Cross-Sections", December 2, 1549, 19. LA-1391, B. Carlson, "Multi-Velocity Serber-Wilson Neutron Diffusion Calculations", March 24, 1952. 20, LEXP-1, "“Nuclear Powered Flight", September 30, 1948. 2l. Memo PLH-3, P. L. Hofmann, "Multi-Group Procedures as used at KAPL", March 27, 1952. 22, MonP-4%28, G. Young and L. Noderer, "Distribution Fufictions and Fission Product Poisoning', November 6, 1947, 23. NAVEXOS P-733, R. N. Lyon, "Ligquid Metals Handbook", June 1, 1950. 24, NDA Memo-15B-1, V. F., Weisskopf, "A Preliminary Dis- cugsion of the Ratio of Capture to Fission", July 23, 1952. . 25. NYO0-636, B. T. Feld, et al, "Final Report of the Fast Neutron Data Projeet®, January 31, 1951. 26. ORNL-1099, S. Glasstone and M. C. Edlund, "The Elements of Nuclear Reactor Theory - Part I", October 26, 1951. 27. ORNL-51-9-126, S. Glasstone and M. C. Edlund, ¥The Elements of Nuclear Reactor Theory - Part iI“, October 19, 1951. 28. ORNL-51-9-127, S. Glasstone and M. C., Edlund, "The . Elements of Nuclear Reactor Theory - Part III", October 26, " 19510 - 29. ORNL-51-9-128, S. Glasstone and M. C. Edlund, "The Elements of Nuclear Reactor Theory - Part IV, Elements of Nuclear Reactor Theory - Part V", October 30, 1951. —-—/ AV S 31. 32. 33. 3k, 35- 36. 37. 38. - 39. SEo e *——-—-—“ _263_ R-233, G. Safonov, "Notes on Multi-Group Techniques for the Investigation of Neutron Diffusion'. RM-852 . Safonov, "A Note of Fast Breeding in Mixtures of U238 and Th232, February 19, 1952. SENP-I, "Science and Engineering of Nuclear Power", Ygéfig§ I - edited by C. Goodman, Addlison-Wesley Press TID-70(Rev), "Journal of Metallurgy and Ceramics", January, 1951. TID-389, R. L. Shannon, "Nuclear Breeding, A Literature Search", November, 1§50. TMS-5, T. M. Snyder, "Fast Neutron Radiative Capture Cross-Sections of Some Reactor Materials', Y-F10-98, W. K. Ergen, "Physics Considerations of Circulating Fuel Reactors", April 16, 1952. H. Hurwitz, Nucleonics, 5, 61 (July 1949). E. P. Steinberg and M. S. Freedman, "Summary of Fission- Yield Experiments”, pp. 1378-1389. Book 3 - "Radio- chemical Studies™ by C. D. Coryell and N. Sugarman. MeGraw-Hill Book Co. (1951). o s -264- S ————————— ACKNOWLEDGMENTS The short-term nature of this project made i1t imperative that we draw on exlsting information at other installations wherever possible. We are very grateful for the active co- operation in this which we encountered in every case. The assistance of the New York Operations Office of the AEC in many phases of our work proved invaluable. The Divlision of Reactor Development and the Operations Analysis Staff of the AEC in Washington contributed much-needed advice and data. We also wish to express our appreclation of the generous amounts of time and information extended to members of our staff at the following AEC installations and Contractors! offices: Argonne National Laboratory Dow Chemical Company Battelle Memorial Institute Iowa State College Brookhaven National Laboratory Knolls Atomic Power Lab. Detroit Edison Company North American Aviation Co. University of California Advance reactor information obtained from other groups saved a great deal of time here and served to expedite comparison with project reactors. We are indebted to the Washington Office of the AEC for data on the Jumbo and Aqueous Homogeneous Reactors and to the KAPL staff for information about their Pin-Type Fast Reactor. The members of the Project's Advisory Committee were Harvey Brooks, John Chipman, Charles D, Coryell, Edwin R. Gilliland, and Harold S. Mickley. This committee, in addition to its assistance 1n shaping the general course of the Project, made many valuable, specific contributions to the work. Among the Project's consultants, Henry W. Newson prcvided many useful suggestions, particularly in the field of reactor controls. Warren M. Rohsenow spearheaded the mechanical design activities. David D. Jacobus helped ecrystallize structural designs of the reactors. George 2724 -265 % i - S ————— — E ~265 T T —— Scatchard and Walter Schumb contributed to and coordinated the work in chemistry. Herbert H. Uhlig provided valuable interpretation of corrosion problems. _ We are grateful to Karl Cohen of the Walter Kidde Nuclear Laboratories for allowing two physicists from his staff to join the Project for the summer. Lee Haworth of the Brookhaven National Laboratory also helped at a critical time when he made it possible for Dr. Jacobus to spend a week with the Project. As Executive Officer, William E. Ritchie facilitated the work of the Project in many ways and helped it to work effectively.