THE UN IV ER SIT Y OF MI CHI GAN COTLLEGE OF ENGINEERING Department of Nuclear Engineering Technical Report THE EFFECT OF THE PORT VOID IN THE PREDICTION OF THERMAL NEUTRON BEAM PORT CURRENT Sanford C. Cohen O R'A Proe'. t'.' -,:' 78: under contract with: NATIONAL SCIENCE FOUNDATION RAT NO. GP-1032 WASHINGTON. C. administered throughOFFICE OF RESEARCH ADMINISTRATION ANN ARBOR January 1964

This report was also a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in The University of Michigan, 19630

ACKNOWLEDGMENTS I would like to express my appreciation for the numerous helpful discussions with the members of the doctoral committee during the course of the research. In particular, the encouragement and suggestions of Professors John S. King and Paul F. Zweifel are gratefully acknowledged. The complete cooperation of Mr. J. B. Bullock and the staff of the Ford Nuclear Reactor is appreciated. I am grateful to Mro Hilding 01son and the staff of the Phoenix Memorial Laboratory for their cooperation. Utilization of The University of Michigan Computing Center and the services of the staff are acknowledged. A U. S. Atomic Energy Commission special fellowship in nuclear science and engineering provided financial support during most of my years of graduate work. A Phoenix Memorial Project fellowship made possible by the Ford Motor Company helped me to continue my work. For these financial grants I am most grateful. Finally, the support provided under NSF Grant GP-1032 for the preparation of this manuscript is acknowledged. ii

TABLE OF CONTENTS Page LIST OF TABLES v LIST OF FIGURES vi NOMENCLATURE x ABSTRACT xiii I. INTRODUCTION 1 A. Statement of the Problem and Motivation for the Study 1 B. Related Published Studies 2 C. Outline of This Study 3 II. ANALYSIS 6 A. Orientation 6 B. Infinite Half-Space Limit 8 C. Diffusion Theory Boundary Condition 13 D. Angular Flux Distribution 16 E. Derivation of j+B 17 F. Derivation of j[ 19 G. Scalar Flux Distribution Along the Lateral Surfaces 22 IIIo EXPERIMENT 28 A. The Ford Nuclear Reactor 28 B. Insertion and Positioning of Ducts 28 C. Normalization 33 D. Detectors 34 E. Counting Equipment 37 F. Unperturbed Flux Distributions for the Water Reflector 38 G. Preliminary Duct Experiments 46 H. Duct Experiments in Water as a Function of Duct Diameter 50 I. Thermal Spectrum 55 Jo Measurements in Complete Graphite Reflector 59 iii

TABLE OF CONTENTS (Concluded) Page IV. DISCUSSION 68 Ao Unperturbed Thermal Flux Distributions 68 B. 1-1/2-in. Duct Experiments 69 Co Thermal Flux Attenuation Within the Empty Ducts 70 D. Radial Flux Distributions at the Duct Mouth 71 E. Fast Flux Perturbation 72 Fo. Thermal Spectrum 75 G. Scalar Flux at the Duct Mouth 76 Ho Scalar Flux Distribution Along the Duct Walls 79 I. Reactivity Predictions 86 V.ow CONCLUSIONS 88 APPENDIX A. A SURVEY FOR THE ENHANCEMENT OF THERMAL NEUTRON LEAKAGE FLUX 90 1. Introduction 90 2, Three-Dimensional Simulation 91 3. Streaming Flux 94 4. Configurational Effects 95 5, The Effect of Reflector Materials lo09 6. Final Remarks 132 7. Subroutine STREAM 135 B. DESCRIPTION OF THE CODES 140 C. COMPOSITIONS AND GROUP CONSTANTS 144 D. COMPARISON OF CURRENT BOUNDARY CONDITION AND PSEUDO BOUNDARY CONDITION AT VACUUM INTERFACE 150 E. DUCT MOUTH CONTRIBUTION TO THE RADIAL CURRENT AT THE WALL 154 F. NORMALIZATION FOR TIME VARYING FLUX 158 Go COMPUTER PROGRAM FOR EVALUATION OF ~/r | R 160 REFERENCES 163 iv

LIST OF TABLES Table Page III.1 Duct Specifications 32 III.2 Normalization Data 35 III.3 Foil Characteristics 36 IIIo.4 Prediction of Activation at Exit of 1-l/2-In. Duct in Water 49 III.5 Neutron Temperature Measurement 58 IVol First Estimate of the Radial Component of the Gradients for Boundary Condition in Two-Dimensional Calculation 85 IVo2 Duct Reactivity-Measured and Predicted 87 AoI Transverse Bucklings 93 AoII Critical Volume, Area, and Relative Leakage for Bare Reactors of Three Geometries 97 A.III Relative Leakage from a Face for Bare Parallelepipeds of Various Shapes 100 A.IV Cold-Clean H20 Reflected Parallelepiped Cores of Various Shapes 106 A.V Infinite Reflector-Cylindrical Geometry; Various Reflector Materials 116 AoVI Two-Dimensional Calculations; Fixed Sized Core; Inserts of Various Materials 120 A VII Three and Six Inch Reflectors of Various Materials 129 Co I Compositions 147 C oII Thermal Group Constants 148 C oIII Fast Group Constants 149 v

LIST OF FIGURES Figure Page 2.1 Void configuration. 6 2.2 Infinite half-space geometry. 9 2.3 Normalized angular flux for c = 1; exact and 12 P-1 approximation. 2.4 Linear extrapolation at vacuum interface. 15 2.5 Vector diagram at vacuum interface. 16 2.6 Configuration for calculation of jA. 19 2.7 Current balance geometry. 23 2.8 Current balance geometry including front interface. 26 3.1 Ford Nuclear Reactor core configuration. 29 3.2 Streaming duct, foil holder, and duct support with normalizer. 30 3.3 Photograph of 1-1/2-in, duct encased in 4-in, graphite insert. 31 3.4 Decay of a gold foil counted in the well counter. 39 3.5 Thermal and epicadmium flux out to four feet in water reflector and along axis of 1-1/2in. duct, 40 3.6 Thermal and epicadmium flux in the core and the water reflector, 42 3.7 East-west traverse of thermal flux in the fuel element bordering on the east of the control rod element, 43 3~8 Unperturbed thermal and epicadmium flux in water reflector. 44 359 Reaction rate at exit of 1-l/2.in. duct in water, 47 vi

LIST OF FIGURES (Continued) Figure Page 3.10 Thermal reaction rate in 4-in. graphite insert and at the exit of 1-1/2-in. duct encased in 4-ino graphite insert. 48 3.11 Perturbed radial flux distributions at duct mouths. 51 3.12 Perturbed axial flux distributions at duct mouths, 52 3.13 Collimator design. 53 3.14 at collimator exit. OBth 3.15 Radial variation of flux at 5-in, duct exit, 56 3.16 Thermal flux attenuation within 1-1/2- and 5-in, ducts. 57 3.17 Ratio of reaction rate of Lu176 to reaction rate of 1/v absorber. 60 3.18 Graphite insert. 61 3.19 Unperturbed thermal and epicadium flux in graphite insert, 64 3.20 Radial flux distribution in graphite insert. 65 3.21 Perturbed thermal flux at center of duct mouths in graphite. 66 3.22 Axial distribution of thermal flux at mouth of 5-in, duct in graphite. 67 4.1 Angular distribution of fast neutrons. 73 4.2 Effect on predicted thermal flux of fast flux depression. 74 4.3 Predicted Bth for graphite. 78 th 4,4 Radial thermal flux behavior at four positions along 5-in. duct wall. 80 4.5 Axial thermal flux distribution along walls of 5-ino duct. 82 vii

LIST OF FIGURES (Continued) Figure Page 4.6 Cylindrical geometry approximation to core-duct configuration. 84 A.1 Core geometries for H20 reflected configuration studies (cold-clean fuel). 101 A.2 Computed flux distributions; H20 reflected core; 3 elements transverse. 102 A.3 Computed flux distributions; H20 reflected core; 4 elements transverse. 103 A.4 Computed flux distributions; H20 reflected core; 5 elements transverse. 104 A.5 Computed flux distributions; H20 reflected core; 7 elements transverse. 105 A.6 Computed flux distributions; comparison of depleted fuel in center and on circumference of core. 108 A,7 Computed flux distributions; infinite H20 reflector; cylindrical geometry. 110 Ao8 Computed flux distributions; infinite graphite reflector; cylindrical geometry. 111 A,9 Computed flux distributions; infinite BeO reflector; cylindrical geometry. 112 A.10 Computed flux distributions; infinite D20 reflector; cylindrical geometry. 113 A.11 Computed flux distributions; 7.5 cm graphite-remainder H20; cylindrical geometry. 114 A.12 Computed flux distributions; 2,5 cm H20-remainder graphite; cylindrical geometry'. 115 Ao13 Core geometries for material insert studies, 118 A.14 Core geometry for 6 in. BeO reflector. 119 viii

LIST OF FIGURES (Concluded) Figure Page Ao15 Two-dimensional computed flux distributions; HaO reflected. 121 A.16 Two-dimensional computed flux distributions; graphite insert. 122 Ao17 Two-dimensional computed flux distributions; BeO insert. 123 A,18 Two-dimensional computed flux distributions; D20 insert. 124 A.19 Two-dimensional computed flux distributions; 6-in. BeO reflector. 125 A.20 Core geometries for 3- and 6-in, reflectors of various materials. 127 A.21 Core geometries for H20 reflected configuration studies (equilibrium Xe-8,4 percent burnup fuel). 128 A.22 (S4-S3)max for 3- and 6-ino reflection. 130 A.23 (S4-S1)max for 3- and 6-ins reflection. 131 A.24 Measured and computed flux distributions along axis of cylindrical DO0 insert. 133 C.o Wigner-Wilkins and Maxwellian thermal spectra for cold-clean FNR core at 300~K. 146 E.1 Duct configuration for calculation of mouth contribution to wall radial current. 155 ix

NOMENCLATURE c Number of neutrons produced per collision d Linear extrapolation distance h Duct length j Neutron current k Multiplication constant ~ Neutron mean free path r Radial position r Position vector s Neutron source t Time v Neutron velocity z Axial position Zo Extrapolated endpoint B2 Buckling D Neutron diffusion coefficient E Neutron energy L Neutron diffusion length R Duct radius S Surface area T Neutron temperature I Neutron streaming flux a Largest eigenvalue of diffusion equation

NOMENCLATURE (Continued) K Inverse of diffusion length (1/L) A Radioactive decay constant Cosine of angle between neutron direction and z axis v Average number of neutrons born per fission Average logarithmic energy decrement a Microscopic neutron cross section T Fermi age Neutron flux X Fraction of neutrons born in energy group i Z Macroscopic neutron cross section Unit vector denoting direction of travel Subscripts a Absorption act Activation as Asymptotic eff Effective epi Epithermal f Fission g Geometric m Material o Scalar s- Scattering t Total xi

NOMENCLATURE (Concluded) th Thermal tr Transport R Removal T Transverse TC Thermal cutoff Superscripts f Fast q Source th Thermal M Milne N Normalized xii

ABSTRACT A method is presented for the prediction of the exit thermal neutron current from a reactor beam port. The results of duct experiments at the Ford Nuclear Reactor afford confidence in the validity of the analytical technique. The motivation for the study is provided by a prevalent desire to enhance thermal neutron beam intensity for spectrometer investigations. The analysis employs elementary P-1 theory, which is inspired by an examination of the infinite half-space limit. A return current boundary condition at the source plane of the port is developed by summing current contributions from the lateral surfaces. An idealization of the scalar flux distribution along the lateral surfaces of the port permits a diffusion theory solution for the source plane angular flux as a function of the port radius and the unperturbed flux distribution in the medium. Activation measurements of the scalar flux distributions at the source planes of 1-1/2-, 3-, and 5-in. diameter aluminum ducts were performed in light water and graphite. Measurements of the exit current from the ducts provided with an internal collimator to eliminate wall leakage contributions were carried out in light water. The largest duct perturbation was observed for the 5-ino duct in water. In this case the source plane scalar flux was depressed by 50 percent from the unperturbed flux, and the angular flux in the direction of the duct exit was 61 percent of the unperturbed value. Comparison of the data with analytical results indicates that the exit current is predicted very well, The source plane scalar flux is predicted with somewhat less accuracy. A background survey for the enhancement of the thermal neutron beam intensity at the Ford Nuclear Reactor is presented in an appendix. Variations in reflector materials and core-reflector configurations were investigated. The results suggest the employment of a ID20reflected core in a slab configuration, xiii

I. INTRODUCTION A. Statement of the Problem and Motivation for the Study This dissertation develops a method for predicting the exit thermal neutron current from a reactor beam port. The necessary removal of a portion of the diffusing medium to accommodate a beam port exerts a significant influence on the behavior of the thermal neutrons in the vicinity of the port. Inasmuch as these neutrons comprise a source for the beam, the beam port perturbs the intensity and spectrum of the exit beam. Experimental results are presented which test the analytical method under realistic reactor conditions. The motivation for this study arises from a demand for beam intensity enhancement. An estimate of the influence of the beam port on the exit current intensity is necessary for any thorough enhancement study, although the problem has a great deal of intrinsic interest as well. The increasing importance of neutron spectrometer experiments has revealed the need for the examination of problems associated with neutron beam extraction.l Roughly, the neutron flux, ~(h), at the exit of a beam port of length h and cross sectional area A is given by o /(O) A > where /(O) is the neutron flux at the source plane of the port. The 1

2 cross-sectional area of the beam is genera.lly limilted by the resolu.tion requirements of the experimento Th-us the intense neutron densities available in the reactor may be attenuated by orders of magnitude at the beam port exit. Bo Related Published Studies Studies for the enhancement of thermal neutron beam intensity have emphasized the reduction of fast neu.tron and ga-mma ray backgroun.d. The split core concept,2 resulting in the design of the National Bureau of Standards Reactor93 involves the removal of fuel from the midplane region of a D20 moderated and' reflected core, The tangential beam port concept, applied to the Brookhaven Hig:h Flu.x Beam Reactor4 JHFBR.) involves placement of the beam ports tangential to the core. Furthermore. the D20 moderated and reflected HFBR is undermoderated, so as to produace exceptionall.ly high tiherrr.aI flt.,x peaki:ng in the reflector. The effects of d.uc.t intlArod-iUctior.n. into a react.or system ha've been examined primarily:fromT t.:e standpojrint of the inf l1uence on the multiplication constant of tPe reactor. Behrens7 revised the magnitLde of L the diffusion area of a m.edium containing gas,-ccoolant passages, to account for the additional streaming leakage. Reynolds et a.6 investigated the reactivity effects of large voids in tole nrefl.ector of the Pool Critical Assembly at Oak Ridge National LaboratoryO In a theoretical examination of the effect on reactivity of a reflector d;cnt Baraff et a1O7

3 treated the problem by ordinary perturbation theory. These studies do not yield information concerning the perturbed scalar flux in the vicin. ity of the void or the current within the void. Shielding considerations have prompted the examination of the transport of neutrons down long ducts.8 The influence of the duct walls on the attenuation of neutrons by ducts was studied by Simon and Clifford9 and they concluded that the direct streaming contribution predominates for h/R >> 1o PierceylO attempted to separate the contributions to the streaming flux at various distances along the dusct axis in measurements at the LIDO reactor. In none of these investigations, however, was consideration given to the duct influence on the streaming source. C. Outline of This Study In Chapter II a simple analytical method is de-veloped to predict the th.erma1, neutron current at the exit of an evacuated duct from a knowledge of the unperturbed scalar flux distribution. in. the medi-um0 We consider a long:radial* duct introduced in-to the reactor reflector) presenting a small reactivity perturbation to the mulwrt:iplying system. The infinite half-space problem is reviewed from the standpoint of the anisotropic limit of the finite void. This permits comparison between exact solutions and the P-1 approximations and motivates a PIL approach to the present problem. A pseudo boundary condition on the scalar flux A railal duct refers to a duct whoseaxis lies along a radius of the reactor core0

4 is developed from the consideration of a finite return current to the duct mouth. Utilization of this boundary condition. permits a diffusion theory solution for the perturbed scalar flux distribution at the duct mouth. The diffusion theory solution is used in conjunction with the P-1 angular distribution to predict the exit current. The return current to the mouth is evaluated. by summing contributions from the lateral surfaces of the duct. Finally, some consideration is given to the scalar flux distribution along the lateral surfaces of the duct; although the realistic problem is not solved, a reasonable boundary condition is derived from idealizations of the physical situation. A number of activation experiments were conducted outside of the core of the Ford Nuclear ReactorO These are described in Chapter III. Initial comments concern the reactor, the positioning of ducts, the normalization procedure, and the foil detection techniques. Measurements of the unperturbed scalar flux distributions in the water reflector and reactor core are presented. Preliminary l-1/2-ino duct experiments in water and a small graphite insert are described0 The scalar flux data at the mouth and the collimated exit current for 1-1/2-, 3-, and 5-in0 ducts in water are displayed. Data are-presented for the attenuation of the thermal flux within the i-1/2- and 5-ino-ducts. A measurement of the neutron temperature perturbation at the mouth of the 5-ino duct is described, The final section describes the scalar flux measurements at the mouth of the ducts in the graphite reflector~ All unperturbed scalar flux measurements are compared with group-diffusion calculations~ ScaLar

flux data at the duct mouths and exit currents are compared with predictions based upon the analysis presented in Chapter IIo The discussion in Chapter IV considers the assumptions inherent in the analysis. The limitations of diffusion theory for the prediction of the unperturbed scalar flux in the reflector are pointed out, and the seriousness of.these limitations from the point of view of the present analysis is considered. An attempt to predict the realistic scalar flux distributions along the duct walls is discussedo Duct reactivity evaluations obtained from two-dimensional group-diffusion calculations based upon initial estimates of the wall gradients are compared with experiment. In addition9 items from previous chapters which require further consideration or clarification are discussed in Chapter IV. A survey directed toward enhancement of the thermal neutron beam intensity at the Ford Nuclear Reactor is presented in Appendix A. The investigation considers variations in reflector materials and core-reflector configurations for radial beam portso The results are relegated to an appendix because the work was in the nature of a background study preliminary to the main body of this dissertation~ In this survey, consequently. the beam port influence on the exit current is neglected. This can be considered a conservative omission from the standpoint of the reflector materials study, since the longer diffusion lengths possessed by the most desirable reflectors give rise to lower beam port current depressionso The method described in Chapter II, however, can be used as a calculational refinement for the conigurati.ons of particlar interest and a specified beam port diameter,

II. ANALYSIS A. Orientation The configuration of interest is a long cylindrical void inserted into a diffusing medium. This is sketched in Figure 2.1. Neutrons Ri A`~ B Figure 2.1. Void configuration. streaming from all points on the interface between the diffusing medium and the void form a beam at B far from interface A.* Assume that the beam is collimated, so that a detector placed on the axis at B optically views only the front interface at A. We are interested in the number of neutrons *Henceforth interface A will often be referred to as the mouth of the duct and B as the duct exit. 6

per unit time passing through a unit area normal to the axis at B. This quantity will be called JB(E), the partial current at B. JB(E) is equivalent to the integral over the surface, SA, of the angular current with energy, E, leaving z dSA directed into solid angle d2 about Q intercepting dSB: (2.1) j+(E) = dSA. A(r,Q,E) dS- (2.1) B dSB Then for any fragment of the energy spectrum: ~~+ B1 A- — ~~d2 jB(EoHE1) -dE dSA Z-Q ~A(r,,E) -, (2.2) dSB or the thermal current can be expressed as: j =Bth = dE dSA z. -A(rQ E) d = dSA Z ) Bth dSB dSB The problem then reduces to that of solving for the angular current at interface A. The formulation is initiated by writing the steady state Boltzmann equation for the angular flux, ~(r,Q,E), in space, angle, and energy: Q.v ~(r,,E) + Z(r,E)X(r,s,E) - s(r,QE) d2' dE' d(r,Q',E')Zs(r,Q'+Q,E'+E). (2.4)

8 The conventional symbols used in reactor physics are employed, and are defined in the Nomenclature. Hypothetically a solution for the geometry under consideration can be obtained after the source and scattering kernel are specified. In light of the complexity of the geometry, the pursuit of an exact solution for ~(r,,QE) is nearly futile. A numerical approach, utilizing a high speed digital computer is feasible, but besides lacking in generality, would be prohibitively time-consuming. Rather, we seek a technique which yields the angular flux to a sufficiently high degree of approximation to provide realistic predictions for jBth and which is simple enough to be useful for practical computationso To this end, P-1 theory is examined. B. Infinite Half-Space Limit In order to assess the validity of a P-l, or diffusion theory, approach consider the case which brings about the highest degree of anisotropy at interface A, that of a full. vacuum boundaryo Allowing R., the void radius, to go to infinity and considering monoenergetic neutron diffusion in an infinite half-space of fixed composition, exact solutions can be obtained for the steady state transport equation (assuming isotropic scattering):.. + I s(z,l) + s) d+I (z,iL), (2.5)

9 and comparisons may be drawn with P-1 theory results. Two cases are particularly well known. These are the classical Milne problem, s = 0, and the case of constant production, s = constant. We shall examine the angular flux in the case of constant production for a unit isotropic source per unit volume. The coordinate frame is sketched in Figure 2.2. Davison expresses the angular flux at the Figure 2.2. Infinite half-space geometry. boundary in the constant production problem in terms of the Milne solution: 9(oti)= tL([lz iM(o,1),for [ <. (2.6) ( ijM.l

10 For c = 1:* IjM(0) = o(), (2.7) and thus: [ q((o,)=] = (OL,), for,u < 0. (2.8) where [~(0,1)]N is the tabulatedl2 Milne angular distribution normalized to unit scalar flux at the boundary. The P-1 equations for isotropic scattering and a constant isotropic source are: dz (2.9) 3 d. o(Z) + Zt sd(z) = o 3 dz The angular flux is given by: 1 (Z) = 2n+l in(z)Pn(G). (2.10) The solution for the scalar flux is expressed as: Sl (Z)= Ce KZ + s (2.11),,.l *Our concern with reflector materials indeed directs attention to values of c close to 1. For light water, c =.994.

11 where K Zt -a L (212) Applying the Marshak boundary condition. 1 L 0,(O~ )dp 0 ~ (2.13) 0 and setting the source equal to -unity., one obtains for the P1i angular distribution at the boundary: N pl() -2 ( 2 1 4) ~' O 2 ~p_1(0.,tl) is the familiar form of the normalized Pdl angular distribution, and is given by: rPl (~90) -t Z 3 (2.15)`p-1 4 The normalized P1 and Milne angular distributions are compared in Figure 2035 A comparison of the scalar fluxes is obtained "by integrating Eqso (208) and (2o14) over all angles. yielding: -(0) 9 -1.a=. 155 (21 6) L~

N 1.4 ANGULAR FLUX N (0,pL(NORMALIZED TO UNIT SCALAR FLUX) VS. 1.3 1.2 EXACT; MILNE PROBLEM AND CONSTANT PRODUCTION (C=I) 1~2~,,P-1 THEORY i. i7 7 1.0.97.8 =t.7~ ~ ~ ~ ~.7 701.6 7.5.4.3.2 7 -1.0 -9 -.8 -.7 -.6 -5 -.4 - 3 -2 -.1 0.1.2.3.4.5.6.7.8.9 1.0 Figure 2.5. Normalized angular flux for c = 1; exact and P-i approximation.

13 angular flux close to A = -1o Setting 1t = -1 in Eqs. (2.8) and (2.14) we obtain: - ~' 1 =.993 (217) The same result is obtained for the Milne problem (s = O) if both the P-1 solution and the exact solution are normalized to the exact value of the scalar flux at the boundary. Thus the P-1 approach gives rise to an angular flux for very small angles which differs by less than 1 percent from the exact solution. It should be pointed out that the P-l1 solution is free of transport corrections. C. Diffusion Theory Boundary Condition Since a P-l1 approach provides an. excellent approximation for the angular flux in the limiting case of the infinite half-space, we shall examine the finite void using diffusion theory. The current returning to the mouth of the void originates from neutrons leaving the lateral surfaces. This forms a basis for a boundary condition at A which is physically realistic and consistent with the general formulation of diffusion theoryo In addition to the assumption of low capture in the medium, c' 1, and small angles to the detector, (R/h)2 << 1, the following two assumptions

are made" lo The perturbed scalar flux is flat over the entire duct mouth. This permits us to neglect the radial dependence at the mouth. 20 The fast flux is unperturbed by the. void. As a corollary to this the assumption is made that the overall reactor fission source perturbation is negligible. Specification of the partial current at A, coupled with the continuity requirements at the boundary between the multiplying medium and the reflector completely specifies the diffusion theory solution for the scalar flux in the medi.um~ It is convenien.t however. to transform the current condition into one on the scalar flux.9 o0 Proceeding analogously to the familiar treatment of the limiting full vacuium interface1 and referring to Figure 2o4o jA 4 6 dz(28) A.Assuming a linear decay of the scalar fl.ux past A. OgZ) Zi d A 9 (2.19) dzA we solve for the distance, d, past the interface, at which o0(d) = 4 JA 4 jA =a d I + 1o A0 (2.20)

15 d4'0 (Z) d+Oid A Figure 2.4. Linear extrapolation at vacuum interface. Combining Eqs. (2.18) and (2.20) we obtain: ( ) = 4 A (2.21) 3 ~tr 4 jA the well-known diffusion theory extrapolation condition, applied here to a vacuum boundary possessing a finite return curent. It is shown in Appendix D that the application of this pseudo boundary condition on the scalar flux rather than the actual one involving the partial current introduces negligible error for c Z- 1. Furthermore, the employment of a linear extrapolation distance, d, rather than the extra11 polated endpoint, zo, is valid for values of c close to one:

16 d _c 1/2 for c 1 (2.22) zo D. Angular Flux Distribution The P-1 approximation to the angular flux is needed to calculate duct currents. It is instructive to derive it directly from the integral form of the transport equation. In the constant cross section approximation the angular flux is given by (Eq.(4-16) of Reference 11):,(r, 0) = r- 4+ s(r -r) e dr). (2.23) Figure 2.5 is a vector diagram pertinent to Eq. (2.23). The source term in the equation is defined per unit solid angle. Figure 2.5. Vector diagram at vacuum interface. Expanding any function, f(r-Q), about the point r in a Taylor series

gives~ f(r( a) -= f(r) + ( k)f(r)+V ) f) + (2 24) Accordingly expanding the scalar flux and source in the integral equation, performing the integration over r, and retaining two terms yields: C. /(rQ) &[Oo(r) ~ 2-\7,o(r)] + I[s(r)- m>Vs(r)] (2-25) 4j-c Eo Derivation of j Using the angular distribution presented in Eqo (2o25) and diffusion theory solutions for the source and scalar flux distributions at the mouth, we are in a position to derive an expression for jB, the partial current at the exit of a long collimated duct. Restricting ourselves to the geometry illustrated in Figure 201, and assuming polar symmetry: V r + z (2026) ar az Q r + (227) and: dQ d SB Q (2 28)

The angular flux at A in the direction of a detector on the axis at B is expressed as: ~A(rQ)'- o (r) +' { _ zI r ajs: (2.29) Assume that,0o s, and their derivatives are constant over the emitting surface, so that: A(r- L ) + - s _ n ds. (2.30) Lrto p dz A dz Substituting this expression for the angular flux into Eq. (2-3) and integrating over the surface at A results in: B 42+ 3 ( ) d0 A R2 2 ds + g2 s h ds 231 For a long duct, R2/h2 ~ 1, SO that the expression reduces to: jB @h dzj [s X A (2052)

19 The slowing-down source can be computed from the fast flux obtained in a few-group analysis. Assuming fast flux isotropy: f f S ZR o 4ir F. Derivation of JA Equation (2.21) specifies the diffusion theory boundary condition to be used in obtaining the distribution of the perturbed scalar flux in the vicinity of the duct mouth. It is necessary to sum contributions from all lateral surfaces of the duct in order to derive an expression for jA in terms of the scalar flux distribution along the walls of the duct. The geometry is illustrated in Figure 2.6. Figre2..0 on f co Figure 2.6. Configuration for calculation of jA.

20 Equation (2.25) is employed once again for the angular flux along the lateral surfaces of the duct. Assuming that the slowing-down source contribution is negligible, and that c' 1: d(z,R,n) - 4Lo(z,R) - ~ V BoI ~ (2.34) Since the mouth is treated in one dimension, it will suffice to solve for the return current at the center. Referring to Figure 2.6: = -r z (2355) -dQ dS, (2.36) = R + z (2.37) and: dS = 2T Rdz. (2~38) The return current at the mouth is expressed as: jA = dS ( ))( R.,~ Q )(2239) = dSA Substituting the angular flux: ~(z,R,Q) - o(z,R) + + z, (2~40) 4~~~~~C _ ar'

21 into Eqg (2-39), and supposing that the void extends to infinity, we obtain for the return current at the mouth: r0 2C 00 z 4E j- =,, dz Z(Rz) +I dz A e R d dz 2 A 2 (R2z2)2o (+) (R2+z2) 92 (2. 41) In order to assess the validity of this expression, let us examine the limits as R-+O and R-oo. It is first convenient to transform variables such that: o00 Z 00 z nLR _ _ of z~o(R, Rz) )r Rz. =dz Z(o~+2)2)+ I dz A R Qt dz (12 A 2 (l0z)0 (i+z52 (1+z (2042) When R becomes small, the scalar flux and its derivatives are slowly varying with respect to the remainder of the integrand, and can be taken outside the integral sign. The integrations yield: lim JA - o(,0) + IR i ~o A (2.43) lmjA 4 which is the partial current at A in the absence of the void0 which is the partial current at A in the absence of the void. In the limit of large R the rest of the integrands are slowly varying with respect to >o and its derivatives, and accordingly:* *We assume here that the scalar flux is monotonically decreasing with z; the unperturbed flux is of the form: S(Rz) = CieaiRz ii

22 lim jA = 0, (2.44) R->oo the vacuum interface condition. Equation (2.41), then, has the desired limits. G. Scalar Flux Distribution Along the Lateral Surfaces The final task, and evidently the most formidable, is the evaluation of the scalar flux distribution along the lateral surfaces of the void. We expect the average perturbation of the angular flux along the duct walls contributing most heavily to the return current to be less than that at mouth. This is because of the considerably more favorable view factors along the walls. For small void diameters it is reasonable to assume that the angular flux along the walls is unperturbed. A sound approach to the problem, however, consists of a current balance over all surfaces. Such a procedure has been carried out for a void of finite length along the axis of a cylindrical reactor.l4 Considering for the moment a long cylindrical void, the radial component of the gradient at a point (R,zl) on the surface is obtained by equating the outgoing (in the positive radial direction) partial current to the contributions from all surfaces of the void. The angular distribution is given by Eq. (2.34). Figure 2.7 illustrates the geometry. Carrying out

23 Figure 2.7. Current balance geometry. the angular integrations and solving for the radial component of the gradient, one obtains: ~~~/~rl R z _/r _ l + 3 | dz o(R,z) fi(p) + dz o/Rz f2(p) Xo(Rnzj) 2 2R (Rz) (Rz) dz (Rz ) f ) 2.45) where: f(P) = {2(2+p )(lp2, (2.46) f2(p) -= k[2(l-p4) + 3(2-p2))E(P2) - (1_p2)(8+p2)K(p2fl 3p (2.47) _(p) = 2(lp)l/ (2+p2)K(p2) - 2(l+P )E(p), (2.48)

24 2 2l/2 zj-z pi(z,-z)2 + ~ p;) = PW 1+/2 (2.49) and K and E are the complete elliptic integrals of the first and second kinds o15 The integral equation is not easily soluble~ Zimmerman assumed a separable solution, in which case Eqo (2459) reduces to. (25o0) rigor, it possesses the correct limits (for the void along the axis of a cylindrical reactor) o The trivial case of the infinite cylinder with a constant flux along the walls2 although. not directly applicable to our problem, is instructive and does permit a separable solution~ For this case ~o(Z)/~o(z = 1,S and o/z = 0, and the i ntegrations are performed:from -ca to c The numerator in Eqo (200) vanishes, and we obtairn,e 8,/ z.4~..z = o 0, (2o51) for all zthe axis of ThIn the spirital case of the infinite cylinder with a constant flux of

25 the form ~o(z) = a+bz also leads to a zero gradient condition for the infinitely long cylinder. This can be shown by examining the point zl = 0. For this case ~o(z)/o(zl) = l+cz and o Z) = c, and the numerator of the right-hand side of Eq. (2.50) becomes: 1 dz(l+cz)fl(p) + dz cf3(p) (2.52) -00 -00 The terms multiplied by the constant are odd functions, and their interals are zero. The remaining terms are identical to the constant flux case, giving: R I 6/6/ra o 0 (2.53) O(R,O) This example, however, is somewhat of a mathematical artifice, in that the result is not the same for zl f 0; and so the assumption of separability is not strictly valid (moreover, the linear expression permits negative fluxes). The actual void geometry consists of a semi-infinite cylinder bounded by an interface at z = 0 (See Figure 2.8). The current balance at any point zi > 0 now contains additional terms from the contribution of the mouth. The isotropic and z component of the gradient contributions are derived in Appendix E and are of the form: 1 zl2 2 g o ) 2;0.' l+ R2 Z2 6, z2 /,Riz. d,L~2 Z('O~~~'"'I~B; " " "r''"'u

26 R Figure 2.8. Current balance geometry including front interface. These additional terms must be added to the right-hand side of Eq. (2.45). Postulating as before a constant flux everywhere along the boundaries such that O/kzo = o and ~o(O) = 1, and assuming separability, one finds that the contribution from the mouth is equivalent to that from the negative half of the infinite cylinder studied previously. This leads once again to the condition: R = O, for all zl >. (2.55) ~O ZZ We have not solved the physically interesting problem, however. In fact the argument for the R -+ 0 limit of Eq. (2.41) demonstrates that a constant flux along the lateral surfaces does not give rise to a duct perturbation at the mouth at all. The realistic unperturbed scalar flux is likely to be a decaying exponential or sum of exponentials, in which

27 case a z dependent radial component of the gradient is expected along the duct walls. Positive near the mouth, the radial component of the gradient becomes negative at an axial distance from the mouth such that the streaming contributions exceed the unperturbed flux in the medium. In light of the complexity of the actual physical situation, the pursuit of an accurate representation of the scalar flux distribution at the duct walls will be avoided. Instead we assume that the zero gradient boundary condition is a reasonable approximation. This is equivalent to direct utilization of the unperturbed scalar flux if the unperturbed flux distribution is radially flat in the vicinity of the duct. The validity of this zero order perturbation treatment will be assessed by its success in predicting the results of experiment.

III. EXPERIMENT A. The Ford Nlfuclear Reactor The experiments were conducted on the south face of the Ford Nuclear Reactor~ The FNR is a swimming pool facility similar in design. to the 16 Bulk Shielding Reactor at Oak Ridge National Laboratory~ The fuel is of the MTR type. Regular elements contain 140 grams of U235 in 18 plate 3 ino x 3 ino subassemblies~ Partial. elements_ which house the poison 235 rods9 contain 71 grams of U3 in 9 plate subassemblies. A more complete description of the facility is contained in the literature.17 The core configuration adopted for the experiments is illustrated in Figure 3lo. Occasional minor variations in the fuel loading were necessitated by criticality considerations, bult changes were confined to the north extremities of the core. The specific elements on the south face were left runchanged during the period of experimentso B, Insertion anad Positioning of Ducts A duct. the duct support, and a foil holder are sketched in Figure 3.2. Variable sized collars mounted on the aluminum plate of the support accommodate the ducts for positioning~ The weighted plug bolted -to the bottom of the plate fits into a south matrix hole of the grid plate which is furnished with an offset dowel pin to guarantee angular placement. 28

29 BEAM TUBE 1 2 1R W I (d PNEUMATICA ( N / I~,-,-,,,,,- t,-,,-E -/\ N I )' A,BC SHIM RODS CR CONTROL ROD G GRAPHITE FC FISSION CHAMBER Figure 3.1. Ford Nuclear Reactor core configuration.

30 it(Q,~)o I L"W~~~ ~4'.1 Streaming Duct Foil Holder Duct Support with Normalizer Figure 3.2. Streaming duct, foil holder, and duct support with normalizer.

31 Figure 3.3 is a photograph of the 1-1/2-in. duct encased within the 4-in. graphite insert positioned adjacent to the core. 4-in. graphite insert.

32 Table III. 1 is a list of the dimensions of the three alumi:num ducts employed in the water runso Also tabulated are the reactivity worths of the ducts positioned 4 in. below the centerline of the core in the column pictured in Figure 3 l. The ducts were evacuated to eliminate attenuation of neutrons by air and water vapor.'TABLE III o 1 DUCT SPECIFTICATIO NS Wall Front Reactivity bk/k 0OD.o Thick- Window Length., Axial. Distance from Core i.n. ness9 Thick- ft 0 76 6 6 1 166 ino ness in. ____: crm cm. cm 1-1/2 00625 O020 4 0....7 007 5 o083,O.40 O.... o ooo 5 o187 0 0 6 2 4 1.% o o0%j < ol. Runs were made at vJarious low power le'vels in the range of 500 watts to 70 kilowatts Th.e duration of each run was ch.osen so that duct insertion and withdrawal, occupied a -time interval, of less than. 1-1/2 percent of -the total i rradiation. Spacing of runs was limited'by the induced acti.vity of the m.etal wnich would h.ave decayed in. less th.an one-h.alf h~our had t~h.e material been pure alumilnumo Th~e presence of ailloyi.ng con.

33 stituents in the aluminum extended the cooling time considerably for high power activationso C. Normalization Because the experiments were performed at variable reactor power on different days, it was necessary to employ a precise and reliable procedure to normalize the runs to one another. First attempts at normalization using the pneumatic tubes on the west face of the core proved unsatisfactory~ Although the activations in the pneumatic system agreed within ~ 2 percent with a fixed monitor viewing the west face at the exit of a beam port, the proportionality between the flux intensity on the west face and at the duct position was destroyed by rod manipulations, xenon buildup, and core movements, Normalization was finally accomplished with the aid of the pinwheel frame shown in Figure 3~2~ Foils were mounted at the four corners of the diamond which were each 6-1/2 ino from. the duct axis and 1/2 ino from the reactor core. That the duct itself exerted no influence on the normalizing foils was verified by comparing the normalization of consecutive 5 in. void and no-void runs with. the fission chambers positioned as in Figure 3o1. The infl.u.ence of poison rods was minimized by holding B shim. rod 100 percent withdrawn and the control rod greater than 40 percent withdrawn du.ring an irradiationc Furtherm.ore, the duct axis was positioned 4 ino below the core midplaneo That the spectrum was not changing

54. significantly for the pul.rpose of normali.zation. was confirm.ed by irradiating two cadmi.um co vered foi l.s si.mui.lta.rneou-.sl.y wit:',t the bare normali zing foils Typical. normalization data are displayed in Table T1::I:2 Com.parison. of'the roo't-mean.-.squ.are devi.ation between. the four positions set a normalization error i.pper Limiit, of approximately: 13 percentL Repeated ru.ns were reprod-u1cible within. t.nere lm.i. tso D, De't e tcto rs:Mevtallic foil acti vators were used for all of the measurements. Goldd, copper9 man.ganese. and luteti-rnf. were each -u.tili zed to some extent. Table'T1To3 iJS a list of the perti rnenrt foil characteristicso Gold was used:for most of t"he mea....re.mients, f.t could be obtained chemically' p;ure and possesses a hi.gh actiration cross section together with. a long half-1lifeo In. addition., te large 409 e-' resonance cross sectieon permi.ts epithe'rmal a... a r.c. a a well-defined en.ergyo bRare an-d cadm. iu.-'er.; d -1/ 6 i.n.x 1/1 6 in x fOC;1 i,,:foils were employed at th:e duletl c olit:ho wit;h rno s.Tignifi carn.t pertim.bratio n.iduced in larger foi l iLrradiated at the exi.t of2 th e di.n-t, The small. foils at the mou.th. could he coun.ted in. a well. co;aftter soo: 1fter i.r.radi.ati.on. at the flulx Level.s necessary to acttivate thie foil s at the e:xito Weigh:i.ng was accomplished on ar eiec tre osta tic a ba la.e. Epi2thermial flu.xes were miea.ured by completely su:r:rosindirng the foi ls with 20 mnils of cadmftnu Icomplete; cadmrm,.um enclosuare was shown. to lead

TABLE III.2 NORMALIZATION DATA E-2 4-ww Nominal Duct Rms Deviation Run Control Rod Reactor Mon. 2 Mon. 4 No Date S ize, Mon. 1..2 Mon. 3.....m..eviatio DNo. te inze'vPosition, Power Bare Thermal Mon Bare Thermal } 32 9/4 1-1/2 40 5 1.0 1.0 1.0 1.0 1.0 1.0 1.0 _ 28 9/4 1120 42 1.200.204.201.210.207.211.205 2.5 31 9/4 H20 40 0.5.099.0995.097.099.099.102.99 1.5 39 9/12 1-1/2 58 70 10.75 10.9 11.05 11.2 10.6 10.9 1l1.0 1.5 40 9/12 1-1/2 40 70 8.35 8.25 8.13 8.32 8.40 8.15 8.12 2.0 44 9/14 5 80 35 4.15 3.96 3.81 3.90 4.o09 4.19 4.01 3.5 61 12/4 5 70 1.75 1.69 1.64 1.82 1. 71 1.73 3.5 *As recorded by the reactor instrumentation.

TABLE IIL5 FOIL CHARACTERISTICS Ref. 18 Ref. 19 Princ, Abundance, Thickness, A) f.act(E) EP Detector i.Activity gact(.025 eN epi E Pur ity 4& ~~in. atepi B Counted b. b. Au 100.001.41 Mev 98 1558 Excellent 7 Mn 100.002.845 Mev 15.5 11.8 Poor 7 Cu and Ni Impurities CU63 69 Nat., 0oo5.~5 Mev 4-3 4L4 Excellent Abund. Annih. y from B Lu176 2.6 Nat..o1o.208 Mev 360 b. Excellent Abund. in 7 at.14 2.3 ev Lu-Al Alloy Reson.

37 to large errors. Cadmium cutoff energies are derived by Dayton and Pettuso20 A 20 mil covering of cadmium around any of the foils listed in Table III3 affords a cutoff energy for an equivalent perfect filter of ~448 ev in an isotropic flux and o342 ev in a monodirectional flux. The region between these two energies is estimated to contribute less than 1/2 percent to the total activation in an assumed Maxwellian spectrum fitted with a 1/E tail at o18 evo The foils were washed in acetone, weighed, and covered with a thin layer of paper and tapeO it was determined that the covering material introduced no perturbation in the foil activation by carrying out several trial irradiations with various thicknesses of paper and tapeo Bare and cadmium-covered foils were placed at least one inch apart at the mouth of the ducts. The perturbation induced by the cadmium on the bare foil at the mouth and on the one at the exit of the duct was estimated to be well within the normalization erroro E. Counting Equipment Except for occasional utilization of the multichannel analyzer, counting was done in a well counter equipped with a 3 in. x 3 in. NaI (T1) scintillation crystal backed with a Dumont No. 6292 photomultiplier tubeo A Radiation Instrument Development Laboratory Model 4951 scaler provided the high voltage to the photomultipliera The discriminator was set well above the noise level and the high voltage was maintained in the center

of the counting plateau at 1275 volts. Reproducible positioning of foils in the well was accomplished with the aid of a test tube device incorporating a centering rod~ Stability of the system over long intervals of time was important. Figure 3~4 illustrates the measured decay of a gold foil over a 7-day periodo Subsequent count rates were kept below the maximum in the figure to eliminate dead time corrections and to avoid jamming of the mechanical register. An absolute activity determination on the multichannel analyzer set an efficiency of 52 percent for the.41, Mev gold gamma rays~ Fo Unperturbed Flux Distributions for the Water Reflector The first experiment was done to verify that the current at the exit of the evacuated 1-1/2 in. duct was far in excess of the unperturbed flux in the vicinity of the duct exit. If this were the case, activations measured at the duct exit could be attributed solely to streaming contributions from'the region close to the mouth of the duct. Thermal* and epicadmium gold reaction rates were measured in the water bordering the south face of the reactor core out to four feet. The measurements were performed at the midplane of the core one row east of the position displayed in Figure 3l.o The relative reaction, rates are plotted in Figure *The thermal reaction rate is taken to mean the difference between bare and cadmi m-covered measurements.

59 106 Activity versus Time For a Gold Foil in the Well Counter 105 2 3 4 5 6 Time In Days Figure 3.4. Decay of a gold foil counted in the well counter.

40 7 Relative Gold Activation versus Distance from Core 10 \ 0 Thermal Reaction Rate in Water \ x A Epithermal Reaction Rate in Water XThermal Reaction Rate Along Axis of 1 1/2" Duct 6 I0 \ IQ5 I0 \ 0 U -0 3 >I 0 L \ r \ 0 10 20 30 40 50 Distance From Core (in.) 10

Thermal reaction rate data for foils placed within the 1-1/2 in. duct positioned 5/16 in. from the reactor core are shown in the same figure. The bare and cadmium-covered measurements were made separately to prevent severe mutual shielding. The results demonstrate that the streaming current in the duct exceeds the unperturbed flux in the medium by greater than two orders of magnitude 34 in. from the duct mouth. Of peripheral interest is the flux distribution in the core. Foil insertions were made between the plates of the fuel using 10 mil aluminum venetian blind slats for positioning. The disadvantage factor in the fuel plates of the FNR is nearly unity,22 and for this reason measurements in the channels between plates are representative of the homogenized medium. Since the gold cadmium ratio is low within the core, copper foils were used for this measurement. The data are shown in Figure 3.6. The flags indicate the uncertainty in position and normalization. The results for groups three and four of a four-group, one-dimensional machine calculation are normalized to the data. The flux peak in the two off-center fuel elements can be explained by the close proximity to the water channels serving the poison rods, which were nearly 100 percent withdrawn during the course of the experiment. Since access to the partial element was limited, the peaking effect in the vicinity of a water channel is demonstrated in Figure 3.7 by an east-west flux distribution in the element bordering the east side of the control rod element (see Figure 3.1).

Flux Distribution in Reactor Core OMeas. Thermal Rcn. Rate I Meas. Epicadmium Rcn. Rate (X5) 3.0 3.0 N Predicted by 2-Gp. I IN 1-D. Group Diffusion Calculation o! I 2.O I.0 H20 Reflector F.E.#34 F.E. 31 F.E.#29 F.E.433 F.E.# 413 Graphite Reflector Core Figure 3.6. Thermal and epicadmium flux in the core and the water reflector.

43 1.50 Cu Reaction Rate versus Distance From West Boundary of Fuel Element 1.40 - @ 1.30 1.20' 1.10 1.00 9Q' i I 0 I 2 3 Distance From West Boundary of Fuel Element (in.) Figure 3.7. East-west traverse of thermal flux in the fuel element bordering on the east of the control rod element. Most of the streaming measurements in water were performed with the duct axis positioned 4 in. below the core midplane in the column illustrated in Figure 3.1. This position was chosen to minimize the perturbing influence of the control rod. The unperturbed duct axis flux distribution at this position is plotted in Figure 3.8 out to 22 cm from the core. The data are normalized at 2.16 cm to the results of a two-group, two-dimensional group-diffusion code. The data in Figure 35.8, as well as most of the subsequent results, are plotted on the basis of detector reaction rates. This presents the

UNPERTURBED FLUX VERSUS DISTANCE FROM CORE IN WATER 70 0 MEASURED THERMAL REACTION RATE o 0 A MEASURED EPICADMIUM REACTION RATE 0 60 0 CALCULATED BY 2-GROUP DIFFUSION F | 0 0 CODE (NORM. AT 2.16 CM.) 03 50 ESTIMATED POSITIONING ERROR ~ 3mm ESTIMATED REACTION RATE ERROR+2%' 40 z30 0 < 20 b.I o 10 2 4 6 8 10 12 14 16 18 20 22 DISTANCE FROM CORE (CM.) Figure 3.8. Unperturbed thermal and epicadmium flux in water reflector.

45 data in its raw form, yet it is a convenient form for interpretation in the present study. The normalization technique discussed earlier permits direct comparison of the results of measurements employing identical detectorso Corrections for flux perturbations induced by the foils must be made before the ratios of epithermal to thermal fluxes can be evaluated. 23 From the results of Dalton and Osborn we estimate a low correction to the thermal flux measured'by small l mil gold foils in water: ~'th th = lo04 o (3.1) meas. The self-shielding induced by the 4~9 ev gold resonance brings about a 24 considerably larger correction to the epicadmium reaction rate: L-epi= 22 (3A2),/ oa(E) dE epi measo for 1 mil gold foils. Applying these corrections, a comparison can be drawn with the resulting ratios of epithermal to thermal flux predicted by the four-group codeo Assuming a 1/E epicadmium spectrum and adopting the epithermal absorption integral in Table IIIo3 we get:

46 5.53 Kev.625 iev (E)dE o.625 ev /(E)dE at 2.16 cm in the water reflector (Figure 3.8). The four-group code predicts that ~3/'4 =.195, a 5 percent deviation from the measured ratio. G. Preliminary Duct Experiments With the evacuated 1-1/2-in. duct positioned at various distances from the core, bare and epicadmium reaction rates were measured at the duct exit. The data are presented in Figure 3.9. The thermal data are compared to predictions for the angular flux, obtained by using the unperturbed flux distribution in Eq. 2.41 to compute the boundary condition, a group-diffusion calculation to obtain the perturbed flux distribution at the duct mouth, and Eq. 2.30 for the angular flux. It is noteworthy that the peak occurs at a duct position closer to the core than the position of the unperturbed flux peak in the medium. The same measurement was performed in a 4 in. x 4 in. x 28 in. graphite insert bored with a 1-1/2-in. axial hole to accommodate the duct. The insert was positioned flush against the reactor core as shown in Figure 3.3, and the duct was moved in steps of 1 in. by introducing graphite plugs into the hole in front of the duct. The thermal flux data in the solid insert as well as the streaming data at the duct exit are plotted in Figure 3.10. The dotted lines represent the previous water data for comparison.

47 MEASURED REACTION RATE AT 1 1/2" DUCT EXIT VS. DISTANCE OF DUCT MOUTH FROM CORE o MEASURED THERMAL REACTION RATE (CADMIUM- BACKED) a MEASURED EPICADMIUM.008 REACTION RATE 0 _PREDICTED THERMAL ANGULAR FLUX -..007 (NORMALIZED AT 1.16 CM.) E.006.005-.0 0 H.004.0035 -J 0.002 -~..001 I i I I I I 1.0 2.0 3.0 4.0 5.0 6.0 7.0 DISTANCE OF DUCT MOUTH FROM CORE (CM.) Figure 3.9. Reaction rate at exit of 1-1/2-in. duct in water.

48 60 6 Unperturbed Thermal Flux versus Distance From Core //'O In Graphite Insert 5 0 / X ___.-Previous Water Data of Figure 3.8 50 / \ O! \\ I x 40 - E 0 0 20 oI I U 0 I0 10 I I I I I I Thermal Reaction Rate at 1 1/2"Duct Exit versus.007 O Distance of Duct IMouth From Core O In Graphite Insert - -- Previous Water Data of Fi gure 3.9.00 6 t j Ax (Shifted for no cadmium backing) E E.:.005 E\'.004 0 o.003 _., I I I I, 2 3 4 5 6 7 Distance From Core (in.) Figure 5.10. Thermal reaction rate in 4-in. graphite insert and at the exit of l-l/2-in. duct encased in 4-in. graphite insert.

To compare the data with predictions for the actual magnitude of j+Bt at the duct exit it was necessary to measure the leakage contriBth butions from the duct walls and also the effect of activation from backscattering at the exit. Measurements with the duct positioned at 2.16 cm in water resulted in the compilation of Table III-4. A 23 percent backTABLE IIIo4 PREDICTION OF ACTIVATION AT EXIT OF 1-1/2 INCH DUCT IN WATER Exit Measurement Bare reaction rate 0O0691 x 1010 gm- -min10 It Cdo-backed reaction rate o00531 x 10 Cd.-covered reaction rate o00142 x 1010'I.0I x101 0 Thermal monodirectional reaction rate.00389 x 10 Reaction rate from walls of duct.00082 x 101 Net thermal monodirectional reaction rate from mouth of duct.00307 x 1010 Net thermal monodirectional reaction rate from mouth of duct'B (corro for cntgo 1/4 in. foil) o00322 x 10 Mouth Prediction Direct streaming from ~6875 in.of radius of duct oO0286 x 10 1 Transmission through wall, from last 10.0625 inoof radius o00035 x 10 Net direct streaming prediction. from mouth Bth2.00321 x 1010 scattering contribution was revealed by backing the bare foils with cadmium. A wall contribution of 21 percent was measured by covering the mouth of the duct with cadmium A 5 percent correction originated from

53 47.61 24. 61 4',, z1/26 1 1/2.252",SPACER 426 S; CADMIUM DUCT MOUTH CENTERER POSITION Figure 3.13. Collimator design. neutron detector at the exit to view.905 sq. in. of the mouth regardless of the duct diameter. The high epithermal content of the spectrum at the collimator exit necessitated employment of a detector possessing a relatively low resonance integral. Manganese combined this attribute with a high activation rate.* The bare and cadmium-covered foils were irradiated simultaneously by offsetting the covered foils by 3/8 in. from the centerline. A copper impurity in the manganese required that the foils be counted in the single channel well counter within one manganese half-life. The results of the collimator measurements are plotted in Figure 3.14. *The reactivity perturbations induced by the introduction of the larger ducts caused power fluctuations in the reactor system. Since the measuring foils and normalizing foils possessed different decay constants, the transient was not automatically corrected for by the normalizing foils. It was necessary to evaluate the normalization by the analysis presented in Appendix F.

51 PERTURBED FLUX AT DUCT MOUTH VS DISTANCE FROM DUCT AXIS IN WATER AT 2.16 CM. x NO DUCT 680Co 80 1/2" DUCT o + 3" DUCT x A 5" DUCT E 70 -PREDICTED THERMAL ot) FLUX AT MOUTH 60 x IX Xxxl xe5 X X xxx oxx I ~II I I <. +1 o o 50 al i 0 50A 1 0/2" PRED. 0 + I THERMALI I +I a 1 1+ I 5" PRED. C) 3 I' Ir I A A -3 -2 -I 0 2 3 EA ST WEST DISTANCE FROM DUCT AXIS (IN.) Figure 3.11. Perturbed radial flux distributions at duct mouths.

52 DUCT INTERFACE THERMAL / 50- -' -I5~01 A~d,/dz =.08 CM-, PREDICTED =.I I CMZ>9 + =.24 CM T70~~~~~~~~ ~PREDICTED =.26 CM z k d /dz 60M Q 30.60 CM PREDICTED = 42 CM-I 0 ct3, /X/1PERTURBED FLUX VS DISTANCE FROM CORE 9 IN WATER 20 EPICADMIUM'z _,9 _+~~~ O 1-1/2" DUCT 10 — z + 3'DUCT ~z~*A 5'DUCT / /UNPERTURBED FLUX.2 4.6.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 DISTANCE FROM CORE (CM.) Figure 3512. Perturbed axial flux distributions at duct mouths.

53 24. 61 " XI I 1/2 1/252 SPACER 46" CADMIUM FOIL DUCT MOUTH CENTERER POSITION Figure 3.13. Collimator design. neutron detector at the exit to view.905 sq. in. of the mouth regardless of the duct diameter. The high epithermal content of the spectrum at the collimator exit necessitated employment of a detector possessing a relatively low resonance integral. Manganese combined this attribute with a high activation rate.* The bare and cadmium-covered foils were irradiated simultaneously by offsetting the covered foils by 3/8 in. from the centerline. A copper impurity in the manganese required that the foils be counted in the single channel well counter within one manganese half-life. The results of the collimator measurements are plotted in Figure 3.14. *The reactivity perturbations induced by the introduction of the larger ducts caused power fluctuations in the reactor system. Since the measuring foils and normalizing foils possessed different decay constants, the transient was not automatically corrected for by the normalizing foils. It was necessary to evaluate the normalization by the analysis presented in Appendix F.

1.00 |Bth AT COLLIMATOR EXIT (FIXED SOURCE PLANE AREA) VS. VOID DIAMETER. NORMALIZED TO I FOR NO.90 [ \ VOID. IN WATER AT 2.16 CM.80.70 - +m w.60 -I I n-.50 0 RELATIVE REACTION RATE AT.40 COLLIMATOR EXIT PREDICTED RELATIVE jBth.30 I I I I I I I 1.0 2.0 3.0 4.0 5.0 6.0 70 8.0 VOID DIAMETER (IN.) Figure 3.14. JBth at collimator exit.

55 The solid line presents the prediction for jBth The data and the predictions for j are normalized to a value of unity for no void. Bth 3. Thermal Flux Attenuation Within the Empty Ducts The behavior of the flux within the empty duct is related to the main topic of this study, and a series of experiments was performed to examine the attenuation of the forward component of the thermal flux. Bare and cadmium-covered measurements were performed separatelyo The detectors consisted of 1/16 ino square 1 mil gold foils. The bare foils were backed with cadmium, which posed the formidable problem of mutual shieldingo In order to avoid the laboriousness of separate runs for each foil position, the five foils were positioned to avoid shadowing by spacing them 600 apart. Investigations within the 5 ino duct demonstrated that the radial flux was reasonably flat near the exit of the 5 in. duct (Figure 3.15)o The thermal reaction rates within the 1-1/2 in. and 5 in. ducts as a function of axial distance from the mouth are plotted in Figure 3.16. The measured slopes at 44 in, are recorded in the figure. I The rmal Spectrum A question of some concern was the influence of the void on the thermal spectrum, and consequently the spectral effect on the apparent perturbations measured. Results published by Walton et alo25 indicate no

56.07.06 Flux versus Distance From Axis at Exit of 5" Duct 0 Measured Thermal Reaction Rate.05 A Measured Epicadmium Reaction Rate o. E *.04 E c CLI_ o.03 I A.02.0 I I -1.5 -1.0 -.5 0.5 1.0 1.5 Radial Distance (in.) Figure 3.15. Radial variation of flux at 5-in. duct exit.

57 1.0 Thermal Flux versus Distance From Mouth of Duct O 5 "Duct A 11/2" Duct.1 - Slope -2.48 i, ro o a U 0 I o.01 Slope= -2.70.001 I 20 30 40 50 Axial Distance From Mouth of Duct (in.) Figure 3.16. Thermal flux attenuation within 1-1/2- and 5-in. ducts.

58 hole effect on the spectrum in polyethylene with voids up to 2 in. in diameter. The effect of the 5 in. duct on the thermal neutron temperature at the mouth was measured using the.14 ev resonance in the Lul76 absorption cross section as a spectral index. Bare and cadmium-covered 10 mil, 2-1/2 percent Lu-Al foils were irradiated simultaneously with manganese* at the mouth of the voided 5 in. duct and at the same positions with the duct filled with water. The activities were measured on a multichannel analyzer corrected for drift with a Co59 standard. The resultant activities are listed in Table III.5. TABLE III.5 NEUTRON TEMPERATURE MEASUREMENT H20 Medium Bare Lul76 21.82 x 105 gm-l-min-1 Cd. -covered Lu176... Bare Mn 11250 x 105 " Cd.-covered Mn 356 x 105 Ath (Lul76 )/Ath(Mn).00200 5 in. Void Bare Lul76 10.99 x 105 gm-l-min-1 Cd. -covered Lu76... Bare Mn 5561 x 105 Cd.-covered Mn 276 x 105 Ath (Lu76 )/Ath(Mn).00208 *Manganese was used as the l/v detector because Lu175 possesses a large resonance integral, giving a cadmium ratio of only 1.59 for the voided duct.

59 The ratio of the reaction rate of Lu176 to that of l/v detector is calculated using the form of the resonance given by Schmid and Stinson:26 a(E\FE =Const. (3.4) 1 + 1108(E-.142)' The calculated ratio, normalized to unity at 0~C, is plotted up to 100~C in Figure 3.17. In this range of temperatures, the observed ratios suggest a temperature rise at the mouth of the voided duct of 10~ + 5~C. A 10OC temperature rise at 3000K would account for a 1.7 percent depression in the average activation cross section of a 1/v detector. J. Measurements in Complete Graphite Reflector Graphite was chosen as a second medium in which to examine duct effects. The selection of graphite was based upon diffusing properties widely different* from those of light water, as well as accessibility at the FNR. The south face of the core illustrated in Figure 3.1 was covered with three rows of 3 in. x 3 in. graphite reflector elements, except for two columns at the center, which were left open to accommodate the solid graphite (reactor grade) insert sketched in Figure 3.18. The axis of the 5 in. hole bored through the graphite was along the centerline of the reactor core. Graphite plugs were designed to accommodate 1-1/2 and 3 in. aluminum ducts. *The thermal diffusion coefficient for light water is.155 cm and for pure graphite.915 cm.

6o 1.50 Ratio of Reaction Rate of Lu176 to Reaction Rate of a (I/V) Absorber versus Neutron Temp 0~- 100~ C (Norm. to 1 at T= 0~ C) 1.40 - 1.30 I _ /._.2 I. IO 00 i I I I I I I0 20 30 40 50 60 70 80 90.100 Temperature ~ C Figure 3.17. Ratio of reaction rate of Lu176 to reaction rate of l/v absorber.

61 6" - NORMALIZERS HANDLE-' 9" CORE INTERFACE MATRIX PLUGS Figure 3.18. Graphite insert.

62 Measurements were made with the insert positioned against the south face of the reactor core. The side clearance was such as to allow no more than a 0050 ino water gap between the insert and the columns of reflector elements. Time limitations allowed only scalar fluxes at the duct mouth to be measured~ For -this reason, the duct lengths were limited to 1-1/2 ft, which is effectively a void of infinite length from the standpoint of the perturbation at the mouth of the largest duct. The ducts were evacuated and positioned 3 in, from the core-graphite interfaceo It was anticipated that water absorption might be a serious problem. To avoid the consequences of water penetration the graphite insert was coated with a thin layer of an Epoxy resin. It was verified that thlis was an effective sealer by soaking coated and uncoated samples; of graphite in water and weighing shortly after removal.o Since the insert represented a large amount of reactivity, *the reactor was brought critical with the insert in place on the south face of the core. The normalization was accomplished with four foils extending into the water from lucite brackets. Comparing the normalization results with counts from the fission chamibers revealed that normalization was not influenced by the presence of the ductso Inasmuch as the thermal neutron mean free path is long in graphite, simultaneous bare and cadmium-covered measurements were avoided. A cadmium ratio of 25 in t:he graphite permitted the use of bare foil measure

ments exclusively. If the epicadmium flux had been perturbed by the 5 ino duct to the same extent as the thermal, flux, such a procedure could lead to an error of no more than 1-1/2 percent. Figure 3o19 presents the results of unperturbed flux measurements in the solid graphite insert. The results are normalized at 7~62 cm to the distributions obtained by a two-group, one-dimensional diffusion calculation. A radial traverse is shown in Figure 3020~ The flux measurements at the duct mouth. were made at two positions 3/8 in. from the centerline, and agreed within 1 percent. The data are plotted in Figure 3o210 The solid line is the predicted scalar flux at the mouth obtained byusing the measured unperturbed flux distribution to compute the'boundary condition~ Figure 3~22 displays the axial distribution at the mouth of the 5 ino duct and compares the data with. the diffusion theory resultO

64 24 GRAPHITE H2 0 22 UNPERTURBED FLUX VERSUS DISTANCE FROM CORE IN GRAPHITE o MEASURED THERMAL REACTION RATE 1 8_ \n z MEASURED EPICADMIUM REACTION RATE (XIO) -CALCULATED BY r16\\ t 2-GP DIFFUSION o \ CODE(NORM. AT 7.62 CM.) _ 14t\ X ESTIMATED POUJ \ \ SITIONING ERROR 1o I —'- 2mm 12 ESTIMATED REAC_ 0 \o \TION RATE ERROR < 10 \ 0 8 w cr bJ w 0 a. u 4 o 2 0Io 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 DISTANCE FROM CORE (CM.) Figure 3. 19. Unperturbed thermal and epicadmium flux in graphite insert.

65 Unperturbed Flux versus Distance from Duct Axis In Graphite 2 2 0 Measured Thermal Reaction Rate A Measured Epicadmium Reaction Rate 20 18,o 16 C) * 14 E 12 o U 10 0 U 8 6 4 2 Ah AA AA -3-I0 2 3 Distance From Axis (in.) Figure 3.20. Radial flux distribution in graphite insert.

66 24 23 22 21 MEASURED THERMAL REACTION RATE 0....- PREDICTED THERMAL FLUX X 19 (3 18 17 wI 16cr 15 zo 14 13Cr 12 0L o PERTURBED THERMAL FLUX VS. VOID DIAMETER 10 9 8 6 1.0 2.0 3.0 4.0 5.0 6.0 70 VOID DIAMETER (IN.) Figure 3.21. Perturbed thermal flux at center of duct mouths in graphite.

Perturbed Thermal Flux versus Distance From Core 5" Duct in Graphite 0 Measured 22 ___Predicted. 20 18 c 16 -. 14 0 12 o u u s F 1 0 U 8 6 I2 3 4 5 6 7 8 Distance From Core (Cm) Figure 3.22. Axial distribution of thermal flux at mouth of 5-in. duct in graphite.

IV. DISCUSSION A. Unperturbed Thermal Flux Distributions The measured unperturbed thermal flux distributions, rather than the computed ones, were used to obtain the predictions presented in Chapter III. Utilization of the computed unperturbed thermal flux distributions, however, leads to no more than 3 percent disparity in the prediction of the duct mouth thermal flux at the positions studied. This is significant in that it suggests that duct perturbations can be predicted from diffusion theory flux distributions without the need for experiment. It should be pointed out, however, that the unperturbed flux shape in the immediate vicinity of the duct mouth must be accurately described by theory~ It is generally recognized that few-group diffusion theory has limitations when used to predict accurate flux distributions in reflectorso The cross section averaging procedure27 and the high degree 28 of fast flux anisotropy give rise to deviations'between theory and experiment. Theory and experiment are compared in Figures 3~8 and 319. The percentage deviation increases with distance from the core, although for the graphite reflector the situation is complicated by the existence of.interfaces between highly dissimilar media. Utilization of more than two groups in the few-group scheme does not improve the agreement No attempt has been made to include spectral. effects close to the core68

69 reflector interface, nor has the curvature of the fuel element, which introduces up to 1/4 in. of water in front of the graphite insert, been considered. Infinite medium, Maxwellian-averaged cross sections were used in the reflector region. Presumably accurate predictions for the flux distribution far into the reflector region can be attained provided that appropriate transport and multigroup techniques are utilized. A thorough examination of this problem is beyond the scope of this study. B. 1-1/2-In. Duct Experiments The agreement between the experimental data shown in Figure 3.9 and the prediction based upon the perturbed angular flux at the duct mouth is good. It should be recalled, however, that unlike the calculations the measurements include the leakage contribution from the walls of the duct. This implies that the wall leakage contribution in the region examined is either a small or a constant percentage of the mouth contribution. Measurements with the duct at 2016 cm indicate a 21 percent contribution from the walls. This number compares well with the results of 10 similar measurements reported by Pierceyo Summing contributions from the lateral surfaces based upon the unperturbed thermal flux in the medium, a wall leakage contribution of 15 percent is predicted. The disparity of 6 percent could be accounted for by an albedo* contribution *The albedo contribution is taken to mean neutrons of all energies which, having entered the duct through the mouth, make at least one scattering collision in the wall before they diffuse back into the duct and are detected as thermal neutrons.

70 of the fast neutrons passed by the cadmium. Subtraction of the wall leakage contribution permits the prediction f Bth The excellent agreement presented in Table III.4 suggests that any thermal neutron albedo contribution from the mouth at the exit of the 1-1/2 in. duct lies within the experimental error. The 1-1/2 in. duct was provided with the internal collimator sketched in Figure 3..i3 to completely eliminate duct wall and albedo considerations-. A measurement was performed with identical copper foils at the duct mouth and the collimator exit. This permitted a convenient comparison between the experimental value and the prediction for the mouth contribution to the exit thermal activation. The difference between the two was within the uncertainty in the collimator dimensions. Unfortunately no data:comparable to the.uncollimated data of Figure 3.9 were obtained. C. Thermal Flux Attenuation. Within the Empty Ducts The "duct streaming" measurements within the empty 1-1/2 in. and 5 in. ducts were motivated more out of academic interest than from any direct bearing on the main theme of this study. It was of interest to determine whether the thermal flux far from the mouth of the duct obeyed the l/r2 geometrical attenuation predicted by Simon and Clifford.9 The cadmium-backed foils were allowed to view all surfaces of the duct directly in front of them in order to include thet albedo contribution. For the 1-1/2 in. duct h/R = 60 at the farthest measured distance from the

71 mouth., and for the 5 in. duct h/R = 18o The experiment did not correspond to the configuration studied by Simon and Clifford in that they treated a point isotropic source at the duct mouth. The distributed source available experimentally at the duct mouth does not alter the results of their analysiso The presence of direct streaming sources in the duct walls, however, is estimated to gire rise to a slightly more rapid attenuation than l/r2o If we assume an. attenuation of the form 1/rn, the measurements (Figure 3516) yield exponents of n 2 2.7 for the 1-1/2 in. duct and n ~ 2.5 for the 5 ino duct. The large value of the exponent for the 1-1/2 in. duct is surprising, since the thermal albedo contribution from the mouth was estimated from other measurements (see Section IV. B) to be insignificant. In spite of the precautions taken, however, mutual shadowing between foils might have taken place. D. Radial Flux Distributions at the Duct Mouth We have assumed in the analysis that the flux is flat over the entire duct mouth. Examination of the distributions in Figure 31.11 reveals that this is not entirely correct, but that most of the recovery in the flux occurs close to tlhe walls of the ducto The skewing of the perturbed thermal flux distributions is caused by the asymmetry of the unperturbed radial distribution. In any case, the analytical technique treats only the center of the duct moutho The degree of accuracy achieved

72 in applying the analysis to the entire source plane, then, depends upon the extent of collimation utilized in a practical situation~ The average perturbed scalar flux over the entire duct mouth. is approximately 10 percent higher than the flux minimum at the center. Eo Fast Flux Perturbation In assuming a negligible fast flux perturbation at the duct mouth it was anticipated that even significant changes would affect the thermal flux i.n only a minor way. It is conceivable that the epithermal region of the spectrum (for example, ~625 ev to 5 Kev) could be treated by the analytical method applied to the thermal flux, but no attempt was made to predict duct perturbations in. the epithermal regionO Epicadmium gold activations, which primarily measure the 4.9 ev flux, indicate a 26 percent scalar flux depression at the mouth of the 5 in. duct, Epicadmium manganese activations, constitutinhg absorption in a 340 ev resonance plus a significant portion of the 1/v portion of the spectrum, indicate a 22 percent scalar flux depression. It is reasonable to assume that the duct perturbation tothe higher energy, more nearly monodirectional* components of the spectrum is smallero The integrated effect from the point of view of slowing down sources into the thermal region is difficult to *Figure 4o1 displays angular distributions for the very fast flux (~82 to 10 Mev) derived from an S8 calculation.,

73 2100 2000 190~ 1700 1600 1 50~ 150~ 1600 170~ 1800 190~ 200~ 2100 2200 Angular Distribution of Fast Neutrons 1400 1400 --— (.82 to 10 Mev) Calculated by S8 Approx. 2200 (a) 1.5 Cm. from Core i (b) Core - H2 0 Interface (c) 5 Cm. into H2 0 Reflector 2300 130~ 1300 2300 2400 1200 1200 2400 ~~~~~~~~~~2500 1~~~~~~~~~ I l1100 1100 250 2 6 O' 1000 I00~ 2600 2700 g 900 2700 2800 80~ 800 280~ 2900, 1 700 700 1 290~ 3000!) ~. 60~ 60~0 - 300~ 3100o ((b) 50 50' I 310~ 3200 - 400 400 32 00 3300 3400 3500 O 10~ 20' 30 o 300 200 10~ 3500 3400 330~ Figure 4.1. Angular distribution of fast neutrons.

74 evaluate. Figure 4.2 presents the calculated effect of assuming various group 1 (fast) flux depressions at the mouth on the group 2 (thermal) flux for the 5 in. duct. This is computed on the basis of a constant thermal return current to the duct mouth.The results suggest that a 25 percent depression in the integrated fast flux at the duct mouth only leads to a 3 percent depression in the thermal flux. Computed Thermal Flux and Fast Flux.34 - versus Distance From Core For 5" Duct at 2.16 cm in Water O5U Thermal FluxforjAth constant.32 ---- Fast Flux.3 Unperturbed Fast Flux.\ 4.81 O/ \.30 - \\ \.28.66 \ \.2 - 6a \ \ \ \ \ \.24 E. 22 \%.8\\.16 db/dZ 2 -b =.515.14 -( 5 - d/dZ 445 02(R) =.153 d C/dZ.12 42(R)=.149.388 -I 0 I 2 3 Z- Distance From Core (Cm.) Figure 4.2. Effect on predicted thermal flux of fast flux depression.

75 The analysis is inapplicable to ducts which appreciably perturb the overall fission source in the multiplying mediumo The assumption of small fission source perturbations should. of course, be realized for small ratios of the void area to the core face area, or for voids inserted sufficiently far from the reactor core (i.eo. for small reactivity effects). The cross-sectional area of the 5 in. duct in the experimental configuration is of the order of 5 percent of the total area of the reactor face normal to the duct axis~ Epicadmium flux measurements along the walls of the 5 ino duct indicate no significant depression in the fast flux. Ducts which appreciably perturb the reactor fission source may be amenable to an iteration analysis involving one and two-dimensional group-diffusion codest Fo Thermal Spectrum The 100~ 5~C hardening of -the thermal spectrum measured at the mouth of the 5 in. duct is insignificant in the present study (the effect was estimated to lower the average activation cross section of a l1/v detector by less than 2 percent). The fact that a measurable effect was observed, however, suggests that spectral hardening induced by the void may be important in experimental determinations of the neutron spectrum at the exit of a beam port. Measurements at the Ford Nuclear Reactor using a crystal spectrom~29 eter at the exit of the 6 ino beam port sketched in Figure 301 determined

76 a neutron temperature of 40~ + 5~C. The temperature of the pool water is 28~C at a reactor power level of one megawatt. It is reasonable to assume that the 3 in. of graphite and 1-1/2 in. of water between the mouth of the port and the core are sufficient to bring the core neutrons into equilibrium with the diffusing medium. Using the Brown prescription30 for the hardening effect in water: Teff = To +.876 (T ~)(41) we expect a neutron temperature in the unperturbed medium of less than 320C. The difference of 8~ + 50C between the estimated unperturbed neutron temperature and the measured value is presumably due to spectral hardening induced by the port void. G. Scalar Flux at the Duct Mouth The thermal flux measurements at the center of the 5 in. duct mouth lie approximately 7 percent below the predicted values in both media, as seen in Figures 3.11 and 3.21. Assuming for the moment that the unperturbed flux is a correct representation for the distribution at the duct walls, attenuation by the aluminum walls of the duct, which is difficult to evaluate, in addition to the fast flux depression and hardening of the thermal spectrum, which have been discussed earlier, all act to depress slightlythe measured thermal flux at the mouth. The current measurement

77 at the exit of the collimated duct inwater, however, shown in Figure 3.14, agrees well with the predicted valueo The arguments presented in Section II.B indicate that in the limit of the infinite half-space (jAth = 0) P-1 theory affords an adequate prediction for the angular current at the boundary (for A = -1), but the P-1 scalar flux prediction is 15 percent higher at the boundary than the exact quantity. Since the experimental data conform to this trend (in water jA is less than 50 percent of the th unperturbed return current to the 5 in. duct mouth), it is reasonable to ascribe the discrepancy in the scalar flux to measurable deviations from diffusion theory. Further evidence for deviations from diffusion -theory is afforded by the comparison of the measured and predicted gradients at the mouth. In water the measured gradient is 40 percent higher than the predicted value from diffusion theory, and in graphite the measurement is 70 percent higher. Examination of the Milne problem reveals that the exact solution for the gradient possesses a logarithmic infinity at the moutho A measurement of jBth was not performed in the complete graphite reflectoro However, we infer from the results of the perturbed scalar flux measurements at the duct mouths that the analysis applied to the water medium should be equally valid in graphite. Figure 4.3 is a plot of the prediction for j. in graphite, It should be pointed out that since the Bth constant cross section approximation used to obtain Eq. (2.25) is not strictly valid here, this equation is not rigorously applicable. The results of a numerical integration of Eqo (2.23), however, agree to within

1.0 JBtk versus Void Diameter.9 Prediction in Graphite.8.7 +d.6.5.4.3 1.0 2.0 3.0 4.0 5.0 6.0 Void Diameter (in.) Figure 4.I5. Predicted JBth for graphite.

79 one percent with predictions based upon the constant cross section approximation. H. Scalar Flux Distribution Along the Duct Walls An idealization of the scalar flux distribution along the duct walls has been incorporated into the framework of the analysis. Predictions within this framework have compared favorably with the experimental results. Nevertheless, it would have been more satisfying to use the realistic flux distribution in Eq. (2.41) to compute the return current. Up to this point, however, an examination of the behavior of the realistic scalar flux distribution along the duct walls has been avoided. Information concerning the realistic duct wall flux distribution was accessible experimentally. The configuration which gave rise to the largest flux perturbation at the mouth, the 5 in. duct in water, was examined. Foil strips fastened to thin lucite brackets were mounted perpendicular to the duct wall at 1, 3, 5, and 8 cm from the mouth. The relative thermal reaction rates for strips mounted on the east and west sides of the duct are plotted versus radial distance from the duct wall in Figure 4.4. The ratio of the radial component of the gradient at the wall to the scalar flux,, is recorded by each curve.* Figure 4.5 is a plot of the perturbed scalar flux distribution along *The fact that the gradients are higher on the east side of the duct originates from the asymmetry of the unperturbed radial distribution.

1 cm. from Mouth 3 cm. from Mouth 5 cm. from Mouth 8 cm. from Mouth East East East East /2.0- Thermal Reaction Rate versus Radial 2.0 // Distance From Duct Wall 1.8 / At 1, 3, 5, 8 cm. from Mouth of 5"Duct in Water 1.6 - 1.4 - d#/dr 1.2 - d dr @1 ~~~~~~~~.24 d/r_ _ _ _ _ do/dr 1.0 4, Eg d~~~~~~~~~~~~~~~~~d/dr.8 d / -.04 We st West West West.02.0 1.8 / 1.6 - 1.4 d4/dr.3 2 do =.3 1.2 -46 d//dr = -.606 1.0 4,dd4d/dr i~~~o C I I ~~~~d4,/d r Q, -.0 10.84,-o. _ 0 1.0 2.0 0 1.0 2.0 0 1.0 2.0 0 1.0 2.0 Radial Distance From Wall of 5"Duct (Cm.) Figure 4)4. Radial thermal flux behavior at four positions along 5-in, duct wall.

the 5 in.o duct wall compared with the unperturbed scalar flux distribution in the medium out to 8 cm from the mouth. As anticipated, the scalar flux is initially depressed close to the mouth, but recovers rapidly and eventually exceeds the unperturbed flux where the radial component of the gradient has become negative. Unforititnately, no data were obtained beyond 8 cmo However, assuming for the moment that the perturbed distribution is identical to the unperturbed distribution beyond 7 cm, utilization of the perturbed distribution in the calculation of JAth predicts a value for the duct mouth scalar flux which is 10 percent lower than that obtained from the unperturbed distribution4 Since we have concluded earlier that the unperturbed distribution predicts the correct Pl1 value of the duct mouth scalar flux, the enhancement of the perturbed distribution beyond 7 cm must compensate for the depression close to the mouth, An attempt to predict the realistic behavior of the thermal flux distribution along the duct walls was made by the following method. An initial estimate ofrthe quantity ~/ R was obtained for several disz crete axial positions along the duct wall using the unperturbed axial flux distribution in Eq4 (2050) (revised to include the two terms given in expression (2454) to account for the mouth contribution), the separated form of the current balance relationship4 A computer program written to evaluate the initial estimate of / R is presented in Appendix Go The initial estimate of the relative radial component of the gradient at the mouth, -I R was obtained in the usual wayo The relative gradients t~~~~~~~~~

Thermal Flux versus Distance From Duct Mouth O Unperturbed Reaction Rate XMeasured Reaction Rate Along Duct Walls -70 ~For 5" Duct 2.16 cm From Core in Water ~ 60 50 40 _ 30 20 I 2 3 4 5 6 7 8 9 10 Distance From Duct Mouth (Cm.) Figure 4.5. Axial thermal flux distribution along walls of 5-in. duct.

were used for a z varying boundary condition along the water-duct interfaces in a two-dimensional group-diffusion code.* The cylindrical geometry approximration to the actual confi.guration is sketched in Figure 4~6. The calculations were performed with two energy groups, and the fast flux was assumed unperturbed. It was intended that the resulting perturbed flux distribution be used in the unseparated current balance relationship, Eqo (2o45) (revised to include the terms in expression (2o54))X to obtain improved estimates for g |rRo The procedure forms a basis for an iteration i lz technique which was expected to converge rapidlyo The initial estimates for R however, did not permit convergence of the group-diffusion Z code for the 5 ino or 5 in. ducts. Even when the computation was forced into convergence by diminishing the absolute magnitudes of / R far Z from the mouth, the computed flux distributions were unrealistic, tending to increase with distance from the coreo The initial estimates and altered values (to achieve code convergence) of R for the 1-1/2, 3 and 5-in. ducts i.n water are recorded in Table IVoilo Comparisons with the measured gradients at the walls of the 5 in. duct recorded in Figure 404 *The TWENTY GRAM code, described in Appendix B, provides for a logarithmic-derivative or rod-regiond' boundary condition, specified by for energy group io Ci is supplied as input data for any region specified as a "rod-region r O

6".H20 I, \ I I~~~~~~d Ii' II:'i I Fuel Q, Graphite Core Plane of Symmetry 2.16 cm. Figure 4.6. Cylindrical geometry approximation to core-duct configuration.

85 TABLE IV.1 FIRST ESTIMATE OF THE RADIAL COMPONENT OF THE GRADIENTS FOR BOUNDARY CONDITION IN TWO-DIMENSIONAL CALCULATION z 1-1/2 In. Duct 3 Ino Duct 5 In'. DuctDistance Alrt ReAltered Altered from - IR 8Z/6r R 6: | //r Duct Mouth c cm l Z (cm) 1 cm -1 cm -1 (cm) cm cm cm 0.120 Unaltered.240.240.400.400 1.150 3 40 3 40 547.547 2.104.265.265.450.450 3.051.16 164.310.310 4.000.057.057.149.o 149 5 -.047 -.050 -o050 -.020 -.020 6 -.090 -.154 -.154 -.193 -.193 7 -.130 -.254 -.254 -.365 -.365 8 -.165 -.350 -.350 -.536 -.450 9 -.198 -.442 -.396 -.704 -~450 10 -.531 -o396 -.870 -o450 11 -1..033 -~450 12 -lo 195 -.450 13 -1o355 -.450 14 -1.516 -.450 15 -1.678 -.450 16 -lo 841 -.450 17 -2.006 -.450 18 -2175 -o 450

86 indicate that the estimated gradients close to the duct mouth are not unreasonable. The apparent deadlock is attributed to the large negative estimates of the gradients far from the duct mouth. Either these values or the treatment of the negative gradients by the code (as a negative absorption in the duct region) is unrealistic. I. Reactivity Predictions A far more encouraging prediction emerges from an examination of the eigenvalues computed by the two-dimensional code. The duct reactivity calculated by using the altered estimates of the duct wall gradients listed in Table IV.1 compare favorably with experiment. This is not surprising, since the fundamental eigenvalue of the diffusion equation is relatively insensitive to the flux distribution. However, inasmuch as the altered gradient estimates for the 5 in. duct provided a highly unrealistic flux distribution, an additional calculation was performed that utilized only the positive gradients (FiR was set equal to zero past 4 cm). Considering the degree of approximation to the realistic geometry, the result of this calculation was also in reasonable agreement with experiment. The reactivity results are presented in Table IV.2o

87 TABLE IV.2 DUCT REACTIVITY-MEASURED AND PREDICTED DUCTS AT 2.16 CM FROM CORE IN WATER bk/k bk/k Predicted Predicted Predicted using Duct bk/k using usint Diameter (in.) Measured, altered positive z, I.....(Zero past 4 cm), 1-1/2 -. 007 -.007 53 -.03. -.04 ~~~~~5 -.10 -. 11 -. 16

Vo CONCLUSIONS The main results of the analysis and measurements discussed in Chapters II, III, and IV may be summarized as followso l P-1 theory can be used to predict adequately the thermal. neutron current at the exit of a collimated beam port. The source plane scalar flux can be predicted using Phi theory with somewhat less accuracy. The method has been applied by computing the return current at the duct source plane in terms of the measured di.stribution of the unperturbed scalar flux in the reflector. The return current is used in. a few-group diffusion calculation in the form of a pseudo boundary condition on the scalar flux. Predicted exit thermal neutron. currents for ducts up to 5 in.. in diameter in the water reflector at the Ford Nuclear Reactor agree within + 3 percent with activation measurements. Predicted source plane scalar fluxes in water and graphite reflectors agree within 7 percent with the measurements 2. The method can be applied satisfactorily if the calculated distributiorl of the unperturbed scalar f7.L:Ux in the reflector as obtained from few-group diffusion theory is used to compute the return current. The unperturbed scalar flux distributions in the reflector deviate significantly from the few-group diffusion theory results close to boundaries and far from the reactor core. Predictions for the beam port exit currents obtained from the calcul.ated distributions, however, deviate by only 3 percent from those obtained from the measured distributions. 88

89 3. An iteration technique was proposed in an attempt to predict the realistic distributions of the scalar flux along the lateral surfaces of the ducts. Unfortunately, the initial estimates of the radial component of the gradients (obtained by separation of variables) failed to produce convergence of the two-dimensional group-diffusion code. Further work is needed to explore this difficulty~ 4. The results of a survey to optimize thermal neutron leakage flux from the Ford Nuclear Reactor clearly indicate thato a, The optimum geometry is a slab reactor with the beam port external to the core face normal to the smallest core dimension. b. An increase in power density by improved reflection is more desirable than an increase in integrated power because of a reduction in the ratio of fast to thermal flux. co D2O is the most effective reflector material for this purpose.

APPENDIX A A SURVEY FOR THE ENHANCEMENT OF THERMAL NEUTRON ILEAKAGE FLUX lo Introduction This survey is directed toward an optimization of the thermal neutron flux intensity and the neutron spectrum in a reflector region of the Ford Nuclear Reactor (FNR) for beam port applications. The FNR was briefly described in Chapter III. Although the results of the investigation pertain to a swimming pool reactor facility of the FNR type, they should apply at least qualitatively to a more general type of thermal reactor~ For purposes of efficiency, as well as compliance with economic and physical limitations at the existent reactor facility, the following constraints were established: (a) The core composition was held constant, For -this the FNR regular fuel element was adopted (partial elements and poison rods were disregarded). (b) The integrated reactor power was held constant. (c) The vertical core dimension was fixed at 24 in., the height of the "meat" portion of the FNR fuel element~ (d) The beam port was aligned with a radius of the core outside of the fuel, region~ 90

91 (e) The beam port influence on the exit current was neglected in. this phase of the studyo In keeping with these constraints, the survey considers variations in reflector materials and core-reflector configurations. The computer codes utilized for the numerical computations are described in Appendix B. Compositions and group constants are tabulated in Appendix C. Four-group diffusion theory was adopted. The four-group scheme, with breakpoints listed in Appendix C, has'been highly successful in treating small, enriched, light-water reactors. Moreover, a four-group study affords some useful information concerning the neutron spectrum. 2o Three-Dimensional Simulation Three-dimensional group-diffusion calculations would be prohibitively time-consuming for an inquiry of this nature. Two-dimensional calculations were performed whenever possible, but they were limited in usefulness, since criticality search routines were not included~ Therefore most of the computations were performed using one-dimensional simulation of the three-dimensional geometry. This was accomplished by holding two of the dimensions, y and z, constant and searching for the critical size and flux distribution in a variable third dimension, x. Leakage in the y and z directions was incorporated as a transverse buckli ng

92 The transverse buckling was obtained in the following manner. Onedimensional flux distributions inthe y and z directions were acquired for each of the four groups for constant y and z dimensions (leakage in in the x direction was accounted for by allowing the code to search for the critical transverse buckling). Then for each group the bucklings transverse to the x direction were derived from the following relationships: 1i(y)d, <(B2)>i core (A.1) / i (y)dy core and: *- v2i(z)dz <(B )>i core (A 2) 0' "(z)dz core Transverse buckling components which were used in the acquisition of the subsequent results are listed in Table A.Io* The total buckling transverse to the x-direction for the ith group is given by: <(B <(Bt)>i + <(B>i )>i (A.3) *Transverse bucklings obtained for the core were used in all regions Refinements to some of the subsequent results could be achieved by using region-dependent as well as group-dependent bucklingsO

TABLE A. I TRANSVERSE BUCKLINGS (cm-2) Cold-Clean Core. Equil. Xe, b.4% BU Core (B) T2 (B2)T2 (B3)2 (B4)T2 (BT2 B2)T (B. T (B4) T Vertical.00296.00253.00153 -.00472.00298.00253.oo56 -. oo448 3 Rows Horiz..00933. 00oo493.00o499 -.00026 | - Total.01229.00746.00652 -. 00oo498 4 Rows Horiz. oo635337.00341 -.000202.00651.00346.o00348 -.000057 Total.00931.oo00590..00oo494 -.00492.00949.00599.00504 -.00oo454 5 Rows Horiz..00oo464.00248.00252 -. 00016..00254.00255 -.000042 Total.00760.00501.0oo405 -. oo00488 00773.0007.00411 -.00452 7 Rows Horiz..00283.00152.00155 -.000117.00289.00155.00156 -.000035 Total.00579.00oo405.oo308. 00oo484.00587.00oo408.00312 -.00451 8 Rows Horiz. --- --- --- ---.00235.00126.00127 -.000032 Total --- --- --- ---.00533.00379.00283 -.o00o451

94 3. Streaming Flux The scalar flux distributions, which together with the fundamental eigenvalue constitute the principal output of the group-diffusion codes, are not the quantities of foremost interest from the point of view of the present study. Rather, we are interested in the angular flux at the source plane in the direction of the beam port exit. The angular flux is given by Eq. (2.23) in the constant cross section approximationo If the constant cross section restriction is removed: oo.s rl -(r-_ 1 / df[Zs(r-r)Wo)(r-Q9) + s(r —RQ)]e (A. 4) s(r-RQ) is defined here as the source integrated over all angles. For long ports, (h/R)> 1, it is the forward flux, close to i = -1, which is detected at the exito In the limit of a collimated port of infinite length, Eq. (A.4) can be approximated by the one-dimensional integration: XI X d Z (x-x" )dx" (X=1) _ 4; d/x [s (x-x')Oo(x-x,)+ s(x-x')]e (A.5) Then for any position x in the reflector and energy group i, we define the streaming flux:

95 J'(x)o~~ =~ -~ CZ (x xlT)dx" (dx'[Z5(x-x'V)(x-xT) + si(x-x?)]e 0 (Ao6) The streaming flux is emphasized in the subsequent presentation of results. Since beam port perturbations were neglected in this survey, the unperturbed scalar flux and source distributions were used to compute this quantity. Slowing down sources were neglected. Subroutine STREAM, written to compute Ji(x) and the partial current, ji(x), in conjunction with the FOG code, is listed together with the revised main program of FOG in Section A.7. 4. Configurational Effects BARE REACTOR EXAMINATION In addition to allowing a simple analysis, the bare reactor exhibits fast neutron leakage behavior which is similar to that of the reflected core. Examination of the behavior of fast neutron leakage should provide considerable insight into the influence of the reactor configuration on the thermal neutron. flux intensity in a reflected systemn inasmuch as the removal of fast neutrons constitutes a thermal neutron source in the reflector. The flux in a bare homogeneous reactor satisfies the Helmholtz equation:

96 72 + Bg = oo (A.7) The total leakage is given by: D(r)VY2(r)d3r Bg2 D(r)(.r)d3r, (A.8) and the integrated power may be expressed as: Power = consto tZf (r)(r)d3r (A.9) For a fixed integrated power and spatially independent properties: Total Leakage B2 (A lO) A critical bare reactor, however, must satisfy the condition: 2 k~ e k i(A. 1) keff 1 (A+ L2B) 2 The material buckling, 9 is a function only of the reactor material composition, and for a, critical. system it must be equal to the smallest eigenvalue, the geometrical buckli.ng, of Eqo (Ai?7). Thus: 2 2 Total Leakage Bg Bm B (A l2) which is a constant for a critical bare reactor of homogeneous composition operating at a fixed power level, regardless of the core geometry~ The average leakage per unit area, however, is a function of the

97 total surface area of the core. The minimum critical volume and surface area for bare reactors of three geometries are listed in Table AoIIo TABLE A.II CRITICAL VOLUME, AREA9 AND RELATIVE LEAKAGE FOR BARE REACTORS OF THREE GEOMETRIES Minimum Critical Relative Average Geometry Critical Volume Surface Area Leakage/Area Sphere 130/B3 125/B2 1.0 Finite Cylinder 148/B3 155/B2.806 Rectangular Parallelepiped 161/B3 179/B2.698 The fourth column of the table does not express the complete story, however. The leakage is strongly peaked at the center of a face of the parallelepiped, whereas it is uniform over the entire surface of the sphere. Using Gauss's theorem, the total leakage from the reactor can be expressed as: - D d3r V2 (r) - D dS nv _ _ (A.1l).vo s urf r R The flux distribution in a bare cube of linear dimension c is given by (assume the reactor is large enough so that c cd )

98 ((x,yz) =A cos cos Y cos T (A.14) c c c The average leakage per unit area is given byo D c/2 c/2 [D D 44DA dy dz (x y) Z (A.l1) -c/2 -C-c/2 c x = c/2 (CC whereas the peak leakage per unit area occurs at the center of the face, and is equal to: - D n V (r - R - D a (x,y,z DAK (A.16) z o = c/2 Thus for a cube, Peak Leakage/area 2~ 46, and Average Leakage/area Peak Leakage/area for a cube This result indicates that Average leakage/area for a sphere the bare parallelepiped possesses an advantage over the more efficient reactor geometry from the point of view of maximum leakage. Now let us estimate the effect of varying two of the dimensions of a rectangular parallelepiped, holding the third dimension constant. We designate the fixed dimension by c, and the variable dimensions by a and b. For a bare critical parallelepiped operating at constant power, the geometrical buckling is given by: Bg = +a () +() = constant = 3 A.7)

99 which is the buckling for a cube. The total leakage from one face of the reactor is expressed by: 4DA bc - c 2 b2. (A. 18) cdz dy b/ 6 (xyz) I~ a -c/2 -b/2 x a/2 Using Eq. (A.17), we obtain for the leakage out of a face normal to dimension a: 4~DAc 1 a >.707 Leakage = 1 > 707 ~ (A.l9) fCa2 1/2 c Dividing by the area of the face, the average leakage/area is given by: Average Leakage 4DA 1 (A.20) Area xIc (a/c) Table A.III is a list of the total leakage and average leakage per unit area(normalized to unity for a cube) from a face of the parallelepiped normal to dimension a.

100 TABLE A.III RELATIVE LEAKAGE FROM A FACE FOR BARE PARALLELEPIPEDS OF VARIOUS SHAPES a/c Total Leakage Leakage/Area.8 1.89 1.25 ~9 1.27 1.11 1.0 1.00 1.00 1.1.84.909 1.2 73.83 4 Beam extraction from a face which is normal to the slender core dimension is suggested. THE WATER REFLECTED PARALLELEPIPED One group of neutrons is insufficient to study the reflected parallelepiped, thus it is convenient to solve the four-group equations on the computer. The configurations examined are shown in Figure A.1. Reactor cores with 3, 4, 5, and 7 transverse elements were considered. The critical x dimensions recorded in Figure A.1 were obtained from dimension searchs using the three-dimensional simulation technique discussed earlier. The fuel (cold-clean composition) was surrounded by light water except for symmetric rows of graphite normal to the y-axis. Figures A.2-A.5-display the x-axis scalar flux and streaming flux distributions. Three of the four computed groups are presented in these and subsequent plots. Table A.IV is a list of pertinent information from Figures A.2-A.5.

101 z 3 Elements 9/ 1 4 Elements 12' 24" /- 19.3 "-./ 13" (a) (b) H2 0 Surrounding All Cores 5 Elements 15" 7 Elements 21" / 1~ e/945 (c) (d) Figure A.1. Core geometries for O20 reflected configuration studies (cold-clean fuel).

102 1.2 1.0 - - Core H20 0.9 - -- Computed Flux versus Distance from Core Midpoint _ Slab Geometry ~~~~0.8 - - 3 Elements Transverse Scalar Flux 0 -.625 ev 0.7.625 ev - 5.53 Key v.821 Mev - 10 Mev Streaming Flux 06 0 - 1.625 ev 0.6~mmm. —. 625 ev - 5.53 Kev - -.821 Mev - 10 Mev 0.5 0.4 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Distance from Core Midpoint (Cm) Figure A.2. Computed flux distributions; H20 reflected core; 3 elements transverse.

103 Core H O 1.0 0.9b Geometry 0.8 0.1.625 F ACo mputed l Fluxdsr uin H relc t or 4eElements transverse. I N O -.625 ey \ A _ 0.5 BW I.625 ev - 5.53 Kev "821 Mev 10 Me \ | 0. I'.625 ev.821 Mev 1 1 1,,, I Mev. 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Distance from Core Midpoint (Cm) 4 elements transverse.

104 1.2 Core H20 0.8 0.7 0.C 0.5 0.4 Computed Flux versus 0.3 Distance from Core Midpoint \ Slab Geometry I 5 Elements Transverse Scalar Flux \ 0.2 0 -.625 ev - - ~ ~.625 ev - 5.53 Kev.821 Mev - 10 Me O. I Streaming Flux inin O0 -.625 ev I_ -i.,..625 e v - 5.53 Kev ---—.821 Mev- 10 Mev 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Distance from Core Midpoint (Cm) Figure A.4. Computed flux distributions; H20 reflected core; 5 elements transverse.

105 1.2 Core H20 1.0 -- 0.9 0.8 \ 06 I I V; 17 Elements Transverse it~~~~L \: O -.625 eV.625 ev;- 5;53 Ke| Computed Flux versus l I\ - \.625 ev - 5.53 Key -4 - < - - - _ - --.821 Mev - 10 Me 0.5 \ Streaming Flux - \.- __[.1 lMev-1 OMev. 0.4 0.3 0.2 0.1 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Distance from Core Midpoint (Cm) Figure A.5. Computed flux distributions; H20 reflected core; 7 elements transverse.

1o06 TABLE A. IV COLD-CLEAf BH20 REFLECTED PARALLELEPIPED CORES OF VARIOUS SHAPES 1/v ISm 1 (s4_s3) (S4max S )m/ (S3 x) (SI/Simax) in 3 ax max Slmx.. in. -- 3 Elemo Trans~ old-Clean 448 38 256 202 7 00024 Core 4 Elemo Trans",.811 ~647 0478,2`0 434.00027 Cold-Clean4 27 Core 5 Elem Trans. Cora1s l.o6 0794 o r99,o 180 o 3 & 00025 Cold-Clean 3 Core 7 Elem. Trans~ 1.21:: 208.432.00021 Cold-Clean995 2 Core 7 Elem. Trans~ Eq. Xe-.965 770 96.215 9422.ooo16 Core The quantities chosen.here and in t:hle u:bsequent studies to be of major interest are~ (i) the maximum. value of the thermal streaming flux.j; max (ii) tthe maximum. difference between the thermal. and epithermal and the thermal and fast streaming fl'uxes, (W )max and (i - )max; (iii) the ratio of epithermal. to thermal and fast to thermal. streaming fluxes at the location of t:he maximum thermal streaming flux (1 3 4ma and (/1 /Imax); (iv) the inrerse volume of the core, 1/V, a measure of the average power density of the reactor~ Arn additional listing in, Table AoIV

107 compares the cold-clean, seven-element transverse core with a core containing equilibrium Xe and 844 percent fuel burnupo Table AoIV strongly suggests beam extraction adjacent to a face which is normal to a slender dimension of the core~ Quantitative justification is presented for deforming the core to the optimum configuration (maximum power density), and going past this point if possible. LOCALIZED CORE ALTERATIONS It was of interest to examine the effects of an uneven fuel distributiono Figure Ao6 presents the results of a calculation comparing the leakage flux of a core with clean fuel in the center and 8.4 percent fuel burnup on the outside with that of a core endowed with the converse loading. In the latter core the clean and burned-up fuels were loaded in approximately even proportions, while in the former core the clean fuel was moved to the central zone while enough burned-up fuel was loaded on the circumference to maintain criticality4 The calculation was accomplished in cylindrical geometry, and the burned-up composition is that listed in Table CoI (minus the xenon). The slightly reduced size of the configuration with the clean fuel in the center gives rise to higher thermal streaming flux in the reflector; however, the increase is only 5 percent. Localized fuel additions in the vicinity of a beam port would merely act to depress the thermal flux while increasing the fast fluxo The effect of small water gaps at the center of the core was explored, but

o08' ~Clean Fuel 8.4 % B. U. Fuel H20 *O 10 Computed Flux versus Distance from Core Center Cylindrical Geometry u....005'Clean Fuel in Center Scalor Flux 2 416220-.625e 1 1 1 1.625 ev - 5.53 Ke C' l \\ -.821 Mev - 10 Mev Streaming Flux v C-le 0 -.625 e v _ -. _.62 5 ev - 5.53 Kev -.821 Mev - 10 Mev 2 4 6 8 10 1 2 14 16 18 20 22 24 26 28 30 32 34 i...8.4 % B.. U, Fuel Clean Fuel (Cm i \.0100 Computed Flux versus Distance from Core Center x 005L I Cylindrical Geometry \ 005 Clean Fuel on Circumference Scalar Flux 0..625 ev.625 ev - 5.53 Ke.821 Mey - 10 Mey Streaming Flux.-.625 ey - 5.53 Kev. —.-..821 Mey - 10 Me'y 2 4 6 8 10 12 14 16 18 20- 22 24 26 28 30 32 34 Radial Distance (Cm) Figure A.6. Computed flux distributions; comparison of depleted fuel in center and on circumference of core.

109 yielded no net improvement. (A 3 in. gap, while giving rise to a 250 percent local flux peak, would increase the fuel loading by 12 percent, and decrease the reflector thermal flux by 14 percent.) 5. The Effect of Reflector Materials NEARLY INFINITE REFLECTORS IN CYLI[TDRICAL GEOMETRY The initial examination of reflector materials was carried out in cylindrical geometry for a cold-clean core composition. The core was surrounded by a thickness of at least one migration length of each material to achieve the effect of a nearly infinite reflector. Group independent transverse Bucklings were used for simplicity3* The results of the computations are plotted in Figures Ao7-A.12. Results for H20, graphite, BeO, D20, and combinations of graphite and H20 are presented. BeO results are presented because the calculations indicated that this material is at least as effective as pure Be. Table A.V, composed of significant information from Figures Ao7-Ao.l2 points to the near linear dependence of the thermal streaming flux on the average reactor power density (tabulated as the inverse core volume). BeO, clearly the best reflectors increases the thermal streaming flux by a factor of 3,2 while raising the average power density by a factor of B =.00165 cm'2 was adopted here for the core and BS =.00125 cmr2 for the reflectoro These numbers were obtained from epicadmium flux traverses in the FNR17:

110.020 Computed Flux versus Distance from Core Center Cylindrical Geometry I Infinite H20 Reflector Scalar Flux 0 -.625 ev.625 ev - 5.53 Kev.821 Mev - 10 Mev.015 Streaming Flux iR0 - s.625e v -- -' —.625 ev - 5.53 Kev - - — _.821 Mev- 10 Mev H20 Core 2 4 6 8 I 0 12 14 16 18 20 22 24 26 28 30 32 34 Radial Distance (Cm) Figure A.7. Computed flux distributions; infinite H20 reflector; cylindrical geometry.

111 Core Graphite.020 Computed Flux versus Distance from Core Center Cylindrical Geometry.005 60 cm Graphite Reflector Scalar Flux 0 -.625 ey v.625 e v - 5.53 Key.-. —-- -.821 Mev - 10 Mev " Streaming Flux._ __...625ev - 5.53 Key ----' —.821 Mev - 10 Mev 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32'34 Radial Distance (Cm) Figure A.8. Computed flux distributions; infinite graphite reflector; cylindrical geometry.

112 \ Core BeO.020- -\\\.010 - - Cylindrical Geometry\.005 40 cm BeO Reflector _ Scalar Flux 0 -.625\ e 4 -.625 ev - 5.53 Ke \.821 Mev - 10 Me\ Streaming Flux.625 ev - 5.53 Key.0 -0 40.821 Mev - 10 Mev 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Radial Distance (Cm) Figure A.9. Computed flux distributions; infinite BeO reflector; cylindrical geometry.

113 Core D20.020.015 /1 X I,2 waft.010o! I Computed Flux versus Distance from Core Center \ I Cylindrical Geometry I.005 190 cm D20 Reflector Scalar Flux 0 -.625 ev.625 ev - 5.53 Key.821 Mev - 10 Mev Streaming Flux ininm 0 -.625 ey._ m -_.625 ev - 5.53 Kev — ~ -—.821 Mev - 10 Mev 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Radial Distance (Cm) Figure A.O10. Computed flux distributions; infinite D20 reflector; cylindrical geometry.

114 Core Graphite H20.020.0 15.010- \ \ Computed Flux versus O -.625 e\.625 ev - 5.53 Ke Y inderi.821 Meyv 10 Mev o Streaming Flux.625 ev - 5.53 Key - - -..821 Mev - 10 Mev 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Figure A Computed flux distribtions; 7. cm graphite

"5.020 Core H20 Graphite1.0 1 5 I I I I I I 1 11/-,,O.1.1 I' Computed Flux versus Dista ce from Core Center: 2.5 cm H20 Remainder Graphite \.005 I Cylindrical Geometry Scalar Flux 0 -.625 ev.625 eRv - 5.53 Kev ( -.821 Mey - 10 Me \ Streaming Flux m ~~ 0 -.625 ev \ -. ---,..625 ev - 5.53 Kev -- -.821 Mey - 10 Me v 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 Radial Distance (Cm) Figure A.12. Computed flux distributions; 2.5 cm H20remainder graphite; cylindrical geometry.

116 TABLE A. V INFINITE REFLECTOR —CYLINDRICAL GEOMETRY VARIOUS REFLECTOR MATERIALS SM| (S4-S3) (S4_S1) S I3 q(S l/S41ax) I (si/SmYax) ino.s Ho20 Rfo.0119.0095 o006i. 218. 546.00025 Refl. Graph. Reflo o 0151 0071 o og8 o 642 6o 69 0 o 00oo46 BeO R 0379 0258.0309. 368. 258. ooo83 Reflo _ D20 Refio o 0255.0191 o 0224 o279 o 183.00050 7~5 cm Graphder0118.0088.0070 305.4 32 o00034 Remainder] H20 2.5 cm H20 Remainde o 0161 o0108 oo0068.329.609.00033 D20 in extending the thermal streaming flux peak farther from the core than in H20 is a reason for the improved spectral. ratifoK~/4 This fact presents a strong motivation for increasing the average reactor power density by improved reflection rather than integrated power boost. Infinite reflection by D20 increases the average power density by a factor of 2 over the H20 reflected core, whereas (J ) goes up by a factor of 35.7 Infinite graphite reflection shows up poorly in this set of calculationso Graphite used in conjunction with water demonstrates some im

117 provement. The effect of 1 in. of water located between the core and the graphite is conspicuous. This configuration gives rise to significant gainsins in 4 and (f - )max max max REFLECTOR INSERTS The effect of reflector inserts of graphite, BeO, and D20 iwas examined for the equilibrium-.,Xei. 8-4 percent, fuel burnup composition. The geometry is sketched in Figure A.13. The fundamental core dimensions shown in Figure A.13a were established from a one-dimensional simulation study. Calculations were performed in XY geometry, and the vertical transverse bucklings recorded in Table A.I were used to account for leakage in the third dimension. The insert widths were chosen to enable the inserts to accommodate a six inch port. The reactivity contributions,* 6k/k, are recorded in Table A.VI along with the other significant quantities. Figures Aol5-A.18 present the scalar flux and streaming flux distributions along the x-axes of the inserts. Figure A.o19 and the bottom row of Table A.VI present the results of a calculation of the core displayed in Figure A.14, possessing full 6 in. layers of BeO, The insert study permits a comparison of the effects of H20, graphite, D20, and BeO independent of power density influences (a somewhat artificial situation). Light water excels from the standpoint of suppression *The core dimensions were held constant, therefore all of the inserts, possessing better reflector properties than H20, constituted positive reactivity perturbations. Enhancement of the thermal flux arising from an increase in power density, then, does not appear here.

118 I —' 19.2" - 12E g ox 3 — rI (a) H20 Surrounding all Cores 73.94 X BeO ~r~t~xruSO L~n"N ~ Be.~ - 7.08"- -o x (b) 5.50" D20 i' ~a~F~&S~jt~S~f~S~f~ ~D 20 7.08" — x Graphite Graphite (c) Figure A.13. Core geometries for material insert studies.

119 3/, x CORE: BeO-H2 12" -BeO CORE BeO GRAPHITE Figure A.14. Core geometry for 6 in. BeO reflector.

TABLE A. VI TWO-DIMENSIONAL CALCULATIONS;FIXED SIZED CORE-INSERTS OF VARIOUS MATERIALS 1/V Smaxa ( (4_S3) max (S4-S) maX (S3/Sjax) (S /Slax) 6k/k in -3 Full 0.0316.0234.0173.266.506 ---.00018 H20 Graph..0288.0225.0211.271.347 +1.92%.00018 Insert BeO l Insert.0288.0205.0212.389.375 +2.3 %.00018 Insert D420 Insert.0361.0279.0289. 266.266 +1.86%.00018 6 in..o980.o649 50759 I.330.212 ---.00041 BeO J J

.05 Core H2 0.04.03 x~~~~~~~~~~~~~~~~~~~~ Computed Flux versus k.~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~02.02 Distance from Core Midpoint Two - Dimensional Calculation Full H20 Reflector Scalar Flux 0-.625 ev -.625 ev - 5.53 Key _ _.O I.821 Mev - 10 Mev Streaming Flux ~ -.625 e6 v -- " —.625 ev - 5.53 Key - -—.821 Mev - 10 Me, 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Distance from Core Midpoint (Cm) Figure A.15, Two-dimensional computed flux distributions; H20 reflected.

Core Graphite H 20.04.030.02 Computed Flux versus __ Distance from Core Midpoint Two Dimensional Calculation 14 cm x 18 cm Graphite Insert Scalar Flux 0 -.625 eyv.O I L.625 ev - 5.53 Kev.821 Mev Y-10 Mev Streaming Flux I "" 0 - 0. 625 e v._ -- ~1,.625 ~v - 5.53 Kev --—' —.821 Mev - 10 Mev 2 4 6 8 I 0 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Distance from Core Midpoint (Cm) Figure A.16. Two-dimensional computed flux distributions; graphite insert.

.05.......Core BeO H20.04.03.02 Computed Flux versus Two-Dimensional Calculation I t L \ IO cm x 18 cm BeO Insert Scalar Flux.01 1 *.625 eY - 5.53 Kev Streaming Flux 0 -.625 ev.625 ev - 5.53 Key ~821 Mevy- 10 Mev 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Distance from Core Midpoint ( Cm ) Figure A.17. Two-dimensional computed flux distributions; BeO insert.

.05 Core D20 H20.04.01 Computed Flux versus 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 40 Distance from Core Midpoint (Cm) Figure A.18. Two-dimensional computed flux distributions; D20 insert. Figure A.18. Two-dimensional computed flux distributions; D20 insert.

.12 Core BeO H20. 1 I _ _ _ _ _ Computed Flux versus J1O Distance from Core Midpoint Two-Dimensional Calculation.09 _ ___ 6" BeO Reflection Scalar Flux.08 0 -.625ev - II 1.625 ev - 5.53 Key.821 Mev - 10 Mev.077 Streaming Flux U-I i I I I I I ~' - i.5.625 e3.06 Y - —.~~~~~~~~~~~~~~~~~~~~~~~~~~~625 ev - 5.53 Key ---, —.821 Mev - 10 Mev.05.04.03.02.0 I 2 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 34 36 38 Distance from Core Midpoint (Cm) Figure A.19. Two-dimensional computed flux distributions; 6 in. BeG reflector.

126 of the epithermal content of the spectrum, exemplified by comparison of the quantities (2>/1 ) and _J ) for all four materials. max max Every reflector insert diminishes the ratio, (S1/S 4ax)O As before, we attribute this effect to the longer diffusion lengths possessed by the insert materials, extending the high thermal flux region farther into the reflector. The D20 insert results are particularly conspicuous. In addition to a 65 percent increase in the quantity ( - 4 )max, the D20 insert brings about a 14 percent enhancement in the thermal flux. REFLECTOR LAYERS IN PARALLELEPIPED GEOMETRY The final study constitutes a comparison of the streaming flux for 3 in. and 6 in. layers of reflector materials completely covering two faces of a rectangular parallelepiped reactor core. Figure A.20 is an illustration of the geometry. The fundamental water reflected core is sketched in Figure A.20ao Equilibrium Xe, 8.4 percent fuel burnup was chosen for the core composition. Once again, H20, graphite, BeO, and D20 were examined. The configurations were simulated with one-dimensional computations. The results are tabulated in Table A.VII. Together with the quantities previously tabulated, an additional dimension, d, is listed, which is the distance from the core to the reflector location appropriate to the quantity alongside it o Figures A.22 and Ao23 present the quantities (J4 - )max and ( 4 1)max plotted versus

127 the dimension a.* Dimension a is an inverse measure of the relative average power density. 19.2 I x,/- X (a) 12/ C C0RE 3, 3" (b) a A 6", H20 Surrounds all cores. (c) Figure A.20. Core geometries for 3 in. and 6 in. reflectors of various materials. *Additional points for 4-1/2 in. of the reflector materials were computed to obtain the shapes of the curves.

128 The consequence of shrinking dimension a by extension of the transverse dimension, illustrated in Figure A.21, is compared in Table A.VII and Figures A.22 and A.23 with the results of reflection by 3 in. and 6 in. layers of reflector materials. Although this calculation was presented in Section A.4 for the cold-clean core composition, it is of interest to compare this configurational effect with the effect of reflection for the fuel composition presently under consideration. 4N' I Cu4,{a)(c) (d) Figure A.21. Core geometries for H20 reflected configuration studies (equilibrium Xe-8.4 percent burnup fuel). The results appearing in Table A. VII are too numerous to discuss in detail. The geometry considered in this portion of the study has practical application for ports located 180~ apart. Graphite makes a far better showing here than in the cylindrical geometry study. The maximum thermal streaming flux, (S ), in this configuration increases more rapidly than the average power density with improved reflection. The results for reflection by 6 in. of D20 are striking.* Figures A.22 *Due to the calculational simplification of region-independent bucklings (thermal, in particular), the results for the 6 in. reflectors may be highly unrealistic.

129 TABLE A.VII THREE AND SIX INCH REFLECTORS OF VARIOUS MATERIALS S4 d s3/S4 (S4-S3) d (4 1) d a/V max S max i.max max ( m m -3 in. in. in. in. in. 3 in. Graph. Refl..614 3.254.324.44.4o8 7 13.5.000223 4 Elem. Trans. n in. Graph. Refl. 1.28 4.3.186.174 1.08 5.6 1.09.6 lo.9.000317 4 Elem. Trans. 3 in. BeO Refl..773 2.1.333.307.582.0.6oo0 3.0 12.9.000268 4 Elem. Trans. 6 in. BeO Refl. 1.65 3.1.250.140 1.29 3.9 1.6.9 8..ooo4o6 4 Elem. Trans. 5 in. Do20 Refl..853 3.254.232.636 3.66 14.4.00024 4 Elem. Trans. 6 in. D20 Refl. 3.4 4.3.144.074 2.99 4.9 3.22 4.9 5.45 ooo635 4 Elem. Trans. H20 Refi..485.9.216.464.385 1.1.282 1.3 19.2.00018 4 Elem. Trans. H20 Refl. 5 Ele.714.9.216.444.569 1.1.428 1.3 15.0.00018 5 Elem. Trans. H20 Refl. Ele..965.9.216.422.770 1.1.596 1.3 12.2.0001ooo63 7 Elem. Trans. H20 Refl. 8 Ele. 1.039.9.216.416.830 1.1.648 1.3 11.6.00015 ElemTrans. Trans.

(54_53 Max versus Core Width 3.0 For Graphite, BeO, D2 0, H2 0 EquiI.Xe-8.4 % B. U. Core Slab Geometry 4 Elements Transverse GRAPHITE 2.0O 2-0 ~~~~~~~~~~~~~~~~~~~~~~~~~BeO 0 H2 RefI. @ 3" Refl. * 6" Refl. 0 4 1/2"RefI. 0 1.0 ZI~~~~hI 10 20 30 40 50 WIDTH OF CORE (Cm.) S4 3 Figure A.22. (SS )max for 5- and 6-in. reflection.

(S4 -S )Max versus Core Width 3.0 For Graphite, BeO, D2 0, H 2 0 Equil. Xe - 8.4 % B. U. Core Slab Geometry 4 Elements Transverse 2.H 2.0 - - ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~Graphite BeO 0 H20 Refl. M 3" RefI. H * 6" Refl.. 4 1/2"Refl. 1.0 10 20 30 40 50 WIDTH OF CORE (Cm.) Figure A.23. (S'KS1)max for 3- and 6-in, reflection.

132 and A.23 compare the reflectors on the basis of constant average power density (excluding the curves for H20 reflection obtained by altering the ratio of the core dimensions). On this basis, D20 excels, followed in order by graphite and BeO. 6. Final Remarks Although the group constants used in the calculations were obtained in a consistent manner, a question may be raised regarding the validity of the cross sectionso The source of the cross section library is cited in Appendix C. Much of the analysis reported in the literature3 is based upon this set of cross sections or slight variations of ito The reasonable agreement between the predicted and measured distributions of the thermal and epicadmium fluxes presented in Figures 3.6, 3o8, and 3519 tends to corroborate the validity of the group constants for H20, graphite, and the coreo In order to examine the validity of the D20 group constants, a measurement was made with a limited quantity of D20, and the experimental results were compared with a group-diffusion calculation. A flux traverse was performed in a cylindrical D20 insert 5 in. long and 5-1/2 in. in diameter placed against the center of the south face of the FNR.32 The two-dimensional analysis utilized 4 groups in cylindrical geometry. A comparison of the measured and calculated thermal and epit~hermal f lux distributions along the axis of the insert is presented in Figure A2 24)

8.0 Flux in D2 0 Insert versus Axial Distance From Core X Measured Relative Thermal Activation in D2 0 7.0 OMeasured Relative Epicadmium Activation in D2 0 i \ | Computed Flux Distributions in D2 0 Insert /|~~~ ~ ~~~~~~~~~ \ | ~~~Thermal 6.0 x X \|Epithermal \r0 | X a Computed Flux Distributions in H2 2. - _ _ _Thermal 5.0 2'\ D20 I H20 4.0 3.0 2.0 0 1.00 0 2 4 6 8 10 12 14 16 18 20 22 Distance From Core'"m.) Figure A.24. Measured and computed flux distributions along axis of cylindrical D20 insert.

134 Both the measured and calculated distributions were normalized to distributions obtained along the same axis in the complete light water medium (also plotted in Figure A.24)o Many of the results reported here were based upon the technique described in Section Ao2, simulating three-dimensional geometry using one-dimensional computations. An examination of the validity of this method is afforded by comparing the results of actual two-dimensional calculations with the results of one-dimensional simulations. Since four-group, two-dimensional computer studies are extremely time-consuming, few comparisons were made. The first comparison was carried out for the core illustrated in Figure A.14, possessing symmetric 6 in. layers of BeOo The two-dimensional calculation resulted in an eigenvalue (See Eqo B.1).6 percent lower than that obtained in the onedimensional simulation. The water reflected core of Figure Ao13a was examined next, the two-dimensional calculation yielding an eigenvalue 3 percent lower than the one-dimensional simulationo Finally, the deviation for the cold-clean, water reflected core shown in Figure Aolb was 2 percent. None of the discrepancies are serious, considering the disagreement between the results of group-diffusion calculations and precise criticality determinations for clean, homogeneous assemblies reported by Feinero33 It is curious, however, that the smaller cores yield better agreemento It was anticipated that the deviation between the one -dimensional simulation results and the two-dimensional results would be inversely proportional to the core size, owing to corner effectso

135 Apparently a reflector composition influence exists, as wello A question may be raised concerning the accuracy of the calculated fast flux magnitudes in the reflector, i1 and _2o P-1 theory tends to overestimate the attenuation of the forward-peaked, high energy components of the spectrum. Experimental fast flux determinations are scarceo Since the reflector regions of greatest interest lie within 1 to 3 fast neutron mean free paths from the core, P-1 errors in the fast flux are presumed to be small. Most of the core alterations examined in this appendix involved increases in the reactor power density~ Practical considerations call for a remark concerning core heat removal capabilitieso A cursory examination of this problem was carried out with an existent computer 34 program, originally written to evaluate the FNR two megawatt performanceo The results of the calculation, assuming a constant coolant mass flow rate through the core and an integrated reactor power of one megawatt, indicate that in going from 24 to 4 regular fuel elements a maximum fuel cladding temperature rise of only 7~F is anticipated~ 7o Subroutine STREAM Subroutine STREAM, used in conjunction with the FOG group-diffusion code (control program 3), computes the partial current, defined by. J (x) 4 a, A2dx1)

136 and the streaming flux, defined by Eq. (Ao6), at each mesh point in a designated region. The streaming flux integral is performed over the designated region plus the region before it. Thus for reasonable accuracy the width, w, of the region immediately before the designated region must be defined sufficiently large so that w >> 1 for each energy group. The Fortran program listing of the revised main program of FOG and subroutine STREAM are recorded hereo In order to call the subroutine L63 of the input data must be set different from zero. Then a master card and one card for each energy group is expected at the end of the FOG input deck. The card arrangement is as follows: Master Card (Format 4I12) N(1) = 0 Bypass Subroutine = 1 Utilize Subroutine N(2) - 0 Do not compute j -1 Compute jo N(3) = 0 Do not compute M = 1 Compute / N(4) = I I is the FOG region designated for the computation. Cross Section Card* (Format 2F12.0) F(l) = ES Scattering cross section in region I F(2) = 7,s Scattering cross section in region I-1 *One card is required for each energy group.

C FOG CODE MAIN PROGRAM - WRITTEN AT NORTH AMERICAN AVIATION C REVISED AT UNIVERSITY OF MICHIGAN, 1962 $COMPILE FORTRANPRINT OBJECTgPUNCH OBJECT.DUMP*EXECUTE MAIN3000 C CONTROL PROGRAM NO.3. THIS PROGRAM CONTAINS THE CRITICALITY SEARCHFOG23080 C ES AND ADJOINT FLUX CALCULATIONS. FOG23090 C FOG2 3100 DIMENSION LM(40) LB(40) NPO(3) LBS(40) MR(40) L(250),A(5000),PHI (FOG23110 1239,4),APHI(239,4),SOU(239)SOP(40),SOUP(239)gBSA(40,4),SOPP(40) FOG23120 2,R(239),RI(239),NN(5),C(5),D(239),DP(40)tG(239),GP(40)G1l(239),G1PFOG23130 3(40),BETA(239),DELTA(239),DELR(40),T(4),DIF(40,4),TRANS(40, FOG23140 44), SIGT(40,4),CIH(4),OMEGA(4),OMEG1(4),FLP(239),BUCK(40),BUCK2(4FOG23150 50,4),VUSIG(4094),SIGPT(40)tFL(239),SIGA(40,4),ACl(40),AC2(40),AC3(FOG23160 640,4),AC4(40,4),AC5(4094),BUCK3(4),AC6(40,4),AC7(40,4), AC8(40O4),FOG23170 7AC9(40),AC10(40),CHI(40,4)-,BL4B2( 4),83(4).B4(4),B5(4),RW(40) FOG23180 DIMENSION NFU(40),SOUS(239),SOPS(40),SIP(4) FOG23190 COMMON L,A,APHI,D,G,G1,AC3,AC4,AC5,AC6,AC7,AC8,DPNFU,C,NN FOG23200 EQUIVALENCE (L(249),N)t(L(248),LOB) FOG23210 EQUIVALENCE (L(1),NOG) (L(2),N1) (L(3),N2),(L(4),N3) (L(5)- N4) (L(FOG23220 16),N5),(L(7),-M),(L(8),LM)(L(48),N6),(L(49),N7),(L(50),N8),(L(51),FOG23230 2N9),(L(52),NlO),(L(53),Nll),(L(54),Nl2)9(L(55),N13),(L(56)N14)(LFOG23240 3(57),N15),(L(58),N16),(L(59),N17),(L(60),NPO),(L(63)tN18).(L(64),NFOG23250 419),(L(65),N20),(L(66)9N21),(L(67),LBS),(L(107),N22)*(L(108),N23),FOG23260 5(L(i09),MA),(L(110).,MR),(L(150)tMICT),(L(151),MICT2),(L(152).LB) FOG23270 EQUIVALENCE (A(4956),LB1),(A(4957),LB2),(A(4958),LBT),(A(4959),LB3FOG23280 1),(A(4960),SOPS),(A(3340),SIP),(A(3344),LB5),(A(3345),GAM1). FOG23290 EQUIVALENCE (A(1),ESP1),(A(2),ESP2),(A(3.),E. IGEN2),(A(4),ESP3),(A(5FOG23300 1),THET),(A(6),FMUL),(A(8),RW)t(A(48),C1),(A(49),C2),(A(50),BUCK,BUFOG23310 2CK1,'BUCK3),(A(368)9C3),(A(369),C4),(A(395),CIH),(A(400),BUCK2), FOG23320 3(A(560),TRANS,SOP),(A(720),AC1),(A(760),AC2),(A(800).VUSIG),(A(960FOG23330 4),DELR),(A(1000),SIGA),(A(1160),AC9),(A(1200),DIF),(A(1360),GlP),(FOG23340 5A(1560),AC1O),(A(1400),CHI),(A(1600),OMEGA).(A(1604).OMEG1),(A(160FOG23350 68), T),(A(16'12),SIGPT),(A(1652),PG), 9(A(1653),PG2),(A(1654),PG3),(AFOG23360 7(1655),PG4),(A(165&),N100),(A(1657),EIGEN),(A(1658),EIGEN1) FOG23370 EQUIVALENCE (A(1659),P),(A(1660),GP),(A(1700)SOU),(A(1939),A1),( FOG23380 1A(1940),SOPP),(A(198Q),LT1),(A(1981),LT2),(A(1982),B1),(A(1983),BLFOG23390 2),(A(1987) B2),(A(1991) B3),(A(1995) B4) (A(1999),W)(A(2085),B5)FOG23400 3(A(2009),W1),(A(2004),W2),(A(2014),PSI1),(A(2019),PSI2),(A(2024),DFOG23410 4ET),(A(2025),PIM),(A(2026),ERR),(A(2027).SUM),(A(2028),GAM),(A(202FOG23420 59),NM1)(A(2034),NM2),(A(2038),MM1),(A(2039),NAA),(A(2044),NSKP), FOG23430 6(A(2049),ICT),(A(2054),ICT1),(A( 2055),ICT2),(A(2056.),ICT3),(A(205FOG23440 77),DELRT),('A(2058),BUCKT),(A(2059),C10),(A(2064),SGPT) FOG23450 EQUIVALENCE (A(2600),SOUP),(A(2850),R),(A(3100),RI),(A(3350)tFL),(FOG23460 1A(3600),FLP),(A(3840),SIGT),(A(4000),PHI),(APHISOUS.BSA),(GBETA)FOG23470 2,(G1,DELTA),(A(2068),CON),(A(2069),CON1),(A(2074),FAC) FOG23480 CALL ZERO MICT=35 FOG23490 MICT2=10 FOG23500 5 READ INPUT TAPE 7,6 FOG23510 6 FORMAT (72H FOG23520 1 FOG23530 WRITE OUTPUT TAPE 6,6 FOG23540 10 CALL DATIN FOG23550 20 CALL OMCAL FOG23560 IF (N100-1) 30,595 FOG23570 30 NSKP=O FOG23580 40 CALL DAPR FOG23590 IF (N22-1) 50,50,110 FOG23600 50 CALL CONS FOG23610 60 CALL FLUX FOG23620 IF (N20) 70,90,65 FOG23630 65 IF (N20-3) 70,70,80 FOG23640 70 CALL CRIT1 FOG23650 GO TO 90 FOG23660 80 CALL CRIT2 FOG23670 90 CALL ANPR FOG23680 100 IF (N22) 120,120,110 FOG23690 110 CALL AFLUX FOG23700 120 IF (N18) 140,1409130 FOG23710 130 CALL CHECK FOG23720 CALL REPR FOG23730 CALL STREAM FOG23685 140 IF (L(250)) 5,5,150 FOG23740 150 CONTINUE FOG23750 END FOG23760

138 C SUBROUTINE STREAM WRITTEN AT U OF M TO BE USED IN CONJUNCTION WITH FOG S COMPILE FORTRAN9PRINT OBJECT.PUNCH OBJECT SUBSTREAM SUBROUTINE STREAM 1 DIMENSION LM(4Q),LB(40)tNPO(3),LBS(40).MR(40),L(250),A(5000).PHI( 1 1239,4),APHI(239.4)tSOU2 39) P(40)0UP(239)*BSA(404)SOPP(40) 2 2.R(239),RI(239).NN(5)C(5).D(239).DP(40)tG(239)tGP(40).Gl(239)GIP 3 3(40).BETA(239),DELTA(239)DELR(40),T(4)DIF(40,4)tTRANS(40. 4 44), SIGT(404).CIH(4).OMEGA(4).OMEG1(4).FLP(239).BUCK(40).BUCK2(4 5 50.4).VUSIG(40,4),SIGPT(40).FL(239).SIGA(40.4)AC1(40).AC2(40).AC3( 6 640,4).AC4(40,4).AC5( 40,4),BUCK3(4).AC6(40.4),AC7(40.4). AC8(40.4). 7 7AC9(40).AC10(40),CHI(40,4),BL(4).82(4) B3(4).84(4)B85(4).RW(40) 8 DIMENSION CURR(239).NNN(4) AA(2).SUMM(239.4) DIMENSION NFU(40),SOUS(239).SOPS(40),SIP(4) 10 COMMON LA.APHID,G,GG1AC3,AC4,A C5.AC6,AC79AC8tDPNFU.CNN 11 EQUIVALENCE (L(249),N)*(L(248),LOB) 12 EQUIVALENCE (L(1),NOG),(L(2),N1),(L(3),N2),{L(4),N3) (L(5),N4)*(L( 13 16) NS5).(L(7).M),(L(8),LM) (L(48)9N6).(L(49).N7).(L(50).N8)t(L(51)# 14 2N9),(L(52),NlO), (L(53), Nll),(L(54),N12),(L(55),N13) (L(56),N14),(L 15 3(57) N15),(L(58),N16).(L(59).N17),(L(60) NPO).(L(63) N18),(L(64),N 16 419)*(L(65).N20).(L(66).N21)t(L(67?).LBS).tL(107) N22).(L(108)N23). 17 5(L(109),MA),(L(11O).MR). (L(150).MICT),(L(151),MICT2).(L(152).LB) 18 EQUIVALENCE (A(4956),LB1),(A(4957),LB2),(A(4958),LBT),(A(4959),LB3 19 1).(A(4960),SOPS).(A(3340).SIP).(A(3344),LB5).(A(3345),GAMl) 20 EQUIVALENCE (A(1),ESP1),(A(2).ESP2).(A(3),EIGEN2).(A(4),ESP3) (A(5 21 1).THET).(A(6).FMUL).(A(8).RW).(A(48). Cl1)(A(49).C2).(A(50),BUCKBU 22 2CK1.BUCK3). (A(368)tC3).(A(369).C4) (A(395)*CIH)*(A(400)BUCK2), 23 3(A(560),TRANSSOP)9(A(720).AC1)*(A(760).AC2)*(A(800)*VUSIG).(A(960 24.4),DELR) (A(1000) SIGA) (A(1160) AC9)t(A(1200)tDIF)*(A(1360),G1P).( 25 5A(1560),AC10),(A(1400),CHI),(A(1600),OMEGA),(A(1604)*OMEG1),(A(160 26 68).T).(A(1612),SIGPT),(A(1652).PG1).(A(1653))PG2),(A(1654),PG3),(A 27 7(1655) PG4).(A(1656).N100),(A(1656)A1657)EIGEN)*(A(1658),EIGEN1) 28 EQUIVALENCE (A(1659),P),(A(166Q),GP),(A(1700)*SOU).(A(1939),Al).( 29 1A(1940)SOPP A198981)LT2)(A(1982)B)(A(1983)BL 30 2),(A(1987).B2),(A(1 91),B3),(A(1995),B4),(A(1999).W),(A(2085),B5)t 31 3(A(2QO9).W1).(A(2004).W2).(A(2014).PSI1)*(A(2019),PSI2),(A(2024).D 32 4ET),(A(2025).PIM),(A(2026),ERR),(A(2027).5UM).(A(2028).GAM),(A(202 33 59).NMlI,(A(2034).NM2).(A(2038).MM1),(A(2039).NAA),(A(2044).NSKP), 34 6(A(2049),ICT)9(A(2054),ICT1),(A( 2055),ICT2),(A(2056),ICT3)9(A(205 35 77),DELRT),(A(2058)sBUCKT),(A(2059),C10O)(A(2064),SGPT) 36 EQUIVALENCE (A(2600),SOUP),(A(2850),R),(A(3100),RI),(A(3350),FL),( 37 1A(3600),FLP),(A(3840),SIGT),(A(4000),PHI)9,APHISOUSBSA),(GBETA) 38 2,(G1,DELTA)9(A(2068),CON)9(A(2069),CON1)*(A(2074).FAC) 39 READ INPUT TAPE 7,59,(NNN(I),I=14) 101 59 FORMAT (4112) 201 N24=NNN(1) 301 N25=NNN(2) 401 N26=NNN(3) 501 N27=NNN(4) 601 IF (N24) 19,19.1 2 1 IF(N25) 9,9,30 30 DO 31 J=1,NOG NF=1 4 LOL=1 5 DO 5 I-1,M 6 NF=NF+LM(I) 7 DO 2 K=LOLNF 8 IF (I-M) 6,7.7 9 6 CURR(K) = (PHI(K.J)/4.0)-((DIF(I,J)/2.O)*(PHI(K+1,J)-PHI(KJ 1))/DELR(I)) 11 GO TO 2 12 7 CURR(K) = (PHI(KJ)/4.O) - ((DIF(I*J)/201) * (PHI(K.J)-PHI(K-1.J 1))/DELR(I)) 14 2 CONTINUE 15 IF (I-1) 3,3,4 16 3 LOL=LOL + LM(I) + 1 17 GO TO 5 18 4 LOL z LOL+LM(I) 19 5 CONTINUE. 20 WRITE OUTPUT TAPE 6,60,J 60 FORMAT(35H RADIUS PLUS CURRENT GROUPI3) 23 WRITE OUTPUT TAPE 6,61,(R(L),CURR(L),L=1,NF) 24 61 FORMAT (2E16.8) 25 31 CONTINUE

139 9 IF (N26) 19,19,10 26 10 I=N27 WRITE OUTPUT TAPE 6,629I 62 FORMAT (55H1 STREAMING FLUX IN REGION 113) DO 18 J=1,NOG READ'INPUT TAPE 7964,(AA(K),K=1,2) 64 FORMAT(2F12*0) 226 SIGMS1=AA(1) 326 SIGMS2=AA(2) 426 WRITE OUTPUT TAPE 6,659SIGMS1.SIGMS2,J 65 FORMAT(23H SIGMS IN REGION I-1 ISE16,8, 626 121H SIGMS IN REGION I ISE16.8.8H GROUPI3) LOTT = O 0 28 I2=I-2 128 IF(I2)28.28,27 27 DO 20 N=11I2 29 LOTTT = LM(N) 30 LOTT = LOTTT + LOTT 31 20 CONTINUE 32 28 LOT = LOTT + 1 3 33 I3=LM(f) 133 DO 17 K=1,I3 34 SUMM(K#J)= 0.0 14=LOT+LM(I-1) 135 DO 13 L=LOT,I4 36 FL2=K 236 FL3=LM(I-1)+LOT-L 336 IF(((SIGA(I-i 1J)+SIGMS1)*DELR(I-1)*FL3+(SIGA(I,J)+SIGMS2)* 436 1DELR(I).*FL2)-500)23,13,13 536 23 IF(L-14)25,11911 63625 IF (L-LOT) 11,11,12 37 11 STR=(((SIGMS1*PHI(LJ)+SOUP(L)*CHI(I-1,J))/2.O)*DELR(I-1))*EXP( 1-((SIGA(I-1,J) + SIGMS1) * DELR(I-1) * FL3 39 2) - ((SIGA(IJ)+SlGMS2 ) * DELR(I) * FL2)) 40 GO TO 21 41 12 STR = ((SIGMS1*PHI(L,J)+SOUP(L)*CHI(I-1,J))*DELR(I-1))*EXP(-(( ISIGA(I-1,J-) +'IGMS1 )*DELR(I-1)*FL3)-((SIGA(19J)+SIGMS2 2 )*DELR(I)*FL2)) 44 21 SUMM(K.9J) = SUMM(KJ) + STR 13 CONTINUE 46 N=LOT+LM(I-1) 47 N1=N+K 147 DO 16 L=NN1 48 FL4=N+K-L 148 IF(((SIGA(IJ)+SIGMS2)*DELR(I)*FL4)-50.O)24916916 248 24 IF(L-N)15,15,26 348 26 IF (L-N-K) 14915,15 49 14 STR=((SIGMS2*PHI(LJ)+SOUP(L)*CHI( IJ))*DELR(I))*EXP(-(SIGA(IJ)+S lIGMS2 )*DELR(I)*FL4) 51 GO TO 22 52 15 STR = ((SIGMS2*PHI(LJ)+SOUP(L)*CHI(IJ))/2.0) * DELR(I) 1*EXP(-(SIGA(IJ)+SIGMS2)*DELR(I)*FL4) 153 22 SUMM(KJ) = SUMM(KJ) + STR 16 CONTINUE 55 17 CONTINUE 56 18 CONTINUE 61 IF (NOG-3) 35,35,32 32 WRITE OUTPUT TAPE 6,33 33 FORMAT(69HO PT. IN REGION GROUP 1 GROUP 2 GROUP 3 1 GROUP 4) WRITE OUTPUT TAPE 6,34,(K,SUMM(K,1),SUMM(Kt2),SUMM(K,3),SUMM(K,4), 1K=1,13) 34 FORMAT(I7,E22.8,3E14.8) GO TO 19 35 IF (NOG-2) 39,39,36 36 WRITE OUTPUT TAPE 6,37 37 FORMAT(55HO PT. IN REGION GROUP 1 GROUP 2 GROUP 3) WRITE OUTPUT TAPE 6,38,(KSUMM(K1),SUMM(K,2),SUMM(K,3),K=1,13) 38 FORMAT (I7,E22.8,2E14*8) GO TO 19 39 WRITE OUTPUT TAPE 6.40 40 FORMAT(41HO PT. IN REGION GROUP 1 GROUP 2) WRITE OUTPUT TAPE 6,41,(K,SUMM(K,1),SUMM(K,2),K:II3) 41 FORMAT(I7tE22.8,EE14.8) 19 RETURN 62 END 63

APPENDIX B DESCRIPTION OF THE CODES Computer calculations were performed on the IBM-709, and later on the IBM-7090. Some revision of the Fortran language codes were necessary for compatibility with The University of Michigan executive system. Program revisions involved tape number changes, function name revisions, memory zeroing, and occasional function deletions. In addition, since the executive system monitor occupies several thousand locations of the 32 K memory, it was often essential to shorten or break down programs written for the scope of the entire memory. Group-diffusion codes solve the following set of coupled equations: DiV2/i + [Di B i i i+ Z.Ei ~= xi (V4)- + E (B.1) Finite-difference methods along with a variety of convergence accelerators are employed in obtaining solutions. Flexibility is generally built into a code in the form of an assortment of boundary conditions and specific geometrical configurations~ FOG is a one-dimensional few-group diffusion code providing for as many as four energy groups and 239 space points. The code permits a choice of three geometries, nine sets of boundary conditions, and five T40

141 types of criticality searches. Downscattering between adjacent groups only is allowed. TWENTY GRAND37 is a two-dimensional group-diffusion code based upon the Equipoise convergence technique. Three thousand mesh points are accommodated, and the code was revised to handle as many as four energy groups (the original form accommodated six energy groups). Both XY and RZ geometries are available and neutron transfer between any of the groups is permitted. Zero flux, zero derivative, and logarithmic-derivative boundary conditions are utilizable. 38 Thermal neutron group constants were computed with TEMPEST, the Fortran version of the SOFOCATE39 code. Cross sections, supplied in the form of a library deck, can be averaged over a thermal spectrum determined from (i) the Wigner-Wilkins equation for light moderators, (ii) the Wilkins equation for heavy moderators, or (iii) a Maxwellian distribution. FORM,40 the Fortran version of MUFT-IV,41 a Fourier transform slowing-down code, was used to obtain fast group constants. Fast spectra are generated from the spatially Fourier-transformed sl owing-down distribution by the P-1 or B-1 approximation, with or without the SelengutGoertzel approximation. Cross section data are supplied on tape in the form of a 54-group cross section library from o625 ev to 10 Meva The memory size limitations of The University of Michigan system required alteration of the main program, a listing of which appears at the end of this appendix~

142 C FORM (FORTRAN MUFT) PROGRAM - WRITTEN AT NORTH AMERICAN AVIATION C REVISED UNIVERSITY OF MICHIGAN, 1962' $COMPILE FORTRAN, PRINT OBJECT, PUNCH OBJECT MAIN1000 DIMENSION FILE3.(7776) PROB(325,18) D3(644)g D1(599) D2(1641), 0 1DUMMY(5215) COMMON FILE 3,D 1,PROB,D2 NSD, IP, ANED.D3,NADD,DUMMY NFIRST CALL ZERO REWIND 3 80 NFIRST=1 90 CALL DELSET 100 CALL SEQPGM END C SUBROUTINE DELSET $ASSEMBL E PUNCH-OBJECT ZERO 0000 ENTRY ZERO ZERO LXA *+4,1 8STZ 777'77 1 TIX -* s -1 + 1 TRA 1,4 OCT 526-61 END $COMPILE FORTRAN, PRINT OBJECT, PUNCH OBJECT MAIN2000 DIMENSION FILE3(7.776),PROB(325.,18),D3(644), D1(599),D2(1641), 0 1DUMMY(5215) C QMON F I.LE3D1 PROB,D2 NSDI P A,NED,D3,NADD,DUMMYNFIRS T 1NTIMENTESTgIPtNSD.NDNSNINULNALPHANINNRA 1 CALL INPUT 110 CALL INPi 120 IF (NFIRST-1 )22,4 130 2 NTEST=O 140 NTIME 1 150 CALL READLB(IPNSDNTIMENTEST) 160 IF(NTEST)12,3,12 170 12 CALL SELPGM(l) 3 iF(NADD)13913,11 180 11 CALL ADD 190 13 WRITE TAPE 3,LFILE3(I),I1=,7776)-. 200 CALL SELPGM(4) 4 READ TAPE 3t-F-ILE3(I),I=1,7776[ — 220 CALL SEQPGM END C SUBROUT I NE INPUT C SUBROUTINE INP1 C -SUBROUTINE READLB C SUBROUTINE ADD $COMPILE FORTRAN, PRINT OBJECT. PUNCH OBJECT MAIN3000 DI-MENSION F ILE3(7776 ) PROB(325,18),D3 (644), D( 599 ) D2(1641 ) 0

143 1DUMMY( 5215) COMMON FILE3,D1,PROBD2,NSD, IP,ANEDD3,NADDDUMMY.NFIRS.T, 1NTIMENTEST CALL GRUCON (NF IRST) CALL SEQPGM END C SUBROUTINE GRUCON $COMPILE FORTRAN, PRINT OBJECT, PUNCH OBJECT MAIN4000 DIMENSION FILE3(7776),PROB(325,18),D3(644), Dl(599),D2(1641), O 1DUMMY( 5215 ) COMMON FILE3,DI,PROB8D29NSD, IP,ANED,D3,NADDDUMMYtNFIRST, 1NTIMENTEST, IPNSD*NSNI NUL.NALPHANINtNRA IF(NFIRST-1 )6,6,7 240 6 NTIME=2 250 CALL READLB lPNSD,N TIME -NTEST)' 260 IF(NADD/10) 15915,14 270 14 CALL ADD 280 15 WRITE TAPE 3,PROB 290 7 CALL SEQPGM END C -SUBROUTINE READLB C SUBROUTINE ADD $COMPILE FORTRAN9 PRINT OBJECT, PUNCH OBJECT MAIN5000 DIMENSION FILE3(7776),PROB(325,18), D3(644)t Dl(599),D2(1641), 0 1DUMMY(5215) COMMON FILE3,D1 PROB D2*NSD, IPA,NEDD3,NADDDUMMYg NFIRST CALL SLODON(NFIRST) 300 REWIND 3 310 CALL EDIT12 320 IF(NED-2) 10,108 330 8 CALL EDIT3 340 IF(NED-3) 1091099 350 9 CALL EDIT4 360 10 NFIRST=NFIRST+I 370 CALL SELPGM(3) 12 CALL SELPGM(1) 06000390 END 400 C SUBROUTINE SLODON C SUBROUTINE EDIT12 C SUBROUTINE EDIT3 C SUBROUTINE EDIT4

APPENDIX C COMPOSITIONS AND GROUP CONSTANTS Table CoI is a listing of all compositions considered for this study. Two classes of core compositions were examined~ The first is the cold-clean initial fuel loading of 140 gm/element. The second is appropriate to a core with equilibrium xenon and 8f4% fuel burnup, representative of the FER operating at one megawatt during the fall of 1962 Core compositions are presented for a homogenized regular fuel element with the water between subassemblies averaged into the composition. The homogenization of the subassembly is valid since the disadvantage factor of the fuel is o988-22 - The existence of partial elements and poison rods was neglectedo Thermal group constants were evaluated by averaging cross section data over spectra computed by the TEMPEST code from 0-~625 evo The TEMPEST library of cross sections, most recently revised in February, 1961, was used throughout for consistency~ A temperature of 3000K and a material buckling of o0130 cm-2 were used as input data. Core constants were evaluated using the Wigner-Wilkins light moderator approximationo Figure Col shows the Wigner-Wilkins spectrum computed for the cold-clean core composition compared to a Maxwellian spectrum at 3000K. Both spectra were normalized in the following manner: ~625 ev S /(E)dE = o. (Cdl) 144

145 Reflector constants were averaged over a Maxwellian spectrum at 300OK (the buckling was taken to be zero). Table C.II lists the thermal group constants for the compositions in Table C.I. Fast group constants, evaluated by the FORM code, are entered in Table C.III. The FORM library tape, updated to May, 1962, was obtained directly from the North American Aviation Corporation, and was used as the source of all cross section data. The four-group scheme had the following breakpoints: o625 ev-5.53 Kev; 5o53 Kev-.821 Mev; o821 Mev10 Mev. Constants were evaluated for a full U235 fission source spectrum using the Bl approximation and the self-consistent age approximation. A buckling of.0130 cm-2 was used to obtain core constants and.00001 cm-2 for the reflector. Resonance self-shielding factors (Lfactors) of unity were used for both U235 and U238o

146 100 Thermal Spectrum O(E) versus E As Calculated by Tempest - Cold Clean FNR Core Wigner-Wilkins 300~K Maxwellian 3000 K 10 0.1 10 -3 10-2 10-1 Energy (ev) Figure C.1. Wigner-Wilkins and Maxwellian thermal spectra for cold-clean FNR core at 300~K.

TABLE C. I COMPOSITIONS Reflector Cold-Clean Equil. Xe Refector Core 8.4% BU Core Homogenized Graph. H20 D20 BeO Be Graph. Volume Nuc. 1024 Volume Nuc. x 24 Volume Nuc. 1024 Nuclei x 1024 Fract ion cm3 Fract ion cm3 Fract ion cm3 cm3 H20.5845 ---. 5845 ---.0694 H ---.0392 ---.0392 ---.00464 ---.067 --- 0 ---.0196 ---.0196 ---.00232 ---.0335.0331.0728 A1.4133.0249.4133.0249.0774.00oo466 -- - --- U235.00197 9.46 x 10-5.00181 8.67 x 105 --- --- --- --- --- --- u238 2.26 x 10-4 1.08 x 10-5 2.26 x 10-4 1.08 x 105 --- B10'* _ ---.20 x.27x10-6 --- --- Xel35 --- --- __.85 x 10-9.. C --- --- ---.84.68.80 --- --- --- D20 - -- --- --- --- _ _ - _.0331 --- EBe to ac n -— c for n X' fission.0725.1236 *Equivalents to account for non-Xel35 fission product poisoning.

TABLE C. II THERMAL GROUP CONSTANTS Reflector Cold-Clean Equil. Xe Homognize Homogenized Core 8.4% BU Core Graph. 1H20 D20* Be BeO < Za > cm1.0629.0622.00256.000263. 0195.000029.00108.000651 < D > cm.2993.2989.693.915.1546.831.4155.4354 < VZf > cm-1.0996.0915 0.0 0.0 0.0 0.0 0.0 0.0 < Ztr > cm- 1.377 1.378.490.364 2.485 ---.802.7655 < Zs > cm 1.313 1.313 ---.384 3o6.375.741.720 *Not available in TEMPEST library; obtained by averaging BNL-325 cross section data over a Maxwellian spectrum.

149 TABLE C. III FAST GROUP CONSTANTS Reflector Cold-.Clean Equil. Xe* Homogenized Core BU Core Graph. H20 D20 Be BeO Graph..... e < Za >1 cm 1.o000898.000885.000056.oooo000003.00138.oo00144.00453.00449 < Za >2.000285.000267.000018 0.0.000013 0.0.000002.000001 < Ea >3.00oo485.oo454.000124 0.0.ooo00954 o0.0o. 000034. 1ooool < a >.0019.00173.000073 0.0.000787.000148.ooo000729.00ooo549 < ZR >1 cm-1.07429.07445.0321.0275.1087.0835.0419.0462 < ZR >2.08509.o86.0199.0113.1494.0372.0275.0230 < ZR >3.08252.0804.0161.00657.1520.0196.0169.0134 < ZR >.0254.0236.00734.00372.0o495.0115.00885.00746 < VZf >1 cm-'.000306.000281 0.0 0.0 0.0 0.0 0.0 0.0 < Vif >2.000380.000346 0.0 0.0 0.0 0.0 0.0 0.0 < vwf >3.00593.00539 0.0 0.0 0.0 0.0 0.0 0.0 < vkf >.00207.00180 0.0 0.0 0.0 0.0 0.0 0.0 < D >1 cm 2.017 1.920 2.369. 2.410 2.048 2.175 1.908 1.442 < D >2 1.156 1.144 1.085 1.049 1.057 1.230.581.549 < D >3.905.907.950.934.595 1.242.486.502 < D > 1.365 1.341 1.245 1.122 1.245 1.334.7364.6287 < Zs >1 cm-1.201.201 ---.163.261.178.209.285 < ZS >2.597.597 ---.342.761.248.636.641 < Zs >3.910.910 ---.377 1.385.240.736.695 Subscripts: 1-.821 Mev to 10 Mev 2-5.53 Kev to.821 Mev 3-.621 ev to 5.53 Kev Non-subscripted constant denotes.621 ev to 10 Mev. *Xel35 cross sections were not available on the library tape, thus it was neglected here.

APPENDIX D COMPARISON OF CURRENT BOUNDARY CONDITION AND PSEUDO BOUNDARY CONDITION AT VACUUM INTERFACE We desire to compare solutions to the two-group diffusion equations utilizing (i) the realistic boundary condition on the par/o H tial current and (ii) the pseudo boundary condition on the scalar flux. We examine a system which is similar to the one studied in the text but deviates sufficiently to O x allow an analytic solution. Consider a slab of width a, with fast neutrons entering the medium at x = 0. We write the two-group diffusion equations: 2, di 2 - 1/ 0 (D.1) dx d2 2 D2 - ZaO2 + ZRi 0 (D.2) The solution for the flux in group 1 is: Ae + Be ~l(x) = Ae-lx + Be' (D.3) and for group 2: 2(x) = Ce2x + De 2x + Sll(x); (D.4) 150

151 where: 2 ZRD 1 = (D.5) D1 K 2 =__(D.6) D2 and: S Z = R 22 (D.7) D2 ~C2 21 Adopting the approach of Chapter II, we assume that the fast flux decays as it would in the infinite medium unperturbed by the void at a, allowing us to set B = O. Then the thermal flux is given by: A2(X) = Ce -2X + DeK2X + SiAe-Klx, (DO8) where the constant A is determined by the source strength, qo. Another constant is eliminated by a boundary condition on the thermal flux at x = O. For the sake of argument assume that: () = 2(0) tr2 d/2 =0 (Do9) j2(o) = 6 ax which specifies the thermal flux within one arbitrary constant, C;giving: 1 + 2tr2K2 Kz2x 2(X) = C e-K2X - e (l - 3 trz2) f + S1A - i1 K2 o (D.10)

152 Applying the realistic boundary condition at x = a: jz(a)(a) (a) dx za jA (D.ll) the constant becomes: 2 - a + 32 ~tr2K ( 2 KZa 4jA-SA - 3 tr2 e a - (1-2/3 tr2) + ~tr2 2 e Ci 3=/3 Itr2 3 2 ( - tr23 ee K2a 2 (+ trK2 2 e(2a (l-3 2tr22) (D.12) Applying the pseudo boundary condition: B(a+d) = 4jA' (D.13) the constant becomes: 42rl(a+d 4jASA-] e- l-a d 3tr2rL eK2(a+d) ii-Ae -(~(1 2/3 ~tr2K2) e-K2(a+d) _ (1 + tr2K2 eK2(a+d) (D.14) dt nc e i n tr2K2 he Inasmuch as we have used the uncorrected diffusion theory extrapolation distance in the analysis: ~ (a+d) + K+ 2 a + e-<~ = e 37 e e tr3 tr (D.15) Then for ~tra << 1: e+~(a+d) _ ea e~ 5a~ trK) (D.16)

1553 It follows that:* Ci - Cii (Do17) *For light watei 2/3 ~tr2z2 = 0095 and 2/3 ~tr2a1 = ~O52~

APPENDIX E DUCT MOUTH CONTRIBUTION TO THE- RADIAL CURRENT AT THE WALL Consider the contribution to the partial current in the radial direction at (R,zl) from the mouth of a cylindrical void. The number of neutrons per second passing through dS1 emanating from the mouth is given by: j (Rz1)dS1 = SdSz-(~(r~0n)dO o (Eol) SA The angular flux at the mouth, neglecting source contributions, is expressed by: p(r,O, Q) 1 "O7(rn)-12 Vo r (Eo2) Assuming polar symmetry: Vo lr = l (Es3) Referring to Figure Eol: dSA = rdrdp, (Eo4) dS1 = 2Rdgdz, (Eo5) and: Sh2p 1a. (E.6) 154

i~! Z155 Figure E.1. Duct configuration for calculation of mouth.contribution to wall radial current. From geometrical considerations: n =r cos 29 + y sin 29, (E7) p = R sin 2, (E.8) and' h = r + R cos 2. (E.9) Accordingly: dS1 ~ dS1 nd = S-n-) S (R+r cos 2, (E.7) 4r2 ~

156 and: 2 2 2 h2 2 R2 2 = zl2+p + = z + 2rR cos 2 + r (Ell) Using Eqs. (E.3) and (E.6), the angular flux takes on the form: 0(r,0,Q) & o(r, ) — - z (E12) and the current is obtained by substituting the above expressions into Eq. (E.1): j (Rzl) = 2dRdzdr d d rdr 0 1j!(E13) _ X 0L1 O h aL1]zi 2Rdz (R+r cos 2O) 6 zr 6az rr tr 0 0 Assume that -'i is negligible in comparison with 0o(r,O) and ar r _ |, which are taken to be constant over the entire mouth. Then Eq. az r (E.o3) reduces to: R T/2 + (R,zl) = (O) rdr dO (Rtr Qos 2) j (Rz): o(r 0 [zl+R2+r2+2rR cos 20]2 0 0 (Eo14) -- r rr d0 [z1+R2+r2+2rR cos 205 o2 - Or equivalently: R R + (R,zl) = Q(0) rd rdri1(r,R,zl)-zl rdrI2(rRz ) (Eo 5)

157 The integration to obtain I1 is easily performed, giving: 2 2 -r2 Il(r,R,zl) = R (zl+R -r) ) (E.16) 2 a( )/+R +r2 -4r The second integration is considerably more difficult, but finally reduces to: I2 4 -(R+rR2zl) 2 K( 2) I2(Tr,R,zj) =96(rR)5/2(l )2 (_q2 )R+r) cI (R+3r) i +2 - 2(R+r)q4-2(2R+r)q+2r] E(2)} (E.17) where: q~ 4rR 2 2 )2, (E.18) and K and E are the complete elliptic integrals of the first and second kinds.15 The first integral with respect to r, giving the isotropic contribution to the current at (R,zl), can be done analytically, yielding: R i z1 4R2 r rdrIl(rRnzl) - 4zR 2 + *-. The second integral, representing the contribution of the z component of the gradient to the current, must be done numerically. Combining terms, we write the final form for the contribution to the partial current at (R,z1) from the mouth: 2 1+ f 4R2 2' Z1 + Rz1 1~R Z ___ ~ o rdrI2(rR,zl)

APPENDIX F NORMALIZATION FOR TIME VARYING FLUX Consider the normalization of an experiment conducted in a time varying flux, /(t). For purposes of discussion assume that the appropriate integrations over energy have been performed. The time rate of change of the concentration of a species i is expressed as: dCi i dC1 Ri(t) - Xci (F.1) dt where R(t) is the time dependent reaction rate: Ri(t) = Zi~(t). (F.2) The activity at time t, the termination of an irradiation, is obtained by solving Eq. (F.1), and is given by: t xi(t -t) Ai(t) = Ai dt'Ri(t' )ek (t'-t) = ii dt')(tt)e ~~.0 0(F.3) We define the time averaged value of the flux: t <tf >(t')dt' (F.4) Setting the time dependent flux equal to the product of a steady state flux magnitude and a function of time: /(t) = Of(t), (F.5) 158

159 Eq. (F.3) becomes: A (t) = A fdt'f(t')e (t-t) (F.6) We desire to obtain the average value of the flux measured by a detector of species a normalized to the average flux measured at another position and at the same time by a detector of species b. Then: 1 ~a ~ f(t')dt' < / >a <$a=l (F.7) a 1 $b f(t')dt' t and substituting from Eq. (F.6): A (t)\a~a dtlf(t')e < ________pa_______. (~F.8) N tb Ab (t ) Xb> dtlf~t~(t,_t) 0o Since we are interested only in relative normalized quantities: t~?t' < N Ab(t) be ( a b)t (F.9) < a A a (t) e~ Xb) dt'f(t' )e (F9 For the trivial case of a and b identical Eq. (F.9) reduces to: a Aa(t) <'N Ab(t) (F.)

APPENDIX G COMPUTER PROGRAM FOR EVALUATION OF R a / 1r The Fortran program for the computation of R along the duct wall is listed here. The calculation assumes separation of variables in the current balance relationship and evaluates Eq. (2.50) (revised to include the terms in Expression (2X54) to account for the duct mouth contribution) at discrete points along the duct wall for an unperturbed axial flux distribution of a designated shape. Exponential, cosine, linear, and flat flux shapes are permissible. The input data are of the following form (Format I2, 8F8.4): N(1) = 1 Linear flux shape = 2 Flat flux shape = 3 Cosine flux shape = 4 Exponential flux shape F() F(2) F(53)F() 01 Parameters for flux shape F(4) F(5) = zo Position of duct mouth (cm) F(6) = R Duct Radius F(7) = ~ Neutron mean free path in medium F(8) = 5 Interval of computation 160

161 C WALL INTERFACE SEPARABLE CALCULATION - WRITTEN U OF M t 1962 $COMPILE FORTRAN,PRINT OBJECT.PUNCH OBJECTEXECUTEDUMP BOUND C COMPUTATION OF SEPARABLE BOUNDARY CONDITION FOR TUBE IN REFLECTOR 1 DIMENSION TEG(3)9 SUM(3)9 GET(3),TEG1(3) 2 C DEFINE FUNCTIONS USED IN PROGRAM 3 PHI1F(ZZ) = AA+A*ZZ 4 PHI3F(ZZ) = AA*COS(A*ZZ+B) 5 PHI4F(ZZ) = AA*FXP(A*ZZ) + BB*EXP(B*ZZ) 6 DPHI3F(ZZ) = -AA*A*SIN(A*ZZ+B) 7 DPHI4F(ZZ) = AA*A*EXP(A*ZZ) + BB*B*EXP(B*ZZ) 8 PAINF(ZZ) = 1*O/SQRT(1O+(Z2-ZZ)*(Z2-ZZ)/(4O0*R*R)) 9 QUEERF(Y) = SQRT(4.0*R*Y/((Z2-ZO)*(Z2-ZO)+(R+Y)*(R+Y))) 10 C READ AND PRINT INPUT DATA 11 C NOTE. INDIC FOR FLUX. 1lLINEAR. 2'FLAT, 3'COS, 4'EXP 12 1 READ INPUT TAPE 7.2tINDICAAABBBtZOR.AMBDAtDECRE 13 2 FORMAT(I2,8F8.4) 14 WRITE OUTPUT TAPE 6,3 15 3 FORMAT(66H1COMPUTATION OF SEPARABLE BOUNDARY CONDITION FOR TUBE IN 114 1 REFLECTOR) 115 WRITE OUTPUT TAPE 694eAAgABBPB 16 4 FORMAT(26HOPARAMETERS FOR FLUX SHAPE/4HOAA=F8B.44H A=F8B4,4H BB=F 17 18*494H B=F8*4) 18 WRITE OUTPUT TAPE 695*.Z 19 5 FORMAT(23HOPOSITION OF ZO IN CM,=F8,4) 20 WRITE OUTPUT TAPE 6,6,R 21 6 FORMAT(23HORADIUS OF TUBE IN CM*=F8~4) 22 WRITE OUTPUT TA.PE 6.79 AMBDA, DECRE 23 7 FORMAT(32HOTRANSPORT M.F6Po LAMBDA IN CM.=F8*4/ 24 127HOINCREMENT FOR INTEGRATION=F8.4) 25 C SET Z2,COMPUTE PHIZ2, PHIZO, DPHIZOt'PRINT FLUX SHAPE 26 Z2=ZO+DECRE 27 8 IF(INDIC-3) 9,1011O 28 9 IF(INDIC-1) 12,12t13 29 10 PH-IZ2=PHI3F(Z2) 30 PHIZO=PHI3F(Z0) 31 DPHIZO=DPHI3F(ZO) 32 IF(Z2-(ZO+DECRE)) 14,14,24 33 11 PHIZ2=PHI4F(Z2) 34 PHIZO=PHI4F(ZO) 35 DPHIZO=DPHI4F(ZO) 36 IF(Z2-(ZO+DECRE)) 15,15,24 37 12 PHIZ2=PHI1F(Z2) 38 PHIZO=PHILF(ZO) 39 DPHIZO=A 40 IF (Z2-(ZO+DECRE)) 16,16924 41 13 PHIZ2=1.0 42 PHIZO=1.O 43 DPHIZO=OO 44 IF(Z2-(ZO+DECRE)) 17,17.24 45 14 WRITE OUTPUT TAPE 6918 46 GO TO 22 47 15 WRITE OUTPUT TAPE 6,19 48 GO TO 22 49 16 WRITE OUTPUT TAPE 6.20 50 GO TO 22 51 17 WRITE OUTPUT TAPE 6921 52 18 FORMAT(24HOFLUX IS COSINE IN SHAPE) 53 19 FORMAT(29HOFLUX IS EXPONENTIAL IN SHAPE) 54 20 FORMAT(24HOFLUX IS LINEAR IN SHAPE) 55 21 FORMAT(22HOFLUX IS FLAT IN SHAPE) 56 22 WRITE OUTPUT TAPE 6923 57 23 FORMAT(72HO Z2 DENOM Fl F2 F3 F4 L.D 58 IPHI/PHI DPHI/PHI) 59 C BEGIN COMPUTATION OF DENOMFlF2 60 24 Zl=ZO 61 J=3 62 DO 25 I 1,3 63 TE'G( I) =OO 64 25 CONTINUE 65 GO TO 27 66 26 Z1=Z1+DECRE 67 27 DO 28 I=1.3 68 GET( I)=TEG( I) 69 28 CONTINUE 70

162 29 IF(INDIC-3) 30,31,32 71 30 IF(INDIC-1) 33,33,34 72 31 PHIZ1=PHI3F(Z1) 73 DPHIZ1=DPHI3F(Z1) 74 GO TO 35 75 32 PHIZ1=PHI4F(Z1) 76 DPHIZ1=DPHI4F(Z1) 77 GO TO 35 78 33 PHIZI=PHI1F(Z1) 79 DPHIZ1=A 80 GO TO 35 81 34 PHIZ1=1iO 82 J=2 83 35 X=PAINF(Z1) 84 CALL IEFl(15707XEFFG) 85 IF(G-1.0) 102,102a100 86 100 WRITE OUTPUT TAPE 6,101,Z1 87 101 FORMAT(47H ELLIPTIC INT. OUTSIDE OF RANGE *Zl=F8*4) 88 E=l1O 188 F=1-0.O 288 102 TEG(1) = PHIZ1*((2O0*(1O-X*X*X*X)+3.0*(2O0-X*X))*E-(l.o-X*X) 89 1*(8.0+X*X)*F)/(PHIZ2*X) 90 TEG(2) = PHIZ1*(2.0-(2.O+X*X)*SQRT(1*0-X*X))/PHIZ2 91 IF(INDIC-2) 36,37,36 92 36 TEG(3) = (DPHIZ1*(Z2-Z1)*((2e0+X*X)*F-2.0*(11*+X*X)*E))/ 93 1(PHIZ2*SQRT((Z2-Z1)*(Z2-Z1)+4*O*R*R)) 94 37 IF(Z1-ZO) 38938,40 95 38 DO 39 1=1,3 96 TEG1(I) = TEG(I) 97 SUM(I) = 0*50*TEG(I) 98 39 CONTINUE 100 GO TO 42 99 40 DO 41 I=1,3 101 SUM(I) = SUM(I) + TEG(I) 102 41 CONTINUE 103 42 DO 44 I=1.J 104 IF(ABSF(TEG(I)I-A8SF(GET(I))f 43,26,26 105 43 I.F (-A&ISF(-TEG(I))-O0005*ABSF( tEG( f )) 44,26,26 106 44 CONTINUE 107 DEINOM.= 0-:666.-*.-(l1.O+DECRE*SUM(1)/(6.2832*R)) 108 F1 = (10-DECRE*SUM(2)/(4.0*R))/DENOM 109 - 2- =.DECRE*SUM03)/(9*4248*R*DENOM) 110 C COMPUTE F3 AND- F-4- 111 F3 = (PHIZO/(2.0*R*PHIZ2*DENOM)* (( IZ2-ZO)*(Z2-ZO)+2.0*R*R)/ 112 1SQRTL4._Q-*tRL*R+IZ2-ZO)*(Z2-ZO) )-(Z2-ZO)) 113 DEL = R/20.0 114 SUM = 0.0 115 Y=O.O 116 TEG=O.O 117 IF (INDIC-2) 47,45,47 118 47 Y=Y+DEL 119 Q=QUEERF(Y) 120 CALL IEF1(1.5707, Q.EFtG) 121 IF (G-1.O) 105.105,103 122 103 WRITE OUTPUT TAPE 6,104.Y 123 104 FORMAT(48H ELLIPTIG INT. OUTSIDE OF RANGE,RAD=F8.4) 124 105 TEG = (Q*Q*Q/(96O0*SQRT(Y*Y*Y*Y*Y*R*R*R*R*R)*(1lO-Q*Q)*(l.O-Q*Q))) 125 1*Y*(((R+Y)*Q*Q*Q*Q-(R+3.0*Y)*Q*Q+2.0*Y)*F-(2.O*(R+Y)*Q*Q*Q*Q-2.0* 2(2*0*-+Y)*Q*Q+2.O*Y)*E) 127 IF ((R-Y)-0.O0001) 45,45,46 128 46 SUM = SUM + TEG 129 GO TO 47 130'45 SUM = SUM+O*50*TEG 131 F4 = ( 4O*(Z2-ZO)*(Z2-ZO)*DPHIZO*DEL*SUM)/(3e1416*PHIZ2*DENOM) 132 C COMPUTE FINAL VALUE AND PRINT 133 BOUND = F1 + AMBDA*F2 - F3 + AMBDA*F4 134 GRAD = BOUND/AMBDA 135 WRITE OUTPUT TAPE 6,48,Z2,DENOM,F1,F2,F3,F4,BOUND GRAD 136 48 FORMAT (8F9.4) 137 IF (INDIC-2) 50,49,50 138 49 IF ((Z2-ZO)-12.0) 51.52.52 139 50 IF (PHIZ2-O0.25*PHIZO) 52,51,51 140 51 Z2 = Z2 + DECRE 141 GO TO 8 142 52 GO TO 1 143 END 144 $DATA

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164 REFERENCES (Continued) 13. Glasstone, S., and Edlund, M. C., "The Elements of Nuclear Reactor Theory," D. Van Nostrand Company, Inc. (1952). 14. Zimmerman, E. L., "Boundary Values for the Inner Radius of a Cylindrical Annular Reactor," ORNL-2484 (1958). 15. Jahnke, E. and Emde, F., "Tables of Functions," Dover Publications (1945). 16. "Research Reactors," Chapter 2, U. S. Atomic Energy Commission, McGraw-Hill Book Company, Inc. (1955). 17. Shapiro, J. L., et al., "Initial Calibration of the Ford Nuclear Reactor," MMPP-110-l (1958 ). 18. Hughes, D. J., and Schwartz, R. B., "Neutron Cross Sections," BNL-325, Second Edition (1958), 19. Macklin, R. L, and Pomerance, H. S., "Resonance Capture Integrals," First Intern. Conf. Peaceful Uses of Atomic Energy, Geneva, 1955P/833. 20. Dayton, I. E., and Pettus, W. G., "Effective Cadmium Cutoff Energy," Nucleonics, 15, No. 12, 86 (1957). 21. Dalton, G. R., private communication (1963). 22. Reynolds, A. B., "Reactivity Effects of Large Voids in the Reflector of a Light-Water-Moderated and Reflected Reactor," thesis, AECU-4391 (1959)5 235 Dalton, G. R., and Osborn, R. K., "Flux Perturbations by Thermal Neutron Detectors," Nucl. Sci. Eng., 9, 198 (1961). 24~ Fastrup, B., "On Cadmium Ratio Measurements and Their Interpretation in Relation to Reactor Spectra," Riso' Report No, 11 (1959). 25. Walton, R. B., et al.,, "Measurements of Neutron Spectra in Water, Polyethylene, and Zirconium Hydride," Proc. Symp. on Inelastic Scattering of Neutrons, IAEA (1960). 26. Schmid, L. Co, and Stinson, W. Po, "Calibration of Lutetium for Measurements of Effective Neutron Temperatures," Nucl. Sci, Enge Letters, 7, 477 (1960).

165 REFERENCES (Continued) 27. Klahr, C,, "Limitations of Multigroup Calculations," Nucl, Scio Eng., 1, 253 (1956). 28. Cantwell, M., and Goldsmith, M,'"The Effect on Calculated Clean Critical Activation Shapes of the Use of Transport Approximations in the Fast Groups," Nucl. Sci, Eng., 12, 490 (1962). 29. Donovan, J. L., King, Jo S., and Zweifel, P. F., "A Thermal Neutron Spectrum Measured by a Crystal Spectrometer," Univ. of Mich., ORA Report 03671-3-T, Ann Arbor, January, 1963. 30. Brown, H. D., "Neutron Energy Spectra in Water," DP-64 (1956), 31. Goldsmith, M,, et al., "Theoretical Analysis of Highly Enriched Light Water Moderated Critical Assemblies," Second Intern. Conf. Peaceful Uses of Atomic Energy, Geneva, 1958 P/2376. 32. Daniels, Eo, forthcoming master's thesis, The University of Michigan (1963). 33. Feiner, Fo., et al., "Precise Criticality Determinations in the Solid Homogeneous Assembly," Trans. Am. Nucl, Soc., 5 No. 2, 344 (Nov. 1962). 34. Bullock, J. B., "Calculation of the Maximum Fuel Cladding Temperatures for Two Megawatt Operation of the Ford Nuclear Reactor," unpublished, Phoenix Memorial Laboratory Memo Report No. 1, The University of Michigan (1962). 35. "University of Michigan Executive System for the IBM-7090 Computer," unpublished, The University of Michigan (1963). 36. Flatt, H. P., "The Fog One-Dimensional Neutron Diffusion Equation Codes," unpublished, North American Aviation Corp. (1961). 37. Tobias, M. L., and Fowler, T. B., "The Twenty Grand Program for the Numerical Solution of Few-Group Neutron Diffusion Equations in Two Dimensions," ORNL-3200 (1962). 38. Shudde, R. H,, "Tempest II," unpublished, North American Aviation Corp. (1960), 39. Amster, Ho, and Suarez, R., "The Calculation of Thermal Constants Averaged Over a Wigner-Wilkins Flux Spectrum; Description of the Sofocate Code," WAPD-TM-39 (1957).

166 REFERENCES (Concluded) 40. McGoff, D, J., "Form, A Fourier Transform Fast Spectrum Code for the IBM-709," NAA-SR-Memo-5766 (1960). 41, Bohl, H., Gelbard, E. M., and Ryan, G. H., "Muft-4; Fast Neutron Spectrum Code for the IBM-704," WAPD-TM-72 (1957).

UNIVERSIT OF MICHIGAN 3 9015 02827 4705 THE UNIVERSITY OF MICHIGAN DATE DUE