3 ' - OAK RIDGE NATIONAL LABORATORY operated by | UNION CARBIDE CORPORATION for the U.S. ATOMIC ENERGY COMMISSION ‘ORNL- TM— 378 . /-w COPY NO. - //5 DATE - November 5, 1962 TEMPERATURES IN THE MSRE CORE DURING STEADY-STATE POWER OPERATION | e o i\ : " J. R. Engel and P. N. Haubenreich L on I e \‘fi_‘;’h AR r ABSTRACT Over=-gll fuel and graphite temperature distributions were calculated for a detailed hydraulic and nuclear representation of the MSRE. These temperature distributions were weighted in various ways to obtain nuclear and bulk mean temperatures for both materials. At the design power level of 10 Mw, with the reactor inlet and outlet temperatures at 1175°F and 1225°F, respectively, the nuclear mean fuel temperature is 1213°F. The bulk average temperature of the fuel in the reactor vessel (excluding .-the volute) is 1198°F. For the same conditions and with no fuel per- meation, the graphite nuclear and bulk mean temperatures are 1257°F and 1226°F, respectively. Fuel permeation of 2% of the graphite volume raises these values to 1264°F and 1231°F, respectively. The effects of power on the nuclear mean temperatures were combined with the temperature coefficients of reactivity of the fuel and graphite to estimate the power coefficient of reactivity of the reactor. If the reactor outlet temperature is held constant during power changes, the power coefficient is - 0.018% 2%/Mw. If, on the other hand, the average of the reactor inlet and outle’ ‘ce@eratures is held constant, the power coefficient is - 0.04T% é%/m, T ’ imile Price $§ .4 " ,/// = MicrofiMpPrice $ / ” Available fr N Office echnical ices De ment of Commerce ashington 25, D. C. NOTICE This document contains information of a preliminary nature and was prepared primarily for internal use at the Oak Ridge National Laboratory. It is subject to revision or correction and therefore does not represent a final report, The information is not to be abstracted, reprinted or othgrwise given public dis- semination without the approval of the ORNL patent branch, Legal and Infor- mationn Control Department, p" ¥, LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commissien: A. Makes any warranty or representation, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparotus, method, or process disclosed in this report may net infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclossd in this report. As used in the above, '‘person acting on behalf of the Commission’ includes ony employse or contracter of the Commission, or employes of such contractor, to the extent that such employae or contractor of the Commission, or employse of such contracter prepares, disseminates, or provides access to, any information pursuant to his employment or contract with the Commission, or his employment with such contracter, CONTENTS Page ABSTRACT & & o o o o « o o . 1 LIST OF FIGURES & & & v v v 6 v o v s v e v o 5 LIST OF TABLES & « + o s « = 4 v & o o o o o o o o o 6 INTRODUCTION . . » . . 7 DESCRIFTION OF CORE 8 Fuel Channe-ls « a & e < @ . o & . o o & o & ° ¢ ® 4 ® ll Hydraulic Model . . . . . &« o ¢ ¢ o ¢ o 6 o o o « &« « o 12 Neutronic Model . ¢« « .+ o o & o ¢ o o & o s o o o & o = 15 NEUTRONIC CAICUIATIONS . . & o o o o o & & & o o o o o « » - 18 Flux Distributions . . « ¢ « v o ¢ ¢ o v « 5 « 5 o o » 19 Power Density Distribution . . . . ¢« . « . + « o ¢ . . 22 Nuclear Importance Functions for Temperature . . . . . 22 FUEL TEMPERATURES « .= &+ ¢ 4 o s o 4 v « o o s o o + o 2 o« . 26 Over-all Temperature Distribution . . . . « + + &« .+ + . 30 Nuclear Mean Temperature . . « ¢ ¢« ¢« « &+ o o o = « o 33 Bulk Mean Temperature . « « o+ + o o o « 4+ & = « « « « 36 GRAPHITE TEMPERATURES . & « v & o v v o o o o o o o s o o« 36 ILocal Temperature Differences . . « o « « « « » o « . 36 Over-all Temperature Distribution . . . . . . . . . . . 421 Nuclear Mean Temperature . . . . . + +« + 4 o « o o+ « . 4k Bulk Mean Temperature . . o« « « + v « o &+ o o+ o « « o« . hk POWER COEFFICIENT OF REACTIVITY . & & v v v o o v o o o o o 45 DISCUSSION o o « & o o o « o o o o o » o « o o o o o o « » o b7 Temperature Distributions . . . . . . . « . « . « . « . 47 Temperature Control . . + & v ¢ ¢ o o & o « « &+ + & « k7 APPENDIX + « v o v o o o e e e b e e e e e e e e e e e kO Nomenclature o « & o+ v & o s o 4 o 4 o« s o o =+« .« bo Derivation of Equations . . + v v « v v v 4 o o o + o 51 N LIST OF FIGURES Cutaway Drawing of MSRE Core and Core Vessel MSRE Control Rod Arrangement and Typical Fuel Channels Nineteen-Regicn Core Model for Equipcise 3A Calcu- lations. See Table 3 for explanation of letters. Radial Distribution of Slow Flux and Fuel Fission Density in the Plane of Maximum Slow Flux Axial Distribution of Slow Flux at a Position 7 in. from Core Center Line Radial Distribution of Fluxes and Adjoint Fluxes in the Plane of Maximum Slow Flux Axial Distribution of Fluxes and Adjoint Fluxes at a Position 7 in. from Core Center Line Axial Distribution of Fuel Fission Density at a Position 7 in. from Core Center ILine Relative Nuclear Importance of Fuel Temperature Changes as a Function of Position on an Axis Iccated 7 in. from Core Center Line Relative Nuclear Importance of Graphite Tempera- ture Changes as a Function of Position on an Axis Iocated 7 in. from Core Center Line Relative Nuclear Importance of Fuel and Graphite Temperature Changes as a Function of Radial Position in Plane of Maximum Thermal Flux Channel Outlet Temperatures for MSRE and for a Uniform Core Radial Temperature Profiles in MSRE Core Near Midplane Axial Temperature Profiles in Hottest Channel of MSRE Core (7 in. from Core Center Line) 10 16 20 21 23 2k 25 27 28 29 3k he 43 i..-l LAY N LIST OF TABLES Fuel Channels in the MSRE Ccre Regions Used to Calculate Temperature Distributions in the MSRE Nineteen-Regicn Core Model Used in Eguipoise Calculations for MSRE Flow Rates, Powers and Temperatures in Reactor Regions Jocal Graphite-Fuel Temperature Differences in the MSRE Nuclear Mean and Bulk Mean Texperatures of Graphite Temperatures and Associated Pcwer Coefficients of Reactivity nge 11 1 17 31 Lo Ly Lé INTRODUCTION This report is concerned with the temperature distribution and zp- propriately averaged temperatures of the fuel and graphite in the MSRE reactor vessel during steady operation at power. The temperature distribution in the reactor is determined by the heat production and heat transfer. The heat production follows the over- all shape of the neutron flux, with the fraction generated in the grephite depending on how much fuel is soaked into the graphite. TFuel channels tend 1o be hottest near the axis c¢f the core because of higher pover densities there, but fuel velocities are equally important in'determining fuel tem- peratures, and in the fuel channels near the axis of the MSRE core the velocity is over three times the average for the core. Variations in velocity also occur in the outer regions of the core. Graphite tempera- tures are locally higher than the fuel temperatures by an amount which varies with the power density and also depends on the factors which gov- ern the heat transfer into the flowing fuel. The mass of fuel in the reactor must be kinown for inventory calcu- lations, and this requires that the mean density of the fuel and the graphite be known. The bulk mean temperatures must therefore be calcu- lated. The temperature and density of the graphite and fuel affect the neutron leakage and absorption (or reactivity). The reactivity effect of a local change in temperature depends on where it is in the core, with the central regions being much more important. Useful quantities in reactivity analysis are the "nuclear mean" temperatures of the fuel and graphite, which are the result of weighting the local temperatures by the local nuclear importance. The power coefficient of reactivity is a measure of how much the control rod poisoning must be changed to obtain the desired temperature control as the power is changed. The power coefficient depends on what temperature is chosen to be controlled and on the relation of this tem- perature to the nuclear mean temperatures. This report describes the MSRE core in terms of the factors which govern the temperature distribution. It next presents the calculated temperature distributions and mean temperatures. The power coefficient of reactivity and its effect on operating plans are then discussed, An appendix sets forth the derivation of the necessary equations and the procedures used in the calculations. DESCRIPTION OF CCRE Figure 1 1s a cutaway drawing of the MSRE reactor vessel and core. Circulating fuel flows downward in the annulus between the reactor vessel and the core can into the lower head, up through the active region of the core and into the upper head. The major contribution to reactivity effects in the operating reactor comes from a central region, designated as the main portion of the core, where most of the nuclear power is produced. However, the other regions within the reactor vessel also contribute to these effects and these con- tributions must be included in the evaluation of the reactivity behavior. The main portion of the core is comprised, for the most part, of a regular array of close-packed, 2-in. square stringers with half-channels machined in each face to provide fuel passages. The regular pattern of fuel chan- nels is broken near the axis of the core, where control-rod thimbles and graphite samples are located (see Fig. 2), and near the perimeter where the graphite stringers are cut to fit the core can. The lower ends of the vertical stringers are cylindrical and, except for the five stringers at the core axis, the ends fit into a graphite support structure. This structure consists of graphite bars laid in two horizontal layers at right angles to each other, with clearances for fuel flow. The center five stringers rest directly on the INOR-8 grid which supports the hori- zontal graphite structure. The main portion of the core includes the lower graphite support region. Regions at the top and bottom of the core where the ends of the stringers project into the heads as well as the neads themselves and the inlet annulus are regarded as peripheral regions. The fuel velocity in any passage changes with flow area as the fuel moves from the lower head, through the support structure, the channels, and the channel outlet region into the upper head. The velocities in most chainnels are nearly equal, with higher velocities near the axis and near the perimeter of the main portion of the core. For most passages, the UNCLASSIFIED ORNL-LR-DWG 61097 AIR INLET AIR QUTLET FLEXIBLE CONDUIT TO CONTROL ROD DRIVE (3) SMALL GRAPRITE SAMPLE AGCESS PORT CONTROL ROD COOLING AIR INLETS CONTROL ROD COOLING AIR QUTLETS COOLING JAGKET AfR INLETS COOLING JACKET AIR QUTLETS AGCCESS PORT COOLING JAGKETS REACTOR ACCESS PORT FUEL OUTLET SMALL GRAPHITE SAMPLES CONTROL ROD THIMBLES {3) HOLD DOWN ROD LARGE GRAPHITE SAMPLES (5) CORE CENTERING GRID FLOW DISTRIBUTOR GRAPHITE-MODERATOR STRINGER REACTOR CORE CAN REACTOR VESSEL ANTI-SWIRL VANES MODERATCR SUPPORT GRID Fig. 1. Cutaway Drawing of MSRE Core and Core Vessel 10 'l ===l # J .\.. Rt N %"\ \i/ _”\ GUIDE BAR o . N7, -, 2 Y //J//, ) | Fig. 2. MSRE Control Rod Arrangements and Typical Fuel Channels Il greater part of the pressure drop occurs in the tortuocus path through the horizontal supporting bars. This restriction is absent in the central passages, resulting in flows through these channels being much higher than the average. The variations in fuel-to-graphite ratic and fuel velocity have a significant effect on the nuclear characteristics and the {temperature distribution of the core. In the temperature analysls reported here, the differences in flow area and velocity in the entrance reglons were neg- lected; i.e., the flow passages were assumed tc extend frcm the top to the bottom of the main portion of the core without change. Radial variations were taken inte account by dividing the core intc concentric, cylindrical regions according to the fuel velocity and the fuel-to-graphite ratio. Fuel Channels The fuel channels are of severesl types. The number of each in the final core design is listed in Table 1. The full-sized channel is the typical channel shown in Fig. 2. The half-channels occur near the core perimeter where faces of normal graphite stringers are adjacent to flat- surfaced stringers. The fractional channels are half-channels extending to the edge of the core. The large annmulus is the gap between the graphite and the core can. There are three annuli around the contrcol-rod thimbles. The graphite specimens, which occupy only the upper half of one lattice space above a stringer of normal cross section, were treated as part of a full-length normal stringer. Table 1. Fuel Channels in the MSRE Channel Type Number Full-size 1120 Half-size 28 Fractional 16 Large annulus 1 Thimble annuli 3 12 Hydraulic Model Hydraulic studies by Kedll on a fifth-scale model of the MSRE core showed that the axial velocity was a function of radial pesition, pri~ marily because of geometric factors at the core inlet. As a result of these studies, he divided the actual core chamnels intc several groups according to the velocity which he predicted would exist. This division was based on a total of 1064 channels with each of the control-rod-thimble annuli treated as four separate channels. He did not attempt to define precisely the radial boundaries of some of the regions in which the chan- nels would be found. Since the total number of fuel channels in the core is greater than the number assumed by Kedl, and since radial position is important in evaluating nuclear effects, it was necessary tc make a modified division of the core for the temperature analysis. For this purpose the core was divided into five concentric annular regions, as described below, based on the information obtained by Kedl. The fuel velocities assigned by Kedl to the various regions are used as the nominal velocities. Region 1 This region consists of the central 6-in. square in the core, with all fuel channels adjacent to it, plus one-fourth of the area of the graphite stringers which help form the adjacent channels. The total crcss~-sectional area of Region 1 is 45.0 in.2 and the equivalent radius is 3.78 in. The fuel fraction (f) for the region is 0.256. This region contains the 16 channels assigned to it by Kedl plus the 8 channels which he classified as marginal. Because of the cylindrical geometry around the control-rod thimbles, six of the channels which were marginal in Kedl's model have the same flow velocity as the rest of the region. It was not considered worthwhile to provide a separate region for the two remaining channels. The nominal fluid velocity is 2.18 ft/sec. lE. S. Bettis et al., Internal Correspondence. Region 2 This region covers most of the core and contains only normal, full- sized fuel channels. All the fuel channels which were not assigned else- where were assigned to this region. On this basis, Region 2 has 940 fuel channels, a total cross-sectional area of 1880 in.e, a fuel fraction of 0.224, and equivalent outer radius of 24.76 in. (the inner radius is equal to the outer radius of Region 1), and a nominal fluid velocity of 0.66 ft/sec. Region 3 This region contains 108 full-sized fuel channels as assigned to it > by Kedl. The total cross-sectional area is 216 in.”; the fuel {raction is 0.224; the effective outer radius is 26.10 in.; and the nominal fluid velocity is 1.63 f£t/sec. Region 4 This region was arbitrarily placed outside Region 3 even though it contains marginal channels from both sides of the region. All the half- channels and fractional half-channels were added to the 60 full-sized channels assigned to the region by Kedl. This gives the equivalent of 78 full-sized channels. All of the remaining graphite cross-sectional area was also assigned to this region. As a result, the total cross- sectional area is 245.9 in.g; the fuel fraction is 0.142; the effective outer radius is 27.58 in., and the nominal fluid velocity is 0.90 ft/sec. Region 5 The salt annulus between the graphite and the core can was treated as a separate region. The total area is 29.55 in.a; the fuel fraction is 1.0; the outer radius is 27.75 in. (the inner radius of the core can), and the nominal fluid velocity is 0.29 ft/sec. Bffective Velocities The nominal fluid velocities and flow areas listed above result in a total flow rate through the core of 1315 gpm at 1200°F. All the veloci- ties were reduced proportionately to give a total flow of 1200 gpm. Table 2 lists the effective fluid velocities and Reynolds numbers for the various regions along with other factors which describe the regions. Table 2. Core Regions Used to Calculate Temperature Distributions in the MSRE Number of Total Crosse Full-sized sectional Area Region Fuel Channels {;nflfi} 1 12* 45.00 2 o 1880. 3 108 216.0 4 78 245.9 5 o** 29.55 Fuel Fraction T T Y R L TR T A " TR B M S e, 0.256 0.224 0.22k 0.142 1.000 Effective Effective Flow Outer Radius Fluid Velccity Reynolds Rate (in:) (£t /sec) Number (gpm) 3.78 1.99 3120 72 24 .76 0.60 o5 791 26.10 1.49 2360 22k 27.58 0.82 1300 89 27.75 0.26 Yol 24 1200 * Plus 3 control-rod-thimble annuli. % Annulus between graphite and core shell. HT 15 Neutronic Model The neutronic calculations upon which the tempereature distributions are based were made with Iquipoise 3A,2’3 a8 2-group, 2-dimensional, nulti- region neutron diffusion program for the IBM-7090 computer. Because of the limitation to two dimensions and other limitations on the problemn size, the reactor meodel used for this calculation differed somewhat fromn tlie hydraulic model. The entire reactor, including the reactor vessel, was represented in cylindrical (r,z) geometry. Three basic materials, fuel salt, graphite and INOR were used in the model. A total of 19 cylindrical regions with various proportions of the basic materials was used. Figure 3 is a vertical half-section through the model showing the relative size and location of the various regions. The region composi- tions, in terms of volume fractions of the basic materials, are summa- rized in Table 3. Regions J, L, M, and N comprise the main portion of the core. This portion contains 98.7% of the graphite and produces 87% of the total power. The central region (L) has the same fuel and graphite fractions as the central region of the hydraulic model. However, the outer boundary of the region is different because of the control-rod-thimble representation. bince it was necessary to represent the thimbles as a single hollow cyl- inder (region K) in this geometry, a can containing the same amount of INOR as the three-rod thimbles in the reactor and of the same thickness was used. This established the outer radius of the INOR cylinder at 3 in. which also is close to the radius of the pitch circle for the three thim- bles in the MSRE. The central fuel region was allowed to extend only to the inside radius of the rod thimble. The portion of the core outside the rod thimble and above the horizontal graphite support region was homogenized into one composition (regions J and M). The graphite-fuel 2T. B. Fowler and M. L. Tobias, EQUIPOISE~3: A Two-Dimensional, Two=- Group Neutron Diffusion Code for the IBM-7090 Computer, ORNL~3199 (Feb. 7, 1962 ). 3. . Nestor, Jr., FQUIPOISE-3A, ORNL-3199 Addendum (June 6, 1362). 16 Uneclassified ORNL-LR-DWG. T16k42 Mge 3. Ninetcen-Region Corve Nodel Zor Eguipoise Calculations. See Table 2 for explanstio: ’ Table 3. Nineteen-Region Core Model Used in EQUIPOISE Calculations for MSRE radius Z Composition (in.) (in.) (Volume percent ) Region Region inner outer bottom top fuel graphite INOR Represented A 0 29.% Th.92 76 .04 0 0 100 Vessel top B 29.00 29.56 - 9.14 Th .92 0 0 100 Vessel sides C 0 29.56 -10.26 - 9.14 0 0 100 Vessel bottom D 3.00 29.00 67 .47 Th.92 100 0 0 Upper head E 3.00 28 .00 66.22 67 .47 93.7 3.5 2.8 F 28 .00 29.00 0 67 -47 100 0 0 Downcomer G 3.00 28.00 65.53 66,22 k.6 5.4 0 H 3.00 27-75 6h .59 65.53 63.3 36.5 0.2 I 27.75 28.00 0 65.53 0 0 100 Core can J 3.00 27.75 5.50 6L .59 22.5 T7.5 0 Core K 2.94 3.00 5.50 Th.92 0 0 100 Simulated thimbles L 0 2.94 2.00 64 .59 25.6 T4 .4 0 Central region M 2.94 27.75 2.00 5 .50 22.5 77.5 0 Core N 0 27.75 0 2.00 23.7 76.3 0 Horizontal stringers 0 0 29.00 - 1.41 0 66.9 15.3 17.8 P 0 29.00 - 9.14 -1.41 90.8 0 9.2 Bottom head Q 0 2.94 66.22 T4 .92 100 0 0 R 0 2.94 65.53 66 .22 89.9 10.1 0 S 0 2.94 64 .59 65.53 43.8 56 .2 0 LT 18 mixture containing the horizontal graphite bars at the core inlet was treated as a separate region (I). The main part of the core, as described in the preceding paragraph, does nct include the lower ends of the graphite stringers, which extend beyond the horizental graphite bars, cor the pointed tops of the stringers. Trhe bottom ends were included in a single regicn (O and the mixture at the top ¢f the core was approximated by 5 regicns E, G, H, R, and 8). The thickness of the material contained in the upper and lower heads (regions O, 8, snd P) was adjusted so that thes amcunt of fuel salt was 2guel O The amcunt ccntalned in the reactor heads. As a result of this a fugtment the over-all height of the neutronic model of the reactor is U! not exactly the same as the physical height. The upper and lower heads themseives (regions A and C) were flattened cut and represented as metal disecs cf the nominail thickness of the reactcr material. iIr the radial directicon ocutside the main part of the core, the core can (region I), the fuel inlet annulus (regios F) and the reactor vessel \region B) were included with the actual physical dimensicns of the re- actor applied to them. The materials withir each separate regicn were treated as homogeneous - mixtures in the calculations. As g resulht, the calculations give only zhe overeall shape of the flux in the regicas where inhomogeneity exists beuzuse of the presence of two cr more of the basi: materials. The first step in the nsutronisc caleulatlons was the establishment required for criticality. A carrier salt cortaining TC mel % IiF, %5% BeFo and 5% ZrF4 was assumed. The LiF and ZriFg voncentrations were keld zonstant and the BeF- concentration was reduceda as th2 concentraticn of UF4 was increased in the criticality e U“ji, 5% U238, 1% U23”, searck. The uranium was assumed ts censist of 93% 236 and 1% Thi [/ isotepic ccmposition is bypical of the material ex- pected to be avgilable for the reactor. The lithium to be used in the marufaciure of the fuel szlt contairs 73 ppm Llf\° This value was used in the calsulations. No chemizal impurities were considered in the fuel mix- ture. Al of the calculations were made for an iscthermal system at 1200°F. The criticality search was made with MODRIC, a one=-dimensicnal, multi- group, multiregion neutron diffusior. program. This established the criti- cal concentration of UF, at 0.15 mol %, leaving 24 .85% BeF,, program was used for radial and axisl traverses cf the model used for the The sare Fquipcise 3A calculation to provide the 2-group cconstants for that program. The spatial distributions cf the fluxes and fuel power density were cb- tained directly from the results of the Fguipcise 3A calculation. The same fluxes were used to evaluate the nuclear impertance functions for temperature changes. Flux Distributions The radial distribution cf the slow neutron flux calculated for the MSRE near the midplane is shown in Fig. 4. This plane contains the nraxi- mum value of the flux and is 35 in. above the bottom of the main part of the core. The distortion of the flux produced by abscrptions in the sinu- lated control-rod thimble; 3 in. from the axial centerline, is readily apparent. Because of the magnitude of the distortion this simplified representation of the control-rod thimbles is probably not adequate for an accurate description of the slow flux in the immediate vicinity of the thimbles. However, the over~all distribution is probably reasonably accurate. The distorticn of the flux at the center of the reactor alsc precludes the use of a simple analytic expression tc describe the radial distribution. Figure 5 shows the calculated aexial distribution of the slow flux along a line vwhich passes through the maximum value, 7 in. from the verti- cal centerline. The reference plane for measurements in the axial direc- tion is the bottom of the horizontal array of graphite bars at the lower end of the main portion of the ccre. Thus, the outside of the lower heed is at -10.26 in.; the top of the main portion of the core is at 64.59 ir.; and the outside of the upper head is at 76.04. The shape of the axisl distribution in the main portion is closely approximated by a sine curve extending beyond it at both ends. The equation of the sine approximetion shown in Fig. 5 is UNCLASSIFIED ORNL-LR-DWG 75777 0.9 0.8 0.7 0.6 0.5 FRACTION OF MAX. VALUE 0.4 0.3 0.2 0.1 0 5 10 15 20 25 30 RADIUS, in. Fig. 4. Radial Distribution of Slow Flux and Fuel Fission Density in the Plane of Maximum Slow Flux FRACTION OF MAXIMUM VALUE -10 0 i0 20 30 40 50 60 Z,in Fig. 5. Axial Distribution of Slow Flux at a Position 7 in. Core Center Line 70 from UNCLASSIFIED ORNL-LR-DWG 75823 80 12 22 Egzj = Sin[?%[ 7 m (z + 4.36)} , witl the linear dimension given in inches. The relation of the slow neutrcn flux to the other fluxes (fast, fast adjoint and slow adjeint) is shown in Figs. 6 and 7. These figures rresent the absolute values for a reactcr power level cof 10 Mw. Power Density Distributicn A function that is of greater interest than the flux distributicn, from the standpoint of its effect on the reactor temperatures, is the Gistribution of fissicn power density in the fuel. For the fuel compo- sition considered here, only 0.87 of the total fissions are induced by thermal neutrens; the fraction of thermal fissions in the main part of the core is 0.90. 1In spite of the relatively large fraction of nonthermal fissions, the over-all distribution of fuel fission density is very simi- lar to the slow-flux distribution. The radial distribution is shown on Fig. & for a direct comparison with the thermel flux. The axial fission density distribution (see Fig. 8) was fitted with the same analytic ex- Pression used for the axial slow flux. The quality of the fit is about the same for both functions. Nuclear Importance Functions for Temperature The effects urcn reactivity cf local temperature changes in fuel and graphite have been derived from first-order two-group perturbation theory and are reported elsewhere.l+ This analysis produced weighting functions or nuclear importance functions, G(r,z) for the local fuel and graphite temperatures. The weighted average temperatures of the fuel and graphite obtained by the use of these functions may be used with the appropriate temperature ccefficients of reactivity to calculate the reactivity change associated with any change in temperature distribution. uB. E. Prince and J. R. Engel, Temperature and Reactivity Coefficient. Averaging in the MSRE, ORNL TM=-379 {Octcber 15, 1962). 23 UNCL ASSIFIED ORNL-LR-DWG 73611 (x1013) | \—Lfi 5 | 8 | - \FAST FLUX 12 o £ < \\\\\ c o 5 \\\ 2 10 =z = o S > 8 7 SLOW FLUX Lid \/‘ — —— = ,r"— ~~ & T~ \\ o« SLOW ADJOINT \\ E 6 e - —— N & - I o S~e r e v FAST ADJOINT ~~w £ 4 > - — L 2 0 0 5 10 15 RADIUS (in.) Fig. 6. Radial Distribution of Fluxes and Adjoint Fluxes in the Plane of Maximum Slow Flux 24, UNCLASSIFIED ORNL-LR-DWG 73612 (x1019) \ | 14 L l FAST FLUX | | . 12 - | 10 . i SLOW FLUX 8 o | \ | N FLUX FOR REACTOR POWER OF {0 Mw (neutrons/cmz-sec) ~ . . SLOW ADJOINT T NN - - N -y - -'h‘ - -~ - FAST ADJOIN ~ ™ | ! | 20 30 40 Fig. 7. Position 7 AXIAL POSITION (in.) 80 AxZal Distribution of Fluxes and Adjoint Fluxes at a in. frcm Core Center ILine FRACTION OF MAXIMUM VALUE 7 in. Fig. 8. Axial Distribution of Fuel from Core Center Line Fission Density at a Position UNCLASSIFIED ORNL-LR-DWG 75824 Gc The temperature weighting functions are evaluated in terms of the * * * * four flux products: ¢l ¢l’ ¢l ¢2, ¢2 ¢1 and ¢2 ¢2. The coefficients applied to the varicus terms depend on the material being considered, its lccal volume fraction and the manner in which temperature affects its physical and nuclear properties. Since the weighting functions are evalu- ated separately for fuel and graphite, a particular weighting function exists cnly in those regions where the material is represented. Figures 9 and 10 show the axial variation in the weighting functions fer fuel and graphite at the radial pcsition of the maximum fuel power censity. For both materials the sxial variaticn in the main part of the core is very closely approximated by a sine-squared. The functicn: .. 2] = 1 sin [7§Tfl-(z + 4071)J is plctted on each ¢f the figures for comparison. The wide variations at the ends resuit primarily from the drastic changes in veolume fraction and the apparent discontinuities are the result of dividing the reactor into discrete regioas for the Equipcise 3A calcuiation. The radial temperature weighting functicns for both fuel and graphite near the core midplane are shewn in Fig. 11. Because of the peculiar shape of the curves; nc attemrt was made tc fit analytic expressions to these functinns. FUEL TEMPERATURES The fuel temperature distribution in the reactor has an cver-all shape which is determined by the shape c¢f the power density and the fuel flow pattern. Within the main part cf the core;, where the fuel flows in dis- crete channels. temperature variations across each individual channel are superimpczed on this cver-all shape. These lccal effects will be touched on later (page 37 ) in the section cn graphite-fuel temperature differences. The present secticn deals with the over-all shape c¢f the fuel temperature distributicn and average temperatures. These were calculated for a reactor power level of 1C Mw with inlet and outlet temperatures at 11795 and 1225°F, respectively. The total fuel flow rate was 1200 gpm. 27 UNCLASSIFIED ORNL-LR-DWG 73618 1. ° | 1 | | ~ EQUIPOISE 3A RESULT : \ 08 sin2 ["/79.4 ( 2+ 471 ] // . ) \ 0.4 / _/ O ™ — - —— b \k&—mwMAiN PORTION OF CORE-——"——————“——*fl RELATIVE NUCLEAR IMPORTANCE 1 - T ] -0.2 | oy } -0.6 : : -20 -10 0 10 20 30 40 50 60 70 80 z, AXIAL POSITION (in.) Fig. 9. Relative Nuclear Importance of Fuel Temperature Changes as a Function of Posgition on an Axis Located 7 in. from Core Center TLine 28 UNCLASSIFIED ORNL-LR-DWG 73619 IMPORTANCE RELATIVE NUCLEAR 1.0 | , —— EQUIPOISE 3A RESULT ——sin2[T/r0.4 (24 a.71)] 0.8 - 0.6 / 0.4 / \ \ ‘ \ 0.2 / S \\ \ i <~ MAIN PORTION OF CORE — 4-4 —0.2 —— : | | . —0.4 1 ' -10 0 10 20 30 40 50 60 70 80 z,AXIAL POSITION (in.) Fig. 10. Relative Nuclear Impcrtance of Graphite Temperature Changes as & Function of Position on an Axis Located 7 in. from Core Center Line RELATIVE NUCLEAR IMPORTANCE 1.0 0.9 0.8 0.7 0.6 0.5 0.4 © W 0.2 0.1 29 UNCLASSIFIED ORNL-LR-DWG 73617 ] T N\ GRAPHITE\ N N\ \ \ \ \ A\ \\ \ 5 10 15 20 25 30 RADIUS (in) Fig. 11. Relative Nuclear Importance of Fuel and Graphite Temperature Changes as a Function of Radial Position in Plane of Maximum Thermal Flux 30 Over-all Temperature Distribution In describing the over-all fuel temperature distribution it is con- venient to regard the reactor as corsisting of a main core, in which most cf the nuclear heat is produced, and a number of peripheral regions which, tegether, contribute only a small amcunt to the fctal power level. Of the 19 reactcr regicns used for the neutronicicalculations (see Fig. 3), 14 coentain fuel and 10 ¢of these were regarded as peripheral regions. The four remaining fuel-bearing regions (J, L, M, and N) were combined to represent the main part of the ccre. On this basis, the main portion ex- tends radially tc the inside cof the core shell and axially from the bottom of the herizontal graphite bars to the top of the uniform fuel channels. This portion accounts for 87% of the total reactor power. The volute on the reactor inlet was neglected in the temperature calculatjons because cf its physical separaticn from the rest of the reactor. Peripheral Regions Since only 13% cof the reactor power is produced in the peripheral regions, the temperature variations within them are smell and the details cf the temperature distributions within these regions were not calculated. The mean temperature rise for each region was calculated from the fraction of the toctal power produced in the region and the fraction of the tectal flow rate through it. The inlet temperature to each region was assumed constant at the mixed mean cutlet of the preceding regicn. An approximate buliremean temperature, midway between the region inlet and outlet tempera- tures, was assigned tc each peripkeral region. Table 4 summarizes the flow rates, powers and fuel temperatures in the varicus reactor regions. Msin Ceore The wide variaticns in fuel temperature, both radially and axially, in the main part c¢f the core necessitate a more detailed description of the temperature distribution. The average temperature of the fuel in a channel at any axial position is equal to the channel inlet temperature plus a rise proportional to the sum of the heat generated in the fuel and that transferred to it from the adjacent graphite as the fuel moves from the channel inlet to the specified pcint. The heat prcoduced in the fuel fcllows very closely the radial and 31 Table 4. Flow Rates, Powers and Temperatures in Reactor Regions o Flow Powerb Tempo Riseb Av. Temp.c Region (gpm) (kw) (°F) (°F) D 1157 45k 2.4 1224 .7 E 1157 151 0.8 1223.1 F 1200 251 1.3 1175.6 G 1157 105 0.5 1222.4 H 1157 11k 0.6 1221 .9 J 1157 8287 43 a L 43 159 22 d M 1157 192 1.0 a N 1200 68 3 a 0 1200 69 0.3 1177.1 P 1200 134 0.7 1176.6 Q 43 12 1.7 1200.9 R 43 2 0.3 1199,9 S 43 2 0.2 1199.7 aRegions not containing fuel are excluded. bAt 10 Mw “with T, = 1175°F; T_ . = 1225°F out dActual temperature distribution calculated for this region. See text. axial variation of the fission power density. Since the heat production in the graphite is small, no great error is introduced by assigning the same spatial distribution to this term. Then, if axial heat transfer in the graphite ig neglected, the net rate of heat addition to the fuel has the shape of the fuel power density. The fuel temperature rise is in- versely proportional to the volumetric heat capacity and velocity. Thus ‘2 (Qf)m Tf (I‘,Z) = Tf (Z =0) + éf W Pf(r,z) dz , (l) 32 where Qf is an equivalent specific power which includes the heat added to the fuel from the graphite. The channel inlet temperature, Tf(z =0), is assumed constant for all channels and its value is greater than the reactor inlet temperature because of the peripheral regions through which the fuel passes before it reaches the inlet to the main part of the core. The volu- metric heat capacity, (pcp)f, is assumed constant and only radial variations in the fuel velocity, u, are considered. It is further assumed that the radial and axial variations in the power-density distribution are separable: P(r,z) = A(r) B{(z) . (2) Then Tf(r,z) = 'I‘f(z =0) + r—éajl)l—; fi%%f { B(z) dz . (3) If the sine approximetion for the axial variation of the power density (Fig. 8) is substituted for B(z), equation (3) becomes Tpr,z) = Tp(z =0) + Xxééig-{}os a - COS[T;¢7 (z + 4.36)} }-. (%) In this expression, X is a collection of constants, (Q.) - 17 fm 5 pf and _ e a = = (0 + 4.36) . (6) The limits within which (4) is applicable are the lower and upper boundaries of the main part of the core, namely, O €z < 64.6 in. It is clear from this that the shape of the axial temperature distribution in the fuel in any channel is proportional to the central portion of the curve [1 - cos B). (The axial distribution for the hottest channel in the MSRE is shown in Fig. 14, where it is used to provide a reference for the axial temperature distribution in the graphite.) D !._j_) The radial distribution of the fuel temperature at the outlet of the channels is shown in Fig. 12 for the reference conditions at 10 Mw. This distribution includes the effects of the distorted power-density distri- bution (Fig. 4) and the radial variations in fuel velocity. At the refer- ence conditions the main-core inlet temperature is 1177.3°F and the mixed mean temperature leaving that region is 1220.8°F. The additional heat re- quired to raise the reactor outlet temperature to 1225°F is produced in the peripheral regions above the main part of the core. Also shown on Fig. 12 for comparison is the cutlet temperature distributicn for an ideal- ized core with a uniform fuel fraction and uniform fuel velocity. For this case the radial power distribution was assumed to feollow the Bessel func- tion, JO(E.h r/R), which vanished at the inside of the core shell. The distribution was normelized to the same inlet and mixed-mean outlet tem- peratures that were calculated for the main portion of the MSRE core. Nuclear Mean Temperature At zero power the fuel temperature is essentially constant throughout the reactor and, at power it assumes the distribution discussed in the pre- ceding section. The nuclear mean temperature of the fuel is defined as the uniform fuel temperature which would produce the same reactivity change from the zero-power condition as that produced by the actual fuel tempera- ture distribution. The nuclear average temperature of the fuel is obtained by weighting the local temperatures by the local nuclear importance for a fuel tenpera- ture change. The importance functicn, G(r,z), includes the effect of fuel volume fraction as well as all nuclear effects (see p. 22 et seq.). There- Tore « J; Tf(r,z) G(r,z) dv Tf = (7) f G(r,z) av v In carrying out the indicated integration for the MSRE, both the numerator aznd denominator of (7) were split irto 11 terms, one for the main part of the core and one for each of the fuel-begring peripheral regions. For the . . . . . peripheoral reogicns, the fuel tomperaturce in each reogicn was assumcd constant TEMPERATURE, °F Core 1290 1275 1250 1225 1200 1175 Fig. 1-20 UNCLASSIFIED ORNL-LR-DWG 75775 RADIUS, in. Channel Outlet Temperatures for MSRE and for a Uniform (e A at the average temperature (see Table 4). The volume integrals of the welghting functions were obtained by combining volume integrals of the flux products, produced by the Equipoise 3A calculation, with coefficients derived from the perturbation calculations. For the main part of the core, the temperatures and temperature weighting functions were combined and integrated analytically (in the axial direction) and numerically (in the radial direction) to produce the required terms. The net result for the fuel in the MSRE reactor vessel is Te = Ty +0.762 Ty = Ty) 6) A similar analysis was performed for a uniform cylindrical reactor with uniform flow. In this case the fuel power density was assigned a sine distribution in the axial direction and a JO Bessel function for the radial distribution. Both functions were allowed to vanish at the reactor boundaries and no peripheral regions were considered. A tempera- ture weighting function equal to the product of the fuel fraction and the square of the thermal flux (or power density since both have the same shape in the ideal case) was used. For this case: T. = T, +0.838 (r . -T ) (9) The principal reason for the lower nuclear average fuel temperature in the MSRE is the large volume of fuel in the peripheral regions at the inlet to the main bart of the core and the high statistical welight assigned to these regions because of the high fuel fraction. The existence of the small, high-velocity, low-temperature region around the axis of the core has little effect on the nuclear average temperature. Actually this "short~circuit' through the core causes a slight increase in the nuclear average temperature. There is lower flow and larger temperature rise in the bulk of the reactor than if the flow were uniform, and this effect outweighs the lower temperatures in the small region at the center. # See appendix for derivation of equations for the simple case. 36 Bulk Mean Temperature The bulk mean fuel temperature is obtained by integrating the local, unweighted temperature over the volume of the reactor. As in the case of the nuclear mean temperature, the large vclume cf low-temperature fuel at the inlet to the main part of the MSRE ccre makes the bulk mean tempera- ture for the MSRE lower than for a simple, uniform core MSRE: T Tt 091;6(TGut - Tin) (10) i Uniform Core: T, T, * 1/2 (Tout - Tin) (11) GRAPHITE TEMPERATURES During steady=-state power operaticn, the mean temperature in any graphite stringer is higker than the mean temperature cof the fuel in the adjacent channels. As g result, both the nuclear mean and the bulk mean temperatures of the graphite are higher than the corresponding mean tem- peratures in the fuel. In general, it is convenient tc express the graphite temperature in terms of the fuel temperature and the difference between the graphite and fuel temperatures. That is: Tg = T, + AT , (12) where AT is a positive number at steady-state. Nearly all of the graphite in the MSRE (98.7%) is contained in the regions which are combined to form the main part c¢f the core. Since the remainder would have only a very smell effect on the system, only the main part of the core is treated in evalvating grarhite temperatures and dis=- tributions. Iocal Temperatiure Differences In order to evaluate the local graphite-fuel temperature difference, it is necessary to consider the ccre in terms of a number of unit cells, each containing graphite and fuel and extending the length of the core. 37 In calculating the local temperature distributions, it is assumed that no heat is conducted in the axial direction and that the heat generation is uniform in the radial direction over the unit cells. The temperature distributions within the unit cells are superimposed on the cover=-all re- actor temperature distributions. The difference between the mean graphite and fuel temperatures within an individual cell can, in general; be btroken down into three parts: 1. the Poppendiek effect, which causes the fuel near the wall of a channel tc be hotter than the mean for the channel; 2. the temperature drop across the graphite-fuel interface, resulting from heat flow out of the graphite; and 3. the difference between the mean temperature in a graphite stringer and the temperature at the interface, which is necessary to conduct heat produced in the graphite to the surface. These parts are treated separately in the fcllowing paragraphs. Poppendiek Effect As may be seen from the Reynolds nmumbers in Table 2 (page 1), the fluild flow in most of the core is clearly in the laminar regime and that in the remainder of the core is in the transiticon range where flow may be either laminar or turbulent. 1In this analysis, the flow in the ertire core is assumed to be laminar to provide conservatively pessimistic esti- nates of the temperature effects. Fquations are available for directly computing the Poppendiek effect 5,6 (i.e., for laminar flow in circular channels or between infinite slabs; the rise in temperature of fluid near the wall asscciated with interral heat generation and relatively low fluid flow in the boundary layer). Since the fuel channels in the MSRE are neither circular nor infinite in two dimensions, the true Poppendiek effect will be between the results predicted by these two approaches. The method used here was to assume 5H. F. Poppendiek and L. D. Palmer, Forced Convection Heat Transfer in Pipes with Volume Heat Sources Within the Fluids, ORNL-1395 (Nov. 5, 1953 ). H. F. Poppendiek and L. D. Palmer, Forced Convection Heat Transfer Between Parallel Plates and in Annuli with Volume Heat Sources Within the A L. o ~ T grrea dweo 09 N Fiulds, ORNL=L7CL May 1i, 1954 ). 38 circular channels with a diameter such that the channel flow ares is equal to the actual channel area. This slightly overestimates the effect. For circular channels with laminar flow and heat transfer from graphite to fuel, the following equation applies: ’ : ' p p 2 [11(1+Pf4;w q_w)-BJ | - This equation is strictly applicable only if the power density and heat flux are uniform along the channel and the length of the channel is in- finite. However, it is applied here to a finite channel in which the power density and heat flux vary along the length. It is assumed thet the temperature profile in the fluid instantaneously assumes the shape corresponding to the parameters at each elevation. This probably does not introduce much error, because the heat generation varies continuously and not very drastically along the channel. If it is further assumed that the heat generation terms in the fuel and in the graphite have the same over-all spatial distribution, the Poppendiek effect is directly proportional to the fuel power density. This simple proportionality results from the fact that, in the absence of axial heat conduction in the graphite, the rate of heat transfer from graphite to fuel, q.,> is directly proportional to the graphite specific power. Temperature Drop in Fluid Film Since the Poppendiek effect in the core is calculated for laminar Tlow, the temperature drop through the fluid immediately adjacent to the chennel wall is included in this effect. Therefore, a separate calculation of film temperature drop is not required. lemperature Distribution in Graphite Stringers The difference between the mean temperature in a graphite stringer and the temperature at the channel wall cannot be calculated analytically for the geometry in the MSRE. Therefore, two approximations were calcu- lated as upper and lower limits and the effect in the MSRE was assigned & value between them. One approximation to the graphite stringer is a cylinder whose cross- sectional area is equal to that of the more complex shape. This leads to 2 cylinder with a radius of 0.9935 in. and a surface-to-volume ratio of 2.01 ing"lo If only the fuel-channel surface (the surface through which 21l heat must be transferred from the graphite) of the actual graphite stringer is considered, the surface-to-volume ratio is 1.8k in.“l. There- fore, the cylinder approximation underestimates the mean graphite tenpera- ture. The second approximation is to consider the stringer as a slab, cooled on two sides, with a half-thickness of 0.8 in. (the distance from the stringer center line to the edge of the fuel channel). This approxi- mation has a surface-to-volume ratic of 1.25 ino_l which causes an over- estimate of the mean graphite temperature. The value assigned to the difference between the mean graphite temperature and the channel-wall temperature was obtained by a linear interpolation between the two ap- proximations on the basis of surface-to-volume ratio. For the cylindrical geometry with a radially uniform heat source in the stringer, the difference between the average temperature in the stringer and the channel wall temperature is given by: 2 \ 1 P rs T - T = L8 » (lll') g W 8 k g For the slab approximation: . . P £ T - T = 3 £ (15) Application of unit cell dimensions for the MSRE and linear interpolation between the two approximations leads to y F T «T = 9.97 x 10 E& (16) g Net local Temperature Difference The two temperature effects described by Fquations (13) and (16) mey be combined to give the local mean tempersature difference between grephite and fuel. Lo 2 . - 2 11 (} + b ‘) -8 B -l E& Pele Fe Ty AT = 9.97 x 10 kg + X, I3 . (17) Since both terms in this equation are directly proportional to the power density, the local temperature difference may also be expressed in terms of a maximm value and the local, relative power density. AT (r, z) = BT Plr, z) | (28) P m Maximum Iocal Values The maximum values of the local graphite-fuel temperasture difference may be obtained by applying the appropriate specific powers to Equation (17).* Because of the dependence on specific power, the temperature dif- ferences are strongly influenced by the degree of fuel permeation of the graphite which affects primarily the graphite power density. The effects were eXamined for O, 0.5, and 2.0% of the graphite volume occupied by fuel. In all cases it was assumed that the fuel in the graphite was uniformly distributed in the stringers and that the specific powers were uniform in the transverse direction for individual graphite stringers and fuel chan- nels. Table 5 shows the maximum graphite-fuel temperature differences for the three degrees of fuel permeetion, at the 10-Mw reference condition. Table 5. Iccal Grapkite-Fuel Temperature Differences in the MSRE Graphite Permeation Maximum ILocal by Fuel Temperature Difference % of graphite volume ) (°F) 0 62.5 0.5 65.8 2.0 15.5 *In these calculations, it was assumed that 6% of the reactor power is produced in the graphite in the absence of fuel permeation. This value is based on calculations of gamma and neutron heating in the graphite by C. W. Nestor, (unpublished). The thermal conductivities of fuel and graph- ite were assumed to be 3.21 and 13 Btu/hr £t°F, respectively. In the above computations, it was assumed that fuel which soaked into the graphite was uniformly distributed. This is not the worst possitle case. Slightly higher temperatures result if the fuel is concentrated near the perimeter of the stringers. To obtain an estimate of the in- creased severity, a stringer was examined for the case where 2% of the graphite volume is occupied by fuel. It was assumed that the salt con- centration was 15% of the graphite volume in a layer extending intdé the stringers far enough (0.05 in.) to give an. average.concentration of 2%. The salt concentration in the central portion of the stringers was assumred to be zero. The specific power in the graphite was divided into two parts: the contribution from gamma and neutron heating which was assumed uniform across the stringer, and the contribution from fission heating which was confined to the fuel-bearing layer. This resulted in an increase of 2.0°F in the maximum graphite-~fuel temperature difference at 10 Mw. This in- crease was neglected because other approximations tend to overestimate the temperature difference. Over-all Temperature Distribution The over-all temperature distribution in the graphite is obtained by adding the graphite-fuel temperature difference to the fuel temperature. Figure 13 shows the radial distribution at the midplane of the MSRE for 10-Mw power operation with no fuel soakup in the graphite. The fuel tem- perature distribution, which is a scaled~down version of the corresponding curve in Fig. 12 to allow for axial position, is included for reference. Figure 14 shows the axial temperature distributions at the hottest radial position for the same conditions as Fig. 13. The fuel temperature in an adjacent channel is included for reference. The continuously increasing fuel temperature shifts the maximum in the graphite temperature to a position considerably above the reactor midplane. The distributions shown in Figs. 13 and 14 are for the mean tempers- ture within individual graphite stringers. The local temperature distri- butions within the stringers are superimposed on these 1n the operating reactor. TEMPERATURE (°F) 42 UNCLASSIFIED ORNL-LR-DWG 73616 1290 1280 GRAPHITE 1270 1260 {250 1240 1230 1220 1210 {200 {190 1180 ll———mfl—-—-—_‘ ! 1 ! | ’ j Fig. 13. 5 10 15 20 RADIUS (in.) 5 30 Radial Temperature Profiles in MSRE Core Near Midplane TEMPERATURE (°F) 43 UNCLASSIFIED ORNL—LR— DWG 73615 TN / ~ GRAPHITE P 1260 - 1300 1240 /// 1/// 1220 / ///, FUEL 1200 ’////// 1180 [ 1160 0 10 c0 30 40 SO 60 70 DISTANCE FRCM BOTTOM OF CORE (in.) Fig. 1l4. Axial Temperature Profiles in Hottest Channel of MSRE Core (7 in. from Core Center Line) bl Nuclear Mean Temperature The nuclear mean graphite temperature is obtained by weighting the local temperatures by the graphite temperature weighting function. In general j' T (r, 2) + AT(r, 2) | G (r, z) av Tg* = - [ - - ] - (17) J; Gg(r, z) dv Substitution of the appropriate expressions leads to an expression of the form T = T. + b n o i T - Tin) + D AT (18) l( out vhere bl and b2 are functions of the system. The nuclear mean graphite temperature was evaluated for three degrees of fuel permeation of the graphite. Table 6 lists the results for the reference condition at 10 Mw. Bulk Mean Temperatures The bulk mean graphite temperature is obtained in the same way as the nuclear mean temperature, but without the weighting factor. The bulk mean graphite temperature for various degrees of graphite permeatiocn by fuel are listed in Table 6. Table 6. Nuclear Mean and Bulk Mean Temperature of Grephite® Graphite Permeation by Fuel Nuclear Mean T Bulk Mean T % of graphite volume) (°F) (°F) 0 1257 1226 0.5 1258 1227 2.0 1264 1231 ®At 10 Mw reactor power, T, = 175°F, T_, = 1225°F. 45 POWER COEFFICIENT OF REACTIVITY The power coefficient of a reactor is a measure of the amount by which the system reactivity must be adjusted during a power change tc maintain a preselected contrcl parameter at the contrcl peint. The reactivity adjust- ment is normally made by changing control rod positicon(s) and the control parameter is usually a temperature cr a combination of temperatures in the system. The reactivity effect cof interest is only that due tc the change in power level; it does not include effects such as fuel burnup or changing poison level which follcw a power change. The power coefficient of the MSRE can be evaluated through the use of the relaticnships between reactor inlet and ocutlet temperatures and the fuel and graphite nuclear average temperatures develcped above. These relationships may be expressed as follows for 10-Mw power operaticn with no fuel permeation: Out = Tin + 500F # (19) * m o ~ Tf = .‘l.out - ll 09 F 3 (CO ) * . o T, = T v 3L6F (21) Although these equations were develcped for fuel inlet and outlet tem- veratures of 1175 and 1225°F, respectively, they may be applied cver a range of temperature levels withcut significant error. In addition to the temperature relations, it is necessary tc use temperature coefficients of reactivity which were derived on a basis compatible with the temperature weighting functions. The fuel and graph- ite temperature coefficients of reactivity from first-order, two-group - Fy - o - verturbation theoryT are -h4.b x 1077 °F L and ~7.3 x 10 2 °F l, respectively. 'i'hus - * - * Ak = -{4k.4 x 10 5) AT, - (7.3 x 10 5) ATg . (22) T B. E. Prince and J. R. Engel, Temperature and Reactivity Ccefficient Aversging in the MSRE, ORNL TM=-379 (Oct. 15, 194% ;. 46 The magnitude of the power coefficient in the MSRE is strongly depend- ent on the choice of the control parameter (temperature). This is best il- lustrated by evaluating the power coefficient for various choices of the controlled temperature. Three conditions are considered. The first of these is a reference condition in which no reactivity change is imposed on the system and the temperatures are allowed to compensate for the power- induced reactivity change. The other choices of the control parameter are constant reactor outlet temperature (To = const ) and constant average of ut inlet and outlet temperatures (& T, * T = const). In all three cases the initial condition was assumed to be zeggtpower at 120C°F and the change associated with a power increase to 10 Mw was determined. The results are summarized in Table 7. It is clear that the power ccefficlient for the ref- erence case must be zero because no reactivity change is permitted. In the other cases the power coefficient merely reflects the amount by which the general temperature level of the reactor must be raised from the reference condition to achieve the desired control. Fuel permeation of the graphite has only a small effect on the power coefficient of reactivity. For 2% of the graphite volume occupied by fuel, the power coefficient based on a constant average of reactor inlet and out- let temperatures is -0.052% %E/Mw (vs. -0.047 for no fuel permeation}. Table T. Tem.peraturesa and Associated Power Coefficients of Reactivity * * T T T T Power Control ocut ic T g Coefficient Parameter (°F) (°F) (°F) (°F) (% Ak/k/Mw) No external control 1184.9 1134%.9 1173.0 1216.5 0 cf reactivity Tout = const 1200 1150 1188.1 1231.6 - 0.018 Twn Tout - = const 1225 1175 1213.1 1256.6 - 0.047 ®At 10 Mw, initially isothermal at 1200°F. k7 DISCUSSION Temperature Distributions The steady-state temperature distributions and average temperatures rresented in this report are based on a calculational model which is cnly an approximation of the actual reactor. An attempt has been made tc in- 2lude in the model those factors which have the greatest effect on the temperatures. However, some simplifications and approximations have been made which will produce at least miror differences between the predicted temperatures and those which will exist in the reactor. The treatment of the main part of the core as a series of concentric cylinders with a clearly defined, constant fuel velocity in each region is one such simplifi- cation. This approximation leads tc very abrupt temperature changes at the radial boundaries of these regions. Steep temperature gradients will un- doubtedly exist at these points but they will not be as severe as thcse indicated by the model. The neglect of axial heat transfer in the graphite also tends to produce an exaggeration of the temperature profiles cover those to be expected in the reactor. It is obvious that the calculated axial temperature distribution in the graphite (see Fig. 14) can not exist without producing some axial heat transfer. The effect of axial heat trans- fer in the graphite will be to flatten this distribution somewhat and reduce the graphite nuclear-average temperature. Some uncertainty is added to the calculated temperature distributicns by the lack of accurate data on the physical properties of the reactor nate- rials. The properties of both the fuel salt and the graphite are based on estimates rather than on actual measurements. A review of the temperature calculations will be required when detailed physical data become available. However, no large changes in the temperature pattern are expected. Temperature Control It has been shown that the power coefficient of the MSRE depends upon the choice of the temperature to be controlled. Of the two modes of cor- trel considered here, control of the reactor outlet temperature appesrs to 48 control-rod motion for a given power change. However, the magnitude of the power coefficient is only one of several factors to be considered in selecting a control mode. Another important consideration is the quality of the control that can be achieved. Recent studies of the MSRE dynamics with an analog computer8 indicated greater stability with control of the reactor outlet temperature than when the average of inlet and outlet was controlled. 8 S. J. Ball, Internal correspondence. +3 I--SICiq 3] kg APPENDIX Nomenclature specific heat of fuel salt, volume fraction of fuel, Total circulation rate of fuel, Temperature weighting function, heat produced per fission, thermal conductivity or neutron multiplication factor, equivalent half-thickness of a graphite stringer treated as a slab, length of the core, relative specific power, equivalent specific power (absolute), rate of heat transfer per unit area, radius, equivalent radius cf a fuel channel, equivalent radius of a graphite stringer treated as a cylinder, radius of the core, tinme, temperature, local transverse mean temperature in a single fuel channel or graphite stringer fuel nuclear mean temperature, graphite nuclear mean temperature, bulk mean temperature of fuel in the reactor vessel bulk mean temperature of the graphite, fe R @ %a* @ FAM T 2% Subscrigts f g in out Nomenclature - cont'd velocity, volume of fuel in the core, 2.405 r R 2 axial distance from inlet end cf main core, dimensionless radius, defined by (6), fraction of core heat originating in fuel, defined by (5), density of fuel salt, macroscopic fission cross section, neutron flux, adjoint flux, local temperature difference between mean temperature across a graphite stringer and the mean temperature in the adjacent fuel, fuel, graphite, fuel entering the reactor mixed mean of fuel leaving the reactor, wall, or fuel-graphite interface, fast neutron group, thermal neutron group, maximum value in reactor. Derivation of Fguations for a Simple Model Fuel Temperature Distribution Consider a small volume of fuel as it moves up through a channel in the core. The instantaneous rate of temperature rise at any point is S ° BT al S(DCP ) In this expression it is assumed that heat generated in the graphite adjacent to the channel is transferred directly into the fuel (i.e., is not conducted along the graphite;, which accounts for the fraction 6. In the channel being considered, the fuel moves up with a velocity, u, which is, in general, a function of both r and z. e w) w2 The steady-state temperature gradiernt along the channel is then ar _ arfot g(r, z) = = a3) dz z/dt e(pCp)f u(r, z) (a3 let us represent the neutron flux by _ 2.405 ¢ \ . gz #(r, z) = qm I, (; = ,> sin T : (ak) Assume further that the channels extend from z = 0 to z = L (ignoring the entrance and exit regions for this idealized case). The temperature at z = 0, for all channels, is Tin' The temperature up in the channel is found by integration 52 T(r, z) = T Jfiz HZ%¢m - <é e ) sin <7~ > dz . (a5) in T Y e(pc e ulr, Z) For convenience in writing, let us define HZ, ¢ @), = L2 (26) m = g;&%2;£ ’ (a7) Then Qp), L J,(x) T. + 0 :rr(pcp)f U.(X, z) T{x, z) = - cos %&1) s (a8) and the temperature at the outlet of the channel, 2(Qp), L J_(x) Tin * fl(pcp)f u(x, L) (29) T(x, L) = Now let us calculate the mixed mean temperature of the fuel issulng from all of the channels. This requires that we weight the exit tempera- ture by the flow rate. The total flow rate is, in general; F = JHR 2 f(r, z) u(r, z) dr . (a10) o For the idealized case where f and v are uniform throughout the core: F = sRfu . (a1l) The general expression for mixed mean temperature at L is then R m - = ji o f(r, L) u(r, L) T(r, L) ar . (al2) out F 0 Substitution of (a9) for the temperature gives 23 h(Qp) L f R out Tin + §T5§;7; 5 f(r, L) r Jo(x) dr H3 § 4(Q.) L 2 .2.405 T'm R Doy F(pcp)f <2.1+05) _£ f(x, L) x I (x) ax . (al3) 0 For the ideal case where f is constant, integration gives l*(Qf)ml'f "RV T L= T Floc ), ETE,) (2.4) 3, (2.4) (allt) From (alh) we find that (T - T, ) F(pC_ ). (2.4) (Qf)m = out 1n2 P T (315) 4 LfR Iy (2.4) which may be rewritten as (F(pc_ ). (T . - T, ) B p’'f ‘Tout in n| |_2.405 Qg = L RAL eJ %LeJlZz.uoflJ ‘ (a16) The first term in (al6) is simply the average effective value of the fuel specific power (including heat transferred in from the graphite). The second and third terms are, respectively, the maximum-to-average ratios for the sine distribution of specific power in the axial direction and the JO distribution in the radial direction. Substituting (al6) in (a8) gives (217) T (x, 2) = T. + - cos — 2.405 © (Toue = Tin) 950 1 nz AR R® by Jl(2.h05) u(x,z) L .) This equation will be used in subsequent developments because it leads to expression of the average temperatures in terms of the reactor in- let temperature and the fuel temperature rise across the reactor. Fuel Nuclear Mean Temperature The nuclear mean temperature is obtained by weighting the local tem- perature by the nuclear importance, which is also a function of position, and integrating over the volume of fuel in the core. For the idealized case the importance varies as the square of the neutron flux. Thus the nuclear mean temperature of the fuel is given by R 1, J‘ jf T (r, z) ¢2 (r, z) 2r £(r, 2) dr dz © O Tp = . (a18) R .L jfl Jfl ¢ s z) 2w f(r, 2} dr 4z G C when (aht) is substituted for ¢(r, z), the denominator, which is 0 @ Ve, » becomes ~, ", R L : ¢2 V, = 2fl¢m2 J ‘é f{r, z) r JO2 (:2;&92;5') sin” £§:> ar dz . o fe R L (219) For constant f, integration gives .7;5 V, = Iif:f&fijiffi 3.2 (2.4505) (a20) fc 2 1 ’ ’ With the substituticn of (alt), (all), (al7) and (a20), and assum- ing £ and u %o be ccnstant, {al8) becomes * (r., -T. ) 2.405 L T, = T, + cut 31n > Jr X JO3(x) sin” %E * 2.405 Jq (2.405)L ° © (i - cos %E‘) dx dy (a21) Integration with respect to z results in ¥* (Tout = Tin ) . 2 -11-05 3 T, o= T+ J o oxIl) ax . (s22) 2){2.405) Jl3 (2.405) o 25 Graphical integration with respect to x yields j"g'ho5x JO3(X) ax = 0.564 . (a23) o Thus, the final result for the idealized core is * Tp o= T, 0+ 0.638 (TOut - Tin) . (a2l ) el Bulk Mean Temperature The bulk mean temperature of the fuel in the core is R .L 11 Jf Tf(r,z) 2qr f(r,z) dr dz o "o T = R L . (225) £ jfl Jfi 2qr f(r,z) dr dz O © The denominator is the volume of fuel in the core. Considering only the idealized case again - e - Vfc = g RTLE (az6) Substituting (all) and (al7) into (225) for constant f and u gives o = T + 2.405 nt (Tout - Tii’l) ‘]'R fo.J (.2_'&9..5.5.) * of in 2 Vg, J,(2.505) o Yo o R c 1 <1 - cos £ Jar az (a27) which is readily integrated to give (T T, ) o out - “in Tp= T, + 5 (a28) The equations for evaluating the nuclear and bulk mean temperatures of the fuel have been presented to illustrate the general method which must be employed. An idealized core was used in this illustration to reduce the complexity of the mathemetical expressions. This technique may be applied to any material in any reactor by using the appropriate temperature distributions and weighting functions. The results obtained by applying this method to the fuel and graphite in the MSRE are pre- sented in the body of the revort. geb = o - = O O~ O oo MSRP Director's Office, Rm. 219, 9204-1 G. M. Adamson L. G. Alexander S. E. Beall . Bender . Bettis . Billington . Blankenship . Bohlmann . Bolt . Borkowski . Brandon . Bruce antor . Cole Conlin Cook Corbin Cristy Crowley Culler DeVan Ditto Donnelly Douglas Dunwoody Engel . Epler . Brgen . Fraas . Frye . Gabbard Gallaher Greenstreet Grimes . Grindell Guymon . Harley Harrill . Haubenreich . Hise . Hoffman . Holz . dJarvis Jordan . Kasten Kedl . Kelley FEGUHnygoanEgS = - CHEPPHOsEHQO P GE QW - * * - - EWtflfl%"fltflt’d"flfi"fitfl?’flbflDUOQP&EUC-{ZUEIJU)QMC—IQ HGUnoyYzazsZzuhndoagrwmonYy="RYdDEPQNGH Distribution IWEBEG AR E 2 38 B 62 PO I':-HL;IPdet:di‘JUbfl"—ibc-li—i"db@}gc—iOUC-amegfi—ctdflbtfl"UEH-BQEUMPO:I: SRR \Q =3 EHEEORREOG Y ! - - - - . Kirslis . Krewson Iane . Lindauer . Lundin Lyon . MacPherson . Maienschein . Mann . McDonsld . MeDuffie . McGlothlan Miller . Miller Moore . Moyers . Northup o - mmhoumwwanszbzm . R. Osborn Patriarca . R. Payne M. Perry . B. Pike . E. Prince L. Redford . Richardson . C. Robertson . W. Savage . W. Savolainen E. Savolainen Scott . Secoy . Shaffer . Skinner . Slaughter . Smith Smith mzzumm . Splewak . A. Swartout . Taboada . Tallackson . Thoma . Trauger Ulrich Weaver Weaver . Webster Weinberg White Wilson - - - < QEHH"unQugi s 98, 99. 100-101. 102-103. 10k, 105-107 . 108. 109-110. 111. 112. 113. 11k, 115-129. 50 Distribution -~ contd C. H. Wodtke R. Van Winkle Reactor Division Library Central Research Library Document Reference Section Iaboratory Records ORNL-RC External D. F. Cope, Reactor Division, AEC, ORO A. W. larson, Reactor Division, AEC, ORO H. M. Roth, Division of Research and Development, AEC, ORO W. L. Smalley, Reactor Division, AEC, ORO Jd. Wett, AEC, Washington Division of Technical Information Extension, AEC