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/tJL At I &1L iAj UNCLASSIFIED a UNCLASSIFIED BNL1344 Subject Category: PHYSICS UNITED STATES ATOMIC ENERGY COMMISSION TEMPERATURE COEFFICIENTS OF REACTIVITY OF REACTORS By Jack Chernick January 9, 1953 Brookhaven National Laboratory Upton, New York Technical Information Service, Oak Ridge, Tennessee Work performed under Contract No. AT302Gen16. Date Declassified: December 2, 1955. This report has been reproduced directly from the best available copy. Issuance of this document does rot constitute authority for declassification of classified material of the same or similar content and title by the same authors. Printed in USA, Price 25 cents. Available from the Office of Technical Services, Department of Commerce, Wash ington 25, D. C. GPO 988112 This report was prepared asa scientific account of Govern mentsponsored work. Neither the United States, nor the Com mission, nor any person acting on behalf of the Commission makes any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the in formation contained in this report, or that the use of any infor mation, apparatus, method, or process disclosed in this report may not infringe privatelyowned rights. The Commission assumes no liability with respect to the use of,or from damages resulting from the use of, any information, apparatus, method, or process disclosed in this report. TIMPERATR COEFFICIIETB OF IWACTIT= OF FACTORS By Jack Chernick Introduction The subject of temperature effects on the reactivity of a nus lear reactor is of cmnaiderable practical importance since the stability of a reactor under operating condition depends upon its temperature coef ficients. The actual operating temperature coefficient of a reactor in Wolves its particular structure and cooling system too much to lead it self to a general discussion. Instead we shall confine our remarks to the socalled unrform teamerature coefficient, i.e., the reactivity change per degree rise in temperature of all the reactor constituents in some limited and defined temperature range. Early Theoretical Resulto The physical effects contributing to the uniform temperature coefficient of a reactor are nov generally understood. It is custom ary to separate the coefficient into two parts, the nuclear temperature coefficient which is determined ly the change in nuclear crosssections with temperature, and the density temperature coefficient which is due to thermal expansion of the reactor. The density temperature coefficient was discussed by Soodak (M2966) who pointed out that the neutron leakage from a reactor varied inversely with the 4/3 power of the density. He thus derived the formula (1) ( ~~ T d for the density temperature coefficient in term of the amltiplication factor k., and the coefficient of volume expansion 4. of the system. For graphite moderated, normal uraniiu reactors the density tam perature coefficient is of the order 10 'C and can be neglected In com parison with the nuclear temperature coefficient which is of the order 10o/oC. On the other hand, enriched fluid fuel reactors have density temperature coefficients which range from 10 4,C to 10l3C. The neutron leakage from a reactor is also affected by the change in neutron crosssections with temperature since the migration area of the reactor is then altered. For a 1/v absorber, the diffusion area of the reactor increases with the square root of the absolute temperature while the age to theraal energy is slightly, decreased. hdis leads to the formula (Glasstone: Elemonts of Nuclear Reactor Theory, p.341), (2) ( ( I T k. To 2 3 s o for the contribution of the neutron leakage to the nuclear temperature 2 coefficient of the reactor. In equation (2), B2 is the geometrical buckling of the reactor, 1 the diffusion area evaluated at the neu tron temperature To. For a graphiteoraniu reactor, To 385fK at room 4mperature. The increase in neutron leakage is due mainly to the increase in diffusion area rather than in the neutron age and, near critical, the reacetivity coefficient runs about 3 x ID'/DC. The nuclear temperature coefficient of a reactor depeads on changes in k, as well as on changes in neutron leakage. In the early literature two such effects were recognisedt 1. Leveliwn or flattening of the iUtracell flux dis tribution. This effect occurs with increase in tem perature since the diffusion of neutrons through the moderator is eased, thus enhancing the probability of absorption in the fuel. Since the thermal utilisation is increased, one now obtains a positive contribution to the reactor temperature coefficient. The leveling effect is in general due to increase in the absorption mean free path with temperature. However, in waterwod erated reactors, the effect is caused by the large change in the scattering mean free path in the thermal region. The thermal utilization may also vary with tampers ture in the presence of aonI/v absorbers. For graphite, natural uranium reactors, the departure from the 1/v lai is aall and is generally neglected. 2. Doppler broadening of ;he resonance bends in uranmur. The resonance absorption of neutrons in U2" increases with temperature due to the broadening of the resonance bands. We have here the first contribution to the uni form temperature coefficient which depends primarily an the temperature of the fuel rather than that of the mod erator. The effect is important to the stability of graphiteuranium reactors since the fuel temperatures are the first to respond to any sudden power rise. The West Stas Reactor. The first measurement of the uniform temperature coefficient of a reactor is mentioned by Fermi in an early (January 15, 1943) pro gress report (CP416). By the simple experiment of opening the windows, the temperature of the West Stands reactor was brought down from 22~OC to 14C. The reactor was then brought back to room temperature, the en tire operation consuming about three weeks. From the shift in the criti cal position of a control rod, and after correcting for barometric effects, a uniform temperature coefficient of 3.8 x 10/DC was obtained. Since this result was approximately that expected from increased neutron leakage alone, Fermi concluded at the time that the Doppler broad ening effect must be large enough to cancel the leveling effect, i.e., about 6 x 10c5/C. In a February 6, 1943 progress report (CP455), however, we find that the fuel temperature coefficient was measured and led to a value of only 1 x 105/C. The experiment consisted of raising the temperature of insulated metal lumps in the central portion of the reactor about 300C and measuring the change in reactivity. This result meant that they had to look elsewhere for a large temperature effect and Fermi suggested izedietely that it might be found in the variation of 'I with neutron energy. Following this experiment a theoretical discussion of temperature effects in graphiteuranium reactors vas given in a short paper by Mor rison (CP478). He considered both lumped cubic lattices and rod lat tices. On the basis of theoretical results of Teller and Metropolis (CP387) on chemical binding effects in graphite, he estimated the mean temperature of these lattices in the neighborhood of 4000K. This value is in good agreement with later experimental determinations of the neu tron temperatures. Early estimates of the best value to use had ranged from 3000K to 4500K. Morriaon's estimates of the temperature coefficients of reactivity of those lattices is given in the following tables latteie T3pe T 1l 8" cubic lattice, 6 lb lumps 4000K #2 12" cubic lattice, 48 lb lumps 4100 #3 n8" rod lattice, 1.6 cm radius 412 Temperature Coefficient of Reactivity T+4 le akaUe e velig Doppler Broadening Eta Effect ,_UI fl 4.5xl0~ /C 10.axlO5/C 1.3 x 10 /oc xl0/oc 4x05/C #2 2.9 15.3 5 6 #3 2.7 4.6 9.7 910X5/OC The first table gives Harrison's estimate of the neutron temperature of these lattices. His estimate of the 1coefficient of reactivity was based on the experimental value of the uniform temperature coef ficient of the '"est Stands reactor (lattice 1l). The large value of the *)coefficient is at variance with earlier estimates and Is a re flection of the large value assumed by Morrison for the leveling ef fect of this lattice. Morrison also considered for the first time the hardening of the neutron spectrum in the fuel elents at least insofar as it af fected the calculation of the coefficient. The hardening of the neutron spectrum is caused by the preferential absorption of low en ergy neutrons as they penetrate into the fuel. The opposite effect of hardening on the thermal utilization was not considered by Morri son. Be came to the conclusion, sh b by the above table for lattice v2, that a reactor with large metal elements would be unstable with regard to temperature. The Branmdom es1arshall lkerimt An attempt to determine the dependence of on temperature was carried out by Dragdon, Hughes and Marshall early in 1944. By using folls of normal and depleted uranium located at the center of a spherical cavitr in a themal oolu, they measured the ratio of thermal fission in 8 to radiative captures in S. In order to obtain the sae ratio at a higher neutron temperature, they them sow rounded the foil with a spherical shell of silver of 108 mil thickneas. 6 The silver acted as a filter for low energy neutrons and Bragdon, Hughes and Marshall es timated the effective neutron temperature change at 1600C. They obtained a value of 1.0211 t 0.0046 4fr the ratio of capture to fission at the higher neutron tempera ture compared to the same ratio at the neutron temperature of the thermal column. The change in ) with temperature was then esti mated as Sd = 5.2 x 105 . At the time, the authors assumed that the radiative capture cross section of U235 was zero. However, reanalysis of their work by Friedman and Kaplan at Brookhaven (BNL152) on the basis of a fin ite but temperature independent value of X= (25)/'f(25) results only in a slight reduction in the value of the "temperature coef ficient. Attempts have since been made by Wigner (DPW2071) and by Sampson and Hrvwits in August 1951 (KAPL511) to determine the tem perature dependence of C in the thermal region by the use of approxi mate formulas for the temperature dependence of r,(28) and 0(25). Wigner obtained a value of 22 x 105/OC on the basis of the c T BragdonHughesMarshall experiment which he believed to be unlikely Sampson and Hurvitz concluded that the best value indicated by the Bragdon experiment was "~ 10 a 5 x 105/o 7 Sampson and thrvits also attempted to obtain A fro neutron spee d T tromter data which they thought indicated a positiw value of about 10 x 105/0C. They then obtained a value of A a 29 z 107C5/ which d T is in complete diaagrpemnt with the results of the early reactor e  per ants uhich w have already daecussed. However, even for # 0, T the crosseotion formulas assumed of these authors led to a value of 1 x 10o5/C for . Crossseetion data available to ua at took ) T haven through D.J.Bugh s and his group lead to more reasonable values of 2U However, it appears that at present we must still turn to a T reactor temperature coefficient experinant for our beat asti.atei of the variation of ) with temperature. Theory of the Donler Broadeninr gEfect In January 1942, Vigner (C4) pointed out that resonamee ab sorption in uranium lattices could, in goad approxiation, be repre sented by a volume absorption and a surface abaorptioe term. atew a peiaents 1by Cruts (C116), Hughee (cP35oo), Haelhawise (AXL.4323) @ Risser ani others ((FML958) have borne out this division of the resoe ance absorption and the method is universally used in resonamee escape factor calculations. In addition, Wipger gave the formulas for the shape of the Doppler broadened resonance lines. Thus if the level atrue ture were know, the Doppler temperature coefficient could be readily computed. Since the actual resonance structure was largely unknown, Viper and later Dancoff (CP1589) worsad with various simple models of the resonanee struetare to obtain the temperature eoeffielmt of the volume absorption term. 8 According to Wigner's method the effective resonance integral is given by A + B surfle mass where A and B are respectively the volume and surface absorption inte grals. Wigner obtained a value of 3 x 10"'/'C for the value of I dT. A dTo Dancoff, in refining the method on the basis of experimental data on the low lying resonances estimated that the temperature coefficient of volume absorption lay between lo2 and 1.7 x 104/C. For graphite uranium reactors, Kaplan and Friedman (BIL152) showed that this esti mate leads to values of ,1 2 1.4 to 2.0 x I10/ C for the effect of the Dopplor broadening on the resonance escape factor. Thore also exists some U29 activation measurements by Creuts (0110) and Mitchell (CP597) at elevated temperatures which yielded values of 1 dA of 1.7 x lO/0C and 1.1 x IO4/MC respecLively. The A dT spread of the experimental resulLts almost coincides with Daneoff's theoretical results. BExoriments on the X10 Reactor During the startup !P early operation of the Oak Ridge graphite uranium reactor, an extensive study of the reactivity coefficients was undertaken by L.B.Borst (obnPo60 and Ch. 13 of the National 'clear Energy Series IV5). Several attempts vere made to obtain the barometric coefficient of the reactor. Of these, the most accurate value was determined in a long run at constant power during which a change of atmosphric pres sure of 5.8 = Rg occurred within a period of 6 hours. From these em poriments a value of 0.4 0.05 inh/m Hg was established for the barometric coefficient. In subsequent msnon experiments, it was found that values ranging from 0.3 to 0.5 inb/a Hg vere required in order to correct satisfactorily for slow reactivity changes. In this con nection, it should be pointed out that, while the barometer will re spond to humidity as well as partial nitrogen pressure, the poisoning effect on a reactor is almost wholly due to the nitrogen content of the air. Thus barometric pressure alone can not be an exact indication of the nitrogen poisoning effect. An attempt was made to obtain the uniform temperature coefficient of the 110 reactor by placing steam radiators in the inlet air chamber in order to increase the equilibrium temperature of the reactor. A coefficient of 0.75 inhour/OC was obtained which was quite low compared to data obtained in later experiments. Similarly, an effort was made to estimate the metal temperature coefficient of the reactor from data on a heated slug. Fermi had suggested following the transient temperature changes in a metal slug as a method of power calibration cnd from the associated control rod movements, Borst fourd a uniform metal temperature coefficient of 1.2 inhoura/0C which appeared rather high. Finally, Borst relied on various metal and graphite temperature transients in order to obtain more accurate values of the reactivity coefficients. On the basis of extensive operating experience, he concluded that the uni 10 form tempeature coefficient of reactivity of the XIC reactor (in vacuo) vas 5.9 x IO10AC, of which only 1.0 x 10"5/O vcs attributed to the metal teaerature coefficient. It. is of interest to compare Borst's resutts vith Fermi's etimate of Fabruary 1944 (CP13U9) of the uniform temperature ooef ficiant of the Clinton raaclor. He split the various effects up as follcus: leveling: 6.5 x lC05/OC leakage: 3.1 x 10"5/C Doppler Broadening: 2 x IC5/OC EtaEffect: 5.5 x 105/C kff: 4.1 x 105/oc The estian',e of the Tffect was based on the experimental results of Bradcn, Hughes and Marshall. In early design calculations for the BKL reactor, K.plen and Frled.n (BNL152) obtained considerably smaller values fcr the leveling effect by using elementary diffusion thecry. These were re.; ectively 3.4 x 105/oC for the Oak Ridge re actor and 2.6 x 125/C fcr the DBL reactor. On this basis, the Oak Ridge results are consisent idth those of the Bra3donHughes'iarshall ex7 ardent. Prookhaven R.actor Exro Ericnta During the Etartur cf the ML reactor a considerable program of experiments of use in recctor evaluation were carried out. The 11 organisation of these experiments under the direction of L.B.Borst was so efficient thet all the low power work, including mch of the loading of the reactor to criticality, was carried out in about two weeks. We shall discuss here only those experiments which are per tinent to our present topic. The Baroetric Coefficient of the BNL Reactor The first opportunity to obtain the barometric coefficient of the BEL reactor occurred. on the night of August 23, 1950, while the reactor was temporarily idle at a slightly suberiticel loading. The effective multiplication factor of the reactor, measured by: its free period was 0.9997 at the time. The experiment arose from a sug gestion of the theoretical group who noted that at such a loading the steady state neutron level of the reactor, being inversely proportional to lkeff, could be expected to vary by Pbout 10% during the night be cause of changes in barometer alone. The flux level of the free re actor was therefore followed throughout the night by means of neutron counters. The curves were compared with accurate barometric pressure end partial nitrogen pressure data supplied by the Meteorology group. Graphite and metal temperatures at various points in the reactor were read frequently but no detectable changes in the thermocouple rend ings were recorded, possibly because of the insensitivity of these instru ments to small temperature changes. A total barometric change of 1.5 m Hg occurred during the night. The correlation of the neutron counting rate with barometric pressure was fairly good, yielding a barometric coefficient of 0.4 0.1 12 inhours/m Eg. The correlation with :artial nitrogen pressure was not nearly as good, thus indicating that there was little actual mixing of the outside air with that inside the reactor. If more time were avail able, it vwuld have been qciite interesting to pursue this study for longeT reriods of, time end for loadings closer to criticality in order tc learn more about the terorol behavior of a free reactor which is just sufficently suberitcrl to be stable a&rinst diurnal barometric ai thhernal fluctuations. Efforts to determine the barometric coefficient of the BNL re etcr ;', control rod chians necessary to maintain constant power during siz *er.thr changes led to no great iirroverent in the precision of the bcrcetric coefficient. It was therefore decided to use the reactor fors to simnlete the barometric effect. By sealing off the inlet air ducts :nd "errting one or more fens, a uniform pressure dror ranging fron 15 to 53 mm g was maintained over the reactor. These pressure changes were much greater than could be provided by the most severe weather conditions and the entire experiment was completed in about 3 hours. The results are ahovn in the following table. Barometer Coefficient of ENL Reactor Critical Barometric Position Pressure Drop Change in change in Coefficient of #9 Rod (=m BE) Pressure (am HN) Reactivitz (Ih) (Ih/mHa) 425.2 ea 0 /45.5 15.4 15.4 5.6 0.36 47C.8 38.3 22.9 7.0 0.31 490.7 53.2 14.9 5.5 0.37 419.8 C 53.2 19.7 0.37 13 The mean value obtained was 0.35 0.014 inh/m g*. Extensive control rod calibrations were carried out, both previous to sad during the vs periment. It was estimated that errors in the sensitivity of the #9 control rod were about 4%. A possibly more serious source of error of about .03 nlh/m g is indicated by the fact that the final critical position of the control rod differed somewhat from its initial position. A change of 1C in the over all reactor temperatures, which were not reed during the xperimant, could account for the discrepancy. lifoas Tamarature C oeffiset of the ~L Ractor The uniform temperature coefficient of the BEL reactor was meas ured on two different occasions at essentially sero power. The method used was that of drawing in the cold night air into the reactor by use of the fans and following the reactivity changes with a calibrated con trol rod. Graphite, metal and air temperatures and barometric pressures were recorded as a function of time, the former by means of thermocouples placed in representative parts of the reactor. We used several thermo couples since we were cognizant of the difficulties experienced at Oak Ridge in similar experiments of this type. As expected, the major changes in the metal temperatures occurred during the early part of these runs, while the reverse was true of the graphite temperatures. Temperature changes during a given interval of time wre found to vary appreciably at different points in the reactor. The uniform temperature coefficient of the reactor uncorrected to vaum was found to be 1.26 0.09 inhours/C. Of this coefficient, 24 the contribution of the metal coefficient was 0.78 0.16 inhours/OC while that of the graphite coefficient was 0.48 0.14 inhours/C. After correcting the latter for barometer we find a uniform tempera ture reactivity coefficient of 5.7 x 105/oc of which 2.0 x 105/C is attributable to the metal temperature coefficient. On the basis of operating experience, we would tend to increase the graphite coefficient somewhat. In any case there is good agreement with theory and in par ticular with the result of the BragdonHugheaMarshall experiment. The Metal TeMDerature Cbefficient Our first attempt to determine the metal temperature coefficient oat he BIL reactor involved the use of insulated metal cartridges heated to about 450C and then placed in three of the central channels of the reactor. With the hot cartridges in place, the control rods were ad justed to produce a slightly falling reactor period at a negligible power level. As the cartridge temperature decayed, the reactivity of the reactor increased until a positive period was observed. The control rod was them adjusted to bring the reactor slightly below critical. This procedure as repeated several times during the experiment which ran for about 4 hours. The technique used was suggested by Borst and proved to be a very sensitive one for measuring the exceedingly small changes of reactivity which were involved. Under the condition of the experiment a reactivity coefficient of 0.0093 0.0C04 inhours/'C was observed. The use of statistical weight factors, however, yields a metal tempera ture reactivity coefficient of 1.4 x 105/0C, which is about 30% lower 15 than the values ob nined in other experiments. Teporetu're F3ash Am'eriments The most accurate value of the metal temperature coefficient of the BNL reactor was obtained in temperature flneh experiments during uhich the reactor was permitted to rum away and brought under control by the increase in metal temperatures alone. The method used was simle in conception. The reactor cooled to ambient teaperatTre, was started up without fans from a noZ igible power level with an initial period 'Aich ranged from 36 sec to 4 minutes. There was little change in metal tempertures until the integrated power output of the reactor hid risen to the 10 kw min level. Thus the initial reactivity of the reactor could be determined from the galvqnometer period. At the time of mxiamn gelvenoncter reading, the reactivity would be zero and the corresponding metal temperature at the position of maximnu flux eould be used to obtain a metal temperature coefficient for the reactor. Of considerable interest in these ereriments was the fact that the rover output and reactor temperature oscillated through several dis tinct maxima and minima before damping out. The grarhite thermocourles registered no increase in temperature until one or more such oscillaticns had occurred. This was expected beecuse of the nmch larger heat capacity and 'might of the moderetor relative to the fuel and the fact that the maximum power output reached was only of the order of a W. For the low temaorrture flashes, the tedium of waiting for the reactor to reach sig nificant o'. r levels was reduced by bringing the reactor up rapidly to the XE region and then setting the controls to the proper position. Since 16 the uniform temperature coefficient of a reactor depends on the dis tribution of temperature as well as flux, readings of available metal thermocouples other than the one at maximum flux were occasionally re corded. It was found that during the run, the temperature rise at points in the same rlane parallel to the reactor air gap was closely proportional to the theoretical flux distribution. In the first temperature flash which took place on October 10i 1950, the metal thermocounles were attached not to uranium but to the midpoint of the upper aluminum fins. Thus it was possible for an appreciable temperature gradient to exist between the uranium surface and the fins. In the second set of experiments (Novenber 10, 1950), thermocouples were placed on the fuel surface as well. It was found during the run that the difference in temperature between the uranium end adjacent Lluminum fins was actually negligible and the pre vious value obtained for the metal temperature coefficient was confirmed. Because of the absence of xenon and graphite temperature effects, the temperature flesh experiment appears to be the best possible method of determining the metal temperature coefficient of a grphiteurmniun reactor. Despite the fact that under the different initial control settings the maximum metal temperatures ran from 600C to 2000C, the metal temperature coefficients varied very little wdth temperEture. An average vali.e of 0.49 inhour/OC rise in maximum metal temperature was obtained whichh corresronds to a uniform reactivity coefficient of 17 2.0 x la/oC. Uider strictly adiabatic conditions, the theoretical value of the mazimm overshoot in temperature in a reactor temperature flash experiment is twice the equilibrium value and this value was nearly attained when the reactor was started up with an initial period of 36 seconds. Temieratur Coeffiielant, of Natural Uranium Water Latticea Before leaving the subject of temperature coefficients of re actors we should like to quote some of the results obtained some years ago at Oak Ridge in water, normal uranium, exponential experiments. The uniform temperature coefficients of these assemblies were found to be negative for tight lattices and positive at large water to metal ratios. The crossover from negative to positive values of reactivity oeourred at a watermetal volume ratio of 1.6. Experimental values at other geometrical arrangements are given in the following table: Water Metal Ratio Reactivitr Coefficient 1.2 1 x 104/oC 2.1 +1 x I C4/c 3.1 +2 xIO4C The experimental results were based on measured buckling of the lat tices at about 25C and 75C00. Changes in the migration area with den sity were calculated. The experiamnts show the tendency of leveling effects in water le tioes to produce positive temperature coefficients. Thoory of To1,erature Dependent Reactor Kinetics We should now like.to consider briefly some reactor problems which involve temperature effects. To begin with, reactor transients 18 under ordinary and emergency conditions are of great importance both in the design and sefe onerption of a reactor. When temperature ef fects are considered, the equations governing the system are nonlin ear and generally must be solved by numerical methods. All of the AEC * laboratories have done considerable work on these problem as they af fect their porticuler reactor types. For small deviations from steady state conditions, perturbation methods such as those outlined by Nordheim (MDDC35) are useful. For large deviations from equilibrium, more exact methods are required and mzny of the problems have been solved only with the aid of high speed electronic computing machines. The damped, almost periodic nature of the temperature and flux curves obtained in the BNL temperature flash experiments aroused our in terest in the nonlineer asnects of the problem. If the delayed neutrons are assumed to act only to increase the effective neutron generation time as is the case if the reactor periods are not too short, then the transient equations take the simple form dn/dt Akmexn ka (kex) ST dT/dt = fn AT where 1/ is the averge neutron generation time, ,e is the temperature coefficient of the reactor and X is the relaxation constant for heat removal. In the ease that (kge)c, the excess k introduced by a control rod is constant, the temperature equation mBy be reduced to the dimension less nonlinear form 19 W + (10)6 6e e(0 + @2) In the single parameter e 4 (k)g)/A . Solutions of this equation, obtained by the method of isoclines, are ubown in Slide No. 11163 (Unclassified) The first elide shows a phase diagram (rate of change of tem perature vs temperature) when the reactor control setting is above the cold critical setting. If the reactor has a positive temperature coef ficient, the solution curves all diverge with time, regardless of initial conditions. There are two possible equity ibrium positions for a reactor with a negative temperature ooefficient. One occurs at zero flux and sero, i.e., ambient temperature and is unstable in the same sense that a in verted pendulum is in unstable equilibrium. The second equilibrium point is a stable one. The equilibrium point is a spiral if the reactor cool ing rate is sufficiently low. On a time basis, the solutions conveaging to the stable equilibrium point are then of damped, almost periodic type. At higher cooling rates, critical damping occurs and the equilibrium point, in the language of nonlinear mechanics, becomes a node. The limiting spiral connecting both equilibrium points corresponds to the ease of the reactor temperature flash experiments which we have previously discussed. Slide No. 11153 (Unclassified) Thie slide aso what happens when one attempts to shut down a reactor. The control setting is below the cold critical position and the 20 solutions for a reactor with a negative teoper.ture coefficient are very uninLeresting from the mathematical viewpoint. Regardless of initial conditions, everything converges to the stable equilibrium point. A reactor with a positive temperature coefficient, however, has fto possible equilibrium positions one which is an unstable sad dle point. One of the two curves which cross the saddle point is very important since it separates the phase plane into stable and un stnble regions. Only in the stable region do the solution curves con verge to the stable equilibrium point. In BNL173, we have investigated the nature of the solutions of nonlinoar equations of the present type. Of practical interest is the fact thnt the special solutions which separate the stability regions of a reactor are of relatively sisnie form. Statics of a Poactor vith V&riable Local ul.tinlleatiqn In a reactor oerating at power, the fuel and moderator tem persturos ns wo1 as poison concentrations vary with position. The local mult!plic.tion factor is therefore variable and the flux dis tribution asscirtod with the clean reactor is distorted. Rei ctors mna be capable of several quasisteady states which are fr rly well separated with time. Thus during the startup of the BML reactor after a long shutdown, the fuel temperatures omne into equilibrium in a few minutes, the moderator temperatures then come in to equilibrium and finally the xenon production becomes important level ing off after about a day's operation. The variation in the local multi 21 plication factor is different for each of these approziaate equili brium positions. In BNL126, the statics of a reactor with variable local multiplication have been considered on the basis of one and two group methods for certain cases and the results were compared with nertur bation theory approximaticns. Thus the solution of the onegroup equation for a critical slab reactor with Sk local proportional to flux was shoun to be given by an elliptic integral of the first kind. The xenon problem with burnup was solved exactly for various one dimensional cases. Equilibrium temperature conditons for a raeator in which the inpor tnt variation of temperature occurs only along a cooling channel vs reduced to a Integrodifferential equation which can be solved by numerical methods. In general, it ws. found that except in very extreme cases, the usual statistical weight formulas gave very good agreement with exact solutions. The statistical weight formulas for the effect of xenon at flux levels were burnup is important "rre considered in Canadian re ports by Goldstein and CGugnhein (FT2:a) and PRennie (CRT272). Ir ENL126, we have checked their formulas over a wide range of flux levels by an alternateo methodd rnd obtained good agreement. 22 PHASE PLANE DIAGRAM OF TEMPERATURE DEPENDENT REACTOR KINETICS (CONTROL SETTING ABOVE COLD CRITICAL) Slide No. 11163 23 PHASE PLANE DIAGRAM OF TEMPERATURE DEPENDENT REACTOR KINETICS (CONTROL SETTING BELOW COLD CRITICAL) k \\\\ \ \\i z I/II Slide No. 11153 S UNIVERSITY OF FLORIDA 3 1262 08229 985 9 
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