I t? it IJ 
 
 ^, Ail: VI 
 
 6hJL- IM4 
 
 UNCLASSIFIED 
 
 UNCLASSIFIED 
 
 BNL-1344 
 
 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. AT-30-2-Gen-l6. 
 Date Declassified: December 2, 1955. 
 
 This report was prepared as a scientific account of Govern- 
 ment-sponsored 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 privately owned rights. The Commission assumes 
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 from the use of, any information, apparatus, method, or process 
 disclosed in this report. 
 
 This report has been reproduced directly from the best 
 available copy. 
 
 Issuance of this document does not 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 
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TO4PKBAT0HE CCEFFICIHJTS OF KEA.CTI7TTS OF HEA.CTQRS 
 By Jack Cheraick 
 
 introduction 
 
 The subject of temperature affects on the reactivity of a nuc- 
 lear reactor is of considerable practical importance since the stability 
 of a reactor under operating conditions depends upon its temperature coef- 
 ficients. The actual operating temperature coefficient of a reactor in- 
 volves its particular structure and cooling system too such to lend it- 
 self to a general discussion. Instead we shall confine our remarks to 
 the so-called uniform temperature coefficient, i.e., the reactivity change 
 per degree rise in temperature of all the reactor constituents in some 
 limited and defined temperature range. 
 Earlj Theoretical Resn3,fri 
 
 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 
 
 -1- 
 
coefficient which is determined by the change in nuclear cross-sections 
 with temperature, and the density temperature coefficient which is due 
 to thermal expansion of the reactor. 
 
 The density temperature coefficient was discussed by Soodak 
 (M-2966) who pointed out that the neutron leakage from a reactor Tar led 
 inversely with the 4/3 power of the density. Ha thus derived the formula 
 
 (1) <^'a " - t^H 
 
 for the density temperature coefficient in terms of the multiplication 
 factor k^ and the coefficient of volume expansion «c of the system. 
 
 For graphite moderated, normal uranium reactors the density tem- 
 perature coefficient is of the order 10 fc and can be neglected in com- 
 parison with the nuclear temperature coefficient which is of the order 
 10""V°C. °n *be other hand, enriched fluid fuel reactors have density 
 temperature coefficients which range from -10 7*C to -10" 3 /°C. 
 
 The neutron leakage from a reactor is also affected by the change 
 in neutron cross-sections with temperature since the migration area of 
 the reactor is then altered. For a l/r absorber, the diffusion area of 
 the reactor increases with the square root of the absolute temperature 
 while the age to thermal energy is slightly, decreased. This leads to the 
 formula (Glasstone: Elements of Nuclear Reactor Theory, p. 341), 
 
 (2) {-*£-) » - P 2 
 
 2 ^H.V" 
 
 for the contribution of the neutron leakage to the nuclear temperature 
 
 -2- 
 
coefficient of the reactor. In equation (2), B 2 is the geometrical 
 
 2 
 
 buckling of the reactor, 1^ the diffusion area evaluated at the neu- 
 tron temperature T Q . For a graph! te-uraninm reactor, Tq ~ 385°I at 
 rooa temperature. The increase in neutron leakage is due mainly to 
 the increase in diffusion area rather than in the neutron age and, 
 near critical, the reactivity coefficient rune about -3 x 1D~*/°C. 
 
 The nuclear temperature coefficient of a reactor depend* en 
 changes in k^ as wall as on changes in neutron leakage. In the early 
 literature tvo such effects were recognised: 
 
 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 water-mod- 
 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 tempera- 
 ture in the presence of non-l/r absorbers. For graphite, 
 natural uranium reactors, the departure from the l/v lav 
 is small and is generally neglected. 
 
 -3- 
 
2. Pe ppier broadening of ohe resonance bajods in uraniur. 
 The resonance absorption of neutrons in U^ s increases 
 with tenperature due to the broadening of the resonance 
 bands. We have here the first contribution to the uni- 
 form temperature coefficient which depends primarily on 
 the temperature of the fuel rather than that of the Mod- 
 erator. The effeet is important to the stability of 
 graphite-uranium reactors since the fuel temperatures 
 are the first to respond to any sudden power rise. 
 The West Stands 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 (CP-*U-6). By the simple experiment of opening the windows, 
 the temperature of the West Stands reactor was brought down from 22£-°C 
 to 14°C. 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 aftor correcting for barometric effects, 
 a uniform temperature coefficient of -3.8 x 10~V° C 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 10" 5 /°C. 
 
 In a February 6, 1943 progress report (CP-4-55), however, we find 
 that the fuel temperature coefficient was measured and led to a value of 
 
only -1 x 10" Vc. The experiment consisted of raising the temperature 
 of Insulated metal lumps In the central portion of the reactor about 
 300°C and measuring the change In reactivity. This result meant that 
 they had to look elsewhere for a large temperature effect and Feral 
 suggested immediately that it might be found in the variation of °? with 
 neutron energy. 
 
 Following this experiment a theoretical discussion of temperature 
 effects In graphite-uranium reactors was given in a short paper by Mor- 
 rison (GP-478). He considered both lumped cubic lattices and rod lat- 
 tices. On the basis of theoretical results of Teller and Metropolis 
 (CP-387) on chemical binding effects in graphite, he estimated tee mean 
 temperature of these lattices in the neighborhood of 400°K. 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 300°K to 450°K. 
 
 Morrison's estimates of the temperature coefficients of reactivity 
 
 of these lattices Is given in the following tables 
 
 Lft j foe e Tjze Jf_ 
 
 #1 B* cubic lattice, 6 lb lumps 400°K 
 
 #2 12" cubic lattice, 48 lb lumps 410° 
 
 #3 8 n rod lattice, 1.6 cm radius 412° 
 
 Temperature Coefficiect of Reactivity 
 
 kffilce leakage leveling ^mlffT Pr?nf?W , rg Eta Effect Total 
 
 #1 -4.5xl<r 5 /°C 10.dxl0-5/°C -1.3 x 1CK>/°C -9x10" V°C -4xlO-5/°G 
 
 #2 -2.9 15.3 -5 6 
 
 ii -2.7 4.6 -9.7 -9xl<rV°C 
 
 -5- 
 
The first table fires Morrison's estimate of the neutron temperature 
 of these lattices. His estimate of the ^-coefficient of reactivity 
 was based on the experimental value of the uniform ten^erature coef- 
 ficient of the Vest Stands reactor (lattice #1). The large value of 
 the °7 -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 la'-tice. 
 
 Morrison also considered for the first time the hardening of 
 the neutron spectrum in the fuel elements at least insofar as it af- 
 fected the calculation of the 'h -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. He came to the conclusion, shown by the above table for lattice 
 f 2, that a reactor with large metal elements would be unstable with 
 regard to temperature. 
 The prafldca-ft?rt»g-ftirshall Scpertment 
 
 in atterpt to determine the dependence of ^ on temperature 
 was carried out by Bragdon, Itighes and Marshall early in 1944. By 
 using foils of normal and depleted uranium located at the center of 
 a spherical cavity In a thermal column, they measured the ratio of 
 thermal fission in IUjj *° radiative captures In Ugg. In order to 
 obtain the seme ratio at a higher neutron temperature, they them sur- 
 rounded the foil with a spherical shell of silver of 106 mil thickness. 
 
The silver acted as a filter for low energy neutrons and Bragdon, 
 Hughes and Marshall estimated the effective neutron temperature 
 change at 160°C. They obtained a value of 
 
 1.0211 * 0.0046 
 for 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 "7 with temperature was then esti- 
 mated as 
 
 ■X Ji = .5.2 x io-5/bc . 
 
 At the time, the authors assumed that the radiative capture cross- 
 section of U23 5 was zero. However, re-analysis of their work by 
 Friedman and Kaplan at Brookhaven (BNL-152) on the basis of a fin- 
 ite but temperature independent value of Ji « ^(25)/ ^f(25) results 
 only in a slight reduction in the value of the °? -temperature coef- 
 ficient. 
 
 Attempts have since been made by Uigner (DJV-2071) and by 
 Sampson and Hurwits in August 1951 (KAPL-511) to determine the tem- 
 perature dependence of <£ in the thermal region by the use of approxi- 
 mate formulas for the temperature dependence of <r c (28) and <^(25). 
 Wigner obtained a value of -|£ - -22 x 10~ 5 /°C on the basis of the 
 Bragdon-Hughea-Mar shall experiment which he believed to be unlikely „ 
 Sampson and Hurwitz concluded that the best value indicated by the 
 Bragdon experiment was 
 
 i£ in * k ~ irv-5 
 
 d 
 
 Y ~ -10 * 5 x icrV°c 
 
 -7- 
 
Sampson and Hurvits also attempted to obtain »£• fro* neutron speo- 
 
 d T 
 
 troneter data which they thought indicated a positive value of about 
 
 10 x lCrV'C. They then obtained a value of 4-21 - -29 i KTty^C which 
 
 d T 
 
 is In complete disagreement with the results of the early reactor ex- 
 peri wants which we have already discussed. However, even for -££ * 0, 
 the cross-section formulas assumed of these authors led to a value of 
 
 -18 z 10~V°C ior "^ • Cross-section data available to us at Brook- 
 * T 
 
 haven through D.J. Hughes and his group lead to more reasonable values 
 
 of -^2 . However, it appears that at present we must still torn to 
 ^ T 
 
 reactor temperature coefficient experiments for our best estimate* of 
 the variation of -») with temperature. 
 Theory of the Doppler Broadening Effect 
 
 In January 1942, Vlgner (C-4) pointed out that resonance ab- 
 sorption in uranium lattices could, in good app roximation, be repre- 
 sented by a volume absorption and a surf ace absorption term. Later ex- 
 periments by Grouts (C-llo), Hughee (CP-3580), Ifaelhanse (AKW323) and 
 Rlsser and others (GRIL-958) have borne out this division of the resom 
 anee absorption and the method is universally used in resonance escape 
 factor calculations. In addition, Vigner gave the formulas for the 
 shape of the Doppler broadened resonance lines. Thus if the level struc- 
 ture were known, the Doppler temperature coefficient could be readily 
 computed. Since the actual resonance structure was largely unknown, 
 Vigner and later Dancoff (CP-1589) worked with various simple models 
 of the resonance structure to obtain the temperature coefficient of 
 the volume absorption 
 
According to Wigner's method the effective resonance integral 
 
 la given by 
 
 A + B 3u*frV?« 
 nass 
 
 where A and B are respectively the volume and surface absorption inte- 
 grals. Wigner obtained a value of 3 x 10"V°C for the value of A AA. 
 
 A dT 
 
 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 betveen 1 2 and 1.7 x 10~*/°C. For graphite- 
 uranium reactors, Kaplan and Friedman (BIL-152) shoved that this esti- 
 mate leads to values of 
 
 A H$ - -1J> to -2.0 x 10~ 5 /°C 
 p ax 
 
 for the effect of the Dopplor broadening on the resonance escape factor. 
 
 There also exists some u activation measurements by Craut* 
 
 (C-110) and Mitchell (CP-597) at elevated temperatures which yielded 
 
 values of i 34 of 1.7 x 10"^/°C and 1.1 x 1C~*/°C respectively. The 
 A dT 
 
 spread of the experimental results almost coincides with Dancoff *s 
 theoretical results, 
 frwrfaents on the I-J.Q Reactor 
 
 During the start-up end early operation of the Oak Ridge graphite- 
 uranium reactor, an extensive study of the reactivity coefficients was 
 undertaken by L.B.Borst (!*bnP-60 and Cfc. 13 of the National Unclear 
 Energy Series TV-5). 
 
 Several attempts were made to obtain the barometric coefficient 
 of the reactor. Of these, the most accurate value was determined In a 
 
 -9- 
 
long run at constant power daring which a change of atmospheric pres- 
 sure of 5.8 bb& Hg occurred within a period of 6 hours. From these ex- 
 periments a value of -0.4 * 0.05 inh/ma Hg was established for the 
 barometric coefficient. In subsequent xenon experiments , it was found 
 that values ranging from -0.3 to -0.5 inh/n«i Hg were 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 i3 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 ma^e to obtain the uniform temperature coefficient 
 of the X-10 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/°C was obtained which was quite low compared 
 to data obt-.inod in later experiments . Similarly, an effort was made to 
 estimate the metal temperature coefficient of the reactor from data on a 
 heated slug. Fend, had suggested following the transient temperature 
 changes in a metal slug as a method of power calibration and from the 
 associated control rod movements, Borst fourd a uniforn metal temperature 
 coefficient of -1.2 inhours/^C 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 unl- 
 
 -10- 
 
form temperature coefficient of reactivity of the X-13 reactor (in 
 vacuo) was -5.9 x lO^AC, of which only 
 -1.0 x 10r 5 /°C 
 
 was attributed to the metal temperature coefficient. 
 
 It is of interest to compare Borst's results with Fermi's 
 estimate of February 1944. (OP-1329) of the uniform temperature coef- 
 ficient of the Clinton reactor. He split the various effects up as 
 
 f ollcus : , 
 
 leveling: 6.5 1 1C~V°C 
 
 leakage: -3.1 x 10" 5 /°C 
 
 Doppler Broadening: -2 1 1C~ 5 /°C 
 
 Bta-Effect: -5.5 x HrV°C 
 
 kgff : -4.1 x 10-5/°C 
 
 The estimate of the "*j -effect vas based on the experimental results 
 of Brazen, Hughes and Marshall. In early design calculations for 
 the BKL reactor, Kcplan and Friedman (BHL-152) obtained considerably 
 smaller values fcr the leveling effect by using elementary diffusion 
 theory. These were respectively 3.4 x 10~V°C for the Oak Ridge re- 
 ector and 2.6 x 1C~V°C fcr the BUL reactor. On this basis, the Oak 
 Ridge results are consistent with these of the Bra^ori-Hughes-icarshall 
 expert rent. 
 Brookhaven Reactor Exrc-rirents 
 
 During the start-up of the BHL reactor a considerable progrea 
 of experiments of use in reactor evaluation were carried out. The 
 
 -11- 
 
organisation of these experiments under the direction of L.B.Borst 
 was so efficient tht.t all the low power vork, including much of the 
 loading of the reactor to critieelity, was carried out in about two 
 weeks. We shall discuss hare only those experiments which are per- 
 tinent to our present topic. 
 The Barometric Coefficient of the EWL Reactor 
 
 The first opportunity to obtain the barometric coefficient 
 of the BHL reactor occurred on the night of August 23, 1950, while 
 the reactor was temporarily idle at a slightly subcrltiod loading. 
 The effective multip] ication 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 losing the 
 steady stnte neutron level of the reactor, being inversely proportional 
 to 1-koff , could be expected to vary by rbout 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 me ana of neutron 
 counters. The curves were corappred with accurate barometric pressure 
 and partial nitrogen pressure data supplied by the Meteorology group. 
 Or ophite and metal temperatures at various points in the reactor were 
 read frenuently but no detectable changes in the thermocouple readings 
 were recorded, possibly because of the insensltivity of these instru- 
 ments to small temperature changes. A total barometric change of 1.5 
 mm Hg occurred during the night. 
 
 The correlation of the neutron counting rate with barometric 
 pressure was fairly good, yielding a barometrio coefficient of 0„4.±0.1 
 
 -12- 
 
inhours/aB Hg. The correlation with partial nitrogen pressure was not 
 nearly as good, thus indicating that there ve.s little actual ariTing of 
 the outside air with that inside the reactor. If more time were avail- 
 able, it would have been q-.iite interesting to pursue this study for 
 longer periods of tine end for loadings closer to criticalitr in order 
 tc learn mere about the terporrl behavior of a free reactor which is 
 just sufficiently subcriticrl to be stable against diurnal barometric 
 end therrsal fluctuations. 
 
 Efforts to determine the barometric coefficient of the BUL re- 
 eetcr by control rod chants necessary to msintein constant power during 
 shsrp weather changes led to no great irrrrovarent in the precision of 
 the barometric coefficient. It was therefore decided to use the reactor 
 fans to sinuate the b?jrometric effect. By sealing off the inlet air 
 d'Jcts zrti operctirg one or more fens, a uniform pressure drop ranging 
 frra 15 to 53 ma Hg 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 shown in the following table. 
 Barometer Coefficient of EML Reactor 
 
 Critical 
 
 
 
 
 Barometric 
 
 Position 
 
 Pressure Drop 
 
 Change in 
 
 Chsnge in 
 
 Coefficient 
 
 of #9 Sod 
 
 (nan He) 
 
 Pressure [m %) 
 
 Reactivity (lb.) 
 
 (Hi/no He) 
 
 425.2 c« 
 
 
 
 
 445.5 
 
 15.4 
 
 15.4 
 
 5.6 
 
 0.36 
 
 470.8 
 
 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 Man Yalue obtained was 0.35 ± 0.014 inh/nm Hg. Extensive eontrol 
 rod calibrations vera carried out, both previous to and during the ex- 
 periment. It was estimated that errors in the sensitivity of the #9 
 control rod vsre about i%. A possibly more serious source of error of 
 about .03 lah/aa Hg in Indicated by the fact that the final critical 
 position of the eontrol rod differed somewhat fron Its Initial position. 
 1 change of 1°C In the over all reactor temperatures, which were not read 
 during the experiment, could aeeount for the discrepancy. 
 
 DnlfW. Ty-n^tarr* fluffs art —t nf th« EWI. Barter 
 
 The uniform teaperature coefficient of the BJTL reactor was meas- 
 ured on two different occasions at essentially zero 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, netal 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 cognisant 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 were found to vary appreciably at different points In the reactor. 
 
 The uniform temperature coefficient of the reactor uncorrected 
 to vacuum was found to be -1.26 ± 0.09 inhours/^C. Of this coefficient, 
 
 -U- 
 
the contribution of tee metal coefficient was -0„78 ± 0.16 inhours/°G 
 while that of the graphite coefficient was -0.48 ± O.lt inhours/^C. 
 After correcting the latter for barometer we find a uniform tempera- 
 ture reactivity coefficient of -5.7 x 10 -5 /°C of which -2.0 x 10~ 5 /°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 Bragdon-Hughes-Marshall experiment. 
 The Metal Temperature Coefficient 
 
 Our first attempt to determine the metal temperature coefficient 
 of the BIL reactor Involved the use of insulated metal cartridges heated 
 to about 450°C 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. "Die control 
 rod was then adjusted to bring the reactor slightly below critical. This 
 procedure was 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 o 0093 * 0.0C04 inhours/'C was observed. 
 The use of statistical weight factors, however, yields a metal tempera- 
 tare reactivity coefficient of -1„A x 10~fy°C, which is about 30% lower 
 
 -15- 
 
than the values obi.rined in other experiments. 
 T error fiture Flesh Experiments 
 
 The most accurate velue of the metal temperature coefficient 
 of tho BHL reector vas obtained in temperature flash experiments during 
 which the reactor \ms permitted to run away snd brought under control 
 by the increase in metal temperatures alone. The method used vas simple 
 in conception. The repctcr cooled to ambient teraperatTe, was started 
 up without fans from a negligible '-over level with an initial period 
 which renged from 3& sec to <t minutes. There was little change in 
 metal temperatures until the integrated power output of the reactor 
 hrd risen to the 10 lew min level. Thus the initial reactivity of the 
 reactor co .Id be determined from the galvanometer period. At the time 
 of maximum gplvpnometer reading, the reactivity would be zero and the 
 corresponding metal temperature at the position of maximum flux could 
 be used to obtain a metal temperature coefficient for the reactor. 
 
 Of considerable interest in these experiments was the fact thot 
 the rower output and reactor temperature oscillated through several dis- 
 tinct maxima and minima before damping out. The grarhite thermocouples 
 registered no intense in temperature until one or more such oscillations 
 had occurred. This was expected because of the much larger heat capecily 
 and weight of the moderator relative to the fuel End the fact that the 
 maximum power output reached was only of the order of a Md. For the low 
 temperature flashes, the tedium of waiting for the reactor to reach sig- 
 nificant "O"or levels wes reduced by bringing the reactor up rapidly to 
 tlie SW region and then sotting the controls to the proper position. Since 
 
 -16- 
 
the uniform temperature coefficient of a reactor depends on the dis- 
 tribution of temperature a3 well as flux, readings of available metel 
 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 plane parallel to the reactor air gap was closely 
 proportional to the theoretical flux distribution. 
 
 In the first temperature flash which took place on October 10^ 
 1950, the metal thermocouples 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 (Bovember 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 and adjacent aluminum 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 flash experiment appears to be the best possible method of 
 determining the metal temperature coefficient of a gr.'phite-ursfiiura 
 reactor. Despite the fact that under the different initial control 
 settings the maximum metal taraperetures ran from 60°C to 200°C, the 
 metal temperature coefficients varied very little with temperature. 
 An average vali;e of -0.4.9 inhour/°C rise in maximum metal temperature 
 was obtained which corresponds to a uniform reactivity coefficient of 
 
 -17- 
 
-2.0 x 10v°C. Under strictly adiabatie conditions, the theoretical 
 
 value of the marl rums overshoot in temperature in a reactor temperature 
 
 flash experiment is tvrioe the equillbritna value and this value vas nearly 
 
 attained when the reactor was started up with an initial period of 36 
 
 seconds. 
 
 Temperature Coaff icienta of Betmral pr^ma Water lattices 
 
 Before leaving the subject of temperature coefficients of re- 
 actors we should like to quote sone 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 cross-over from negative to positive values of reactivity 
 oecurred at a water-metal volume retio of 1.6. Experimental values 
 at other geometrical arrangements ere given in the following table: 
 WatW *etel Ifrtto ReaffWtt ftefficlenA 
 
 lo2 -1 x wrVc 
 
 2.1 +1 x 1CH-/°C 
 
 3.1 +2 x HK/°C 
 
 The e x perimental results were based on measured buddings of the lat- 
 tices at about 25° C and 75°C. Changes in the migration area with den- 
 sity were calculated. The experiments show the tendency of leveling 
 effects in water le tices to produce positive temperature coefficients. 
 Theory of Temperature Dependent Reactor Kinetics 
 
 Ve 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 safe operation of a reactor. When temperature ef- 
 fects are considered, the equations governing the system are non-lin- 
 ear and generally must be solved by numerical methods. All of the 1EC 
 laboratories have done considerable work on these probleas as they af- 
 fect their pcrticuler reactor types 
 
 For small deviations from steady state conditions, perturbation 
 methods such as those outlined by Bordheim (JDDC-35) are useful. For 
 large deviations from equilibrium, more exact methods are required and 
 m=ny of the problems have been solved only with the aid of high speed 
 electronic computing machines. 
 
 Tho damped, almost periodic nature of the temperature and flux 
 curves obtained in the BHL temperature flash experiments aroused our In- 
 terest in the non-linear aspects of the problem. 
 
 If the delayed neutrons are assumed to act only to increase the 
 effective neutron generation time as is the ease if the reactor periods 
 are not too short, then the transient equations take the simple form 
 dn/dt * ^k^ 
 
 Kx ■ (*ex>c "* T 
 dT/dt « «n -AT 
 
 where l/£ is the average neutron generation time, fi is the temperature 
 coefficient of the reactor and 1 is the relaxation constant for heat 
 removal. In the case that (k ex ) e , the excess k introduced by a control 
 rod is constant, the temperature equation may be redueed to the dimension- 
 less non-linear form 
 
 -19- 
 
e + (i-c)e - oe - -«(©e + e 2 ) 
 
 In the single parameter e ■ ^(t^x^e/A . 
 
 Solutions of this equation, obtained by the method of Isoclines, 
 are shot*i In Slide No. 1-116-3 (Unclassified) 
 
 The first slide shovs a phase diagram (rate of change of tem- 
 perature vs temperature ) when the reactor control setting is aboTe the 
 cold critical setting. If the reactor has a positive tenperature coef- 
 ficient, the solution curves all diverge with tins, regardless of initial 
 conditions. 
 
 There are two possible equilibrium positions for a reactor with 
 a negative temper;- tore coefficient. One occurs at aero flux and aero, 
 i.e., ambient temperature and is unstable in the same sense that an in- 
 verted pendulum is in unstable equilibrium The second equilibrium point 
 is a stable one. The equilibrium point la a spiral if the reactor cool- 
 ing rate is sufficiently low. On a time basis, the solutions converging 
 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 non-linear 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. 1-115-3 (Unclassified) 
 
 This slide shows what happens when one attempts to shut down a 
 reactor. The control setting is below the cold critical position and the 
 
 -2CU 
 
solutions for a rena^or with a negative tenpercture coefficient are 
 very uninteresting from the mathematical viewpoint. Regardless of 
 initial conditions, everything converges to the stable equilibrium 
 point. A reactor with e ro3itivo temperature coefficient, however, 
 has two possible eouilihrium 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- 
 stable regions. Only in the stable region do the solution curves con- 
 verge to the stable equilibrium point. 
 
 In BHL-173, we have investigated the nature of the solutions 
 of non-linoar equations of the present type. Of practical interest 
 is the fact thnt the snecinl solutions which separate the stability 
 regions of a reactor ara of relatively simule form. 
 Statics of a Reactor with vVjiahle Local TfuJ ti-llcation 
 
 In a reactor operating at rower, the fuel and moderator tea- 
 peraturos as woll ?s poison concentrations vary with position. The 
 local multiplication factor is therefore variable find the flux dis- 
 tribution associftod with the clean reactor is distorted. 
 
 Reactors nay be capable cf sovernl quasi-steady stntes which 
 are fairly well separated with time. 'Pius during the start-up of the 
 BHL reactor after a long shutdown, the fuel tenporatures come 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 vnriation in the local molti- 
 
 -21- 
 
plioatlon factor is different for each of these approxiaate eouill^- 
 hriua positions. 
 
 In BNL-126, the statics of a reactor with verieble local 
 multiplication have been considered on the basia of one and two group 
 methods for certain cases and the results \/ere compered with pertur- 
 bation theory approximations. Thus the solution of the one— group 
 equation for a critical slab reactor with Sk local proportional to 
 flux was shown to be given by an elliptic integral of the first kind. 
 The xenon problem with burn-up was solved exactly for various ono- 
 dlmenaional cases. Equilibrium temperature condltons for a rnaotor 
 in which the important v.-jriation of temperature occurs only along a 
 cooling channel w.s reduced to a integro—diff erential equation which 
 can be solved by numerical methods. 
 
 In general, it wrs found that except in very extrene cases, 
 the usual statistical weight formulas gave very good agreeaent with 
 exact solutions. 
 
 The statistical weight foraulas for the effect of xenon at 
 flux levels were burn-up is important "tre considered in Can«3ian re- 
 ports by Goldstein and Guggenheim (KT-2C-G) and Rennie (CST-272). In 
 EIJL-126, we heve checked Lhoir formulas over a wide range of flux levels 
 by an alterna .q method f.nd obtained good r.greemento 
 
 -22- 
 
PHASE PLANE DIAGRAM OF TEMPERATURE DEPENDENT 
 REACTOR KINETICS 
 (CONTROL SETTING ABOVE COLD CRITICAL) 
 
 NEGATIVE TEMPERATURE 
 COEFFICIENT 
 
 -23- 
 
PHASE PLANE DIAGRAM OF TEMPERATURE DEPENDENT 
 
 REACTOR KINETICS 
 (CONTROL SETTING - BELOW COLD CRITICAL) 
 
 Slide No. 1-115-3 
 
 -2k- 
 
UNIVERSITY OF FLORIDA 
 
 3 1262 08229 985 9