T TLEGAL NGTicE .. This report was prepared &5 an secount of Government sponsored work. Keither the Usited t Btates, nor the Commiseion, nor any person acting on behzlf of the Commission: . A. Mzkes any warranty or representution, expregsed or implied, with regpect o the acou- i raey, completeness, or usefulness of the informstion contained in thig report, or that the use - of any information, apparatus, method, or process diselosed in this report may not infringe o privately owned rigits; or B. Assumes any Habilities with respect ‘o the use of, or for demages resulting frem the T use of any information, apparatus, method, or process disclesed in this report. B As used in the sbove, ‘‘person acting on behaif of the Commission® includes any em- . ployee or contractor of the Commission, or employee of such vontractar, to the extent that o such employee or contractor of the Cemmission, or employee of such contractor prepares, : disseminates, or provides access to, any information pursusnt to his emplovment or coentract = with the Commission, or his emplo'yment with guch contractor. LT IRCULATION: T s ot fl'!e Comml £510H, Re? 4Ry person Gc?mg o bfl%u%? of fhe Ccm i | LEGAL MOTICE :fepsri‘ wns prepared ag an ncc@un? of chemm@n‘? s;;eonsm*ed wark Nelfl’!ef ?he Umteé Srmes L '-A -Makes any wqrran?y & reflresenmhe“ o compleleness ar o mf@rmaiz:s' a-p'quus, m&‘rho& - prwéfely @wneé rightsyor . Assumas any iminir : flfiy “information; agp '-:As u&eé T the ohows, s cen?racrar @f flxe Csmzm' e canh'ucfck @-t" ?he Cam_ ‘Q i ORNL-TM-1626 Contract No. W-T4(C5-eng-26 REACTOR DIVISION PERIOD MEASUREMENTS ON THE MOLTEN SALT REACTOR EXPERIMENT DURING FUEL CIRCULATION: THECRY AND EXPERIMENT E. B. Prince NNCUNCEMENT L e e B LEGAL MOTICE Thie repor{ wae prepared as an account of Government spongored werk. Neither the United States, nor the Commisgion, nor any person acting on behalf of the Commisasion: 4, Mzkes sny warranty or represeitation, expressed or implied, with respect i the accu~ racy, completeness, or usefulness of the information centained in thie repert, or that the use = of any information, appsratus, methoed, or process disclosed in thiz report may not infrings privately owned righis; or B, Aspumes any liabilitics with respect to the use of, or for damuges resulting {rom the uze of any information, apparatus, method, or process disclesed in this report, Az used in the sbove, ‘‘person zcting on behzif of the Cemmiseion® ircludes any em- 8 ployee or contractor of the Commission, or employee of such couiractor, te the cxtent that such employee or contractor of the Commigsion, or employes of such contractor prepares, disseminates, or provides access to, any information purauant to his employment or contract with the Commigsion, or his employment with such contractor, AR RO, i t—fl b o= i rai o v e #’ l-:! R S RN S AN OCTOBER 1966 OAK RIDGE NATTONATL, LABORATCRY Oak Ridge, Tennessee operated by UNION CARBIDE CORPCRATION for the U.5. ATOMIC ENERGY COMMISSION | e - CONTENTS Page ABSTRACT . v v v v v v e b 0 e e e e e e e e e e e e e e e e e 1 INTRODUCTION . o v v v v v v v e vt v e e e e e e e e e s e e e 1 THECRY OF ZERO POWER KINETICS DURING FUEL CIRCULATION . . . . . . 2 NUMERICAL AND EXPERIMENTAL RESULTS . . . + « « « « ¢ v v o o « o 27 Numerical GroundwOrK .« o v « « o v o o s v e e 6 e e e e e e 17 Experimental Results . . . . . « « « ¢ ¢ v v v o o s s . 25 DISCUSSION OF RESULTS . & © v v ¢ v v v v v e o o o o s v e o v s 29 Theoretical Model Verification . . . . . . « « + « & « « « & 29 Recommendations for the MSRE . . . . . ¢ + . « . « « « « . . 31 ACKNOWLEDGMENT e e e e e e e e e e e e e e e e e e e e e e 32 REFERENCES . .+ ¢« v« v v 0 v o e v v e o 6w o v 6 o o« v o 33 @ PERICD MEASUREMENTS ON THE MOLTEN SALT REACTOR EXPERIMENT DURING FUEL CIRCULATIOCN: THECRY AND EXPERIMENT B. E. Prince ABSTRACT As an aid in interpreting the zero-power kinetics experi- ments performed on the MSRE, a theory of perlod dependence on the fuel circulaticon is developed from the general space de- pendent reactor kinetics equations. A procedure for evaluat- ing the resulting inhour-type eqguation by machine computation is presented, together with numerical results relating the reactivity tc the observed asymptotic period, hoth with the fuel circulating and with it stationary. Based on this analy- sis, the calculated reactivity difference between the time independent flux conditions for the noncirculating and the circulating fuel states is in close sgreement with the value inferred from the MSRE rod calibration experiments. Rod-bump period measurements made with the fuel circulating were con- verted to differential rod worth by use of this model. These results are compared with similar rod sensitivity measurements made with the fuel stationary. The rod sensitivities measured under these two conditions agree favorably, within the limits of precision of the period measurements. Due to the problem cf maintaining adeguate precision, however, the period-rod sensitivity measurements provide & less conclusive test of the theoretical model than the reactivity difference between the time independent flux conditions. Suggestions are made for improving the precision of the experiments to provide a more rigid test of the theoretical model for the effects of cir- culation on the delayed neutron kinetics. 1. INTRODUCTION It is well recognized that the circulation of fuel in a liquid fueled reactor introduces some unigue effects into its observable kinetic behavior. Foremost in this category is the phenomenon of emission of delayed neutrons in the psrt of the circulating system external to the regctor core where they do not contribute to the chain reaction. The literature in reactor kinetics contains several theoretical studies of this effect,? 223 but there have been few copportunities for parallel experimental studies. The Molten Salt Reactor Experiment, hereafter Mo referred to as the MSRE, has presented a urnique chance to develop and test some aralytical models for the zero power kinetics of a circulating fuel reactor. The experimental measurements discussed in this repcort were made as part of an overall program to calibrate the control rods in the MSEE. In the first secticn fellowing, we develop the thecry of periocd dependence on fuel circulation. The second section gives an account of the results of applying this theory to the experimental measurements mede during MSRE Run No. 3. Finally, in the third section, we discuss the results of this work in terms of the general problem of kinetics analysis for circulating fuel reactors Several questions are lelft open @ by the present study, and these are discussed in that section. 2. THEORY OF ZERO POWER KINETICS DURING FUEL CIRCULATION At negligible reactor power, the special effects produced by fuel circulatior reduce essentially to (a) the transport of delayed neutron precursors tc that part of the lcop external to the core where they subsequently decay, and (b) the skhift of the spatial distribution of delayed neutron precursors in the direction of circulation within the reacter core, relative to the prompt neutron production. Although we skall not attempt to review all eariier studies pertalning to these effects, twe fairly recent studies made by Wolfe® and Haubenreich® are pertinent to the present work., Wolfe employs a perturbation approach anc obtzins an irhour-type eguaiion for ar infinite sliab reactcr, through which 6 “uel ecirculates in the direction of variation cf the neutron flux. Wolfe's = sppreach, while valusble, ig alwmost entirely formal, and requires some modificaticr in corder to cbtain a procedure useful for the guantitative analysis of experimental messurements. In the second of the above men- Tioned works, Haubenreich considers an explicit analytical model rep- reserting the MSRE in the circuiating, Jjust-critical conditicn. By means of & modal analysis, he obtalns effective values for the delayed neutron fractiong for this conditicn. He uses a bare cylinder approximation to f ] L For a comprehensive early study, see Ref. 1. L represent the MSRE core, with boundaries corresponding physically to the channelled region of the actusl core. We have combined and extended thege analyses to include the contri- bution to the chain reaction of the delayed neutrons emitted while the fuel 1s 1n the plenums Jjust sbove and below the graphite core, and o include the case of the flux varying exponentially with a stable asymp- totic period. It will be seen that an inhour-type equation results which relates the period of a circulating fuel reactor to the static reactivity of the same reactor configuration, but in which the fuel is not circulating. The static reactivity, p_, is defined by the relation w2 P, = — = 9 (1) in which v 1s the physical, energy-averaged number of reutrons emitted per fission, and V. is the fictitious wvalue for which the reactor with the same geometric and material configuration would be critical with the fuel stationary. One finds that as a result of this definition, the reactivity for a critical, circulating fuel condition is greater than zero (here, criticality derotes the condition of time independence of the neutrcn and precursor concentrations). However, since the static reactivity is the quantity normally obtained in reactor calculation programs, it 1s most convenient to relate this quantity directly to the asymptotic period., The theory following 1s developed by beginning along generasl lines and delineating special assumptions as they are introduced. Several of these simplifying approximaticns are suitable for MSRE anzlysis, both in the neutronics model and the flow model for the circulating fuel. The starting point of the analysis is the general time dependent reactor equations, written fo include the transport of delayed neutron precursors by fuel motion in the axial direction: 6 L@’é-(lmB)fP@f'!‘--!?‘,f C:v“lé?i (2) T/ "p /071 Tdi i ot ’ Q/ ! i ot ’ 1 =1,2, « .« , 6 . (3) 9 _ By B8 - G - 55 (VCy) = The symbeols ¢ and C represent the neutron flux and delayed neutron pre- cursor densities. The operator L represents net neutron loss (leakage, absorption, and energy transfer by scattering), and P represents the production processes by fissicn. The explicit representation of these operatcrs depends on the model used in analysis. ITf these processes are given their most general representation in terms of the Boltzman transport equation, the neutron flux, ¢, will be a function of poesition, energy, direction, and time variables. In order to provide a framework for discussion which can ke directly related to application, we shall agsume that the angular variables have been integrated from the equation, i.e., that a model such as multigroup diffusion thecry or its continuous energy counterpart prcvides an adequate description of the neutron popu- letion. In Eq. 3, azbove, P¢ is taken as zero in that part of the cir- culating loop which is external to the chain reacting regions. The remaining symbols in Egs. 2 and 3 are (1 - BT}, the fraction of all neutrong from fission which are prompt, and Bi and Ki, the producticn fraction and decay constant for the ith precursor group. The quantities f and f&i are energy spectrum operators which multiply the total volu- metric production retes of prompt and delayed neutrons to obtain neutrons of 2 specific energy. Finally, the symbols v and V represent the neutrcn velocity and the fiuld velocity, respectively. As applied to a fluid fueled reactor with a heterocgeneous structure such as the MSRE {where the axizl fuel channels are located in a matrix of solild graphite moderator}, the usual cellular homogenization must be made on the neutron production and destruction rates. Thus, for example (1 - BT) P% = local rate of production of prompt fission neutrons, per unit cell volume Xi C, = local rate of production of ith group of delayed neutrons, per unit cell volume. If we assume that the operators L and P are time independent, corre- ponding to a fixed rcd position, we can investigate the conditions under which the flux and precursor densities in the reactor and the external . . . A wt , . - circulating system vary in time as e . Assuming solutions of Egs. 2 and 3 exist of the form: 8(x, E, t) = #(x, B; v) & (1) wt Ci(Zi: t) = Ci(.}i; W) e 3 (5) then, 6 2 4 Lfi+(1msT)prg+l>L+fdj_c1:v wid (6) i=1 B. PZ - h.c mi(vc):wc i = 1,2 6 (7) =4 i"i T 3z i i’ T e s ’ ‘ As previously stated, cur primary objective is to relate the observed It stable reactor period, w , to the static reactivity of the given reactor configuration. The static reactivity defined by Eg. 1 is the algebra- ically largest eigenvalue of the equation: w,*(L-p ) TR A =0 (8) where; & e — - % o / R D L T (9) i=1 Rather than attempting to solve the reactor eguations 6 and 7 directly, we will make use of a procedure which is often useful for spatial reactor kinetics problems. Several sources give details of similar analyses for stationary fuel reactor systems (see, for example, Ref. 4). We multiply the "local" equation for the neutron population by an appropriate weight- ing function and integrate over the position and energy variables of the neutron populaticn in order tc obtain a relation involving only “globsl" or integral quantities. As seen below, by using the static adjoint flux, + . . . ‘ ] g Qé, the sclution of the adjoint eguation corresponding to Eq. 8 as the weighting function, the resulting relstion admits a direct physical inter- pretation. The adjoint flux, also obtained in most reactor physics calcu- lation programs in common usage, is the solution of LT d (-0 )FR)T A =0, (10) + - F where I, and (fP) the ssme algebraically largest eigenvalue of Eq. 8. With appropriately are operators adjoint to L and P of Eg. 8, and p, is prescribed boundary conditions on the allowable functions on which these operators are defined, the following relation® can be used to define the * sdjoint operator; o0 = (8 A7) (11) + (7, A8) = (2 Here A represents abstractly either of the operators L and TP and (Q:, A7) denotes the scalar product, i.e., the multiplication of A by Qg and integration over the position and energy varisbles of the neutron population. On forming the scalsr product of Eg. 6 with Q:} we obtain + -1 + , + ) (g, v = (1) + (1 - B, 2 e ¢ Similarly, forming the scalsr product of Eg. 10 with @ and making use of Eq. 11 gives ’+ A = (8., 18) « {1 - p M@, TPF) =0 . (13) Combining Eq. 12 and Eg. 13 gives ¥For the purposes of reactor physics studies, we need to consider oniy real valued functions. + -l A+ + W v B = - (1 - 0 WF,, TRO) + (1 - B, £ P0) + S b { + + \ZJ f\,i(QfS? fdi Ci) . i=1 By making use of Eg. 9, we may rewrite this equation as . 6 6 + -1 ' 4 : - 7 7 oy / = 3\ ® i: i::l P, =W + . (1h) S + o= A = (7, £ PF) (7, T PF) The first term on the right hand side of Eg. 1L is simplified in appear- ance 1f we make a conventional type of definition of +he proxpt neutron generation time; (7, v 9) A = e (7, T PF) [ =3 : (15) ) 1 We may also note that the second term on the right hand side of Eq. 14 appears as the difference between the weighted total production of de- layed neutrons and the weighted production of delayed neutrcone from precursor decay within the resctor. In accordance with ifts definition . by Egs. 1 and 8, the static reactivity is completely determined by the geometric and material configuration of the reactor, whether or not the fuel is circulsting in the actual reactor. The relationship betwsen 0 and w expressed by Eqg. 1k, therefore, has an expliclt physical inter- pretation. For example, if the fuel composition and contrel rod position are such that the flux is time independent when the fuel is circulating, w =0 1in Egs. 6 and 7. Eguation 14 then shows that the static reactivity for the Jjust critical reactor is numericelly egusli to the net difference in the producticn of delayed neutrons described above. In the nore gen- k- eral case when the flux is varying in time according toc a stable periocd, the first term on the right hand side of Eq. 14 will differ from zero, and also the effective decrement in production of delayed neutrons will differ numerically from the time independent case (ci and ¢ depend on ®w through Egs. 6 and 7. Eguation 14 is an inhour type relation which can te used as & foundaticn for an appr?ximate determination of Py given an cobserved asymptotic period, w . One may observe that it in- cludes the usual inhour relation for the stationary fuel reactor as a special case, simply by setting V = 0 in Eq. 7. Before discussing the practical use of Eq. 1k, we can exhibit the previous concepts in an explicit algebraic manner. From Eg. 7 we have, ep = (w+ )7 (B0 - (Ve . (16) Inserting this relationship into Ec. 14 gives if we define B, =8B, —————— (18) Equaticon 17 may be written as ......... 6 = — 6 T (81 - 7]"_) — g =Wl AA Lo w ot Ay " §j 1o (20) 121 * =] This equatiorn has the appearance of the ordinary inhour relation for the 4 stationary fuel reactor,” except that the effective prcauction fractions, §i are reduced by a quantity'7i which depenrnds on the circuistion rate and also on w. In addition, the last fterm on the righlt hand side of Eq. 20 appears because the zeroc point of the static reactivity was chesen To correspond to the cowmposition and geometry of the just-critical stationary el reactor. The usual inhour relation for the stationary fuel reactor is obtained by letting V —» O and 71 - 0 in Eg. 20. Althougkh Eq. 20 is ingstructive in discussing the net effects of fuel circulaticn, Iits sim- plicity is somewhat deceptive since ;£ depends in a complicated way on w, through Eq. 7. Therefcore, it will be mcre converient tc discuss the use of the inkcur relation starting with the form given in Eq. 1k. At first appearance, in order to use Eq. 14 we are required to solve Egs. 6 and 7 for g, Ci(é; w), and ¢(x, E; o), and also to solve Eg. 10 for p_ and é;(g, E; Qs)" Both are eigenvalue prcblems in which the algebraically largest real elgenvalues, ¢ ana Py are cf immediate interest. Not only does Eq. 14 reduce tc an identity if this procedure is ueed, but it is precisely the explicit solution of Egs. & and 7 that we wish to avoid. The great usefulness of Eg. 1l is In the basis it pro- vides for approximating the relation between P and the cbserved stable period @ t. The basic simplification results from assuming that the shape of the asymptotic flux distribution, ¢, is sufficiently well approximated by the static fiux distribution, éss From & physical stand- point, the validity of this appreximation is & consequence of the small- ness of the delayed neutron fraction, £ If this approximation 1s made, o and ¢ is substituted for ¢ in Eq. 7, this equation car be integrated arcund the circulating path of the fuel to cbtain the distributions of gdelayed precursors, c. . Before completing the analysis, however, we shall make a geccnd approximation to simplify the computation of the integrals occurring in Eq. 14. It can be assumed that the correction 10 for the difference in energy spectra for emission of prompt and delayed reutrons appearing in Eq. 14 can be calculated spproximately as a sepa- rate step. This is done by reducing the age for the ith group of de- layed neutrons from that of the prompt reutrons and modifying the static delayed neutron fractions, Bi’ by the relative non-leaksge prcobability factors approprizte to a bare reactor which approximates the actual core. Such an appreximetion can alsc be justified from an objective standpoint, since the correction for the differing "energy effectiveness” of delayed neutronsg relative to prompt neutrons i1s small compared to the effect of the spatial transport of precursors under considerstion. For the MSRE, calculations of the former effect are given in Ref. 2. The net energy correction changes the effective value of ST for B35 from 0.0064 to 0.00666 . Orne further remark should be made concerning Eq. 14. A given static resgctivity, ps’ corresponding to some time independent resactor configuration is also related through Eg. 14 to gll physicaliy allowable transient mcdes present ir the neutron flux after the final resctor con- figuration has been established. In this study, we have chosen to emphasize only the relstion between the circulation and the asymptotic mode. An spproximete analysis indicates that the remaining elgenvalues end eigenfunctions of Egs. € and 7 differ in certain fundamental respects from those of stationery fuel rezactors. For the purpose of this report, we ghsll not attempt to demonstrate this. Except for a brief return to this topic in a later seciicon, we will restrict attention entirely to the analysis of the stable asymptotic period measurements. The ccmpletion of the required anslysis consists of the integration of Bg. 7 arcund the circulating psth of the fuel. Obviously, even if @ is replaced by QS calculated from Eg. 8, further simplifications in the flow model are necegsary before the problem becomes amenable to practical computation. The approximationsg we have used in representing the circu- lating loop of the MSERE are shown schematically in Fig. 1. The model consists of a three region approximation to the actual core, representing the lower cr entrance plenum, the graphite modersted region, and the upper, or exit plenum, respectively. The core is represented as a right cylinder with volumes in the upper and lower plenums equal to those of V. 11 ORNL-DWG 66-5561 e s —— T o | ~—EXTERNAL PIPING {INCLUDING FUEL 1 I PUMP AND HEAT IN REACTOR QUTLET EXCHANGER) z=H UPPER TOP GF 3 INLET FLOW PLENUM ) 10 - PR ROD 7 DISTRIBUTOR E 2% He ° T ANC DCWNCOMER 2 09 7 _ | / £ F | 08 1 —-CONTROL ROD e L ——CONTROL. ROD PP THIMBLES € o7 THIVMBLE x (TYPICAL} £ 06 7 } 3 CHANNELED W 05 [ REGION w0 & & o4l 5 203} o DRIVEN ROD z 02 LOWER LIMIT e SCRAMMED ROD g ¢l LOWER LIMIT x \J o BOTTOM QF MOST GRAPHITE 220 LOWER PLENUM (@) MSRE CORE GEOMETRY {SCHEMATIC) (6} MODEL FOR NEUTRON:CS CALCULATIONS Fig. 1. Gecmetry of MSRE Core and Three Region Core Model Used in Physics Calculations. 1 Tthe actual MSRE ccre. This three-region model is a simplified version of that used for all previous MSRE core physice computations.? Fluld dynamics studies with the MSRE core mockup have indicated that the fuel velocity withian the graphite moderated regicn is very negrly constant over & large part of the core. Higher velocities occur in a small region about the core axis and near the outer radius. In the upper plenum, the flow is nearly laminar, whereas in the bottom head the flow distribution is complex due to the reversal in the flow direction between the peripheral downcomer and the graphite moderated region. In the present study, we have assumed the flow velccity in each region except the lower plenum to be constant (plug flow)} with the magnitude cf the linear velccity determined by Ve =5 (21) wWieTe Lk is the axial length of the kth region, and tk is the residence time of the fluid in the kth region. It will be seen lster that the precursor censitles in the regions of zero neutron flux (external loop) depenc only on the fluld residence time in these regions. Although the lower plenum is in a region of relatively low neutron lmportance so that several approximations are allowable, rather than assign a linear velocity tc this region it was considered more realistic s from 2 physical standpoint Lo treat the region as = "well stirred tank” with an average neutren flux and importance (adjoint flux) assigned to Within the graphite moderated region, the primary difference in the spatial distributions of prompt and delayed neutrons can be expected to be 1n the direction of fuel salt flow. As g first approximation, we have agsumed the velocity profile to be flat in the radisl direction across the entire core, and have neglected the radial averaging of the neutron production rates implied in the scalar product integrals in Eg. 14. This is equivalent to sssuming radial and axial separability of the neutron producticn rates and adjoint fiuxes. Thus, if we preaverage the fluxes over the radial coordinate, and comsider only the axial (Z) dependence of PQ% in the integration of Eq. 7, the problem reduces entirely to a one-dimensional calculation (Line mo@.el).‘)é With these simplifications, the required integration of Eg. 7 can ncew be completed. As written in the preceeding formulas, the precursor densities and neutron reaction rates sre the homogenized values, 1.e., normalized to a unit cell volume of the reactor. To integrate Eq. 7. account must be taken of the variation in the fiuid volume fracticn over the path of flow throcugh the reactor. IFf o is the volume fraction of fuel in the kth region and superscript (o) is used to indicate the pre- cursor densities and prompt neutron production rate in the fuel; c.(z) 1 1l o el(z) (22) fl o PP (z) = a P& (2) . (23) The values of o used for numerical calculations with the model of the MSRE core shown in Fig. 1b were 0.225 for the graphite moderated region, and 1.0 and C.91 for the top and bottom plenums, respectively. The explicit forms of Eq. 7 for the various regions become: a) Craphite modersted region o o 9 _ = s < < ! B, P2 (2) — (O + ) c; =V, 53 OE (Ai W) ¢, = 3T 0=t < t, In the sbove equations, Hc 1s the height of the graphite moderated region, Hwfic is the thickness of the upper plenum, and L-H is the effective length of the region representing the external piping and heat exchanger (see Fig. ib). As descrited above, the lower plenum region is treated by means cf & well mixed tank approximation, in contrast to the pilug flow model for L] i the remaining regions. In Eg. 27, tg is the average residence time and (PQfié)E is the average fission production rate for each element of fluid in the lower plenum. The boundary conditions for each region reqguire that the precursor concentrations in the salt, Cg} are continuous along the path of flow. Ls applied tc Eq. 27, these conditions are; cf(z = Q) = cg(z =Ly , (28) ci(t =t,) =cl(z=0) . (29) These conditions, Jogether with the continuity conditions given sbove, corpletely determine the solution of Ras. 24 through 27. The results cf iategrating the =bove differentiel eguatiocns are: Z Z _(Rg TW) s . (A, w)( 7 ) S o, - e P O , c dz' - e (2) = ¢(0)e £y B (20)e 7 - (30) 0 c H <2z < H; C 2=, z-7" n o+ W) - - o o '\}l + W)( 1;‘,7'1-\‘1 ) ?Z o ()\‘l + w)(vu )dzF e (z) = c/(H Je 1 B (2 )e + » (31) 1 1 C !JH S / H= 2 <1 z-H o) - ome T Ve c;(z) = ci(H)e , (32) =L =0 0 0 " +W)tfi o, Fi - e_(Kjfiw{)f}i’-a Ci(o) = Ci(L)@ + fii(P Qé)g “”““x;“$f;““““ 5 (33) 0. O {h, W)t (“3} _ Bl(P Qé)fi . O(L) _ Bi(P ¢é>£ 1—-e U ¢ (3h) Ciig ™ NoFw Ici X (h, + W)T, ° 3 The last of the equations given above results from averaging the precursor concentration obtained from solving Eg. 27 over the residence time tg' By use of the continuity conditions for the entrance and exit concentrsa- tions in each region, the above equations may be used to solve for c?(o), The result is- H «z' C . . f 5, . E(Ki+w)( v, * tu+tex+tfi) g e, o) = j B.F Qg(z')e T 1 o . H-27 E . -(y ) 7 byt . * J BiP fié(z‘)e v H u c § - 1 - e“(hi+w)tfl "(K1+W)tcirc +L-° . [ o + Bi(P Qé)fl | Xi o 1L —e > (35) where the following definitions of the residence times have been used: Hc ‘:C:{F’} (36) C H - HC o= s (37) Q L —H tex‘“ ~ (38) ex % =4+t o+ . cire c * tu tex * tfl (39) The computational procedure developed in this section may be sum- marized as follows: 1. Calculate spproximations to the static flux and adjoint flux axlal distributions by standard techniques of core physics analysis. 2. From a specified assymptotic inverse period, w, and the static flux distribution, € ., corresponding to the same reactor state, calculate ci(@;w) using Fg. 35. Note that the absolute normalization of the flux e arbitrary since Eg. l4 is independent of the flux normelization. 3. Calculate the axial distritution of precursor densitles in the salt, cg(z;w)? by means cof formulas 30 through 34. FEither a numerical integration procedure or analytical approximations for £ (z) can be used in evaluating the integrzls. The former method was used in the work described in the fcollowing section. L. Calculate Py by performing the integrations in Eg. 14, t is obvicus that the only means of practical calculation with this scheme is the digital computer. A calculation program based on this scheme was written for the IBM-7090. Specific details concerning the numerical resuite, and the application of this analysis to the MSRE experiments are 2s -t given in Section 17 3. NUMERICAL AND EXPERIMENTAL, RESULTS Numerical Groundwork Our intention in this section is to apply Eq. 14 to the analysis of a series of rod bump-stable period measurements made with the MSRE during fuel circulation. These measurements were taken at various con- trol rod insertions during the course of enrichment of the fuel salt in Run Ko. 3. It has been seen from the analysis presented in Section 2 that the quantities required to relate the measured asymptotic periocd to reactivity are the static distributions of the precursor concentration and the fission production rate together with the asgscciated adjoint flux distribution. We have already introduced several approximations in order to simplify calculations with Eg. 14. We now make explicit use of the group diffusion medel for the neutron fluxes. The flux distributions we shall use were obtained from standard core physics analysis with one- dimensional multigroup and two-dimensional, few group diffusion models for the neutronic behavior. In performing a "first round" analysis of the experimental measurements, the attempt was made to maintain as much computational simplicity as possible without discarding the essential features of the reactor and fuel circulation models. It has been shown that the static distribution PO¢S and @Q occurring in Eq. 1k are those appropriate to the asymptotic state, 1.e., the reactor-rod configuration during the period measurement. However, the three control rods in the MSRE are in a highly localized cluster about the center of the core (Fig. 2), and the insertion of a single rod produces a perturbation in the thermal flux distribution which is fairly well localized in the radial direction. Some calculational results in support of this conclusion are shown in Fig. 3. These are plots of the radial variation of the thermal flux, taken at an azimuthal angle half way between control rods 2 and 3. The calculsations were made with the two dimensional diffusion program EXTERMINATOR,® using an R-6 model of the MSRE core gecmetry. The perturbed fluxes with rod No. 2 inserted and with all three rods inserted are compared with the fluxes when all 18 OREL TWG 64-8214 TYPICAL FUEL PASSAGE MOTE: STRINGERS NOS. 7.60 AND &t {(FIVE! ARE REMOVABLE. 74—~ CONTROL ROD ! 77 9, \GUIDE BAR NN REACTOR vJgfg’c‘:Em'ERLmIE i ™ 2 | THREE GRAPHITE AND INOR-8 REMOVABLE SAMPLE BASKETS s AN /| A | 5 NN REACTOR CENTERLINE Fig. 2. Iattice Arrangement of MSRZ Control Rods. 19 CRNL-DWG 66-8563 _S><-ROD NO. 2 INSERTED » - i SAMPLE HOLDER . _| - | CONTROL RCD AND CONFIGURATICN FOR FLUX CALCULATIONS LTHREE RODS INSERTED @ THERMAL. FLUX AT 10 MW (normatized to axial position of maximum flux) N Fig. 3. lated Curves). 20 30 40 50 RADIUS (cm) &0 70 80 Radial Distribution of Thermel Flux in MSRE Core (Calcu- SN ot rods are fully withdrewn.* In g1l cases, the fluxes are normalized to a core power level of 10 Mw. It may be seen that the shape of the flux is relatively undisturbed over a large fraction of the radial cross secticn of the core. One may also note that the total neutron production rates in the numer- ator and dencminator of Eq. 1li are multiplied by the energy spectrums of neutrcn production. Hence, when performing the integration over energy in evalusting the scalar products in Eg. 14, only the high erergy portion (vhere T(E} # 0) contributes tc the result. We therefore need only approximate the fast group adjoint flux distribution. Based on the preceeding observetions, as a final set of simplifying . approximations we have used only the unperturbed distributions of fisslon production and fast adjeint flux, corresponding to the case of the rods fully withdrswn from the core. This simpliification is consistent with the neglect cf the radial variations of flux and precursor ccuncentrations over the core which was discussed in Sectlion 2. The calculated unperturbed axisl distributions of fission production rates and fast adjoint flux are given in Fig. 4. These were used for the calculstion of the axial distributicns of the six delayed neutron precursor groups, in accordance with the scheme developed in Section 2. The Individual precursocor ccncentratious, cg(z)g were multiplied by Ki and summed to obtszin the total locsl rete of emission of delsyed neutrons along the flow path through the resctor core. These resulting totsl dis- tributions are shown in Fig. 5 © # for the particular cases of w = 0 (circulating critical) and w = O.1 SGCHA(EO sec stable period), and also for the case when the fuel is not circulating. All three distributions are normalized to the same total production rate, Pofién The bottom pienum is omitted from Fig. 5 since the well stirred tank model was used to represent the delayed neutron production in this region. Baced on these sarme unperturbed fluxes, the relationship between the stablie inverse period, wfilg and the increment in static reactivity corresponding to the rod bump is shown in Fig. 6. For comparison ¥The "dip" in the thermal flux when the three rods are fully ithdrawn is caused by the presence of the INOR-8 rod thinmbles. ORNL-DWG 66-8582 b T 3 | 12 _EOTTOM "~ PLENLM | PLENUM CHANNELED REGION - TOPmB{ 0% 08 06 } AMPLITUDE (normalized) 04 Fig. 4. 20 40 &0 80 100 120 140 160 180 AXIAL DISTANCE FROM BOTTOM OF GRAPHITE {cm) Axial Distributions of Fission Density in Fuel Salt and Fast Groups Adjoint Flux (Calculated Curves). 200 M Do ORNL-DWG 65-i0654R 5.5 ; T : T T T T e ——— - CHANNELED — |_——’+ TOP —f==- ' REGION ! ‘ PLENUM 50 ------ —_ e T—-- - e, | 4.5 __I —e __ L 4}_ _l— ,,,,,, j \ i / STATIONARY l ! | so| Lo N ? | | I 30— e A | CIRCULATING, w =0 " | T 2.8 koo e e T T Nt _%_ / / . ) i — 40 b e A N e | | % NG N , : _/ CIRCULATING, w = 01 sec™! \ \ l .5 — f e S i e by } TOTAL. DENSITY OF DELAYED NEUTRON EMISSION IN SALT {arbitrary units) 0 2C 40 €C 80 100 420 140 10 180 200 DISTANCE FROM BOTTOM OF GRAPHITE (cm) FPig. 5. Axial Distribution of Net Scurce of Delayed Neutrons in MSRE (Calculated Curves). ORNL-DWG 65-10 100 10f 107" INVERSE PERIOD 7 (sec™) 10°¢ nverse Period of MSRE (Calculated table T Reactivity vs S Fig. 6. Curves ). purposes, & calculated curve is also given for the standard inhour rels- tior*?8when the fuel is stationary in the core. All of these calculations are based on the static velues of Bi and. Ai and the fuel residence times given in Table 1. Table 1. Static Delayed Neutron Precursor Characteristics and Tuel Residence Times used in Calculations Delsyed Group¥ 1 2 3 L 5 6 ) 10t B, (n/10% n} 2.23 1L.57 13.07 26 .28 7.66 2.80 Ag (Secml) 0.0124 0.0305 0.111hk ©.3013 1.140 3.010 Fuel Residence Times {sec)® Core (graphite mcderated region) G.h Upper plenum 3.9 External system 8.1 Lower plenum 3.8 Total 25.2 #* B, corrected for the difference between the statlc energy spectrums of prompt and delsyed neubron emission.® As the curves in Fig. & illustrate, because the margin between the prompt and delayved neutron sources has been increased by the spatial transport of the precursors, the same static reactivity increment produces a faster rate of rise of both neutron and precursor densities. The calculated curves extend somewhat beyond the region of practical interest in experi- mental measurements of the stable period. They are presented mainly for the purpoeses of illustraticon. In analyzing the experimentsl data, it proved to be most expedient tTo carry out machine calculations of the reactivity increments corresponding to each observed stable period. M2 A Experimental Results In the chronology of the MSRE zero power experiments, the measure- ments of interest in this study began at the point of reaching the mini- mum #3250 concentration required for criticality with circulation stopped and all control rods fully withdrawn. A summary of the zero power experi- ments 1s given elsewhere,iO and we consider only those details pertinent to the teopic of this report. Once this first basepcoint 2357 concentration had been reached, the concentration was further increased by the addition of capsules of en- riched fuel salt containing approximately 85g amounts of 23%U. The amount of control rod insertion required to compensate for these additions was meagsured, and period measurements about these new critical rod posi- tions were made. Care was taken to determine the minimum extra addition of #3%U required to reach criticality with the fuel circulating and all rods fully withdrawn. Once this second basepoint was reached, the critical rod position and period messgurements were made first with the pump stopped then with the fuel circulating. The measurements were terminated when enough ®2%U had been added to calibrate one rod over its entire length of travel. Most of the period measurements were made in pairs. The rod whose sensitivity was to be measured was first adjusted to make the reactor critical at about 10 watts. Then it was pulled a prescribed distance and held there until the power had increased by sbout two decades. The rod was then inserted to bring the power back to 10 watts and the measure- ment was repeated at a somewhat shorter stable period. Two fission cham- bers driving log-count-rate meters and & two-pen recorder were used to measure the period. The stable period was determined by averaging the slopes of the two curves (which usually agreed within about 2%). Periods observed were generally in the range of 30 to 150 sec. Prior to pulling the rod for each period measurement, the attempt was made to hold the power level at 10 watts for at least 3 minutes, in an effort te assure initial equilibrium of the delayed neutron precursors. Generally, however, it was difficult to prevent a slight initial drift in the power level as observed on a linear recorder, snd corrections were therefore introduced for this initial period. The difference between the reactivity during the transient and the initial reactivity, as com- puted from the in-hour relstion, was divided by the rod mevement and this sensitivity was ascribed to the rod at the mean position. The results of the rod sensitivity measurements made with the fuel stationary are shown in Fig. 7. Since the rod resctivity worth Is affected by the 235U concentration, theoretical corrections have been applied to the raw data to account for the fact that the 23517 mass was continually being increasgsed during the course of these experiments. The rod sensitivities shown in Fig. 7 have been normalized to & single 235U concentration, that at the beginning of the calibration measurements. Tne calibration curve in Fig. T constitutes a convenient reference from which the static reactivity egquivalent of the 23%U additions can be determined, simply by relating the integral under the curve between the fully withdrawn position and the critical rod position to the 23U nmass. By applying this egquivalence to the minimum ®25U increment between criti- cality with the fuel stationary and with it circulating, one obtains an experimental determination of the resctivity loss increment due to cir- culation. In addition, the curve in Fig. 7 is a convenient compariscn curve for the rod sensitiviiy data taken while the fuel was circulating. In the first of the two above cases, the reactivity increment cbtained from experiment was 0.212 * 0.004% 6k/k. This was found to compere with 0.222% cslculated directly from Eq. 1k, with w = 0. In the second case, the rod sensitivities deternmined from period measurements during circulation and Eg. 14 are shown in Fig. 8. Note that the solid curves in Figures 7 and 8 are identical. The experimental points were found to be distributed fairly closely about the reference curve, but the scatter of points relative to this curve is somewnat larger than observed with the data points of Fig. 7. Suggestions concerning the possible sources of this scatter, and methods for improving the precision of these measurements are included in the following section. @ 0.C7 l»/ 27 0.06 | 005 - ———- Tyflf———l S——— 0 e 1B e B io g e ®le o £ ™ S = “) 3 I —' 004 g Eo | 1 = 003 .% . = / | E # J z / ‘ | u 0.02/ booom L L b I ! @ % ootp o ] 0 5 g 12 16 20 24 28 32 36 40 44 48 52 ROD POSITION (in.) Fig. 7. Differential Worth of MSRE Control Rod No. 1, Measured with Fuel Stationary. (Normalized to initial critical 227U loading). Do o 0.07 (— : - e — : P S __ 006 Lo o +#_ %\ 008 i———— AR L L NG e ————y o T 004 — o O = 4 CO3 e 4 2 FUEL STAT:ONARY = Lt & ooplf LTl NG L L o 201k "o 4 8 2 6 20 24 28 2 36 40 44 48 52 R3OD POSTTION (im) Fig. 8. Differential Worth of MSRE Contrecl Rod Ne. 1, Measured with Fuel Circulating. (Nermalized to initial critical ?3°U loading). N N L, DISCUSSION OF RESULTS Thecretical Meodel Verification In keeping with the order of presentation of the preceeding part of this memorandum, we shall first discuss those aspects of the results which relste directly to the theoretical model. Following this, some suggestions concerning the MSRE experiments will be made. As a test of the theoretical model for determining the effects of fuel circulation on the delayed neutron precursors, quite satisfactory results were obtained for the reactivity increment between the just- critical states with the pump stopped aend with the fuel in steady circu- lation. Here, the principal difference in our model and an earlier calculation by Haubenreich® is the specific inclusion of the top and bottom reactor plenums as additional sources of delayed neutron emission which contribute to the chain reaction. As was shown in Fig. 5, the combined effect of fuel flow and radioactive decay cof the precursors is to "skew" the distribution of delayed emission toward the upper core and the top plenum. It is important to notice that this relative increase in emission of delayed neutrons in the top plenum is further accentuated by the much larger fuel volume fracticn in this region (the effective neutron preduction is the product of the source in the fuel salt times the fuel volume fraction). Because Haubenreich's model accounted for the ultimate contribution to fission of only those delayed neutrons emitted in the channelled region of the core, he obtained a somewhat larger effective loss in delayed neutrons due to circulaticn. Although these results seem to constitute an adeguate test of the model for the steady state condition, we cannot extend this claim to the stable asymptotic periods measured over the entire range of rod travel, due primarily to the lack of precisiocn in the period measurements. How- ever, until more precise measurements can be made, this calculational model gppears to be an adequate description of the delayed neutron pre- cursor transport due to circulation. | Mcre generally, one may pose the guestion of the need for sensitive S e tests and models of the delsyed neutron kinetics during fuel circulation at zero power. For the rod calibration work, it is always possible to stop the fuel pump, perform the tests with the fuel stationary in the core, and use simpler, well developed methods for the analysis of stable periods. Alsc, the questions of ultimgte safety and stability of the reactor generally involve large reactivity sdditions which take place on a time scale short compared tc the core transit time, and greatly in- fluence the kinetics of the prompt neutron generation. These questions also involve nonlinesr interacticn effects with temperature at high power. In addition to the theoretical interest, there are several contri- butions & good understanding of the effect of circulation on the delayed neutron kinetice can make from the viewpoint of resctor operation. For example, the observed differences between the fuel stationary and circu- lating critical conditions can comstitute a sensitive method of determin- ing fuel circulaticn rate, amount of circulating gas bubbles, or any gpecial characteristic due to the circulation. Necessary control system sensitivities and response times for normal operation can be specified with less uncertainity in the desigr of other molten salt reactors. Although muck of this study coverlaps that of earlier work, it is useful to re-emphasize here that the kinetics of circulating fuel systems belong to a mathematical category walch is fundamentally different from that of stationary fuelied reacters. This is particularly important in the description of reasctor transients cccurring on a time scale com- parable with the fuel transit times. It is appropriate to refer to the circulating fuel systems as "time-lag"’ systems, governed by partial differential equations for the delayed emitter ccncentrations, rather than by ordinary differential equations. For many purposes, spproximate reductions of the eguations to a form similer to the kinetics equations of the staticnary fueled systems are adequate (e.g., the replacement of the individual delayed fractions, Si, by effective reduced values, Biff; in the conventional reactor kineties equations)a This reduction, however, does not seem tTo be fully Jjustified for the gnalysis of an arbitrary transient. In this study, the only conditions we have fully examined are the Jjust critical state and the state of flux changing with a stable asymptotic period. For a complete thecretical understanding of the %U- L coupling of the circulation with the delayed neutron kinetics, we need slso to analyze the transient delayed neutron modes, mentioned briefly in Section 2. This transient coupling determines the effective time to establish the asymptotic period, and should slsc provide a basis for approximations made in the analysis of arbitrary transients. Scome pre- timinary work has been done on this problem, and we hope to make this the subject of s future memorandum. Recommendations for the MSRE One reason for the lack of precision of the rod bump-period measure- ment of control rod sensitivity 1s that the latter is the ratio of two amall quantities. For best results, this requires the experimenter to maintain high precision in both the measurement of the increment in rod position and the slope of the log n vs time curve. Early in the course of thesge experiments it was decided that only the regular reactor instru- mentation would be used for the rod sensitivity measurements. Determi- nation of the period in each measurement involved laying a straight-edge along the pen line record on the log n chart and reading the time interval graphically along the horizontal scale which corresponded to a change of severagl decades in the neutron level. OSince these charts, together with the pen speeds, are subject to variations, this is a probable source of error in the rod sensitivity measurements. A second possible socurce of error, probably less important, lles in the measurement of the increments in the rod pcsitions. It has been seen that the scatter in the experi- mental points was larger for the sensitivity mesasurements made during circulation. This is expected, since similar increments in rod with- drawal result in a shorter stable period under these conditions. Although a complete error analysis has not been carried out, these considerations would seem to be obvious starting polnts in any future attempt to obtaln more precise measurements of the rod sensitivities. At the time these experiments were performed, the MSRE on-line computer was unaveilable for the automatic recording of experimental data. It would be useful, during future MSRE operation, to repeat some of the period measurements while at zero power, using the computer in the data logging mode to sutomatically record the log n and time intervaels. The shim rods could be adjusted to vary the initiasl critical position of the regulating rod for the period messurements. The immediate need for more precise experiments is not great, since the precision of the rod cali- bration cobtained from the measurements with the fuel stationary is Judged adequate for further operation. ACKNOWLEDGMENT The cooperation of J. R. Engel and various members of the MSRE operating staff in performing the experiments and aiding in analysis is gratefully acknowledged. Also, the author is indebted to J. L. Lucius cf the Central Data Processing Facility for programming the machine computatvions to implement the analysis. 33 REFERENCES 1. J. A, Fleck, Jr., Thecory of Low Pcwer Kinetics of Circulating Fuel Reactors With Several Groups of Delayed Neutrons, USAEC Report BNL-334 (7-57), Brookhaven National Labecratory, 1955. 2. B. Wolfe, Reactivity Effects Produced by Fluid Mction in a Reacter Core, Nucl. Sci. Eng., 13(2): 80-90 (June 1962). P. N. Haubenreich, Prediction of Effective Yields of Delayed Neutrons in MSRE, USAEC Report ORNL~TM-380, Oak Ridge Naticral Laboratory, Octover 13, 1962, A i. T. Gozani, The Concept of Reactivity and Its Application to Kinetics Measuremerts, Nukleonik, 5(2): 55 (1963). 5. B. Friedman, Principles and Techniques of Applied Mathematics, Chapter 1, Jchn Wiley & Sons, Inc., New York {1956). 6. T. B. Fowler et al., EXTERMINATOR — A Multigroup Code for Solving Neutren Diffusion Problems in One and Twe Dimensions, ORNL-TM-342, February 1965. 7. P. N. Hauvenreich et al., MSRE Design and Operaticns Report, Part ITT: DNuclear Analysis, USAEC Report ORNL-TM-73C, Cak Ridge National Laboratory, February 3, 1964. 8. S. Glasstone and M. C. Edlund, The Elements of Nuclear Reactor Theory, Chapter 10, Van Nostrand, New York {(1952). 9. R. C. Robertson, MSRE Design and Operations Report, Part I: Degcription of Reactor Design, ORNL-TM ra% Janudrg 1965, p. 102. 1C. P. N. Hauvenreich et al., MSRE Zero Power Physics Experiments, Ozk Ridge Nat iocnal Laboratory Report in preparation. ........... Paiepazasaraaial 5 o [ I 17. s O 20. 21. 22. 23. ~ < " & e 26, o7, o8, 29, 30. 31. 52 . 33. 34, 35. 36. ~ ° 39. |_| OND -3 O\ o D [ Al 41, Lo, L3, P — =y ° gqtugdgj ?’fii?fifii?1?49303C4h§¢4(3£fl H o O EEBE ;= o mwy:y:gn»agwrdt—wa 58. 59. 100-101. 102-103. 104-106. 107. GREHPHY SRRSO M ORE YRR ECG T OQUEPEERPEEE QDN R R ° Lundin Liyon MacPherson MacFherson Martin McCoy . McCurdy McDuffie Miller Moore Nephew Payne Perry Piper Pitkanen Podeweltz Preskitt . Prince L. Redford Richardson C. Robertson C. Roller W. Rcsenthal W. Savolainen Scott J. Skinner L. Smith G. Smith Spiewak steffy . Tatlackson . Thoma . Thomas Tobieas . Trauger . Tsagaris . Ulrich . Vondy H. Webster M. Weinberg G. Welfare D. Whitman V. Wilson HrEREHOEOEHGEH O E QS WOEWBHRERRIDOQ Central Research Library Document Reference Section Labcratory Records Laboratory Records (ILRD~RC) Lo J EXTERNAT, DISTRIBUTION 108-122. Division of Technical Information Extension (DTIE) 123. Research and Development Division, ORC 124-125. Reactor Division, ORC o