- OAK RIDGE NATIONAL LABORATORY operated by UNION CARBIDE CORPORATION for the U.5. ATOMIC ENERGY COMMISSION ORNL- TM~- 380 ' COPYNO. - /9 &£ DATE — October 13, 1962 PREDICTION OF EFFECTIVE YIELDS OF DELAYED NEUTRONS IN MSRE P. N. Haubenreich ABSTRACT Equations were developed and calculations were made to determine the ef- fective contributions of delayed neutrons in the MSRE during steady power oper- ation. Nonleakage probabilities were used as the measure of relative importance of prompt and delayed neutrons, and the spatial and energy distributions of the prompt and delayed neutron sources were included in the calculation of these probabilities. Data which indicate a total yield of 0.0064 delayed neutron per neutron were used to compute total effective yields of 0.0067 and 0.0036 for the MSRE under static and circulating conditions respectively. The effective fractions for the individual groups of delayed neutrons will be used in future digital calculations of MSRE kinetic behavior. NOTICE This document contains information of a preliminary nature ond was prepared primarily for internal use at the Oak Ridge National Laboratory. It is subject to revision or correction and therefore does not represent a final report, The information is not to be abstracted, reprinted or otherwise given public dis- semination without the approval of the ORNL patent branch, Legal and Infor- mation Control Department, LEGAL NOTICE This report was prepared as an account of Government sponsored work. Neither the United States, nor the Commission, nor any person acting on behalf of the Commission: A. Makes any warranty or representation, expressed or implied, with respect to the accuracy, completeness, or usefulness of the information contained in this report, or that the use of any information, apparatus, method, or process disclosed in this report may not infringe privately owned rights; or B. Assumes any liabilities with respect to the use of, or for damages resulting from the use of any information, apparatus, method, or process disclosed in this report. As used in the above, *‘person acting on behalf of the Commission’ includes any employee or contractor of the Commission, or amployse of such contractor, to the extent that such employee or contractor of the Commission, or employee of such contractor prepcres, disseminates, or provides access to, ony information pursuant to his smployment or contract with the Commission, or his employment with such contractor, CONTENTS INTRODUCTION APPROACH TO THE PROBLEM DERIVATION OF EQUATIONS Steady~-State Concentrations of Precursors Normalized Source Distributions Expansion of Sources in Series Form Fraction Emitted in the Core Effective Fraction in Noncirculating System RESULTS OF MSRE CALCULATIONS APPENDIX Expressions for Nonleakage Probabilities Data for MSRE Calculations MSRE Dimensions Precursor Yields and Half-Lives Neutron Energies and Ages Neutron Diffusion Length Nomenclature o \osl-slwwm 10 11 12 13 18 P e3 2k 2k INTRODUCTION The kinetics of the fission chain reaction in a circulating-fuel resctor are influenced by the transport of the delayed neutron precursors. This fact makes a rigorous treatfient quite complicated.l’2 The approach generally followed is to drop the transport term from the precursor equa- tion (making the kinetics equations identical with those for stationfiry reactors) and make an approximate allowance for the precursor transport bz replacing the delayed neutron fractions, fii, with "effective" values, Bi. This approximation is used in Z0RCH, the computer program recently developed for analysis of the kinetics of the MSRE.,3 The purpose of the work reported here is to obtain values for 3§ to be used in the MSRE analyses. APPROACH TO THE PROBLEM The importance of delayed neutrons is enhanced in fixed-fuel reactors because the energy spectra of the delayed neutrons lie at much lower en-~ ergies than that of the prompt neutrons. 'he differences in energy spectra make the delsyed neutrons more valuable because they are less likely to escape from the reactor in the course of slowing down to thermal energies. This effect is, of course, also present in circulating-fuel reactors. Of greater importance, however, in these reactors is the gpatial distribution of the delayed neutron sources. Many of the delayed neutrons are emitted outside the core and contribute nothing tc the chain reaction. Further- more, those delayed neutrons which are emitted in the core are, on the aversge, produced nearer to the edges of the core than are the prompt neutrons, which tends to further reduce the contribution of a particulaxr group of delajéd neutrons. lJ. A. Fleck, Jr., Kinetics of Circulating Reactors at Low Power, Nucleonics 12, No. 10, 52-55 (1954 ). 2B. Wolfe, Reactivity Effects Produced by Fluid Motion in a Reactor Core, Nuclear Sci. and Eng. 13, 80~90 (1962). 3¢. W. Nestor, Jr., ZORCH, an IEM-TO90 Progrem for Analysis of Simu- lated MSRE Power Transients With a Simplified Space-Dependent Kinetics Model, ORNL TM -345 (Sept. 18, 1962). The contribution of delayed neutrons during a power transient in a circulating-fuel reactor 1s affected by the continual change in the shape of the spatial source distribution. Thus the use of an "effective" frac- tion for a delay group in analyzing such transients is, in itself, an epproximation. If this approximation is made, and a single set of "ef- fective" fractions is to be used in the analysis of a variety of trans- ients, it would seem that the values should be the fractional contributions of the various groups to the chain reaction under steady-state conditions. The problem at hand is to calculate these contributions. It has been the practice in the analyses of circulating-fuel reactors to take /B, to be just the fraction of the ith group which is emitted inside the core. This implies that the importance of the delayed neutrons is equal to that of the prompt neutrons; which would be true if the in- crease in importance due to lower source energies exactly offset the de- crease due to the distortion of the spatial distribution in the core. The further approximation is usually made in computing the fraction emitted in the core that the precursor production is uniform over the core volume. We shall seek to improve the evaluation of 5; by calculating more accu=- rately the spatial distribution of precursors and by taking into account more explicitly the effect of the spatial and energy distributions on the importance of the delayed neutrons. In a discussion of fixed-fuel reactors, Krasik quotes Hurwitz as de- fining fi:/fli as "essentially the probability that a delayed neutron of the ith kind will produce & fission divided by the probability that a prompt neutron will produce a fission," and adds that "for a simple re- actor this probability is given by the ratio of the nonleakage probabil- ities of the respective types of neutrons."lL Let us adopt the definition of fi;/si as the ratio of the nonleskage probabilities. Suppose that the nonleakage probability for prompt neutrons is Ppr and for the delayed neutrons which are actually emitted in the core it is Pi' Of s particular group, only the fraction Gi is emitted in the core so the nonleakage prob- ability for all neutrons of the ith group is eiPi. Therefore hso Krasik, "Physics of Control," p 8-10 in Nuclear Engineering Handbook, ed. by H. Etherington, McGraw-Hill, New York, 1958. B,0,P B, = () 1 pr The nonleakage probabilities can be expressed in a simple form if the reactor is treated as a bare, homogeneous reactor. The source of prompt neutrons in the reactor is proportional to the fission rate, which follows closely the shape of the fundamental mode of the thermal neutron flux. In a homogeneous, cylindrical, bare reactor this is: #(r,z) = ¢0J0(2,h r/R) sin(s=z/H) (2) For the prompt neutrons, with this spatial source distribution P = ..E:E?L_ (3) Pr 1 4+ 128° where £ - (22) () 2 The spatial distributions of the delayed neutron sources are not the same as that of the prompt neutrons because of the transport of the pre- cursors in the circulating fuel. The source distributions can be calcu- lated from power distribution, fuel velocities, system volumes, etc. (This is done for a simple cylindrical reactor in the next section.) It is convenient, for the purpose of calculating leakage probabilities, to gxpand each source function in an infinite series: oo QO s, (r,2) = mg n:Zl A, 3 (3.r/R) sin(am/H) (5) where j 1s a root of‘Jo(x) = 0. (See Appendix for derivation of non- leakage probabilities.) For neutrons with a source distribution S = AL Jo(er/R) sin(nmz/H) (6) the nonleakage probability is P = _.‘?.......2_...2.,.. (7) where - (2) (8 For the ith group of delayed neutrons, then RrH m nrz %%Ai ffoj;) <_R sin( >1+L232 2mr dr dz (9) P = Jn J~ S, (r,z) 2 dr dz 0o o Note that the age, Ty is the appropriate value for neutrons with the source energy distribution of group i.) If Sir,z) is normslized to one fission neutron, then the dencmlnator in (9) is B;6;+ Thus, from (1), the numerator in (9) is just Bi or The numerator can be integrated to give 5 ' =B~ T J, (3 ) m i BB, = RT ) ) A _ 2B e 5 (10) 1P m=l n=1,3.. 2dn 1418 (Only odd values of n remain in the summation because the contribution of all even values of n to the integral is zero.) The approach we shall follow in calculating effective delayed neutron fractions is then as follows: calculate the steady-state source distri- butions, S (r,z), in a bare, homogenized approximation of the MSRE core; evaluate the coefficients, A, ; compute B from (lO) By "bare, homo- genlzed approximation" we mean a reactor in which the flux is assumed To vanish at the physical boundary and in which the composition is uniform 5It is assumed here that the fuel volume fraction, £, is not a function of r. This is also implied in (2) and (3). so that (2) applies.6 We shall also assume that the fuel velocity is uniform over the entire core. DERIVATION OF EQUATIONS Steady=-State Concentrations of Precursors Let us derive the formula for the steady-state concentration of the precursor of a group of delayed neutrons as a function of position in the core. Begin by considering an elemental volume of fuel as it moves up through a channel in the core. The precursor concentration in the fuel as it moves along is governed by de T - 5vz:f¢(r,z) - Ac (11) The fuel rises through the channel with a constant velocity v so de _ defdz _ 1 dc iz = d'b/ a& - 7 at (12) With the substitution of (2) and (12), equation (11) becomes de —-————szf% g (82E e B Do (13) dz v o R H v Along any channel r is constant and at steady state when ¢0 is not changing, (13) can be integrated to give 2. hr BvE BN (5 A/ ,, c(r,z) = R sin 2 - & (cos 3§ - 'Rz/v + c e Az /v 5 5 H H o AT+ (H - (1h4) where s is the concentration in fuel entering the core at z = 0. The concentration at the outlet of a channel is given by (14) with 6In the actual MSRE core, the flux deviates from (2) because of the depression around the rod thimbles and because the flux does not vanish at the physical edge of the core. \ B1"2’f‘¢0(l‘ + e-7\H/V) Jo(?fl%a ) -7\H/v c(r,H) = +ce Lo (2] (15) The concentration of precursors in the fuel leaving the core is the mixed mean of the streams from all of the channels. fR c{r,E)v(r)f(r) 2mr dar c, = & (16) fR v(r)f(r) 2 ar ) We have assumed that f and v are constant across the core. With this assumption, substitute (15) in (16) and integrate to obtain =At - £ ¢ . 7‘tc 2Bv2f¢0 9(1 + e ) Jl(a 405) c; = c_e + 7\tc 5 (aT) 2.405 « [l + ( ——-—) } T where tc has been substituted for H/v. The precursors decay during the time tx required for the fuel to pass through the external loop. Thus 'ch ¢, = cqe (18) Equations (17) and (18) can be solved for cye =-7\tc -7\tx vzt (1 + e ) e J. (2.405) * 2.405x [1 + (7-\-‘;9-)2} {1 - ewk(tcfiutx)] When this is substituted into (1%) the desired expression for c(r,z) is obtained. Normaldzed Source Distributions As explained on page 6 it is desirable to normalize the delayed neutron source to one fission neutron so that the integral over the core volume will equal BO. 'The rate of neutron production is R H N = Vfv}:f;éa.vf_,c = vE g ofaf Jo(aél*r)sin (Z) fomaraz (20) fc Assuming, as before, that the fuel volume fraction is not a function of r, integration gives 2 N = L4HR“f v2f¢o Jl(z.h05)/2..1+05 - (21) for the total tate of neutron production. The normalized source distribution is Si(r,z) = Kici(r,z)/fl (22) Substitution of (14), (19) and (21) into (22) gives for each group At _z/H - At z/H s(r,z) = 5.8 ¢ + [Sl sin gfi - 8, COs §5-+ 8,8 ¢ ] JQ’(?-il-t%z-:E ) (23) where =N\t -\t (L+e S)e X (ktc/n) 8 = B (21"') ° 5 At N2 A, + t) 25121’[14-(-—;94)][1-:-3 "] 2 2.405 (At /x) s, = B . (25) I EROf [1 + (7—\;-"-)2%1(2-1&95) 10 2.405 (Atc/u) = B (26) 4 ER°f [1 + (7—\;9- )2 } Jl(2.1+05) 8o Expansion of Sources in Series Form For the purpose of calculating the nonleakage probability let us represent S(r,z) by an infinite series which vanishes at 2 = 0, z = H and r = R. @ © S(r,2) = )y ) Ay Ioar) sin(rz) (27) m=l n=1 The condition that S(r,z) venish at r = R is satisfied if a_= ,jm/R vhere j is the mth root of J c)(x) = 0. The boundary conditions at the ends are satisfied by = nsr/L. Expansion of the functions in {23} g:’ures7 - Qo - t .. 7\th/1- hs 22 [1 + (P c:l i (f‘:’f..) o w_) c sl A% Iy l(j ) \t o R H 2 () n S, 5 (2_%(:){_2_1_-_) io [1 + (-1)? ]n sip B2 (28) 2.#02’ r 2z JO ( R ) cos a n=2 - 1 i (29) 2405 Y MA/E 405 r 1+ ()™ e -Mc ' Bz S2JO§[n+(-——-) T (30) 7For a discussion of expansion in series of Bessel functions and half-range sine series expansions see, for instance, C. R. Wylie, Jr., Advanced Engineering Mathematics, 2d ed., p 432-37 and 253=57, McGraw-Hill, New York, 1960, Thus for m = 1 and n = 1 2 Ss 0 } 1l+e +s (31) > Ay = ¥&2+&M5QQA®) Form=1and all n > 1 At _fn 28, 14+ (1) e © 1+ (-1)% A7 115273 105 J. (2.405) RN 2 - 5, 5 n 1[ ° l o 2+ c n “l Form >1 and all n b = 3G , (7"% (33) n~ +( — Fraction Emitted in the Core The equations derived in the foregoing sections permit the evaluation % ‘ of B without the explicit calculation of ei. Furtheér insight may be ob- tained by calculating ei;'and this can be done most easily by using the relation Qe = c ) 1 0 where Q is the volumetric circulation rate of the fuel. This is given by 2 - Q = % R°%v (35) When (35), (21), (19), and (18) are substituted in (3%), the result simplifies to —Kitc ~Ritx 9__.1._%[ l- )2][(l+e )@ - e )] (36) -Ai(tc + ’cx) 1l =@ 12 It is of interest to compare this relation, which takes into account the spatial distribution of the precursor production, with the relation obtained when the precursor production is assumed to be flat over the core volume. The latter relation is . At : At _ 1 l-e (L - e ) . 6= 1-5% [ I T T ] (37) 1 C 1 - e 1 C X The digital programs for MSRE kinetics calculations (MURGATROID9 and ZORCHlO) have as an integral part the computation of delayed neutron fractions from precursor yields and decay constants and the reactor residence times, all of which are input numbers. The fraction computed and used for each group is B gi where § is given by equation (37) Therefore, in order to have the kinetics calculations done with B for the fractions, it is necessary to put in a fictitious value of Bi, equal tos/g Effective Fraction in Noncirculating System The change in the effective delayed neutron fraction between non- circulating and circulating conditions 1s a factor in determining control rod requirements. In the noncirculating core, the source of delayed neutrons has the same shape as the source of prompt neutrons and 2 . Pty Pig = Py 3 (38) -B T o 1l pr 8P R. Kasten, Dynamics of the Homogeneous Reactor Test, ORNL-2072 (June 7, 19%6). 9C. W. Nestor, Jr., MURGATROYD, An IBM~70900 Program for the Analysis of the Kinetics of the MSRE, ORNL-TM-203 (Apr. 6, 1962). loC W. Nestor, Jr., ZORCH, an IBM=7090 Program for Analysis of Simu- T lated MSRE Power Transients with a Simplified Space-Dependent Kinetics Model, ORNL-TM-345 (Sept. 18, 1962]. ’ “\ 13 It may also be of interest to compare the magnitudes of the delayed neutron source distributions under static and circulating systems. In the static system 2.4 r T VIP I (=) sin(=) 8y, (r,z) = P2l g (39) N N is given by (21), and (39) reduces to B. 2.405 Sis(r,z) = = o (2%%#5 ) sin(3%0 (+0) LHRPP 3, (2.405) The same result is obtained if one substitutes tc = tx = o in equations 23-26. RESULTS OF MSRE CALCULATIONS The equations derived in the preceding section were used in calcu- lations for a simplified model of the MSRE core. (See Appendix for data used.) Results are summarized in Table 1. The table shows that the core residence time, in units of precursor half-lives, ranges from 0.2 for the longest-lived group to 41 for the shortest-lived group. Becsuse of this wide range, the shapes of the delayed neutron sources vary widely. Figure 1 shows axial distributions at the radius where Jo(z.h r/R) has its average value, 0.4318. The source densities were normalized to a production of one fission neutron in the reactor. For the longest=-lived group, the SO term, which is flat in the radial direction, is by far the largest. This term is relatively insignificant for groups 3-6. For the very short=lived groups, the Sl term predominates, i.e., the shape ap- proaches that of the fission distribution. Figure 2 shows the twofold effect of circulation in reducing the contribution of the largest group of delayed neutrons. The reduction in the number of neutrons emitted in the core is indicated by the dif- ference in the areas under the curves. The higher leakage probability with the fuel circulating is suggésted by the shift in the distribution, which reduces the average distance the neutrons would travel in reaching the outside of the core. ORNL~LR-Dwg. 75652 LIIETT Tl e e 1k e b o ORNL~LR-Dwg. 75653 Unclagsified 15 Table 1. Delayed Neutrons in MSRE Group 1 2 3 h 5 6 t_é_ (sec) 55.7 22.7 6.22 2.30 0.61 0.23 t,/ty 0.17 0.4 1.5 4.07 15 k1 tx/tl 0.30 0.72 2.6k 7.15 27 72 s./s, 65.7 9.24 0.30 3.6 x 107 9 x 10710 2 x 10723 se/s:L 27.0 10.99 3.01 1.11 0.29 0.11 6, 0.364 0.371 0.458 0.709 0.960 0.994 P i/PPr 0.676 0.718 0.868 0.906 1.010 1.031 8. /B, 0.246 0.266 0.398 0.672 0.970 1.025 £ 0.364 0.370 0.448 0.669 0.906 0.965 is/Ppr 1.055 1.039 1.043 0.948 1.010 ~1.031 10* By 2.11 1h.02 12.54 25.28 7 .40 2.70 10* B, 0.52 3.73 %.99 16.98 7.18 2.77 10“ B:s 2.23 14 .57 13.07 26 .28 7.66 2.80 ; « { 9T 7 Totals for all six groups taken from Table 1 are listed in Table 2 for ease of comparison. Table 2. Total Delayed Neutron Fractions in MSRE Actual yield, ZB:L * Effective fraction in static system, ZBis Fraction emitted in core, circulating, Zaiei T % Effective fraction, circulating, Zfli 0.006405 0.006661 0.003942 0.003617 18 APPENDIX Expressions for Nonleakage Probabilities It is desired to calculate the probability that a neutron from a distributed source of specified shape and initial energy will be absorbed as a slow neutron in a cylindrical, bare reactor. Consider the neutrons to be born at energy Eio with a spatial distri- bution Si(r,z). Use age treatment to describe slowing down, i.e., 'V2q (r;z,7) = %;‘ (r,z,'r) (al) I Q(r:z:'rio) Si(r:Z) (a2) The steady-state equation for the thermalized neutrons is D v°g, (r,2) - 5.8, (r,2) + q,(r,z,7y,) = O (a3) The probability to be calculated is R H f Za¢i(r,z) 2mr dr dz P, = (akt) fRfH Si(r;,z) 2 dr dz © 0 O O If (al) is solved with the condition that q vanish at the boundaries and be finite everywhere inside the reactor, the solution is @ Jr . -Be('c-"t) q, = Z Z a, J (-—E—)sine m 1o (a3) - i imm "o R H m=1 n=1 where 5 jg"e n:r\e . - (B) (F) () It is possible a priori to represent Si(r,z) and ¢i(r,z) by series ¥ 19 D- 20 - “ s, (r,2) = A g (J’L)sm nrz (a7) iv? 0ol el imn "o R H .) | 00 CO J I‘ ' ¢i(r,z) ?mg r;-l Fom o( ) sin (—-—-—) (a8) From (a2), (a5), and (&7) it is evident that 8y zAimn (a9) And if @ is represented by (a8), o0 QO J T 2 DV'f= - D . ( )s:.n(—-—-— (a10) mz; n=l mn 1mn O Substitute (a5), (29) and (alO) into (a3). Because of the ortho- gonality of the functions, one has for each mn 2 m i . DanF ):a F + A e = 0 (ail) vhere, for convenience, Ten = Yio is represented by 2nrdrdz z mn( )ofof o\ R H m*ln—l P, = i R H [ [ s,(ryz) 2 ar az o O (al13) This is the expression for the nonleakage probability given in the text as equation (9). Now consider a special case where the neutron source is proportional to the flux, namely the fission neutrons born at the site of the fission. 20 8, (r,z) = kzéfii(r,z) (ak) - If k is not a function of position, insertion of (alli) into (ak) gives L Pi = E (33-5) Let us find an expression for k as follows: Substitute (a7) and (a8) for S, and ¢i in (al%). This results in the relation Ai‘mn - k%a Fimn (a16) Substitute this in (al2) B> 1, ke 7 - (a17) F = F, al imn im 1+ L2 Be mn This is satisfied for all mn for which F, = 0. IfF.__ # 0, it lmn imn must be that & -Qin Ti L Lo —= 1 (a18) 1+ L B mn Because Ein is, in general, different for each mn, (al8) can only be true for one mn and therefore Fimn must be zZero for all but that mn. It can be shown by consideration of the time-dependent neutron equations, that in a reactor free of extraneous sources, the steady-state flux corresponds to the fundamental mode, i.e., m =n = 1. (See Glasstone and Edlund, 12.37 = 12.41.) Therefore 2 1 e"B:LJ. Ty (at0) P = =-= -~ al9 ik 4, Lzflfl This is the expression used in the text as the nonleakage probability for the prompt neutrons. Data for MSRE Calculations This section presents the data which were used in calculations for the MSRE. It includes dimensions and properties of the reactor and data on delayed neutrons. MSRE Dimensions Reactor dimensions which are required are H, R, f, v (or tc) and t_. For R let us use 27.75 in., the inside radius of the INOR-8 can around the core. Assigning velues to H and v is not simple, because the fission dis- tribution extends past the limits of the graphite core into the uppéf and lower heads. The axial distribution of the fission rate closely follows sin(mz/H) where H is 77.7 ian. The longest graphite stringers are 68.9 in. long, and the channel region is only 62 in. Further com=- plicating the situation is the fact that outside of the channeled region, the fuel velocity is lower than in the channels (because the volume fraction of fuel is much higher in the end regions). There are also radial variations in the fuel volume fraction and chamnel velocity. (In the central channels the fuel velocity is over three times the 0.60 ft/sec which is found in more than three-fourths of the channels.) Let us use the following simplifications. Consider the "core" to be bounded by the horizontal planes at the extreme top and the extreme bottom of the graphite. This gives H = 68.9 in. Enclosed by these boundaries is & total volume of 96.k4 fts, of which 25.0 ft3 is occupied by fuel. Thus £ = 0.259. The residence time of fuel in the "core,” at a flow rate of 1200 gpm, is 9.37 sec. The velocity, H/tc, is 0.61 ft/sec. The total volume of circulating fuel is 69.1 fb3, for a total circuit time of 25.82 sec. Thus t, = 25.82 -~ 9,37 = 16.45 sec. Precursor Yields and Half-Lives For yields and half-lives of the delayed neutron groups, let us use the data of Keepin, Wimett and Zeigler for fission of U235 by thermal neu- trons.ll These are given in Table A-l. Hg, Krasik, "Physics of Control,"” p. 8=4 in Nuclear Engineering Handbook, ed. by H. Etherington, McGraw-Hill, New York, 1950. 22 Table A-l. Precursor Data Group 1 2 3 L 2 6 Half-life (sec) 55.7 22.7 6.22 2.30 0.61 0.23 Decay constant 0.0124 0.030% 0.1114% . 0.3013 1.140 3.010 K (sec™l) Fra.c ional yle 2.11 1k.02 12.54 25,28 T.40 2.70 10 B;» (n/10 23 Neutron Energies and Ages The age of neutrons is given by E _ c D dE T = Ef ol . = (a20) The age of prompt neutrons, which have an initial mean energy of about 2 Mev, is about 292 cm? in the MSRE core at 1200°F.12 Let us estimate the age to thermal energy of the delayed neutrons as follows. E '\ E — D o g@.: —]-3-— ....9. a2l '