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^ n CLASSIFIED I UNCLASSIFIED CF5312108 Subject Category: PHYSICS UNITED STATES ATOMIC ENERGY COMMISSION THE INHOUR FORMULA FOR A I ^ .^  *' ,' ,. AL. 1 1 156 "1 S' ., ~ ~ ' , December 22, 1953 Oak Ridge National Laboratory Oak Ridge, Tennessee Technical Information Service, Oak Ridge, Tennessee Date Declassified: December 20, 1955. This report has been reproduced directly from the beet 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 15 cents. Available from the Office of Technical Services, Department of Commerce, Wash ington 25, D. C. This report was prepared asa scientific account of Govern mentsponsored work. Neither the United States, nor the Com mission, nor any person acting on behalf of the. Commission makes any warranty or representation, express or implied, with respect to the accuracy, completeness, or usefulness of the in formation contained in this report, or that the use of any infor mation, apparatus, method, or process disclosed in this report may not infringe privatelyowned rights. The Commission assumes no liability with respect to the use of,or from damages resulting from the use of, any information, apparatus, method, or process disclosed in this report. CF5312108 The Inhour Formula for a CirculatingFuel Nuclear Reactor with Slug Flow. W. K. Ergen Work performed under Contract No. W7405Eng26 December 22, 1953 OAK RIDGE NATIONAL LABORATORY Operated By CARBIDE AND CARBON CHEMICALS COMPANY POST OFFICE BOX P OAK RIDGE, TENNESSEE The Inhour Formula for a CirculatingFuel Nuclear Reactor with Slug Flow As pointed out in a previous paper, the circulatingfuel reactor differs in its dynamic behavior from a reactor with stationary fuel, be cause fuel circulation sweeps some of the delayedneutrons precursors out of the reacting zone, and some delayed neutrons are given off in locations where they do not contribute to the chain reactions. One of the consequences of these circumstances is the fact that the inhour formula usually derived for stationaryfuel reactors,2 requires some modification before it becomes applicable to the circulatingfuel re actor. The inhour formula gives the relation between an excess multi plication factor, introduced into the reactor, and the time constant T of the resulting rise in reactor power. If the inhour formula is known, then the easily measured time constant can be used to determine the ex cess multiplication factor, a procedure frequently used in the quanti tative evaluation of the various arrangements causing excess reactivity. Furthermore, the proper design of control rods and their drive mecha nisms depends on the inhour formula. Frequently, the experiments evaluating small excess multiplica tion factors are carried out at low reactor power, and the reactor power will then not cause an increase in the reactor temperature. This case will be considered here. In that case, the time dependence of the reactor power P can be described by the following equation: dP/dt = (1/t) (kexB)P + B D(s) P(ts) ds (1)* William Krasny Ergen, The Kinetics of the CirculatingFuel Nuclear Reactor, J. Appl. Phys. (in print) SSee for instance S. Glasstone and M. C. Edlund, The Elements of Nuclear Reactor Theory, D. Van Nostrand Co., Inc., 1952, p. 294 ff. * Some authors, for instance Glasstone and Edlund, loc. cit., write the equations corresponding to (1) in a slightly different form. The difference consists in terms of the order k (Z/T) or ke , which are negligibly small. V is the average lifetime of the prompt neutrons and k the excess multiplication factor (or excess reactivity). The meaning of p and D(s) for a circulatingfuel reactor has been discussed in some detail in ref. We approximate in the following the actual arrangement .y a reactor for which the power distribution and the importance of a neutron are constant and for which all fuel elements have the same transit time S10 through the outside loop. In this approximation, p D(s) is simply the probaltlity that a fission neutron, caused by a power burst at time zero, is a delayed neutron, given off inside the reactor at a time be tween s and s+ds. D(sas normalized so that D(s) ds = 1 (2) If the fuel is stationary, pD(s) is the familiar curve obtained by the superposition of 5 exponentials: 5 _Ais PD(s) = f B..e (3) i=1 The i are the decay constants of the 5 groups of delayed neutrons, and the .i are the probabilities that a given fission neutron is a delayed th neutron of the sroup. For the circulatingfuel reactor we first consider the fuel which was present in the reactor at time zero. At any time s, only a fraction of this fuel will be found in the reactor. This fraction is denoted by F(s), and by multiplying the right side of (3) by F(s), we obtain the function PD(s) for the circulatingfuel reactor. Since 01 is the total time required by the fuel to pass through a complete cycle, consisting of the reactor and the outside loop, it is clear that at s = no1 (n = 0, 1, 2 ...), F(s) is equal to 1; at s = no1+9 (n = 0, 1, 2 ...), F(s) is equal to zero. (We assume 01 k 20 so that the fuel under consideration has not started to reenter the reactor when the last of its elements leaves the reacting zone). Between a = no1 and s = nq+0, F(s) decreases linearly, and hence has the value (nQ,+Qs)/O. At s = no1Q(n= 1, 2, 3 ...), F(s) is zero, but since the fuel under consideration reenters the reactor be tween this moment and s = no1, F(s) increases linearly: F(s) = (snO1+ 9)/Q. For BD(s) we thus obtain: no0l+s 5 s OD(s) = E1 p )e i i=l snO +Q 5 BD(s) = 1 1~_1?iie i PD(s) = 0 Eq. (4) is now substituted into Pe t/T T (kex)Poet/T+ pi i=1 for no s =l+Q, n = 0, 1, 2, ..., for nO1Osnol, n = 2, 3, ., (4) for no1+Q!s= (n+l) 10, n = 0, 1, 2,.., eq. (1), and for P we set P = P e/T. Then o n +0 EO n1 a P e(ts)/T d nO1 + 1 n=1 no e snQ +Q 8 sn 1 +0 )is (tP)/T 1 SPe The common factor Pet/T cancels out. The substitution e= nO1+0s transforms Th comnfatrP (n81+Qs)exp{Ci+(l/Tq s ds "1 into exni hi)+(1/Ti ex0p hi+ (1/T Je [+(1/T dr and the substitutiond= anQl+ 9 transforms nO1 J I nQ1+0) exp{[i+(l/T)J sI ds into expi n[Ji+(l/T ex1 .+(1/T j exp .f [i1+(l/T]')d . The geometric series exp^n [w.+(1/TI 01 can now be summed, and the in tegrals over evaluated by elementary methods. After performing all these operations, one obtains the following inhour formula: ex T+ 5 pie1 15 fi o1 + tPie i 9i ,I + 1)+e ('l" (5) 2=1 2 i = i + (l/T). (6) P is evaluated by means of eq. (2): S= pD(s) ds. If PD(s) is substituted into th~s integral, expressions are obtained which are of the same type as the ones just discussed, and which can be evaluated by the same methods. The result is Q ,(,o) S0 1 + e e (ho + l)+e ii(1 e ) In spite of the formidable appearance of eqs. (5), (6), and (7), it is easy to findfor any given 0 and vl,the value of kex which produces a given time constant T. Furthermore, the following reasoning describes the general fea tures of the equations. Consider first the dependence of kex on T. If T =t, ,1, = ., and the sum on the right of (5) is equal to p, kex is equal to zero. This corresponds to the state in which the reactor is just critical. If T becomes very small, the/u become very large and in the fraction on the right of (5) the numerator is dominated bytu i and the bracket in the denominator by 1. Hence the fraction tends to zero like Qui, as T goes to zero. If 4 is very small, as it is in practice, T will be small as soon as k exceeds 0 by a small amount. Then the ex complicated sum on the right of (5) is of little importance, and T is determined by Z'/T = kex 0, that is the reactor period is inversely proportional to the excess of the reactivity over the reactivity corre sponding to the "prompt critical" condition. In the stationaryfuel reactor, the k which make the reactor ex prompt critical is given by pi. Wit) circulating fuel, the reactor is prompt critical if kex = 0, which is less thanspi, that is it takes less excess reactivity to make the circulatingfuel reactor prompt critical than to do the same thing to a stationaryfuel reactor. This is physically evident because the fuel circulation renders some of the de layed neutrons ineffective. That P<(Pi can also be verified mathe matically. For T intermediate between very small positive valuesand + o we consider again the analogy to the stationary fuel reactor. Here the inhour formula reads: k = i Z) (8) kex T = 1 + iT For every positive ke there is one, and only one, positive time con stant T and vice verse. This in a consequence of the fact that kex is a monoton decreasing function of T, for if this monotony did not exist, there could be several positive T value corresponding to a given value of k ex or vice versa. In the circulatingfuel reactor the situation is qualitatively the same; the fractions under the sum in eq. (5) are essentially of the form x ax (al)x x 1+ e x e (x+l) + e(a (x = a = 01/) x2Cie and an expression of this form can be shown to be a monoton decreas 2 ing function of positive x ; the expression is thus a monoton in creasing function of T (see eq. (6)), and as T increases k decreases ex monotonically, according to eq. (5); for every positive k there is one, ex and only one, positive T, and vice versa. This, of course, does not preclude that there exist for a given k several negative T values, ex in addition to the one positive value. Negative T correspond, how ever, to decaying exponentials, which are of no importance if the rise in power is observed for a sufficiently long time. Consider now the behavior of eq. (5) with variation of 9, the transit time of the fuel through the reactor, and of 91, the transit time of the fuel through the whole loop. If 9 (and hence also 0 ) is very large compared to all 1/Ai and 1ui, eq. (5) reduces to eq. (8), the inhour formula for the stationary fuel reactor. The circu lation is so slow that the reactor behaves as if the fuel were See Glasstone and Edlund, loc. cit. p. 301, eq. 10.29.1. See also preceding footnote. 2 W. K. Ergen, The Behavior of Certain Functions Related to the Inhour Formula of Circulating Fuel Reactor:, Oak Ridge National Laboratory Memo CF 5411 Jan. 15, 1954. 8 stationary, inasmuch as all delayed neutrons, even the ones with the longlived precursors, are given off inside the reacting zone, before much fuel reaches the outside. On the other hand, if 91 and hence also 0 is small compared to all 1/)i, that is if the transit time of the fuel through the complete loop is small compared to the mean life of even the shortlived delayedneutrons precursors, then for T> 0 5 k i: Z (9) ex T 1i=1 1 +)hiT This is the same as the inhour formula for the stationary fuel reactor, except that all the fission yields 1i are decreased by the factor 9/091 This is physically easy to understand, since 9/0, is just the prob ability that a given delayed neutron is born inside the reactor. Of interest is the intermediate case, in which 9 is smaller than the mean life of the longlived delayedneutron precursors, and larger than the mean life of the shortlived precursors. In that case, the longdelayed neutrons act approximately according to eq. (9) and are reduced by the factor 0/81. On the other hand, the neutrons with the shortlived precursors behave approximately like in eq. (8) and are not appreciably reduced. Hence, a small excess reactivity enables the re actor to increase its power without "waiting" for the not very abundant longdelayed neutron. The reactor goes to fairly short time constants with surprisingly small excess reactivities. However, to make the re actor prompt critical, that is to enable it to exponentiate without even the littledelayed neutrons, takes a substantial excess reactivity because of the almost undiminished amount of the latter neutrons. GPO 822202 a It UNIVERSITY OF FLORIDA 3 1262 08905 5445IIIIII I ill 3 1262 08905 5445 
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