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 UNCLASSIFIED 
 
 CF-53-12-108 
 
 Subject Category: PHYSICS 
 
 UNITED STATES ATOMIC ENERGY COMMISSION 
 
 THE INHOUR FORMULA FOR A 
 CmCULATINcCTu^TjF^jC^A^ 
 REACTOR WITH/§rftf™&fe 
 
 By 
 
 W. K. Ergen " — ^££^OS/r, 
 
 SJiC^^s 
 
 A ,j? 
 
 
 [ APK 2 1956 
 
 V V 
 
 
 December 22, 1953 
 
 Oak Ridge National Laboratory 
 Oak Ridge, Tennessee 
 
 Technical Information Service, Oak Ridge, Tennessee 
 
Date Declassified: December 20, 1955* 
 
 This report was prepared as a scientific account of Govern- 
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 ington 25, D. C. 
 
CF -53 -12 -108 
 
 The Inhour Formula for a Circulating-Fuel 
 Nuclear Reactor with Slug Flow. 
 
 W. K. Ergen 
 
 Work performed under Contract No. W-7405~Eng-26 
 
 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 Circulating -Fuel Nuclear 
 Reactor with Slug Flov 
 
 As pointed out in a previous paper, the circulating-fuel reactor 
 differs in its dynamic behavior from a reactor with stationary fuel, be- 
 cause fuel circulation sweeps some of the delayed-neutrons 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 - 
 
 p 
 formula usually derived for stationary-fuel reactors, requires some 
 
 modification before it becomes applicable to the circulating -fuel 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/ f ) r (k ex -(3)P + P |d(s) P(t-s) dsl (1)* 
 
 William Krasny Ergen, The Kinetics of the Circulating -Fuel Nuclear 
 Reactor, J. Appl. Phys. (in print) 
 
 See for instance S. Glasstone and M. C. Edlund, The Elements of 
 Nuclear Reactor Theory, D. Van Nostrand Co., Inc., 1952, p. 29k ff . 
 
 Some authors, for instance Glasstone and Edlund, loc . cit., write 
 the equations corresponding to (l) in a slightly different form. 
 The difference consists in terms of the order k (£/t) or k p, 
 which are negligibly small. ex ex 
 
U 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 circulating-fuel reactor has "been discussed in some detail 
 in ref . We approximate in the following the actual arrangement jy 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 
 0..-0 through the outside loop. In this approximation, B D(s) is simply 
 the probability 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(s)is normalized so that 
 
 f D(s) ds = 1 (2) 
 
 If the fuel is stationary, BD(s) is the familiar curve obtained by the 
 
 superposition of 5 exponentials: 
 
 5 _^ 8 
 
 BD(s) = Z B.X.e i (3) 
 
 i=l 
 
 The ^. are the decay constants of the 5 groups of delayed neutrons, and 
 
 the B. are the probabilities that a given fission neutron is a delayed 
 
 1 th 
 neutron of the i group. 
 
 For the circulating -fuel 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 BD(s) for the circulating-fuel reactor. 
 
 Since 6.. 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 = n©.. (n = 0, 1, 2 . ..), F ( 8 ) is equal to 1; 
 
 at s = n0.+0 (n = 0, 1, 2 ...), F(s) is equal to zero. 
 (We assume ©.. s 2© so that the fuel under consideration has not started 
 to re-enter the reactor when the last of its elements leaves the reacting 
 zone). Between s = n©, and s = n©,+©, F(s) decreases linearly, and 
 hence has the value (n©,+©-s)/©. At s = n© 1 -©(n= 1, 2, 3 . ..), F(s) is 
 zero, hut since the fuel under consideration re-enters the reactor be- 
 tween this moment and s = n©. , F(s) increases linearly: F(s) = (s-n© 1 + 
 ©)/©. For £D(s) we thus obtain: 
 
 PD( 8 ) = n9l ^" S £ P ± ^«"V f ° r ^s&Q^, n = 0, 1, 2, ..., 
 
 i=l 
 s-n© +© 5 
 
 -> 
 
 PD(s) = gi— i^iV iS for nG 1 - Q ^s^n© 1 , n = 1, 2, 3, ••., 
 
 <*> 
 
 pD(s) = 
 
 for n© +©£s£ (n+l)0 -0, n = 0, 1, 2, 
 
 t/T 
 
 Eq. (k) is now substituted into eq. (l), and for P we set P = P e~'~. Th« 
 
 TPe t / T = (k -p)P e^ + T p.>. 
 m o v ex w ' o *- p i i 
 
 T 1=1 
 
 n©,+© 
 
 ? nO.+O-s -\s . 
 
 n 1 P -Vt 
 
 P e' 
 o 
 
 ■s)/T 
 
 ds 
 
 S - n9 l +6 c -Vp e (t-s)/T 
 
 © o 
 
 ds 
 
 The common factor P Q e /T cancels out. The substitution = n© +©-s transforms 
 
f 
 
 0^9 
 
 (n9 1 +0-6)exp{-^ i +(l/T5 s } ds 
 
 ™1 
 
 and the substitution^"^ s-nO.,+ transforms 
 y (s-n0 1+ e) expf-j^+Cl/TjJ sj ds 
 
 no 1 -e 
 
 into g 
 
 exp£nd,|> i+ (l/T)] j exp^> i+ (l/T]) j (^ expf. j> 1+ (l/Tj<NJd <T . 
 
 The geometric series exp£n [X.+(l/T)"| ©,\can now he summed, and the in- 
 tegrals over 6*" evaluated by elementary methods. After performing all 
 these operations, one obtains the following inhour formula: 
 
 k ex =T + P 
 
 i^ fr"" i r h* + ^ r , (5 ) 
 
 5 A p i*i r — -yu.err 
 
 i=l 
 
 1± |_l-e _ 
 
 /!.=>.+ (l/T). (6) 
 
 P is evaluated by means of eq. (2): 
 
 P = J pD(s) ds. 
 If PD(s) is substituted into this 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 
 
p 
 
 ->.o -Vo. -X (©,-©) 
 
 5 ^0 - 1 + e -e ^O + 1)^ (?) 
 
 = la p. 7T-Q 
 
 0A.(1 - e ) 
 
 In spite of the formidable appearance of eqs. (5), (6), and (7), it is 
 easy to f indoor any given © and ,the value of k which produces a 
 given time constant T. 
 
 Furthermore, the following reasoning describes the general fea- 
 tures of the equations. Consider first the dependence of k on T. If 
 T =<ad,yU. = V, and the sum on the right of (5) is equal to p, k 
 is equal to zero. This corresponds to the state in vhich 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 byyu. and 
 the bracket in the denominator by 1. Hence the fraction tends to zero 
 like Q/jo.., as T goes to zero. If U is very small, as it is in practice, 
 T will be small as soon as k exceeds p by a small amount. Then the 
 complicated sum on the right of (5) is of little importance, and T is 
 determined by (, /T = k -p, 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 stationary-fuel reactor, the k which make the reactor 
 
 ' ex 
 
 prompt critical is given by £ p . With circulating fuel, the reactor 
 is prompt critical if k gx = p, which is less than £ p , that is it 
 takes less excess reactivity to make the circulating -fuel reactor prompt 
 critical than to do the same thing to a stationary-fuel reactor. This is 
 physically evident because the fuel circulation renders some of the de- 
 layed neutrons ineffective. That p<£p can also be verified mathe- 
 matically. 
 
 For T intermediate between very small positive values and + *° 
 we consider again the analogy to the stationary fuel reactor. Here the 
 inhour formula reads: 
 
k _ Z+ £ _!i *) (8) 
 
 ex _ T g,_ 1 + \? ' 
 
 For every positive k there is one, and only one, positive time con- 
 stant T and vice versa. This in a consequence of the fact that k 
 is a monoton decreasing function of T, for if this monotony did not 
 exist, there could he several positive T value corresponding to a 
 given value of k , or vice versa. In the circulating-fuel reactor 
 the situation is qualitatively the same; the fractions under the sum 
 in eq. (5) are essentially of the form 
 
 x - 1 + e "* -e ""(*+!) + e" (a ~ l)x (x yi.9. a = 9^9) 
 
 F^r 
 
 and an expression of this form can be shown to he 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 
 mono tonic ally, according to eq. (5); for every positive k there is one, 
 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, 
 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 9.. , the transit 
 time of the fuel through the whole loop. If 9 (and hence also 9 ) 
 is very large compared to all l/)>. and ~LJai., 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 Reactors, Oak Ridge National Laboratory 
 Memo CF 5^-1-1 Jan. 15, 195U. 
 
8 
 
 stationary, inasmuch as all delayed neutrons, even the ones vlth the 
 long-lived precursors, are given off inside the reacting zone, before 
 much fuel reaches the outside. On the other hand, if ©.. and hence 
 also is small compared to all l/\., that is if the transit time of 
 the fuel through the complete loop is small compared to the mean life 
 of even the short-lived delayed-neutrons precursors, then for T» 
 
 k 
 
 = r + ° e 
 
 h 
 
 ex 
 
 T Q i=l 
 
 1 +> i T 
 
 (9) 
 
 This is the same as the inhour formula for the stationary fuel reactor, 
 except that all the fission yields p. are decreased by the factor ©/©,. 
 This is physically easy to understand, since ©/©, is just the prob- 
 ability that a given delayed neutron is born inside the reactor. 
 
 Of interest is the intermediate case, in which is smaller than 
 the mean life of the long-lived delayed-neutron precursors, and larger 
 than the mean life of the short-lived precursors. In that case, the 
 long-delayed neutrons act approximately according to eq. (9) and are 
 reduced by the factor ©/© . On the other hand, the neutrons with the 
 short-lived 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 
 long-delayed 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 little-delayed neutrons, takes a substantial excess reactivity 
 because of the almost undiminished amount of the l-atter neutrons. 
 
 DPO B222DZ 
 
UNIVERSITY OF FLORIDA 
 
 3 1262 08905 5445 
 
 *