The inhour formula for a circulating-fuel nuclear reactor with slug flow

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Title:
The inhour formula for a circulating-fuel nuclear reactor with slug flow
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8 p. : ; 28 cm.
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English
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Ergen, W. K
Oak Ridge National Laboratory
U.S. Atomic Energy Commission
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Subjects / Keywords:
Nuclear fuels -- Measurement -- Mathematical models   ( lcsh )
Nuclear reactors   ( lcsh )
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bibliography   ( marcgt )
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Statement of Responsibility:
W.K. Ergen.
General Note:
"CF-53-12-108". "Work performed under contract no. W-7405-Eng-26."
General Note:
"Subject category: Physics."--Cover.
General Note:
"December 22, 1953."
General Note:
"Date declassified: December 20, 1955."--P.2 of cover.

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-^ n CLASSIFIED


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CF-53-12-108

Subject Category: PHYSICS



UNITED STATES ATOMIC ENERGY COMMISSION


THE INHOUR FORMULA FOR A


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AL. 1 1 156 "1

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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
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disclosed in this report.






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 Flow


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-
formula usually derived for stationary-fuel reactors,2 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/t) (kex-B)P + B D(s) P(t-s) ds (1)*



William Krasny Ergen, The Kinetics of the Circulating-Fuel 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 circulating-fuel 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
S1-0 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 _A-is
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 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 PD(s) for the circulating-fuel 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 re-enter 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,+Q-s)/O. At s = no1-Q(n= 1, 2, 3 ...), F(s) is
zero, but since the fuel under consideration re-enters the reactor be-
tween this moment and s = no1, F(s) increases linearly: F(s) = (s-nO1+
9)/Q. For BD(s) we thus obtain:


no0l+-s 5 s
OD(s) = E1 p )e i
i=l
s-nO +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 nO1-Osnol, n = 2, 3, ., (4)

for no1+Q!s= (n+l) 1-0, n = 0, 1, 2,..,
eq. (1), and for P we set P = P e/T. Then
o
n +0

EO n1 a P e(t-s)/T d


nO1

+ 1
n=1-
no -e


s-nQ +Q 8
s-n 1 +0 -)is (t-P)/T 1
SPe


The common factor Pet/T cancels out. The substitution e= nO1+0-s transforms
Th comnfatrP








(n81+Q-s)exp{-Ci+(l/Tq s ds
"1
into
exni hi)+(1/Ti ex0p hi+ (1/T Je [+(1/T dr

and the substitutiond= a-nQl+ 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 o-1 + t-Pi-e 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 stationary-fuel 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 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<(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 circulating-fuel reactor
the situation is qualitatively the same; the fractions under the sum
in eq. (5) are essentially of the form

-x -ax -(a-l)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 54-1-1 Jan. 15, 1954.




8



stationary, inasmuch as all delayed neutrons, even the ones with the
long-lived 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 short-lived delayed-neutrons 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 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 0/81. 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 latter neutrons.


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