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Pulsed operation of a fast reactor
Reactor
Science
and T’c&mology
(J.N.E.
Par11 A/B).
1961. Vol.
14. pp. 55 lo 58.
PULSED OPERATION
Fwgamon
Pms
Ltd.
Printed
in Northern
Ireland
OF A FAST REACTOR*t
I. I. BONDARENKOand Yu. YA. STAVISSKII Abstract-Formulae are given for the duration of the power pulses and for the variation of power with time. Possible parameters are given for a pulsed fast reactor that could be used for nuclear physics research. FORmany physical experiments using nuclear reactors the deciding factor is not the mean reactor power over long time intervals but the power during the short interval of time necessary for the experiment. In such cases pulsed operation of the reactor proves to be most advantageous, i.e. operation in which the power of the reactor increases by a large factor during a short interval of time. The advantage of pulsed operation shows itself most clearly in a fast reactor because in it the lifetime of the neutrons is very small compared to that in reactors of other types. A pulsed reactor can be used as follows: (1) In experiments using a time-of-flight neutron velocity selector; (2) In the study of short-lived activities (delayed neutrons,(i~~) short-lived isomers, etc.); (3) In various investigations in which the main difficulty is due to background not connected with the operation of the reactor. In 1955 Blokhintsev proposed a pulsed reactor in which the power pulse arises from the motion of a part of the core which is fixed on a rotating disk. In such an arrangement the reactor remains sub-critical over a prolonged period and becomes super-critical only during short intervals of time, namely, when the moving part of the core passes through the main fixed section of the core. The variation of reactor power with time during a pulse can be expressed with sufficient accuracy by the following equation :
-w(t)@) s + Ty dt. (1) 1 ( 7 Here w(t) is the instantaneous value of the reactor power (the number of fissions per set), T is the mean lifetime of the prompt neutrons;, v is the number of secondary neutrons, S is the strength of the external neutron source and e(t) is the excess multiplication coefficient (super-criticality) of the prompt neutrons dw(r) =
*Translated by N. KIMMERfrom Atomnaya Energiya 7, 417 (1959). t This article reports on work performed in 1956. Some of the data presented in it were also obtained by T. N. ZU~MEV. .$ For a determination of the mean lifetime of prompt neutrons see the paper by USACHEV.‘*)
in the reactor. In this equation the influence of delayed neutrons has not been taken into account since their lifetime in the case considered is much greater than the duration of the power excursion. In order to obtain a short power burst one must make e(t) positive over short time. We take as the origin of time (t = 0) the moment when e(t)‘has its maximum value (see Fig. 1). We shall assume that the function e(t) is symmetric with respect to the point t = 0. This condition is evidently always satisfied if the variation in the reactivity is effected by a uniform displacement’ of a moveable part of the core. At an instant of time far removed from the beginning of the pulse e(t) has a constant negative value equal to eI. During this period the power of the reactor is also constant:
w,=_L. &IV
Subsequently c(t) begins to increase.
i
As long as the
t
7-m-I-
FIG. l.-Schematic development of the power pulses in the pulsed operation of a fast reactor. 52
56
1.1. B~NDARENKO
and YLLYA.
rate of increase of ~(1) is sufficiently small and its value still remains sufficiently far from zero, the process has a quasi-static character. During this period of time the reactor power at any instant is near its equilibrium value S u(t) = - . (3) e(0v This state of affairs persists as long as the condition (4) is satisfied. Subsequently the power begins to increase more slowly than would follow from equation (3). Calculations show that as e(t) becomes zero the reactor power reaches the value 1.25 s r(‘(--to) R5-, +Y
M)
where y = -
dt
(5)
After this e(t) becomes I1.p-10
STAVISSKII
Thus near the maximum the pulse has a Gaussian form with a half-width 0 given by 0 N 2.35 $1; . Further away from the maximum. the shape of the pulse differs from the Gaussian form. Its increase-is somewhat slower and its drop more rapid. However, numerical calculations from equation (1) show that for large power pulses significant deviations from Gaussian form are observed only at the edges of the pulse and contribute only slightly to the total energy of the pulse. In reactors the maximum value of e(t), namely E,, is always small compared to unity. Therefore the dependence of E on the position, X, of the moveable part of the core may be given approximately in parabolic form E(X) = E, - aXa, (10) where a is a quantity depending on the parameters of the moving part of the core and of the reactor. This constant characterizes the steepness of the variation of the activity with displacement of the moving part of the core. The dependence of E on time is expressed in terms of the magnitude of a and the speed u of the motion of the moving part of the core as follows:
positive and the required increase of power begins. In describing the variation of power in this period the second term of the r.h.s. of equation (1) can be neglected. This does not lead to significant errors if one is dealing with pulses during which the reactor power increases by a large factor, which is the case &= E, - austa. of interest here. Then Using (7) and (10) we obtain an expression for the c(t) dt total number of fissions during a pulse dw(t) -=-* (6)
w(t)
7
E = ;K(E,,,); ’ The maximum value of the power, w,, reached at the instant t = t, (when e(t) again takes the value 0), is
(11) Kh,J =
w
tn =
w(-to)
exp [/Tt0 F] (7)
It is also of interest to know the dependenceof the duration of the pulse on the basic quantities characterizing the reactor (T, a, 0). From the relations given above one can obtain the following expression:
Let us examine the shape of the pulse near the maximum. At times t near to to the dependence of e(t) can be represented as follows: Thus the duration
’40 = -y(t
-
[
-
0 - 27OY
(12)
of the pulse decreases with de-
(8) creasing T and with increasing a and V. The duration
to).
We then obtain from equation (6) w(t) = w,exp
a = 1.74&4.
1.
depends also on the energy of the pulse (more precisely on the ratio of the pulse energy to the strength of the extraneous neutron sources). However, over a wide range of pulse energies this: dependence is very
57
Pulsed operation of a fast reactor
weak. In equation (12) the dependence is contained in the coefficient A: A N
8E y,p3q2:3c4/3 S
In _ [1
1 . -(l/4)
Until now we have investigated single power pulses. We now consider the case when the reacto. operates periodically, giving n pulses in one second. Under such conditions sources of delayed neutrons will accumulate in the reactor. If the reactor power is at all appreciable, then the sources of delayed neutrons accumulated during the previous impulses,
s= 2
CiAi,
(13)
Let us denote the E, which satisfies equation (16) by ~0. Thus when the reactor operates with e,,, = E,, a self-sustaining chain reaction is realized. which corresponds to the operation of an ordinary reactor under stationary conditions with Ken = 1. For F,,, > so the chain reaction dies away, It is interesting to note that the time-dependence of the mean power of a pulsed reactor after a small perturbation (for e,,, near to co) is similar to the time-dependence of the power in ordinary fast reactors. To show this let us write the equations for ordinary reactors as follows : w(t) _
z:nit,(r)
I
I’
will be significant extraneous sources relative to the pulse. Here ci is the concentration of the predecessors of the delayed neutrons in the i-th group and Aj is the corresponding decay constant. If the interval between pulses is small compared to the lifetime of the delayed neutrons the mean power of the reactor W(r) can be expressed as W(r) == E(r)n -jE(t)/] f . &I
(14)
Here the second term represents the power generated during the interval between pulses. In the cases of greatest interest, practically the whole power must be generated during the pulses. For this to be so it is necessary that
We shall consider that what follows. Taking into account of the delayed neutrons the following equation the mean power:
this condition
does hold in
dci -=
dt
-
1
(l7a)
jG$
AiCj(t) +
(1%)
w(t)yBi*
It is here assumed that the lifetime of the prompt neutrons is zero. For reactors with a short life-time of prompt neutrons (fast reactors) this assumption is always justified. In conditions of delayed criticality it is easy to show that, in the case of small perturbations, the system of equations (I 5) describing a pulsed reactor coincides with the system (17) if one denotes
K&J ME ,.) I
by P*-
Thus the time-dependence of the mean power of a pulsed reactor proves to be the same as for an ordinary the accumulation and decay reactor, for a certain effective fraction of delayed we obtain from equation (I I) neutrons. Thus a pulsed reactor may be controlled describing the variation of using the same type of regulating system as for ordinary reactors. Here the displacement of the w(t) = z: ~iWk controlling mechanism leads to a change of P,. (ISa) z K(e,)n; For large perturbations the kinetics of a pulsed 7’ reactor differ from those of ordinary reactors. This difference shows itself for instance in the fact that for dci(t) = -Aici(t) + W(/)~l$li. (15b) ordinary reactors, as distinct from pulsed reactors. dt the regulating influence of delayed neutrons disappears for E > p, and this essentially changes the reactor Here pi is the yield of delayed neutrons of the i-th kinetics. group, 2 pi = 8. From these equations it is easy to As an example let us state the result of calculations for one variant of a fast reactor with pulsed operation. obtain ‘a condition for quasi-stationary reactor In this case the core contains plutonium and the operation (pulse amplitude independent of time): moving part of the core is z35U. The value of the K(eJn/9 = I. (16) basic reactor parameters determining its operation 4A-(4 PP.)
58
1. I. BONDARENKO and Yv. Ya. Sr~vlssKn
under pulsed conditions is a = 7.5 x lO-*ems, T = 10-s set and eI = O-2. At a velocity of motion of the moving core section of 300 m/set and a pulse frequency of 10 per second, the duration of a pulse will be 13 x 10-6sec at an equilibrium value of the instantaneous critical excess se = 2-6 x 10-a. The effective proportion of delayed neutrons in these conditions is 24 x 10-p. At a mean reactor power of 10 kilowatts the power in pulsed operation reaches -lo5 kilowatts.
Acknowle&ements-In conclusion the authors wish to express their deep indebtedness to D. I. BL~KHINTSEV, A. I. LUPUNSKII and 0. D. KAZACHKOVKY for constant attention and advice and also to F. I. and F. V. KRUSNARYOV for valuable discussion. REFERENCES
1. HOFFMANNF. DE, FELD B. T. and STEINP. R., Pkys. Rea. 74, 1330, (1948). 2. PIN H. C., Nucleonics 13, No. 10, 48 (1955). 3. USACHEV L. N_, Reactor Cons?ruction and 77wory of Reactors. Moscow, Acad. Sci. U.S.S.R. (1955).
Science
and T’c&mology
(J.N.E.
Par11 A/B).
1961. Vol.
14. pp. 55 lo 58.
PULSED OPERATION
Fwgamon
Pms
Ltd.
Printed
in Northern
Ireland
OF A FAST REACTOR*t
I. I. BONDARENKOand Yu. YA. STAVISSKII Abstract-Formulae are given for the duration of the power pulses and for the variation of power with time. Possible parameters are given for a pulsed fast reactor that could be used for nuclear physics research. FORmany physical experiments using nuclear reactors the deciding factor is not the mean reactor power over long time intervals but the power during the short interval of time necessary for the experiment. In such cases pulsed operation of the reactor proves to be most advantageous, i.e. operation in which the power of the reactor increases by a large factor during a short interval of time. The advantage of pulsed operation shows itself most clearly in a fast reactor because in it the lifetime of the neutrons is very small compared to that in reactors of other types. A pulsed reactor can be used as follows: (1) In experiments using a time-of-flight neutron velocity selector; (2) In the study of short-lived activities (delayed neutrons,(i~~) short-lived isomers, etc.); (3) In various investigations in which the main difficulty is due to background not connected with the operation of the reactor. In 1955 Blokhintsev proposed a pulsed reactor in which the power pulse arises from the motion of a part of the core which is fixed on a rotating disk. In such an arrangement the reactor remains sub-critical over a prolonged period and becomes super-critical only during short intervals of time, namely, when the moving part of the core passes through the main fixed section of the core. The variation of reactor power with time during a pulse can be expressed with sufficient accuracy by the following equation :
-w(t)@) s + Ty dt. (1) 1 ( 7 Here w(t) is the instantaneous value of the reactor power (the number of fissions per set), T is the mean lifetime of the prompt neutrons;, v is the number of secondary neutrons, S is the strength of the external neutron source and e(t) is the excess multiplication coefficient (super-criticality) of the prompt neutrons dw(r) =
*Translated by N. KIMMERfrom Atomnaya Energiya 7, 417 (1959). t This article reports on work performed in 1956. Some of the data presented in it were also obtained by T. N. ZU~MEV. .$ For a determination of the mean lifetime of prompt neutrons see the paper by USACHEV.‘*)
in the reactor. In this equation the influence of delayed neutrons has not been taken into account since their lifetime in the case considered is much greater than the duration of the power excursion. In order to obtain a short power burst one must make e(t) positive over short time. We take as the origin of time (t = 0) the moment when e(t)‘has its maximum value (see Fig. 1). We shall assume that the function e(t) is symmetric with respect to the point t = 0. This condition is evidently always satisfied if the variation in the reactivity is effected by a uniform displacement’ of a moveable part of the core. At an instant of time far removed from the beginning of the pulse e(t) has a constant negative value equal to eI. During this period the power of the reactor is also constant:
w,=_L. &IV
Subsequently c(t) begins to increase.
i
As long as the
t
7-m-I-
FIG. l.-Schematic development of the power pulses in the pulsed operation of a fast reactor. 52
56
1.1. B~NDARENKO
and YLLYA.
rate of increase of ~(1) is sufficiently small and its value still remains sufficiently far from zero, the process has a quasi-static character. During this period of time the reactor power at any instant is near its equilibrium value S u(t) = - . (3) e(0v This state of affairs persists as long as the condition (4) is satisfied. Subsequently the power begins to increase more slowly than would follow from equation (3). Calculations show that as e(t) becomes zero the reactor power reaches the value 1.25 s r(‘(--to) R5-, +Y
M)
where y = -
dt
(5)
After this e(t) becomes I1.p-10
STAVISSKII
Thus near the maximum the pulse has a Gaussian form with a half-width 0 given by 0 N 2.35 $1; . Further away from the maximum. the shape of the pulse differs from the Gaussian form. Its increase-is somewhat slower and its drop more rapid. However, numerical calculations from equation (1) show that for large power pulses significant deviations from Gaussian form are observed only at the edges of the pulse and contribute only slightly to the total energy of the pulse. In reactors the maximum value of e(t), namely E,, is always small compared to unity. Therefore the dependence of E on the position, X, of the moveable part of the core may be given approximately in parabolic form E(X) = E, - aXa, (10) where a is a quantity depending on the parameters of the moving part of the core and of the reactor. This constant characterizes the steepness of the variation of the activity with displacement of the moving part of the core. The dependence of E on time is expressed in terms of the magnitude of a and the speed u of the motion of the moving part of the core as follows:
positive and the required increase of power begins. In describing the variation of power in this period the second term of the r.h.s. of equation (1) can be neglected. This does not lead to significant errors if one is dealing with pulses during which the reactor power increases by a large factor, which is the case &= E, - austa. of interest here. Then Using (7) and (10) we obtain an expression for the c(t) dt total number of fissions during a pulse dw(t) -=-* (6)
w(t)
7
E = ;K(E,,,); ’ The maximum value of the power, w,, reached at the instant t = t, (when e(t) again takes the value 0), is
(11) Kh,J =
w
tn =
w(-to)
exp [/Tt0 F] (7)
It is also of interest to know the dependenceof the duration of the pulse on the basic quantities characterizing the reactor (T, a, 0). From the relations given above one can obtain the following expression:
Let us examine the shape of the pulse near the maximum. At times t near to to the dependence of e(t) can be represented as follows: Thus the duration
’40 = -y(t
-
[
-
0 - 27OY
(12)
of the pulse decreases with de-
(8) creasing T and with increasing a and V. The duration
to).
We then obtain from equation (6) w(t) = w,exp
a = 1.74&4.
1.
depends also on the energy of the pulse (more precisely on the ratio of the pulse energy to the strength of the extraneous neutron sources). However, over a wide range of pulse energies this: dependence is very
57
Pulsed operation of a fast reactor
weak. In equation (12) the dependence is contained in the coefficient A: A N
8E y,p3q2:3c4/3 S
In _ [1
1 . -(l/4)
Until now we have investigated single power pulses. We now consider the case when the reacto. operates periodically, giving n pulses in one second. Under such conditions sources of delayed neutrons will accumulate in the reactor. If the reactor power is at all appreciable, then the sources of delayed neutrons accumulated during the previous impulses,
s= 2
CiAi,
(13)
Let us denote the E, which satisfies equation (16) by ~0. Thus when the reactor operates with e,,, = E,, a self-sustaining chain reaction is realized. which corresponds to the operation of an ordinary reactor under stationary conditions with Ken = 1. For F,,, > so the chain reaction dies away, It is interesting to note that the time-dependence of the mean power of a pulsed reactor after a small perturbation (for e,,, near to co) is similar to the time-dependence of the power in ordinary fast reactors. To show this let us write the equations for ordinary reactors as follows : w(t) _
z:nit,(r)
I
I’
will be significant extraneous sources relative to the pulse. Here ci is the concentration of the predecessors of the delayed neutrons in the i-th group and Aj is the corresponding decay constant. If the interval between pulses is small compared to the lifetime of the delayed neutrons the mean power of the reactor W(r) can be expressed as W(r) == E(r)n -jE(t)/] f . &I
(14)
Here the second term represents the power generated during the interval between pulses. In the cases of greatest interest, practically the whole power must be generated during the pulses. For this to be so it is necessary that
We shall consider that what follows. Taking into account of the delayed neutrons the following equation the mean power:
this condition
does hold in
dci -=
dt
-
1
(l7a)
jG$
AiCj(t) +
(1%)
w(t)yBi*
It is here assumed that the lifetime of the prompt neutrons is zero. For reactors with a short life-time of prompt neutrons (fast reactors) this assumption is always justified. In conditions of delayed criticality it is easy to show that, in the case of small perturbations, the system of equations (I 5) describing a pulsed reactor coincides with the system (17) if one denotes
K&J ME ,.) I
by P*-
Thus the time-dependence of the mean power of a pulsed reactor proves to be the same as for an ordinary the accumulation and decay reactor, for a certain effective fraction of delayed we obtain from equation (I I) neutrons. Thus a pulsed reactor may be controlled describing the variation of using the same type of regulating system as for ordinary reactors. Here the displacement of the w(t) = z: ~iWk controlling mechanism leads to a change of P,. (ISa) z K(e,)n; For large perturbations the kinetics of a pulsed 7’ reactor differ from those of ordinary reactors. This difference shows itself for instance in the fact that for dci(t) = -Aici(t) + W(/)~l$li. (15b) ordinary reactors, as distinct from pulsed reactors. dt the regulating influence of delayed neutrons disappears for E > p, and this essentially changes the reactor Here pi is the yield of delayed neutrons of the i-th kinetics. group, 2 pi = 8. From these equations it is easy to As an example let us state the result of calculations for one variant of a fast reactor with pulsed operation. obtain ‘a condition for quasi-stationary reactor In this case the core contains plutonium and the operation (pulse amplitude independent of time): moving part of the core is z35U. The value of the K(eJn/9 = I. (16) basic reactor parameters determining its operation 4A-(4 PP.)
58
1. I. BONDARENKO and Yv. Ya. Sr~vlssKn
under pulsed conditions is a = 7.5 x lO-*ems, T = 10-s set and eI = O-2. At a velocity of motion of the moving core section of 300 m/set and a pulse frequency of 10 per second, the duration of a pulse will be 13 x 10-6sec at an equilibrium value of the instantaneous critical excess se = 2-6 x 10-a. The effective proportion of delayed neutrons in these conditions is 24 x 10-p. At a mean reactor power of 10 kilowatts the power in pulsed operation reaches -lo5 kilowatts.
Acknowle&ements-In conclusion the authors wish to express their deep indebtedness to D. I. BL~KHINTSEV, A. I. LUPUNSKII and 0. D. KAZACHKOVKY for constant attention and advice and also to F. I. and F. V. KRUSNARYOV for valuable discussion. REFERENCES
1. HOFFMANNF. DE, FELD B. T. and STEINP. R., Pkys. Rea. 74, 1330, (1948). 2. PIN H. C., Nucleonics 13, No. 10, 48 (1955). 3. USACHEV L. N_, Reactor Cons?ruction and 77wory of Reactors. Moscow, Acad. Sci. U.S.S.R. (1955).