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Reactivity feedback with short delay times
Journal
of Nuclear
Fhergy,
Vol. 23, pp. 569 10 574.
REACTIVITY
Perpamon
Press 1969.
Printed
in Northern
Ireland
FEEDBACK WITH SHORT DELAY TIMES WALTER H. K~~HLER*
Institut fiir Neutronenphysik und Reaktortechnik Kernforschungszentrum Karlsruhe, 75 Karlsruhe, Postfach 3640, Germany (First received 2 May 1969 and infinal form 30 June 1969) Abstract-Analytical expressions based on the point reactor model are presented for the increase in peak power, temperature and energy release--caused by short time delays in the temperature feedback of reactivity-in superpromptcritical excursions. A criterion is presented to decide whether the delay time ought to be included in the analysis or not. Two specific cases, slowing-down delay and heat transfer delay of Doppler feedback in Na cooled fast breeder reactors are discussed. 1. INTRODUCTION
SAFETYconsiderations require with much greater emphasis for fast than for thermal reactors the existence of a prompt, inherent shut-down effect. In oxide or carbide fueled fast breeder reactors the negative Doppler reactivity effect satisfies this need. One is concerned, however, about possible short delays in the Doppler feedback because of the extremely short reactor periods in superpromptcritical excursions. Such a delay can be caused by the finite time of heat transfer from the PuO, grains in the fuel (whose Doppler contribution is negligible) to the UO, grains in the fuel (that are responsible for the negative Doppler coefficient of reactivity). Another cause of delay is the slowing-down of neutrons to those energies (about 1 keV) where most of the Doppler feedback is generated. Especially the slowing-down delay is short, in the order of microseconds, and has led to this investigation of short time delays in superpromptcritical excursions. The aim of this investigation is to show with a simplified model under what conditions short time delays may be neglected in the analysis of the excursion. Should the model indicate that the delay time cannot be neglected, then a more detailed analysis, within the usually more refined excursion models, becomes necessary. 2. THEORY
The FUCHS(1946) model for superpromptcritical excursions is extended to include a delay time T in the temperature reactivity feedback: Ii(t) = py n(t) - n, = (mc)
p(t) = ps * On leave from Texas A + M University. 1
569
n(t)
T(t)
cqt -
(2) T)
(3)
W. H. K~HLER
570
where ps = n = A = /I = m =
c= -a
= T =
step reactivity in $ above prompt critical power prompt neutron generation time fraction of delayed neutrons fuel mass fuel specific heat Doppler coefficient of reactivity fuel temperature increase over steady state conditions.
We proceed to obtain approximate solutions for the peak power, n,, the final temperature increase Tf and the total energy Et for the case where the delay time T is much shorter than the reactor period, or
For the opposite case, delay time much longer than reactor period, has given an approximate solution. Developing T(t - T) into a TAYLOR series, we obtain:
FORBES
(1958)
Since no p, we see that we can neglect the third term on the r.h.s. of equation (5) because of equation (4). For n, > no we obtain the following expressions:
n(p) =
$$
’- ” _z
1 ln
(p, _ ,I
l-;p8
(WA
nnz = -
1
ln
Ga/3 (
I-;p,
1
(6)
I --
TB A ps I
(74 (8) (9) In the last three formulae the terms in front of the bracket are identical with the solution for n, > n,. Equation (7) gives an expression for the peak power, n,, that is always above the exact solution because of the truncation of the Taylor series, equation (5). Comparison of equations (7a) to (9) with numerical solutions of equations (l)-(3) including the exact treatment of delayed neutrons, have shown, that these equations give good results for ~/3p*/A Q 0.5. FUCHS
Reactivity feedback with short delay times
571
While we do not have an explicit solution, n(t), we see that the delay time T causes a smaller increase in the energy released than in the peak power. This means a narrower and unsymmetric pulse-shape, a behavior that is also exhibited in other non-linear energy models (NYER (1964)). If we are willing to accept a 10 per cent error in the energy release due to the delay time T (mainly because uncertainties in the model and in physical parameters can lead to larger errors) then a criterion for neglecting the delay time would be
To restate this criterion in terms of a reactivity ramp, a, in ($/set), we make use of a correlation by NYER (1964), saying that from the point of maximum inverse period, and final quenching of the burst are essentially the same for a W max. 3 the development reactivity ramp as for a step transient whose initial inverse period is equal to co,,, of the ramp. Quantitatively this statement gives the equivalence
(11) Numerical calculations have shown that c is about 36 for typical rapid superpromptcritical excursions in fast breeder reactors with an uncertainty of about 20 per cent. Numerical calculations have also shown that the correlation, equation (ll), still holds in the presence of short delay times. Before restating the criterion in terms of a ramp the uncertainty of 20 per cent in the constant c will be discussed. A 10 per cent error in ps would, as long as equation (10) is valid, lead to at most a 1 per cent error in the contribution of the delay time term to the total energy released, equation (9). Equation (11) shows that ps is proportional to &, hence a 20 per cent error in c would imply approximately a 10 per cent error in ps and would thus lead to a maximum error of about 1 per cent in the contribution of the delay time term to the energy release. In other words, when we now restate the criterion for neglecting the effect of the delay time in terms of the reactivity ramp a, T is determined in such a way that the acceptable error in E, due to the delay time is less than 10 f 1 per cent:
(12) We do not wish to imply that when T exceeds the limit of equation (12) a serious additional safety hazard results; we merely say one has to investigate the safety implications when T exceeds this limit. This investigation which will normally be a numerical treatment should also make use of more refined models. 3. APPLICATION
TO FAST
BREEDER
REACTORS
3.1 Magnitude of delay times (i) Slowing-down delay. Calculations are performed for typical Na cooled fast breeder reactors to obtain the slowing-down delay as T, = 2 Dci~,~ i
(13)
W. H. K~=R
572
DC'
c-_-
10 %
1%
FrG. l.-Slowing
18 b6.5
/ 17 100
215
16
15
465eV
14
13
12
2.15 465
1.0
11
10 '-----
10.0 21.5 46.5 ke
down delay times and relative contribution to Doppler for typical fast breeder with oxide fuel.
coefficient
where DCi is the contribution to the Doppler coefficient of reactivity in energy group i and T,$ is the slowing-down time into group i which is evaluated according to BORGWALDT (1968) as Pm i_ 7x-
J 0
#(t)t
dt
(14)
m q+(t) dt
s0
q+(t) is the first generation flux (after fission) in energy group i. In Fig. 1 we show the histogram of the ~,i and DCi in the most relevant energy groups. In general we find 7,/A to be about 5 for typical fast oxide breeders with an increase of roughly 20 per cent when Na is removed from the core (although T, stays about constant in these two cases). (ii) Heat transfer delay. Models for treating the heat transfer delay time, TV, which characterizes the delay in attaining an asymptotic temperature distribution in TABLEI.--CQMPARISON
OF NUMERICAL AND ANALYTICAL RESULTS
7
A
t, (msec) numerical
n, x lo-’ numerical
Factor * numeric.
Factor * analyt.
0 10 A 20h 100A
0 0.0728 0.1456 0.728
1.094 1.095 1.096 1.11
1.521 1.597 1.681 2.66
1.050 1.105 1.75
1.048 1.096 1.497 1.74 217
200h
1.456
1.138
5.106
3.36
@P,
Et X 1O-2
numeric.
Factor* numeric.
Factor* analyt.
1.456 1.493 1.534 2.060
1.025 1.054 1.415
1.024 1.048 1.25
3-569
2.45
Further data: ps = $2.0; A = 3.60 x lo-‘; ,!I = 3.64 x 1O-s; a/(m) = 1.33 x 1O-a; n, = 1.0 x 10-z; t, is the time to reach the power maximum. * The factors are the ratio of the solution for 7 # 0 to the solution for 7 = 0, were only the first two terms in equation (7a) are used for n,. t The analytical approximation, equation (7), for n, is always above the exact solution. For ~pp,/h = 0.728 there is slow convergence of the expansion of the logarithm in equation (7) that leads to equation (7a). The three factors quoted are for equation (7a) with 2 terms, equation (7a) with 3 terms and equation (7). The last value is above the exact (numerical) result as it should be.
Reactivity feedback with short delay times
513
n(t)
.
10'
\ \
\ I;
\
\ \ \ \ \ \ \ \ \ \ \
lo6
lo5
I \ \ \ lo4
I \ \
\ \ \ \ lo3
\
‘\
l-___
~
102
0.5
FIG. 2.-Power
1.0
1.5
burst - - - - T = 0; -
2-o
7=
200
m
A.
the 23*U02 fuel grains after fissions occur in the 23sPu02 grains have been proposed by FRAUDE (1963), PETERSON(1964) and FISCHER and KELLER (1966). Based on the last reference BRAESSet al. (1967) cite the following pairs (Th(psec), d = diameter of fuel grains (,u)): (50, 40), (130, 60), (340, 90) for an enrichment of 14.3 per cent. The delay times decrease with increasing enrichment. For a grain size of 10~ and below the delay time is essentially zero since the range of fission products in U02 is about 10,~. Taking 40,~~as a reasonable grain size for mixed oxide fuels we would get 7,/A SW100.
574
W. H. K~~HLBR
3.2 Numerical and analytical results
A code developed by H~BEL (1968) that treats the delayed neutrons exactly and is based on a method by FEHLBERG (1960) was used to check the range of validity of the approximate results presented in Section 2. Table 1 gives the results of such a comparison. We conclude that for r/?p,/A G 0.5, the analytical formulation presented in Section 2 still gives usable results. In Fig. 2 the power burst obtained for the last case in Table 1 is shown. We see clearly the influence of the delay time: increased peak power, narrower pulse width and asymmetric pulse shape around the power peak. Using the criterion given by equation (12) with a reasonable upper bound of a = 100 ($/set) and B/A = 5 x lo3 for typical fast breeder reactors we conclude that the slowing-down delay time, TV,may be neglected in the analysis of the Doppler limited burst, whereas a typical heat transfer delay time, TV,of the order of 50 psec to 100 ,usec yields rdap/A = O-035 to O-070 which is in the range were the delay time effect has to be included in the analysis. 4. CONCLUSION
Analytical expressions based on the point reactor are presented for the increase in peak power, final temperature and energy release-caused by short time delays in the temperature reactivity feedback-in superpromptcritical excursions. This leads to a criterion to decide whether it is necessary to include the delay time in the burst analysis or not. A method for calculating the slowing-down delay time for Doppler feedback in fast breeder reactors is indicated. The results show that slowing-down delay may be neglected whereas heat transfer delay in the mixed oxide fuel needs to be considered in the burst analysis. Acknowledgement-A discussion with K. Orr regarding slowing-down delay in Doppler feedback has stimulated this work. REFERENCES BORGWALDT H. (1968) Private communication. BRAESSD. et al. (1967) Proc. Intl Co@ on the Safety of Fast Reactors. Aix-en-Provence. FEHLBERG E. (1960) Z. angew. Math. Mech. 40,252 and 449. FISCHERE. A. and KELLERK. (1966) Nukleonik 8,471. FORBESS. G. (1958) IDO-16452. FRAUDEA. (1963) Private communication. FUCHSK. (1946) L. A. 596. H~BELW. (1968) Private communication. NYERW. E. (1964), In Technology of Nuclear Reactor Safety (T. J. THOMPSON and J. G. BECKERLY, Editors), Vol. 1, p. 417. The M.I.T. Press, Cambridge.
of Nuclear
Fhergy,
Vol. 23, pp. 569 10 574.
REACTIVITY
Perpamon
Press 1969.
Printed
in Northern
Ireland
FEEDBACK WITH SHORT DELAY TIMES WALTER H. K~~HLER*
Institut fiir Neutronenphysik und Reaktortechnik Kernforschungszentrum Karlsruhe, 75 Karlsruhe, Postfach 3640, Germany (First received 2 May 1969 and infinal form 30 June 1969) Abstract-Analytical expressions based on the point reactor model are presented for the increase in peak power, temperature and energy release--caused by short time delays in the temperature feedback of reactivity-in superpromptcritical excursions. A criterion is presented to decide whether the delay time ought to be included in the analysis or not. Two specific cases, slowing-down delay and heat transfer delay of Doppler feedback in Na cooled fast breeder reactors are discussed. 1. INTRODUCTION
SAFETYconsiderations require with much greater emphasis for fast than for thermal reactors the existence of a prompt, inherent shut-down effect. In oxide or carbide fueled fast breeder reactors the negative Doppler reactivity effect satisfies this need. One is concerned, however, about possible short delays in the Doppler feedback because of the extremely short reactor periods in superpromptcritical excursions. Such a delay can be caused by the finite time of heat transfer from the PuO, grains in the fuel (whose Doppler contribution is negligible) to the UO, grains in the fuel (that are responsible for the negative Doppler coefficient of reactivity). Another cause of delay is the slowing-down of neutrons to those energies (about 1 keV) where most of the Doppler feedback is generated. Especially the slowing-down delay is short, in the order of microseconds, and has led to this investigation of short time delays in superpromptcritical excursions. The aim of this investigation is to show with a simplified model under what conditions short time delays may be neglected in the analysis of the excursion. Should the model indicate that the delay time cannot be neglected, then a more detailed analysis, within the usually more refined excursion models, becomes necessary. 2. THEORY
The FUCHS(1946) model for superpromptcritical excursions is extended to include a delay time T in the temperature reactivity feedback: Ii(t) = py n(t) - n, = (mc)
p(t) = ps * On leave from Texas A + M University. 1
569
n(t)
T(t)
cqt -
(2) T)
(3)
W. H. K~HLER
570
where ps = n = A = /I = m =
c= -a
= T =
step reactivity in $ above prompt critical power prompt neutron generation time fraction of delayed neutrons fuel mass fuel specific heat Doppler coefficient of reactivity fuel temperature increase over steady state conditions.
We proceed to obtain approximate solutions for the peak power, n,, the final temperature increase Tf and the total energy Et for the case where the delay time T is much shorter than the reactor period, or
For the opposite case, delay time much longer than reactor period, has given an approximate solution. Developing T(t - T) into a TAYLOR series, we obtain:
FORBES
(1958)
Since no
n(p) =
$$
’- ” _z
1 ln
(p, _ ,I
l-;p8
(WA
nnz = -
1
ln
Ga/3 (
I-;p,
1
(6)
I --
TB A ps I
(74 (8) (9) In the last three formulae the terms in front of the bracket are identical with the solution for n, > n,. Equation (7) gives an expression for the peak power, n,, that is always above the exact solution because of the truncation of the Taylor series, equation (5). Comparison of equations (7a) to (9) with numerical solutions of equations (l)-(3) including the exact treatment of delayed neutrons, have shown, that these equations give good results for ~/3p*/A Q 0.5. FUCHS
Reactivity feedback with short delay times
571
While we do not have an explicit solution, n(t), we see that the delay time T causes a smaller increase in the energy released than in the peak power. This means a narrower and unsymmetric pulse-shape, a behavior that is also exhibited in other non-linear energy models (NYER (1964)). If we are willing to accept a 10 per cent error in the energy release due to the delay time T (mainly because uncertainties in the model and in physical parameters can lead to larger errors) then a criterion for neglecting the delay time would be
To restate this criterion in terms of a reactivity ramp, a, in ($/set), we make use of a correlation by NYER (1964), saying that from the point of maximum inverse period, and final quenching of the burst are essentially the same for a W max. 3 the development reactivity ramp as for a step transient whose initial inverse period is equal to co,,, of the ramp. Quantitatively this statement gives the equivalence
(11) Numerical calculations have shown that c is about 36 for typical rapid superpromptcritical excursions in fast breeder reactors with an uncertainty of about 20 per cent. Numerical calculations have also shown that the correlation, equation (ll), still holds in the presence of short delay times. Before restating the criterion in terms of a ramp the uncertainty of 20 per cent in the constant c will be discussed. A 10 per cent error in ps would, as long as equation (10) is valid, lead to at most a 1 per cent error in the contribution of the delay time term to the total energy released, equation (9). Equation (11) shows that ps is proportional to &, hence a 20 per cent error in c would imply approximately a 10 per cent error in ps and would thus lead to a maximum error of about 1 per cent in the contribution of the delay time term to the energy release. In other words, when we now restate the criterion for neglecting the effect of the delay time in terms of the reactivity ramp a, T is determined in such a way that the acceptable error in E, due to the delay time is less than 10 f 1 per cent:
(12) We do not wish to imply that when T exceeds the limit of equation (12) a serious additional safety hazard results; we merely say one has to investigate the safety implications when T exceeds this limit. This investigation which will normally be a numerical treatment should also make use of more refined models. 3. APPLICATION
TO FAST
BREEDER
REACTORS
3.1 Magnitude of delay times (i) Slowing-down delay. Calculations are performed for typical Na cooled fast breeder reactors to obtain the slowing-down delay as T, = 2 Dci~,~ i
(13)
W. H. K~=R
572
DC'
c-_-
10 %
1%
FrG. l.-Slowing
18 b6.5
/ 17 100
215
16
15
465eV
14
13
12
2.15 465
1.0
11
10 '-----
10.0 21.5 46.5 ke
down delay times and relative contribution to Doppler for typical fast breeder with oxide fuel.
coefficient
where DCi is the contribution to the Doppler coefficient of reactivity in energy group i and T,$ is the slowing-down time into group i which is evaluated according to BORGWALDT (1968) as Pm i_ 7x-
J 0
#(t)t
dt
(14)
m q+(t) dt
s0
q+(t) is the first generation flux (after fission) in energy group i. In Fig. 1 we show the histogram of the ~,i and DCi in the most relevant energy groups. In general we find 7,/A to be about 5 for typical fast oxide breeders with an increase of roughly 20 per cent when Na is removed from the core (although T, stays about constant in these two cases). (ii) Heat transfer delay. Models for treating the heat transfer delay time, TV, which characterizes the delay in attaining an asymptotic temperature distribution in TABLEI.--CQMPARISON
OF NUMERICAL AND ANALYTICAL RESULTS
7
A
t, (msec) numerical
n, x lo-’ numerical
Factor * numeric.
Factor * analyt.
0 10 A 20h 100A
0 0.0728 0.1456 0.728
1.094 1.095 1.096 1.11
1.521 1.597 1.681 2.66
1.050 1.105 1.75
1.048 1.096 1.497 1.74 217
200h
1.456
1.138
5.106
3.36
@P,
Et X 1O-2
numeric.
Factor* numeric.
Factor* analyt.
1.456 1.493 1.534 2.060
1.025 1.054 1.415
1.024 1.048 1.25
3-569
2.45
Further data: ps = $2.0; A = 3.60 x lo-‘; ,!I = 3.64 x 1O-s; a/(m) = 1.33 x 1O-a; n, = 1.0 x 10-z; t, is the time to reach the power maximum. * The factors are the ratio of the solution for 7 # 0 to the solution for 7 = 0, were only the first two terms in equation (7a) are used for n,. t The analytical approximation, equation (7), for n, is always above the exact solution. For ~pp,/h = 0.728 there is slow convergence of the expansion of the logarithm in equation (7) that leads to equation (7a). The three factors quoted are for equation (7a) with 2 terms, equation (7a) with 3 terms and equation (7). The last value is above the exact (numerical) result as it should be.
Reactivity feedback with short delay times
513
n(t)
.
10'
\ \
\ I;
\
\ \ \ \ \ \ \ \ \ \ \
lo6
lo5
I \ \ \ lo4
I \ \
\ \ \ \ lo3
\
‘\
l-___
~
102
0.5
FIG. 2.-Power
1.0
1.5
burst - - - - T = 0; -
2-o
7=
200
m
A.
the 23*U02 fuel grains after fissions occur in the 23sPu02 grains have been proposed by FRAUDE (1963), PETERSON(1964) and FISCHER and KELLER (1966). Based on the last reference BRAESSet al. (1967) cite the following pairs (Th(psec), d = diameter of fuel grains (,u)): (50, 40), (130, 60), (340, 90) for an enrichment of 14.3 per cent. The delay times decrease with increasing enrichment. For a grain size of 10~ and below the delay time is essentially zero since the range of fission products in U02 is about 10,~. Taking 40,~~as a reasonable grain size for mixed oxide fuels we would get 7,/A SW100.
574
W. H. K~~HLBR
3.2 Numerical and analytical results
A code developed by H~BEL (1968) that treats the delayed neutrons exactly and is based on a method by FEHLBERG (1960) was used to check the range of validity of the approximate results presented in Section 2. Table 1 gives the results of such a comparison. We conclude that for r/?p,/A G 0.5, the analytical formulation presented in Section 2 still gives usable results. In Fig. 2 the power burst obtained for the last case in Table 1 is shown. We see clearly the influence of the delay time: increased peak power, narrower pulse width and asymmetric pulse shape around the power peak. Using the criterion given by equation (12) with a reasonable upper bound of a = 100 ($/set) and B/A = 5 x lo3 for typical fast breeder reactors we conclude that the slowing-down delay time, TV,may be neglected in the analysis of the Doppler limited burst, whereas a typical heat transfer delay time, TV,of the order of 50 psec to 100 ,usec yields rdap/A = O-035 to O-070 which is in the range were the delay time effect has to be included in the analysis. 4. CONCLUSION
Analytical expressions based on the point reactor are presented for the increase in peak power, final temperature and energy release-caused by short time delays in the temperature reactivity feedback-in superpromptcritical excursions. This leads to a criterion to decide whether it is necessary to include the delay time in the burst analysis or not. A method for calculating the slowing-down delay time for Doppler feedback in fast breeder reactors is indicated. The results show that slowing-down delay may be neglected whereas heat transfer delay in the mixed oxide fuel needs to be considered in the burst analysis. Acknowledgement-A discussion with K. Orr regarding slowing-down delay in Doppler feedback has stimulated this work. REFERENCES BORGWALDT H. (1968) Private communication. BRAESSD. et al. (1967) Proc. Intl Co@ on the Safety of Fast Reactors. Aix-en-Provence. FEHLBERG E. (1960) Z. angew. Math. Mech. 40,252 and 449. FISCHERE. A. and KELLERK. (1966) Nukleonik 8,471. FORBESS. G. (1958) IDO-16452. FRAUDEA. (1963) Private communication. FUCHSK. (1946) L. A. 596. H~BELW. (1968) Private communication. NYERW. E. (1964), In Technology of Nuclear Reactor Safety (T. J. THOMPSON and J. G. BECKERLY, Editors), Vol. 1, p. 417. The M.I.T. Press, Cambridge.