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Ragheb-Ch8-2011-DecayHeatGeneration.PDF
Chapter 8
DECAY HEAT GENERATION IN FISSION REACTORS
© M. Ragheb
3/22/2011
1. INTRODUCTION
After a reactor core is shut down, through the insertion of its control rods, heat
continues to be generated by the decay of the fission products, even though the fission
power would stop to be generated.
The fission products heat generation, also called afterheat or decay heat, would
have to be extracted from the system; otherwise it would lead to fuel damage, steamcladding interaction leading to hydrogen generation, melting or even vaporization of the
core. It depends on the design of the nuclear power plant as shown in Fig. 1, particularly
on its power density.
Light Water Reactors, LWRs as Pressusized Water Reactors, PWRS and Boiling
Water Reactors, BWRs offer the same type of response.
Figure 1. Power in thermal kilowatts per unit mass of reactor fuel in metric tonnes for
different reactor designs. LWR: Light Water Reactor, AGR: Advanced Gas-cooled
Reactor, Magnox: Magnesium alloy cladding reactor.
The decay heat power generation decreases rapidly as a function of time, but
cooling provisions for dissipating it immediately after shutdown must be incorporated
into the reactor’s design. The power release from such a process must be estimated and
accounted for in the safety design of nuclear power plants.
2. SOURCE TERMS
Two source terms contribute to the decay heat rates. One results from the rate of
negative beta particles emission by the fission products given by:
 particles 
Rβ (T=
) 3.8 × 10−6 T −1.2 
 sec.fission 
(1)
The second results from the rate of gamma photons emissions by the fission
products, given by:
 photons 
Rγ (T=
) 1.9 × 10−6 T −1.2 
 sec.fission 
(2)
where: T is the time after the fission event in days.
3. BETA AND GAMMA ENERGY RELEASES
We consider the mean energies of the beta and gamma particles as:
Eβ = 0.4 MeV ,
Eγ = 0.7 MeV .
where the average energy of the gammas is about twice that of the betas.
This suggests that the energy release from the beta emissions slightly exceeds the
energy release from the gamma emissions from the fission products.
The rate of emission of beta and gamma ray energy can be written as:
=
E (T ) Rβ (T ) Eβ + Rγ (T ) Eγ
= [( 3.8 × 10−6 × 0.4 ) + (1.9 × 10−6 × 0.7 )]T −1.2
= [(1.52 × 10−6 ) + (1.33 × 10−6 )]T −1.2
(3)
 MeV 
= 2.85 × 10−6 T −1.2 
 sec.fission 
These simple empirical expressions are accurate within a factor of 2 or less [2].
4. DECAY POWER AFTER SHUTDOWN
If we want to take into consideration the operating time of a given core, and
calculate the heat release after shutdown, we consider the time scale shown in Fig. 2 and
adopt the following analysis.
Figure2. Time sequence after reactor shutdown for decay heat calculation. Reactor
power is P0 in Watts.
The energy produced by fissions in the interval dT at time τ is:
Eτ (T ) =2.85 × 10−6 (τ − T ) −1.2 [
MeV
]
sec.fission
(4)
For a reactor operating at a power of P thermal Watts (Wth), and considering that
the energy release per fission event is 200 MeV of which 10 MeV are carried away by the
antineutrinos associated with process negative beta decay, leading to a recoverable
energy per fission event of 190 MeV / fission, one can write for the fission rate dF/dt:
dF
1
MeV
sec
 Watt  (Joules/sec) 1 fission
= P [MWth] .106 
[
].
[
].
[
].86,400[
]
-13

dt
Watt
190 MeV 1.6 x 10
Joule
day
 MWth 
fissions
= 2.84 × 1021 P [
]
day
Notice that if we consider that fission’s recoverable energy is 200 MeV, we get
instead a value of 2.69 x 1021 fissions per day per MWth of power, a difference of 5
percent in the estimate. Some sources use the 200 MeV value and hence underestimate
the decay heat power generation by 5 percent.
This leads to the equivalence:
 fissions 
2.84 × 1021 
 ⇔ 1 MWth of power
 day 
We can deduce that the number of fissions occurring in the interval dT is:
=
N 2.84 × 1021 P0dT [ fissions ]
(5)
where: P0 is the reactor power level in MWth,
dT is an interval of time in days.
Thus the rate of emission of beta and gamma energy at time τ , due to fissions in
the interval dT is:
R(T ) = Eτ (T ). N
MeV
] × 2.84 x1021 P0dT [fissions]
sec.fission
MeV
8.094 × 1015 P0 (τ − T ) −1.2 dT [
]
=
sec
=2.85 × 10−6 (τ − T ) −1.2 [
(6)
We integrate over the irradiation period from “zero time” to “time T0” the reactor
shut down time to get:
P=
8.094 × 10 P0
15
T =T0
∫
(τ − T ) −1.2 dT
T =0
T0
 (τ − T ) −1.2+1  MeV
]
= 8.094 × 1015 P0 
 [
 +0.2  0 sec
(7)
Substituting the lower and upper limits:
 MeV 
P
 sec  =4.05 × 1016 [(τ − T ) −0.2 − τ −0.2 ]
0
P0 [ MWth ]
Using the power in MWth units,
(8)
P [ MWth ]
MeV
Watt.sec −6 MWth
[(τ − T0 ) −0.2 − τ −0.2 ]
10
= 4.05 × 1016 × 1.6 × 10−13
P0 [ MWth ]
sec
MeV
Watt (9)
=6.48 × 10−3[(τ − T0 ) −0.2 − τ −0.2 ]
Notice that the time after shutdown is t = (τ − T0 ) , and the reactor operation time
is T0. Figure 3 shows the decay power percentage of the total power as a function of time
for a typical reactor.
In term of the time after shutdown t:
P (t )= 6.48 × 10−3 P0 [t −0.2 − (t + T0 ) −0.2 ] [ MWth ]
(9)’
EXAMPLE
At 1 second after shutdown for a reactor that operated for one year the decay
power ratio would be:
P (t )
= 6.48 × 10−3[t −0.2 − (t + T0 ) −0.2 ]
P0
1
1
) −0.2 − (
+ 365) −0.2 ]
24 × 60 × 60
24 × 60 × 60
−3
−0.2
=
6.48 × 10 [(0.0000157) − (365.0000157) −0.2 ]
=
6.48 × 10−3[(
=
6.48 × 10−3[9.13734 − 0.30729]
= 6.48 × 10−3 × 8.83005
= 57.218 × 10−3
= 0.057218
≈6 %
At 1 minute after shutdown for a reactor that operated for one year the decay
power ratio would be:
P (t )
= 6.48 × 10−3[t −0.2 − (t + T0 ) −0.2 ]
P0
1
1
) −0.2 − (
+ 365) −0.2 ]
24 × 60
24 × 60
6.48 × 10−3[(0.000694) −0.2 − (365.000694) −0.2 ]
=
=
6.48 × 10−3[(
=
6.48 × 10−3[4.28280 − 0.30728]
= 6.48 × 10−3 × 4.52072
= 29.29 × 10−3
= 0.02929
≈ 3%
Figure 3. Decay heat power and energy release after shutdown as a function of time.
5. TOTAL HEAT GENERATION AFTER SHUTDOWN
An expression for the total energy release from decay heat generation after reactor
shutdown can be derived by use of Eqn. 9. Let us consider the variable:
=
t time after shutdown= (τ − T0 )
thus the energy release after shutdown can be written as:
t
E (t ) = ∫ P(t )dt [ MWth.day ]
(10)
0
By substituting from Eqn. 9 for the power P, we get:
t
E (t ) =6.48 × 10−3 P0 ∫ [(τ − T0 ) −0.2 − τ −0.2 ]dt
0
Substituting for:
=
t (τ − T0 )
we get:
t
E (t )= 6.48 × 10 P0 ∫ [t −0.2 − (t + T0 ) −0.2 ]dt
−3
0
Carrying out the integration yields:
E (t ) = 8.1 × 10−3 P0 [t 0.8 − (t + T0 )0.8 + T00.8 ]
(11)
where t is the time after shut down, and T0 is the reactor operation time. This relationship
is shown in Fig. 3.
A procedure is shown for the estimation of the decay heat power and energy
release after shutdown for a typical Pressurized Water Reactor (PWR), as well as the
decay heat power and energy release for a 3,000 MWth PWR with different operational
times.
!
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decayheat.f90
Procedure generating the decay heat power and integrated energy.
release for a Pressurized Water Reactor (PWR)
Decay heat power and integrated energy release after shutdown
for a constant reactor power P0, and for different operational
times t0.
Program saves output to file : toutput
This output file can be exported to a plotting routine, e.g. Excel
M. Ragheb, Univ. of Illinois at Urbana-Champaign
!
program decayheat
real t(11),lt(11),t0(4)
real power(4,11),energy(4,11)
real lpower(4,11),lenergy(4,11)
P0 is steady state reactor thermal power in MWth
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100
real :: p0=3000.0
Initialize time scale in seconds
tttt=1./(24.*60.*60.)
Generate time scale ticks
t(1)=0.1*tttt
write(*,*) t(1)
do i=2,11
t(i)=t(i-1)*10.0
write (*,*) t(i)
end do
pause
Open output file for Excel plotting
open(unit=10,file='toutput.xls')
Define exponents in decay heat formula
xx=-0.2
x=0.8
Initialize initial reactor operating times T0 (days)
T0=1 week
t0(1)=7.0
T0=1 month
t0(2)=30.
T0=1 year
t0(3)=365.0
T0= 10 years
t0(4)=3650.0
do i=1,4
do j=1,11
t is time after shutdown (days)
tt=t(j)+t0(i)
power is decay power, lpower is natural logarithm of decay power
power(i,j)=6.48e-03*p0*((t(j)**xx)-(tt**xx))
lpower(i,j)=log(power(i,j))
write (*,*)lpower(i,j)
energy is integral decay heat energy release
lenergy is natural logarithm of decay energy
energy(i,j)=7.60e-03*p0*((t(j)**x)-(tt**x)+(t0(i)**x))
lenergy(i,j)=log(energy(i,j))
calculate natural logarithm of energy
lt(j)=log(t(j))
write(*,*)lt(j),lpower(i,j),lenergy(i,j)
end do
end do
Write results on output file
do j=1,11
write(10,100)lt(j),(lpower(i,j),lenergy(i,j), i=1,4)
format(9(E11.5,1x))
end do
write(*,*)'End of program run, Please enter return to continue'
pause
stop
end
Figure 4. Procedure for the estimation of the decay heat power and energy release.
Figure 5. Decay heat power release for a 3,000 MWth PWR for different operational
times.
Figure 6. Decay energy release after shutdown for a 3,000 MWth PWR for different
operational times.
6. SAFETY IMPLICATIONS
If the decay heat is not successfully extracted after a core shutdown, damage to
the core in terms of fuel damage, hydrogen production, melting or even vaporization
would occur. This unique feature characterizes nuclear power plants from other heat
engines, which would stop heat generation once their operation is stopped.
The misunderstanding of this particular feature of nuclear power plants has led
uninformed operators to commit serious human errors partially contributing to serious
reactor accidents such as the Three Mile Island accident.
For instance, if one considers a reactor operation time of T0 = 1 year, and consider
a reactor power of P0 = 3,000 MWth, substitution in Eqn. 11 yields for the energy release
one day after shutdown:
E (1 day ) = 7.6 × 10−3 × 3,000[10.8 − (1 + 365)0.8 + 3650.8 ]
= 22.8[10.8 − (366)0.8 + 3650.8 ]
(12)
= 17.20[ MWth.day ]
and for 1 month after shutdown:
E (1 month)= 22.8[300.8 − (395)0.8 + 3650.8 ]
=179.64[ MWth.day ]
(13)
These are substantial amounts of energy release. As shown in Fig. 4, this heat
release leads to the rise of the reactor temperature. In the hypothetical adiabatic heating
accident, where no heat extraction is assumed, the rise in reactor temperature would
eventually lead to melting of the fuel (Fig.5) and a subsequent fission product release
from the central void in the fuel pellets and results in the danger of a plant hazard.
If this were a feature of all nuclear reactor designs, then it may be argued that the
risk involved can be justifiably borne to enjoy the benefit of nuclear electricity.
However, this is not necessarily the case, since nuclear reactors can be designed to be
forgiving and inherently safe.
Figure 7. Consequences of unrestricted core heat up in different reactor systems. HTGR:
High Temperature Gas-cooled Reactor, PWR: Pressirized Water Reactor.
Typical operational temperatures that should not be exceeded to avoid fission
products release are shown in Table 1.
Table 1. Operational temperature of different reactor fuels.
Reactor concept
Magnesium alloy cladding (Magnox)
AGR stainless steel cladding
Boiling Water reactor (BWR)
Pressurized Water Reactor (PWR)
Liquid Metal Reactor (LMR), Na cooled
Temperature
Degrees Celsius
450
750
300
320
750
The time at which damage occurs is particularly short in a PWR system at about
1.5 x 104 seconds = 4.2 hours, compared with the High Temperature Gas Cooled Reactor
(HTGR) or the district heating low temperature SECURE system which can be extended
to 30 days.
Systems that can recover from an increase in core temperature in an accident
situation are designated as “inherently safe,” forgiving,” or “passive” reactor designs, and
are an active area of safety research and development. They ought be adopted as
replacements of aging earlier design power plants instead of extending their operational
lives.
Figure 8. Fuel element temperature profile for PWR fuel.
7. DECAY HEAT POWER AND INTEGRATED POWER FROM THE
SYSTEM ANALYSIS HANDBOOK
The decay heat power ration following the shutdown of a reactor is shown in Fig.
9. A reactor that operated for a period of 1013 seconds is considered to have operated for
an infinite time.
Figure 9. Decay heat power ratio for U235 fuel as a function of time after shutdown.
The energy release after shutdown for different periods of operation is shown in
Fig. 9. The graphs are based on the ANSI / ANS 5.1 standard of 1979. The decay heat
power involves the combined release from U235 fission products and actinides U239 and
Np239 decay. The constants for the actinides and 23 fission groups from the standard are
incorporated into the thermal hydraulics code RELAP5 / MOD 1.6 to obtain the decay
power and energy release.
The data do not include the contribution from the thermal fission of Pu239 or the
fast fission of U238, which would add about 2 percent to the power for light water
reactors, but may be substantial in a fast reactor spectrum. The effect of fuel burnup on
the fission products is not included. The fission energy release from delayed neutrons
fission depends on the negative reactivity at shutdown and could account for 1-2
MWth.sec per MWth of initial reactor power.
Both the decay heat and energy release for an arbitrary power history can be
estimated by linearizing the power history into time intervals of constant power levels,
computing the contributions from each time interval and summing them to obtain the
total values.
EXAMPLE
Consider a reactor that operated for 100 hours at a steady power level of 1,000
MWth, then shut down for 80 hours, and then operated again for 100 hours at a steady
state power of 500 MWth, before being shut down. We wish to estimate the decay heat
power 20 hours after shutdown.
The first operating interval contributes:
P1 1,000[ MWth ] × 4.5 × 10−4 [
=
MWth
]
MWth
(14)
read from Fig. 9 at 100 hours operation at 1,000 MWth, and at 200 hours after the first
shutdown.
The second operating interval contributes the following decay power:
=
P2 500[ MWth ] × 3.00 × 10−3[
MWth
]
MWth
(15)
read at 100 hours operation at 500 MWth and 20 hours after shutdown.
Adding the contributions from the two operational periods from Eqns. 14 and 15,
we get:
Ptotal= P1 + P2
= 1,000[ MWth ] × 0.00045[
MWth
MWth
] + 500[ MWth ] × 0.003[
]
MWth
MWth (16)
= 0.45 + 1.5
= 1.95 MWth
Notice that the contribution from the second operational interval is larger than
from the first one. The decay heat generation 20 hours after shutdown is a small fraction
of the operational power level.
Correction factors accounting for neutron capture in the fission products, and
actinide correction factors can be optionally applied if the production rate of U239 / U235
fissions is available.
Figure 10. Decay heat energy release for U235 fuel as a function of time after shutdown.
7. OTHER REPRESENTATIONS
Way and Wigner derived and expression for the beta and gamma energy from the
fission products as:
P
= 6.22 × 10−3 t −1.2
P0
(17)
where t is the time after irradiation in seconds.
The estimated total decay heat power is double the value in Eqn. 12.
A more accurate representation of the decay heat power for short cooling times is
given by the empirical equation [7]:
P
= 0.1{(t + 10) −0.2 − (t + T0 + 10) −0.2
P0
(18)
− 0.87[(t + 2 x107 ) −0.2 − (t + T0 + 2 x107 ) −0.2 ]}
where the time now is in seconds instead of days. This expression includes an allowance
for the heat produced by the beta decay of U239 and Np239 resulting from the radiative
capture of neutrons in U238.
The heat generated from the U235 fission products alone can be obtained by
subtracting the approximate expressions for the heat generated by the decay of U239 and
Np239 as:
t +T0
t
−(
−(
)
)
PU 239
2,040
2,040
= 0.0025[e
−e
]
P0
t +T0
t
PNp239
−(
)
−(
)
290,000
290,000
= 0.0013[e
−e
]
P0
(19)
8. DISCUSSION
The decay heat generation decreases rapidly after shutdown and becomes a small
fraction of the operational fission power level.
However, the safety design of a nuclear power plant must include provisions in
terms of pumping facilities and decay heat exchange equipment that could accommodate
the decay heat generation immediately after shutdown, which would amount to about 6
percent of the operational power level at one second after shutdown.
EXERCISES
1. Prove that 1 Watt(th) of power corfresponds to 2.84x1015 [fissions/day], starting from
the consideration that the fission of an atom of uranium releases 190{MeV/fission] of
enrgy, or from the burning of about 1.112 gms of U235 per day generates a power output
of 1 MWth.
2. Modify the computer procedure listed in the Appendix to generate plots of the decay
heat power and integral energy release for a 1,500 MWth reactor that operated for a
period of 1 year.
Compare the results to those of a reactor that operated for 10 years.
3. A reactor has the following power history:
a) Operation at a power level of 3,000 MWth for 1 year.
b) Operation at a power level of 2,000 MWth for 6 months, followed by a scram.
Determine the decay heat power at the following times using the Systems Analysis
Handbook data:
i)
Six minutes after shutdown.
ii)
One day after shutdown,
iii)
One month after shutdown.
4. The relation for the decay heat power versus time P(t) from the fission products
assuming an infinite irradiation period is given in the reference: “Decay Heat Power in
Light Water Reactors,” ANSI/ANS-5.1, published by the American Nuclear Society
(ANS) as:
P(t )
= A.t − a
P0
where t is the time after shutdown in seconds., and:
A = 0.0603, a= 0.0639 for 0<t<10 s
A = 0.0766, a= 0.1810 for 10<t<150 s
A = 0.1300, a= 0.2830 for 150<t<4x106 s.
1. Derive an expression for the total energy release between the times t1 and t2.
2. For a power reactor producing P0 = 3,000 MWth, calculate the total energy release
from the decay heat within 10 second, 150 seconds and 4x106 seconds after shutdown in
MegaJoules (MJs).
REFERENCES
1. V. H. Ransom et. al., “RELAP5/MOD1 Code Manual Volume 1: System Models
and Numerical Methods,” EGG-2070, November, 1980.
2. W. Lyon, “WREM: Water Reactor Evaluation Model, Revision 1,” Nuclear
Regulatory, Commission, May 1975.
3. American National Standards Institute, “American National Standard for Decay
Heat Power in Light Water Reactors,” ANSI/ANS 5.1, August 1979.
4. G. Breit and E. P. Wigner, Phys. Rev., 49, p. 519, 1936.
5. Samuel Glasstone and Alexander Sesonske, “Nuclear Reactor Engineering,” D.
Van Nostrand and Company, 1967.
6. K. Way and E. P. Wigner, Phys. Rev. 70, p.1318, 1948.
7. S. Untermeyer and J. T. Weills, USAEC Report ANL-4790, 1952.
8. Harold Etherington, ed., “Nuclear Engineering Handbook,” McGraw-Hill Book
Co., 1958.
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