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Energy Release in Fission
In the steady state, when atoms undergoing fission are in equilibrium with their radioactive fission products, the energy released per fission is distributed approximately as in Table 2.10.
In a short burst of nuclear energy, such as in a fission bomb or in a rapid rise in reactor power, the total energy released is the sum of the first four terms, 182 to 191 MeV. When a reactor is shut down after reaching steady state, or when fuel from such a reactor is discharged, the energy of beta and gamma decay of the fission products, 13 MeV in all, is released gradually over a long period of time. The neutrino energy is not available. An average of 200 MeV of recoverable energy per fission is used in this text.
The rate of heat release and the intensity of radiation from the fuel are important factors in the design of emergency cooling systems for reactors, casks for shipping discharge fuel, fuel reprocessing plants, and facilities for storing fission-product wastes. These depend on the rate of fission of the fuel when it was in the reactor, the length of time the fuel was in the reactor, and the length of time the fuel was allowed to “cool” before shipping and processing. The exact calculation of these relationships is very tedious because of the large number of nuclides contributing to heat and radiation release, and large digital computers are required [B2]. An approximate statistical correlation by Way and Wigner [W2] provides simple equations suitable for quick approximations.
At a time t in days after fission, the products of a single fission undergo beta decay at a rate 0(f) given by
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Table 2.9 Percent fission yield by mass number’*’ Fission by slow neutrons Fission by fast neutrons*
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(See footnotes on page 56.)
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Table 2.9 Percent fission yield by mass number (Continued)
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tData from [B3. Dl, Gl, Kl, W1J.
ї235и and 23,Pu yields are for a fasl-reactor neutron speclrum; 233 Th and 25*U yields are for fission-spectrum neutrons.
and release energy in the form of beta particles, gamma rays, and neutrinos at a rate E(t) given by
E(t) = 3.9Г1’2 + 11.7Г1-4 eV/s (2.86)
The above equations apply after about 1 min after fission has taken place. Approximately one-fourth of this energy is due to gamma radiation and one-fourth to beta.
In a case of practical interest, a fuel sample will have been in a reactor liberating heat at some constant rate for T days, and will then have been cooled for t days. The rate of disintegration of fission products in the fuel sample in curies per watt of reactor power will be
rT+t
[№ disintegrations/^-fission)] (86,400 s/day) (dt days)
_______ Jt_________________________________________________________
(200 MeV/fission) [1.60 X 10"13 (W-s)/MeV] [3.7 X 1010 disintegrations/(s-Ci)]
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О. = L9[f-o.2 -(7-+ f)-o.2]
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235U Fission
Nuclide I4Se —^Sr — gr —* gzr |
Half-life Short 1.9s 32.3s 153s 28.8yr 64.1 h
for radioactive
decoy
Figure 2.14 Fission-product decay chain for mass 90.
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Similarly, the ratio of the rate of beta — and gamma-energy release P<j(T, r) to the rate of heat release in fission Pf is
Pd(T’ f) = 0.0042 [f 0,2 — (T + Г)’0’2] + 0.0063[r°-4 — (T + O’0,4] (2.88)
Equation (2.88) can also be written as
Pg(T, Q _ Pd(~, Q _ Pd(~ T + Q
where the two quantities on the right-hand side are calculated from Eq. (2.88) for infinite irradiation time and for cooling times of t and T +1, respectively.
A more accurate estimate of the decay energy from fission products can be obtained from the ANS Standard [A2]. The data are presented here as the decay-heat rate F(°°, t) at cooling time t from fission products produced at a constant fission rate of unity, over an infinitely long operating period and without neutron absorption in the fission products. Values of F(°°, t) for the fission of 235 U by thermal neutrons are presented in Table 2.11. Data for the fission — product decay-heat rate from thermal fission of 239Pu and from the fast fission of 238U are also given in the ANS Standard [A2]. These data are applicable to light-water reactors containing 235U as a major fissile material and 238 U as the fertile material. The time domain of the official ANS Standard extends from cooling times of 1 to 104 s.
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The fission-product decay-heat rate F(T, t) per unit fission rate for finite irradiation time T can be synthesized from
where уj and X,- are empirical constants. Values of 7,- and X,- for 235 U thermal fission are listed in Table 2.12. The data in Table 2.11 for infinite irradiation time can be constructed from Eq. (2.90) by choosing T = 10’3 s.
Alternatively, F(T, t) can be obtained from the data in Table 2.11 by
F(T, t) = F(°°, t) — F(«, T+t) (2.91)
Data in Table 2.11 for cooling times greater than 104 s can be used in Eq. (2.91) to synthesize values of F(T, t) within the time domain (1 to 104 s) of the ANS Standard.
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Table 2.11 Decay-heat power from fission products from thermal fission of 23SU and for near-infinite reactor operating time’*’
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(See footnotes on page 61.)
Table 2.11 Decay-heat power from fission products from thermal fission of Ms U and for near-infinite reactor operating time (Continued)
Decay-heat power Time after F(°°, t),
reactor shutdown, (MeV/s)/ Percent
s (fissions/s) uncertainty
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6.0 |
X |
106 |
2.983 |
X |
10’1 |
2.0 |
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8.0 |
X |
106 |
2.680 |
X |
10’1 |
2.0 |
|
1.0 |
X |
107 |
2.457 |
X |
10’1 |
2.0 |
|
1.5 |
X |
107 |
2.078 |
X |
10*1 |
2.0 |
|
2.0 |
X |
107 |
1.846 |
X |
10’1 |
2.0 |
|
4.0 |
X |
107 |
1.457 |
X |
10"1 |
2.0 |
|
6.0 |
X |
107 |
1.308 |
X |
10’1 |
2.0 |
|
8.0 |
X |
107 |
1.222 |
X |
10*1 |
2.0 |
|
1.0 |
X |
10s |
1.165 |
X |
10’1 |
2.0 |
|
1.5 |
X |
108 |
1.082 |
X |
10"1 |
2.0 |
|
2.0 |
X |
108 |
1.032 |
X |
10’1 |
2.0 |
|
4.0 |
X |
10s |
8.836 |
X |
10‘2 |
2.0 |
|
6.0 |
X |
108 |
7.613 |
X |
10’2 |
2.0 |
|
8.0 |
X |
108 |
6.570 |
X |
10’2 |
2.0 |
|
1.0 |
X |
109 |
5.678 |
X |
10’2 |
2.0 |
^For irradiation time of 1013 s. Calculated for no neutron absorption in fission products.
Source: American Nuclear Society Standards Committee Working Group ANS-5.1, “American National Standard for Decay Heat Power in Light Water Reactors,” Standard ANSI/ANS-5.1, American Nuclear Society, La Grange Park, 111., 1979. With permission of the publisher, the American Nuclear Society.
The total decay-heat power Pd(T, t) for fission products from a reactor operating at constant total thermal power Pf, and neglecting neutron absorption in fission products, is given by the following simplified method, from the ANS Standard:
P’d(T, t) = 1.02 МШЛ (2.92)
where F(T, t) is evaluated from 235U data, using Eq. (2.90) or (2.91), and Q is the thermal energy per fission. The factor 1.02 corrects for the greater heat generation per fission from 238 U fission products during the period of about 100 s after reactor shutdown. The ratio PdIPfQ of fission-product decay heat rate at cooling time t to reactor power prior to shutdown is plotted as a function of T and t in Fig. 2.15.
Neutron absorption in fission products has a small effect on decay-heat power for t < 104 s and is treated by a correction factor G. The corrected total decay-heat power is given by the ANS Standard, in terms of thermal-neutron flux (in neutrons/cm2-s), reactor operating time T (in s), and cooling time t (in s) as
P(T, t) = P'(T, t)G
The parameter ф is the total number of fissions after irradiation time T per initial fissile atom, calculated by techniques described in Chap. 3. Equation (2.94) applies for operating times T< 1.2614 X 10® s (4 years), shutdown times? < 104 s, and ф < 3.0. A more detailed technique for calculating fission-product decay-heat power from an arbitrary time-dependent fission power, including contributions from the fission of 235 U, 238 U, and 239 Pu, is given in the ANS Standard [A2].
To predict the decay-heat rate from fission products after cooling times of several years, additional corrections must be made for absorption of neutrons in long-lived fission products, particularly the absorption of neutrons in stable 133 Cs to form 2.05-year 134 Cs. Computer codes such as ORIGEN [B2] and CINDER [El] are particularly useful for this purpose.
Estimated maximum values of the ratio G of fission-product decay-heat rate, with neutron absorption in fission products considered, to the decay-heat rate in the absence of neutron absorption in fission products are given in Table 2.13 [А2]. The data are calculated for 235 U — 238 U fuel irradiated for 4 years in a light-water reactor. For cooling times of < 104 s, the
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Table 2.12 Decay-heat parameters for fission products from thermal fission of 235 U
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Source: American Nuclear Society Standards Committee Working Group ANS-5.1, “American National Standard for Decay Heat Power in Light Water Reactors,” Standard ANSI/ANS-5.1, American Nuclear Society, La Grange Park, III, 1979. With permission of the publisher, the American Nuclear Society.
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correction is less than a 6 percent increase. For cooling times of about 3 years, neutron absorption causes the fission-product decay-heat rate to increase by about 60 percent.
Decay of the actinides formed by neutron capture is another source of decay heat, although during cooling times of less than a few hundred years it contributes much less decay heat than do the fission products. The actinide nuclides that contribute appreciably during the first few days after reactor shutdown are 23.5-min 239U and 2.35-day ^’Np. The quantities of these actinides at the time of reactor shutdown can be calculated using the techniques described in Chap. 3, and their rate of decay after shutdown can be predicted from Eqs. (2.13) and 2.14). The decay-heat rate due to these two species can then be estimated as a function of T and t by multiplying the decay rates by the average thermal energy released per decay [A2]:
U = 0.474 MeV/decay
Np = 0.419 MeV/decay
For longer cooling times additional decay heat will be liberated by longer-lived actinides formed by neutron capture in the fuel material, e. g., 237U, 238Pu, 239Pu, 240Pu, 241 Pu, 241 Am, 242 Cm, ^Cm, etc., and by radionuclides formed by neutron reactions with fuel structural material, such as metal cladding. Methods and illustrative data that can be used in estimating the concentrations of such radionuclides and their contributions to decay heat are discussed in Chaps. 3 and 8.