NPTEL – Chemical Engineering – Nuclear Reactor Technology
FBR Neutronics: Breeding potential,
Breeding Ratio, Breeding Gain and
Doubling time
K.S. Rajan
Professor, School of Chemical & Biotechnology
SASTRA University
Joint Initiative of IITs and IISc – Funded by MHRD
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NPTEL – Chemical Engineering – Nuclear Reactor Technology
Table of Contents
1 BREEDING ....................................................................................................................................... 3 1.1 BREEDING RATIO ........................................................................................................................................ 4 1.2 BREEDING GAIN .......................................................................................................................................... 4 1.3 DOUBLING TIME (DT) ............................................................................................................................... 5 2 REFERENES/ADDITIONAL READING ..................................................................................... 7 Joint Initiative of IITs and IISc – Funded by MHRD
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NPTEL – Chemical Engineering – Nuclear Reactor Technology
In this lecture, we shall discuss about the calculation of breeding ratio and doubling
time
At the end of this lecture, the learners will be able to
(i)
(ii)
(iii)
define conversion or breeding ratio
determine breeding potential and breeding gain
determine the time required for doubling the fuel inventory (doubling
time)
1 Breeding
Breeding in nuclear reactors refers to the process in which significantly amount of
fertile materials are converted to fissile materials by nuclear transmutation. This
requires the fertile isotope to have large cross section for neutron capture. Since the
main purpose of a nuclear reactor is to produce electricity, breeding is considered as
an off-shoot of excess neutrons produced during fission above the ones required for
sustenance of chain reaction.
The possibility of breeding in a nuclear reactor, taking into account of the type of
fissile material used, depends on the number of neutrons produced for every neutron
absorbed in the fuel. This is denoted by reproduction factor ‘η’. This is related to the
cross sections as follows:
η=
σf
ν
σa
(1)
The probability of breeding is enhanced when the value of ‘η’ exceeds two by a large
fraction. For example, the value of ‘η’ for U-235, Pu-239 and U-233 when
bombarded by thermal neutrons is 2.07, 2.11 and 2.30 respectively. Breeding with
thermal neutrons using U-235 and Pu-239 fuels is virtually impossible due to neutron
absorption in structural materials and moderator. However, with U-233 fuel, it is
possible to achieve breeding using thermal neutrons.
The scenario is different for the case of bombardment with fast neutron. The value of
‘η’ for U-235, Pu-239 and U-233 when bombarded by fast neutrons is 2.3, 2.7 and
2.45 respectively. Hence, breeding is possible with all the above fissile nuclei due to
bombardment with fast neutron.
Conversion ratio is a widely used term to denote the ability of a reactor to convert
fertile to fissile material. Conversion ratio is defined as the ratio of number of fissile
nuclei produced to the number of fissile nuclei consumed.
When a reactor is operated without any fertile material, the conversion ratio is zero. In
this case, the reactor is referred to as ‘burner’ as it burns all the fuel without
producing any fissile material.
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NPTEL – Chemical Engineering – Nuclear Reactor Technology
It may be recalled that the uranium fuelled thermal reactors either use natural uranium
or enriched uranium. In both these cases, isotopic abundance of U-238, a fertile
material, is very high. It is possible to achieve a higher conversion ratio by facilitating
higher neutron capture in U-238 relative to neutron absorption in U-235. This must be
carried out without the loss of criticality. To ensure that sufficient neutrons are
available for chain reaction, the neutron losses in structural elements and moderator
must be reduced. Also, the use of moderators like heavy water and carbon require
more collisions for neutron thermalization and hence require larger core. This
improves the contact between neutron and U-238 contact, thereby improving the
conversion ratio.
The conversion ratio can be predicted approximately as follows:
Conversion Ratio (CR) = η235ε-1-l
(2)
‘η235’ corresponds to reproduction factor of pure U-235. Hence to improve the
conversion ratio, leakage of neutrons (l) must be reduced. For most thermal reactors,
CR is between 0.4 and 0.7 and these reactors are called ‘converters’ (0<CR<1). ‘ε’ is
the fast fission factor that indicates the contribution due to fast neutrons in a thermal
reactor.
1.1 Breeding Ratio
The reactor with conversion ratio greater than 1 is called a breeder reactor. This is
the reactor that produces more fuel than that it consumes. For breeder reactors, the
term ‘breeding ratio (BR)’ is used.
The breeding ratio is maximum when the leakage of neutrons (l) is zero. This is called
maximum breeding ratio (BRmax) and is also called Breeding Potential of the fuel.
Breeding Potential = BRmax= η-1
(3)
Please note that in Eq. (3), ‘ε’ the fast fission factor is taken as unity as most of the
breeders are fast breeders. The reproduction factor is to be calculated for Pu-239, the
predominant fissile isotope in fast reactors.
1.2 Breeding Gain
Another term widely used in a breeder reactor is ‘Breeding Gain (BG)’.
relationship between BG and BR is
The
BG = BR – 1
(3)
This represents the extra fissile material produced for every atom of fuel (fissile
isotope) consumed.
For a Pu-239 fuelled fast reactor,
BR = η239-1-l
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(4)
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NPTEL – Chemical Engineering – Nuclear Reactor Technology
⎛ σ f ⎞
CR= BR = ⎜⎜ ν ⎟⎟ − 1 − l
⎝ σ a ⎠ 239
(5)
Example -1: Determine Breeding Ratio for Pu-239 fuelled fast reactor. Take ν =
2.975; σ f = 1.850; σ a = 2.11 and l = 0.405.
1.85
⎞
Using Eq. (5), BR = ⎛⎜
2.975 ⎟ − 1 − 0.405
⎝ 2.11
⎠
BR = 1.203
Example – 2: Consider a fast breeder reactor operating with Breeding Ratio of
1.3. If it is desired to accumulate an additional 1500 kg of fissile material,
determine the amount of pure Plutonium fuel to be burnt.
By definition of Breeding Ratio,
BR = number of fissile nuclei produced/number of fissile nuclei consumed
BR = mass of fissile nuclei produced/mass of fissile nuclei consumed
Let ‘x’ be the mass of fissile nuclei consumed, then BR = (x+1500)/x=1.3
Solving the above for ‘x’ gives x = 5000 kg
Hence 5000 kg of Plutonium must be burnt to produce an additional 1500 kg of fissile
material.
1.3 Doubling time (DT)
It is defined as the time required to accumulate a mass of fuel equal to that loaded
initially in a reactor system. When the initial inventory of fissile material is low,
doubling time is reduced. In other words, doubling is achieved in a shorter period of
time with initial lower loading of fissile material.
Let us derive an expression for calculation of Doubling Time (DT).
The reactor power per unit mass of fuel (P’) may be related to the number of fuel
nuclei per unit mass of the fuel (Nf) as
P’=EfNfφ-σf
(6)
The above equation may be rewritten as:
Power per unit mass of fuel (W/kg)= Energy released per fission (J) * Number
fissions per unit time per unit mass of fuel (s-1kg-1)
The number of fissions per unit time per unit mass of fuel represents the rate of
consumption of a unit mass of fuel.
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NPTEL – Chemical Engineering – Nuclear Reactor Technology
Therefore, If MF is the mass of fuel loaded, then the rate of consumption (RC) of fuel
of mass MF is given by
RC = MFNfφ-σa (atoms/s)
(7)
If ‘BR’ is the Breeding Ratio, then the rate of production (RP) of fissile mass is given
by
RP = BR*MFNfφ σa (atoms/s)
(8)
−
The net increase in fuel = Rate of Production – Rate of consumption
Net increase = Rnet =(BR-1)*MFNfφ σa (atoms/s)
(9)
−
If DT is the Doubling Time, then
DT*Rnet=MFNf
(10)
Therefore,
DT =
MFN f
Rnet
=
MFN f
(BR − 1)M F N f φ
−
σa
=
1
(BR − 1)φ −σ a
(11)
An expression may be obtained for DT In terms of reactor power (P’) as follows:
Recall, P’=EfNfφ-σf
φ- = P’/ (NfEfσf)
(12)
Substituting the above in the equation for Doubling Time, we get
DT =
N f Efσ f
(BR − 1)σ a P '
(13)
Now let’s discuss the factors that influence Doubling Time.
(i) Breeding Ratio: Higher the Breeding Ratio, greater is the amount of fissile material
produced. Hence, the time required for doubling the mass is reduced at higher
Breeding Ratios. We have seen earlier that Breeding Ratio depends on the
reproduction factor η, which inturn is influenced by absorption cross section,
fission cross sections and ‘ν'. As the cross sections are functions of neutron
energies, appropriate choice of neutron energy and steps to minimize neutron
leakage will result in increased Breeding Ratio (BR).
(ii) Power per unit mass of the fuel: When this quantity is high, the average neutron
flux is also high among other variables. As evident from Eq. (11), with increased
neutron flux, the Doubling Time is reduced. Hence the Doubling time decreases
with increased reactor Power per unit mass of the fuel.
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NPTEL – Chemical Engineering – Nuclear Reactor Technology
Example – 4: Determine the doubling time of a Pu-239 fast breeder reactor. The
reactor is operated at 400 MW/tonne Pu-239 with a Breeding Ratio of 1.2. The
absorption and fission cross section are 2.16 and 1.81 b respectively. The number
of Pu-239 per unit mass is 2.52x1021 (atoms/g).
Data: Ef = 3.2x10-11J; Nf = 2.52x1021; P’ = 400 MW/tonne = 400 J/gs;
σa = 2.16 b; σf = 1.81 b
Substituting the above in Eq. (13),
DT =
N f Efσ f
(BR − 1)σ a P '
=
2.52 x10 21 * 3.2 x10 −11 *1.81
(1.2 − 1)2.16 * 400
DT = 9776 days ~ 27 years
2 Referenes/Additional Reading
1. Nuclear Energy: An Introduction to the Concepts, Systems, and Applications of
Nuclear Processes, 5/e, R.L. Murray, Butterworth Heinemann, 2000 (Chapter 13).
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