2008 September 1

The 4-Factor Formula for k

B. Rouben

McMaster University

EP 4D03/6D03

Nuclear Reactor Analysis

2008 Sept-Dec

2008 September 2

Contents

We derive the 4-factor formula for the reactor multiplication constant.

2008 September 3

Two Energy Groups Up to now we have considered in detail the neutron-

diffusion equation in one energy group. In the 1-group model, all neutrons are treated as if

they had the same energy. However, we know that in fact the neutron energy in

thermal reactors spans many orders of magnitude: from the MeV range for neutrons born in fission to small fractions of an eV for thermal neutrons.

Although the one-group formalism helps to understand the underlying principles, an improved methodology must be used for the most accurate reactor calculations.

cont’d

2008 September 4

Two Energy Groups (cont’d)

Because a very large fraction of the fissions is in the low-energy range in thermal reactors, a 2-energy-group diffusion equation is often appropriate for the calculation of the flux shape in the reactor.

In the 2-group formalism, the 2 groups are: Group #1, the “fast” group (actually, the

“slowing-down” group) Group #2, the “thermal” group

The energy boundary which separates the two groups depends somewhat on the computer code used in the analysis; however, the value 0.625 eV is often used.

cont’d

2008 September 5

The Four-Factor Formula

An early formalism for analyzing the multiplication factor in two groups is the 4-factor formula for k, the infinite-lattice multiplication constant.

We derive it here, by following what happens to neutrons as they follow the cycle from one neutron generation to another.

Because the four-factor formula is for the infinite lattice, there is no need to consider leakage.

2008 September 6

The Four-Factor Formula (cont’d)

Imagine N neutrons born in thermal fission. There is a small number of fast-neutron fissions which

occur (neutron-induced fission in U-238 has a neutron-energy threshold of ~0.7 MeV). The number of fast fissions is a small fraction of all fissions (a few percent).

A factor is defined which is the ratio of total number of fission neutrons to the number born in thermal fissions.

is called the fast-fission ratio (1) Therefore total number of fission neutrons = N (2) cont’d

2008 September 7

The Four-Factor Formula (cont’d)

In thermal reactors, these fission neutrons are thermalized by the moderator.

However, during slowing down, a number of neutrons will be lost in resonance captures.

A factor p is defined which gives the probability of neutrons not being captured in the resonance-energy range.

p is called the resonance-escape probability (3) Therefore the total number of neutrons surviving to

the thermal-energy range = Np (4)

cont’d

2008 September 8

The Four-Factor Formula (cont’d) The thermal neutrons can be absorbed in fuel, or in other

components of the lattice. The neutrons absorbed in materials other than fuel are

simply lost, as far as the chain reaction is concerned. A factor f is defined which gives the fraction of thermal

neutrons absorbed in fuel. f is called the thermal or fuel utilization (5) Therefore the total number of neutrons which survive to the thermal-energy range and are absorbed in the fuel = Npf (6)

cont’d

2008 September 9

The Four-Factor Formula (cont’d)

Some (not all) of the thermal neutrons which are absorbed in the fuel will induce fission in the fissile nuclides (U-235, Pu-239, Pu-241).

The factor is defined as the number of fission neutrons produced per thermal-neutron absorption in fuel.

is called the reproduction factor (7) Therefore the total number of fission neutrons which

are born in the next generation = Npf (8)

2008 September 10

The Four-Factor Formula (cont’d)

The infinite-lattice multiplication factor is the ratio of fission neutrons in one generation to the number in the previous generation.

Therefore the 4-factor formula for k results:

)9(

pfN

pfNk

2008 September 11

Interactive Discussion/Exercise

The derivation in the past few slides started from the identification of the fast-fission factor, , in the neutron cycle .

Show that the four-factor formula can also be derived by starting at any of the different points in the cycle, i.e., starting from p, f, or .

2008 September 12

Typical Values of Four Factors in CANDU

Because the basic lattice has complex geometry, and because it has materials with high absorption cross section (the fuel) as well as materials with high scattering cross section (the moderator), diffusion theory cannot be used for calculation within the cell.

We must use transport-theory codes, e.g. WIMS-IST or DRAGON, or empirical codes (i.e., codes built upon the results of careful measurements), e.g., POWDERPUFS-V, to model the “microscopic” flux distribution within the basic lattice, its reactivity, and the quantitative trend (depletion) with burnup.

[The nuclear properties thus obtained can then be used in diffusion codes to calculate the “macroscopic” flux distribution in the core.]

Typical values of the 4 factors, obtained from POWDERPUFS-V lattice calculations, are shown in the next Table.

2008 September 13

Typical Values of Four Factors in CANDU

Fuel Burnup (MW.d/Mg(U))

Reproduction Factor

()

Fast-Fission Factor

()

Resonance-Escape

Probability(p)

Fuel Utilization

(f)k-infinity

0.00 1.2431 1.0265 0.9060 0.936 1.08199

391.69 1.2395 1.0265 0.9060 0.937 1.07974

1203.80 1.2438 1.0264 0.9061 0.939 1.08624

1621.77 1.2408 1.0264 0.9061 0.940 1.08464

2044.75 1.2357 1.0264 0.9061 0.941 1.08098

2899.11 1.2214 1.0264 0.9062 0.942 1.06983

3756.39 1.2044 1.0263 0.9063 0.942 1.05578

5033.40 1.1767 1.0263 0.9064 0.943 1.03244

6288.67 1.1489 1.0262 0.9065 0.944 1.00860

6701.17 1.1398 1.0262 0.9065 0.944 1.00081

7110.56 1.1310 1.0262 0.9065 0.944 0.99316

7516.81 1.1223 1.0262 0.9066 0.944 0.98567

7919.95 1.1138 1.0262 0.9066 0.944 0.97836

2008 September 14

Graphing the Four Factors in CANDU

0.85

0.90

0.95

1.00

1.05

1.10

1.15

1.20

1.25

0 2000 4000 6000 8000 10000

Fuel Burnup (MW.d/Mg(U))

Reproduction Factor

Fast-Fission Factor

Resonance-EscapeProbability

Fuel Utilization

k-inf

2008 September 15

The Four Factors (cont’d) From the Table and graphs, we can see that 1.026, and p 0.906, and that they change very little (< 1 part per

thousand, i.e., < 1 mk) with fuel burnup (or irradiation). This is understandable since and p originate with U-238, and the U-238 fraction in the fuel changes very little.

f 0.940. It increases by a few parts per thousand (a few mk) with burnup. This is due mostly to the additional absorption in the accumulating fission products in the fuel.

The biggest change with burnup is in (range 1.11-1.24): At first (up to the plutonium peak, ~1,200 MW.h/Mg(U)), increases a

bit. This is due to the initial high rate of Pu-239 production. It then decreases more substantially with burnup, beyond the plutonium

peak. This is due to the overall decrease in fissile inventory (U-235 depletes, while Pu-239 keeps building, but at a slower net rate).

The total change in is a decrease of about 30 parts per thousand from fresh fuel to exit-burnup (~8,000 MW.d/Mg(U)) fuel.

In summary, it is mostly the change in , and less so the change in f, which drives the change in k-infinity.

2008 September 16

Need the Diffusion Equation in 2 Groups The 4-factor formula can give us information about

the reactor multiplication constant (or system reactivity), but it does not provide any information about the flux shape in the reactor.

To compute the flux distribution in the core, we need the diffusion equation, in two energy groups (or more).

2008 September 17

END