NUKEFACT #69

THE NEUTRON BALANCE EQUATION

***

The First Reactor Kinetics Equation

last update August 7, 2004

INTRODUCTION

The fundamental explanation of reactor behavior lies within a set of balance equations consisting of a neutron balance equation and six precursor balance equations, one for each precursor group. Collectively these equations and their coupled solution are recognized as defining "point reactor kinetics", meaning that during transients the reactor’s entire neutron population responds in lock step as a single entity. Commercial reactors of large physical size sometimes exhibit spatial effects where uniform response in all regions of the core does not occur. The balance equations allow for determination of important basic properties and behavioral characteristics of a nuclear reactor, including the delayed neutron population fraction, the source multiplication factor, the prompt jump factor, the reactor period equation, and the classic in-hour equation.

This essay develops several forms of the neutron balance equation for "point kinetics". These equations are valid for all possible reactor behavior. A future essay will illustrate numeric application.

THE NEUTRON BALANCE EQUATION

In simplest form, a balance equation is an accounting tool which can track a population, or an inventory, over a period of time by using systematic additions and subtractions that reflect cumulative change to the initial quantity. Mathematically this may be expressed as:

69.1

_{where:

N₂
=
number of units in inventory at time-2

N₁
=
initial number of units in inventory at time-1

delta-N
=
change in number of units in inventory over time interval t₁ to t₂

Since the two sides of the Equation 69.1 are equal, the equation can be said to be in balance. If delta-N = 0 over a series of time intervals, then N₂ = N₁ . The number of units is constant with time and the inventory or population is said to be in a steady-state condition. If delta-N is unequal to zero over a series of time intervals, then the number of units is changing with time and the inventory or population is said to be in a transient-state.

In a nuclear reactor, a simplified accounting process for the neutron population can be established by breaking time in to a series of increments, each being equal to the prompt neutron lifetime. Fast neutrons existing at the start of a lifetime interval are taken as the reference population. As an accounting rule, fast neutrons produced in that lifetime interval, less losses, are taken as the starting population for the subsequent lifetime interval. If the reactor is critical, this reference population will reproduce itself to start the next lifetime. The production of neutrons is in balance with the loss of neutrons. If the reactor is off-critical, this reference population will undergo change to start the next lifetime. The change results from an imbalance between production and loss of neutrons.

Neutron Loss: As fast neutrons slow down during a prompt neutron lifetime, leakage and absorption diminish the number of neutrons in the population until finally, at the end of the lifetime interval the remaining neutrons cause fission. At this point, all of the reference fast neutrons starting the interval have expired. None of the original fast neutrons are carried over to the next interval. The total neutron losses in the interval are equal to the number of fast neutrons starting the interval.

Neutron Production: The fission events at the end of the prompt neutron lifetime produce prompt neutrons and precursor atoms. Other fast neutrons are produced during the interval by precursor decay and by non-fission source emission. These three production mechanisms constitute the total neutron gains during the prompt lifetime interval.

The production and losses during the lifetime are not tallied until the start of the next lifetime interval. The number of fast neutrons starting the second lifetime is simply the reference population of lifetime-1 plus the fast neutron production in the first interval, minus the neutrons lost during the first interval, or:

69.2

Likewise, the precursor balance equation, treats the precursor inventory in the same two prompt neutron lifetimes. The relationship of the precursor inventory starting the second lifetime to that starting the preceding lifetime is a function of additional precursor atoms produced in the first interval minus precursor atoms lost during the first interval, or:

69.3

The balance equations are "coupled", meaning that none can be solved independently of the others. This is akin to the simultaneous solution of two or more algebraic equations. The solution must satisfy all equations in the set. Physically, neutrons cause fission in a nuclear reactor and a few of the fissions produce precursor atoms. In turn, precursor atoms release delayed neutrons which initiate chain reactions consisting of fission neutrons. Hence, the behavior of neutrons and precursor atoms is intimately linked, through the fission process.

As developed herein, the balance equations use the prompt neutron lifetime as the basic time unit. The prompt neutron lifetime is dependent on reactor design, but in large commercial PWR’s and BWR’s is extremely short, with a typical value being 0.0001 seconds. In order to reduce the number of equations that must be dealt with, the six precursor balances are coalesced into a single effective precursor group with a decay constant of lambda_eff. The precursor balance equation will be developed in the next essay.

BALANCE EQUATION FOR FAST NEUTRONS

A balance on fast neutrons is a logical beginning because it is the fast neutrons that appear at the start of a prompt neutron lifetime interval. Restating the word equation, 69.2, for the neutron balance in more definitive terms, the number of neutrons starting a particular l_p interval is equal to the number of neutrons starting the prior l_p interval, plus neutrons produced in the prior interval and minus neutrons lost in the prior interval. In equation form, this becomes:

69.4

where:

N₂^'
=
fast neutrons at start of interval l_p2

N₁^'
=
fast neutrons at start of interval l_p1

Subscripts identify the prompt neutron lifetime interval to which the neutron population applies or in which the production/loss events occur. Prime notation is used to indicate that the neutron term represents the number of neutrons appearing in a particular lifetime, and not the rate of production of those neutrons.

There are three contributors to fast neutron production, namely production from fission, from precursor decay, and from non-fission source emission. These are quantified as:

69.5

where:

k = k_eff
=
the effective multiplication factor

beta
=
the precursor yield fraction

(1 - beta)
=
the fraction of fissions that produce prompt neutrons

Note that the effective multiplication factor, k_eff, will henceforth be represented as "k". For the short duration of a prompt neutron lifetime interval, the effective multiplication factor is taken to be constant with time. The thermal fission events in interval l_p1 produce {fast} prompt neutrons that appear at the start of l_p2.

The production of (fast) delayed neutrons from the decay of precursor inventory is:

69.6

where:

lambda_eff
=
the effective precursor decay constant, sec^-1

C₁
=
the precursor inventory at start of interval l_p1

l_p
=
the prompt neutron lifetime, seconds

The production of (fast) non-fission neutrons also contribute to the total neutron population, as:

69.7

As the prompt neutron lifetime, l_p1, expires, thermal neutrons causing fission eliminate the remainder of the original N₁^’ reference population. The loss of fast neutrons in a prompt neutron lifetime includes all fast neutrons that started the interval, or:

69.8

Substitution of the production and loss components into Equation 69.4 gives the fast neutron balance equation as:

69.9

The square brackets on the right-hand-side of Equation 69.9 enclose the three neutron production terms. Equation 69.9 is the balance equation for fast neutrons over two successive prompt neutron lifetimes.

BALANCE EQUATION FOR THERMAL NEUTRONS CAUSING FISSION

Two considerations focus attention to the thermal neutrons causing fission at the end of the prompt lifetime interval. First, it is the thermal fission events that produce precursor atoms in the precursor balance equation. And secondly, it is the thermal fissions that determine the reactor power level.

To convert the balance on fast neutrons to one on thermal neutrons causing fission, both sides of Equation 69.9 are multiplied by k /nu.

69.10

which reduces to:

69.11

where:

n₂^'
=
N₂^' x (k/nu) = thermal neutrons causing fission in l_p2

n₁^'
=
N₁^' x (k/nu) = thermal neutrons causing fission in l_p1

k/nu
=
the fraction of fast neutrons causing thermal fission

The prompt neutron production term now represents the number of prompt neutrons that cause thermal fissions in l_p2. The delayed neutron production term represents the number of delayed neutrons that cause thermal fissions in l_p2. Likewise, the non-fission neutron production term represents the number of non-fission neutrons that cause thermal fissions in l_p2.

By moving the first n₁^' on the right-hand-side of the Equation 69.11 to the left-hand-side and combining prompt neutron production from fission with the loss term, the thermal neutron balance equation becomes:

69.12

The left-hand-side of Equation 69.12 represents the change in the number of thermal neutrons causing fission between intervals l_p1 and l_p2, due to the imbalance between production and loss of thermal neutrons causing fission, as expressed by the terms on the right-hand-side of the equation.

An important distinction exists between Equation 69.11 and Equation 69.12. Equation 69.11 defines the neutron population based on the population in the previous lifetime as modified by the difference between production and loss in that lifetime. Equation 69.12 defines the change in the neutron population in two successive lifetimes based on the difference between production and loss in the lifetime. Expressing the balance equation in terms of change facilitates its application to transient situations.

The first term on the right-hand-side of Equation 69.12 is no longer a simple production component. Rather, it represents the difference between the production of prompt neutrons causing thermal fission in l_p2 and total loss of thermal neutrons causing fission in l_p1, or the capability of the prompt neutrons to propagate the chain reactions. Since, for any k_eff < 1.0065 prompt neutrons fail to make up the total losses (losses which accrue through leakage, absorption, fission, and precursor formation) from the preceding lifetime, this term, for operational purposes, is always negative, meaning that operationally the reactor is always subcritical on prompt neutrons. As will be illustrated in a future essay, the prompt neutrons are in deficit for even the super prompt-critical conditions.

Equation 69.12 can also be expressed as:

69.13

where:

delta-n
=
n₂^' - n₁^'

delta-k
=
k - 1

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N₂	=	number of units in inventory at time-2
N₁	=	initial number of units in inventory at time-1
delta-N	=	change in number of units in inventory over time interval t₁ to t₂

	N₂^'	=	fast neutrons at start of interval l_p2
	N₁^'	=	fast neutrons at start of interval l_p1

k = k_eff	=	the effective multiplication factor
beta	=	the precursor yield fraction
(1 - beta)	=	the fraction of fissions that produce prompt neutrons

lambda_eff	=	the effective precursor decay constant, sec^-1
C₁	=	the precursor inventory at start of interval l_p1
l_p	=	the prompt neutron lifetime, seconds

n₂^'	=	N₂^' x (k/nu) = thermal neutrons causing fission in l_p2
n₁^'	=	N₁^' x (k/nu) = thermal neutrons causing fission in l_p1
k/nu	=	the fraction of fast neutrons causing thermal fission