NUKEFACT #70

THE PRECURSOR BALANCE EQUATION

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The Second Reactor Kinetics Equation

last update August 17, 2004
INTRODUCTION

Of the two balance equations necessary to define reactor behavior, the neutron balance equation was developed in Nukefact #69. The precursor balance equation, as developed herein, allows for determination of the precursor inventory and its rate of change for all possible reactor behavior. The precursor inventory is important because the number of precursor atoms present at any time determines the production rate of delayed neutrons, which act as source neutrons. It is the source neutrons that are multiplied to produce reactor power. For off-critical conditions, it is the ongoing change in precursor inventory that is the driving force (prime mover) of ongoing power change. As with the neutron balance equation, the precursor balance equation applies to "point kinetics", meaning that during transients the precursor inventory throughout the reactor core responds by the same factor of change.

GENERAL CONSIDERATIONS OF THE PRECURSOR BALANCE

As indicated in Nukefact #69, 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:

70.1
where:
C2 = number of units in inventory at time-2
C1 = initial number of units in inventory at time-1
delta-C = change in number of units in inventory over time interval t1 to t2

Since the two sides of the Equation 70.1 are equal, the equation can be said to be in balance. If delta-C = 0 over a series of time intervals, then C2 = C1 . The number of units is constant with time and the inventory or population is said to be in a steady-state condition. If delta-C 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.

As with neutrons, a simplified accounting process can be established by breaking time in to a series of increments, each being equal to the prompt neutron lifetime. Precursor atoms existing at the start of a lifetime interval are taken as the reference inventory. As an accounting rule, precursor atoms produced in that lifetime interval, less losses, are added to the reference inventory and taken as the starting population for the subsequent lifetime interval. Unlike neutrons, not all precursor atoms are lost at the end of a lifetime interval. A large fraction of the precursor atoms are carried over to the subsequent lifetime interval. If the reactor is critical, this reference inventory will reproduce itself to start the next lifetime. The production of precursor atoms is in balance with the loss of precursor atoms. If the reactor is off-critical, this reference inventory will undergo change to start the next lifetime. The change results from an imbalance between production and loss of precursor atoms.

Precursor Loss: During a prompt neutron lifetime, a number of the radioactive precursor atoms will decay and emit fast delayed neutrons. This decay amounts to a small portion of the large precursor inventory. Since precursor atoms that decay will emit no more neutrons, these atoms are no longer a part of the precursor inventory. The precursor atoms that decay are counted as precursor losses during the lp interval. Conversely, a large part of the precursor inventory survives from lifetime to lifetime.

Precursor Production: A small fraction of the fission events at the end of the prompt neutron lifetime produce precursor atoms. This precursor production constitutes the total precursor gains during the prompt lifetime interval.

Applying the same accounting rule as for neutrons, the precursor production and loss during the lifetime are not tallied until the start of the next lifetime interval. The number of precursor atoms starting the second lifetime is simply the reference inventory of lifetime-1 plus the precursor atoms produced in the first interval, minus the precursor atoms lost during the first interval, or:

70.2
In order to reduce the number of equations that must be dealt with, the six precursor balances are coalesced into a single effective group with a decay constant of lambdaeff.

THE BALANCE EQUATION FOR PRECURSOR ATOMS

A balance on precursor atoms is made in the same manner as for neutrons. Restating the word equation 70.2 for the precursor balance in more definitive terms, the number of precursors starting a particular lp interval is equal to the number of precursors starting the prior lp interval, plus precursors produced in the prior interval and minus precursors lost in the prior interval. In equation form, this becomes:

70.3
where:
C2' = precursor atoms at start of interval lp2
C1' = precursor atoms at start of interval lp1
Subscripts identify the prompt neutron lifetime interval to which the precursor inventory applies or in which the production/loss events occur. Prime notation is used to indicate that the precursor term represents the number of precursors appearing in a particular lifetime, and not the rate of production of those atoms.

The number of precursor atoms produced in lp seconds is:

70.4
where:
n1' = thermal neutron fissions in lp1
nu = total fission neutron yield, neutrons/fission
beta = the preursor yield fraction

The number of precursor atoms lost by radioactive decay in lp seconds is:

70.5
where:
lambdaeff = the effective precursor decay constant, sec-1

Substitution of the precursor production and loss components into Equation 70.3 gives the precursor balance equation as:

70.6
The precursor production and loss terms on the right-hand-side of Equation 70.6 are enclosed in parentheses. Equation 70.6 is the balance equation for precursor atoms over two successive prompt neutron lifetimes.

By moving C1' from the right-hand-side of the Equation 70.6 to the left-hand-side, the precursor balance equation becomes:

70.7
or:
70.8
where:
delta-C = C2' - C1'

The left-hand-side of Equation 70.8 represents the change in the number of precursor atoms between intervals lp1 and lp2, due to the imbalance between production and loss of precursors, as expressed by the terms on the right-hand-side of the equation.

An important distinction exists between Equation 70.6 and Equation 70.8. Equation 70.6 defines the precursor inventory based on the inventory in the previous lifetime as modified by the difference between production and loss in that lifetime. Equation 70.8 defines the change in the precursor inventory 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. Equation 70.8 is used to derive the transient precursor inventory.

To express the precursor balance in terms of real time, both sides of Equation 70.8 are divided by lp:

70.9
where:
n1 = the thermal neutron fission rate in lp1

Prime notation on n1 is no longer required because dividing by lp converts the numeric counts in a lifetime (e.g. the number of fissions occurring in a particular lifetime) to a fission rate in that lifetime, i.e. thermal neutrons causing fission per second.

On multiplying the left-hand-side of Equation 70.9 by C1/C1, we have:

70.10
Which, on rearranging terms on the left-hand-side, becomes:

70.11
The bracketed term on the left-hand-side is the fractional rate of change in the precursor inventory over two successive prompt neutron lifetimes. Because of the coupling of the neutron balance and the precursor balance, this fractional rate is identical to that of the neutron balance equation. Conventionally, the inverse of this ratio is known as the reactor period, T, with units of seconds. Using period notation, Equation 70.11 becomes:

70.12
Using conventional notation for real time, Equation 70.12 in its most general form becomes:

70.13
Change in Precursor Inventory Precursor Production Rate Precursor Loss Rate
where:
T(t) = the reactor period at time "t", seconds
n(t) = n1/lp = the thermal neutron fission rate at an instant in time, "t", fissions/second
C(t) = the precursor inventory at time "t", atoms
lambdaeff(t) = the effective decay constant at time "t", sec-1

In this general expression, the four parameters containing the time notation, (t), can be changing with time. For specific applications of Equation 70.13 in future essays, some of these parameters may be set as constant with time.

PRECURSOR BALANCE EQUATION FOR REACTOR POWER

For certain operational applications it is helpful to convert the fission rate to reactor power. Dividing both sides of Equation 70.13 by the power conversion factor, namely 3.1x1010 (fissions/second)/watt and by the neutron yield factor nu gives:

70.14
On combining terms, Equation 70.14 becomes:
70.15
Change in Reactor Power Precursor Production Power Precursor Loss Power
The precursor production rate represents the fraction of power production that delivers precursor atoms. The bar over the precursor inventory denotes that "C" is proportional to precursor inventory but is modified so that the product represents the power produced by precursor neutrons, i.e. by the precursor atoms being lost by decay. The need for expressing the precursor balance in units of watts is so that the units for the first and second kinetics equations are common. Equation 70.15 is sometimes referred to as the second reactor kinetics equation.

REMARKS

Since neutron behavior is the primary interest, the first reactor kinetics equation is usually the focus of what limited attention is given the balance equations. The precursor balance equation is usually considered of lesser import. This is unfortunate. At the very least the two balance equations are of equal importance, especially in light of the fact that it is the precursor atoms, acting as the neutron source, that are the prime mover of off-critical reactor power.

As with the derivation of the neutron balance equation, the differential nature of the precursor balance equation has been suppressed by the substitutuion of "1/T" for the fractional rate of change of the precursor inventory. The period for the precursor inventory is of identical value to that of the neutron population. The reason for this is that it is the changing precursor inventory that is the prime mover of the transient. The delayed neutrons emitted by the precursors are multiplied almost instantly by the prompt neutrons produced by fissions in the chain reaction. Nevertheless, a future essay will demonstrate the equivalence of "T" in the two balance equations.

In general, Nukefact essays aim at minimizing mathematical derivations and maximizing the physical meaning of equations. However, an exception has been made in the dealing with the balance equations. Nukefact #69 contained 22 equations and this essay contains 15. There are several reasons for the greater number of equations:

  1. The balance equations are the bed rock from which everything known about reactor behavior evolves.

  2. The balance equations are the direct link between the physical model of the chain reaction and the mathematical description of that process.

  3. The details of developing the various forms of the balance equations presented in these two essays are not contained in any other training resource.

  4. The balance equations provide a remarkably easy path to much valuable insight into reactor behavior.

SUMMARY

  1. A balance equation is an accounting tool for tracking a population, or an inventory, over a period of time by using systematic additions and subtractions that reflect cumulative change to the initial quantity

  2. Two balance equations are necessary to define reactor behavior with time, namely a neutron balance equation and an "effective precursor" balance equation.

  3. The two balance equations are coupled, meaning that every solution must satisfy both equations.

  4. The balance equations as developed herein define "point reactor kinetics", meaning that during transients the precursor inventory throughout the reactor core responds by the same factor of change.

  5. The balance equations apply to both the steady state and the transient state
    *If precursor production and loss are in balance, then the precursor inventory exhibits steady state.
    *If precursor production and loss show an imbalance, then the precursor inventory exhibits transient behavior.

  6. The basic time unit of the balance equations is the prompt neutron lifetime
    *precursor inventory exists at the beginning of an lp interval
    *precursors are lost by radioactive decay during the interval
    *precursors are produced by thermal neutron induced fissions at the end of lp interval

  7. Precursor atoms produced during a lifetime interval are added to the starting population and precursor atoms lost during a lifetime interval are subtracted from the starting population to establish the precursor inventory starting the next lifetime interval.

  8. The majority of precursor atoms in inventory carry over from lifetime to lifetime.

  9. Simple algebraic solution of the balance equations yields the precursor inventory for both steady-state and transient conditions.

  10. The balance equations provide the only legitimate starting point for investigating the basics of nuclear reactor behavior.
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