NUKEFACT #69

THE NEUTRON BALANCE EQUATION

CONTINUED

On multiplying the right-hand-side of Equation 69.13 by n1’/n1’ and lp/lp we have:

69.14
Which, on rearranging terms on the left-hand-side, becomes:

69.15
The bracketed term on the left-hand-side is the fractional rate of change in the neutron population over two successive prompt neutron lifetimes. Conventionally, the inverse of this ratio is known as the reactor period, T, with units of seconds. Using period notation, Equation 69.15 becomes:

69.16
where:
1P/T = the fractional change in thermal neutrons causing fission between lp1 and lp2
As for Equation 69.12, the left-hand-side of Equation 69.16 represents the change in the number of thermal neutrons causing fission between intervals lp1 and lp2, 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.

On dividing both sides of Equation 69.16 by the prompt neutron lifetime, lp, the fissions in prompt lifetime number one convert to fission rates and source emissions also become rates.

69.17
where:
n1 = n1'/lp = the thermal neutron fission rate in lp1, fissions/second
S1 = S1'/lp = the neutron emission rate of the non-fission source in lp1 , neutrons/second

Prime notation on n1 and S1 is no longer required because dividing by lp converts the numeric counts in a lifetime (the number of fissions occurring in a particular lifetime) to rates in that lifetime, e.g. thermal neutrons causing fission per second.

Using conventional notation for real time, Equation 69.17 in its most general form for a transient situation becomes:

69.18
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
delta-k(t) = the delta-k value at time "t"
k(t) = the value of keff at time "t"
lambdaeff(t) = the effective decay constant at time "t", sec-1
C(t) = the precursor inventory at time t, atoms
S = S1 = the non-fission source neutron emission rate at time t, neurons/second

In this general expression, four parameters [other than n(t) and C(t)] containing a time variable, (t), can be changing with time, namely the reactor period, keff, delta-k, and lambdaeff. For specific applications of Equation 69.18 in future essays, some of these parameters may be set as constant with time.

To express the neutron balance in terms of reactivity , both sides of Equation 69.18 are divided by keff(t):

69.19
to give:

69.20
Change in Fission Rate Prompt Neutron Fission Rate - Total Fission Rate Delayed Neutron Fission Rate Non-Fission Neutron Fission Rate
where:
l*(t) = lp/k(t)
rho(t) = [k(t) - 1]/k(t)

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 69.20 by the power conversion factor, namely 3.1x1010 (fissions/second)/watt gives:

69.21
which becomes:

69.22
Change in Reactor Power Prompt Neutron Power - Total Power Delayed Neutron Power Non-Fission Neutron Power
where:
P(t) = n(t)/(3.1x1010), watts
C(t)-bar = C(t)/(nu x 3.1x1010), watt-sec
S-bar = S/(nu x 3.1x1010), watts

Equation 69.22 is sometimes referred to as the first reactor kinetics equation.

REMARKS

Even though nearly all expressions relevant to defining and understanding reactor behavior with time derive directly from the balance equations, operational instruction rarely gives more than passing recognition to the existence of these equations. The primary reason for neglect is probably that the balances are always expressed in the form of differential equations, an automatic turnoff to the prospective operator in the class room. Another reason for so little attention is that the means for numeric application is not obvious due to the interdependence of several parameters and the coupling of the equations.

The key to avoiding the differential barrier is a simple substitution introduced in Equation A-16, namely letting "1/T" represent the fractional rate of change of the neutron population. The substitution of reactor period suppresses the differential aspect of the neutron balance equation with a term that represents a familiar concept. The neutron balance now appears to be a straightforward algebraic expression, which, as an added benefit, is capable of yielding a wealth of information about reactor behavior. Algebraic solution of the balance equations in this form yields important unadulterated reactor behavioral properties and characteristics, including the delayed neutron population fraction , the source multiplication factor, the prompt jump factor, the reactor period equation, and the in-hr equation.

Future essays will be devoted to developing these relations and to demonstrating numeric applications. Such examples aid in understanding the underlying physical process.

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 neutron population as a whole responds by the same factor of change.

  5. The balance equations apply to both the steady state and the transient state. If production and loss of neutrons are in balance, then the neutron population exhibits steady state. If production and loss of neutrons show an imbalance, then the neutron population exhibits transient behavior.

  6. A balance can be made on any particular set of neutrons. The three balances presented are:
    * the fast neutron balance
    * the thermal neutrons causing fission balance
    * the reactor power balance - the first kinetics equation

  7. A balance on thermal neutrons causing fission is important because these fissions determine the rate of precursor production in the precursor balance equation and because these fissions determine the reactor power level

  8. The basic time unit of the balance equations is the prompt neutron lifetime
    * a fast neutron population appears at the beginning of the lp interval
    * fast neutrons are produced during the interval and neutrons are lost during the interval
    * thermal neutrons cause fission at the end of lp interval

  9. The fast neutron population starting a prompt neutron lifetime interval diminishes throughout the interval, and expires at the end of the lp interval as the remaining thermal neutrons cause fissions

  10. Production of fast neutrons during a lifetime interval is by three reactions
    * prompt neutrons from fissions
    * delayed neutrons from precursor decay
    * non-fission neutrons from an installed neutron source

  11. Neutron losses - all neutrons starting a lifetime interval expire by the end of the lp interval, these neutrons are not carried over from one lp interval to the next lp interval.

  12. Fast neutrons produced during a lifetime interval are added to the starting population and neutrons lost during a lifetime interval are subtracted from the starting population to establish the fast neutron population starting the next lifetime interval.

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