NUKEFACT #72

Solving the Balance Equations

THE GENERAL EQUATION for PRECURSOR INVENTORY

***

last update September 17, 2004
INTRODUCTION

There are important reasons for being able to define the magnitude of the precursor inventory. First, it is apparent from the general expression for reactor power, as developed in Nukefact #71, that delayed neutrons are an important determinant of power. Secondly, employing a single effective precursor delay group as an aid to understanding requires a knowledge of all six group precursor inventories. In turn, key operational equations contain the effective precursor decay constant in their makeup, including the in-hr equation, the reactor period equation, and the source multiplication equation.

Nukefact #30 presented an equation for precursor inventory that is available in some conventional training materials. The equation was used to generate a lambda-effective diagram. Whereas a comparable expression for reactor power has been lacking prior to Nukefact #71, that precursor definition applies to all reactor conditions from shutdown to full-power, and beyond, for the steady-state and for the transient state.

As with reactor power, both the first reactor kinetics equation (Nukefact #69) and the second reactor kinetics equation (Nukefact #70) can be solved to yield an expression for reactor precursor inventory.

PRECURSOR INVENTORY from the PRECURSOR BALANCE EQUATION

Equation 70.13 in Nukefact #70 presented a general form of the precursor balance for the transient situation:

72.1
where:
1/T(t) = the fractional rate of change of the precursor inventory at time "t", sec-1
T(t) = the reactor period at time "t", seconds
n(t) = 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
nu = the average neutron yield (prompt + delayed) from a fission event
beta = the precursor yield fraction

In the transient state, at least two parameters will be changing with time, namely the thermal neutron fission rate and precursor inventory. If k-effective is changing with time, or a significant non-fission source is present, then the reactor period and the precursor decay constant will also be changing with time. Herein, precursor inventory change with time is derived for the condition of a stable reactor period, meaning that the period as constant, lambda-effective is constant, and k-effective is constant.

Equation 72.1 then reduces to:
72.2
Moving all terms containing C(t) to the left-hand-side of the equation, and combining these terms gives:

72.3
Equation 72.3 states that in the transient state, for increasing inventory, the precursor production rate must not only make up for the decay loss rate but must produce enough excess precursor atoms to account for the rate of inventory increase. For decreasing inventory, the precursor production must fail to make up the decay losses by an amount that accounts for the inventory decrease.

Solving for C(t) gives:
72.4
With n(t) representing the fission rate, reactor power is obtained by dividing both sides of Equation 72.4 by the power conversion factor 3.1x1010 (fissions/second)/watt:

72.5
PRECURSOR INVENTORY from the PRECURSOR BALANCE EQUATION
where:
P(t) = n(t) / 3.1x1010 = reactor power, watts

Finally, dividing both sides of Equation 72.5 by nu and combining terms gives:

72.6
where:
C-bar(t) = C(t)/(nux3.1x1010), watt-sec

Equation 72.6 indicates that the precursor inventory is essentially a function of reactor power and the rate of change in precursor inventory. Per the precursor balance equation, the rate of change in precursor inventory depends on the mismatch between precursor production and precursor loss. Contrary to the situation for reactor power, 1/T is significant for operational application. Equation 72.6 is the precursor equation presented in Nukefact #30.

Equations 72.4 and 72.6 are also useful for determining inventories for the individual precursor groups by substitution of group constants, to give:
72.7
where:
C-bari(t) = Ci(t) /(nu x 3.1x1010 ), watt-sec = inventory of the ith precursor group at time "t"
betai = the precursor yield fraction for the ith precursor group
lambdai = the decay constant for the ith precursor group, sec-1

A future Nukefact essay will revisit the precursor inventory to illustrate numeric applications and present a decay model that parallels that of the neutron chain reaction.

PRECURSOR INVENTORY from the NEUTRON BALANCE EQUATION

Equation 69.17 in Nukefact #69 presented a general form of the neutron balance for a transient situation:

72.8
where:
T(t) = the reactor period at time "t", seconds
n(t) = the thermal neutron fission rate at an instant in time, "t", fissions/second
delta-k(t) = keff(t) - 1
keff(t) = the effective decay constant at time "t"
lambdaeff(t) = the effective decay constant at time "t", sec-1
C(t) = the precursor inventory at time, "t", atoms
S = the non-fission source neutron emission rate at time "t", neutrons/second

In the transient state, at least two parameters will be changing with time, namely the thermal neutron fission rate and precursor inventory. If k-effective is changing with time, or a significant non-fission source is present, then the reactor period and the precursor decay constant will also be changing with time. Herein, precursor change with time is derived for the condition of a stable reactor period, meaning that the period as constant, lambda-effective is constant, k-effective is constant, delta-k is constant, and the non-fission source is insignificant.

Equation 72.8 then reduces to:

72.9
Moving all terms containing n(t) to the left-hand-side of the equation, and combining these terms gives:

72.10
On solving for C(t) we have:

72.11
PRECURSOR INVENTORY from the NEUTRON BALANCE EQUATION

Dividing both sides of Equation 72.11 by (nu x 3.1x1010} and combining terms gives:

72.12
where:
C-bar(t) = C(t) / (nu x 3.1x1010), watt-sec
P(t) = n(t) / 3.1x1010 = reactor power, watts

Equation 72.12 can also be expressed as:

72.13
where:
rho = delta-k/k
l* = lp/k

Nukefact #71 demonstrated that the multiplying factors on reactor power in Equations 72.6 and 72.13 are equivalent. Of the two derivations, Equation 72.6 is most commonly used for determining the precursor inventory.

REMARKS

This essay has detailed the derivation of an equation that defines precursor inventory in terms of pertinent core parameters for all possible reactor conditions involving a constant k-effective, including equilibrium subcritical multiplication, criticality, and supercriticality. Some conventional training resources do present this important expression but rarely grasp its importance due to a failure to recognize delayed neutrons as being source neutrons.

The fact that Equation 72.6 has existed in a number of texts for many years lends support to the form of the power equation presented in Nukefact #71, namely Equation 71.6. The appearance of (1/T) in both the power equation and the precursor equation is an indicator of consistency in representing the physical process. To eliminate (1/T) prior to completing the derivation of reactor power, because it is insignificant in the operational range, which then creates the long standing mystery of why both the power equation and the period equation break down at prompt criticality and beyond, is just inexcusable.

The natural inclination is to focus on the neutron balance equation because the neutrons cause the fissions that produce reactor power. The fact that it is always the source neutrons that determine the magnitude of the neutron population, and hence of reactor power, is a serious oversight. In seeking understanding of reactor behavior, the importance of the precursor balance equation is equal to that of the neutron balance equation.

SUMMARY

  1. A general definition of precursor inventory is provided by both the precursor balance equation and the neutron balance equation.

  2. The definition of precursor inventory applies to all possible reactor conditions having a constant k-effective, including subcriticality, criticality, and supercriticality.

  3. The two definitions of precursor inventory are equivalent, as would be expected since the balance equations are coupled.

  4. The definition of precursor inventory obtained from the precursor balance equation is more useful in understanding the physical process.
HomePage | NukeFacts | ExamBank | PopQuiz | Visuals | NRC | Rx TRN | Xe TRN | FeedBack | E-mailXpress |