NUKEFACT #71

Solving the Balance Equations

THE GENERAL EQUATION for REACTOR POWER

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last update September 1, 2004
INTRODUCTION

Commercial reactors in the U.S. are used to produce electrical power for general consumption. For those dealing with reactor operation, one of their earliest curiosities concerns just what determines power level. This question has never been satisfactorily answered in conventional training resources. It is the topic of this essay.

Nukefact #3 presented an equation for reactor power that was based on the neutron balance equation. The equation form was one of source multiplication. But contrary to the conventional equation for subcritical source multiplication, which fails for k-effective values equal to and greater than 1.0000, the power equation in Nukefact #3 was valid for k-effective values approaching 1.0065. This essay details that derivation and extends the result to all possible values of k-effective, specifically to values of k-effective greater than 1.0065.

Both the first reactor kinetics equation (Nukefact #69) and the second reactor kinetics equation (Nukefact #70) can be solved to yield a general expression for reactor power. The power definitions apply to all reactor conditions from shutdown to full-power, and beyond, for the steady-state and for the transient state, and quantify reactor power in watts based on the neutron source strength and the value of k-effective.

REACTOR POWER from the NEUTRON BALANCE EQUATION

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

71.1
where:
T(t) = the reactor period at time "t", seconds
lp = the prompt neutron lifetime, seconds
n(t) = 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"
beta = the precursor yield fraction
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, reactor power 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, and delta-k is constant.

Equation 71.1 then reduces to:

71.2
The non-fission source term is retained because it is an important contributor at equilibrium subcritical multiplication, where a stable rate exists, namely T = infinite seconds.

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

71.3
Solving for n(t) gives:
71.4
With n(t) representing the fission rate, reactor power is obtained by dividing both sides of Equation 71.4 by the power conversion factor 3.1x1010 (fissions/second)/watt:
71.5
REACTOR POWER from the NEUTRON BALANCE EQUATION
where:
P(t) = n(t) / 3.1x1010 = reactor power, watts

Dividing the numerator and denominator on the right-hand-side by k and combining terms, restates the power equation in terms of reactivity:

71.6
where:
C-bar(t) = C(t)/(nux3.1x1010), watt-sec
S-bar = S/(nux3.1x1010), watts
l* = lp/k

There is much to be said about Equation 71.4 and its variations with respect to the physical meaning of the mathematical expression. We will begin that discussion in the next Nukefact.

For operational purposes, the l*/T term in the denominator of Equation 71.6 is insignificant and can be omitted, making the operational approximation:

71.7
Equation 71.7 is the power equation presented in Nukefact #3.

REACTOR POWER from the PRECURSOR BALANCE EQUATION

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

71.8
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
nu = the number of neutrons released per fission
beta = the precursor yield fraction
C(t) = the precursor inventory at time "t", atoms
lambdaeff(t) = the effective decay constant at time "t", sec-1

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, reactor power change with time is derived for the condition of a stable reactor period, meaning that the period is constant, lambda-effective is constant, and k-effective is constant.

Equation 71.8 then reduces to:

71.9
Moving all terms containing C(t) to the left-hand-side of the equation, and combining these terms gives:
71.10
Solving for n(t) gives:
71.11
With n(t) representing the fission rate, reactor power is obtained by dividing both sides of Equation 71.11 by the power conversion factor 3.1x1010 (fissions/second)/watt:
71.12
REACTOR POWER from the PRECURSOR BALANCE EQUATION
where:
P(t) = n(t) / 3.1x1010 = reactor power, watts

Equation 71.12 can also be expressed as:

71.13
where:
C-bar(t) = C(t) / (nu x 3.1x1010), watt-sec

In the following section the equivalence of the two power equations, one from the neutron balance and the other from the precursor balance, is demonstrated.

POWER EQUATION EQUIVALENCE

Taking the non-fission source to be negligible, Equation 71.6, the equation for reactor power as derived from the neutron balance equation, can be written as:

71.14
And Equation 71.13, the equation for reactor power as derived from the precursor balance equation, can be written as:
71.15
For Equations 71.14 and 71.15 to be equivalent, it must be shown that the respective multiplying factors on the right-hand-side are identical. This will be done by transforming the multiplier of Equation 71.15 into that of Equation 71.14. First we refer to the in-hr equation for a single precursor group:
71.16
Moving the first term on the right-hand-side of Equation 71.16 to the left-hand-side gives:
71.17
Substitution into the multiplying bracket of Equation 71.15 gives:
71.18

71.19
Which, on expanding the expression for reactor period on the left-hand-side is:
71.20
Canceling reactivity and creating a common denominator gives:
71.21
which is:
71.22
Thus, the multiplying factors for the two power equations are identical, meaning that the two equations for power, Equations 71.6 and 71.13, are identical. The two coupled balance equations provide a consistent definition for reactor power. From the standpoint of understanding the physical process, the power definition from the neutron balance will be seen as the more useful.

The following numeric example verifies the algebra:

The factor from the neutron balance is:

And the factor from the precursor balance is:
The small difference between the two values is due to round-off error in the neutron balance factor. The correct value is 1342.

REMARKS

This essay has developed an equation that defines reactor power in terms of pertinent core parameters for all possible reactor conditions involving a constant k-effective, including equilibrium subcritical multiplication, criticality, and supercriticality. Conventional training resources have failed to provide this important expression. At best, the conventional definition determines reactor power for conditions of subcritical equilibrium multiplication. Conditions of criticality and supercriticality must resort to smoke and mirrors. Convention leaves the power level at criticality, and beyond, as arbitrary. One only need pick it out of thin air.

The reason for this omission in conventional training is that delayed neutrons are not recognize as source neutrons. Thus, once the non-fission source becomes negligible, the conventional equation for source multiplication breaks down. Any operator who has ever started up a reactor, or even a simulator, knows full well that there is a continuity of reactor power between the Sub-Critical region and the Delayed-Critical region. It is the equations in this essay that explain this continuity. Chain reactions constitute the physical process while the reactor is subcritical and chain reactions constitute the physical process when the reactor is critical and supercritical. The underlying physical process within a reactor does not change when the reactor reaches criticality.

SUMMARY

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

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

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

  4. The definition of reactor power obtained from the neutron balance equation is more useful in understanding the physical process.

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