NUKEFACT #34

INVERSE MULTIPLICATION and FUEL LOADING

PART I - FUNDAMENTALS

last update November 27, 1998
Inverse multiplication has a long history of use in graphically monitoring core nuclear status during fuel loading and also as control rods are withdrawn during approach to criticality. In early nuclear experiments fuel loadings were carried to the point of criticality. Today, in loading fuel assemblies into large commercial reactors, inverse multiplication is used as a safety measure to ensure that criticality does not occur during the fueling operation. For initial reactor startup, or startup after refueling, inverse multiplication is often used to extrapolate to the critical control rod position.

In derivation, a major attribute of inverse multiplication is its linear relationship to k-effective. This allows for a straight-line extrapolation of the inverse multiplication plot to the exact condition of criticality, well before the criticality is approached. In application, inverse multiplication exhibits anything but linear behavior. The reason for this is that inverse multiplication plots are constructed versus the number of fuel assemblies loaded, or versus the control rod position, neither of which has a one-to-one correspondence with k-effective. The first, and foremost, reason for non-linearity in the inverse multiplication plot is this disconnect from k-effective. In Nuclear Training Centers much is then made of the source-detector-fuel geometry to explain the non-linear shape of the inverse multiplication plot. The fact of the matter is that inverse multiplication plots against anything other than keff are likely to be non-linear ... regardless of geometry. This is not to say that source-detector-fuel geometry cannot affect the shape of the inverse multiplication plot, even significantly, but rather that if the inverse multiplication plot is against anything other than keff, it is almost certain to be non-linear. We will present a means to construct a linear inverse multiplication plot during fuel loading and startup.

The next four NUKEFACT essays will address the subject of source multiplication and its applications ... or misapplications. Herein, in Part I, we review the fundamentals of source multiplication and the definition of inverse multiplication.

NON-FISSION SOURCE MULTIPLICATION

In NUKEFACT #3, Source Multiplication is NOT Limited to the Subcritical Reactor, a set of chain reactions was used to model the physical process of subcritical source multiplication in the Generation Time Model, as follows:


Figure 34.1 - The Chain Reaction Model

where:
S' = the number of non-fission neutrons emitted into each generation
keff = the effective multiplication factor, which must be less than 1.0

The non-fission source neutrons emitted into each successive generation time interval initiate a set of chain reactions. Each chain, in proceeding from one generation to the next, is multiplied by keff, and is finite in length. The terms in each vertical column sum to the total fast neutrons, N', in the generation. From these few generations, a pattern can be detected; the number of fast neutrons in generation "i" is:

34.1

Number of Fast Neutrons in Generation "i"

With keff < 1 as "i" approaches infinity, the power series in keff sums to a limit, which is:

34.2

A Mathematical Equivalence

On substitution of Equation 34.2 into Equation 34.1, and on dividing both sides of the equation by the generation time, the source strength, S, and the fast neutron production, N, to be expressed in terms of neutrons produced per second, for the steady state condition, called equilibrium subcritical multiplication.

34.3

Number of Fast Neutrons at Equilibrium Subcritical Multiplication

where:
N = the steady state fast neutron production rate, neutrons/second
S = the non-fission source strength, neutrons/second
keff = the effective multiplication factor
M = the non-fission source multiplication factor = -1/(keff -1)

Equation 34.3 defines the equilibrium subcritical neutron population in the entire reactor. Because of certain mathematical substitution involved in its derivation, per Equation 34.2, it is valid only for values of keff that are less than one ... i.e., only for the subcritical reactor. In the process of establishing the equilibrium neutron population for a given source strength and keff, the important non-fission source multiplication factor, M, has been defined.

34.4

The Non-Fission Source Multiplication Factor "M"

From Equation 34.3, the ratio of the total fast neutron production, N, to the non-fission source strength, S, is also a measure of the non-fission source multiplication factor. Thus, the simple model of Figure 34.1 yields the definition of non-fission source multiplication for the subcritical steady state condition. This is the factor by which the source emissions are multiplied by fission neutrons to yield the total fast neutron population. And even though derived from a simple chain reaction model, this source multiplication factor is valid for the actual equilibrium subcritical reactor.

Table 34.1 lists the Source Multiplication Factor, M, for keff values that increase progressively from 0.0000 to 0.9000.

keffective

 MULTIPLICATION M = -1/(keff - 1)
0.00001.0000
0.10001.1111
0.20001.2500
0.30001.4286
0.40001.6667
0.50002.0000
0.60002.5000
0.70003.3333
0.80005.0000
0.900010.0000

TABLE 34.1 - Non-Fission Source Multiplication Factor for Specific k-effectives

Column 1 lists keff values. Column 2 lists the non-fission source multiplication factor, M, for each keff value. The minimum keff value of 0.0000 represents the reactor prior to the first fuel assembly loading. The maximum keff value of 0.9000 represents the reactor with all fuel assemblies loaded. Since the fuel assemblies are loaded with control rods and other poison present, the maximum keff is the normal shutdown keff, assuming the loading has proceeded as expected. The magnitude of this shutdown keff depends on the reactor design. Figure 34.2 is a graphical display of this relationship:


FIGURE 34.2 - M versus keff

The vertical axis is the Source Multiplication Factor, M, and the horizontal axis is keff. Figure 34.2 illustrates that M is non-linear with respect to keff ... and in fact ascends rapidly toward infinity as keff exceeds 0.9000 and approaches 1.0000. This behavior is not particularly helpful in identifying the point of potential criticality during fuel loading.

INVERSE MULTIPLICATION

Inverse multiplication is the reciprocal of the non-fission source multiplication factor. Whereas non-fission source multiplication is non-linear with keff, inverse multiplication is shown to be linear, going to zero as criticality is approached. Hence, the behavior of inverse multiplication is potentially more useful in monitoring nuclear status. The expression for inverse multiplication is:

34.5

Inverse Multiplication

where:
IM = 1/M = Inverse Multiplication

Equation 34.5 is expressed in the form shown, rather than the simpler IM = 1 - keff, because in this form it relates directly to the standard equation for a straight line, which is:

34.6

Equation of a Straight Line

where:
y = the y-coordinate of a point on the straight line
x = the x-coordinate of a point on the straight line
a = the slope of the straight line
b = the y-axis intercept of the straight line

On comparison of the expression for inverse multiplication, Equation 34.5, with the equation for a straight line, Equation 34.6, it is seen that the inverse multiplication expression is an equation of a straight line. Inverse multiplication, as defined by Equation 34.5 is a linear function of keff. The slope of Equation 34.5 is a = -1, the coefficient of keff. And, the y-axis intercept is b = 1. The x-axis intercept is at keff = 1, or at reactor criticality.

Table 34.2 lists the inverse multiplication, 1/M, for the same keff values given in Table 34.1.

keffective
SOURCE MULTIPLICATION M = -1/(keff - 1)
 INVERSE MULTIPLICATION 1/M = -keff + 1
0.00001.0000 1.0000
0.10001.11110.9000
0.20001.25000.8000
0.30001.42860.7000
0.40001.66670.6000
0.50002.00000.5000
0.60002.50000.4000
0.70003.33330.3000
0.80005.00000.2000
0.900010.00000.1000

TABLE 34.2 - Inverse Multiplication for Specific k-effectives

Columns 1 and 2 are from Table 34.1. Column 3 is the inverse multiplicaion value, or the reciprocal of Column 2. The IM value prior to loading the first fuel assembly, i.e. with keff = 0.0000, is 1.0000. The IM value after all fuel asseblies are loaded, i.e. with keff = 0.9000 has decreased to 0.1000. Figure 34.3 is a graphical display of the relationship between 1/M and keff:


FIGURE 34.3 - 1/M versus keff

The vertical axis is inverse multiplication and the horizontal axis is keff. This graphic is known as an inverse multiplication diagram. Here we see Equation 34.5 as a graphical display, the IM plot is a straight line with a slope of -1, a y-axis intercept of IM = 1.0, and a projected x-axis intercept of keff = 1.0. The fact that the IM plot (a straight line) projects to, or anticipates, criticality is the principal reason for its use during fuel loading. In this case, if additional fuel assemblies of equal worth remain to be loaded, the extrapolation indicates that criticality would be attained. This would be cause to terminate the fueling operation.

THE PHYSICAL MEANING of INVERSE MULTIPLICATION

Having established the linearity between IM and keff, we need now return to Equation 34.4, where much valuable information remains. In inverted form we have:

34.7

Inverse Multiplication ... A Second Look

or:

34.8

Equation 34.8 displays two important relationships, the first being:

34.9

Inverse Multiplication ... A Direct Measure of Fuel Status

In Equation 34.3 "M" was introduced to represent -1/(keff -1). That arbitrary substitution, on inversion, leads to the relationship given in Equation 34.9. The numeric value of inverse multiplication, which is always positive, is nothing less than the subcritical delta-k value without a minus sign. Thus, inverse multiplication provides direct indication as to the nuclear status of the core ... in terms of delta-k. The inverse multiplication diagram of Figure 34.3 is more meaningful when considered as a plot of -delta-k versus keff.

The second important relationship from Equation 34.8 is:

34.10

The Worth of the Non-Fission Neutron Source

This relationlship derives directly from the chain reaction model in Figure 34.1 and can be obtained by rearranging Equation 34.3, the expression for equilibrium source multiplication. At equilibrium multiplication, the delta-k worth of the non-fission source is equal in magnitude and opposite in algebraic sign to the core delta-k because the non-fission source neutrons exactly make up for chain reaction losses and produce the steady state condition of equilibrium subcritical multiplication. In Part II Equation 34.10 provides the basis for measuring inverse multiplication, and therefore delta-k during fueling. And one would expect a plot of -delta-k versus keff to be a straight line.

CORE NUCLEAR STATUS and FUEL ASSEMBLY WORTH

In example fuel loadings that follow, we refer to the individual fuel assembly worth. This section defines that worth in terms of delta-k. Contrary to statements made in NUKEFACT 25, TWO MANY PARAMETERS FOR DEFINING NUCLEAR STATUS, concerning the preference toward reactivity instead of delta-k, delta-k is the form that must be used in dealing with non-fission source multiplication and inverse multiplication.

In the process of loading individual fuel assemblies, there will be a particular keff value associated with the partial core configuration following each fuel assembly loading. This keff value represents the nuclear status of the partially fueled core. We use the superscript "i" to denote that the kieff value applies after "i" fuel assemblies have been loaded. Delta-ki = kieff - 1 also represents the core fuel status after a particular number of assemblies are loaded, in terms of deviation from, or proximity to, criticality.

Let the nuclear status with "i - 1" fuel assemblies loaded be represented as:

34.11

Delta-k with "i-1" Fuel Assemblies Loaded

where:
delta-ki-1 = the proximity to criticality with "i-1" fuel assemblies loaded
ki-1eff = the effective multiplication factor with "i-1" fuel assemblies loaded

Then on loading the "ith" fuel assembly, the nuclear status becomes:

34.12

Delta-k with "i" Fuel Assemblies Loaded

where:
delta-ki = the proximity to criticality with "i" fuel assemblies loaded
kieff = the effective multiplication factor with "i" fuel assemblies loaded

The worth of a particular assembly is then equal to the difference between the nuclear status before the assembly was loaded, ki-1eff and the nuclear status after the assembly was loaded, kieff. The worth of a fuel assembly is referred to as delta(delta-k), which is:

34.13

The Worth of a Single Fuel Assembly Loading

where:
[delta(delta-k)]i = the worth of the "ith" fuel assembly loading

Equation 34.13 was first introduced in NUKEFACT #24, THE PARADOX IN EXPRESSING NUCLEAR STATUS: Delta-K versus Rho. On rearranging Equation 34.13, we see how the fuel assembly worth is added to the core keff to obtain the new nuclear status.

34.14

Adding the "ith" Fuel Assembly Worth to Core ki-1eff

It is important to recognize the important distinction between delta-k, which represents a particular nuclear status, and delta(delta-k), which represents the difference between two specific nuclear states. When reference is made to fuel assemblies of equal, or unequal, worth in the following sections, it is in terms of delta(delta-k) ... or on the fuel assemblies effect (change) on the partial core nuclear status existing before the fuel assembly was loaded.

SUMMARY

TO BE CONTINUED

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