NUKEFACT #35

last update May 25, 2001

For purposes of simplicity, the examples employ only a few fuel assemblies. Typically, the number of fuel assemblies loaded into large commercial reactors range between 150 and 250 units. In BWRs each fuel assembly contains a fully inserted control rod, while PWRs use a combination of control rods and B-10 poison in the moderator. In both, when the loading is completed, the core is in a shutdown condition. Criticality is not attained during the fuel loading, in fact it is to be avoided. In each of the following examples, the final fuel assembly loading produces k_eff = 0.9000.

Loading Fuel Assemblies of Equal Worth

With inverse multiplication and fuel assembly worth defined we can now take a hypothetical fuel loading and generate a fuel loading 1/M diagram. Table 35.1 contains data for loading nine fuel cells of equal worth, i.e. where delta(delta-k) for each cell is identical at, delta(delta-k) = +0.1000. Note that per Equation 34.14 the k_eff value is obtained by adding the individual assembly worth to the previous k_eff value.

Column 1 lists the number of fuel assemblies loaded. Column 2 is the individual fuel assembly worth, delta(delta-k). Column 3 is the k_eff value after the fuel cell of Column 1 is loaded. Column 4 is the inverse multiplication value based on the value of k_eff, using Equation 34.5. Figure 35.1 is a graphical display of this data:

The 1/M data are typically plotted versus the number of fuel assemblies (cells) loaded, as shown in Figure 35.2.

Loading Fuel Assemblies of Decreasing Worth

As a second example we proceed to a case more representative of an actual fuel loading. Table 35.2 contains data for loading seven fuel assemblies of unequal worth. Typically, the first assembly loaded contributes the largest worth, in this case delta(delta-k) = +0.3000. Thereafter, the worth of each assembly loaded gradually decreases. For this example the worth of the second assembly is +0.2000 and assemblies three through five are worth +0.1000 each. Assemblies six and seven are worth +0.0500 each.

Column 1 lists the number of fuel assemblies loaded. Column 2 is the individual fuel assembly worth, delta(delta-k). Column 3 is the k_eff value after the fuel cell of Column 1 is loaded. Column 4 is the inverse multiplication value based on the value of k_eff, using Equation 34.5. Figure 35.3 is a graphical display of this data:

Then, as by current convention, the 1/M data are plotted versus the number of fuel assemblies (cells) loaded, as shown in Figure 35.4.

This form of loading curve is considered to be conservative because it initially projects criticality to occur with fewer assemblies than the intended full core loading. However, as the loading nears completion, the slope of the loading curve decreases and the projection to criticality becomes more realistic.

Loading Final Fuel Assemblies of Minimal Worth

As a continuation of the above example, suppose that two additional fuel assemblies are loaded which have very small worth. This too is representative of commercial reactor fuelings where the total number of assemblies is large. Table 35.3 contains data for loading two fuel assemblies of minimal worth, following the seven assemblies loaded in the previous example. The worth of the eighth and ninth assemblies loaded is +0.0010 for each.

Column 1 lists the number of fuel assemblies loaded. Column 2 is the individual fuel assembly worth, delta(delta-k). Column 3 is the k_eff value after the fuel cell of Column 1 is loaded. Column 4 is the inverse multiplication value based on the value of k_eff, using Equation 34.5. For 1/M versus k_eff the loading curve still appears as shown in Figure 35.3, with the 1/M values for assemblies eight and nine overlaying the data point for the seventh assembly.

Figure 35.5 is a graphical display of this data versus number of assemblies loaded:

This example illustrates a loading curve shape that is representative of many, if not most, commercial refuelings. The fact that actual determination of 1/M must be based on Nuclear Instrument count data, as discussed in a future NUKEFACT, introduces statistical variations into the data points of Table 35.3, which results in a loading curve that is not so smooth as shown in this simple example. However, this does not alter the fact that the overall shape of the loading curve is determined by the diminishing worth of the successive fuel assemblies added to the configuration.

Loading Fuel Assemblies of Increasing Worth

As a final example to cover the remaining possibility, we proceed to a case that may be a hypothetical situation, or may occur under very unusual conditions. Table 35.4 contains data for loading seven fuel assemblies of unequal worth. The first two assemblies loaded contribute the least worth, in this case delta(delta-k) = +0.0500 for each assembly. Thereafter, the worth of each assembly loaded gradually increases. For this example the worth of the third through fifth assemblies is +0.1000 each. The worth of the sixth assembly is +0.2000 and the worth of the seventh assembly is +0.3000.

Column 1 lists the number of fuel assemblies loaded. Column 2 is the individual fuel assembly worth, delta(delta-k). Column 3 is the k_eff value after the fuel cell of Column 1 is loaded. Column 4 is the inverse multiplication value based on the value of k_eff, using Equation 34.5. Figure 35.6 is a graphical display of this data:

Then, as by current convention, the 1/M data are plotted versus the number of fuel assemblies (cells) loaded, as shown in Figure 35.7.

This form of loading curve is considered to be non-conservative because it initially projects criticality to require many more assemblies than the intended full core loading. However, as the loading nears completion, the slope of the loading curve increases and the projection to criticality becomes more realistic.

SUMMARY

For the examples given the worth of fuel assemblies have been assumed to change in a particular manner. In practice, such change may vary. The worth of each assembly depends on the order in which it is loaded and its location in the core. It is possible that one assembly may be of less worth than the previous assembly, while the following assembly is worth more.

The potential benefit of the loading curve (1/M vs k_eff) is that it will provide an ongoing linear projection to the critical condition during the loading of fuel assemblies. Unfortunately, this promise is rarely realized because the operational loading curve plots 1/M versus the number of assemblies loaded. As a result, the criticality projections from the first few assemblies loaded are likely to be unreliable, either projecting far fewer or many more assemblies than required to attain criticality. Fortunately, even these loading curves tend toward improved projections as the loading nears completion.

It would seem that there is potential room for improvement in application of the loading curve to commercial reactor fuelings. Since 1/M is nothing more than the absolute value of the subcitical delta-k, k-effective is easily determined for creating a second loading curve, i.e. a linear loading curve that plots 1/M versus k_eff. It is quite likely that, with little extra effort, these two forms of the loading curve would allow more reliable interpretation of the proximity to criticality than the single conventional loading curve. The curve with 1/M versus k_eff always projects to the same end-point (k_eff = 1.0000).

The influence of instrument geometry will be addressed in a future NUKEFACT. Adverse instrument geometry can create non-linearities in the loading curve, of the same form as illustrated for assemblies of unequal worth.