A nuclear reactor operating at a steady power has a neutron multiplication factor of precisely 1.0. In this state, each fission event yields on average one further fission reaction to maintain the chain reaction. If the multiplication factor is <1.0, the nuclear chain reac­tion will die away quickly, while if >1.0, the chain reaction will build up exponentially. A definition of the neutron multiplication factor, usually denoted k, is:

k =(No. of neutrons born in generation j)/

(No. of neutrons born in generation j — 1)

In a reactor operating in the steady state, it is possible to maintain k extraordinarily close to 1 using the inherent negative feedback effects, so that the power output precisely matches the amount of thermal energy extracted from the reactor; thermal power production from the reactor in excess of that removed by the turbine generator raises the temperature of the reactor, and temperature feedback effects reduce the multiplication factor to establish a new balance between reactor output and thermal load.

Some reactors are operated on-load, meaning that fuel can be inserted during full power operation. In such a reactor, it is theoretically possible to start the reactor with just sufficient fissile material (such as 235U) so that k just marginally exceeds 1.0 at full power. As the fissile material is depleted (burns up), fresh fuel is added to compensate and a corresponding quantity of spent fuel is discharged.

However, on-load refueling is not practical in all reactors; many are designed such that access to the fuel is possible only with the reactor shut down and such reactors operate with batch refueling. This means that the reactor is shut down periodically (typically every 12-24 months) and a fraction of the fuel replaced (with an equivalent amount of the highest burnup fuel discharged). In between refueling shutdowns, the reactor operates with no change in the fissile inventory, except for those changes due to fuel depletion and the buildup of fission products and transuranic elements through neutron captures. The next refueling occurs when the fissile content is depleted to the point where, with all control absor­bers removed from the core, the reactor is just capa­ble of maintaining k = 1.0 at full power.

It is useful to note the concept of reactivity at this point, which is a dimensionless quantity usually measured in units of per cent mill or 10—5. It is defined by

p = 105[(k — 1)/k]pcm

A reactor that is just critical has a reactivity of zero and is in steady-state operation. Positive reactivity corre­sponds to a reactor where the neutron flux and power output are increasing with time, while conversely, negative reactivity corresponds with decreasing flux and power. Control rod insertion is said to cause a decrease in reactivity, while their withdrawal is said to increase reactivity.

The upper curve in Figure 1 illustrates the variation of k with irradiation for a typical PWR fuel assembly with an initial 235U enrichment of 4.45 wt% and no burnable poison. At low irradiation, k is around 1.3, while by the time the fuel has reached its discharge burnup of ^55 Gigawatt-days per tonne (1 Gigawatt-days is a convenient unit of thermal energy output for nuclear fuel equivalent to 8.64 x 1013J ton— ; the mass refers to 1 ton of initial heavy metal and the unit is usually abbreviated GWd/tHM), k for the assembly is lower than 1.0. Thus, by the time the fuel is ready for discharge, it is incapable of sustaining a chain reaction by itself. However, the core always contains a mix of fresh fuel in its first cycle of irradiation along with highly irradiated fuel and the fresh fuel effectively donates excess neutrons to the high irradiation fuel such that for the core as a whole, k = 1.0.

Batch refueling causes a problem, however, because if the core is capable of attaining k = 1.0 at the end of the cycle, then by implication, the uncontrolled k must exceed 1.0 at earlier times. In principle of course, control rods can be inserted to control the excess multiplication factor, but in practice, there are limita­tions: First, the control rods have a limited effectiveness and more might be required than is practical within the engineering constraints. Second, inserting control rods distorts the spatial distribution of neutron flux and power within the core, causing localized power peaking that might easily exceed the acceptable performance limits of the fuel. Some means is therefore needed to supplement the control rods.

To an extent, PWRs are able to achieve this with boron as boric acid dissolved in the coolant. This soluble absorber approach has its limitations as well, as will be seen later, and soluble absorber is not an option for most other reactor types. Burnable poisons are used to fill this reactivity control gap. The lower curve in Figure 1 illustrates how a burnable poison (in this case, 24 fuel rods per assembly consisting of 6 wt% Gd2O3 in UO2 with a 235U enrichment of

4.45 wt%) might be used to control the excess reac­tivity of fresh fuel by holding down the multiplication factor in low burnup fuel assemblies. Consider, for example, a 3-batch fueling scheme in which the core is made up of equal quantities of fuel assemblies in their first, second, and third irradiation cycles. At the start of the fuel cycle, the first cycle assemblies will have zero burnup, corresponding to points A and A0 unpoisoned and poisoned, respectively. Similarly, the second and third cycle assemblies will correspond to points B and C. The multiplication factor of the unpoisoned core will be k1 = [k(A) + k(B) + k(C)]/3 at the start of the cycle. At the end of the cycle, the three assembly types will have burned up to points B, C, and D and k will be k2 = [k(B) + k(C) + k(D)]/ 3 = 1.0. The difference in reactivity between the start and end of the cycle is k1 — k2 = [k(A) — k(D)]/3. For the poisoned assembly, the corresponding differ­ence is

k0— k2 = [k(A0) — k(D)]/3.

Since k(A0) < k(A), the change in multiplication factor during the cycle is considerably reduced. Ide­ally, the burnable poison material is depleted during the fuel cycle such that there is none left at the end of the first irradiation cycle indicated; any residual poison material left at the end of the cycle means that the cycle will end prematurely or will require an increased initial 235U loading if the cycle length is to be preserved, both of which equate to an economic
penalty. The small gap between the poisoned and unpoisoned curves in Figure 1 represents just such a residual poison penalty.