The fission process may produce 2, 3 or more free neutrons that are capable of inducing further fissions and so on. This sequence of fission events is known as the fission chain reaction and it is of importance in nuclear reactor physics.
The chain reaction can take place only in the proper multiplication environment and only under proper conditions. It is obvious, if one neutron causes two further fissions, the number of neutrons in the multiplication system will increase in time and the reactor power (reaction rate) will also increase in time. In order to stabilize such multiplication environment, it is necessary to increase the non-fission neutron absorption in the system (e.g. to insert control rods). Moreover, this multiplication environment (the nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is exponential.
On the other hand, if one neutron causes less than one further fission, the number of neutrons in the multiplication system will decrease in time and the reactor power (reaction rate) will also decrease in time. In order to sustain the chain reaction, it is necessary to decrease the non-fission neutron absorption in the system (e.g. to withdraw control rods).
In fact, there is always a competition for the fission neutrons in the multiplication environment, some neutrons will cause further fission reaction, some will be captured by fuel materials or non-fuel materials and some will leak out of the system.
In order to describe the multiplication system, it is necessary to define the infinite and finite multiplication factor of a reactor. The method of calculations of multiplication factors has been developed in the early years of nuclear energy and is only applicable to thermal reactors, where the bulk of fission reactions occurs at thermal energies. This method well puts into the context all the processes, that are associated with the thermal reactors (e.g. the neutron thermalisation, the neutron diffusion or the fast fission), because the most important neutron-physical processes occur in energy regions that can be clearly separated from each other. In short, the calculation of multiplication factor gives a good insight in the processes that occur in each thermal multiplying system.
Infinite Multiplication Factor – Four Factor Formula
In this section, the infinite multiplication factor, which describes all the possible events in the life of a neutron and effectively describes the state of an infinite multiplying system, will be defined.
The required condition for a stable, self-sustained fission chain reaction in a multiplying system (in a nuclear reactor) is that exactly every fission initiate another fission. The minimum condition is for each nucleus undergoing fission to produce, on the average, at least one neutron that causes fission of another nucleus. Also the number of fissions occurring per unit time (the reaction rate) within the system must be constant.
This condition can be expressed conveniently in terms of the multiplication factor. The infinite multiplication factor is the ratio of the neutrons produced by fission in one neutron generation to the number of neutrons lost through absorption in the preceding neutron generation. This can be expressed mathematically as shown below.
It is obvious the infinite multiplication factor in a multiplying system is a measure of the change in the fission neutron population from one neutron generation to the subsequent generation.
- k∞ < 1. If the multiplication factor for a multiplying system is less than 1.0, then the number of neutrons is decreasing in time (with the mean generation time) and the chain reaction will never be self-sustaining. This condition is known as the subcritical state.
- k∞ = 1. If the multiplication factor for a multiplying system is equal to 1.0, then there is no change in neutron population in time and the chain reaction will be self-sustaining. This condition is known as the critical state.
- k∞ > 1. If the multiplication factor for a multiplying system is greater than 1.0, then the multiplying system produces more neutrons than are needed to be self-sustaining. The number of neutrons is exponentially increasing in time (with the mean generation time). This condition is known as the supercritical state.
If there are neutrons in the system at t=0, that is, if n(0) > 0, the solution of this equation gives the simplest point kinetics equation with delayed neutrons (similarly to the case without delayed neutrons):
Let us consider that the mean generation time with delayed neutrons is ~0.085 and k (k∞ – neutron multiplication factor) will be step increased by only 0.01% (i.e. 10pcm or ~1.5 cents), that is k∞=1.0000 will increase to k∞=1.0001.
It must be noted such reactivity insertion (10pcm) is very small in case of LWRs. The reactivity insertions of the order of one pcm are for LWRs practically unrealizable. In this case the reactor period will be:
T = ld / (k∞-1) = 0.085 / (1.0001-1) = 850s
This is a very long period. In ~14 minutes the neutron flux (and power) in the reactor would increase by a factor of e = 2.718. This is completely different dimension of the response on reactivity insertion in comparison with the case without presence of delayed neutrons, where the reactor period was 1 second.
Reactors with such a kinetics would be quite easy to control. From this point of view it may seem that reactor control will be a quite boring affair. It will not! The presence of delayed neutrons entails many many specific phenomena, that will be described in later chapters.
But the infinite multiplication factor can be defined also in terms of the most important neutron-physical processes that occur in the nuclear reactor. For simplicity, we will first consider a multiplying system that is infinitely large, and therefore has no neutron leakage. In the infinite system. There are four factors that are completely independent of the size and shape of the reactor that give the inherent multiplication ability of the fuel and moderator materials without regard to leakage:
The infinite multiplication factor (k∞) may be expressed mathematically in terms of these factors by following equation, usually known as the four factor formula:
k∞ = η.ε.p.f
In reactor physics, k∞ or its finite form keff is the most significant parameter with regard to reactor control. At any specific power level or condition of the reactor, keff is kept as near
to the value of 1.0 as possible. At this point in operation, the neutron balance is kept to exactly one neutron completing the life cycle for each original neutron absorbed in the fuel.
From infinite to finite multiplication factor
The infinite multiplication factor is derived based on the assumption of no neutrons leak out of the reactor (i.e. a reactor is infinitely large). But in reality, each nuclear reactor is finite and neutrons can leak out of the reactor core. The multiplication factor that takes neutron leakage into account is the effective multiplication factor – keff, which is defined as the ratio of the neutrons produced by fission in one neutron generation to the number of neutrons lost through absorption and leakage in the preceding neutron generation.
The effective multiplication factor (keff) may be expressed mathematically in terms of the infinite multiplication factor (k∞) and two additional factors which account for neutron leakage during neutron thermalisation (fast non-leakage probability) and neutron leakage during neutron diffusion (thermal non-leakage probability) by following equation, usually known as the six factor formula:
keff = k∞ . Pf . Pt
Operational factors that affect the fission chain reaction in PWRs.
Detailed knowledge of all possible operational factors that may affect the multiplication factor of the system are of importance in the reactor control. It was stated the keff is during reactor operation kept as near to the value of 1.0 as possible. The criticality of the reactor is influenced by many factors. For illustration, in an extreme case also the presence of human (due to the water, carbon, which are good neutron moderators) near fresh uranium fuel assembly influences the multiplication properties of the assembly.
If any operational factor changes one of the contributing factors to keff (keff = η.ε.p.f.Pf.Pt), the ratio of 1.0 is not maintained and this change in keff makes the reactor either subcritical or supercritical. Some examples of these operational changes, that may take place in PWRs, are below and are described below:
Control rods (insertion/withdrawal) influences the thermal utilization factor. For example, control rods insertion causes an addition of new absorbing material into the core and this causes a decrease in thermal utilization factor.
The thermal utilization factor for heterogeneous reactor cores must be calculated in terms of reaction rates and volumes, for example, by the following equation:
where Σa is the macroscopic absorption cross section, which is the sum of the capture cross section and the fission cross section, Σa = Σc + Σf. The superscripts U, M, P, CR, B, BA and O, refer to uranium fuel, moderator, poisons, control rods, boric acid, burnable absorbers and others, respectively. It is obvious, that the presence control rods, boric acid or poisons causes a decrease in the neutron utilization, which, in turn, causes a decrease of multiplication factor.
In comparison with the chemical shim, which offset positive reactivity excess in entire core, with control rods the unevenness of neutron-flux density in the reactor core may arise, because they act locally.
↑boron ⇒ ↓keff = η.ε.p. ↓f .Pf.Pt
The concentration of boric acid diluted in the primary coolant influences the thermal utilization factor. For example, an increase in the concentration of boric acid (chemical shim) causes an addition of new absorbing material into the core and this causes a decrease in thermal utilization factor.
The thermal utilization factor for heterogeneous reactor cores must be calculated in terms of reaction rates and volumes, for example, by the following equation:
where Σa is the macroscopic absorption cross section, which is the sum of the capture cross section and the fission cross section, Σa = Σc + Σf. The superscripts U, M, P, CR, B, BA and O, refer to uranium fuel, moderator, poisons, control rods, boric acid, burnable absorbers and others, respectively. It is obvious, that the presence control rods, boric acid or poisons causes a decrease in the neutron utilization, which, in turn, causes a decrease of multiplication factor.
When compared with burnable absorbers (long term reactivity control) or with control rods (rapid reactivity control) the boric acid avoids the unevenness of neutron-flux density in the reactor core, because it is dissolved homogeneously in the coolant in entire reactor core. On the other hand high concentrations of boric acid may lead to positive moderator temperature coefficient and that is undesirable. In this case more burnable absorbers must be used.
Moreover this method is slow in controlling reactivity. Normally, it takes several minutes to change the concentration (dilute or borate) of the boric acid in the primary loop. For rapid changes of reactivity control rods must be used.
↑TM ⇒ ↓keff = η.ε. ↓p .f. ↓Pf . ↓Pt (EOC)
This operational change is very difficult to describe, because changes in moderator temperature lead to the change of almost all the coefficients. Major impacts on multiplication of the system arise from the change of the resonance escape probability and the change of total neutron leakage (see thermal non-leakage probability and fast non-leakage probability).
Change of the resononce escape probability. It is known, the resonance escape probability is dependent also on the moderator-to-fuel ratio. All PWRs are designed as undermoderated reactors. As the moderator temperature increases the ratio of the moderating atoms (molecules of water) decreases as a result of the thermal expansion of water. Its density simply decreases. This, in turn, causes a hardening of neutron spectrum in the reactor core resulting in higher resonance absorption (lower p). Decreasing density of the moderator causes that neutrons stay at a higher energy for a longer period, which increases the probability of non-fission capture of these neutrons. This process is one of two processes, which determine the moderator temperature coefficient (MTC). The second process is connected with the leakage probability of the neutrons. The moderator temperature coefficient must be for most PWRs negative, which improves the reactor stability, because a reactor core heating causes a negative reactivity insertion.
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Boron letdown curve (chemical shim) and boron 10 depletion during a 12-month fuel cycle. Change of the thermal utilization factor. The impact on the thermal utilization factor depends strongly on the amount of boron which is diluted in the primary coolent (chemical shim). As the moderator temperature increases the density of water decreases due to the thermal expansion of water. But along with the moderator also boric acid is expanded out of the core. Since boric acid is a neutron poison, and it is expanding out of the core, positive reactivity is added. The positive reactivity addition due to the expansion of boron out of the core offsets the negative reactivity addition due to the expansion of the moderator out of the core. It is obvious this effect is significant at the beginning of the cycle (BOC) and gradually loses its significance as the boron concentration decreases.
- Change of the neutron leakage. Since both (Pf and Pt) are affected by a change in moderator temperature in a heterogeneous water-moderated reactor and the directions of the feedbacks is the same, the resulting total non-leakage probability is also sensitive on the change in the moderator temperature. In result, an increase in the moderator temperature causes that the probability of leakage increases. In case of the fast neutron leakage, the moderator temperature influences macroscopic cross-sections for elastic scattering reaction (Σs=σs.NH2O) due to the thermal expansion of water, which results in an increase in the moderation length. This, in turn, causes an increase of the leakage of fast neutrons.
- For the thermal neutron leakage there are two effects. Both processes have the same direction and together causes the increase in the thermal neutron leakage. This physical process is a part of the moderator temperature coefficient (MTC).
- Macroscopic cross-sections for elastic scattering reaction Σs=σs.NH2O, which significantly changes due to the thermal expansion of water. As the temperature of the core increases, the diffusion coefficient (D = 1/3.Σtr) increases.
- Microscopic cross-section (σa) for neutron absorption changes with core temperature. As the temperature of the core increases, the absorption cross-section decreases.
- For the thermal neutron leakage there are two effects. Both processes have the same direction and together causes the increase in the thermal neutron leakage. This physical process is a part of the moderator temperature coefficient (MTC).
The effect of change in the flow rate through the primary circuit have identical consequences as the effects of the moderator temperature. In reality, when there is an abrupt change (e.g. as a result of a disconnection of the reactor coolant pump) in the flow rate and the reactor power remains the same, the difference between inlet and outlet temperatures must increase. It follows from basic energy equation of reactor coolant, which is below:
P=↓ṁ.c.↑∆t
The inlet temperature is determined by the pressure in the steam generators, therefore the inlet temperature changes minimally during the transient. It follows the outlet temperature must change significantly as the flow rate changes. When the inlet temperature remains almost the same and the outlet changes significantly, it stands to reason, the average temperature of coolant (moderator) will change also significantly. Therefore the effect of change in the flow rate through the primary circuit have identical consequences as the effects of the moderator temperature.
The decrease in flow rate is associated with negative reacivity insertion. Special attention is needed in case of an abrupt increase in the flow rate. At normal operation such increase in the flow rate can not occur, except the controlled reactor coolant pump connection, which can be connected only under specific conditions.
↑burnable absorbers ⇒ ↓keff = η.ε.p. ↓f .Pf.Pt
The amount of burnable absorbers in the nuclear fuel influences the thermal utilization factor. In some cases (espacially in the case of gadolinium absorbers) the presence of burnable absorbers influences all the factors in the four factor formula due to very high self-shielding effects. But at this place we consider only the change in the thermal utilization factor. An increase in the amount of burnable absorbers causes an addition of new absorbing material into the core and this simply causes a decrease in thermal utilization factor.
The thermal utilization factor for heterogeneous reactor cores must be calculated in terms of reaction rates and volumes, for example, by the following equation:
where Σa is the macroscopic absorption cross section, which is the sum of the capture cross section and the fission cross section, Σa = Σc + Σf. The superscripts U, M, P, CR, B, BA and O, refer to uranium fuel, moderator, poisons, control rods, boric acid, burnable absorbers and others, respectively. It is obvious, that the presence control rods, boric acid or burnable absorbers causes a decrease in the neutron utilization, which, in turn, causes a decrease of multiplication factor.
