Point Kinetics Equations
As we have seen in previous chapters, the number of neutrons is multiplied by a factor keff from one neutron generation to the next, therefore the multiplication environment (nuclear reactor) behaves like the exponential system, that means the power increase is not linear, but it is exponential.
It is obvious the effective 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.
- keff < 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.
- keff = 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.
- keff > 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.
But we have not yet discussed the duration of a neutron generation, that means, how many times in a one second we have to multiply the neutron population by a factor keff. This time determines the speed of the exponential growth. But as was written, there are different types of neutrons: prompt neutrons and delayed neutrons, which completely change the kinetic behaviour of the system. Therefore such a discussion will be not trivial.
To study the kinetic behaviour of the system, engineers usually use the point kinetics equations. The name point kinetics is used because, in this simplified formalism, the shape of the neutron flux and the neutron density distribution are ignored. The reactor is therefore reduced to a point. In the following section we will introduce the point kinetics and we start with point kinetics in its simplest form.
In the equation for neutrons, the first term on the right hand side is the production of prompt neutrons in the present generation, k(1-β)n/l, minus the total number of neutrons in the preceding generation, -n/l. The second term is the production of delayed neutrons in the present generation. As can be seen, the rate of absorption of neutrons is the same as in the simple model (-n/l). But a distinction is between the direct channel for prompt neutrons (1-β) production and the delayed channel resulting from radioactive decay of precursor nuclei (λiCi).
In the equation for precursors, there is the balance between the production of the precursors of i-th group and their decay after the decay constant λi. As can be seen, the rate of the decay of precursors is the radioactivity rate (λiCi) and the rate of production is proportional to the number of neutrons times βi, which is defined as the fraction of the neutrons which appear as delayed neutrons in the ith group.
As can be seen, the point kinetics equations include two differential equations, one for the neutron density n(t) and the other for precursors concentration C(t).
Again, the point kinetics equations are often expressed is terms of reactivity (ρ = (k-1)/k) and prompt generation time, Λ, as:
Both forms of the point kinetics equation are valid. The equation using Λ, prompt neutron generation time, is usually better for calculations. This is because most reactivity transients are induced by changes in the absorption cross-section rather than in the fission cross-section. During this transients the prompt neutron lifetime is not constant whereas the prompt generation time remains constant.
The previous equation defines the reactivity of a reactor, which describes the deviation of an effective multiplication factor from unity. For critical conditions the reactivity is equal to zero. The larger the absolute value of reactivity in the reactor core, the further the reactor is from criticality. In fact the reactivity may be used as a measure of a reactor’s relative departure from criticality. According to the reactivity, we can classify the different reactor states and the related consequences as follows:
Approximate Solution of Point Kinetics Equations
Sometimes, it is convenient to predict qualitatively the behaviour of a reactor. The exact solution can be obtained relatively easy using computers. Especially for illustration, the following approximations are discussed in the following sections:
- Prompt Jump Approximation
- Prompt Jump Approximation with One Group of Delayed Neutrons
- Constant Delayed Neutron Source Approximation