## 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

**ρ**<

**) causes at first a sharp change in prompt neutrons population and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response is a result of the small value of prompt neutron generation time in the denominator of point kinetics equation.**

*β*If we are interested in **long-term behaviour** (asymptotic period) and not interested in the details of the prompt jump, we can simplify the point kinetics equations by assuming that the **prompt jump takes place instantaneously** in response to any reactivity change. This approximation is known as the **Prompt Jump Approximation (PJA)** in which the rapid power change due to prompt neutrons is neglected, corresponding to taking ** dn/dt |_{0} = 0** in the point kinetics equations. That means the point kinetics equations are as follows:

From the equation for neutron flux and the assumption, that the delayed neutron precursor population does not respond instantaneously to a change in reactivity (i.e. C_{i,1} = C_{i,2}), it can be derived that the ratio of the neutron population just after and before the reactivity change is equal to:

The prompt-jump approximation is usually valid for smaller reactivity insertion, for example, for **ρ < 0.5 β. **It is usually used with another simplification, the

**one delayed precursor group approximation**.

**prompt jump approximation (PJA)**. This eliminated the fast time scale due to prompt neutrons. In this section we consider that delayed neutrons are produced only by

**one group of precursors**with the same decay constant (averaged) and delayed neutron fraction. Point kinetics equation using PJA and one group of delayed neutrons becomes:

This simplification then leads to:

Assuming that the reactivity is constant and n_{1}/n_{0} can be determined from prompt jump formula this equation leads to very simple formula:

**ρ**<

**) causes at first a sharp change in prompt neutrons population and then the neutron response is slowed as a result of the more slowly changing number of delayed neutrons. The rapid response is a result of the small value of prompt neutron generation time in the denominator of point kinetics equation.**

*β*If we are interested in short-term behaviour and not interested in the details of the asymptotic behaviour, we can simplify the point kinetics equations by assuming that the production of the delayed neutrons is constant and equal to the production at the beginning of the transient. This approximation is known as the **Constant Delayed Neutron Source Approximation** (CDS) in which changes in the amount of delayed neutrons are neglected, corresponding to taking dC_{i}(t)/dt = 0 and C_{i}(t) = C_{i,0} in the point kinetics equations. That means the point kinetics equations are as follows:

The above equation can be solved analytically and assuming that the reactivity is constant the solution is given as:

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