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
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. Ci,1 = Ci,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 < 0.5β. It is usually used with another simplification, the one delayed precursor group approximation.
This simplification then leads to:
Assuming that the reactivity is constant and n1/n0 can be determined from prompt jump formula this equation leads to very simple formula:
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 dCi(t)/dt = 0 and Ci(t) = Ci,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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