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
As can be seen from the solution of exact point kinetics equation, any reactivity insertion (ρ < β) 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.

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:

prompt jump approximation - equation

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:

prompt jump approximation - prompt jump

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.

In the previous section we have simplified the point kinetics equation using 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:

PJA with one group

This simplification then leads to:

solution - PJA - one group

Assuming that the reactivity is constant and n1/n0 can be determined from prompt jump formula this equation leads to very simple formula:

solution - PJA - one group2

As can be seen from the solution of exact point kinetics equation, any reactivity insertion (ρ < β) 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.

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:

constant production of delayed neutrons

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

constant production of delayed neutrons 2

Nuclear and Reactor Physics:

  1. J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
  2. J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
  3. W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
  4. Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
  5. W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
  6. G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
  7. Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
  8. U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

  1. K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
  2. K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
  3. D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2. 
  4. E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above: