- #1

dRic2

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__qualitatively__the response of a reactor to a large step insert of reactivity (e.g. more than 2/3 $) is it allowed to neglect the latent neutrons contribution to simplify the equations ?

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- Thread starter dRic2
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- #1

dRic2

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- #2

Astronuc

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Of course, we have to be concerned about transients without scram, or delayed scram.

- #3

dRic2

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I'm asking this because our professor usually ask to evaluate a simplfied response to a step insert in reactivity (or some other kind of insertion) without the aid of numerical computation. I'm always stuck with a non-linear system of differential equations even if I use the simple mono-group approximation.

Should I linearize the equation? But I knew that linearization works only when you have small changes in the system and I don't think that is the case.

Or maybe the constant precursor approximation is more appropriate ?

I'm really interested only in the transient not in the behaviour of the reactor for large values of time, so maybe I can use some of the above approximation.

Thanks again for the answer

- #4

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1. Solve it numerically. It is pretty easy to solve by writing a simple computer program. There are built-in solvers in Matlab, or you can write your own solver in Python or Fortran.

2. If you are asked to solve it analytically, you are usually only required to use one delayed neutron group. This leads to a system of two equations that can be solved analytically to give two expoential roots. You can do a web search to find examples of this solution technique.

3. If you have to solve it analytically and need to use more delayed neutron groups, then you usually need to use the in-hour equation and are given a plot to use. You just need to look up the "stable period" for a given reactivity.

- #5

dRic2

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Even with mono-group approximation you get a system of 3/4 differential equations and I didn't find a way to solve it analytically without further simplifications

- #6

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Here is a paper that analytically solves the point kinetics equations with adiatic feedback. It can be done, but it is probably more complicated than a homework problem.

A. A. Nahla, "An analytical solution for the point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model," Progress in Nuclear Energy, 51, p124-128 (2009).

- #7

dRic2

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$$\frac {dP}{dt} = \frac {\rho - \beta}{Λ} P + \lambda C$$

$$\frac {dC}{dt} = \frac {\beta}{Λ} P- \lambda C$$

$$ \frac {dT_f}{dt} = \frac P {\tau_f} - \frac k {\tau_f} (T_f - T_c)$$

$$d\rho = \delta \rho_0 + \alpha_f dT_f$$

where ##\tau_f = m_f c_{p_f}##. I'd like to know if there is a way to predict how things will unfold after, for example, a step insertion of reactivity. If ## \rho << \beta## maybe Prompt Jump will help, but what if it is not ? Is there a way to make some predictions at least ?

Back to my original question, I thought that one could neglect the latent neutrons contribution in order to drop an equation, but apparently it is not a very good thing to do. So I guess there are no shortcuts after all! Thanks for the replies anyways! And I'll definitely check the article if I can find it.

- #8

Astronuc

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https://www.sciencedirect.com/science/article/pii/S0149197008000565A. A. Nahla, "An analytical solution for the point reactor kinetics equations with one group of delayed neutrons and the adiabatic feedback model," Progress in Nuclear Energy, 51, p124-128 (2009).

One might be able to obtain the paper through a university or institutional library, otherwise one must purchase it.

- #9

dRic2

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Thank you very much! I can download from siencedirect.com through my university.

- #10

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[1] When looking at very small reactivity insertions in a conventional PWR or BWR, the easiest way to predict the dynamic power response is via the "Power Defect in Reactivity" Method. This is what reactor control systems designers actually use - rather than complicated numerical simulations. It takes advantage of the strong non-linearities in feedback. [I wish I knew how to use LaTex!]

dRho / dPower = [Partial dRho / Partial dTfuel * dTfuel/dPower +

Partial dRho / Partial dTmod * dTmod / dPower]

NOTE: the partial derivative terms are all non-linear in practice.

One takes this expression, and inverting a bit, solve for: dPower / dt yielding:

dPower / dt = dRho /dt * [Partial dRho / Partial dTfuel * dTfuel/dPower +

Partial dRho / Partial dTmod * dTmod / dPower]^-1

NOTE: in a BWR one must add the Void Reactivity term

[2] When looking at very large reactivity insertions -- one must be talking about unconventional reactors, criticality experiments, or TRIGAs. Conventional PWRs and BWRs (at least in the US) are designed such that possible reactivity additions are very small (Rho << Beta). In Bell & Glasstone's "Nuclear Reactor Theory" Chapter 9.6 describes the Fuchs-Hansen Model (p.517) which is applicable in situations where delayed neutrons just don't matter. The dynamics are driven by the prompt neutron populations and Doppler feedback. Other texts refer to it as the Nordheim-Fuchs method such as in Hetrich's "Dynamics of Nuclear Reactors", Chapter 5-5, p.164. Most of these methods are actually old -- as current licensed and operating PWR and BWR reactors are designed such that this issue is almost irrelevant. They barely talk about it in universities any more.

And then we find some experiment gone awry (Chernobyl 1986) where they actually inserted Rho > 2-3* Beta, or in some enriched Uranium chemical processing accident (Tokai Mura, Japan) - which end with the disassembly of the critical system.

- #11

dRic2

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Just to be sure

What is the issue you are referring at here? The large insertion of reactivity or the fact that the behavior of the reactor is dominated by prompt neutrons ?as current licensed and operating PWR and BWR reactors are designed such that this issue is almost irrelevant

- #12

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Hope this clarifies the question.

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