# Reactor dynamics with a large reactivity insert

Gold Member
I was wondering, if I want to understand 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 ?

Astronuc
Staff Emeritus
One would not ignore delayed neutrons, which would give rise to the ramp in power following the initial jump. In reality, two additional things would happen in a reactor. The temperature of the fuel would increase, and via the Doppler effect on resonance absorption, some negative reactivity would be inserted into the core. The second factor would be the activation of the reactor protection system, which should respond in seconds to a high power signal, which would activate a control rod insertion, or scram, which is a lot of negative reactivity in a few seconds, before the longer lived delayed neutron precursors emit neutrons.

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

dRic2
Gold Member
Thank you. But if I can not ignore them, then the equations are pretty difficult to solve (especially if you consider temperature feedback).

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.

You need to use the point kinetics equations to solve a step insertion transient (at a minimum, you could also use 3D kinetics). So how do you do this?

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.

Astronuc
Gold Member
@rpp true if you consider kinetics only. If you throw in even a very simple model for thermohydraulic and feedback coefficients (see for example Hetrick, Dynamics of Nuclear Reactor) it is not so easy to solve...

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

You are right, it can get complicated quickly. It's hard to give more advice unless I know exactly what you are trying to solve.

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).

Gold Member
Well, I'm not asking about a particular homework problem because even if I tell you one, a very small change in the equations will change everything. What I'm interested in is if there are some assumption that can simplify the equations in order to get a qualitative understanding of the phenomena. For example, even if you neglect the changes in the coolant's temperature (you assume it constant) you get a system of equations like this (the most simplified model of PWR I could think of... I don't even know if you can call it a PWR ):

$$\frac {dP}{dt} = \frac {\rho - \beta}{Λ} P + \lambda C$$
$$\frac {dC}{dt} = \frac {\beta}{Λ} P- \lambda C$$
$$\frac {dT_f}{dt} = \frac P {\tau_f} - \frac k {\tau_f} (T_f - T_c)$$
$$d\rho = \delta \rho_0 + \alpha_f dT_f$$

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.

Astronuc
Staff Emeritus
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).
https://www.sciencedirect.com/science/article/pii/S0149197008000565
One might be able to obtain the paper through a university or institutional library, otherwise one must purchase it.

dRic2
Gold Member
Thank you very much! I can download from siencedirect.com through my university.

I am new to Physics Forums but spent the last 44 years of my professional work in this area so I'll throw my 2 cents (of reactivity) in....

[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.

rpp and dRic2
Gold Member
Thank you for the informations.

Just to be sure
as current licensed and operating PWR and BWR reactors are designed such that this issue is almost irrelevant
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 ?

I am referring to the safety concern in current day operating PWRs and BWRs that a large reactivity insertion event could occur that would cause a power surge capable of damaging internal structural elements. The use of relatively low worth individual control rods in combination with appropriate core physics design (which gives favorable Doppler and Moderator feedback coefficients) makes this a significantly less important safety issue than it was in the 1940's and 1950's.
Hope this clarifies the question.

rpp