# Reactor dynamics with a large reactivity insert

#### dRic2

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 ?

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#### 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.

#### rpp

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.

#### dRic2

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

#### rpp

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).

#### dRic2

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).

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.

"Reactor dynamics with a large reactivity insert"

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