Squirrel

"Fork squirrel" to create a new MOOSE-based application.

This application is intended as a light weight and flexible implementation of the PK equation with the ability to capture the change in reactivity caused by the DNP movement and changes in temperature.

If you use the Squirrel app considder citing: 🔗 https://doi.org/10.1080/00295639.2025.2494182

DNP movement

A good derivation of this can be found in Chapter two of Mathematical-Methods-in-Nuclear-Reactor-Dynamics. For a given power $$ P (x,t) = P(t) |\psi(x)\rangle $$ with $|P(x)\rangle$ being the L2 normalized flux $\Phi$:

$$ |\psi(x) \rangle = \frac{\Phi_0}{\int_{x} dx \Phi_0} $$

The squirrel app solves the PK by weighting the DNP $C$ concentration. This is derived in [[Derivation]] [[Derivation old]] $$ \dot P(t) = \frac{\rho-\beta}{\Lambda} P(t) + \sum_I \lambda_I \frac{\langle \psi(x) |C_I(x)\rangle}{\langle \psi(x)| \psi(x)\rangle} $$ with $$ \langle \psi|\phi\rangle = \int dx \psi^* \phi
$$ and for the DNPs: $$ |\dot C_I(x) \rangle = \frac{\beta_I}{\Lambda}P(t)|\psi(x) \rangle - \lambda_I |C_i(x) \rangle + U \nabla C_I(x) $$

we are imposing steady state initial conditions so that $|\dot C_I(x) \rangle = \dot P(t) = 0$ with that we get for $t = 0$ $$ |C_I(x) \rangle = \frac{\beta_I}{\Lambda \lambda_I}|\psi(x) \rangle $$

Temperature feedback

For a given temperature field $T$ we can calculate the difference at each point $i$ from the steady state Temperature $T_{ref}$ .

$$ \delta T_i = T_i - T_{ref, i} $$

We can now calculate the change in $\rho_{temp}$ introduced by a temperature change:

$$ \rho_{temp} = \langle \psi |\delta T \rangle \Gamma_{pcm/K}. $$

with $\Gamma$ being the reactivity change in reactivity caused by a global increase by $1$ Kelvin and $\int dr \psi(r) = 1$

Doppler and Density

It is possible that the reactivity change depends on the temperature $\Gamma(T)$. According to https://www.researchgate.net/publication/361864499 the Doppler effect follows a $\log(T)$ relation, while the Density feedback follows a linear relation.

In addition, the importance function is different.
The Doppler follows $\psi^* \psi = \psi^2$ while the density feedback follows $\psi$

Implementation

AuxKernel

The ScalarMultiplication AuxKernel implements the multiplication of a scalar variable with a variable. $$ P(t) |P(x)\rangle $$ The ScalarMultiplicationPP AuxKernel does the same for a postprocessor

Postprocessor

WeightDNPPostprocessor

implements the $$ A_I = \lambda_I \frac{\langle \psi|C_I\rangle}{\text{Norm}} $$ for the $I$th group.

TemperatureFeedback

postprocessor implements $$ \rho = \sum_i \frac{\psi_i}{\Delta T} \delta T_i $$

TemperatureFeedbackInt

postprocessor implements

$$ \rho = \frac{\langle \psi|\delta T \rangle}{\Delta T} $$

TempDoppler

Implements

$$ \rho = \langle \psi^* \psi| (T_i - T_{ref, i}) \Gamma(T_i) \rangle $$

with $\Gamma(T_i) = a\log(bT_i+c) + d$ . $a$, $b$, $c$ and $d$ are appropriate fit parameters.

TempDensity

Implements

$$ \rho = \langle \psi| (T_i - T_{ref, i}) \Gamma(T_i) \rangle $$

with $\Gamma(T_i) = a T_i+ b$ . $a$ and $b$ are appropriate fit parameters.

TwoValuesL2Norm

implements:

$$ \sqrt{\int_\Omega uv d\Omega} $$

Tests

Sanity checks

In [[Sanity_checks]] simple tests are preformed. They should confirm the correct implementation of the PK solver.

Problems

In [[Problems]] the performance of the PK within a multi physics context is tested. Namely the Lid driven cavity benchmark and the flushing of DNPs in the MSRE.