Point Kinetics Equation with $6$ Delayed Groups

Reactor power equation

$p^{}(t)=\frac{(t)-}{}p(t)+{\displaystyle \underset{k=1}{\overset{6}{}}}\frac{{}_k}{}{}_k(t)$

Precursor balance equation

${}_k^{}(t)={}_{k}p(t)-{}_k{}_k(t),k=1,..,6$

$={\displaystyle \underset{k=1}{\overset{6}{}}}{}_k$

$p(t)$: Reactor Power

$(t)$: Delayed Precursor Concentration

$(t)$: Reactivity $(Driving$ Function$)$

$={}_{\text{e}\text{x}\text{t}}(t)-f(p)$

$(t)$ : Delayed Precursor Concentration

$(t)$: Reactivity $($Driving Function$)$

$$: Life Time $(20s)$

$(t)={}_{\text{e}\text{x}\text{t}}(t)-((t){\displaystyle \underset{0}{\overset{t}{}}}p()d+(1-(t))p(t))$

where

$p(0)=1{10}^{-6}$

${}_k^{}(0)=\frac{{}_k}{{}_k}p(0)$

Task

Describe the physical meanings of the point kinetics equations and the parameters in the equations.

Ex$)$ Prompt neutron, delayed neutron, fission product, beta decay, neutron life time, $\backslash$beta $\_$ and lambda etc.

Write a Python program to solve this problem using the $4$th order Runge$-$Kutta method and obtain the reference solution up to $1$ sec with $\backslash$Delta t$=0\mathrm{.}0001$ sec

Write a PYthon subprogram for the explicit and modified Euler methods, respectively. Solve the problem with delta t$=0\mathrm{.}02,0\mathrm{.}01,0\mathrm{.}005,0\mathrm{.}001($sec$)$ with explicit, modified Euler$$s and $4$th order RK methods.

Plot the solution and examine the dependence of the error in the peak power and time on the time step size.