This model assumes nonlinear feedback from the moderator density and fuel temperature changes. It also assumes linear feedback from the moderator and fuel temperature changes. The reactivity equation is
$$r(t)=\sum_{i=1}^{n_s}r_{si}(t)+\sum_{i=1}^{n_p}[W_{\rho i}R_\rho(\rho_i(t))a_{Mi}\Delta T_{Mi}(t)]\sum^{n_f}_{i=1}[W_{Fi}R_F(T_{Fi}(t))+a_{Fi}\Delta T_{Fi}(t)]$$ Where \(r_{si}(t)\) is a table of reactivity curves, \(R_\rho\) is a table of reactivity as a function of moderator density of the fluid \(\rho_i(t)\). \(W_{\rho i}\) is the density volume weighting factor, \(\Delta T_{Mi}(t)\) is the spatially-averaged temperature of the moderator fluid, \(a_{Mi}\) is the volume fluid temperature coefficient, \(\eta_p\) is the number of thermal fluid volumes in the reactor core, \(R_F\) is a table of the doppler reactivity as a function of the heat structure volume average fuel temperature \(T_{Fi}(t)\). \(\Delta T_{FI}(t)\) is the difference of the volume average fuel temperature, \(W_{Fi}\) and \(a_{Fi}\) are the fuel temperature and heat structure weighting factor, and the heat structure fuel temperature coefficient, and \(\eta_F\) is the number of fuel volumes in the reactor core. Next we use the conservation equations for mass, energy, and momentum in a pipe.
$$\begin{matrix}
\frac{\partial \rho}{\partial t}+\frac{\partial \rho u}{\partial x}=0 \\
\frac{\partial \rho u}{\partial t}+\frac{\partial (\rho u^2+P)}{\partial x} + f\frac{\rho}{2D_h}u|u|=\rho g_x=0 \\
\frac{\partial\rho E}{\partial t}+\frac{\partial(\rho u(E+\frac{P}{\rho}))}{\partial x} +h_wa_w(T_m-T_w)+u(f\frac{\rho}{2D_h}u|u|-\rho g_x)+q_m^{”’}=0
\end{matrix}$$
Where \(\rho\) is the density, \(\rho u\) is the momentum, and \(\rho E\) is the total energy. P is the pressure, \(D_h\) is the hydraulic diameter of the pipe, \(f\) is the wall friction factor, \(g_x\) is the gravity component in the pipe. \(h_w\) is the heat transfer coefficient. \(a_w\) is the heating wall area density per unit of fluid volume. \(T_w\) is the wall temperature, and \(T_m\) is the fluid temperature, and \(q^{”’}_M\) is the external heat source. \(q_M^{”’}\) is the energy deposited into the thermal fluids in the reactor.
[[Equation for fuel temperature feedback]]
[[TRIGA Mk 2 Reactivity]] – similar equation
- zhangPOINTKINETICSCALCULATIONS2013