For the non-linear solver click here.
Equations: 1 group scheme for neutronics, incompressible thermal hydaulic for water.
Power and Precursors:
\frac{dP}{dt} = \frac{\rho - \beta}{\Lambda}P +\lambda C
\frac{dC}{dt} = \frac{\beta}{\Lambda}P - \lambda C
Fuel and coolant energy balance:
c_fM_f\frac{dT_f}{dt} = P - K(T_f - T_c)
c_cM_c\frac{dT_c}{dt} = K(T_f - T_c) -\Gamma c_c(T_{out} - T_{in})
Reactivity:
\rho = \alpha_f(T_f-T^0_f) + \alpha_c(T_c - T^0_c) + \alpha_h(h-h^0)
P: Power, C: Precursors, Tf = fuel temperature, Tc = coolant temperature (water) .
Then the linearized equations are (assuming h as the only input):
\frac{d\delta P}{dt} = \frac{\alpha_f( \delta T_f) + \alpha_c( \delta T_c) + \alpha_h( \delta h)}{\Lambda} P^0 - \frac{\beta}{\Lambda} \delta P +\lambda \delta C
\frac{d \delta C}{dt} = \frac{\beta}{\Lambda} \delta P - \lambda \delta C
c_fM_f\frac{d \delta T_f}{dt} = \delta P - K( \delta T_f - \delta T_c)
c_cM_c\frac{d \delta T_c}{dt} = K( \delta T_f - \delta T_c) -2\Gamma c_c( \delta T_c )
Now considering X: vector of state space variables, U:vector of inputs and Y: vector of outputs we can write the system in this way:
X = (\delta P,\delta C,\delta T_f,\delta T_c)U = (\delta h)
Y = (\delta P, \delta C, \delta T_f, \delta T_c, \delta \rho)
\frac{dX}{dt} = AX +BU
Y = CX + DU
And solve the step response it with matlab:
Errata: there is an error at line 39, correct code is: A(4,4) = -(k/C_c/M_c + 2G/M_c);
download code (matlab)