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Calculation of the total recirculation mass flow rate, $\mbox{$\dot{m}$}_i$

The main function in the hydraulics module hydraul1.c is shown in Figure 10.1. The initialization is carried out in the function init_hydr which among other things reads the input file^10.1 and calculates the computational grid.

The most important action of the function hydraulics, the calculation of the steady-state total recirculation mass flow rate, $\mbox{$\dot{m}$}_i$, is carried out at the line in which a call to secant_method2 is made. As we can see the function main_hydraulics_diff is solved. This function is a very simple function and serves as a link to the main function, main_hydraulics. The function main_hydraulics calculates the pressure drops in the closed loop starting from the liquid outside the separator which is at pressure $P_{\mbox{\protect\scriptsize sys}}$, per definition and returns the pressure difference at the liquid surface for a given total recirculation mass flow rate, $\mbox{$\dot{m}$}_i$. The calculational steps of the main_hydraulics function is shown in Figure 10.2 and as we can see the steps correspond to the building blocks in the electrical analog of the closed loop depicted in Figure 5.4 (see p. ).

The flow chart illustrated in Figure 10.2 assumes that we a priori know both the steam separator outlet mass flow rates and the feedwater mass flow rate, $\mbox{$\dot{m}$}_d$ (see Figure 5.4). These quantities are, however, unknowns and as a consequence we need to perform an iteration inside the main_hydraulics function in order to calculate the correct pressure differential, $p-P_{\mbox{\protect\scriptsize sys}}$. A schematic of the iteration procedure inside the function main_hydraulics is depicted in Figure 10.3.

In steady-state the feedwater mass flow rate corresponds to the total mass flow rate at the steam separator vapor outlet, ie

$$\mbox{$\dot{m}$}_d = \mbox{$\dot{m}$}_{\ell_1} + \mbox{$\dot{m}$}_{g_1}$$ (10.1)

such that the only unknowns which remain are the separator outlet mass flow rates. Since the carry-over and carry-over fractions in practice are small it is the core exit flow quality which controls the separator outlet mass flow rates. This is the reason for choosing the change in the core exit flow quality, $\Delta x_{{\framebox[1.5ex]{\raisebox{-2.2pt}[0pt][0ex]{\scriptsize 1}}},e}$, as the parameter in the stop criterion.

It is out outmost importance to keep the total number of calculations of the pressure changes in the closed loop at an absolute minimum since the core flow calculation is costly to carry out. We have therefore chosen not to iterate on the core exit flow quality inside the main_hydraulics function, ie we choose a very lax convergence criterion of $\varepsilon_{x_e} = 1$. This is possible because in all cases of practical interest $\mbox{$\dot{m}$}_d$ has a weak impact on the core exit flow quality $\mbox{$<\!{x}\!>$}_{{\framebox[1.5ex]{\raisebox{-2.2pt}[0pt][0ex]{\scriptsize 1}}},e}$ and consequently on the steam separator outlet quantities. With the secant method as non-linear solver the total recirculation mass flow rate, $\mbox{$\dot{m}$}_i$, is obtained in typically 7 iterations for an accuracy criterion which reads $p-P_{\mbox{\protect\scriptsize sys}} < 1 {\mbox{ Pa}}$.

We should note that this strategy is not applicable if we choose the Pegasus method in order to solve the non-linear equation for $\mbox{$\dot{m}$}_i$. This is because this method uses a range in which the zero is located and when the pressure difference is not calculated accurately (because of a high $\varepsilon_{x_e}$) the method calculates a range which dose not include the zero and therefore stops with a useless solution.

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