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Main calculational procedure, core_hydraulics

In this section we will present a very short description of the calculational steps involved in the calculation of the core hydraulics^8.2. The details of the calculations are complicated by the pressence of both grid spacers and the void departure phenomenon--this is a good reason for not going into detail with the discussion.

The function which calculates the core flow solution is called, core_hydraulics. The most important steps in the function is illustrated in Figure 8.1.

The only detail we will consider in this section is the problem of calculating the bulk (mixing cup) liquid temperature, $T_\ell$. This quantity has vital importance especially during subcooled boiling where the density of the bulk liquid is calculated from the property function $\rho_\ell(p,T)$. When we have calculated (or estimated) the state at point z_i, $\hspace{0.2ex}\underline{s}{}\hspace{0.15ex}_i$, and the mixure enthalpy at point i, $\mbox{$<\!{h}\!>$}_i$, we can calculate the bulk liquid temperature either from a curve fit of the form

$$T_{\ell,i} = fcn(h_{\ell,i},p_i)$$ (8.1)

where $h_\ell$ is the buld liquid enthalpy [J/kg] or from an equation of state

$$dh = c_p(p,T) dT + (1-\beta T)\frac{dp}{\rho}$$ (8.2)

where c_p is the specific heat at constant pressure [J/(kg$\cdot$K)] and $\beta$ is the thermal expansion coefficient [1/K].

Since we do not have access to a curve fit of the form (8.1) we calculate the temperature on the basis of an equation of state in the implementation. However, as we will see shortly the equation of state method is as accurate as the curve fit method and fully sufficient for our purposes.

According to White [28, Table 1-4] we have

$$\vert 1 - \beta T \vert < 1.23 \qquad {\mbox{for}} \qquad p_{\mbox{\protect\scriptsize sat}} \le 98.6 {\mbox{ bar}}$$ (8.3)

for saturated water which for $dp = 0.5 {\mbox{ bar}}$ implies

$$\vert (1 - \beta T) \frac{dp}{\rho} \vert < 90 {\mbox{ J/kg}}$$ (8.4)

The 90 J/kg is negligible when we compare the specific enthalpy of water at saturation, h_f, which satisfies

$$h_f > 10^6 {\mbox{ J/kg}} \qquad {\mbox{for}} \qquad p_{\mbox{\protect\scriptsize sat}} > 33 {\mbox{ bar}}$$ (8.5)

We can conclude that we can rightfully use the following very accurate approximation for water

$$dh \simeq c_p(p,T) dT$$ (8.6)

or equivalently

$$dT \simeq \frac{1}{c_p(p,T)} dh$$ (8.7)

In the implementation we use (8.7) in the following way

$$T_{\ell,i} = \int\limits_{z_{i-1}}^{z_i} \frac{1}{c_p(p,T)} dh + T_{\ell,i-1}$$ (8.8)

which is approximated by the second order accurate approximation

$$T_{\ell,i}^{(k+1)} = \frac{1}{2} \left( \frac{1}{c_p(p_i,T_... ...\ell,i} - \mbox{$<\!{h_\ell}\!>$}_{\ell,i-1} ) + T_{\ell,i-1}$$ (8.9)

the bulk liquid specific enthalpy, $\mbox{$<\!{h_\ell}\!>$}_{\ell,i}$, is calculated by equation (6.98) (see p. ). Notice that (8.9) defines a fixed point iteration scheme for the solution of $T_{\ell,i}$ with the superscripts are iteration numbers as usual. It can be shown that this scheme is convergent for $T_{\ell,i}-T_{\ell,i-1} \le 100\mbox{${}^\circ\mbox{C}$}$.

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