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Practical form of the conservation equations

In this section we derive a form of the conservation equations which we will use in the analysis of the steady-state core flow. The derivation is based on the two-fluid one-dimensional conservation equations we stated in the previous section.

For simplicity the model we use subsequently is a mixture model which accommodates a slip between the phases through a drift-flux relation. The foundation of the model is the following three mixture conservation equations which are established by adding the phasic conservation equations (6.27)-(6.32) given in section 6.5
Mass (mixture):

$$\frac{\partial}{\partial t}\left[A_c(1-\mbox{$<\!{\alpha}\!... ...\mbox{$<\!{\alpha}\!>$}\mbox{$<\!{u_g}\!>$}_g\rho_g\right] = 0$$ (6.31)

Momentum (mixture):

$$\begin{eqnarray*} \lefteqn{\frac{\partial}{\partial t}\left[ \mbox{$<\!{u_\ell}... ...\alpha}\!>$})\rho_\ell + \mbox{$<\!{\alpha}\!>$}\rho_g \right) \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

Energy (mixture):

$$\begin{eqnarray*} \lefteqn{\frac{\partial}{\partial t}\left[(\mbox{$<\!{e_\ell}... ...box{$<\!{\alpha}\!>$})A_c + q_g'''\mbox{$<\!{\alpha}\!>$} A_c \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

which arise from addition of the phasic equations. In (6.39) we have introduced the void weighted cross-sectional average of the enthalpy of liquid and vapor, $\mbox{$<\!{h_\ell}\!>$}_\ell , \mbox{$<\!{h_g}\!>$}_g$, defined by

$$\begin{eqnarray*} \mbox{$<\!{h_\ell}\!>$}_\ell &\;\hbox{$=$\kern-0.68em\raise1.... ...tstyle\triangle$}}\;& \mbox{$<\!{e_g}\!>$}_g+ \frac{p}{\rho_g} \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

Since we are interested in the steady-state case with constant cross-sectional flow area such that

$$\frac{\partial}{\partial t} \equiv 0$$ (6.32)

and

$$\frac{\partial}{\partial z}\left[A_c\right] \equiv 0$$ (6.33)

we obtain the following conservation equations
Mass (mixture):

$$\frac{d}{dz}\left[(1-\mbox{$<\!{\alpha}\!>$})\mbox{$<\!{u_\... ...\mbox{$<\!{\alpha}\!>$}\mbox{$<\!{u_g}\!>$}_g\rho_g\right] = 0$$ (6.34)

Momentum (mixture):

$$\frac{d}{dz}\left[ \mbox{$<\!{u_\ell}\!>$}_\ell ^2 \rho_\el... ...\alpha}\!>$})\rho_\ell + \mbox{$<\!{\alpha}\!>$}\rho_g \right)$$ (6.35)

Energy (mixture):

$$\begin{eqnarray*} \lefteqn{\frac{d}{dz} \left[(\mbox{$<\!{h_\ell}\!>$}_\ell + \... ...'''(1-\mbox{$<\!{\alpha}\!>$}) + q_g'''\mbox{$<\!{\alpha}\!>$} \end{eqnarray*}$$

$\parbox{1.50cm}{\begin{eqnarray} \end{eqnarray}}$

In order to make the conservation equations (6.43)-(6.45) even more tractable we are going to rearrange the equations and simplify them even further.

Firstly with reference to Appendix A we are safe to assume negligible changes in kinetic and potential energies in the energy equation such that the revised energy equation can be stated as

$$\frac{d}{dz}\left[\mbox{$<\!{h_\ell}\!>$}_\ell \rho_\ell\mb... ...'''(1-\mbox{$<\!{\alpha}\!>$}) + q_g'''\mbox{$<\!{\alpha}\!>$}$$ (6.36)

Furthermore, since in a mixture model we in reality cannot distinguish between heat added to the liquid or vapor we write the right hand side as

$$q' \;\hbox{$=$\kern-0.68em\raise1.1ex \hbox{$\scriptscript... ...\mbox{$<\!{\alpha}\!>$})A_c + q_g'''\mbox{$<\!{\alpha}\!>$}A_c$$ (6.37)

where the functional form of q' [W/m] is determined by the neutronics calculation (see section 1.8).

Finally we will introduce the flow quality, $\mbox{$<\!{x}\!>$}$, into the conservation equations and use the mass conservation to rearrange the conservation of momentum and energy.

In terms of the mass fluxes (6.18)-(6.20) we may write the mass conservation equation (6.43) as

$$\frac{d}{dz}\left[\mbox{$<\!{G_\ell}\!>$}+\mbox{$<\!{G_g}\!>$}\right] = 0$$ (6.38)

and introducing the flow quality (6.21) we may state the energy equation as

$$\mbox{$<\!{G}\!>$} \frac{d}{dz}\left[ \mbox{$<\!{h_\ell}\!>... ...!{h_g}\!>$}_g\mbox{$<\!{x}\!>$} \right] \simeq \frac{q'}{A_c}$$ (6.39)

or

$$\mbox{$<\!{G}\!>$} \frac{d}{dz}\left[ \mbox{$<\!{h}\!>$} \right] \simeq \frac{q'}{A_c}$$ (6.40)

where we have introduced the mixture enthalpy, $\mbox{$<\!{h}\!>$}$, defined by

$$\mbox{$<\!{h}\!>$} \;\hbox{$=$\kern-0.68em\raise1.1ex \hbo... ...\mbox{$<\!{x}\!>$}) + \mbox{$<\!{h_g}\!>$}_g\mbox{$<\!{x}\!>$}$$ (6.41)

We may also express the mixture enthalpy in terms of the specific latent heat of vaporization, $\mbox{$<\!{h_{\ell g}}\!>$}$, defined by

$$\mbox{$<\!{h_{\ell g}}\!>$} \;\hbox{$=$\kern-0.68em\raise1.... ...ngle$}}\;\mbox{$<\!{h_g}\!>$}_g- \mbox{$<\!{h_\ell}\!>$}_\ell$$ (6.42)

such that we can write

$$\mbox{$<\!{h}\!>$} = \mbox{$<\!{h_\ell}\!>$}_\ell + \mbox{$<\!{x}\!>$}\mbox{$<\!{h_{\ell g}}\!>$}$$ (6.43)

Introducing the mass fluxes $\mbox{$<\!{G_\ell}\!>$}$ and $\mbox{$<\!{G_g}\!>$}$ and the flow quality $\mbox{$<\!{x}\!>$}$ into the momentum equation enables us to eliminate the mean velocities $\mbox{$<\!{u_\ell}\!>$}_\ell$ and $\mbox{$<\!{u_g}\!>$}_g$ since we have (see (6.18) and (6.19))

$$\mbox{$<\!{u_\ell}\!>$}_\ell = \frac{\mbox{$<\!{G_\ell}\!>$}}{\rho_\ell (1-\mbox{$<\!{\alpha}\!>$})}$$ (6.44)

and

$$\mbox{$<\!{u_g}\!>$}_g= \frac{\mbox{$<\!{G_g}\!>$}}{\rho_g \mbox{$<\!{\alpha}\!>$}}$$ (6.45)

Utilizing these expressions reveal that the expression which enters the momentum equation

$$\frac{d}{dz}\left[\mbox{$<\!{u_\ell}\!>$}_\ell ^2\rho_\ell(... ...+ \mbox{$<\!{u_g}\!>$}_g^2\rho_g\mbox{$<\!{\alpha}\!>$}\right]$$

can be written as

$\parbox{16cm}{ \begin{eqnarray*} \frac{d}{dz}\left[\mbox{$<\!{u_\el... ...<\!{x}\!>$}^2}{\rho_g\mbox{$<\!{\alpha}\!>$}}\right\}\right] \end{eqnarray*}}$
Rewriting the momentum equation using the above mentioned expression yields

$$-\frac{dp}{dz} - \tau_w \frac{P_f}{A_c} -g( (1-\mbox{$<\!{\... ...ac{\mbox{$<\!{x}\!>$}^2}{\rho_g\mbox{$<\!{\alpha}\!>$}}\right]$$ (6.46)

For convenience we list the conservation equations which have to be solved along the fuel channel
Momentum (mixture):

$$-\frac{dp}{dz} - \tau_w \frac{P_f}{A_c} -g( (1-\mbox{$<\!{\... ...ac{\mbox{$<\!{x}\!>$}^2}{\rho_g\mbox{$<\!{\alpha}\!>$}}\right]$$ (6.47)

Energy (mixture):

$$\mbox{$<\!{G}\!>$} \frac{d}{dz}\left[ \mbox{$<\!{h}\!>$} \right] \simeq \frac{q'}{A_c}$$ (6.48)

where $\mbox{$<\!{h}\!>$}$ is defined by (6.51) (see also (6.53)).

In the momentum equation (6.57) we have introduced the term $-P_{\mbox{\protect\scriptsize loc}}'$ which describes the pressure loss per unit length due to local phenomena. In our case these local phenomena only include the pressure loss due to the grid spacers--the modeling of this pressure loss is treated in depth in section 6.7.2.3.

Note that the conservation of mass does not enter the above description since it is incorporated into the two other conservation equations.

At this point it is appropriate to state the assumptions of our model in regard to the subject of thermal equilibrium. We state that our model belongs to the group of partial equilibrium models since we assume that

• The liquid phase is in a subcooled or saturated state.
• The vapor phase is in a saturated state.

In order to solve the conservation equations we have a number of external relations, which we call constitutive relations. Below we give a brief survey of these relations^6.7

.

Intrinsic:

)

Liquid

.

At saturation: $\rho_f = f(p)$, $\mu_f = f(p)$ and k_f = f(p).

.

Subcooled state: $\rho_\ell = f(p,T_\ell)$, $\mu_\ell = f(p,T_\ell) \approx \mu_f(T_\ell)$, $k_\ell = f(p,t_\ell) \approx k_f(T_\ell)$, $h_\ell = f(p,T_\ell)$, $C_{p,\ell} = f(p,T_\ell)$ and $T_\ell = f_5(p,h_\ell)$.

)

Vapor at saturation: $\rho_g = f(p)$, $\mu_g = f(p)$ and h_g = f(p).

.

External:

)

Local pressure drop per unit length: $P_{\mbox{\protect\scriptsize loc}}' = f_0(z_0,K_{\mbox{\protect\scriptsize sp}},\mbox{$<\!{G}\!>$},\rho_f,\rho_g,\mbox{$<\!{x}\!>$})$.

)

Wall shear stress: $\tau_w = f_1(p,\mbox{$<\!{G}\!>$},\rho_f,\rho_g,\mbox{$<\!{x}\!>$})$.

)

Void fraction model: $\mbox{$<\!{\alpha}\!>$}= f_2(\mbox{$<\!{x}\!>$},\rho_\ell,\rho_g,\mbox{$<\!{G}\!>$})$.

)

Subcooled boiling model:

.

Point of void departure: $\left.\mbox{$<\!{h_\ell}\!>$}_\ell \right\vert _{z=z_d} = f_3(h_f,q_w'',D_H,C_{p,\ell},k_\ell,\mbox{$<\!{G}\!>$})$.

.

Flow quality (profile fit): $\mbox{$<\!{x}\!>$}= f_4(\mbox{$<\!{h}\!>$},h_f,h_{fg},\left.\mbox{$<\!{h_\ell}\!>$}_\ell \right\vert _{z=z_d})$.

In the next section we will investigate the constitutive relations one-by-one.

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