5.3.1. Compressible Volumes Without Cover Gas

A compressible volume with no cover gas is treated as a compressible liquid in an expandable container. The volume \(V\) is assumed to vary linearly with pressure p and temperature \(T\):

(5.3-1)\[V = V_{\text{r}} \left\lbrack 1 + \alpha_{\text{p}}\left( p - p_{\text{r}} \right) + \alpha_{\text{T}}\left( T - T_{\text{r}} \right) \right\rbrack\]

where \(V_{\text{r}}\) is the volume at a reference pressure \(p_{\text{r}}\) and reference temperature \(T_{\text{r}}\). Also the coolant density \(\rho\) is assumed to vary linearly with \(p\) and \(T\):

(5.3-2)\[\rho = \rho_{\text{r}} \left\lbrack 1 + \beta_{\text{p}}\left( p - p_{\text{r}} \right) + \beta_{\text{T}}\left( T - T_{\text{r}} \right) \right\rbrack\]

where

\(\alpha_{\text{p}}\) = the volume pressure expansion coefficient, \(\frac{1}{V} \frac{\partial \text{V}}{\partial \text{p}}\)

\(\alpha_{\text{T}}\) = the volume thermal expansion coefficient, \(\frac{1}{V} \frac{\partial \text{V}}{\partial \text{T}}\)

\(\beta_{\text{p}}\) = the sodium compressibility, \(\frac{1}{\rho} \frac{\partial \mathrm{\rho}}{\partial \text{p}}\)

\(\beta_{\text{T}}\) = the sodium thermal expansion coefficient, \(\frac{1}{\rho} \frac{\partial \mathrm{\rho}}{\partial \text{T}}\)

The mass of the liquid in the compressible volume is

(5.3-3)\[m = \rho V\]

Using Eq. (5.3-1) and Eq. (5.3-2) in Eq. (5.3-3) and dropping second-order terms gives

(5.3-4)\[m = m_{\text{r}} \left\lbrack 1 + \left( \alpha_{\text{p}} + \beta_{\text{p}} \right) \left( p - p_{\text{r}} \right) + \left( \alpha_{\text{T}} + \beta_{\text{T}} \right) \left( T - T_{\text{r}} \right) \right\rbrack\]

which can be rewritten as

(5.3-5)\[\delta p = \frac{\frac{\delta m}{m_{\text{r}}} - \left( \alpha_{\text{T}} + \beta_{\text{T}} \right) \delta T}{\alpha_{\text{p}} + \beta_{\text{p}}}\]

where

\(\delta m = m - m_{\text{r}}\)

\(\delta p = p - p_{\text{r}}\)

\(\delta T = T - T_{\text{r}}\)

Eq. (5.3-5) is a general relationship for the pressure change in a compressible volume with no cover gas as a result of mass and temperature changes.

To obtain expressions for \(b_0 \left( j \right)\), \(b_1 \left( j \right)\), and \(b_2 \left( j \right)\), which characterize this compressible volume \(j\), we apply conservation of mass and conservation of energy principles to the volume with flow in and flow out during a time step. Conservation of mass gives

(5.3-6)\[\Delta m = \Delta t \left\lbrack \sum{{\overline{w}}_{\text{in}} - \sum{\overline{w}}_{\text{out}}} \right\rbrack\]

where \(\Delta m\) is the change in the liquid mass in the compressible volume during the time step \(\Delta t\), \(\sum{\overline{w}}_{\text{in}}\) is the sum of the average mass flow rates into the compressible volume during \(\Delta t\), and \(\sum{\overline{w}}_{\text{out}}\) is the sum of the average mass flow rates out during \(\Delta t\).

Conservation of energy gives

(5.3-7)\[\left( m_{3} + \Delta m \right) \left( T_{3} - \Delta T \right) = m_{3} T_{3} + \Delta t \left\lbrack \sum{{\overline{w}}_{\text{in}} T_{\text{in}} - \sum{{\overline{w}}_{\text{out}} T_{\text{out}}}} \right\rbrack + \frac{Q}{c_{\text{l}}} \Delta t\]

where \(m_3\) and \(T_3\) are the mass and temperature of the liquid in the compressible volume at the beginning of the time step, \(\Delta T\) is the change in temperature of the liquid in the compressible volume during the time step, \(\sum{\overline{w}}_{\text{in}}T_{\text{in}}\) in the sum of the average mass flow rates into the volume multiplied by the incoming temperature, \(\sum{{\overline{w}}_{\text{out}}T_{\text{out}}}\) is the sum of the average mass flow rates out of the volume multiplied by the outgoing temperature, \(Q\) is the heat flow rate from the compressible volume walls and from other components in contact with the compressible volume liquid, and \(c_{\text{l}}\) is the heat capacity of the liquid in the compressible volume. Eq. (5.3-7) expresses the fact that the energy in the liquid in the compressible volume at the end of the time step is the sum of the energy present at the beginning of the time step, the excess of the energy flowing in over that flowing out during the time step, and the energy contributed to the liquid from the walls of the compressible volume during the time step.

Solving Eq. (5.3-7) for the change in the liquid temperature during the time step, gives

(5.3-8)\[\Delta T = \frac{- T_{3} \Delta m + \Delta t \left\lbrack \sum{{\overline{w}}_{\text{in}} T_{\text{in}} - \sum{{\overline{w}}_{\text{out}} T_{\text{out}}}} \right\rbrack + \frac{Q}{c_{\text{l}}} \Delta t}{m_{3} + \Delta m}\]

To first order, the \(m_3 + \Delta m\) in the denominator is approximated as \(m_3\). Inserting Eq. (5.3-6) and Eq. (5.3-8) into Eq. (5.3-5) gives

(5.3-9)\[\begin{split}\Delta p = \left\lbrack \sum{{\overline{w}}_{\text{in}} - \sum{\overline{w}}_{\text{out}}} \right\rbrack \frac{\Delta t}{\alpha_{\text{p}} + \beta_{\text{p}}} \left\lbrack \frac{1}{m_{\text{r}}} + \frac{\left( \alpha_{\text{T}} + \beta_{\text{T}} \right) T_{3}}{m_{3}} \right\rbrack \\ - \frac{\left( \alpha_{\text{T}} + \beta_{\text{T}} \right) \Delta t}{\left( \alpha_{\text{p}} + \beta_{\text{p}} \right) m_{3}} \left\lbrack \sum{{\overline{w}}_{\text{m}} T_{\text{in}} - \sum{{\overline{w}}_{\text{out}} T_{\text{out}} + \frac{Q}{c_{\text{t}}}}} \right\rbrack\end{split}\]

Comparison of this equation with Eq. (5.2-20) shows that for the compressible volume with no cover gas

(5.3-10)\[b_{0} = - \frac{\left( \alpha_{\text{T}} + \beta_{\text{T}} \right) Q\Delta t}{\left( \alpha_{\text{p}} + \beta_{\text{p}} \right) m_{3} c_{\text{l}}}\]

(5.3-11)\[b_{1} = \frac{\Delta t}{\alpha_{\text{p}} + \beta_{\text{p}}} \left\lbrack \frac{1}{m_{\text{r}}} + \frac{\left( \alpha_{\text{T}} + \beta_{\text{T}} \right) T_{3}}{m_{3}} \right\rbrack\]

and

(5.3-12)\[b_{2} = - \frac{\left( \alpha_{\text{T}} + \beta_{\text{T}} \right) \Delta t}{\left( \alpha_{\text{p}} + \beta_{\text{p}} \right) m_{3}}\]

The \(Q \Delta t\) term in Eq. (5.3-10) is calculated in a manner similar to that described in Section 5.4.4, except that the value for \(b_0\) is calculated before the temperatures at the end of the step are calculated, so \(q\) is calculated on the basis of temperatures at the beginning of the time step. In the present version of the code, the reference mass \(m_{\text{r}}\) and the reference temperature \(T_{\text{r}}\) are taken as the mass and temperature at the beginning of the time step.