5.3.2. Compressible Volumes With Cover Gas

A compressible volume with cover gas is treated in a fashion similar to that without cover gas except that the compression of the liquid is neglected compared with the compression of the gas, so that all of the expansion or compression is attributed entirely to the gas. The cover gas is assumed to expand or compress adiabatically, and an increase in the gas volume is equal to the decrease in the liquid volume. The accompanying rise or fall in the level of the gas-liquid interface is taken as the volume change divided by the area of the compressible volume. In addition, the conservation of mass and the conservation of energy principles are observed. The conservation of mass is applied by taking the increase in the mass of the liquid in a compressible volume as the difference between the liquid flowing in and that flowing out during a time step, and the conservation of energy is taken as the increase in the mass of the liquid in the compressible volume times its temperature as the difference between the mass times temperature flowing in and that flowing out during a time step.

The liquid pressure at an elevation, \(z_{\text{r}}\), in the compressible volume is given by

(5.3-13)\[p_{\text{l}} = p_{\text{g}} + \rho_{\text{l}}g \left( z_{\text{i}} - z_{\text{r}} \right)\]

where

\(p_{\text{l}}\) = the pressure in the liquid

\(p_{\text{g}}\) = the pressure of the cover gas above the liquid

\(\rho_{\text{l}}\) = the liquid density

\(g\) = the acceleration of gravity

\(z_{\text{i}}\) = the height of the liquid gas interface

\(z_{\text{r}}\) = the reference height for the compressible volume

and the change in the liquid pressure in the compressible volume is obtained by taking differentials of Eq. (5.3-13):

(5.3-14)\[\Delta p_{\text{l}} = \Delta p_{\text{g}} + g\left( z_{\text{i}} - z_{\text{r}} \right) \Delta\rho_{\text{l}} + \rho_{\text{l}} g \Delta z_{\text{i}}\]

Adiabatic compression of the cover gas is taken as

(5.3-15)\[p_{\text{g}} V_{\text{g}}^{\gamma} = \text{const}\]

where

\(V_{\text{g}}\) = the volume of the cover gas

\(\gamma\) = the ratio of the specific heat at constant pressure to that at constant volume for the cover gas

and in differential form becomes:

(5.3-16)\[\frac{\Delta p_{\text{g}}}{p_{\text{g}}} + \gamma \frac{\Delta V_{\text{g}}}{V_{\text{g}}} = 0\]

The conservation of liquid mass for a compressible volume gives

(5.3-17)\[\Delta m_{\text{l}} = \Delta t \left\lbrack \sum{{\overline{w}}_{\text{in}} - \sum{\overline{w}}_{\text{out}}} \right\rbrack\]

where

\(\Delta m_{\text{l}}\) = the liquid mass increase in the compressible volume during the time step

\(\Delta t\) = the time-step size

\(\sum{\overline{w}}_{\text{in}}\) = the sum of the average liquid mass flow rates into the compressible volumes during the time step

\(\sum{\overline{w}}_{\text{out}}\) = the sum of the average liquid mass flow rates out from the compressible volume during the time step.

The conservation of energy for a compressible volume yields

(5.3-18)\[\left( m_{\text{l}} + \Delta m_{\text{l}} \right) \left( T_{\text{l}} + \Delta T_{\text{l}} \right) = m_{\text{l}} T_{\text{l}} + \Delta t \left\lbrack \sum{{\overline{w}}_{\text{in}} T_{\text{in}} - \sum{{\overline{w}}_{\text{out}} T_{\text{out}} }} \right\rbrack\]

where

\(m_{\text{l}}\) = the liquid mass at the beginning of the time step

\(\Delta T_{\text{l}}\) = the increase in liquid temperature in the compressible volume during the time step

\(\sum{{\overline{w}}_{\text{in}} T_{\text{in}}}\) = the sum of the average liquid mass flow rates times temperatures entering the compressible volume during the time step

\(\sum{\overline{w}}_{\text{out}} T_{\text{out}}\) = sum of the average liquid mass flow rates times temperatures leaving the compressible volume during the time step.

In the present version of the code, \(\Delta m_{\text{l}}\) is neglected in comparison with \(m\), giving

(5.3-19)\[\Delta T_{\text{l}} = \frac{\Delta t}{m_{\text{l}}} \left\lbrack \sum{{\overline{w}}_{\text{in}} T_{\text{in}} - \sum{{\overline{w}}_{\text{out}} T_{\text{out}}}} \right\rbrack\]

In addition to the above equations, we take

(5.3-20)\[\Delta V_{\text{g}} = - \Delta V_{\text{l}}\]

(5.3-21)\[V_{\text{l}} = \frac{m_{\text{l}}}{\rho_{\text{l}}}\]

(5.3-22)\[\Delta z_{\text{i}} = \frac{\Delta V_{\text{l}}}{A}\]

where

\(V_{\text{l}}\) = the volume of the liquid in the compressible volume at the beginning of a time step

\(m_{\text{l}}\) = the mass of the liquid in the compressible volume at the beginning of a time step

\(\rho_{\text{l}}\) = the density of the liquid

\(A\) = the area of the compressible volume.

Differencing Eq. (5.3-21), we have

(5.3-23)\[\Delta V_{\text{l}} = \frac{\Delta m_{\text{l}}}{\rho _{\text{l}}} - \frac{m_{\text{l}}}{\rho_{\text{l}}^{2}} \Delta \rho_{\text{l}}\]

and taking

(5.3-24)\[\Delta \rho_{\text{l}} = \frac{\partial \mathrm{\rho}}{\partial \text{T}} \Delta T_{\text{l}}\]

we can write the following expression for the change in the liquid pressure during a time step as

(5.3-25)\[\Delta p_{\text{l}} = \left( \gamma \frac{p_{\text{g}}}{V_{\text{g}}} + \frac{\rho_{\text{l}}g}{A} \right) \left( \frac{\Delta m_{\text{l}}}{\rho _{\text{l}}} - \frac{m_{\text{l}}}{\rho _{\text{l}}^{2}} \frac{\partial\rho}{\partial \text{T}} \Delta T_{\text{l}} \right) + g\left( z_{\text{i}} - z_{\text{r}} \right) \frac{\partial\rho}{\partial \text{T}} \Delta T_{\text{l}}\]

Inserting Eq. (5.3-17) for \(\Delta m_{\text{l}}\)and Eq. (5.3-19) for \(\Delta T_{\text{l}}\) and then comparing with Eq. 5.2-19, we see that the values of the \(b\)’s for a compressible volume with a cover gas are

(5.3-26)\[b_{0} = 0\]

(5.3-27)\[b_{1} = \Delta t \left[ \frac{\gamma P_{\text{g}}}{V_{\text{g}}\rho } + \frac{g}{A} \right]\]

(5.3-28)\[ b_{2} = \Delta t \frac{\partial\rho}{\partial \text{T}} \left\lbrack \frac{g\left( z_{\text{i}} - z_{\text{r}} \right)}{m_{\text{l}}} - \frac{1}{\rho} \left( \frac{\gamma p_{\text{g}}}{V_{\text{g}}\rho} + \frac{g}{A} \right) \right\rbrack\]