.. _section-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 :math:`V` is assumed to vary
linearly with pressure p and temperature :math:`T`:

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

.. math::
    :label: (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

:math:`\alpha_{\text{p}}` = the volume pressure expansion coefficient,
:math:`\frac{1}{V} \frac{\partial \text{V}}{\partial \text{p}}`

:math:`\alpha_{\text{T}}` = the volume thermal expansion coefficient,
:math:`\frac{1}{V} \frac{\partial \text{V}}{\partial \text{T}}`

:math:`\beta_{\text{p}}` = the sodium compressibility,
:math:`\frac{1}{\rho} \frac{\partial \mathrm{\rho}}{\partial \text{p}}`

:math:`\beta_{\text{T}}` = the sodium thermal expansion coefficient,
:math:`\frac{1}{\rho} \frac{\partial \mathrm{\rho}}{\partial \text{T}}`

The mass of the liquid in the compressible volume is

.. math::
    :label: (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

.. math::
    :label: (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

.. math::
    :label: (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

:math:`\delta m = m - m_{\text{r}}`

:math:`\delta p = p - p_{\text{r}}`

:math:`\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 :math:`b_0 \left( j \right)`, :math:`b_1 \left( j \right)`,
and :math:`b_2 \left( j \right)`, which characterize this compressible volume
:math:`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

.. math::
    :label: (5.3-6)

    \Delta m = \Delta t \left\lbrack \sum{{\overline{w}}_{\text{in}} - \sum{\overline{w}}_{\text{out}}} \right\rbrack

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

Conservation of energy gives

.. math::
    :label: (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 :math:`m_3` and :math:`T_3` are the mass and temperature of
the liquid in the compressible volume at the beginning of the time
step, :math:`\Delta T` is the change in temperature of the liquid in the
compressible volume during the time step,
:math:`\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,
:math:`\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, :math:`Q` is the heat flow rate from the
compressible volume walls and from other components in contact with the
compressible volume liquid, and :math:`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

.. math::
    :label: (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 :math:`m_3 + \Delta m` in the denominator is
approximated as :math:`m_3`. Inserting :eq:`(5.3-6)` and :eq:`(5.3-8)` into :eq:`(5.3-5)` gives

.. math::
    :label: (5.3-9)

    \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

Comparison of this equation with :eq:`5.2-19` shows that for the
compressible volume with no cover gas

.. math::
    :label: (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}}}

.. math::
    :label: (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

.. math::
    :label: (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 :math:`Q \Delta t` term in :eq:`(5.3-10)` is calculated in a manner similar to
that described in :numref:`section-5.4.4`, except that the value for :math:`b_0`
is calculated before the temperatures at the end of the step are
calculated, so :math:`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 :math:`m_{\text{r}}` and the reference temperature :math:`T_{\text{r}}`
are taken as the mass and temperature at the beginning of the time step.