.. _section-12.3:

Interface Velocities
--------------------

The average velocity in a liquid slug is

(12.3‑1)

.. _eq-12.3-1:

.. math:: v_{l} = \frac{W}{\rho_{l}A_{c}}

but the presence of films on the cladding and structure causes the
liquid-vapor interface to move at a somewhat different velocity. :numref:`figure-12.3-1` shows an interface moving at a velocity :math:`v_{i}` for a time
interval :math:`\Delta t`. A film of liquid sodium is present in the vapor region
on both the cladding and structure. The cladding film is of thickness
:math:`w_{fe}`, while that on the structure has thickness :math:`w_{fs}`. The film on
the cladding can move with velocity :math:`v_{fe}`, and the structure film can
have a velocity of :math:`v_{fs}`. In later versions of SASSYS-1, film motion
modeling will be available, and so :math:`v_{fe}` and :math:`v_{fs}` are included in
the equations below for completeness; currently, however, film motion is
neglected, and :math:`v_{fe}` and :math:`v_{fs}` are set to zero.

If the coolant channel volume between :math:`z_{i}\left( t \right)` and
:math:`z_{i}\left(t + \Delta t \right)` is taken as a control volume, then the liquid
volume :math:`V_{l}` in the control volume at time :math:`t` is

(12.3‑2)

.. _eq-12.3-2:

.. math:: V_{l}\left( t \right) = P_{e}v_{i}\Delta t(w_{fe} + \gamma_{2}w_{fs})

where :math:`P_{e}` is the outer perimeter of the cladding and :math:`\gamma_{2}` is
the ratio of the surface area of the structure to the surface area of
the cladding. At :math:`t + \Delta t` the liquid volume in the control volume is

(12.3‑3)

.. _eq-12.3-3:

.. math:: V_{l}\left( t + \Delta t \right) = A_{c}v_{i}\Delta t

Accounting for the liquid added to and subtracted from the control
volume during :math:`\Delta t` gives

(12.3‑4)

.. _eq-12.3-4:

.. math:: V_{l}\left( t + \Delta t \right) = V_{l}\left( t \right) + v_{l}A_{c}\Delta t - P_{e}\left( w_{fe}v_{fe} + \gamma_{2}w_{fs}v_{fs} \right)\Delta t

Substituting Eqs. :ref:`12.3-2 <eq-12.3-2>` and :ref:`12.3-3 <eq-12.3-2>`
into :ref:`12.3-4 <eq-12.3-4>` gives

(12.3‑5)

.. _eq-12.3-5:

.. math:: A_{c}v_{i}\Delta t = P_{e}v_{i}\Delta t\left( w_{fe} + \gamma_{2}w_{fs} \right) + v_{l}A_{c}\Delta t - P_{e}\Delta t(w_{fe}v_{fe} + \gamma_{2}w_{fs}v_{fs})

or

(12.3‑6)

.. _eq-12.3-6:

.. math:: v_{i} = \frac{v_{l} - P_{e}\left( w_{fe}v_{fe} + \gamma_{2}w_{fs}v_{fs} \right)/A_{c}\ }{1 - P_{e}(w_{fe} + \gamma_{2}w_{fs})/A_{c}}`

which is the equation used for computing interface velocities. Note that
this equation is applicable whether the interface is moving in the
positive or the negative axial direction and holds whether the interface
is moving towards the liquid side or the vapor side.

.. _figure-12.3-1:

.. figure:: media/Figure12.3-1.jpeg
	:align: center
	:figclass: align-center

	Geometry Associated with the Liquid-Vapor Interface Velocity
	Calculation