.. _section-16.2:

In-pin Hydrodynamic Model
-------------------------

.. _section-16.2.1:

Physical Models
~~~~~~~~~~~~~~~

The in-pin hydrodynamic model describes the motion of the molten fuel
and fission gas mixture in the cavity formed inside the fuel pins during
a loss of flow accident. As the accident proceeds, the size of the
cavity increases, both radially and axially (:numref:`figure-16.1-2`). Newly molten
fuel and fission gas are added to the moving components in the cavity.
Some of the fission gas is dissolved in the molten fuel, in the form of
small bubbles constrained by surface tension. The effect of this fission
gas is controlled by the input variable PRSFTN. If PRSFTN is less than
:math:`10^{7}`, the volume of the dissolved gas is assumed to be
negligible, and thus it does not contribute immediately to the cavity
pressure. If :math:`\text{PRSFTN} > 10^{7}`, the volume occupied by the dissolved
gas is taken into account in the cavity pressure calculation, as
described in :numref:`section-14.2.6`. The remainder of the fission gas is in the
form of free gas, residing in bubbles that are too large to be
constrained by surface tension. This gas contributes immediately to the
pressurization of the cavity. Because of the continuous coalescence of
the small bubbles, leading to the formation of new larger bubbles, the
originally dissolved gas is continuously released from the molten fuel
and is added to the free fission gas. The continuous heating of the
molten fuel and fission gas leads to the pressurization of the cavity
and eventually to the cladding failure. The fuel and fission gas in the
vicinity of the failure location are ejected into the coolant channel
leading to a local depressurization of the cavity. This depressurization
causes the fuel motion inside the pin toward the failure location.

The in-pin fuel motion is treated as a one-dimensional, compressible
flow with a variable flow cross section. The fuel and fission gas are
assumed to form a homogeneous mixture in thermal equilibrium. However,
if the local fuel volume fraction is less than an input value FNFUAN,
the pressure gradient is assumed to act only on the fuel cross section.
This is an attempt to roughly account for the annular fuel flow regime
which may exist for large void fractions. Fuel vapor pressures are also
included in the in-pin hydrodynamic model. The fuel vapor pressure is
based on the average fuel temperature in any axial cell, with the
assumption that no significant radial temperature profiles can be
maintained after the onset of fuel motion.

As the cladding heats up and the in-pin pressures increase, the cladding
rip can propagate from the original location, allowing the ejection of
fuel and fission gas from the cavity into the channel at new axial
locations. The fuel pin can also be totally disrupted at certain axial
locations where the cladding becomes very weak and the solid pin is
largely molten. The pin disruption leads to the formation of upper and
lower pin stubs with cavities that can eject fuel axially into the
disrupted region.

.. _section-16.2.2:

Method of Solution and General Numerical Considerations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

.. _section-16.2.2.1:

Variables and Mesh Grid Used in Calculations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The independent variables used in the in-pin model are the axial
coordinate :math:`z` and the time :math:`t`. Only one spatial coordinate is necessary,
as LEVITATE models the pin cavity in a one-dimensional geometry. The
dependent variables calculated by the in-pin hydrodynamic model for each
component are the generalized density :math:`{\rho'}`, the enthalpy h (or temperature
T), and the velocity u. The generalized densities have been introduced
in :numref:`Chapter %s<section-14>` and, for the component j in the cavity, are defined as
follows:

(16.2-1)

.. _eq-16.2-1:

.. math::

	{\rho'}_{\text{j,ca,k}} = \rho_{\text{j,ca,k}} \cdot \frac{A_{\text{j,ca,k}}}{\text{AXMX}} \
	= \rho_{\text{j,ca,k}} \cdot \theta_{\text{j,ca,k}}

where

:math:`\rho_{\text{j,ca,k}}` is the physical density of component :math:`j` at the axial
location :math:`k` in the pin cavity

:math:`A_{\text{j,ca,k}}` is the cross sectional area occupied by component :math:`j` at
location :math:`k` in the pin cavity. This area refers to all the pins in the
subassembly.

:math:`\text{AXMX}` is the reference input area

:math:`\theta_{\text{j,ca,k}}` is the generalized volume (or area) fraction of
component :math:`j` at location :math:`k` in the pin cavity

The mass, energy and momentum partial differential equations are solved
using the Eulerian finite difference semi-explicit formulation, as
explained in :numref:`section-16.2.2.2`. A staggered mesh grid is used to obtain
the numerical formulation, with the densities and enthalpies defined at
the center of each cell while the velocities are defined at the
boundaries. Variable flow areas are treated in the in-pin model.
However, the cavity geometry is not as irregular as the coolant channel
geometry, and the conventional single velocities were used at the
boundaries, as opposed to the dual velocities used in the coolant
channel model (see :numref:`section-16.4.2.1`). The components treated in the
cavity model are the molten fuel, the dissolved fission gas and the free
fission gas. They are assumed to form a homogeneous mixture and move
with the same velocity at all axial locations. A full donor cell
formulation was used in the finite difference formulation in order to
improve the stability characteristics of the solution.

.. _section-16.2.2.2:

Description of the Method of Solution and Logic Flow
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

The in-pin hydrodynamic model is solved in the routines LE1PIN and
LE2PIN. First, the cavity enlargement is calculated in LE1PIN, using the
solid fuel temperatures provided by PLHTR. If melting occurs, the
quantities of molten fuel and fission gas to be added to the moving
components are determined. Then the mass conservation equation is solved
for all axial cells, accounting for the changes in fuel and free
fission-gas mass due to convection and melt-in. LE1PIN then solves the
energy conservation equations for the fuel and fission-gas mixture. It
is assumed that the fuel and fission gas remain at the same temperature
in all the axial cells. Using the new masses and temperatures, the new
pressures are then calculated. It is noteworthy that at this point the
pressures have also been updated in the channel (see :numref:`section-16.4.2.2`)
so that we can use a consistent set of pressures for the ejection
calculation, which is the next step in the LE1PIN routine. The ejection
calculation leads to changes in the mass and pressure in all nodes that
are ejecting fuel and fission gas into the coolant channel. The ejection
can take place radially, via the cladding rip or axially, via the open
ends of the fuel-pin stubs when the fuel-pin disruption has already
occurred. Both modes of ejection can be present simultaneously. The
LE1PIN routine also examines the possibility of pin disruption. If an
axial cell is found which has to be disrupted, a flag is set
(IDISR(I)=9), but no other changes are preformed. The actual pin
disruption is performed in the routine LEDISR, as described in :numref:`section-16.5`.

The routine LE2PIN then solves the momentum conservation equation for
all cells, obtaining the new velocities at the end of the time step.
These velocities are obtained by using the new pressures calculated in
LE1PIN, and in this sense, the method of solution is mixed,
explicit-implicit, rather than purely explicit. The routine LE2PIN also
solves the mass conservation equations for the dissolved gas and
calculates the maximum time step acceptable for the in-pin hydrodynamic
model in the next computational cycle.

.. _section-16.2.3:

Finite Difference Forms and Solution Technique, Special Situations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The equations describing the in-pin hydrodynamic and thermal process are
solved in the LE1PIN and LE2PIN routines. The partial differential
equations, as well as the finite difference formulation, are generally
the same in LEVITATE and PLUTO2. The reader is referred to the PLUTO2
chapter (:numref:`section-14.2`) for a detailed description of the equations. Only
the features of the in-pin model that are specific to LEVITATE will be
discussed here.

The main feature of LEVITATE is the fuel-pin disruption mode. The
decision for disruption of a certain axial pin cell is made in the
routine LE1PIN. An undisrupted cell will be disrupted if the molten fuel
cavity covers a large fraction of the original pin cross section.

(16.2-2)

.. _eq-16.2-2:

.. math::

	\frac{R_{\text{ca,k}}}{R_{\text{pin,os,k}}} > \text{FNDISR}

and the cladding is molten or close to melting, i.e.,

(16.2-3)

.. _eq-16.2-3:

.. math::

	T_{\text{cl,os,i}} > T_{\text{se,so}}

and:

(16.2-4)

.. _eq-16.2-4:

.. math::

	\begin{align}
	T_{\text{cl,in,i}} > T_{\text{se,so}} - 50 && \text{if } \Delta R_{\text{cl,k}} > 0.5 \cdot \Delta R_{\text{cl}}^{0}
	\end{align}

or:

(16.2-5)

.. _eq-16.2-5:

.. math::

	\begin{align}
	T_{\text{cl,in,i}} > T_{\text{se,so}} - 150 \text{if } 0 < \Delta R_{\text{cl,k}} < 0.5 \Delta R_{\text{cl}}^{0}
	\end{align}

The disruption of one or more axial pin nodes leads to the formation of
a disrupted region, extending from the cell IDISBT to IDISTP. Note that
the axial position of the in-pin cells is denoted in Eq. :ref:`16.2-1<eq-16.2-1>` by the
subscript :math:`k`, while the subscript used in the channel is :math:`i`. The
correspondence between :math:`i` and :math:`k` is given below:

(16.2-6)

.. _eq-16.2-6:

.. math::

	i = k + \text{IDIFF}

that is, the in-pin cell :math:`k` will have the same axial location as
the channel cell :math:`k + \text{IDIFF}`.

The motion of the material present in this region is calculated by the
coolant channel hydrodynamic model. In the disrupted region, the coolant
channel covers the entire cross sectional area of the subassembly. Only
one disrupted region is allowed in LEVITATE. Thus, if two or more
disjoint disrupted regions appear at any given time (e.g., cells 8 and
10 are disrupted, but not 9), the undisrupted nodes between the regions
will also be forced to disrupt. In this case, a message is printed
indicating that one or more nodes have been disrupted due to the
disruption of neighboring nodes and the formation of large fuel/steel
chunks is likely to occur. Only the decision about disruption is made
LE1PIN. The nodes to be disrupted are flagged (:math:`\text{IDISR} \left( I \right) = 9`), but the
disruption process will only be modeled later in the cycle, in the
routine LEDISR. This process is described in :numref:`section-16.3`.

After the occurrence of the fuel-pin disruption, the model describing
the hydrodynamics of the molten fuel/fission gas in the pin cavity
continues to operate. However, instead of modeling a continuous channel,
it now describes the hydrodynamic behavior of two disjoint channels,
i.e., the cavities remaining in the upper and lower undisrupted pin
stubs. These cavities communicate directly with the channel via the open
ends of the pin stubs. The channel pressure in front of these open ends
is used as the boundary condition for the in-pin hydrodynamic model.