.. _section-16.5:
Freezing, Melting and Heat-transfer Processes Related to Stationary Pin Stubs and Hexcan Wall
---------------------------------------------------------------------------------------------
.. _section-16.5.1:
Physical Models
~~~~~~~~~~~~~~~
The models presented in this section describe a series of physical
processes related to the stationary pin stubs and the hexcan wall.
Except for the heat transfer in the cladding and in the hexcan wall, the
processes described lead to geometry changes that affect directly the
hydrodynamic model of the coolant channel.
One of the most important phenomena during a LOF accident is the
freezing of the initially molten fuel and the formation of the fuel
crusts. The molten fuel ejected from the pin cavity is accelerated in
the coolant channel by the local pressure gradients and begins to move
toward the extremities of the channel. In the process, however, it
exchanges heat with the sodium, cladding and structure, all of which
have a temperature well below the fuel freezing temperature. Eventually
the fuel will begin to freeze, forming stationary fuel crusts on the
cladding and/or structure or leading to the formation of solid fuel
chunks which continue to move in the channel. Steel freezing can also
occur occasionally, particularly at locations where the molten steel is
the dominant component and is in contact with cold cladding or
structure. The freezing of steel leads, in LEVITATE, to a local increase
in the thickness of the cladding and/or structure, rather than to the
formation of a distinct steel crust.
The fuel crusts formed at various axial locations can begin to melt if
the power level increases, thus releasing the fuel again and allowing it
to move in the coolant channel. Another mechanism which could also
release the fuel crust is the crust breakup, which can occur when the
underlying steel support melts and the crust becomes unstable. The
continuous heating of the cladding and hexcan wall leads eventually to
steel melting. The molten steel is ablated, becoming part of the moving
components in the channel, and the thickness of the cladding and/or
structure is reduced, increasing the flow area of the coolant channel.
In a similar manner, the solid fuel and steel chunks generated via the
fuel-pin disruption can begin to melt due to the direct heating and heat
exchange with other components. This will result in a transfer of fuel
from the chunk field to the molten fuel field in the coolant channel.
This transfer of mass is accompanied by a transfer of energy and
momentum as well as a change in the geometry of the solid chunks.
.. _section-16.5.2:
Description of the Method of Solution and Logic Flow
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The heat-transfer equations in the cladding and structure, which are not
directly related to a geometry change, are solved in the LESOEN routine,
i.e., while advancing the solution for the hydrodynamic model in the
channel. All other processes related to the stationary pin stubs and to
the hexcan wall involve a change in the local geometry and the routines
describing them are called after completing the solution for both the
in-pin and coolant channel hydrodynamic models. All melting/freezing
processes modeled here affect the hydrodynamic parameters of the coolant
channel. Changes in the local temperatures, velocities, or pressures are
performed in each of the routines described below, whenever necessary.
The routine LELUME (**LE**\ VITATE CHUNK-\ **LU** **ME**\ LTING) calculates
the melting and the size changes of the solid fuel/steel chunks at all
axial locations. The routine LEFREZ (**LE**\ VITATE **FR**\ EE\ **Z**\ ING AND
MELTING) then models a series of important processes, such as fuel/steel
freezing and crust formation, fuel/steel chunk formation, fuel crust
remelting and breakup. Next called is the routine LEABLA (**LE**\ VITATE
ABLATION) which calculates the gradual melting and ablation of the
cladding and hexcan wall. The routine LEDISR (**LE**\ VITATE
**DISR**\ UPTION) performs the disruption of the fuel pin, whenever a
disrupt node is predicted. The disruption, which can occur in one or
more nodes in any time step, leads to changes in geometry, mass, energy
and pressure for various components present in the respective cell.
Finally, the routine LESRME (**LE**\ VITATE **S**\ T\ **R**\ UCTURE
**ME**\ LTING) calculates the rupture of the hexcan wall due to melting
and/or pressure burst effects.
.. _section-16.5.3:
Fuel/steel Freezing and Crust Formation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The calculation of fuel/steel freezing is performed in the routine
LEFREZ. The fuel flow regimes lead to a situation very different from
the steel flow regimes in terms of freezing and are treated separately.
.. _section-16.5.3.1:
The Freezing Process when a Fuel Flow Regime is Present
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In this case, the molten steel component exists in the form of droplets
imbedded in the molten fuel and has a temperature well above freezing.
Thus, only the freezing of the molten fuel must be considered. The
freezing calculation decides what amount of fuel, if any, will freeze in
each node during the current time step. It also has to decide where the
frozen fuel will go, when leaving the moving molten-fuel field. Three
possibilities exist:
1. The freezing fuel will form (or add to) a crust on the cladding;
(when the cladding has been completely ablated, the fuel can freeze
on the bare fuel pins).
2. The freezing fuel will form (or add to) a crust on the hexcan wall
(:numref:`figure-16.5-1`).
3. The freezing fuel will generate solid fuel chunks, which will be
added to the chunk field
The freezing calculation begins by examining the enthalpy of the molten
fuel:
1. If :math:`h_{\text{fu,i}} > h_{\text{fu,freeze}}`, no fuel freezing occurs. The
enthalpy :math:`h_{\text{fu,freeze}}` is an input parameter, i.e.,
:math:`h_{\text{fu,freeze}} = \text{EGBBLY}`. It has to satisfy the conditions:
.. math::
:label: 16.5-1
h_{\text{fu,so}} < h_{\text{fu,freeze}} < h_{\text{fu,liq}}
1. If :math:`h_{\text{fu,i}} < h_{\text{fu,so}}`, rapid fuel freezing occurs,
leading to the formation of solid fuel chunks only. In order to avoid
numerical problems, only one tenth of the fuel mass
:math:`\left( 0.1 \cdot \rho_{\text{fu,i}} \cdot A_{\text{fu,i}} \cdot \Delta z_{\text{i}} \right)`
in the cell is allowed to freeze in each time step under these circumstances.
2. If :math:`h_{\text{fu,so}} < h_{\text{fu,i}} < h_{\text{fu,freeze}}`, only
partial fuel freezing is allowed to occur. The amount of freezing
fuel is determined as follows:
.. math::
:label: 16.5-2
\Delta {\rho'}_{\text{fu,i}} = {\rho'}_{\text{fu,i}} \cdot \frac{h_{\text{fu,freeze}} \
- h_{\text{fu,i}}}{h_{\text{fu,freeze}} - h_{\text{fu,so}}}
The area occupied by this newly frozen fuel is:
.. math::
:label: 16.5-3
\Delta A_{\text{fu,i}} = \frac{\Delta{\rho'}_{\text{fu,i}}}{\rho_{\text{fu,so}}} \cdot \text{AXMX}
The assumption is made that the fuel looses most of he energy at the
channel boundaries, i.e., by exchanging heat with the cladding and
hexcan wall. Thus, the frozen fuel is first used to form a fuel crust on
the solid boundaries. The distribution of :math:`\Delta {\rho'}_{\text{fu,i}}` between the
clad and hexcan wall is made as follow:
.. math::
:label: 16.5-4
\Delta {\rho'}_{\text{fu,cl,i}} = \Delta {\rho'}_{\text{fu,i}} \cdot \frac{\text{WTCL}}{\text{WTCL} + \text{WTSR}}
.. math::
:label: 16.5-5
\Delta {\rho'}_{\text{fu,sr,i}} = \Delta {\rho'}_{\text{fu,i}} \cdot \frac{\text{WTSR}}{\text{WTCL} + \text{WTSR}}
where the weights WTCL and WTSR are:
.. math::
:label: 16.5-6
\text{WTCL} = A_{\text{cl,i}}^{L} \cdot \left\lbrack \left( T_{\text{fu,i}} - T_{\text{cl,i}} \right) \cdot C_{\text{mfu,cl,i}} \
+ \left( T_{\text{fu,i}} - T_{\text{ffc,i}} \right) \cdot C_{\text{ff,cl,i}} \right\rbrack
.. math::
:label: 16.5-7
\text{WTSR} = A_{\text{sr,i}}^{L} \cdot \left\lbrack \left( T_{\text{fu,i}} - T_{\text{sr,i}} \right) \cdot C_{\text{mfu,sr,i}} \
+ \left( T_{\text{fu,i}} - T_{\text{ffs,i}} \right) \cdot C_{\text{ff,sr,i}} \right\rbrack
Occasionally, one or both of these weights can be set to zero. This
happens for :math:`\text{WTCL}` when the cladding surface is molten and thus no solid
support for freezing exists or when the pins are totally disrupted in
the cell considered. In these cases, :math:`\text{WTCL} = 0`.
Similarly, :math:`\text{WTSR} = 0` whenever the hexcan wall surface is molten and no
crust formation can occur. In this case, we will still use :eq:`16.5-4`
and 16.5-5 to calculate the distribution of the frozen fuel between
cladding and hexcan wall. The situation can arise, however, where the
amount of frozen fuel is too large for the available freezing area. For
example, let us assume that the cladding surface is molten and :math:`\text{WTCL} = 0`.
In this case, we can have:
.. math::
:label: 16.5-8
\Delta A_{\text{fu,i}} > A_{\text{ch,op,sr}}
where
.. math::
:label: 16.5-9
A_{\text{ch,op,sr}} = A_{\text{ch,sr}} - A_{\text{ff,sr}}
.. math::
:label: 16.5-10
A_{\text{ff,sr}} = L_{\text{ff,sr}} \cdot l_{\text{ff,sr}}
.. _figure-16.5-1:
.. figure:: media/image27.png
:align: center
:figclass: align-center
Freezing, Melting and Crust Breakup Processes
and the elements used in :eq:`16.5-7` and :eq:`16.5-8` have been defined in
:numref:`section-16.4.3.9`. If :eq:`16.5-6` is satisfied, only an amount of fuel
consistent with the available area is allowed to freeze on the
structure, while the remaining fuel generates solid chunks:
.. math::
:label: 16.5-11
\Delta {\rho'}_{\text{fu,sr,i}} = \frac{\rho_{\text{fu,so}} \cdot A_{\text{ch,op,sr}}}{\text{AXMX}}
.. math::
:label: 16.5-12
\Delta {\rho'}_{\text{fu,lu,i}} = \Delta{\rho'}_{\text{fu,i}} - \Delta {\rho'}_{\text{fu,sr,i}}
When both :math:`\text{WTCL} = 0` and :math:`\text{WSTR} = 0` all the freezing fuel is used to
generate solid fuel chunks:
.. math::
:label: 16.5-13
\Delta {\rho'}_{\text{fu,} l \text{u,i}} = \Delta {\rho'}_{\text{fu,i}}
Another decision that has to be made in connection with the fuel
freezing is the steel entrapment in the frozen fuel. Based on
experimental evidence obtained in the posttest examination of several
fuel motion tests, it appears that only small steel droplets are trapped
in the frozen fuel. LEVITATE assumes that small droplets exist only when
small amounts of steel are locally present. Otherwise, the steel exists
in large droplets that will tend to separate from the freezing fuel.
Thus, if:
.. math::
:label: 16.5-14
\frac{\theta_{\text{se,i}}}{\theta_{\text{fu,i}}} > 0.1
no steel entrapment occurs. Otherwise, the amount of steel trapped in
the frozen fuel is calculated from:
.. math::
:label: 16.5-15
\Delta {\rho'}_{\text{se,i}} = {\rho'}_{\text{se,i}} \cdot \frac{\Delta {\rho'}_{\text{fu,i}}}{{\rho'}_{\text{fu,i}}}
The actual geometry of the fuel crust (i.e. thickness and area
coefficient) is not changed in LEFREZ when fuel freezing occurs. The
geometry is changed in LEGEOM at the beginning of the next time step.
The geometry of the chunks is changed in LEFREZ due to the condition of
the new chunks. The radius of the resulting chunks is a mass-weighted
average of the chunks being combined. A more detailed discussion about
the chunk geometry can be found in :numref:`section-16.4.3.9`.
.. _section-16.5.3.2:
The Freezing Process when a Steel Flow Regime is Present
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In a steel flow regime, both steel freezing and fuel freezing can occur.
The molten fuel is in the form of droplets imbedded in the molten steel
and can freeze as it approaches the steel temperature. The amount of
fuel freezing in cell :math:`i`, :math:`\Delta {\rho'}_{\text{fu,i}}` is calculated using the same
procedure outline in :numref:`section-16.5.3.1`. However, because no direct
contact between the freezing fuel and the channel boundaries exists,
crust formation cannot occur. All freezing fuel is used to generate
solid fuel chunks:
.. math::
:label: 16.5-16
\Delta {\rho'}_{\text{fu,} l \text{u,i}} = \Delta {\rho'}_{\text{fu,i}}
The steel-freezing calculation follows the same lines as the fuel
calculation, but is keyed to the steel temperature rather than enthalpy:
1. If :math:`T_{\text{se,i}} > T_{\text{se,freeze}}`, no steel freezing occurs.
The temperature :math:`T_{\text{se,freeze}}` is determined using the input
parameter :math:`\text{FRMRSE}`:
.. math::
:label: 16.5-17
T_{\text{se,freeze}} = T_{\text{se,so}} + \left( T_{\text{se,} l \text{q}} - T_{\text{se,so}} \right) \cdot \text{FRMRSE}
1. If :math:`T_{\text{se,i}} < T_{\text{se,so}}`, rapid steel freezing occurs,
leading to the formation of steel chunks only.
2. If :math:`T_{\text{se,so}} < T_{\text{se,i}} < T_{\text{se,freeze}}`, only
partial steel freezing is allowed to occur. The amount of freezing
steel is calculated from:
.. math::
:label: 16.5-18
\Delta {\rho'}_{\text{se,i}} = {\rho'}_{\text{se,i}} \cdot \frac{T_{\text{se,freeze}} - T_{\text{se,i}}}{T_{\text{se,freeze}} - T_{\text{se,so}}}
The steel can freeze on the cladding and/or on the hexcan wall, leading
to an increase in the thickness of these structures. This increase is
uniformly distributed along the perimeter, as opposed to the partial
crust formed initially in the fuel flow regimes (:numref:`figure-16.5-1`). The
distribution of :math:`\Delta {\rho'}_{\text{sei}}`, follows the same procedure presented in
:numref:`section-16.5.3` for fuel. If not all the freezing steel can be
distributed between cladding and structure (e.g., in a cell where the
pins have been disrupted and/or the hexcan wall surface is molten), part
or all of the freezing steel is used to generate solid steel chunks, as
appropriate.
.. _section-16.5.4:
Fuel Crust Breakup and Remelting
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The fuel crust formation on the cladding and the fuel crust formed on
the hexcan wall are completely independent. Either one can be present in
any axial cell, having its own temperature and geometry. The formation
of the crusts was explained in :numref:`section-16.5.3`. Once present, the fuel
crust can disappear in two ways: via breakup and via remelting. These
processes can occur in any flow regime modeled by LEVITATE.
.. _section-16.5.4.1:
Fuel Crust Breakup
^^^^^^^^^^^^^^^^^^
It is assumed that the frozen fuel crust is unstable and is allowed to
break up whenever there is no underlying solid support. In general, this
means that the underlying steel is molten. However, when fuel crusts are
present on bare fuel pins, the underlying support is also fuel. Thus,
the breakup of the crust on the clad will occur when:
.. math::
:label: 16.5-19
T_{\text{cl,os}} > T_{\text{se,} l \text{q}}
In this case, a gradual but fairly rapid breakup occurs. The crust is
reduced each time step by:
.. math::
:label: 16.5-20
\Delta {\rho'}_{\text{ffc}} = {\rho'}_{\text{ffc}} \cdot \text{CIBREK}
where :math:`\text{CIBREK} = 0.01` is a built-in constant, defined in the routine
:math:`\text{LEFREZ}`. Both the thickness :math:`1_{\text{ff,cl,i}}` and the length (i.e., the
area coefficient :math:`C_{\text{ff,c} l \text{,i}}`) are reduced proportionately:
.. math::
:label: 16.5-21
1_{\text{ff,cl,i}} = 1_{\text{ff,cl,i}} \cdot \sqrt{1 - \text{CIBREK}}
.. math::
:label: 16.5-22
C_{\text{ff,cl,i}} = C_{\text{ff,cl,i}} \cdot \sqrt{1 - \text{CIBREK}}
The fuel breaking loose from the crust generates chunks, which are then
merged if necessary with the chunks already present in the cell. If
steel inclusions are present in the crust, they are also reduced
proportionately and added to the steel chunk field. A similar procedure
is used for the structure crust breakup. The process is illustrated in
:numref:`figure-16.5-1`.
.. _section-16.5.4.2:
Remelting of the Fuel Crust
^^^^^^^^^^^^^^^^^^^^^^^^^^^
If the fuel crusts are present in a cell and breakup does not occur, the
crust can still disappear via remelting. Melting of the cladding crust
occurs if:
.. math::
:label: 16.5-23
T_{\text{ffc,i}} > T_{\text{fu,melt}}
where
.. math::
:label: 16.5-24
T_{\text{fu,melt}} = \left( T_{\text{fu,so}} + T_{\text{fu,} l \text{q}} \right) \cdot 0.5
The amount of fuel crust melting in one time step is given by:
.. math::
:label: 16.5-25
\Delta {\rho'}_{\text{ffc,i}} = {\rho'}_{\text{ffc,i}} \cdot \frac{T_{\text{ffc,i}} \
- T_{\text{fu,melt}}}{T_{\text{fu,} l \text{q}} - T_{\text{fu,melt}}}
If steel inclusions are present, they are reduced in the same
proportion. The resulting molten fuel and steel are added to the molten
fuel and steel fields in the channel, respectively.
A similar procedure is used for the remelting of the fuel crust on the
hexcan wall.
.. _section-16.5.5:
Cladding and Hexcan Ablation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
When the coolant channel contains molten fuel, which is in an annular or
bubbly flow regime, the temperature at the cladding surface rises
rapidly, leading to steel melting and ablation. These processes are
modeled in the routine :math:`\text{LEABLA}`. The melting occurs only at the surface,
while the bulk of the cladding can remain below the melting point. To
model this situation, the cladding is divided into two radial cells,
with a thick inner cell and a thin cell at the outer surface (:numref:`figure-16.5-2`). The thin cladding cell at the outer surface has a small thermal
capacity and is thus quite sensitive to variations in the magnitude of
the boundary heat fluxes. This cell is of key importance in the melting
and ablation process. Ablation of the cladding can occur only if no fuel
crust is present on cladding and:
.. math::
:label: 16.5-26
T_{\text{cl,os,i}} > T_{\text{se,ablation}}
where
.. math::
:label: 16.5-27
T_{\text{se,ablation}} = \left( T_{\text{se,so}} + T_{\text{se,} l \text{q}} \right) \cdot 0.5
The fraction of the outer cell mass which is ablated is given by:
.. math::
:label: 16.5-28
F_{\text{ablation,i}} = \frac{T_{\text{cl,os,i}} - T_{\text{se,ablation}}}{T_{\text{se,} l \text{q}} - T_{\text{se,ablation}}}
.. _figure-16.5-2:
.. figure:: media/image28.png
:align: center
:figclass: align-center
Temperature Grid in the Cladding and Structure
An amount :math:`\Delta {\rho'}_{\text{se}}` is removed from the outer clad cell and added
to the molten steel field in the channel. The outer radius of the
cladding is changed to:
.. math::
:label: 16.5-29
R_{\text{cl,os,i}} = \sqrt{R_{\text{cl,in,i}}^{2} - \left( R_{\text{cl,os,i}}^{2} \
- R_{\text{cl,in,i}}^{2} \right) \cdot \left( 1 - F_{\text{ablation,i}} \right)}
Due to the explicit solution technique used in the cladding temperature
calculation, a very thin outer cladding cell can lead to numerical
instabilities in the temperature calculation or alternatively might
require very small time steps. To avoid this difficulty while still
maintaining an explicit method of solution in the temperature
calculation, a lower limit :math:`\Delta R_{\text{min}}` is imposed on the thickness of
the outer cladding cell. Whenever:
.. math::
:label: 16.5-30
R_{\text{cl,in,i}} = R_{\text{cl,os,i}} - R_{\text{cl,in,i}} \leq \Delta R_{\text{min}}
The temperature calculation grid is restructured. The internal radius
:math:`R_{\text{cl,in,i}}` is set to:
.. math::
:label: 16.5-31
R_{\text{cl,in,i}} = R_{\text{cl,os,i}} - 2 \cdot \Delta R_{\text{min}}
and the temperature of the cladding cells is adjusted appropriately.
This process continues until :math:`R_{\text{cl,in,i}} = R_{\text{cl,is,i}}`, at
which time the internal cladding cell disappears completely. The
ablation process can still continue until the outer (and only) cladding
cell reaches the thickness :math:`\Delta R_{\text{min}}`. Afterwards, no ablation is
allowed until the remaining cladding reaches the melting point, when the
cladding is completely removed and the fuel pin begins to transfer
energy directly to the flowing components in the channel.
A similar procedure is used for the ablation of the hexcan wall. The
ablation process is illustrated in :numref:`figure-16.5-1`.
.. _section-16.5.6:
Fuel/Steel Chunk Melting
~~~~~~~~~~~~~~~~~~~~~~~~
The moving solid chunks in the channel, both fuel and steel, can begin
to melt due to heat transfer from the surrounding components and, in the
case of the fuel chunks, due to internal heat generation. This process
is modeled in the routine :math:`\text{LELUME}`. The remelting of the fuel chunks
occurs when:
.. math::
:label: 16.5-32
h_{\text{f} l \text{,i}} > h_{\text{f} l \text{,melt}}
where
.. math::
:label: 16.5-33
h_{\text{f} l \text{,melt}} = \left( h_{\text{fu,so}} + h_{\text{fu,} l \text{q}} \right) \cdot 0.5
The amount of molten fuel is obtained from:
.. math::
:label: 16.5-34
\Delta {\rho'}_{\text{f} l \text{,i}} = {\rho'}_{\text{f} l \text{,i}} \cdot \frac{h_{\text{f} l \text{,i}} \
- h_{\text{f} l \text{,melt limit}}}{h_{\text{fu,} l \text{q}} - h_{\text{f} l \text{,melt limit}}}
where
.. math::
:label: 16.5-35
h_{\text{f} l \text{,melt limit}} = \frac{\left( h_{\text{f} l \text{,melt}} - 0.1 h_{\text{fu,} l \text{q}} \right)}{0.9}
The quantity :math:`h_{l l,\text{melt limit}}` is defined by :eq:`16.5-35` to be
slightly below :math:`h_{l l \text{,melt}}`. When melting occurs, as defined by
:eq:`16.5-34`, the enthalpy of the molten material is :math:`h_{\text{fu,} l \text{q}}`, and the
enthalpy of he remaining chunks is :math:`h_{l l,\text{melt limit}}`. Because
:math:`h_{l l,\text{melt limit}}` is lower than :math:`h_{l l,\text{melt}}`, which is used
in :eq:`16.5-32` to trigger the melting process, the continuous melting of
very small amounts of fuel is avoided. Melting of the fuel chunks in
cell :math:`i` will occur only when their enthalpy again reaches
:math:`h_{l l,\text{melt limit}}`. The size of the chunks is decreased
appropriately, but their number remains unchanged. The molten fuel is
added to the molten fuel in the channel:
.. math::
:label: 16.5-36
{\rho'}_{\text{fu,i}} = {\rho'}_{\text{fu,i}} + \Delta{\rho'}_{\text{fu,i}}
and the energy and velocity of the molten fuel field are adjusted to
reflect the addition of the molten fuel resulting from the remelting of
the chunks.
A similar procedure is used for melting the steel chunks, which occurs
when:
.. math::
:label: 16.5-37
h_{\text{s} l \text{,i}} > h_{\text{s} l \text{,melt}}
where
.. math::
:label: 16.5-38
h_{\text{s} l \text{,melt}} = \left( h_{\text{se,so}} + h_{\text{se,} l \text{q}} \right) \cdot 0.5
The amount of molten steel is obtained from
.. math::
:label: 16.5-39
\Delta {\rho'}_{\text{s} l \text{,i}} = {\rho'}_{\text{s} l \text{,i}} \
\cdot \frac{h_{\text{s} l \text{,i}} - h_{\text{s} l \text{,melt limit}}}{h_{\text{se,lq}} \
- h_{\text{s} l \text{,melt limit}}}
.. _section-16.5.7:
Heat-transfer Calculation for the Steel Cladding and the Hexcan Wall
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
These calculations are performed in the routine LESOEN.
.. _section-16.5.7.1:
Heat-transfer Calculation for the Steel Cladding
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The heat-transfer calculation for the steel cladding is performed using
the two-node mesh illustrated in :numref:`figure-16.5-2`. Two energy equations, one
for each node, are solved explicitly. Due to ablation the geometry of
the two cladding nodes can change in any time step.
The energy conservation equation for the outer cladding node in the
axial cell :math:`i` is:
.. math::
:label: 16.5-40
\rho_{\text{se}} \cdot \frac{\partial \text{h}_{\text{cl,os,i}}}{\partial \text{t}} \cdot \left( R_{\text{cl,os,i}}^{2} - R_{\text{cl,in,i}}^{2} \right) \cdot \pi \cdot \Delta z_{\text{i}} \\
= k_{\text{se}} \cdot \frac{T_{\text{cl,in,i}} - T_{\text{cl,os,i}}}{R_{\text{cl,os,i}} - {R'}_{\text{cl,in,i}}} \cdot 2 \pi R_{\text{cl,in,i}} \cdot \Delta z_{\text{i}} - \sum_{\text{j}}{H_{\text{cl,j,i}}} \\
\cdot \left( T_{\text{cl,os,i}} - T_{\text{j,i}} \right) \cdot \frac{A_{\text{cl,j,i}}^{L}}{N_{\text{pins}}} + Q_{\text{cl,j}} \cdot \frac{R_{\text{cl,os,i}} - R_{\text{cl,in,i}}}{\Delta R_{\text{cl}}^{0}} \\
where the :math:`\sum_{\text{j}}` is performed over all the components in
the channel that are in contact with the cladding and :math:`Q_{\text{cl,i}}` is
the energy source in the cladding in cell :math:`i`. The subscript se refers
here to the solid steel, as opposed to the previous occurrences where it
was followed by :math:`i` (e.g. :math:`\rho_{\text{se,i}}`), when it was referring to the
molten steel in the channel. After integration over :math:`\Delta t`, division by
:math:`\Delta z_{\text{i}}` and rearrangement, e.g., :eq:`16.5-40` becomes:
.. math::
:label: 16.5-41
h_{\text{cl,os,i}}^{n + 1} = h_{\text{cl,os,i}} + \frac{\Delta t}{\rho_{\text{se}} \cdot \pi \left( R_{\text{cl,os,i}}^{2} - R_{\text{cl,in,i}}^{2} \right)} \left\{ \frac{k_{\text{se}} \cdot 2 \pi R_{\text{cl,in,i}}}{R_{\text{cl,os,i}} - {R'}_{\text{cl,in,i}}} \right. \\ \left.
\times \left( T_{\text{cl,in,i}} - T_{\text{cl,os,i}} \right) - \frac{\text{AXMX}}{N_{\text{pins}}} \left\lbrack {H'}_{\text{Na,cl,i}} \left( T_{\text{cl,os,i}} - T_{\text{Na,i}} \right) \right. \right. \\ \left. \left.
+ {H'}_{\text{fu,cl,i}} \left( T_{\text{cl,os,i}} - T_{\text{fu,i}} \right) + {H'}_{\text{se,cl,i}} \left( T_{\text{cl,os,i}} - T_{\text{se,i}} \right) \right. \right. \\ \left. \left.
+ {H'}_{\text{cl,fi,i}} \left( T_{\text{cl,os,i}} - T_{\text{fl,i}} \right) + {H'}_{\text{cl,sl,i}} \left( T_{\text{cl,os,i}} - T_{\text{sl,i}} \right) \right. \right. \\ \left. \left.
+ {H'}_{\text{cl,ffc,i}} \left( T_{\text{cl,os,i}} - T_{\text{ffc,i}} \right) \right\rbrack + Q_{\text{cl,i}} \cdot \frac{R_{\text{cl,os,i}} - R_{\text{cl,in,i}}}{\Delta z_{\text{i}} \cdot \Delta R_{\text{cl}}^{o}} \right\} \\
where :math:`{H'}_{\text{j,cl,i}}` are the generalized heat-transfer
coefficients which have been defined previously.
The energy conservation equation for the inner cladding node in the
axial cell is shown below:
.. math::
:label: 16.5-42
\rho_{\text{se}} \frac{\partial \text{h}_{\text{cl,in,i}}}{\partial \text{t}} \cdot \left( R_{\text{cl,in,i}}^{2} - R_{\text{cl,is,i}}^{2} \right) \cdot \pi \cdot \Delta z_{\text{i}} \\
= - k_{\text{se}} \frac{T_{\text{cl,in,i}} - T_{\text{cl,os,i}}}{R_{\text{cl,os,i}} - R_{\text{cl,in,i}}} \cdot 2 \pi R_{\text{cl,in,i}} \cdot \Delta z_{\text{i}} \\
+ H_{\text{cl,in,pin,i}} \cdot 2 \pi \cdot R_{\text{cl,is,i}} \cdot \Delta z_{\text{i}} \cdot \left( T_{\text{pin,os,i}} - T_{\text{cl,in,i}} \right) \\
+ Q_{\text{cl,i}} \cdot \frac{R_{\text{cl,in,i}} - R_{\text{cl,is,i}}}{\Delta R_{\text{cl}}^{o}}~,
where :math:`H_{\text{cl,in,pin,i}}` is the heat-transfer coefficient between the
inner cladding node and the outer fuel node. It takes into account the
gap heat conductance and is defined as follows:
.. math::
:label: 16.5-43
H_{\text{cl,in,pin,i}} = \frac{1}{\frac{1}{H_{\text{gap}}} + \frac{{R'}_{\text{cl,in,i}} - R_{\text{cl,is,i}}}{k_{\text{se}}}}
After integration over :math:`\Delta t` and rearrangement, :eq:`16.5-42` becomes:
.. math::
:label: 16.5-44
h_{\text{cl,in,i}}^{n + 1} = h_{\text{cl,in,i}} - \frac{\Delta t}{\rho_{\text{se}} \cdot \pi \left( R_{\text{cl,in,i}}^{2} - R_{\text{cl,is,i}}^{2} \right)} \cdot \left\lbrack \frac{k_{\text{se}} \cdot 2 \pi R_{\text{cl,in,i}}}{R_{\text{cl,os,i}} - R_{\text{cl,in,i}}} \right. \\ \left.
\cdot \left( T_{\text{cl,in,i}} - T_{\text{cl,os,i}} \right) - H_{\text{cl,in,pin,i}} \cdot 2 \pi R_{\text{cl,is,i}} \right. \\ \left.
\cdot \left( T_{\text{pin,os,i}} - T_{\text{cl,in,i}} \right) - Q_{\text{cl,i}} \cdot \frac{R_{\text{cl,in,i}} - R_{\text{cl,is,i}}}{\Delta R_{\text{cl}}^{o} \cdot \Delta z_{\text{i}}} \right\rbrack \\
A special situation occurs whenever the inner cladding node disappears
as a consequence of the ablation process, which has been described
previously. This situation is indicated by setting the temperature
:math:`T_{\text{cl,in,i}}`, which is not longer used, to a negative arbitrary
value, i.e., :math:`T_{\text{cl,in,i}} = -100`. In this case, the outer cladding
node exchanges heat directly with the fuel pin and the energy
:eq:`16.5-41` is changed to:
.. math::
:label: 16.5-45
h_{\text{cl,os,i}}^{n + 1} = {h'}_{\text{cl,os,i}} + \frac{\Delta t}{\rho_{\text{se}} \cdot \pi \cdot \left( R_{\text{cl,os,i}}^{2} - R_{\text{cl,in,i}}^{2} \right)} \\
\cdot \left\lbrack - \frac{\text{AXMX}}{N_{\text{pins}}} \cdot \sum_{\text{j}}{{H'}_{\text{cl,j,i}} \left( T_{\text{cl,os,i}} - T_{\text{j,1}} \right)} + H_{\text{cl,os,pin,i}} \cdot 2 \pi R_{\text{cl,in,i}} \right. \\ \left.
\cdot \left( T_{\text{pin,os,i}} - T_{\text{cl,os,i}} \right) + Q_{\text{cl,i}} \frac{R_{\text{cl,os,i}} - R_{\text{cl,in,i}}}{\Delta z_{\text{i}} \cdot \Delta R_{\text{cl}}^{o}} \right\rbrack
where :math:`H_{\text{cl,os,pin,i}}` is the heat-transfer coefficient between the
outer cladding node and the outer pin node (when the inner node was
vanished):
.. math::
:label: 16.5-46
H_{\text{cl,os,pin,i}} = \frac{1}{\frac{1}{H_{\text{gap}}} + \frac{R_{\text{cl,os,i}} - R_{\text{cl,os,i}}}{k_{\text{se}}}}
The energy transferred between the cladding and the pin in each LEVITATE
time step is integrated over the heat-transfer time step and stored
under the name :math:`\text{HFPICL} \left( I \right)`. This quantity is then used in the pin
heat-transfer calculation, in the routine PLHTR, as the pin boundary
condition at :math:`R = R_{\text{pin,os,i}}`.
Another special situation is the case when the cladding has been totally
ablated, and the moving components in the channel are in direct contact
with the fuel pin. For this case, the cladding temperature
:math:`T_{\text{cl,os,i}}` is set equal to :math:`T_{\text{pin,os,i}}`:
.. math::
:label: 16.5-47
T_{\text{cl,os,i}} = T_{\text{pin,os,i}}
This allows all heat transfer to the channel components, as well as the
freezing/melting processes, to be calculated correctly. Freezing of fuel
on the bare fuel pin can still occur, if predicted by the freezing
model. This situation can be identified in the output by the presence of
undisrupted pin nodes, i.e., :math:`\text{IDISR} \left( I \right) \neq 1` with no cladding on them,
i.e., :math:`\text{WICLAD} \left( I \right) = 0`. In these cells, the output will indicate that no
inner cladding node is present, i.e., :math:`T_{\text{cl,in,i}} = -100`, and the
temperature printed under :math:`T_{\text{cl,os,i}}` will represent the outer
temperature of the fuel pin, according to :eq:`16.5-47`.
The procedure outlined above is used in the fuel and blanket region of
the fuel pin. A similar but simplified procedure is used for the
remainder of the pin, where no fuel is present. A zero heat-flux
boundary condition is used in these nodes.
The new cladding temperatures are finally obtained from the new
enthalpies:
.. math::
:label: 16.5-48
\begin{aligned}
T_{\text{cl,os,i}}^{n + 1} = T \left( h_{\text{cl,os,i}}^{n + 1} \right) ; && T_{\text{cl,in,i}} = T\left( h_{\text{cl,in,i}}^{n + 1} \right)
\end{aligned}
.. _section-16.5.7.2:
Heat-transfer Calculation for the Hexcan Wall
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The heat-transfer calculation for the hexcan wall (the hexcan wall is
alternatively referred to as structure) is performed using the two-cell
mesh illustrated in :numref:`figure-16.5-2`. *The outer structure cell is the cell
facing the coolant channel*. This cell is fairly thin, and responds
rapidly to changes in the heat-transfer from the coolant channel. This
is necessary for the correct modeling of the structure ablation process,
which has been described previously. It is assumed that no heat transfer
occurs at the outer boundary of the hexcan wall, i.e., at the boundary
facing the neighboring subassemblies. Because of this condition, and in
order to improve the accuracy of the two-node calculation for the fairly
thick structure, the temperature in the structure is assumed to have a
parabolic variation:
.. math::
:label: 16.5-49
T \left( \Delta R \right) = a \cdot \Delta R^{2} + b \cdot \Delta R + c
where :math:`\Delta R` is the radial coordinate measured from the hexcan boundary
facing the channel towards the pin. The coefficients :math:`a`, :math:`b`, and :math:`c` are
defined each time step by the conditions:
.. math::
:label: 16.5-50
\frac{\text{dT}}{\text{d} \left( \text{AR} \right)} \big\rvert_{\Delta \text{R}} = - \Delta R_{\text{sr,i}} = 0
.. math::
:label: 16.5-51
T \left( 0 \right) = T_{\text{sr,os,i}}
.. math::
:label: 16.5-52
T \left( - \Delta R_{\text{sr,os,i}} - \frac{1}{3} \Delta R_{\text{sr,in,i}} \right) = T_{\text{sr,in,i}}
Using :eq:`16.5-50` through :eq:`16.5-52`, it is found that the temperature
gradient at the boundary between the outer and the inner structure nodes
is given by:
.. math::
:label: 16.5-53
\frac{\text{dT}}{\text{d} \left( \Delta \text{R} \right)} \big\rvert_{\Delta \text{R} = - \Delta R_{\text{sr,os,i}}} \\
= \left( T_{\text{sr,os,i}} - T_{\text{sr,in,i}} \right) \cdot \frac{18 \cdot \Delta R_{\text{sr,in,i}}}{5\Delta R_{\text{sr,i}}^{2} + 8\Delta R_{\text{sr,i}} \cdot \Delta R_{\text{sr,os,i}} - 4\Delta R_{\text{sr,os,i}}^{2}} \\
This temperature gradient expression will be used in the
energy-conservation equations for the structure. The equation for the
outer structure cell is written:
.. math::
:label: 16.5-54
\rho_{\text{se}} \cdot \frac{\partial \text{h}_{\text{sr,os,i}}}{\partial \text{t}} \cdot \Delta R_{\text{sr,os,i}} \cdot \Delta z_{\text{i}} \cdot L_{\text{sr,i}} \\
= - k_{\text{se}} \cdot \frac{\partial \text{T}}{\partial\left( \Delta \text{R} \right)} \big\rvert_{\Delta R = - \Delta R_{\text{sr,os,i}}} \cdot \Delta z_{\text{i}} \cdot L_{\text{sr,i}} \\
+ \sum_{\text{j}}{H_{\text{sr,j,i}} \cdot \left( T_{\text{j,i}} - T_{\text{sr,os,i}} \right) \cdot A_{\text{sr,j,i}}^{L}} \\
where :math:`\sum_{\text{j}}` is performed over all the components in the
channel that are in contact with the structure. Although :math:`\Delta R` is used as
the "radial" coordinate, the hexcan wall is in fact assumed to be flat,
with thickness :math:`\Delta R_{\text{i}}` and perimeter :math:`L_{\text{sr,i}}`.
After integration over :math:`\Delta t`, rearrangement and use of :eq:`16.5-53`, the
above equation becomes:
.. math::
:label: 16.5-55
h_{\text{sr,os,i}}^{n + 1} = h_{\text{sr,os,i}} + \frac{\Delta t}{\rho_{\text{se}} \
\cdot R_{\text{sr,os,i}} \cdot {L'}_{\text{sr,i}}} \cdot \left\lbrack - \frac{k_{\text{se}} {L'}_{\text{sr,i}} \
\cdot 18 R_{\text{sr,in,i}}}{5R_{\text{sr,i}}^{2} + 8R_{\text{sr,i}} \cdot R_{\text{sr,os,i}} - 4R_{\text{sr,os,i}}^{2}} \
\cdot \left( T_{\text{sr,os,i}} - T_{\text{sr,in,i}} \right) \right. \\ \left.
+ {H'}_{\text{Na,sr,i}} \cdot \left( T_{\text{Na,i}} - T_{\text{sr,os,i}} \right) + {H'}_{\text{fu,sr,i}} \
\cdot \left( T_{\text{fu,i}} - T_{\text{sr,os,i}} \right) + {H'}_{\text{se,sr,i}} \cdot \left( T_{\text{se,i}} \
- T_{\text{sr,os,i}} \right) \right. \\ \left.
+ {H'}_{\text{fl,sr,i}} \cdot \left( T_{\text{fl,i}} - T_{\text{sr,os,i}} \right) + {H'}_{\text{s} l \text{,sr,i}} \
\cdot \left( T_{\text{s} l \text{,i}} - T_{\text{sr,os,i}} \right) + {H'}_{\text{ffs,sr,i}} \cdot \left( T_{\text{ffs,i}} \
- T_{\text{sr,os,i}} \right) \right\rbrack
where
.. math::
:label: 16.5-56
{L'}_{\text{sr,i}} = \frac{L_{\text{sr,i}}}{\text{AXMX}}
and :math:`{H'}_{\text{j,sr,i}}` are the generalized heat-transfer
coefficients which were defined previously as follows:
.. math::
:label: 16.5-57
{H'}_{\text{j,sr,i}} = H_{\text{j,sr,i}} \cdot \frac{A_{\text{j,sr,i}}}{\text{AXMX} \cdot \Delta z_{\text{i}}}
The new temperature of the structure outer node is then obtained from
its enthalpy:
.. math::
:label: 16.5-58
T_{\text{sr,os,i}}^{n + 1} = T \left( h_{\text{sr,os,i}}^{n + 1} \right)
The energy equation for the inner structure cell (i.e., the cell which
is not in contact with the coolant channel) is written as follows:
.. math::
:label: 16.5-59
\rho_{\text{se}} \cdot \frac{\partial \text{h}_{\text{sr,in,i}}}{\partial \text{t}} \cdot \Delta R_{\text{sr,in,i}} \cdot \Delta z_{\text{i}} \cdot L_{\text{sr,i}} \\
= k_{\text{se}} \frac{\partial \text{T}}{\partial\left( \Delta \text{R} \right)} \big\rvert_{\Delta \text{R} = - \Delta R_{\text{sr,os,i}}} \cdot \Delta z_{\text{i}} \cdot L_{\text{sr,i}} \\
After integration over :math:`\Delta t` and rearrangement, the above equation becomes:
.. math::
:label: 16.5-60
h_{\text{sr,in,i}}^{n + 1} = h_{\text{sr,in,i}} \\
+ \frac{\Delta t \cdot k_{\text{se}} \cdot 18 \Delta R_{\text{sr,in,i}} \cdot \left( T_{\text{sr,in,i}} \right)}{\rho_{\text{se}} \cdot \Delta R_{\text{sr,in,i}} \cdot \left( 5 \Delta R_{\text{sr,i}}^{2} + 8 \cdot \Delta R_{\text{sr,i}} \cdot \Delta R_{\text{sr,os,i}} - 4\Delta R_{\text{sr,os,i}}^{2} \right)} \\
The new temperature of the structure node is then obtained from its enthalpy:
.. math::
:label: 16.5-61
T_{\text{sr,in,i}}^{n + 1} = T \left( h_{\text{sr,in,i}}^{n + 1} \right)
A special situation occurs when the "inner" structure cell has
disappeared completely as a result of the ablation process. This
situation is indicated by setting :math:`T_{\text{sr,in,i}}` to an arbitrary
negative value, i.e., :math:`T_{\text{sr,in,i}} = -100`. In this case, only the
:eq:`16.5-54` for the outer structure cell is solved in a simplified form.
The term
:math:`k_{\text{se}} \frac{\partial \text{T}}{\partial \left( \Delta R \right)} \big\rvert_{\Delta R = - \Delta R_{\text{sr,os,i}}}` is
set to zero, and the final equation is shown below:
.. math::
:label: 16.5-62
h_{\text{sr,os,i}}^{n + 1} = h_{\text{sr,os,i}} + \frac{\Delta t}{\rho_{\text{se}} \
\cdot \Delta R_{\text{sr,os,i}} \cdot {L'}_{\text{sr,i}}} \cdot \sum_{\text{j}}{{H'}_{\text{j,sr,i}} \
\cdot \left( T_{\text{j,i}} - T_{\text{sr,os,i}} \right)}
When the "inner" structure cell has been removed and the outer cell has
reached both a minimum thickness and the melting point, the hexcan wall
is assumed to be breached. Intersubassembly fuel motion is likely to
begin. Theoretically, this should be the end of the LEVITATE calculation
and the beginning of a transition-phase calculation. However, the code
will only print a warning message and the calculation will continue
assuming that the hexcan wall will maintain the minimum thickness,
although its temperature has risen above the melting range.