.. _section-16.4:
Coolant Channel Hydrodynamic Model
----------------------------------
.. _section-16.4.1:
Physical Models
~~~~~~~~~~~~~~~
.. _section-16.4.1.1:
Geometry
^^^^^^^^
The coolant channel described by LEVITATE is delimited by the hexcan
wall and the outer surface of the fuel pins. Axially, this channel is
limited by the upper and lower sodium plena. Calculations are performed
in a 1-D geometry, with the variables being functions of the axial
height :math:`z`. As illustrated in :numref:`figure-16.1-1`, the area of the coolant channel
can vary widely due to the local disruption of the fuel pins, cladding
and structure ablation, and fuel/steel blockage formation at various
axial locations.
.. _section-16.4.1.2:
Hydrodynamic Models
^^^^^^^^^^^^^^^^^^^
The fluid dynamic models in LEVITATE calculate the motion of the
materials in the channel. The materials that are tracked by LEVITATE
are:
1. liquid fuel
2. liquid steel
3. sodium (liquid, two-phase mixture or superheated vapor)
4. fission gas
5. fuel vapor
6. steel vapor
7. solid fuel chunks
8. solid steel chunks
A separate mass and energy conservation equation is solved for each of
these components. Momentum conservation equations are solved for three
velocity fields, associated with the following component groups:
1. sodium, fission gas, fuel vapor, steel vapor
2. liquid fuel, liquid steel
3. solid fuel chunks, solid steel chunks
Phase transitions can occur, leading to mass, momentum and energy
exchange among various moving components and/or among moving and
stationary components. These exchanges are described in :numref:`section-16.4.1.5`.
.. _section-16.4.1.3:
Structural Models
^^^^^^^^^^^^^^^^^
The structural models included in LEVITATE describe the thermodynamic
behavior of the stationary components that act as boundaries for the
coolant channel. The stationary components modeled in LEVITATE are:
1. steel cladding
2. steel hexcan wall
3. cladding frozen fuel crust (with possible steel inclusions)
4. hexcan wall frozen fuel crust (with possible steel inclusions)
One or more energy equations are solved for each of these components
(e.g., two equations are needed for cladding, that has two radial
nodes). Due to freezing/melting processes, mass and energy exchange can
take place between these stationary components and the moving components
in the channel, as explained in :numref:`section-16.4.1.5`.
.. _section-16.4.1.4:
Fuel/Steel Flow Regimes
^^^^^^^^^^^^^^^^^^^^^^^
As indicated in :numref:`section-16.1`, the interactions between the different
components present in the channel, i.e., mass, energy and momentum
transfer, are largely determined by the local configuration which, in
turn, is determined by the flow regimes used. The fuel/steel flow
regimes modeled in LEVITATE are presented conceptually in :numref:`figure-16.4-1`.
They are a bubbly fuel flow regime, a partial annular fuel flow regime,
an annular steel flow regime, and a bubbly steel flow regime. The bubbly
fuel channel flow is characterized by the presence of large amounts of
liquid fuel, with droplets of molten steel, solid fuel and steel chunks,
and bubbles of fission gas and sodium vapor imbedded. There is no
relative motion between the steel and fuel, and the relative velocity
between the molten fuel and the vapor bubbles is quite low, due to the
large drag and inertial forces acting on the bubbles.
As the volume fraction of fuel decreases, the bubbly flow regime is
changed to the annular fuel flow regime, with the fuel blanketing the
cladding partially or totally and the two-phase sodium/gas mixture
flowing at the center of the channel. A stream of solid fuel chunks,
interacting with both the two-phase sodium/gas mixture and the liquid
fuel streams, can also be present. Molten steel droplets are imbedded in
the fuel film. The relative velocities between the fuel and mixture are
significantly higher than in the bubbly flow regime.
When molten steel is the dominant component at a certain axial location,
the flow regime used can be either an annular or a bubbly steel regime.
The bubbly steel flow regime is characterized by the presence of a large
amount of molten steel, with droplets of molten fuel, solid fuel and
steel chunks, and bubbles of fission gas and sodium vapor imbedded. As
the volume fraction of molten steel decreases, the bubbly steel regime
is changed to the annular steel flow regime. In this regime, the molten
steel totally covers the cladding and the vapor mixture flows at the
center of the channel. Molten fuel droplets can be imbedded in the
molten steel, and solid fuel and steel chunks can interact with both the
vapor mixture and the molten steel. In the Release 1.1 version, a
partial annular steel flow regime has been introduced. The steel is
assumed to cover the cladding only partially moving in the form of
droplets and rivulets. This picture is supported by experimental
evidence that indicates that the steel does not wet the cladding. As the
local amount of molten steel increases, a full annular steel flow regime
can develop. The partial annular steel flow regime is used only when the
chunk model is active, i.e. :math:`\text{ICHUNK} = 1`. If the chunk model is not used,
the full annular steel flow regime is still used. It is expected that,
as the validation effort continues, the partial annular steel flow
regime will be available, regardless of the chunk model option used.
.. _figure-16.4-1:
.. figure:: media/image9.png
:align: center
:figclass: align-center
Continuous Fuel/Steel Regimes Modeled in LEVITATE
Because only very limited information is currently available about the
fuel-steel flow regime transitions in fuel pin bundles, these
transitions are governed in LEVITATE by the fuel-steel volume fraction
present at each axial location.
The continuous fuel flow regimes used in LEVITATE have been shown to
slow down substantially the fuel dispersal, bringing the results of
calculations in close agreement with the experimental data [16-1,
16-11].
.. _section-16.4.1.5:
Mass, Momentum and Energy Exchange (non-convective)
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Mass, momentum and energy exchange can take place among various
components at any given time and location. All these transfers are
strongly dependent on the flow regime prevailing at the time and
location considered. The mass transfers are due to phase changes
(freezing, melting, vaporization, condensation) or to disruption events,
such as the disruption of the solid fuel pins or fuel crust breakup. The
mass transfer is largely independent of the local flow regime and the
allowable transfers are summarized in :numref:`figure-16.4-2`.
The mass transfer in any given computational cell, together with the
changes in the mass of each component due to convection, determine the
mass of each component in the cell. These masses, together with the
energy from the previous time step, determine the local flow regime and
thus the local configuration. The local energy and momentum transfers
are strongly dependent on the local flow regime. The allowable energy
transfer paths among various LEVITATE components are illustrated in
:numref:`figure-16.4-3` to :numref:`figure-16.4-6` for various flow regimes. Similarly, the
allowable momentum paths are illustrated in :numref:`figure-16.4-7` through
16.4-10.
.. _figure-16.4-2:
.. figure:: media/image10.png
:align: center
:figclass: align-center
Mass exchange Possibilities among Various LEVITATE Components
.. _figure-16.4-3:
.. figure:: media/image11.png
:align: center
:figclass: align-center
Energy Exchange Possibilities among Various LEVITATE Components, for the Fuel Annular Flow Regime
.. _figure-16.4-4:
.. figure:: media/image12.png
:align: center
:figclass: align-center
Energy Exchange Possibilities among Various LEVITATE Components, for the Fuel Bubbly Flow Regime
.. _figure-16.4-5:
.. figure:: media/image13.png
:align: center
:figclass: align-center
Energy Exchange Possibilities among Various LEVITATE Components, for the Steel Annular Flow Regime
.. _figure-16.4-6:
.. figure:: media/image14.png
:align: center
:figclass: align-center
Energy Exchange Possibilities among Various LEVITATE Components, for the Steel Bubbly Flow Regime
.. _figure-16.4-7:
.. figure:: media/image15.png
:align: center
:figclass: align-center
Momentum Exchange Possibilities among Various LEVITATE Components, for the Fuel Annular Flow Regime
.. _figure-16.4-8:
.. figure:: media/image16.png
:align: center
:figclass: align-center
Momentum Exchange Possibilities among Various LEVITATE Components, for the fuel Bubbly Flow Regime
.. _figure-16.4-9:
.. figure:: media/image17.png
:align: center
:figclass: align-center
Momentum Exchange Possibilities among Various LEVITATE Components, for the Steel Annular Flow Regime
.. _figure-16.4-10:
.. figure:: media/image18.png
:align: center
:figclass: align-center
Momentum Exchange Possibilities among Various LEVITATE Components, for the Steel Bubbly Flow Regime
.. _section-16.4.2:
Method of Solution and general Numerical Considerations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-16.4.2.1:
Variables and Mesh Grid Used in Calculations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The independent variables used in LEVITATE are the axial coordinate, z,
and the time, :math:`t`. Only one spatial coordinate is necessary, as LEVITATE
models the subassembly in a one-dimensional geometry. The dependent
variables calculated by the hydrodynamic model for each component, are
the generalized density :math:`{\rho'}`, the enthalpy :math:`h` (or temperature :math:`T`), and the
velocity :math:`u`. The generalized densities have been introduced in :numref:`Chapter %s<section-14>`
and, for component :math:`k`, are defined as:
(16.4-1)
.. _eq-16.4-1:
.. math::
{\rho'}_{\text{k,i}} = \rho_{\text{k,i}} \cdot \frac{A_{\text{k,i}}}{\text{AXMX}} = \rho_{\text{k,i}} \cdot \theta_{\text{k,i}}
where
:math:`\rho_{\text{k,i}}` = physical density of component :math:`k` at location :math:`i`
:math:`A_{\text{k,i}}` = cross sectional area occupied by component :math:`k` at location
:math:`i`
:math:`\text{AXMX}` = reference area
:math:`\theta_{\text{k,i}}` = generalized (area) fraction of component :math:`k` at location :math:`i`
The mass, energy and momentum partial differential conservation
equations are solved using an Eulerian finite difference semi-explicit
formulation, as explained below. The mesh grid used for the finite
difference formulation is presented in :numref:`figure-16.4-11`. As indicated in
this figure, the densities and enthalpies are defined at the center of
each cell, while the velocities are defined at the boundaries. Because
of the highly irregular geometry treated by LEVITATE, special attention
was necessary for the treatment of abrupt area changes [16-12]. Fuel
velocities are defined at each cell boundary, with :math:`u_{\text{i}}` being the
velocity just before boundary :math:`i`, and :math:`{u''}_{\text{i}}` the velocity just
after that boundary. The terms "before" and "after" are used in relation
to the positive direction of the axial coordinate :math:`z`.
In order to reduce the numerical diffusion, characteristic of Eulerian
numerical schemes, the boundaries of each region containing a certain
component are tracked separately as they move through the Eulerian
cells.
.. _figure-16.4-11:
.. figure:: media/image19.png
:align: center
:figclass: align-center
Mesh Grid used in the Channel Hydrodynamic Model
.. _section-16.4.2.2:
Description of the Method of Solution and Logic Flow
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
A simplified modular chart of the LEVITATE model is presented in :numref:`figure-16.1-5`. The calculation begins by calculating the changes in the
position of material boundaries during the current time step. The new
interface positions, at the end of the time step, are calculated for all
components, except sodium in the LEIF (**LE**\ VITATE **I**\ NTER\ **F**\ ACE)
routine. The position of the sodium slugs, which determine the
boundaries of the sodium region, is calculated in the LEREZO
(**LE**\ VITATE **REZO**\ NING) routine. This routine can add (or remove)
nodes to the LEVITATE compressible region as the sodium slugs move out
of (or into) the channel. The mass conservation equation is solved next,
for all components and all axial locations. Each equation is solved
explicitly, i.e., the convective fluxes are based on the generalized
densities present in each cell at the beginning of the time step. These
calculations are performed in the LEMACO (**LE**\ VITATE **MA**\ SS
**CO**\ NSERVATION) routine. The new densities are then used in the LEVOFR
(**LE**\ VITATE **VO**\ LUME **FR**\ ACTION) routine to determine the volume
fraction of each component at each axial location. Using these volume
fractions, the LEVOFR routine also determines the flow regime in each
axial cell. This flow regime will be assumed to exist in the cell for
the duration of the time step. It is worth noting that this is an
"implicit type" assumption as the flow regimes are based on the
densities calculated at the end of the time step. The next routine
called is LEGEOM (LEVITATE GEOMETRY) which determines the geometrical
characteristics defining each local configuration. This routine will
change the thickness of the fuel crust as necessary or determine the
fraction of the cladding circumference covered by the liquid fuel in the
partial annular flow regime. In general, LEGEOM calculates the area of
contact between various components in various flow regimes. These areas
will be used later in calculating the energy and momentum transfer
between various components.
The next routine called is LETRAN (**LE**\ VITATE **TRAN**\ SFER) which
calculates the heat-transfer and friction coefficients for all axial
locations and among all components that are in direct contact. Thus, the
code will use the flow regimes present at a certain location to
determine which heat transfer and friction coefficients have to be
calculated. The allowable exchanges for each flow regime and each
component have already been presented in :numref:`figure-16.4-3` through :numref:`figure-16.4-10`.
Once contact areas and the corresponding heat-transfer coefficients have
been calculated, the energy conservation equation can be solved for all
components. The LESOEN (**LE**\ VITATE **SO**\ LID, LIQUID AND STATIONARY
**EN**\ ERGY EQUATION) routine is called to solve the energy conservation
equations for the fuel and steel channels, liquid fuel, liquid steel,
stationary cladding, hexcan wall, frozen fuel on the cladding and frozen
fuel on the hexcan wall. All equations are solved explicitly, i.e., the
convective fluxes are based on beginning of time step densities, thus
allowing the axial decoupling of the equations. The energy equation for
sodium (two-phase or single-phase vapor) and fission gas is solved in
LENAEN (**LE**\ VITATE SODIUM-\ **NA** **EN**\ ERGY). The energy equations for
fuel and steel vapor are solved in LEFUVA (**LE**\ VITATE **FU**\ EL
**VA**\ POR ENERGY) and LESEVA (**LE**\ VITATE **S**\ TEEL **VA**\ POR ENERGY),
respectively. The new temperatures calculated in LENAEN, LEFUVA and
LESEVA are used to determine the new pressure of each of the
compressible components and thus the total new pressure. The
hydrodynamic in-pin model is then used to advance the in-pin solution in
the LElPIN and LE2PIN routines. These routines interact with the channel
model via the fuel injection process, which is modeled in the LElPIN
routine. Molten fuel and fission gas are ejected from the cavity into
the channel, leading to changes in the local pressure. The momentum
equation for each of the three velocity fields is then solved in the
routine LEMOCO (**LE**\ VITATE **MO**\ MENTUM **CO**\ NSERVATION). The method
of solution is still explicit and the equations are uncoupled axially,
but the equations for all three fields are solved simultaneously rather
than independently, as was done in the mass and energy equations. Also
it is important to note that the pressures used in the momentum equation
are the pressures at the end of the time step. The routine called next,
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 **FRE**\ E\ **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
**ABLA**\ TION) 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
disrupted 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\ **RU**\ CTURE
**ME**\ LTING) calculates the rupture of the hexcan wall, due to melting
and/or pressure burst effects.
.. _section-16.4.3:
Differential Equations and Finite Difference Equations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-16.4.3.1:
Mass Conservation Equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^
This section describes the solution of the mass conservation equations
for a generic component :math:`k`, as performed in the routine LEMACO. The
region where the :math:`k` component is present has been previously updated in
the routines LEIF and LEREZO, and is defined by two integers
:math:`i_{\text{k,BT}}` and :math:`i_{\text{k,TP}}`, indicating the number of the bottom
and top node of the region, respectively. A mass conservation equation
is solved for each axial node :math:`i`, with :math:`i_{\text{k,BT}} \leq i \leq i_{\text{k,TP}}`.
The original mass conservation is written as:
(16.4-2)
.. _eq-16.4-2:
.. math::
\frac{\partial}{\partial \text{t}}\left( \rho_{\text{k,i}} \cdot A_{\text{k,i}} \
\cdot \Delta z_{\text{i}} \right) + \left\lbrack \left( \rho A u \right)_{\text{k,i+1/2}} \
- \left( \rho A u \right)_{\text{k,i-1/2}} \right\rbrack = 0
No source terms are considered in LEMACO, as all phase changes and
injections are treated in separate routines, which will be described
later.
After dividing by :math:`\text{AXMX} \cdot \Delta z_{\text{i}}` and using the definition of
generalized densities, we obtain:
(16.4-3)
.. _eq-16.4-3:
.. math::
\frac{\partial}{\partial \text{t}}{\rho'}_{\text{k,i}} = - \left\lbrack \left( {\rho'} u \right)_{\text{k,i+1/2}} \
- \left( {\rho'} u \right)_{\text{k,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}}
Finally, after integrating over the length of the time step, :math:`\Delta t`, we
obtain:
(16.4-4)
.. _eq-16.4-4:
.. math::
\Delta{\rho'}_{\text{k,i}} = - \left\lbrack \left( {\rho'} u \right)_{\text{k,i+1/2}} - \left( {\rho'} u \right)_{\text{k,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}}
or
(16.4-5)
.. _eq-16.4-5:
.. math::
{\rho'}_{\text{k,i}}^{n + 1} = {\rho'}_{\text{k,i}}^{n} - \left\lbrack \left( {\rho'} u \right)_{\text{k,i+1/2}} \
- \left( {\rho'} u \right)_{\text{k,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}}
In relation to the use of the superscripts, it is noted that whenever a
time-dependent quantity such as :math:`\left( {\rho'} u \right)` is written without a superscript,
its value is evaluated at the beginning of the computational time step.
The quantity :math:`\left( {\rho'} u \right)_{\text{k,i-1/2}}` is the mass convective flux of
component :math:`k` at the boundary :math:`i-1/2` and is calculated using an upstream
differencing approach:
If :math:`{u'}_{\text{k,i}} \geq 0`,
(16.4-6a)
.. _eq-16.4-6a:
.. math::
\left( {\rho'} u \right)_{\text{k,i-1/2}} = {\rho'}_{\text{k,i-1}} \cdot {u'}_{\text{k,i}}
If :math:`{u'}_{\text{k,i}} < 0`, then
(16.4-6b)
.. _eq-16.4-6b:
.. math::
\left( {\rho'} u \right)_{\text{k,i-1/2}} = {\rho'}_{\text{k,i}} \cdot {u''}_{\text{k,i}}
The velocities :math:`{u'}_{\text{k,i}}` and :math:`{u''}_{\text{k,i}}` are correlated by the
mass continuity equation across boundary :math:`i-1/2`:
(16.4-7)
.. _eq-16.4-7:
.. math::
{u'}_{\text{k,i}} \rho_{\text{k,i-1}} A_{\text{k,i-1}} = {u''}_{\text{k,i}} \cdot \rho_{\text{k,i}} \cdot A_{\text{k,i}}
The solution of the mass equation is illustrated below for the molten
fuel component. The molten fuel region extends from :math:`i_{\text{fu,BT}} = \text{IFFUBT}`
to :math:`i_{\text{fu,TP}} = \text{IFFUTP}`. The first step in the solution is to
calculate the convective fuel fluxes at all internal cell boundaries. It
is important to note that the convective fluxes defined at boundary
:math:`i-1/2` have the subscript :math:`i`, in the code, i.e., :math:`\text{COFUCH} \left( I \right)`. Similarly,
those defined at the boundary :math:`i+1/2` have the subscript :math:`i+1`.
Note that the velocity :math:`{u''}_{\text{k,i}}` is stored as :math:`\text{UFCH} \left( I \right)`, and the
velocity :math:`{u'}_{\text{k,i}}` used in the previous equations is calculated as:
(16.4-8)
.. _eq-16.4-8:
.. math::
{u'}_{\text{fu,i}} = {u''}_{\text{fu,i}} \cdot \text{CCFU} \left( I \right)
where the coefficient :math:`\text{CCFU} \left( I \right)` is defined by the Eq. :ref:`16.4-7<eq-16.4-7>` as:
(16.4-9)
.. _eq-16.4-9:
.. math::
\text{CCFU} \left( I \right) = \frac{{u'}_{\text{fu,i}}}{{u''}_{\text{fu,i}}} \
= \frac{\rho_{\text{fu,i}} \cdot A_{\text{fu,i}}}{\rho_{\text{fu,i-1}} \cdot A_{\text{fu,i-1}}} \
= \frac{{\rho'}_{\text{fu,i}}}{{\rho'}_{\text{fu,i-1}}}
The convective fluxes through the boundaries of the fuel region are set
to zero, as no fuel is allowed to cross these boundaries in the LEMACO
routine.
(16.4-10a)
.. _eq-16.4-10a:
.. math::
\left( {\rho'} u \right)_{\text{fu,IFFUBT}} = 0
(16.4-10b)
.. _eq-16.4-10b:
.. math::
\left( {\rho'} u \right)_{\text{fu,IFFUTP+1}} = 0
To preserve the accuracy of results in the boundary cells a correction
is applied to the convective fluxes through all boundaries :math:`\text{IFFUBT}+1` and
:math:`\text{IFFUTP}` whenever the corresponding interface (bottom and top) crosses
that boundary during the time step. This correction is explained below
for the case when the top fuel boundary crosses the cell boundary. The
location of the interface at the end of the time step has already been
calculated in the routine LEIF and is shown in :numref:`figure-16.4-12`. The value
of :math:`\left( {\rho'} u \right)_{\text{fu,IFFUTP}}` is calculated according to Eq. :ref:`16.4-7<eq-16.4-7>` and then
used in Eq. :ref:`16.4-5<eq-16.4-5>`. The implicit assumption in Eq. :ref:`16.4-5<eq-16.4-5>` is that the
value of the convective fluxes is constant over the length of the time
step. However, as shown in :numref:`figure-16.4-12` the original location of the
fuel interface was :math:`\text{FUIFTP}^{0} \left( 1 \right)`, and, before the interface
reaches the cell boundary :math:`\text{ZC(IFFUTP)}`, the flux across the boundary is
zero. The fraction of :math:`\Delta t` during which the convective flux is present is:
(16.4-11)
.. _eq-16.4-11:
.. math::
F_{\text{correction}} = \frac{\text{FUIFTP} \left( 1 \right) - \text{ZC}\left( \text{IFFUTP} \right)}{\text{FUIFTP}^{0}\left( 1 \right) \
- \text{FUIFTP}\left( 1 \right)}
.. _figure-16.4-12:
.. figure:: media/image20.png
:align: center
:figclass: align-center
:numref:`figure-16.4-12`: Correction for Material Boundary Crossing a Cell Boundary
In order to maintain the form of Eq. :ref:`16.4-5<eq-16.4-5>`, rather than changing the
value of :math:`\Delta t` associated with :math:`\left( {\rho'} u \right)_{\text{fu,IFFUTP}}` a corrected flux is
defined, such that the product flux \* time is correct:
(16.4-12)
.. _eq-16.4-12:
.. math::
\left( {\rho'} u \right)_{\text{fu,IFFUTP}} = \left( {\rho'} u \right)_{\text{fu,IFFUTP}} \cdot F_{\text{correction}}
A similar correction is applied for the bottom boundary.
Similar mass conservation equations are solved for all other LEVITATE
components, i.e. molten steel, fuel and steel chunks, fission gas,
sodium, steel vapor, fuel vapor, fission gas still present in the chunks
and fission gas present in the molten fuel.
.. _section-16.4.3.2:
Liquid Fuel Energy Conservation Equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The liquid fuel energy equation is solved in the routine LESOEN, for all
cells in the molten fuel region, i.e., :math:`\text{IFFUBT} \leq I \leq \text{IFFUTP}`. We begin with
the energy equation in conservation form:
(16.4-13)
.. _eq-16.4-13:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}}\left( \rho_{\text{fu,i}} \cdot h_{\text{fu,i}} \cdot A_{\text{fu,i}} \cdot \Delta z_{\text{i}} \right) + \left\lbrack \left( \rho A h u \right)_{\text{i+1/2}} - \left( \rho A h u \right)_{\text{i-1/2}} \right\rbrack \\
= Q_{\text{fu,i}} \cdot \rho_{\text{fu,i}} \cdot A_{\text{fu,i}} \cdot \Delta z_{\text{i}} - \sum_{\text{j}}{H_{\text{fu,j,i}} \cdot A_{\text{fu,j,i}} \cdot \Delta T_{\text{fu,j,i}}} \\
\end{matrix}
where:
:math:`Q_{\text{fu,i}}` = fission power source in cell :math:`i` [J/(kg-s)]
| :math:`H_{\text{fu,j,i}}` = heat-transfer coefficient between fuel and
component :math:`j` in cell :math:`i`
| [J/(m\ :sup:`2` - s - K)]
:math:`A_{\text{fu,j,i}}` = heat-transfer area between fuel and component :math:`j` in
cell :math:`i` [m\ :sup:`2`]
After dividing by :math:`\text{AXMX} \cdot \Delta z_{\text{i}}` and using the definition of
generalized densities, we obtain:
(16.4-14)
.. _eq-16.4-14:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}}\left( {\rho'}_{\text{fu,i}} \cdot h_{\text{fu,i}} \right) = - \left\lbrack \left( {\rho'} h u \right)_{\text{i+1/2,fu}} - \left( {\rho'} h u \right)_{\text{i-1/2,fu}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}} \\
+ Q_{\text{fu,i}} \cdot {\rho'}_{\text{fu,i}} - \sum_{\text{j}}{{H'}_{\text{fu,j,i}} \cdot \Delta T_{\text{fu,j,i}}} \\
\end{matrix}
where
(16.4-15)
.. _eq-16.4-15:
.. math::
{H'}_{\text{fu,j,i}} = H_{\text{fu,j,i}} \cdot \frac{A_{\text{fu,j,i}}}{\text{AXMX} \cdot \Delta z_{\text{i}}}
Integrating over :math:`\Delta t` and using the identity:
(16.4-16)
.. _eq-16.4-16:
.. math::
\Delta \left( {\rho'} h \right) = {\rho'}^{n + 1} \cdot \Delta h + h^{n} \cdot \Delta{\rho'}
we obtain:
(16.4-17)
.. _eq-16.4-17:
.. math::
\begin{matrix}
{\rho'}_{\text{fu,i}}^{n + 1} \Delta h_{\text{fu,i}} = - \left\lbrack \left( {\rho'} h u \right)_{\text{i+1/2,fu}} - \left( {\rho'} h u \right)_{\text{i-1/2,fu}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
- h_{\text{fu,i}}^{n} \cdot \Delta{\rho'}_{\text{fu,i}} + {Q'}_{\text{fu,i}} \cdot {\rho'}_{\text{fu,i}} \cdot \Delta t \\
- \sum_{\text{j}}{{H'}_{\text{fu,j,i}} \cdot \Delta T_{\text{fu,j,i}} \cdot \Delta t} \\
\end{matrix}
Finally, dividing Eq. :ref:`16.4-17<eq-16.4-17>` by :math:`{\rho'}_{\text{fu,i}}^{n + 1}`
and expressing :math:`\Delta{\rho'}_{\text{fu,i}}` as:
(16.4-18)
.. _eq-16.4-18:
.. math::
\Delta {\rho'}_{\text{fu,i}} = - \left\lbrack \left( {\rho'} u \right)_{\text{fu,i+1/2}} \
- \left( {\rho'} u \right)_{\text{fu,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}}
from the continuity Eq. :ref:`16.4-4<eq-16.4-4>`, we obtain the change in the fuel
enthalpy:
(16.4-19)
.. _eq-16.4-19:
.. math::
\begin{matrix}
\Delta h_{\text{fu,i}} = \left\{ - \left\lbrack \left( {\rho'} h u \right)_{\text{fu,i+1/2}} - \left( {\rho'} h u \right)_{\text{fu,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}} \\
+ h_{\text{fu,i}}^{n} \left\lbrack \left( {\rho'} u \right)_{\text{fu,i+1/2}} - \left( {\rho'} u \right)_{\text{fu,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}} \\
+ Q_{\text{fu,i}} \cdot {\rho'}_{\text{fu,i}} - \sum_{\text{j}}{{H'}_{\text{fu,j,i}} \cdot \Delta T_{\text{fu,j,i}}} \right\} \cdot \frac{\Delta t}{{\rho'}_{\text{fu,i}}^{n + 1}} \\
\end{matrix}
The new fuel enthalpy is obtained as:
(16.4-20)
.. _eq-16.4-20:
.. math::
h_{\text{fu,i}}^{n + 1} = h_{\text{fu,i}}^{n} + \Delta h_{\text{fu,i}}
The energy convective terms in Eq. :ref:`16.4-19<eq-16.4-19>` are calculated using an
upstream differencing approach, i.e.;
(16.4-21)
.. _eq-16.4-21:
.. math::
\left( {\rho'} h u \right)_{\text{fu,i-1/2}} = \begin{cases}
{\rho'}_{\text{fu,i-1}} \cdot h_{\text{fu,i-1}} \cdot {u'}_{\text{fu,i}} & \text{if } {u'}_{\text{fu,i}} \geq 0 \\
{\rho'}_{\text{fu,i}} \cdot h_{\text{fu,i}} \cdot {u''}_{\text{fu,i}} & \text{if } {u'}_{\text{fu,i}} < 0 \\
\end{cases}
These fluxes are based on the fuel densities, enthalpies and velocities
at the of the time step. They are calculated in the routine LEMACO,
before the calculation of the new densities, and stored in the array
:math:`\text{COFUOS} \left( I \right)`. The sum of the heat-transfer contributions,
:math:`\sum_{\text{j}}{{H'}_{\text{fu,j,i}} \cdot \Delta T_{\text{fu,j,i}}}` is
presented in detail in Eq. :ref:`16.4-22<eq-16.4-22>`.
In the code, Eq. :ref:`16.4-19<eq-16.4-19>` is written as:
(16.4-22)
.. _eq-16.4-22:
.. math::
\begin{matrix}
\text{DEGEOS} = \left\lbrack - \text{COFUOA} + \text{COFUOB} + \text{HSFU} \left( I \right) * \text{DEFUCH} \left( I \right) \\
- \text{HTFUNA}\left( I \right) * \left( \text{TEFUOS}\left( I \right) - \text{TENA} \left( I \right) \right) \\
- \text{HTFUCL}\left( I \right) * \left( \text{TEFUOS}\left( I \right) - \text{TECLOS}\left( I \right) \right) \\
- \text{HTFUFL}\left( I \right) * \left( \text{TEFUOS}\left( I \right) - \text{TELUCH}\left( I \right) \right) \\
- \text{HTFUSL}\left( I \right) * \left( \text{TEFUOS}\left( I \right) - \text{TESELU}\left( I \right) \right) \\
- \text{HTFUSR}\left( I \right) * \left( \text{TEFUOS}\left( I \right) - \text{TESROS}\left( I \right) \right) \\
- \text{HTSEFU}\left( I \right) * \left( \text{TEFUOS}\left( I \right) - \text{TESECH}\left( I \right) \right) \right\rbrack \\
* \text{DTPLU} \big/ \text{DEFUCH}\left( I \right)
\end{matrix}
where
(16.4-23)
.. _eq-16.4-23:
.. math::
\text{COFUOA} = \left\lbrack \left( {\rho'} h u \right)_{\text{fu,i+1/2}} - \left( {\rho'} h u \right)_{\text{fu,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}}
(16.4-24)
.. _eq-16.4-24:
.. math::
\text{COFUOB} = h_{\text{fu,i}}^{n} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{fu,i+1/2}} \
- \left( {\rho'} u \right)_{\text{fu,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}}
:math:`\text{HTFUNA} \left( I \right) = {H'}_{\text{fu,mi,i}}` = generalized heat-transfer coefficient
between fuel and gas mixture in cell :math:`i`
:math:`\text{HTFUCL} \left( I \right) = {H'}_{\text{fu,cl,i}}` = generalized heat-transfer coefficient
between fuel and cladding in cell :math:`i`
:math:`\text{HTFUFL} \left( I \right) = {H'}_{\text{fu,fl,i}}` = generalized heat-transfer coefficient
between fuel and fuel chunks in cell :math:`i`
:math:`\text{HTFUSL} \left( I \right) = {H'}_{\text{fu,sl,i}}` = generalized heat-transfer coefficient
between fuel and steel chunks in cell :math:`i`
:math:`\text{HTFUSR} \left( I \right) = {H'}_{\text{fu,sr,i}}` = generalized heat-transfer coefficient
between fuel and hexcan wall in cell :math:`i`
:math:`\text{HTSEFU} \left( I \right) = {H'}_{\text{fu,se,i}}` = generalized heat-transfer coefficient
between fuel and molten steel in cell :math:`i`
The generalized transfer coefficients :math:`{H'}_{\text{fu,j,i}}` are related to
the heat-transfer coefficients :math:`H_{\text{fu,j,i}}` and the transfer areas
:math:`A_{\text{fu,j,i}}` by Eq. :ref:`16.4-15<eq-16.4-15>`. These transfer coefficients are
calculated prior to the energy equation solution in the routine LETRAN
(**LE**\ VITATE **TRAN**\ SFER). Depending on the flow regime, some of those
coefficients can be zero, as illustrated in the decision arrays :numref:`figure-16.4-3` through :numref:`figure-16.4-6`.
After the energy change is calculated, and the new energy is obtained
according to Eq. :ref:`16.4-20<eq-16.4-20>`, the temperature of the fuel is calculated by
using an external function, :math:`\text{TEFUEG}`, which uses the enthalpy as argument
(16.4-25)
.. _eq-16.4-25:
.. math::
T_{\text{fu,i}}^{n + 1} = T \left( h_{\text{fu,i}}^{n + 1} \right)
Several checks are performed on the final temperature in order to avoid
numerical difficulties. Thus, if only small amounts of fuel are present
(fuel volume less than .1% of the cell volume) the temperature of the
fuel is set equal to the cladding temperature or, if the fuel is
surrounded by molten steel, to the molten steel temperature.
The condensation/vaporization energy sources/sinks are not included
here. They will be introduced later in this chapter, when presenting the
energy conservation calculation for fuel vapor.
.. _section-16.4.3.3:
Liquid Steel Energy Conservation Equation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The liquid steel energy equation is solved in the routine LESOEN, for
all cells in the molten steel region, i.e. :math:`\text{IFSEBT} \leq I \leq \text{IFSETP}`. The
energy conservation equation is similar to the molten fuel equation, but
does not include a fission source term. Thus, the change in the steel
enthalpy is:
(16.4-26)
.. _eq-16.4-26:
.. math::
h_{\text{se,i}} = \left\{ - \left\lbrack \left( {\rho'} h u \right)_{\text{se,i+1/2}} \
- \left( {\rho'} h u \right)_{\text{se,i-1/2}} \right\rbrack \cdot \frac{1}{z_{\text{i}}} \
+ h_{\text{se,i}}^{n} \left\lbrack \left( {\rho'} u \right)_{\text{se,i+1/2}} \
- \left( {\rho'} u \right)_{\text{se,i-1/2}} \right\rbrack \cdot \frac{1}{z_{\text{i}}} \
- \sum_{\text{j}}{{H'}_{\text{se,j,i}} \cdot T_{\text{se,j,i}}} \right\} \cdot \frac{t}{{\rho'}_{\text{se,i}}^{n + 1}}
The new steel enthalpy is obtained as:
(16.4-27)
.. _eq-16.4-27:
.. math::
h_{\text{se,i}}^{n + 1} = h_{\text{se,i}}^{n} + \Delta h_{\text{se,i}}
The energy convective terms in Eq. :ref:`16.4-26<eq-16.4-26>` are calculated using an
upstream differencing approach, similar to Eq. :ref:`16.4-19<eq-16.4-19>`. They are
calculated in routine :math:`\text{LEMACO}` and stored under the name :math:`\text{COSEOS} \left( I \right)`. These
fluxes are based on steel densities enthalpies and velocities at the
beginning of the time step.
If :math:`{u'}_{\text{se,i}} \geq 0`,
(16.4-28a)
.. _eq-16.4-28a:
.. math::
\left( {\rho'} h u \right)_{\text{se,i-1/2}} = {\rho'}_{\text{se,i-1}} \cdot h_{\text{se,i-1}} \cdot {u'}_{\text{se,i}}
If :math:`{u'}_{\text{se,i}} < 0`, then
(16.4-28b)
.. _eq-16.4-28b:
.. math::
\left( {\rho'} h u \right)_{\text{se,i-1/2}} = {\rho'}_{\text{se,i}} \cdot h_{\text{se,i}} \cdot {u''}_{\text{se,i}}
As previously explained, the molten steel and molten fuel share the same
velocity field. Thus :math:`{u'}_{\text{se,i}}` and :math:`{u'}_{\text{fu,i}}` have the same
value, stored in the array :math:`\text{UFCH} \left( I \right)`.
In the code, Eq. :ref:`16.4-26<eq-16.4-26>` is written as:
(16.4-29)
.. _eq-16.4-29:
.. math::
\begin{matrix}
\text{DEEGSE} = \left\lbrack - \text{COSEOA} + \text{COSEOB} \\
- \text{HTSEFU} \left( I \right) * \left( \text{TESECH} \left( I \right) - \text{TEFUOL} \left( I \right) \right) \\
- \text{HTSECL} \left( I \right) * \left( \text{TESECH} \left( I \right) - \text{TECLOS} \left( I \right) \right) \\
- \text{HTSESR} \left( I \right) * \left( \text{TESECH} \left( I \right) - \text{TESROS} \left( I \right) \right) \\
- \text{HTSEFL} \left( I \right) * \left( \text{TESECH} \left( I \right) - \text{TELUCH} \left( I \right) \right) \\
- \text{HTSESL} \left( I \right) * \left( \text{TESECH} \left( I \right) - \text{TESELU} \left( I \right) \right) \\
- \text{HTSENA} \left( I \right) * \left( \text{TESECH} \left( I \right) - \text{TENA} \left( I \right) \right) \right\rbrack \\
* \text{DTPLU} \big/ \text{DESECH} \left( I \right) \\
\end{matrix}
where
(16.4-30)
.. _eq-16.4-30:
.. math::
\text{COSEOA} = \left\lbrack \left( {\rho'} h u \right)_{\text{se,i+1/2}} \
- \left( {\rho'} h u \right)_{\text{se,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}}
(16.4-31)
.. _eq-16.4-31:
.. math::
\text{COSEOB} = h_{\text{se,i}}^{n} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{se,i+1/2}} \
- \left( {\rho'} u \right)_{\text{se,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}}
:math:`\text{HTSEFU} \left( I \right) = {H'}_{\text{se,fu,i}}` = transfer coefficient between steel and
fuel in cell :math:`i`
:math:`\text{HTSECL} \left( I \right) = {H'}_{\text{se,cl,i}}` = transfer coefficient between steel and
cladding in cell :math:`i`
:math:`\text{HTSESR} \left( I \right) = {H'}_{\text{se,sr,i}}` = transfer coefficient between steel and
hexcan wall in cell :math:`i`
:math:`\text{HTSEFL} \left( I \right) = {H'}_{\text{se,fl,i}}` = transfer coefficient between steel and
fuel chunks in cell :math:`i`
:math:`\text{HTSESL} \left( I \right) = {H'}_{\text{se,sl,i}}` = transfer coefficient between steel and
steel chunks in cell :math:`i`
:math:`\text{HTSENA} \left( I \right) = {H'}_{\text{se,mi,i}}` = transfer coefficient between steel and
gas mixture in cell :math:`i`
The generalized heat-transfer coefficients are defined by:
(16.4-32)
.. _eq-16.4-32:
.. math::
{H'}_{\text{se,j,i}} = H_{\text{se,j,i}} \cdot \frac{A_{\text{se,j,i}}}{\text{AXMX} \cdot \Delta z_{\text{i}}}
and are described in detail in :numref:`section-16.4.3.10`.
After the energy change is calculated, the new energy of the steel is
calculated according to Eq. :ref:`16.4-27<eq-16.4-27>`, and the temperature of the steel is
obtained by using an external function, TESEEG, which uses the enthalpy
as argument:
(16.4-33)
.. _eq-16.4-33:
.. math::
T_{\text{se,i}} = T \left( h_{\text{se,i}}^{n + 1} \right)
Several checks are performed during the steel temperature calculation in
order to avoid numerical difficulties. Thus, if only small amounts of
molten steel are present (steel volume less than .1% of the cell volume)
the temperature of the steel is set equal to the cladding temperature
or, if the steel is surrounded by molten fuel, to the molten fuel
temperature.
The condensation/vaporization energy sources/sinks are not included
here. They will be introduced later in this chapter, when the energy
conservation calculation for steel vapor is presented.
.. _section-16.4.3.4:
Fuel and Steel Chunk Energy Conservation Equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
These equations are solved in the routine LESOEN. Separate equations are
solved for the fuel and steel solid chunks, as described below.
The Energy Conservation Equation for the Fuel Chunks
''''''''''''''''''''''''''''''''''''''''''''''''''''
The energy equation is written in conservative form as follows:
(16.4-34)
.. _eq-16.4-34:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}}\left( \rho_{\text{f} l \text{,i}} \cdot h_{\text{f} l \text{,i}} A_{\text{f} l \text{,i}} \cdot \Delta z \right) + \left\lbrack \left( \rho A h u \right)_{\text{f} l \text{,i+1/2}} - \left( \rho A h u \right)_{\text{fl,i-1/2}} \right\rbrack \\
= Q_{\text{f} l \text{,i}} \cdot \rho_{\text{f} l \text{,i}} \cdot A_{\text{f} l \text{,i}} \cdot \Delta z_{\text{i}} - \sum_{\text{j}}{H_{\text{f} l \text{,j,i}} \cdot A_{\text{f} l \text{,j,i}} \cdot \Delta T_{\text{f} l \text{,j,i}}} \\
\end{matrix}
where
:math:`Q_{\text{fl,i}}` = fission power source in the fuel in cell :math:`i` [J/kg-s]
:math:`A_{\text{fl,i}}` = area covered by chunks in cell :math:`i`, when imagined as a
continuum with density :math:`\rho_{\text{fl}}`.
Following the same steps as outlined in Eq. :ref:`16.4-14<eq-16.4-14>` through :ref:`16.4-19<eq-16.4-19>`, we
obtain:
(16.4-35)
.. _eq-16.4-35:
.. math::
h_{\text{f} l \text{,i}} = \left\{ - \left\lbrack \left( {\rho'} h u \right)_{\text{f} l \text{,i+1/2}} \
- \left( {\rho'} \text{hu} \right)_{\text{f} l \text{,i-1/2}} \right\rbrack \cdot \frac{1}{z_{\text{i}}} \
+ h_{\text{f} l \text{,i}}^{n} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{f} l \text{,i+1/2}} \
- \left( {\rho'} u \right)_{\text{f} l \text{,i-1/2}} \right\rbrack \cdot \frac{1}{z_{\text{i}}} \
+ Q_{\text{f} l \text{,i}} \cdot {\rho'}_{\text{f} l \text{,i}} \
- \sum_{\text{j}}{{H'}_{\text{f} l \text{,j,i}} \cdot {T}_{\text{f} l \text{,j,i}}} \right\} \
\cdot \frac{t}{{\rho'}_{\text{f} l \text{,i}}^{n + 1}}
where
(16.4-36)
.. _eq-16.4-36:
.. math::
\begin{matrix}
\sum_{\text{j}}{{H'}_{\text{f} l \text{,j,i}} \cdot \Delta T_{\text{f} l \text{,j,i}} = {H'}_{\text{fu,f} l \text{,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{fu,i}} \right) + {H'}_{\text{se,f} l \text{,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{se,i}} \right)} \\
+ {H'}_{\text{Na,f} l \text{,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{Na,i}} \right) + {H'}_{\text{cl,f} l \text{,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{cl,i}} \right) \\
+ {H'}_{\text{sr,f} l \text{,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{sr,i}} \right) + {H'}_{\text{ffc,f} l \text{,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{ffc,i}} \right) \\
+ {H'}_{\text{ffs,f} l \text{,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{ffs,i}} \right) + {H'}_{\text{f} l \text{,sl,i}} \cdot \left( T_{\text{f} l \text{,i}} - T_{\text{s} l \text{,i}} \right) \\
\end{matrix}
The energy convective terms in Eq. :ref:`16.4-35<eq-16.4-35>` are calculated using an
upstream differencing approach, similar to that used for Eq. :ref:`16.4-21<eq-16.4-21>`.
The convective fluxes are calculated in the routine LEMACO, before the
calculation of the new fuel chunk densities, and stored in the array
:math:`\text{COLVOS} \left( I \right)`. The new fuel chunk enthalpy and temperature are then
calculated:
(16.4-37)
.. _eq-16.4-37:
.. math::
h_{\text{f} l \text{,i}}^{n + 1} = h_{\text{f} l \text{,i}}^{n} + \Delta h_{\text{f} l \text{,i}}^{n}
(16.4-38)
.. _eq-16.4-38:
.. math::
T_{\text{f} l \text{,i}}^{n + 1} = T \left( h_{\text{f} l \text{,i}}^{n + 1} \right)
.. _section-16.4.3.4.1:
The Energy Conservation for the Steel Chunks
''''''''''''''''''''''''''''''''''''''''''''
This equation is very similar to the equation used for the fuel chunks,
but the fission energy source is not present anymore. Only the final
form of the equation is presented here:
(16.4-39)
.. _eq-16.4-39:
.. math::
\begin{matrix}
\Delta h_{\text{s} l \text{,i}} = \left\{ - \left\lbrack \left( {\rho'} h u \right)_{\text{s} l \text{,i+1/2}} - \left( {\rho'} h u \right)_{s l \text{,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}} \\
+ h_{\text{s} l \text{,i}}^{n} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{s} l \text{,i+1/2}} - \left( {\rho'} u \right)_{\text{s} l \text{,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}} \\
- \sum_{\text{j}}{{H'}_{\text{s} l \text{,j,i}} \cdot \Delta T_{\text{s} l \text{,j,i}}} \right\} \cdot \frac{\Delta t}{{\rho'}_{\text{s} l \text{,i}}^{n + 1}} \\
\end{matrix}
where
(16.4-40)
.. _eq-16.4-40:
.. math::
\begin{matrix}
\sum_{\text{j}}{{H'}_{\text{s} l \text{,j,i}} \cdot \Delta T_{\text{s} l \text{,j,i}} = {H'}_{\text{fu,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{fu,i}} \right) + {H'}_{\text{se,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{se,i}} \right)} \\
+ {H'}_{\text{Na,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{Na,i}} \right) + {H'}_{\text{cl,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{cl,i}} \right) \\
+ {H'}_{\text{sr,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{sr,i}} \right) + {H'}_{\text{ffc,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{ffc,i}} \right) \\
+ {H'}_{\text{ffs,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{ffs,i}} \right) + {H'}_{\text{f} l \text{,s} l \text{,i}} \cdot \left( T_{\text{s} l \text{,i}} - T_{\text{f} l \text{,i}} \right) \\
\end{matrix}
The new steel chunk enthalpy and temperature are then calculated as
follows:
(16.4-41)
.. _eq-16.4-41:
.. math::
h_{\text{s} l \text{,i}}^{n + 1} = h_{\text{s} l \text{,i}}^{n} + \Delta h_{\text{s} l \text{,i}}^{n}
(16.4-42)
.. _eq-16.4-42:
.. math::
T_{\text{s} l \text{,i}}^{n + 1} = T \left( h_{\text{s} l \text{,i}}^{n + 1} \right)
.. _section-16.4.3.5:
Sodium and Fission-gas Energy Conservation Equation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The sodium and the fission gas are assumed in LEVITATE to be in
equilibrium at the same temperature :math:`T_{\text{Na,i}}`. The sodium can exist
in the form of superheated vapor or as a two-phase, vapor-liquid
mixture. When required by the thermodynamic conditions the two-phase
sodium component can become subcooled sodium. The two-phase sodium is
assumed to be in thermodynamic equilibrium. The energy conservation
equation is solved simultaneously for sodium and fission gas for all
cells in the LEVITATE interaction region, i.e., :math:`\text{IFMIBT} \leq I \leq \text{IFMITP}`.
Because the fission-gas region extends only between :math:`\text{IFFIBT} \leq I \leq \text{IFFITP}`,
it is possible that some cells contain only sodium, without any fission
gas. In these cells, the energy equation for the sodium-fission-gas
mixture reduces to a sodium-only equation. Because significant
differences exist between the behavior of superheated and two phase
sodium, two separate equations are used.
.. _section-16.4.3.5.1:
The Energy Conservation Equation for Superheated Sodium and Fission Gas
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
The energy equation used for superheated sodium and fission gas is
written in conservative form as:
(16.4-43)
.. _eq-16.4-43:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}} \left( \rho_{\text{Na,i}} h_{\text{Na,i}} + \rho_{\text{fi,i}} h_{\text{fi,i}} \right) \cdot A_{\text{Mi,i}} \cdot \Delta z_{\text{i}} + \left\lbrack \left( \rho A h u \right)_{\text{Na,i+1/2}} \\
+ \left( \rho \text{Ahu} \right)_{\text{fi,i+1/2}} - \left( \rho A h u \right)_{\text{Na,1-1/2}} - \left( \rho A h u \right)_{\text{fi,i-1/2}} \right\rbrack \\
= A_{\text{Mi,i}} \left( \frac{\partial \text{P}_{\text{Na,i}}}{\partial \text{t}} + \frac{\partial \text{P}_{\text{fi,i}}}{\partial \text{t}} \right) \cdot \Delta z_{\text{i}} + A_{\text{Mi,i}} \cdot \left\lbrack u_{\text{Mi,i}} \cdot 0.5 \cdot \left( P_{\text{Na,i}} + P_{\text{fi,i}} \\
- P_{\text{Na,i-1}} - P_{\text{fi,i-1}} \right) + U_{\text{Mi,i+1}} \cdot 0.5 \cdot \left( P_{\text{Na,i+1}} + P_{\text{fi,i+1}} - P_{\text{Na,i}} - P_{\text{fi,i}} \right) \right\rbrack \\
+ \sum_{\text{j}}{H_{\text{j,Mi,i}} \cdot A_{\text{j,Mi,i}} \cdot \Delta T_{\text{j,Mi,i}}} \\
\end{matrix}
where:
:math:`H_{\text{j,Mi,i}}` = heat-transfer coefficient between component :math:`j` and
sodium-fission gas mixture in cell :math:`i` [J/m\ :sup:`2` - s- K]
:math:`A_{\text{j,Mi,i}}` = heat-transfer area between component :math:`j` and the gas
mixture in cell :math:`i` [m\ :sup:`2`]
:math:`\Delta T_{\text{j,Mi,I}}` = temperature difference between component :math:`j` and the
gas mixture in cell :math:`i` [K]
After dividing by :math:`\text{AXMX} \cdot \Delta z_{\text{i}}` and using the definition of
generalized densities, we obtain:
(16.4-44)
.. _eq-16.4-44:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}} \left( {\rho'}_{\text{Na,i}} h_{\text{Na,i}} + {\rho'}_{\text{fi,i}} h_{\text{fi,i}} \right) = - \left\lbrack \left( {\rho'} h u \right)_{\text{Na,i+1/2}} + \left( {\rho'} h u \right)_{\text{fi,i+1/2}} \\
- \left( {\rho'} h u \right)_{\text{Na,i-1/2}} - \left( {\rho'} h u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}} + \theta_{\text{Mi,i}} \cdot \left( \frac{\partial \text{P}_{\text{Na,i}}}{\partial \text{t}} + \frac{\partial \text{P}_{\text{fi,i}}}{\partial \text{t}} \right) \\
+ \theta_{\text{Mi,i}} \cdot \frac{0.5}{\Delta z_{\text{i}}} \cdot \left\lbrack u_{\text{Mi,i}} \cdot \left( P_{\text{Na,i}} + P_{\text{fi,i}} - P_{\text{Na,i-1}} - P_{\text{fi,i-1}} \right) \\
+ u_{\text{Mi,i+1}} \cdot \left( P_{\text{Na,i+1}} + P_{\text{fi,i+1}} - P_{\text{Na,i}} - P_{\text{fi,i}} \right) \right\rbrack + \sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \cdot \Delta T_{\text{j,Mi,i}}} \\
\end{matrix}
where the generalized heat-transfer coefficients are defined by:
(16.4-45)
.. _eq-16.4-45:
.. math::
{H'}_{\text{j,Mi,i}} = H_{\text{j,Mi,i}} \cdot \frac{A_{\text{j,Mi,i}}}{\text{AXMX} \cdot \Delta z_{\text{i}}}
and the sum of the heat-transfer terms
:math:`\sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \cdot \Delta T_{\text{j,Mi,i}}}` is
presented in Eq. :ref:`16.4-54<eq-16.4-54>`. Integrating over :math:`\Delta t` and using the identity
shown in Eq. :ref:`16.4-16<eq-16.4-16>` to express the quantity :math:`\Delta \left( {\rho'} h \right)`, we obtain:
(16.4-46)
.. _eq-16.4-46:
.. math::
\begin{matrix}
{\rho'}_{\text{Na,i}}^{n + 1} \cdot \Delta h_{\text{Na,i}} + {\rho'}_{\text{fi,i}}^{n + 1} \cdot \Delta h_{\text{fi,i}} \\
= - \left\lbrack \left( {\rho'} h u \right)_{\text{Na,i+1/2}} + \left( {\rho'} h u \right)_{\text{fi,i+1/2}} \\
- h_{\text{Na,i}} \cdot \Delta {\rho'}_{\text{Na,i}} - h_{\text{fi,i}} \cdot \Delta {\rho'}_{\text{fi,i}} \\
- \left( {\rho'} h u \right)_{\text{Na,i-1/2}} - \left( {\rho'} h u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
+ \theta_{\text{Mi,i}} \cdot \left( \Delta P_{\text{Na,i}} + \Delta P_{\text{fi,i}} \right) + \theta_{\text{Mi,i}} \cdot \frac{0.5 \cdot \Delta t}{\Delta z_{\text{i}}} \cdot \left\lbrack u_{\text{Mi,i}} \cdot \left( P_{\text{Na,i}} + P_{\text{fi,i}} \\
- P_{\text{Na,i-1}} - P_{\text{fi,i-1}} \right) + u_{\text{Mi,i+1}} \cdot \left( P_{\text{Na,i+1}} + P_{\text{fi,i+1}} - P_{\text{Na,i}} - P_{\text{fi,i}} \right) \right\rbrack \\
+ \sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \cdot \Delta T_{\text{j,Mi,i}} \cdot \Delta t} \\
\end{matrix}
Two sets of equation relating the thermodynamic characteristics of a gas
are used to refine Eq. :ref:`16.4-46<eq-16.4-46>`. These are:
(16.4-47a)
.. _eq-16.4-47a:
.. math::
h_{\text{Na,i}} = 2.5 R_{\text{Na}} \cdot \left( T_{\text{Na}} - T_{\text{sat}} \right) + h_{\text{Na,vap}}
(16.4-47b)
.. _eq-16.4-47b:
.. math::
h_{\text{fi,i}} = 2.5 R_{\text{fi}} \cdot T_{\text{fi}}
and
(16.4-48a)
.. _eq-16.4-48a:
.. math::
P_{\text{Na,i}} = R_{\text{Na}} \cdot {\rho'}_{\text{Na,i}} \cdot T_{\text{Na,i}} \cdot \frac{1}{\theta_{\text{Na,i}}}
(16.4-48b)
.. _eq-16.4-48b:
.. math::
P_{\text{fi,i}} = R_{\text{fi}} \cdot {\rho'}_{\text{fi,i}} \cdot T_{\text{fi,i}} \cdot \frac{1}{\theta_{\text{fi,i}}}
where the constant :math:`R_{\text{j}}` is defined as:
(16.4-49)
.. _eq-16.4-49:
.. math::
P_{\text{j}} = \frac{R}{M_{\text{j}}}
with
:math:`R` = universal gas constant [J/mol - K]
:math:`M_{\text{j}}` = molar mass of gas :math:`j` [kg/mol]
:math:`T_{\text{sat}} = T_{\text{sat}} \left( P_{\text{Na,i}} \right)` ;
:math:`h_{\text{Na,vap}} = h_{\text{Na,vap}} \left( P_{\text{Na,i}} \right)`.
With :math:`T_{\text{Na,i}} = T_{\text{fi,i}}` and :math:`\theta_{\text{Na,i}} = \theta_{\text{fi,i}} = \theta_{\text{Mi,i}}` ,
after expressing the quantity :math:`\Delta T_{\text{j,Mi,i}}` as:
(16.4-50)
.. _eq-16.4-50:
.. math::
\Delta T_{\text{j,Mi,i}} = T_{\text{j,i}} - T_{\text{Mi,i}}^{n + 1} = T_{\text{j,i}} - T_{\text{Mi,i}} - \Delta T_{\text{Mi,i}}
and differencing Eqs. :ref:`16.4-47a<eq-16.4-47a>` and :ref:`16.4-47b<eq-16.4-47b>` to obtain :math:`\Delta h_{\text{Na,i}}`
and :math:`\Delta h_{\text{fi,i}}`, Eq. :ref:`16.4-46<eq-16.4-46>` becomes:
(16.4-51)
.. _eq-16.4-51:
.. math::
\begin{matrix}
\Delta T_{\text{Mi,i}} \left\lbrack 2.5 {\rho'}_{\text{Na,i}}^{n + 1}R_{\text{Na}} \
+ 2.5 {\rho'}_{\text{fi,i}}^{n + 1} R_{\text{fi}} \right\rbrack + \sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \
\Delta T_{\text{Mi,i}} \Delta t} \\
= - \left\lbrack \left( {\rho'} h u \right)_{\text{Na,i+1/2}} \
+ \left( {\rho'} h u \right)_{\text{fi,i+1/2}} \
- \left( {\rho'} h u \right)_{\text{Na,i-1/2}} \
- \left( {\rho'} h u \right)_{\text{fi,i-1/2}} \right\rbrack \frac{\Delta t}{\Delta z_{\text{i}}} \\
- h_{\text{Na,i}} \Delta {\rho'}_{\text{Na,i}} - h_{\text{fi,i}} \Delta {\rho'}_{\text{fi,i}} \
+ R_{\text{Na}} \Delta \left( {\rho'}_{\text{Na,i}} T_{\text{Na,i}} \right) + R_{\text{fi}} \
\Delta\left( {\rho'}_{\text{fi,i}} T_{\text{fi,i}} \right) \\
+ \theta_{\text{Mi,i}} \frac{0.5 \Delta t}{\Delta z_{\text{i}}} \
\left\lbrack u_{\text{Mi,i}} \left( P_{\text{Na,i}} \
+ P_{\text{fi,i}} - P_{\text{Na,i-1}} - P_{\text{fi,i-1}} \right) + u_{\text{Mi,i+1}} \
\left( P_{\text{Na,i+1}} + P_{\text{fi,i+1}} - P_{\text{Na,i}} - P_{\text{fi,i}} \right) \right\rbrack \\
+ \sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \left( T_{\text{j,i}} - T_{\text{Mi,i}} \right) \Delta t}
\end{matrix}
After using the identity:
(16.4-51a)
.. _eq-16.4-51a:
.. math::
\Delta \left( {\rho'} T \right) = T^{n} \Delta {\rho'} + {\rho'}^{n + 1} \cdot \Delta T
and replacing :math:`\Delta {\rho'}_{\text{Na,i}}` and :math:`\Delta {\rho'}_{\text{fi,i}}` by using the
continuity equation, Eq. :ref:`16.4-51<eq-16.4-51>` becomes:
(16.4-52)
.. _eq-16.4-52:
.. math::
\begin{matrix}
T_{\text{Mi,i}} \cdot \left\lbrack 2.5 {\rho'}_{\text{Na,i}}^{n + 1} R_{\text{Na}} \
+ 2.5 {\rho'}_{\text{fi,i}}^{n + 1} R_{\text{fi}} + \sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \cdot t} \
- R_{\text{Na}} {\rho'}_{\text{Na,i}}^{n + 1} - R_{\text{fi}} \cdot {\rho'}_{\text{fi,i}}^{n + 1} \right\rbrack \\
= - \left\lbrack \left( {\rho'} h u \right)_{\text{Na,i+1/2}} + \left( {\rho'} h u \right)_{\text{fi,i+1/2}} \
- \left( {\rho'} h u \right)_{\text{Na,i-1/2}} - \left( {\rho'} h u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \\
\frac{t}{z_{\text{i}}} + \left( h_{\text{Na,i}} - T_{\text{Na,i}} \cdot R_{\text{Na}} \right) \
\cdot \left\lbrack \left( {\rho'} u \right)_{\text{Na,i+1/2}} - \left( {\rho'} u \right)_{\text{Na,i-1/2}} \right\rbrack\
\frac{t}{z_{\text{i}}} + \left( h_{\text{fi,i}} - T_{\text{fi,i}} R_{\text{fi}} \right) \\
\cdot \left\lbrack \left( {\rho'} u \right)_{\text{fi,i+1/2}} - \left( {\rho'} u \right)_{\text{fi,i-1/2}} \right\rbrack \
\frac{t}{z_{\text{i}}} + \theta_{\text{Mi,i}} \cdot \frac{0.5 \cdot t}{z_{\text{i}}} \cdot \left\lbrack u_{\text{Mi,i}} \
\cdot \left( P_{\text{Na,i}} + P_{\text{fi,i}} - P_{\text{Na,i-1}} - P_{\text{fi,i-1}} \right) \\
+ u_{\text{Mi,i+1}} \cdot \left( P_{\text{Na,i+1}} + P_{\text{fi,i+1}} - P_{\text{Na,i}} - P_{\text{fi,i}} \right) \right\rbrack \
+ \sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \cdot \left( T_{\text{j,i}} - T_{\text{Mi,i}} \right) \cdot t}
\end{matrix}
The terms derived from the original
:math:`\frac{\partial \text{P}}{\partial \text{t}}` and,
:math:`u \cdot \frac{\partial \text{P}}{\partial \text{z}}` terms in Eq. :ref:`16.4-43<eq-16.4-43>` make
a negligible contribution to the right hand side of Eq. :ref:`16.4-42<eq-16.4-42>`. The
terms are neglected in the initial release version, but have been added,
for completeness, in the chunk development version. The left-hand-side
coefficient is replaced by:
(16.4-53)
.. _eq-16.4-53:
.. math::
\begin{matrix}
\text{AUXLR} = 2.5 {\rho'}_{\text{Na,i}}^{n + 1} R_{\text{Na}} + 2.5 {\rho'}_{\text{fi,i}}^{n + 1} \cdot R_{\text{fi}} \cdot \sum_{\text{j}}{{H'}_{\text{j,mi,i}} \Delta t} \\
- R_{\text{Na}} {\rho'}_{\text{Na,i}}^{n + 1} - R_{\text{fi}} {\rho'}_{\text{fi,i}}^{n + 1} \\
\end{matrix}
After dividing Eq. :ref:`16.4-52<eq-16.4-52>` by AUXLR, the new mixture temperature is
calculated as follows:
(16.4-54)
.. _eq-16.4-54:
.. math::
\begin{matrix}
\text{TENA}\left( I \right) = \text{TENA}\left( I \right) + \left( - \text{COENCH} + \text{COHELP} + \text{DTPLU} \right) \\
*\left( \text{HTNACL}\left( I \right)*\left( \text{TECLOL}\left( I \right) - \text{TENA}\left( I \right) \right) + \text{HTSENA}\left( I \right)*\left( \text{TESEOL}\left( I \right) - \text{TENA}\left( I \right) \right) \right) \\
+ \text{HTNASR}\left( I \right)*\left( \text{TESROL}\left( I \right) - \text{TENA}\left( I \right) \right) + \text{HTFUNA}\left( I \right)*\left( \text{TEFUOL}\left( I \right) = \text{TENA}\left( I \right) \right) \\
+ \text{HTNAFL}\left( I \right)*\left( \text{TEFLOL}\left( I \right) - \text{TENA}\left( I \right) \right) + \text{HTNASL}\left( I \right)*\left( \text{TESLOL}\left( I \right) = \text{TENA}\left( I \right) \right) \\
+ \text{HTFCNA}*\left( \text{TEFFCO}\left( I \right) - \text{TENA}\left( I \right) \right) + \text{HTFSNA}*\left( \text{TEFFSO}\left( I \right) - \text{TENA}\left( I \right) \right) \big/ \text{AUXLR} \\
\end{matrix}
where:
(16.4-55)
.. _eq-16.4-55:
.. math::
\begin{matrix}
\text{COENCH} = \left\lbrack \left( {\rho'} h u \right)_{\text{Na,i+1/2}} + \left( {\rho'} h u \right)_{\text{fi,i+1/2}} - \left( {\rho'} h u \right)_{\text{Na,i-1/2}} \\
- \left( {\rho'} h u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
\end{matrix}
(16.4-56)
.. _eq-16.4-56:
.. math::
\begin{matrix}
\text{COHELP} = h_{\text{Na,i}} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{Na,i+1/2}} - \left( {\rho'} u \right)_{\text{Na,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
+ h_{\text{fi,i}} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{fi,i+1/2}} - \left( {\rho'} u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
\end{matrix}
:math:`\text{HTNACL} \left( I \right) = {H'}_{\text{cl,Na,i}}` = transfer coefficient between cladding
and mixture in cell :math:`i`
:math:`\text{HTSENA} \left( I \right) = {H'}_{\text{se,Na,i}}` = transfer coefficient between steel
and gas mixture in cell :math:`i`
:math:`\text{HTNASR} \left( I \right) = {H'}_{\text{sr,Na,i}}` = transfer coefficient between hexcan wall
and gas mixture in cell :math:`i`
:math:`\text{HTFUNA} \left( I \right) = {H'}_{\text{fu,Na,i}}` = transfer coefficient between molten fuel
and gas mixture in cell :math:`i`
:math:`\text{HTNAFL} \left( I \right) = {H'}_{\text{fl,Na,i}}` = transfer coefficient between steel chunks
and mixture in cell :math:`i`
:math:`\text{HTFCNA} = {H'}_{\text{ffc,Na,i}}` = transfer coefficient between frozen fuel on cladding
and gas mixture in cell :math:`i`
:math:`\text{HTFSNA} = {H'}_{\text{ffs,Na,i}}` = transfer coefficient between frozen fuel on can wall
and gas mixture in cell :math:`i`
The convective enthalpy fluxes used in Eq. :ref:`16.4-53<eq-16.4-53>` are calculated using
an upstream differencing approach:
If :math:`{u'}_{\text{Na,i}} \geq 0`
(16.4-57)
.. _eq-16.4-57:
.. math::
\left( {\rho'} h u \right)_{\text{Na,i-1/2}} = {\rho'}_{\text{Na,i-1}} \cdot h_{\text{Na,i-1}} \cdot {u'}_{\text{Na,i}}
If :math:`{u'}_{\text{Na,i}} < 0`
(16.4-58)
.. _eq-16.4-58:
.. math::
\left( {\rho'} h u \right)_{\text{Na,i-1/2}} = {\rho'}_{\text{Na,i}} \cdot h_{\text{Na,i}} \cdot {u'}_{\text{Na,i}}
The enthalpy :math:`h_{\text{Na,i}}` is calculated as follows
If :math:`X_{\text{Na,i}} \geq 1` (single-phase sodium vapor),
(16.4-58a)
.. _eq-16.4-58a:
.. math::
h_{\text{Na,i}} = h_{\text{Na,vap}}\left( P_{\text{Na,i}} \right) + 2.5 \cdot R_{\text{Na}} \
\cdot \left( T_{\text{Na,i}} - T_{\text{sat}} \left( P_{\text{Na,i}} \right) \right)
If :math:`0 \leq X < 1` (two-phase sodium), then
(16.4-58b)
.. _eq-16.4-58b:
.. math::
h_{\text{Na,i}} = h_{\text{Na,liq}} \left( T_{\text{Na,i}} \right) \
+ h_{\text{Na,fg}} \left( T_{\text{Na,i}} \right) \cdot X_{\text{Na,i}}
where
:math:`h_{\text{Na,fg}} \left( T_{\text{Na,i}} \right)` = the heat of vaporization of sodium at
the temperature :math:`T_{\text{Na,i}}`
:math:`h_{\text{Na,liq}} \left( T_{\text{Na,i}} \right)` = enthalpy of liquid sodium on the
saturation curve at :math:`T_{\text{Na,i}}`
:math:`h_{\text{Na,vap}} \left( P_{\text{Na,i}} \right)` = enthalpy of sodium vapor on the
saturation curve at pressure :math:`P_{\text{Na,i}}`
If injection of fission gas has taken place in the previous time step, a
correction of the temperature is made to account for the enthalpy of the
injected gas. When the injection is calculated, this gas is assumed to
be at the same temperature as the sodium-fission gas mixture in the
channel in order to avoid recalculating the mixture temperature.
(16.4-59)
.. _eq-16.4-59:
.. math::
T_{\text{na,i}} = T_{\text{Na,i}} + \left( T_{\text{fi,ejected,i}} - T_{\text{Na,i}} \right) \
\cdot \Delta {\rho'}_{\text{fi,ejected,i}} \cdot 2.5 \frac{R_{\text{fi}}}{\text{AUXLR}}
The pressure of the mixture is then calculated as:
(16.4-60)
.. _eq-16.4-60:
.. math::
P_{\text{ch,i}} = P_{\text{Na,i}} + P_{\text{fi,i}}
where :math:`P_{\text{Na,i}}` and :math:`P_{\text{fi,i}}` are calculated using Eqs.
:ref:`16.4-48a<eq-16.4-48a>` and :ref:`16.4-48b<eq-16.4-48b>`. In using this equation for sodium, it is
recognized that in the neighborhood of the saturation curve the behavior
of the sodium vapor will deviate from the perfect-gas behavior. Thus,
the constant :math:`R_{\text{Na,i}}` is allowed to vary so that the pressure
calculated by Eq. :ref:`16.4-48a<eq-16.4-48a>` will match the saturation pressure when the
mixture is right on the saturation curve. The parameter used to decide
if this procedure is necessary is the ratio:
(16.4-61)
.. _eq-16.4-61:
.. math::
r_{\text{i}} = \frac{\rho_{\text{Na,i}}}{\rho_{\text{Na,vap}} \left( T_{\text{Na,i}} \right)}
where
:math:`\rho_{\text{Na,i}}` = physical density of the sodium vapor
:math:`\rho_{\text{Na,vap}}` = physical density of the saturated sodium vapor
If :math:`r_{\text{i}} < r_{\text{min}}` the sodium vapor is far enough from
saturation and the perfect-gas law is satisfactory. Otherwise, the
constant :math:`{R'}_{\text{Na,i}}` is defined as:
(16.4-62)
.. _eq-16.4-62:
.. math::
{R'}_{\text{Na,i}} = \frac{\left\lbrack R_{\text{Na,i}} \cdot \left( 1 - r \right) \
+ R_{\text{Na,sat,i}} \cdot \left( r - r_{\text{min}} \right) \right\rbrack}{\left( 1 - r_{\text{min}} \right)}
where
(16.4-63)
.. _eq-16.4-63:
.. math::
R_{\text{Na,sat,i}} = \frac{P_{\text{Na,sat}} \left( T_{\text{Na,i}} \right) \
\cdot \theta_{\text{Na,i}}}{{\rho'}_{\text{Na,i}} \cdot T_{\text{Na,i}}}
and :math:`r_{\text{min}}` is the ratio obtained from the sodium thermodynamic
properties tables for a wide range of interest; currently :math:`r_{\text{min}} = .067`.
This derivation assumes that the sodium is in the superheated vapor
region for the entire duration of the time step :math:`\Delta t`. Occasionally,
however, the sodium vapor might become saturated and this assumption
would no longer hold. In the two-phase region the temperature changes
are smaller than those in the single-phase region, since much of the
energy lost will lead to condensation, rather than to a change in
temperature. Thus, where necessary, a correction is performed which
reduces the temperature drop in the two-phase region by bringing the
final sodium temperature close to the saturation curve. The correction
is considered necessary whenever the following condition is satisfied:
(16.4-64)
.. _eq-16.4-64:
.. math::
P_{\text{Na,i}} - P_{\text{Na,sat,i}} > \Delta P_{\text{constant}}
where :math:`\Delta P_{\text{constant}}` is a built-in constant, currently set to
:math:`0.05 \cdot 10^{5}` Pa. This procedure is illustrated in :numref:`figure-16.4-13`. The
corrected sodium temperature is calculated as follows:
(16.4-65)
.. _eq-16.4-65:
.. math::
T_{\text{Na,i}} = \frac{\left( T_{1} \cdot \Delta P_{\text{o}} - T_{\text{o}} \
\cdot \Delta P_{1} \right)}{\left( \Delta P_{\text{o}} - \Delta P_{1} \right)}
where
:math:`T_{\text{o}}` = sodium temperature at the beginning of the time step
:math:`T_{1}` = calculated new sodium temperature, from Eq. :ref:`16.4-59<eq-16.4-59>`
:math:`\Delta P_{\text{o}}, \Delta P_{1} = P_{\text{Na,sat,i}} - P_{\text{Na,i}}` at
the beginning and end of time step, respectively.
.. _figure-16.4-13:
.. figure:: media/image21.png
:align: center
:figclass: align-center
Correction for Sodium Vapor Transition to a Two-phase Mixture
.. _section-16.4.3.5.2:
The Energy Equation for Two-phase Sodium and Fission Gas
''''''''''''''''''''''''''''''''''''''''''''''''''''''''
The same considerations as before are made to arrive at the Eq. :ref:`16.4-46<eq-16.4-46>`,
but the compressible term, :math:`\theta \frac{\partial \text{P}}{\partial \text{t}}`,
is not included for the two-phase component. To express the quantity
:math:`\Delta h_{\text{Na,i}}`, we use the identity:
(16.4-66)
.. _eq-16.4-66:
.. math::
h_{\text{Na,i}} = h_{\text{Na,liq,i}} \cdot \left( 1 - X_{\text{Na,i}} \right) + h_{\text{Na,vap,i}} \cdot X_{\text{Na,i}}
Thus:
(16.4-67)
.. _eq-16.4-67:
.. math::
\begin{matrix}
\Delta h_{\text{Na,i}} = \left( 1 - X_{\text{Na,i}} \right) \cdot \frac{\text{dh}_{\text{liq}}}{\text{dT}} \
\cdot \Delta T_{\text{Na,i}} + X_{\text{Na,i}} \cdot \frac{\text{dh}_{\text{vap}}}{\text{dT}} \cdot \Delta T_{\text{Na,i}} \\
+ \left( h_{\text{vap,i}} - h_{\text{liq,i}} \right) \cdot \Delta X_{\text{Na,i}} \\
\end{matrix}
Using the definitions:
(16.4-68a)
.. _eq-16.4-68a:
.. math::
C_{\text{liq}} = \frac{\text{dh}_{\text{liq}}}{\text{dT}}
(16.4-68b)
.. _eq-16.4-68b:
.. math::
C_{\text{vap}} = \frac{\text{dh}_{\text{vap}}}{\text{dT}}
(16.4-68c)
.. _eq-16.4-68c:
.. math::
X = \frac{v - v_{\text{liq}}}{v_{\text{vap}} - v_{\text{liq}}}
(16.4-68d)
.. _eq-16.4-68d:
.. math::
h_{\text{lv,i}} = h_{\text{vap,i}} - h_{\text{liq,i}}
Eq. :ref:`16.4-66<eq-16.4-66>` becomes:
(16.4-69)
.. _eq-16.4-69:
.. math::
\begin{matrix}
\Delta h_{\text{Na,i}} = \left( 1 - X_{\text{Na,i}} \right) C_{\text{liq}} \cdot \Delta T_{\text{Na,i}} + X_{\text{Na,i}} C_{\text{vap}} \cdot \Delta T_{\text{Na,i}} + h_{\text{lv,i}} \frac{\partial \text{X}}{\partial \text{t}} \cdot \Delta t \\
= \left( 1 - X_{\text{Na},i} \right)\ \ \cdot \ \ C_{\text{liq}}\ \ \cdot \ \ \Delta T_{\text{Na},i} + X_{\text{Na},i}C_{\text{vap}}\ \ \cdot \ \ \Delta T_{\text{Na},i} \\
+ h_{\text{lv},i} \cdot \frac{1}{v_{\text{vap}} - v_{\text{liq}}}\frac{\partial v}{\partial t} \cdot \Delta t - h_{\text{lv},i}\frac{v - v_{\text{liq}}}{\left( v_{\text{vap}} - v_{\text{liq}} \right)^{2}} \cdot \frac{dv_{v}}{\text{dT}}\Delta T_{\text{Na},i} \\\
\end{matrix}
By substituting:
(16.4-70)
.. _eq-16.4-70:
.. math::
v = \frac{1}{\rho} = \frac{1}{\left( \frac{{\rho'}}{\theta} \right)}
in Eq. :ref:`16.4-69<eq-16.4-69>`, we obtain:
(16.4-71)
.. _eq-16.4-71:
.. math::
\begin{matrix}
\Delta h_{\text{Na,i}} = \left( 1 - X_{\text{Na,i}} \right) \cdot C_{\text{liq}} \cdot \Delta T_{\text{Na,i}} + X_{\text{Na,i}} \cdot C_{\text{vap}} \cdot \Delta T_{\text{Na,i}} \\
+ h_{\text{lv,i}} \cdot \frac{1}{v_{\text{vap}} - v_{\text{liq}}} \cdot \frac{\theta^{2}}{{\rho'}^{2}}\left\lbrack \frac{1}{\theta} \cdot \frac{\partial {\rho'}}{\partial \text{t}} \cdot \Delta t - \frac{\partial\theta}{\partial \text{t}} \cdot \Delta t \right\rbrack \\
+ h_{\text{lv,i}} \cdot X_{\text{Na,i}} \cdot \frac{1}{v_{\text{vap}} - v_{\text{liq}}} \cdot \frac{\partial v_{\text{v}}}{\partial \text{T}} \cdot \Delta T \\
\end{matrix}
Finally, substituting Eq. :ref:`16.4-71<eq-16.4-71>` into :ref:`16.4-46<eq-16.4-46>` and keeping the terms
containing :math:`\Delta T` on the left-hand side, we have:
(16.4-72)
.. _eq-16.4-72:
.. math::
\begin{matrix}
T_{\text{Mi,i}}\left\{ {\rho'}_{\text{Na,i}}^{n + 1} \cdot \left\lbrack \left( 1 - X_{\text{na,i}} \right) \
\cdot C_{\text{liq}} + X_{\text{Na,i}} \cdot C_{\text{vap}} + h_{\text{lv,i}} \cdot \frac{1}{\nu_{\text{vap}} \
- \nu_{\text{liq}}} \cdot \frac{\partial\nu_{\text{v}}}{\partial \text{T}} \cdot X_{\text{Na,i}} \right\rbrack \
+ 2.5 {\rho'}_{\text{fi,i}}^{n + 1} R_{\text{fi}} \right\} = \\
- \left\lbrack \left( {\rho'} h u \right)_{\text{Na,i+1/2}} + \left( {\rho'} h u \right)_{\text{fi,i+1/2}} \
- \left( {\rho'} h u \right)_{\text{Na,i-1/2}} - \left( {\rho'} h u \right)_{\text{fi,i-1/2}} \right\rbrack \
\cdot \frac{t}{z_{\text{i}}} \\
+ h_{\text{lv,i}} \cdot \frac{1}{\nu_{\text{vap}} - \nu_{\text{liq}}}\left\lbrack \
\frac{\theta_{\text{Mi}}^{n + 1}}{{\rho'}_{\text{Na,i}}^{n + 1}} \cdot {\rho'}_{\text{Na,i}} - \theta_{\text{Mi}} \right\rbrack \\
- h_{\text{Na,i}} \cdot {\rho'}_{\text{Na,i}} - h_{\text{fi}} \cdot {\rho'}_{\text{fi,i}} \
+ \sum_{\text{j}}{{H'}_{\text{j,Mi,i}} \cdot T_{\text{j,Mi,i}} \cdot t}
\end{matrix}
We now rewrite:
(16.4-73)
.. _eq-16.4-73:
.. math::
\Delta T_{\text{j,Mi,i}} = T_{\text{j,i}} - \left( T_{\text{Mi,i}} + \Delta T_{\text{Mi,i}} \right)
(16.4-74)
.. _eq-16.4-74:
.. math::
\Delta T_{\text{j,Mi,i}} = T_{\text{j,i}} - \left( T_{\text{Mi,i}} + \Delta T_{\text{Mi,i}} \right)
(16.4-75)
.. _eq-16.4-75:
.. math::
\Delta {\rho'}_{\text{fi,i}} = - \left\lbrack \left( \rho u \right)_{\text{fi,i+1/2}} \
- \left( \rho u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}}
and, after some regrouping following the same procedure as in the case
in the single-phase sodium, we obtain:
(16.4-76)
.. _eq-16.4-76:
.. math::
\begin{matrix}
T_{\text{Mi,i}}^{n + 1} = T_{\text{Mi,i}}^{n} + \left\{ - \text{COENCH} + h_{\text{lv,i}} \cdot \frac{1}{\nu_{\text{vap}} \
- \nu_{\text{liq}}} \cdot \frac{\theta_{\text{Mi,i}}^{n + 1}}{{\rho'}_{\text{Mi,i}}^{n + 1}} \cdot {\rho'}_{\text{Na,i}} \
- \theta_{\text{Mi,i}} + \text{COHELP} \\
+ T \cdot \left\lbrack {H'}_{\text{cl,Mi,i}} \cdot \left( T_{\text{cl,OS,i}} - T_{\text{Mi,i}} \right) \
+ {H'}_{\text{sr,Mi,i}} \left( T_{\text{sr,OS,i}} - T_{\text{Mi,i}} \right) + {H'}_{\text{fu,Mi,i}} \left( T_{\text{fu,i}} \
- T_{\text{Mi,i}} \right) \\
+ {H'}_{\text{se,Mi,i}} \left( T_{\text{se,i}} - T_{\text{Mi,i}} \right)+{H'}_{\text{f} l \text{,Mi,i}} \
\left( T_{\text{f} l \text{,i}} - T_{\text{Mi,i}} \right) + {H'}_{\text{s} l \text{,Mi,i}} \
\left( T_{\text{s} l \text{,i}} - T_{\text{Mi,i}} \right) \\
+ {H'}_{\text{ffc,Mi,i}} \left( T_{\text{ffc,i}} - T_{\text{Mi,i}} \right) + {H'}_{\text{ffs,Mi,i}} \
\left( T_{\text{ffs,i}} - T_{\text{Mi,i}} \right) \right\rbrack \right\} \big/ \text{AUXLR}
\end{matrix}
where
(16.4-77)
.. _eq-16.4-77:
.. math::
\begin{matrix}
\text{COENCH} = \left\lbrack \left( {\rho'} h u \right)_{\text{Na,i+1/2}} + \left( {\rho'} h u \right)_{\text{fi,i+1/2}} \\
- \left( {\rho'} h u \right)_{\text{Na,i-1/2}} - \left( {\rho'} h u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
\end{matrix}
(16.4-78)
.. _eq-16.4-78:
.. math::
\begin{matrix}
\text{COHELP} = h_{\text{Na,i}} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{Na,i+1/2}} - \left( {\rho'} u \right)_{\text{Na,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
+ h_{\text{fi,i}} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{fi,i+1/2}} - \left( {\rho'} u \right)_{\text{fi,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} \\
\end{matrix}
(16.4-79)
.. _eq-16.4-79:
.. math::
\begin{matrix}
\text{AUXLR} = {\rho'}_{\text{Na,i}}^{n + 1} \cdot \left\lbrack \left( 1 - X_{\text{Na,i}} \right) \cdot C_{\text{liq}} + X_{\text{Na,i}} \cdot C_{\text{vap}} - h_{\text{vi,i}}\frac{1}{v_{\text{vap}} - v_{\text{liq}}} \\
\cdot \frac{\text{dv}_{\text{v}}}{\text{dT}} \cdot X_{\text{Na,i}} \right\rbrack + 2.5 {\rho'}_{\text{fi,i}}^{n + 1} R_{\text{fi}} + \sum_{\text{j}}{{H'}_{\text{j,Mi,i}}} \cdot \Delta t \\
\end{matrix}
The heat-transfer coefficients are the same as those used in the
single-phase energy equation. The pressure of the mixture is then
calculated as follows:
(16.4-80)
.. _eq-16.4-80:
.. math::
P_{\text{ch,i}} = P_{\text{Na,i}} + P_{\text{fi,i}}
where
(16.4-81)
.. _eq-16.4-81:
.. math::
P_{\text{Na,i}} = P_{\text{sat}}\left( T_{\text{Na,i}} \right)
(16.4-82)
.. _eq-16.4-82:
.. math::
P_{\text{fi,i}} = R_{\text{fi}} \cdot {\rho'}_{\text{fi,i}} \cdot \frac{T_{\text{fi,i}}}{\theta_{\text{Mi,i}}}
However, if significant amounts of liquid sodium are present (more than
30% of the volume fraction), the compressibility of the liquid sodium is
taken into account in calculating the partial fission-gas pressure. Eq.
:ref:`16.4-82<eq-16.4-82>` is written in the form:
(16.4-83)
.. _eq-16.4-83:
.. math::
P_{\text{fi,i}} = \frac{R_{\text{fi}} \cdot {\rho'}_{\text{fi,i}} \cdot T_{\text{fi,i}}}{\theta_{\text{Mi,i}} \
+ \Delta\theta_{\text{Na,liq}}} = \frac{R_{\text{fi}} \cdot {\rho'}_{\text{fi,i}} \cdot T_{\text{fi,i}}}{\theta_{\text{Mi,i}} \
+ \theta_{\text{Na,liq}} \cdot P_{\text{fi,i}} \cdot C_{\Delta P,\text{Na}}}
where
(16.4-84)
.. _eq-16.4-84:
.. math::
C_{\Delta \text{P,Na}} = \frac{\left( \frac{\Delta V}{V} \right)_{\text{Na,liq}}}{\Delta P}
The value of :math:`C_{\Delta \text{P,Na}}` is given by the input constant CMNL. By
solving Eq. :ref:`16.4-83<eq-16.4-83>` and retaining only the positive root, we obtain:
(16.4-85)
.. _eq-16.4-85:
.. math::
P_{\text{fi,i}} = \frac{- \theta_{\text{Mi,i}} + \sqrt{\theta_{\text{Mi,i}}^{2} \
+ 4\theta_{\text{Na,liq}} + C_{\Delta \text{P,Na}} \cdot R_{\text{fi}} \cdot {\rho'}_{\text{fi,i}} \
\cdot T_{\text{fi,i}}}}{2 \cdot \theta_{\text{Na,liq}} \cdot C_{\Delta \text{P,Na}}}
.. _section-16.4.3.6:
Fuel Vapor Energy Conservation Equation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The fuel vapor energy conservation equation is solved in the routine
LEFUVA, which also models the fuel vaporization/condensation processes.
These processes are not included in the formulation of the liquid fuel
energy conservation equation, which is solved in the LESOEN routine.
Thus, the mass and temperature of the liquid fuel is corrected in the
routine LEFUVA, when necessary, to account for the mass and energy
sources/sinks due to condensation and vaporization. Because the fuel
vapor effects become dominant only during high-power transients that are
associated with very short time periods, the fuel vapor is assumed *not*
to be in equilibrium with the liquid fuel. The treatment of the
vaporization and condensation processes will be described in detail
later in this chapter.
.. _section-16.4.3.6.1:
Energy Conservation for Superheated Fuel Vapor
''''''''''''''''''''''''''''''''''''''''''''''
First, the energy equation is solved by assuming that all fuel vapor is
initially superheated. This assumption is consistent with the method of
solution, as all condensation and vaporization events are calculated in
the routine LEFUVA, and the fuel vapor remaining in each cell at the end
of this routine is always superheated, or, in the limit, saturated dry.
Condensation or vaporization effects that might occur during the current
time step are ignored during this first step. The energy equation is
written in conservative form:
(16.4-86)
.. _eq-16.4-86:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}} \left( \rho_{\text{fv,i}} h_{\text{fv,i}} \cdot A_{\text{Mi,i}} \cdot \Delta z_{\text{i}} \right) + \left\lbrack \left( \rho A h u \right)_{\text{fv,i+1/2}} - \left( \rho A h u \right)_{\text{fv,i-1/2}} \right\rbrack \\
= A_{\text{Mi,i}} \frac{\partial \text{P}_{\text{fv,i}}}{\partial \text{t}} \cdot \Delta z_{\text{i}} + A_{\text{Mi,i}} \cdot 0.5 \cdot \left\lbrack u_{\text{Mi,i}} \cdot \left( P_{\text{fv,i}} - P_{\text{fv,i-1}} \right) \\
+ u_{\text{Mi,i+1}} \cdot \left( P_{\text{fv,i+1}} - P_{\text{fv,i}} \right) \right\rbrack + \sum_{\text{j}}{ H_{\text{j,fv,i}} A_{\text{j,fv,i}} \cdot \Delta T_{\text{j,fv,i}}} \\
+ Q_{\text{fu,i}} \cdot \rho_{\text{fu,i}} \cdot A_{\text{Mi,i}} \cdot \Delta z_{\text{i}} \\
\end{matrix}
After dividing by :math:`\text{AXMX} \cdot \Delta z_{\text{i}}` and using the definition of
generalized densities, we obtain:
(16.4-87)
.. _eq-16.4-87:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}} \left( {\rho'}_{\text{fv,i}} h_{\text{fv,i}} \right) = - \left\lbrack \left( {\rho'} h u \right)_{\text{fv,i+1/2}} - \left( {\rho'} h u \right)_{\text{fv,i-1/2}} \right\rbrack \cdot \frac{1}{\Delta z_{\text{i}}} \\
+ \theta_{\text{Mi,i}} \cdot \frac{\partial \text{P}_{\text{fv,i}}}{\partial \text{t}} + \sum_{\text{j}}{{H'}_{\text{j,fv,i}}} \cdot \Delta T_{\text{j,fv,i}} + Q_{\text{fu,i}} \cdot {\rho'}_{\text{fv}} \\
+ \theta_{\text{Mi,i}} \cdot \frac{0.5}{\Delta z_{\text{i}}}\left\lbrack U_{\text{Mi,i}} \cdot \left( P_{\text{fv,i}} - P_{\text{fv,i-1}} \right) + U_{\text{Mi,i+1}} \cdot \left( P_{\text{fv,i+1}} - P_{\text{fv,i}} \right) \right\rbrack \\
\end{matrix}
Integrating over :math:`\Delta t` and using the identity in Eq. :ref:`16.4-16<eq-16.4-16>` to express the
quantity :math:`\Delta \left( {\rho'} h \right)`, we obtain:
(16.4-88)
.. _eq-16.4-88:
.. math::
\begin{matrix}
{\rho'}_{\text{fv,i}}^{n + 1} \cdot \Delta h_{\text{fv,i}} = - \left\lbrack \left( {\rho'} h u \right)_{\text{fv,i+1/2}} - \left( {\rho'} h u \right)_{\text{fv,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}} - h_{\text{fv,i}} \cdot \Delta {\rho'}_{\text{fv,i}} \\
+ \theta_{\text{Mi,i}} \cdot \Delta P_{\text{fv,i}} + \sum_{\text{j}}{{H'}_{\text{j,fv,i}} \cdot \Delta T_{\text{j,fv,i}} \cdot \Delta t} + Q_{\text{fu,i}} \cdot {\rho'}_{\text{fv}} \cdot \Delta t \\
+ \theta_{\text{Mi,i}} \cdot \frac{0.5 \cdot \Delta t}{\Delta z_{\text{i}}} \cdot \left\lbrack u_{\text{Mi,i}} \cdot \left( P_{\text{fv,i}} - P_{\text{fv,i-1}} \right) \\
+ u_{\text{Mi,i+1}} \cdot \left( P_{\text{fv,i+1}} - P_{\text{fv,i}} \right) \right\rbrack \\
\end{matrix}
The fuel vapor is only present in calculations for short periods of time
during high overpower transients. In addition, the gas-gas heat transfer
is generally much more efficient than the gas-liquid or gas-surface heat
transfer. For these reasons the only heat-transfer term maintained in
Eq. :ref:`16.4-86<eq-16.4-86>` is the heat transfer between the fuel vapor and the
sodium-fission gas mixture. Furthermore, because of the lack of
experimental data on the gas-gas heat transfer, this term has been
formulated such that it represents a sizable fraction of the
transferable energy, i.e., the temperatures of the fuel vapor and gas
mixture will equilibrate quite fast, within 10-50 time steps. The
enthalpy of the fuel vapor at a certain temperature is:
(16.4-89)
.. _eq-16.4-89:
.. math::
h_{\text{fv,i}} = h_{\text{fv,sat}} \left( P_{\text{fv,i}} \right) + 2.5 \cdot R_{\text{fv}} \
\cdot \left\lbrack T_{\text{fv,i}} - T_{\text{fv,sat,i}} \left( P_{\text{fv,i}} \right) \right\rbrack
A measure of the transferable enthalpy is obtained by assuming the
lowest final temperature to be the temperature of the sodium-fission gas
mixture. Thus:
(16.4-90)
.. _eq-16.4-90:
.. math::
\Delta h_{\text{fv,i}} = 2.5 \cdot R_{\text{fv}} \left\lbrack T_{\text{fv,i}} - T_{\text{Na,i}}^{n + 1} \right\rbrack
The total transferable enthalpy in cell :math:`i`, during the time :math:`\Delta t` is:
(16.4-91)
.. _eq-16.4-91:
.. math::
\Delta h_{\text{fv,i}} \cdot \rho_{\text{fv,i}} \cdot A_{\text{Mi,i}} \cdot \Delta z_{\text{i}} \
= 2.5 \cdot R_{\text{fv}} \cdot \left\lbrack T_{\text{fv,i}} - T_{\text{Na,i}}^{n + 1} \right\rbrack \
\cdot \rho_{\text{fv,i}} \cdot A_{\text{Mi,i}} \cdot \Delta z_{\text{i}}
The fraction transferred is :math:`\text{CFHTAX}`, which is defined as follows:
When :math:`\Delta t \cdot 10^{3} \leq 0.1`
(16.4-92a)
.. _eq-16.4-92a:
.. math::
\text{CFHTAX} = \Delta t \cdot 10^{3}
When :math:`\Delta t \cdot 10^{3} > 0.1`
(16.4-92b)
.. _eq-16.4-92b:
.. math::
\text{CFHTAX} = 0.1
After multiplying by :math:`\text{CFHTAX}` and dividing by :math:`\text{AXMX} \cdot \Delta z_{\text{i}}`, which
was done for the original equation, the heat-transfer term in Eq.
:ref:`16.4-88<eq-16.4-88>` is replaced by:
(16.4-93)
.. _eq-16.4-93:
.. math::
\sum_{\text{j}}{{H'}_{\text{j,f,i}} \cdot \Delta T_{\text{j,fv,i}} \cdot \Delta t \rightarrow \
- 2.5 \cdot R_{\text{fv,i}} \cdot \text{CFHTAX} \cdot \left\lbrack T_{\text{fv,i}} \
- T_{\text{Na,i}}^{n + 1} \right\rbrack \cdot {\rho'}_{\text{fv,i}}}
Substituting Eq. :ref:`16.4-93<eq-16.4-93>` in Eq. :ref:`16.4-88<eq-16.4-88>` and rewriting the
:math:`\theta_{\text{Mi,i}} \cdot \Delta P_{\text{fv,i}}` and :math:`\rho_{\text{fv,i}}^{n + 1}` terms
in a manner similar to that used for the sodium-gas mixture, in Eqs.
:ref:`16.4-47<eq-16.4-47a>` through :ref:`16.4-51<eq-16.4-51>`, we obtain:
(16.4-94)
.. _eq-16.4-94:
.. math::
\begin{matrix}
\Delta T_{\text{fv,i}}\left\lbrack 2.5\left( {\rho'} \right)_{\text{fv,i}}^{n + 1} R_{\text{fv}} \\
- \left( {\rho'} \right)_{\text{fv,i}}^{n + 1}R_{\text{fv}} \right\rbrack \\
= - \left\lbrack \left( {\rho'} h u \right)_{\text{fv,i+1/2}} - \\
\left( {\rho'} h u \right)_{\text{fv,i-1/2}} \right\rbrack \frac{\Delta t}{\Delta z_{\text{i}}} \\
+ \left( h_{\text{fv,i}} - T_{\text{fv,i}} R_{\text{fv}} \right)\left\lbrack \left( {\rho'} u \right)_{\text{fv,i+1/2}} \\
- \left( {\rho'} u \right)_{\text{fv,i-1/2}} \right\rbrack \frac{\Delta t}{\Delta z_{\text{i}}} \\
+ \theta_{\text{Mi,i}}\frac{0.5\Delta t}{\Delta z_{\text{i}}}\\
\left\lbrack u_{\text{Mi,i}}\left( P_{\text{fv,i}} - P_{\text{fv,i-1}} \right) \\
+ u_{\text{Mi,i+1}}\left( P_{\text{fv,i+1}} - P_{\text{fv,i}} \right) \right\rbrack \\
- 2.5 R_{\text{fv}} \text{CHFTAX} \left( T_{\text{fv,i}} - T_{\text{Na,i}}^{n + 1} \right) {\rho'}_{\text{fv,i}} \\
+ Q_{\text{fu}} {\rho'}_{\text{fv}} \Delta t
\end{matrix}
The terms derived from the original
:math:`\frac{\partial \text{P}}{\partial \text{t}}` and
:math:`\frac{\partial \text{P}}{\partial \text{z}}` terms in Eq. :ref:`16.4-86<eq-16.4-86>` make a
negligible contribution to the right hand side of Eq. :ref:`16.4-94<eq-16.4-94>`. These
terms were neglected in the initial release version, but have been added
for completeness, in the chunk development version. The left-hand-side
coefficient is replaced by:
(16.4-95)
.. _eq-16.4-95:
.. math::
\text{AUXLR} = 2.5 \cdot {\rho'}_{\text{fv,i}}^{n + 1} \cdot R_{\text{fv}} - {\rho'}_{\text{fv,i}}^{n + 1} \cdot R_{\text{fv}}
The terms in Eq. :ref:`16.4-95<eq-16.4-95>` have not been combined because the term
:math:`{\rho'}_{\text{fv,i}}^{n + 1} \cdot R_{\text{fv}}` originates
from the :math:`\frac{\partial \text{P}}{\partial \text{t}}` term and was dropped
in the initial release version, together with the terms originating from
the u :math:`\frac{\partial \text{P}}{\partial \text{u}}` term. The new fuel vapor
temperature is calculated as follows:
(16.4-96)
.. _eq-16.4-96:
.. math::
\begin{matrix}
\text{TEFUVA} \left( I \right) = \text{TEFUVA} \left( I \right) + \left( - \text{COENCH} \
+ \text{COHELP} + \text{HSFU} \left( I \right) * \text{DEFVCH} \left( I \right) * \\
\text{DTPL} - \text{DEFVCH} \left( I \right) * \text{CFHTFN} * \left( \text{TEFUVA} \left( I \right) \
- \text{TENA} \left( I \right) \right) \right) \big/ \text{AUXLR}
\end{matrix}
where
(16.4-97)
.. _eq-16.4-97:
.. math::
\text{COENCH} = \left\lbrack \left( {\rho'} h u \right)_{\text{fv,i+1/2}} \
- \left( {\rho'} h u \right)_{\text{fv,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}}
(16.4-98)
.. _eq-16.4-98:
.. math::
\text{COHELP} = h_{\text{fv,i}} \cdot \left\lbrack \left( {\rho'} u \right)_{\text{fv,i+1/2}} \
- \left( {\rho'} u \right)_{\text{fv,i-1/2}} \right\rbrack \cdot \frac{\Delta t}{\Delta z_{\text{i}}}
and
(16.4-99)
.. _eq-16.4-99:
.. math::
\text{DEFVCH} \left( I \right) = {\rho'}_{\text{fv,i}}
(16.4-100)
.. _eq-16.4-100:
.. math::
\text{CFHTFN} = 2.5 \cdot R_{\text{fv}} \cdot \text{CFHTAX}
The new fuel vapor pressure is then calculated using the new
temperatures:
(16.4-101)
.. _eq-16.4-101:
.. math::
P_{\text{fv,i}}^{n + 1} = R_{\text{fv}} \cdot {\rho'}_{\text{fv,i}}^{n + 1} \cdot T_{\text{fv,i}}^{n + 1} \big/ \theta_{\text{Mi,i}}^{n + 1}
.. _section-16.4.3.6.2:
Condensation of the Fuel Vapor
''''''''''''''''''''''''''''''
Under certain circumstances the fuel vapor will enter the two-phase
region and begin to condense. The decision that such a situation has
occurred is made by comparing the pressure
:math:`P_{\text{fv,i}}^{n + 1}` with the saturation pressure in cell :math:`i`.
If:
(16.4-102)
.. _eq-16.4-102:
.. math::
P_{\text{fv,i}}^{n + 1} < P_{\text{fv,sat,i}} \left( T_{\text{fv,i}}^{n + 1} \right)
no condensation will occur during the current time step. Otherwise, some
condensation will take place, and the calculation proceeds as outlined
below.
First, we determine the temperature :math:`T_{\text{fv,sat,i}}` where the fuel
vapor first reaches saturation. This temperature is obtained using the
same method described in :numref:`section-16.4.3.5` to correct the sodium
temperature where it crosses over from the single phase to the two-phase
region. The formula used is similar to Eq. :ref:`16.4-65<eq-16.4-65>`. The enthalpy change
between the original temperature :math:`T_{\text{fv,i}}^{n}` and the
saturation temperature is, by Eq. :ref:`16.4-94<eq-16.4-94>`:
(16.4-103)
.. _eq-16.4-103:
.. math::
\Delta {h'}_{\text{fv,i}} = \left( T_{\text{fv,sat,i}} - T_{\text{fv,i}}^{n} \right) * \text{AUXLR}
The total enthalpy change originally calculated is:
(16.4-104)
.. _eq-16.4-104:
.. math::
\Delta h_{\text{fv,i}} = \left( T_{\text{fv,i}}^{n + 1} - T_{\text{fv,i}}^{n} \right) * \text{AUXLR}
The enthalpy which still has to be removed via condensation once the
vapor has reached the temperature :math:`T_{\text{fv,sat,i}}` is given by:
(16.4-105)
.. _eq-16.4-105:
.. math::
\Delta h_{\text{fv,i}}^{\text{cond}} = \Delta h_{\text{fv,i}} - \Delta {h'}_{\text{fv,i}} \
= \left( T_{\text{fv,i}}^{n + 1} - T_{\text{fv,sat}} \right) * \text{AUXLR}
In order to find the temperature change leading to the enthalpy change
:math:`\Delta h_{\text{fv,i}}^{\text{cond}}`, we observe that at the
final temperature we have to satisfy the condition for dry vapor:
(16.4-106)
.. _eq-16.4-106:
.. math::
P_{\text{fv,sat}} \left( T_{\text{fv,i}} \right) = R_{\text{fv}} \cdot {\rho'}_{\text{fv}} \
\cdot T_{\text{fv,i}} \cdot \frac{1}{\theta_{\text{Mi,i}}}
Also, with the assumption that the heat of vaporization for fuel
:math:`h_{\text{fv,lg}}` is approximately constant for the range of temperatures
of interest, the total enthalpy change due to condensation and
temperature change can be written as:
(16.4-107)
.. _eq-16.4-107:
.. math::
\Delta h_{\text{fv}} = - \Delta {\rho'}_{\text{fv,i}} h_{\text{fv,lg}} \
+ 2.5 {\rho'}_{\text{fv,i}} R_{\text{fv}} \cdot \Delta T_{\text{fv}}
where :math:`\Delta {\rho'}_{\text{fv,i}}` is the *decrease* in generalized density due to
condensation. Because Eqs. :ref:`16.4-106<eq-16.4-106>` and (:ref:`16.4-107<eq-16.4-107>`) cannot be solved
directly for the temperature, we use a trial and error approach to find
the solution. First, assuming the final temperature if
:math:`T_{\text{fv}}^{1} = T_{\text{fv,i}}^{n + 1}`, i.e., the
temperature originally calculated, and the amount to condense
:math:`\Delta {\rho'}_{\text{fv}}^{1}`, Eq. :ref:`16.4-106<eq-16.4-106>` becomes:
(16.4-108)
.. _eq-16.4-108:
.. math::
P_{\text{fv,sat}} \left( T_{\text{fv}}^{1} \right) = R_{\text{fv}} \cdot \left( {\rho'}_{\text{fv,i}} \
- \Delta {\rho'}_{\text{fv}} \right) \cdot T_{\text{fv}}^{1} \cdot \frac{1}{\theta_{\text{Mi}} \
- \frac{\Delta {\rho'}_{\text{fv}}}{\rho_{\text{fu,liq}}}}
and, after solving for :math:`\Delta {\rho'}_{\text{fv}}`:
(16.4-109)
.. _eq-16.4-109:
.. math::
\Delta {\rho'}_{\text{fv}}^{1} = \frac{\left( {\rho'}_{\text{fv,i}} \cdot R_{\text{fv}} \cdot T_{\text{fv}}^{1} \
- \theta_{\text{Mi,i}} \cdot P_{\text{fv,sat}} \left( T_{\text{fv}}^{1} \right) \right)}{R_{\text{fv}} \cdot T_{\text{fv}}^{1} \
- \frac{P_{\text{fv,sat}} \left( T_{\text{fv}}^{1} \right)}{\rho_{\text{fu,liq}}}}
The enthalpy change in Eq. :ref:`16.4-107<eq-16.4-107>` becomes:
(16.4-110a)
.. _eq-16.4-110a:
.. math::
\Delta h_{\text{fv}}^{1} = - \Delta \rho_{\text{fv,i}}^{1} \cdot h_{\text{hv,lg}} \
+ 2.5 \cdot {\rho'}_{\text{fv,i}} \cdot R_{\text{fv}} \cdot \left( T_{\text{fv,i}}^{n + 1} - T_{\text{fv,sat}} \right)
Then, we assume that the final temperature of the fuel vapor is
:math:`T_{\text{fv}}^{2} = T_{\text{fv,sat}}`, and obtain the new
amount of condensate :math:`\Delta {\rho'}_{\text{fv}}^{2}` and the
energy change:
(16.4-110b)
.. _eq-16.4-110b:
.. math::
\Delta h_{\text{fv}}^{2} = - \Delta {\rho'}_{\text{fv}}^{2} \cdot h_{\text{fv,lg}}
These two situations generally will bracket the actual final temperature
of the fuel vapor, because the first assumption is practically
equivalent to very little or no condensation, while the second assumes
maximum condensation with no temperature change. The actual temperature
is obtained by interpolating between
:math:`T_{\text{fv}}^{1}` and :math:`T_{\text{fv}}^{2}`, with the
condition that the final enthalpy change has to be
:math:`\Delta h_{\text{fv,i}}^{\text{cond}}`, calculated before from Eq.
:ref:`16.4-105<eq-16.4-105>`. Thus:
(16.4-111)
.. _eq-16.4-111:
.. math::
T_{\text{fv,i}}^{n + 1} = T_{\text{fv,i}}^{1} + T_{\text{fv,i}}^{2} \cdot \frac{\Delta h_{\text{fv,i}}^{\text{cond}} \
- \Delta h_{\text{fv}}^{1}}{\Delta h_{\text{fv}}^{2} - \Delta h_{\text{fv}}^{1}}
Using this temperature, the actual condensation is calculated from Eq.
:ref:`16.4-109<eq-16.4-109>` and the generalized density of the fuel vapor is updated:
(16.4-112)
.. _eq-16.4-112:
.. math::
{\rho'}_{\text{fv,i}} = {\rho'}_{\text{fv,i}} - \Delta {\rho'}_{\text{fu,i}}^{\text{cond}}
The temperature and generalized density of the liquid fuel is also
modified to account for the addition of
:math:`\Delta {\rho'}_{\text{fv,i}}^{\text{cond}}` at the temperature
:math:`T_{\text{fv,i}}^{n + 1}`:
(16.4-113)
.. _eq-16.4-113:
.. math::
T_{\text{fu,i}} = \frac{\left\lbrack T_{\text{fu,i}} {\rho'}_{\text{fu,i}} \
+ T_{\text{fv,i}}^{n + 1} \cdot \Delta \rho_{\text{fv,i}} \right\rbrack}{\left( {\rho'}_{\text{fu,i}} \
+ \Delta {\rho'}_{\text{fv,i}}^{\text{cond}} \right)}
(16.4-114)
.. _eq-16.4-114:
.. math::
{\rho'}_{\text{fu,i}} = {\rho'}_{\text{fu,i}} + \Delta {\rho'}_{\text{fv,i}}^{\text{cond}}
.. _section-16.4.3.6.3:
Vaporization of Liquid Fuel
'''''''''''''''''''''''''''
The vaporization model used is a quasi-equilibrium model [16-13] which
allows the fuel and fuel vapor to have different temperatures at the
same location. Bulk boiling of fuel is assumed to occur whenever the
local total pressure is lower than the vapor pressure of the liquid
fuel. This process is fast enough to allow a quasi-equilibrium to be
established each time step, and is illustrated in :numref:`figure-16.4-14` by the
original sharp increase in pressure. Thus, the total pressure, including
the partial pressure of the fuel vapor becomes equal to the fuel vapor
pressure corresponding to the liquid-fuel temperature. When the total
pressure exceeds the liquid-fuel vapor pressure, fuel vaporization can
occur only by surface vaporization, which is generally a significantly
slower process than bulk boiling. The efficiency of surface vaporization
is a function of the local flow regime and time-step length. In the
limit, if the surface vaporization is assumed to be very efficient, the
partial pressure of the fuel vapor can become equal, in each time step,
to the liquid-fuel pressure. In this case, the quasi-equilibrium model
becomes equivalent to a thermal-equilibrium model.
.. _figure-16.4-14:
.. figure:: media/image22.png
:align: center
:figclass: align-center
Time Variation of Fuel Vapor Partial Pressure
This calculation is performed for all cells :math:`I`, with :math:`\text{IFFUBT} < I < \text{IFFUTP}`.
First, we calculate :math:`P_{\text{fu,sat,i}} = P_{\text{sat}} \left( T_{\text{fu,i}} \right)`.
Further vaporization of the liquid fuel in cell I will occur only if:
(16.4-115)
.. _eq-16.4-115:
.. math::
P_{\text{fv,i}} + \left( P_{\text{Na,i}} + P_{\text{fi,i}} + P_{\text{sv,i}} \right) \cdot C_{\text{Pr,fu}} < P_{\text{fu,sat,i}}
where :math:`C_{\text{Pr,fu}}` is a coefficient dependent on the local
configuration and the length of the time step as follows:
If the fuel is the continuous component, i.e., in the annular or bubbly
fuel flow regimes
(16.4-116a)
.. _eq-16.4-116a:
.. math::
C_{Pr,\text{fu}} = 1
If the fuel is in the form of droplets, i.e. in the annular and bubbly
steel flow regimes,
(16.4-116b-c)
.. _eq-16.4-116b-c:
.. math::
C_{\text{Pr,fu}} = \begin{cases}
1 - \Delta t \cdot 10^{4} & \text{if } 1 - \Delta t \cdot 10^{4} > 0 \\
0 & \text{if } 1 - \Delta t \cdot 10^{4} < 0 \\
\end{cases}
The case :math:`C_{\text{Pr,fu}} = 0` corresponds to full thermal equilibrium
between liquid and fuel vapor.
The vaporization of liquid fuel will take place until the following
condition is satisfied:
(16.4-117)
.. _eq-16.4-117:
.. math::
P_{\text{fv,i}} + \left( P_{\text{Na,i}} + P_{\text{fi,i}} + P_{\text{sv,i}} \right) \cdot C_{\text{Pr,fu}} = P_{\text{fu,sat,i}}
Note that all pressures change during vaporization, including
:math:`P_{\text{fu,sat,i}}`. Thus, Eq. :ref:`16.4-117<eq-16.4-117>` cannot be solved directly, and we
need to use a trial-and-error approach. First, it is assumed that the
final fuel temperature is:
(16.4-118)
.. _eq-16.4-118:
.. math::
T_{\text{fu,i}}^{1} = T_{\text{fu,i}}
and
(16.4-119)
.. _eq-16.4-119:
.. math::
P_{\text{fu,sat,i}}^{1} = P_{\text{fu,sat}} \left( T_{\text{fu,j}}^{1} \right)
We can now calculate the amount of new fuel vapor generated from Eq.
:ref:`16.4-117<eq-16.4-117>`:
(16.4-120)
.. _eq-16.4-120:
.. math::
\Delta {\rho'}_{\text{fv,i}} = \frac{\theta_{\text{Mi,i}} \left\lbrack P_{\text{fu,sat,i}}^{1} \
- P_{\text{fv,i}} - \left( P_{\text{Na,i}} + P_{\text{fi,i}} + P_{\text{fv,i}} \right) \
\cdot C_{\text{p,vap,fu}} \right\rbrack}{\left\lbrack R_{\text{fv}} \cdot T_{\text{fv}}^{1} \
- P_{\text{fu,sat,i}}^{1} \cdot \frac{1}{\rho_{\text{fu,liq}}} \right\rbrack}
The new fuel enthalpy is:
(16.4-121)
.. _eq-16.4-121:
.. math::
{h'}_{\text{fu,i}} = - \frac{\left( h_{\text{fu,i}} \cdot {\rho'}_{\text{fu,i}} \
- h_{\text{fu,lq}} \cdot \Delta {\rho'}_{\text{fv,i}} \right)}{{\rho'}_{\text{fu,i}} - \Delta {\rho'}_{\text{fv,i}}}
and the new fuel temperature and vapor pressure are:
(16.4-122)
.. _eq-16.4-122:
.. math::
{T'}_{\text{fu,i}} = {T'}_{\text{fu}} \left( {h'}_{\text{fu,i}} \right)
(16.4-123)
.. _eq-16.4-123:
.. math::
{P'}_{\text{fu,sat,i}} = {P'}_{\text{fu,sat}} \left( T_{\text{fu,i}} \right)
A new guess is now made about the final fuel temperature
:math:`T_{\text{fu,i}}^{2} = {T'}_{\text{fu,i}}`, and the above
procedure is repeated, obtaining :math:`P_{\text{fu,sat,i}}^{2}`,
:math:`\Delta {\rho''}_{\text{fu,i}}`, :math:`{T''}_{\text{fu,i}}` and :math:`{P''}_{\text{fu,sat,i}}`. We can now
obtain the real :math:`P_{\text{fu,sat,i}}`, by imposing the condition that the
final saturation pressure should be equal to the assumed saturation
pressure:
(16.4-124)
.. _eq-16.4-124:
.. math::
P_{\text{fu,sat,i}}^{n + 1} = P_{\text{fu,sat,i}}^{i} - \left( P_{\text{fu,sat,i}}^{2} \
- P_{\text{fu,sat,i}}^{1} \right) \cdot \frac{\Delta P^{1}}{\Delta P^{2} - \Delta P^{1}}
where
(16.4-125a)
.. _eq-16.4-125a:
.. math::
\Delta P^{1} = {P'}_{\text{fu,sat,i}} - P_{\text{fu,sat,i}}^{1}
(16.4-125b)
.. _eq-16.4-125b:
.. math::
\Delta P^{2} = {P''}_{\text{fu,sat,i}} - P_{\text{fu,sat,i}}^{2}
Using :math:`P_{\text{fu,sat,i}}^{n + 1}` in Eq. :ref:`16.4-120<eq-16.4-120>`, we
can calculate the new :math:`\Delta {\rho'}_{\text{fv,i}}`. Then, the generalized densities
and enthalpies of the fuel vapor and liquid fuel are updated:
(16.4-126)
.. _eq-16.4-126:
.. math::
T_{\text{fv,i}}^{n + 1} = \frac{\left( \Delta {\rho'}_{\text{fv,i}} \cdot T_{\text{fu,i}} \
+ {\rho'}_{\text{fv,i}} \cdot T_{\text{fv,i}} \right)}{\left( {\rho'}_{\text{fv,i}} + \Delta {\rho'}_{\text{fv,i}} \right)}
(16.4-127)
.. _eq-16.4-127:
.. math::
h_{\text{fu,i}}^{n + 1} = \frac{\left( h_{\text{fu,i}} \cdot {\rho'}_{\text{fu,i}} \
- h_{\text{lq}} \cdot \Delta {\rho'}_{\text{fv,i}} \right)}{\left( {\rho'}_{\text{fu,i}} \
- \Delta {\rho'}_{\text{fv,i}} \right)}
(16.4-128)
.. _eq-16.4-128:
.. math::
{\rho'}_{\text{fv,i}}^{n + 1} = {\rho'}_{\text{fv,i}} + \Delta {\rho'}_{\text{fv,i}}
(16.4-129)
.. _eq-16.4-129:
.. math::
{\rho'}_{\text{fu,i}} = {\rho'}_{\text{fu,i}} - \Delta {\rho'}_{\text{fv,i}}
Finally, the new pressure due to fuel vapor is calculated:
(16.4-130)
.. _eq-16.4-130:
.. math::
P_{\text{fv,i}}^{n + 1} = R_{\text{fv}} \cdot {\rho'}_{\text{fv,i}}^{n + 1} \
\cdot T_{\text{fv,i}}^{n + 1} \cdot \frac{1}{\theta_{\text{Mi,i}}^{n + 1}}
and is added to the total channel pressure:
(16.4-131)
.. _eq-16.4-131:
.. math::
P_{\text{ch,i}}^{n + 1} = P_{\text{ch,i}}^{n + 1} + P_{\text{fv,i}}^{n + 1}
.. _section-16.4.3.7:
Steel Vapor Energy Conservation Equation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The steel vapor energy conservation equation is solved in the routine
LESEVA. This routine is practically identical to the LEFUVA routine,
which solves the fuel vapor energy equation and was described in detail
in :numref:`section-16.4.3.6`. At the end of the LESEVA routine, the new pressure
due to steel vapor is calculated
(16.4-132)
.. _eq-16.4-132:
.. math::
P_{\text{sv,i}}^{n + 1} = R_{\text{sv}} \cdot {\rho'}_{\text{sv,i}}^{n + 1} \
\cdot T_{\text{sv,i}}^{n + 1} \cdot \frac{1}{\theta_{\text{Mi,i}}^{n + 1}}
and is added to the total channel pressure:
(16.4-133)
.. _eq-16.4-133:
.. math::
P_{\text{ch,i}}^{n + 1} = P_{\text{ch,i}}^{n + 1} + P_{\text{sv,i}}^{n + 1}
.. _section-16.4.3.8:
Momentum Conservation Equation
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The momentum conservation equations are solved in the routine LEMOCO. As
indicated previously, the channel hydrodynamic model in LEVITATE
calculates three velocity fields, each of them describing the motion of
a group of material components, as follows:
.. list-table::
:header-rows: 1
:align: center
:widths: auto
* - :math:`u'_{Mi,i}, u''_{Mi,i}`
- the velocity of the gas mixture; described the motion of the two-phase (or superheated) sodium, fission gas, fuel vapor and steel vapor. The code symbol is UMCH.
* - :math:`u'_{fu,i}, u''_{fu,i}`
- the velocity of the liquid fuel and/or liquid steel components. The code symbol is UFCH.
* - :math:`u_{lu,i}`
- the velocity of the solid fuel and/or steel chunks. The code symbol is ULCH.
One is reminded that LEVITATE uses dual velocities to model the motion
of the gas mixture and liquid components. Thus, :math:`{u'}_{\text{Mi,i}}`
represents the velocity of the gas mixture before the :math:`i-1/2` boundary
and :math:`{u''}_{\text{Mi,i}}` represents the velocity of the mixture after the
same boundary. (Before and after velocities are ordered here by the
positive sense of the axial coordinate). As already shown in :numref:`section-16.4.3.1`, the dual velocities at the boundary :math:`i-1/2` are related by:
(16.4-134)
.. _eq-16.4-134:
.. math::
C_{\text{Mo,fu,i}} = \frac{{u'}_{\text{fu,i}}}{{u''}_{\text{fu,i}}} = \frac{{\rho'}_{\text{fu,i}}}{{\rho'}_{\text{fu,i-1}}}
In the code, only the velocity :math:`{u''}_{\text{fu,i}}` is stored in the array
:math:`\text{UFCH} \left( I \right)`. The velocity :math:`{u'}_{\text{fu,i}}` is always obtained from Eq.
:ref:`16.4-134<eq-16.4-134>`, using the coefficient :math:`C_{\text{Mo,fu,i}}`, which is stored in
the array :math:`\text{CCFU} \left( I \right)`. A similar approach is used for the gas mixture dual
velocities, which are related at the boundary :math:`i-1/2` by the
coefficients :math:`C_{\text{Mo,Mi,i}}`, stored in the array :math:`\text{CCMI} \left( I \right)`.
The three momentum equations are solved simultaneously to avoid
numerical instabilities due to the generally low inertia of the gas
mixture. In cells where only two velocity fields are necessary (e.g., no
fuel/steel chunks are present) a system of only two momentum equations
is solved. Finally, in cells where the gas mixture only is present, the
corresponding momentum equation is solved, while the other two velocity
fields remain zero. We first present the derivation of the momentum
equations.
.. _section-16.4.3.8.1:
The Momentum Conservation Equation for the Gas Mixture
''''''''''''''''''''''''''''''''''''''''''''''''''''''
We begin with the equation written in conservative form for the control
volume illustrated in :numref:`figure-16.4-15`:
(16.4-135)
.. _eq-16.4-135:
.. math::
\frac{\partial}{\partial \text{T}}\left\lbrack \rho_{\text{Mi,i-1}} \cdot \frac{z_{\text{i-1}}}{2} \
\cdot A_{\text{Mi,i-1}} \cdot {u'}_{\text{Mi,i}} + \rho_{\text{Mi,i}} \cdot \frac{z_{i}}{2} \cdot A_{\text{Mi,i}} \
\cdot {u''}_{\text{Mi,i}} \right\rbrack + \left\lbrack \left( \rho A u^{2} \right)_{\text{Mi,i}} \
- \left( \rho A u^{2} \right)_{\text{Mi,i-1}} \right\rbrack = \\
- A_{\text{Mi,i-1/2}} \cdot \left( P_{\text{i}} - P_{\text{i-1}} \right) + \sum_{l}{\left( \Gamma_{\text{Mo,i-1}}^{l} \
\cdot \frac{z_{\text{i-1}}}{2} + \Gamma_{\text{Mo,i}}^{l} \cdot \frac{z_{\text{i}}}{2} \right) \\
- \rho_{\text{Mi,i-1}} \cdot \frac{z_{\text{i-1}}}{2} A_{\text{Mi,i-1}} \cdot g - {\rho'}_{\text{Mi,i}} \cdot \frac{z_{\text{i}}}{2} \
\cdot A_{\text{Mi,i}} \cdot g}
where :math:`\sum{\Gamma_{\text{Mo,i-1}}^{l}}` represents the
momentum sources and sinks for the mixture and will be presented in
detail later in this chapter. We now divide Eq. :ref:`16.4-135<eq-16.4-135>` by :math:`\text{AXMX}` and
using the definition of the generalized density, the notation:
(16.4-136)
.. _eq-16.4-136:
.. math::
\begin{align}
\frac{\Delta z_{\text{i}}}{2} = \Delta z_{2} && ; && \frac{\Delta z_{\text{i-1}}}{2} = \Delta z_{1}
\end{align}
and the correlation of :math:`{u'}_{\text{Mi,i}}` and :math:`{u''}_{\text{Mi,i}}`, obtain
(16.4-137)
.. _eq-16.4-137:
.. math::
\begin{matrix}
\frac{\partial}{\partial \text{t}}\left\lbrack {\rho'}_{\text{Mi,i-1}} \cdot \Delta z_{\text{i}} \cdot u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}} + {\rho'}_{\text{Mi,i}} \cdot \Delta z_{2} \cdot u_{\text{Mi,i}} \right\rbrack \\
= - \left\lbrack \left( {\rho'} u^{2} \right)_{\text{Mi,i}} - \left( {\rho'} u^{2} \right)_{\text{Mi,i-1}} \right\rbrack - \theta_{\text{Mi,i-1/2}} \cdot \left( P_{\text{i}} - P_{\text{i-1}} \right) \\
+ \sum\left( {\Gamma'}_{\text{Mo,i-1}}^{l} \cdot \Delta z_{\text{i}} + {{\Gamma'}^{l}}_{\text{Mo,i}} \cdot \Delta z_{2} \right) - {\rho'}_{\text{Mi,i-1}} \cdot \Delta z_{1} \cdot g - {\rho'}_{\text{Mi,i}} \cdot \Delta z_{2} g \\
\end{matrix}
.. _figure-16.4-15:
.. figure:: media/image23.png
:align: center
:figclass: align-center
Control Volume Used for the Solution of the Momentum Equation
where we used
(16.4-138)
.. _eq-16.4-138:
.. math::
u_{\text{mi,i}} = {u''}_{\text{Mi,i}}
in order to simplify the notation and:
(16.4-139)
.. _eq-16.4-139:
.. math::
{\Gamma'}_{\text{Mo,i}}^{l} = \frac{\Gamma_{\text{Mo,i}}^{l}}{\text{AXMX}}
The convective fluxes in Eq. :ref:`16.4-137<eq-16.4-137>` are defined as follows:
(16.4-140)
.. _eq-16.4-140:
.. math::
\left( {\rho'} u^{2} \right)_{\text{Mi,i}} = {\rho'}_{\text{Mi,i}} \
\cdot \frac{\left( {u''}_{\text{Mi,i}} + {u'}_{\text{Mi,i+1}} \right)^{2}}{4}
An optional formulation of the convective fluxes can be obtained by
setting the input parameter :math:`\text{IMOMEN} = 1`, in which case:
(16.4-141)
.. _eq-16.4-141:
.. math::
\left( {\rho'} u^{2} \right)_{\text{Mi,i}} = {\rho'}_{\text{Mi,i}} \cdot u_{\text{Mo,Mi,i}}^{2}
where
(16.4-142a-d)
.. _eq-16.4-142a-d:
.. math::
u_{\text{Mo,Mi,i}} = \begin{cases}
{u''}_{\text{Mi,i}} & \text{if } {u''}_{\text{Mi,i}} \geq 0 \text{ and } {u'}_{\text{Mi,i+1}} \geq 0 \\
\sqrt{{u''}_{\text{Mi,i}}^{2} + {u'}_{\text{Mi,i+1}}^{2}} & \text{if } {u''}_{\text{Mi,i}} \geq 0 \text{ and } {u'}_{\text{Mi,i+1}} \leq 0 \\
0 & \text{if } {u''}_{\text{Mi,i}} < 0 \text{ and } {u'}_{\text{Mi,i+1}} \geq 0 \\
{u'}_{\text{Mi,i+1}} & \text{if } {u''}_{\text{Mi,i}} < 0 \text{ and } {u'}_{\text{Mi,i+1}} < 0 \\
\end{cases}
This option has been added only recently and has not been tested
extensively. It is expected that in future release versions this
formulation will become the basic option. The quantity
:math:`\theta_{\text{Mi,i-1/2}}` is defined differently for expansions and
contractions [16-5]. For an expansion:
(16.4-143a-b)
.. _eq-16.4-143a-b:
.. math::
\theta_{\text{Mi,i-1/2}} = \begin{cases}
\theta_{\text{Mi,i}} & \text{if } {u'}_{\text{mi,i}} \geq 0 \\
\theta_{\text{Mi,i-1}} & \text{if } {u''}_{\text{Mi,i}} < 0 \\
\end{cases}
and for contraction:
(16.4-144)
.. _eq-16.4-144:
.. math::
\theta_{\text{Mi,i-1/2}} = C_{\Delta \text{p}} \cdot \frac{\theta_{\text{Mi,i-1}} \
\cdot \theta_{\text{Mi,i}}}{\theta_{\text{Mi,i-1}} + \theta_{\text{Mi,i}}}
with :math:`C_{\Delta \text{p}}` currently having the value of 1.67. Equation :ref:`16.4-137<eq-16.4-137>`
is integrated over the time interval and then divided by :math:`\Delta t`. Using the
identities:
(16.4-145a)
.. _eq-16.4-145a:
.. math::
\Delta\left( {\rho'} u \right) = {\rho'}^{n + 1} \cdot \Delta u + u^{n} \cdot \Delta {\rho'}
(16.4-145b)
.. _eq-16.4-145b:
.. math::
\Delta \left( {\rho'} u C_{\text{Mo}} \right) = {\rho'}^{n + 1} C_{\text{Mo}}^{n + 1} \Delta u \
+ {\rho'}^{n + 1} \cdot u^{n} \cdot \Delta C_{\text{Mo}} + u^{n} C^{n} \cdot \Delta {\rho'}
we obtain:
(16.4-146)
.. _eq-16.4-146:
.. math::
\begin{matrix}
\Delta u_{\text{Mi,i}} \cdot \left( {\rho'}_{\text{Mi,i-1}}^{n + 1} \cdot C_{\text{Mo,Mi,i}}^{n + 1} \cdot \Delta z_{\text{i}} + {\rho'}_{\text{Mi,i}}^{n + 1} \cdot \Delta z_{2} \right) \cdot \frac{1}{\Delta t} \\
= - \left\lbrack \left( {\rho'} u^{2} \right)_{\text{Mi,i}} - \left( {\rho'} u^{2} \right)_{\text{Mi,i-1}} \right\rbrack - \theta_{\text{Mi,i-1/2}} \cdot \left( P_{\text{i}} - P_{\text{i-1}} \right) \\
- u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}} \cdot \Delta z_{\text{i}} \cdot \frac{\Delta{\rho'}_{\text{Mi,i}}}{\Delta t} - u_{\text{Mi,i}} \cdot \Delta z_{2} \cdot \frac{\Delta{\rho'}_{\text{Mi,i}}}{\Delta t} \\
- {\rho'}_{\text{Mi,i-1}}^{n + 1} \cdot u_{\text{Mi,i}} \cdot \Delta z_{\text{i}} \cdot \frac{\Delta C_{\text{Mo,Mi,i}}}{\Delta t} + \sum_{l} \left( {\Gamma'}_{\text{Mo,Mi,i-1}}^{l} \cdot \Delta z_{1} + {\Gamma'}_{\text{Mo,Mi,i}}^{l} \cdot \Delta z_{2} \right) \\
- {\rho'}_{\text{Mi,i-1}} \cdot \Delta z_{\text{i}} \cdot g - {\rho'}_{\text{Mi,i}} \cdot \Delta z_{2} \cdot g \\
\end{matrix}
We will now present the term
:math:`\sum_{l}{{\Gamma'}_{\text{Mo,i}}^{l}}` in
more detail:
(16.4-147)
.. _eq-16.4-147:
.. math::
\begin{matrix}
\sum_{l}{{\Gamma'}_{\text{Mo,Mi,i}}^{l}} = {\Gamma'}_{\text{Mo,Mi,i}}^{\text{cond}} + {\Gamma'}_{\text{Mo,Mi,i}}^{\text{vap}} + {\Gamma'}_{\text{Mo,Mi,i}}^{\text{friction wall}} + {\Gamma'}_{\text{Mo,Mi,i}}^{\text{drag fuel}} \\
+ {\Gamma'}_{\text{Mo},\text{Mi},i}^{\text{drag chunks}} + {\Gamma'}_{\text{Mo},\text{Mi},i}^{\text{inertial}} \\
\end{matrix}
The momentum sink
:math:`{\Gamma'}_{\text{Mo,Mi,i}}^{\text{cond}}` is due to
possible condensation of steel and/or fuel vapor:
(16.4-148)
.. _eq-16.4-148:
.. math::
{\Gamma'}_{\text{Mo,Mi,i}}^{\text{cond}} = - \frac{\Delta{\rho'}_{\text{sv,i}}^{\text{cond}}}{\Delta t} u_{\text{Mi,i}} \
- \frac{\Delta{\rho'}_{\text{fv,i}}^{\text{cond}}}{\Delta t} \cdot u_{\text{Mi,i}}
Similarly, the vaporization source is defined as:
(16.4-149)
.. _eq-16.4-149:
.. math::
{\Gamma'}_{\text{Mo,Mi,i}}^{\text{vap}} = \frac{\Delta {\rho'}_{\text{sv,i}}^{\text{vap}}}{\Delta t} \cdot u_{\text{fu,i}} \
+ \frac{\Delta{\rho'}_{\text{fv,i}}^{\text{vap}}}{\Delta t} \cdot u_{\text{fu,i}}
The quantities :math:`\Delta {\rho'}_{\text{fv}}^{\text{cond}}` and
:math:`\Delta {\rho'}_{\text{fv}}^{\text{vap}}` are calculated in the
routine :math:`\text{LEFUVA}`. These calculations are presented in :numref:`section-16.4.3.6`.
The quantities :math:`\Delta {\rho'}_{\text{sv}}^{\text{cond}}` and
:math:`\Delta {\rho'}_{\text{sv}}^{\text{vap}}` are obtained in the
:math:`\text{LESEVA}` routine in the same manner.
The momentum sink due to the wall friction has the form:
(16.4-150)
.. _eq-16.4-150:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,Mi,i}}^{\text{friction wall}} = - f_{\text{Mi,i}} \cdot \frac{{\rho'}_{\text{Mi,i}}}{2D_{\text{H,Mi,i}}} \cdot u_{\text{Mi,i}}^{n + 1} \cdot \left| u_{\text{Mi,i}}^{n} \right| \\
= - f_{\text{Mi,i}} \cdot \frac{{\rho'}_{\text{Mi,i}}}{2D_{\text{H,Mi,i}}} \cdot \left( u_{\text{Mi,i}}^{n} + \Delta u_{\text{Mi,i}} \right) \cdot \left| u_{\text{Mi,i}}^{n} \right| \\
\end{matrix}
The wall friction factor :math:`f_{\text{Mi,i}}` will be described in :numref:`section-16.4.3.10`. The hydraulic diameter of the gas mixture :math:`D_{\text{H,Mi,i}}`
will be described in :numref:`section-16.4.3.9`. The fuel/steel-gas mixture drag
source has the form
(16.4-151)
.. _eq-16.4-151:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,Mi,i}}^{\text{drag fuel}} = {C'}_{\text{D,Mi,fu,i}} \left( u_{\text{fu,i}}^{n + 1} - u_{\text{Mi,i}}^{n + 1} \right) \cdot \left| u_{\text{fi,i}} - u_{\text{Mi,i}} \right| \\
= {C'}_{\text{D,Mi,fu,i}} \cdot \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} - u_{\text{Mi,i}} - \Delta u_{\text{Mi,i}} \right) \cdot \left| u_{\text{fu,i}} - u_{\text{Mi,i}} \right| \\
\end{matrix}
The drag coefficient between fuel and mixture has different forms,
depending on the local flow regime, as shown below:
For the annular fuel and steel flow regimes:
(16.4-152a)
.. _eq-16.4-152a:
.. math::
{C'}_{\text{D,Mi,fu,i}} = f_{\text{Mi,fu,i}} \cdot \frac{{\rho'}_{\text{Mi,i}}}{2 \cdot D_{\text{H,Mi,i}}}
(16.4-152b)
.. _eq-16.4-152b:
.. math::
{C'}_{\text{D,Mi,se,i}} = f_{\text{Mi,se,i}} \cdot \frac{{\rho'}_{\text{Mi,i}}}{2 \cdot D_{\text{H,Mi,i}}}
For the bubbly fuel flow regime:
(16.4-153a)
.. _eq-16.4-153a:
.. math::
{C'}_{\text{D,Mi,fu,i}} = \theta_{\text{Mi,i}} \cdot \rho_{\text{fu,i}} \cdot \text{CIA6} \
\cdot \sqrt{\frac{g}{\sigma_{\text{fu}}} \cdot \left( \frac{{\rho'}_{\text{fu,i}} \
+ {\rho'}_{\text{se,i}}}{\theta_{\text{fu,i}} + \theta_{\text{se,i}}} \
- \frac{{\rho'}_{\text{Mi,i}}}{\theta_{\text{Mi,i}}} \right)} \cdot \left( \frac{\theta_{\text{fu,i}} \
+ \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}}} \right)^{2}
where CIA6 is an input constant, currently equal to 0.107. This form of
:math:`{C'}_{\text{D,Mi,fu,i}}` is explained in more detail in Ch. 14.0, :numref:`section-14.4.6.1`, which describes the PLUTO2 model.
For the bubbly steel flow regime:
(16.4-153b)
.. _eq-16.4-153b:
.. math::
{C'}_{\text{D,Mi,se,i}} = \theta_{\text{Mi,i}} \cdot \rho_{\text{se,i}} \cdot \text{CIA6} \
\cdot \sqrt{\frac{g}{\sigma_{\text{se}}} \cdot \left( \frac{{\rho'}_{\text{fu,i}} \
+ {\rho'}_{\text{se,i}}}{\theta_{\text{fu,i}} + \theta_{\text{se,i}}} \
- \frac{{\rho'}_{\text{Mi,i}}}{\theta_{\text{Mi,i}}} \right)} \cdot \left( \frac{\theta_{\text{fu}} \
+ \theta_{\text{se}}}{\theta_{\text{ch,op,i}}} \right)^{2}
The factor :math:`f_{\text{Mi,fu,i}}` which appears in Eq. :ref:`16.4-152a<eq-16.4-152a>` will be
described in :numref:`section-16.4.3.10`.
The chunk-mixture drag source has the form:
(16.4-154)
.. _eq-16.4-154:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,Mi,i}}^{\text{drag chunk}} = {C'}_{\text{D,Mi,} l \text{u,i}} \cdot \left( u_{l \text{u,i}}^{n + 1} - u_{\text{Mi,i}}^{n + 1} \right) \cdot \left| u_{l \text{u,i}} - u_{\text{Mi,i}} \right| \\
= {C'}_{\text{D,Mi,lu,i}} \cdot \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} - u_{\text{Mi,i}} - \Delta u_{\text{Mi,i}} \right) \cdot \left| u_{l \text{u,i}} - u_{\text{Mi,i}} \right| \\
\end{matrix}
The drag coefficient :math:`{C'}_{\text{D,Mi,lu,i}}` is defined as follows:
For the annular steel or fuel flow regime:
(16.4-155)
.. _eq-16.4-155:
.. math::
{C'}_{\text{D,Mi,} l \text{u,i}} = \theta_{l \text{u,i}} \cdot \rho_{\text{Mi,i}} \
\cdot \frac{1}{R_{l \text{u,i}}} \cdot \left( \frac{\theta_{\text{Mi,i}}}{\theta_{\text{ch,i}}} \right)^{\text{CIA5}} \
\cdot C_{\text{DRAG}} \cdot C_{\text{AREA,Mi,} l \text{u,i}}
where CIA4 and CIA5 are input constants, with the values 0.375 and -2.7,
respectively. The coefficient :math:`C_{\text{DRAG}}` is defined as follows:
If :math:`\text{Re}_{\text{lu,i}} > 500`,
(16.4-156a)
.. _eq-16.4-156a:
.. math::
C_{\text{DRAG}} = 0.44
If :math:`\text{Re}_{\text{lu,i}} \leq 500`, then
(16.4-156b)
.. _eq-16.4-156b:
.. math::
C_{\text{DRAG}} = 18.5 \cdot \left( \text{Re}_{l \text{u,i}} \right)^{- 6}
The chunk Reynolds number :math:`\text{Re}_{\text{lu,i}}` used in Eq. :ref:`16.4-156<eq-16.4-156a>` is
defined as follows:
(16.4-157)
.. _eq-16.4-157:
.. math::
\text{Re}_{l \text{u,i}} = 2 \cdot R_{l \text{u,i}} \cdot \left| u_{\text{Mi,i}} \
- u_{l \text{u,i}} \right| \cdot \rho_{\text{Mi,i}} \cdot \frac{1}{\mu_{\text{Mi,i}}}
The coefficient :math:`C_{\text{AREA,Mi,} l \text{u,i}}` is used to take into account the
fact that the chunks are in contact not only with the gas mixture, but
with other components too, such as molten fuel or steel and cladding.
This coefficient will be described in :numref:`section-16.4.3.10`.
For the bubbly steel and fuel flow regimes:
(16.4-158)
.. _eq-16.4-158:
.. math::
{C'}_{\text{D,Mi,} l \text{u,i}} = 0
Finally, the inertial (or apparent mass) momentum source term has the
form:
(16.4-159)
.. _eq-16.4-159:
.. math::
{\Gamma'}_{\text{Mo,Mi,i}}^{\text{inertial}} = {C'}_{\text{IN,Mi,fu,i}} \cdot \left\lbrack \
\frac{\Delta\left( u_{\text{fu,i}} - u_{\text{Mi,i}} \right)}{\Delta t} + u_{\text{Mi,i}}\frac{\Delta\left( u_{\text{fu}} \
- u_{\text{Mi}} \right)}{\Delta t} \right\rbrack
where
(16.4-160)
.. _eq-16.4-160:
.. math::
\frac{\Delta\left( u_{\text{fu}} - u_{\text{Mi}} \right)}{\Delta z} = \begin{cases}
\frac{\left( {u'}_{\text{fu,i}} - {u'}_{\text{Mi,i}} \right) - \left( {u''}_{\text{fu,i-1}} - {u''}_{\text{Mi,i-1}} \right)}{2\Delta z_{1}} & \text{if } u_{\text{Mi,i}} > 0 \\
\frac{\left( {u'}_{\text{fu,i+1}} - {u'}_{\text{Mi,i+1}} \right) - \left( {u''}_{\text{fu,i}} - {u''}_{\text{Mi,i}} \right)}{2\Delta z_{2}} & \text{if } u_{\text{Mi,i}} < 0 \\
\end{cases}
and the generalized inertial coefficient is defined as follows:
For annular steel or fuel flow, no inertial effects are present:
(16.4-161)
.. _eq-16.4-161:
.. math::
{C'}_{\text{IN,Mi,fu,i}} = 0~.
For bubbly steel flow:
(16.4-162)
.. _eq-16.4-162:
.. math::
{C'}_{\text{IN,Mi,fu,i}} = 0.5 \cdot \rho_{\text{se,i}} \cdot \theta_{\text{Mi,i}}~.
For bubbly fuel flow:
(16.4-163)
.. _eq-16.4-163:
.. math::
{C'}_{\text{IN,Mi,fu,i}} = 0.5 \cdot \rho_{\text{fu,i}} \cdot \theta_{\text{Mi,i}}~.
The source terms
:math:`\sum{{\Gamma'}_{\text{Mo,i-1}}^{l}}` in Eq.
:ref:`16.4-147<eq-16.4-147>` are similar to the
:math:`\sum{{\Gamma'}_{\text{Mo,i}}^{l}}` terms that
have been presented in detail, but they *cannot* be obtained from
:math:`{\Gamma'}_{\text{Mo,i}}^{\mathcal{l}}` by simply replacing the
subscript :math:`i` by :math:`i-1`. The velocity :math:`u_{\text{Mi,i}}` in these terms has to be
replaced by :math:`{u'}_{\text{Mi,i}}`, i.e., :math:`C_{\text{Mo,Mi,i}} \cdot u_{\text{Mi,i}}`.
The components of
:math:`\sum{{\Gamma'}_{\text{Mo,i-1}}^{l}}` are presented below:
(16.4-164)
.. _eq-16.4-164:
.. math::
{\Gamma'}_{\text{Mo,Mi,i}}^{\text{cond}} = - \frac{\Delta{\rho'}_{\text{sv,i-1}}^{\text{cond}}}{\Delta t} \
\cdot C_{\text{Mo,Mi,i}}^{n + 1} \cdot u_{\text{Mi,i}} - \frac{\Delta{\rho'}_{\text{fv,i-1}}^{\text{cond}}}{\Delta t} \
\cdot C_{\text{Mo,Mi,i}}^{n + 1} \cdot u_{\text{Mi,i}}
(16.4-165)
.. _eq-16.4-165:
.. math::
{\Gamma'}_{\text{Mo,Mi,i-1}}^{\text{vap}} = \frac{\Delta\rho_{\text{sv,i-1}}^{\text{vap}}}{\Delta t} \
C_{\text{Mo,fu,i}}^{n + 1} u_{\text{fu,i}} + \frac{\Delta\rho_{\text{fv,i-1}}^{\text{vap}}}{\Delta t} \
C_{\text{Mo,fu,i}}^{n + 1} \cdot u_{\text{fu,i}}
(16.4-166)
.. _eq-16.4-166:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,Mi,i-1}}^{\text{friction wall}} = - f_{\text{Mi,i-1}} \cdot \frac{{\rho'}_{\text{Mi,i-1}}}{2 \cdot D_{\text{H,Mi,i-1}}} \cdot \left( u_{\text{Mi,i}} + \Delta u_{\text{Mi,i}} \right) \\
\cdot \left| u_{\text{MI,i}} \right| \cdot \left( C_{\text{Mo,Mi,i}}^{n + 1} \right)^{2} \\
\end{matrix}
(16.4-167)
.. _eq-16.4-167:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,Mi,i-1}}^{\text{drag fuel}} = {C'}_{\text{D,Mi,fu,i-1}} \left\lbrack \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} \right) \cdot C_{\text{Mo,fu,i}}^{n + 1} - \left( u_{\text{Mi,i}} + \Delta u_{\text{Mi,i}} \right) \\
\cdot C_{\text{Mo,Mi,i}}^{n + 1} \right\rbrack \cdot \left| u_{\text{fu,i}} \cdot C_{\text{Mo,fu,i}} - u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}} \right| \\
\end{matrix}
(16.4-168)
.. _eq-16.4-168:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,Mi,i-1}}^{\text{drag chunk}} = {C'}_{\text{D,Mi,} l \text{u,i-1}} \cdot \left\lbrack \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} \right) - \left( u_{\text{Mi,i}} + \Delta u_{\text{Mi,i}} \right) \\
\cdot C_{\text{Mo,Mi,i}}^{n + 1} \right\rbrack \cdot \left| u_{l \text{u,i}} - u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}} \right| \\
\end{matrix}
(16.4-169)
.. _eq-16.4-169:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,Mi,i-1}}^{\text{inertial}} = {C'}_{\text{IN,Mi,fu,i}} \cdot \frac{\Delta\left\lbrack \left( u_{\text{fu,i}} \cdot C_{\text{Mo,fu,i}} \right) - \left( u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}} \right) \right\rbrack}{\Delta t} \\
+ u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}}^{n + 1} \cdot \frac{\Delta\left( u_{\text{fu}} - u_{\text{Mi}} \right)}{\Delta z} \\
\end{matrix}
where the definition of
:math:`\frac{\Delta\left( u_{\text{fu}} - u_{\text{Mi}} \right)}{\Delta z}`
is the same as Eq. :ref:`16.4-160<eq-16.4-160>`. It should be noted that, in the definition
of the source terms, we used the assumption that changes in the
coefficients :math:`C_{\text{Mo,fu,i}}` can be neglected during one time step,
and thus only the new coefficients at time :math:`n+1` have been used.
We can now replace the expression
:math:`\sum_{l}{\left( {\Gamma'}_{\text{Mo,i-1}}^{l} \cdot \Delta z_{\text{i}} + {\Gamma'}_{\text{Mo,i}}^{l} \cdot \Delta z_{2} \right)}` in
Eq. :ref:`16.4-147<eq-16.4-147>` using Eqs. :ref:`16.4-148<eq-16.4-148>` through :ref:`16.4-167<eq-16.4-167>`. We then rearrange Eq.
:ref:`16.4-147<eq-16.4-147>` in the form:
(16.4-170)
.. _eq-16.4-170:
.. math::
\text{DMX} \cdot \Delta u_{\text{Mi,i}} = \text{AMX} + \text{BMX} \cdot \Delta u_{\text{fu,i}} + \text{CMX} \cdot \Delta u_{l \text{u,i}}
All terms containing the time change of the gas mixture velocity
:math:`\Delta u_{\text{Mi,i}}` were moved to the left-hand side of the Eq. :ref:`16.4-170<eq-16.4-170>`,
and after factoring :math:`\Delta u_{\text{Mi,i}}`, put in the form :math:`\text{DMX} \cdot \Delta u_{\text{Mi,i}}`.
Similarly, all terms containing :math:`\Delta u_{\text{fu,i}}` and
:math:`\Delta u_{l \text{u,i}}` were grouped together on the right-hand side. All other
terms were grouped under the coefficient :math:`\text{AMX}`. It should be observed that
the term
:math:`\frac{\Delta\left( u_{\text{fu}} - u_{\text{Mi}} \right)}{\Delta z}`
which appears in Eqs. :ref:`16.4-159<eq-16.4-159>` and :ref:`16.4-169<eq-16.4-169>` does not contain time
changes and thus will be included in the :math:`\text{AMX}` coefficient.
.. _section-16.4.3.8.2:
The Momentum Conservation Equation for the Molten Fuel/Steel Component
''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
Using an integration procedure similar to that used for the gas-mixture
equation, we obtain the following equation:
(16.4-171)
.. _eq-16.4-171:
.. math::
\begin{matrix}
\Delta u_{\text{fu,i}} \left\lbrack \left( {\rho'}_{\text{fu,i-1}}^{n + 1} + {\rho'}_{\text{se,i-1}}^{n + 1} \right) \cdot C_{\text{Mo,fu,i}}^{n + 1} \cdot \Delta z_{1} + \left( {\rho'}_{\text{fu},i}^{n + 1} + {\rho'}_{\text{se}}^{n + 1} \right) \cdot \Delta z_{2} \right\rbrack \cdot \frac{1}{\Delta t} \\
= - \left\lbrack \left( {\rho'} u^{2} \right)_{\text{fu,i}} - \left( {\rho'} u^{2} \right)_{\text{fu,i}} \right\rbrack - \theta_{\text{fu,i-1/2}} \cdot \left( P_{\text{I}} - P_{\text{i-1}} \right) \\
- u_{\text{fu,i}} \cdot C_{\text{Mo,fu,i}} \cdot \Delta z_{\text{i}} \cdot \left( \frac{\Delta{\rho'}_{\text{fu,i-1}}}{\Delta t} + \frac{\Delta{\rho'}_{\text{se,i-1}}}{\Delta t} \right) \\
- u_{\text{fu,i}} \cdot \Delta z_{2} \cdot \left( \frac{\Delta{\rho'}_{\text{fu,i}}}{\Delta t} + \frac{\Delta{\rho'}_{\text{se,i}}}{\Delta t} \right) - \left( {\rho'}_{\text{fu,i-1}}^{n + 1} + {\rho'}_{\text{se,i-1}}^{n + 1} \right) \cdot u_{\text{fu,i}} \cdot \Delta z_{\text{i}} \cdot \frac{\Delta C_{\text{Mo,fu,i}}}{\Delta t} \\
+ \sum_{\text{m}} \left( {\Gamma'}_{\text{Mo,fu,i-1}}^{m} \cdot \Delta z_{1} + {\Gamma'}_{\text{Mo,fu,i}} \cdot \Delta z_{2} \right) - \left( {\rho'}_{\text{fu,i-1}} + {\rho'}_{\text{se,i-1}} \right) \cdot \Delta z_{1} \cdot g \\
- \left( {\rho'}_{\text{fu,i}} + {\rho'}_{\text{se,i}} \right) \cdot \Delta z_{2} \cdot g \\
\end{matrix}
where
(16.4-172)
.. _eq-16.4-172:
.. math::
\left( {\rho'} u^{2} \right)_{\text{fu,i}} = \left( {\rho'}_{\text{fu,i}} + {\rho'}_{\text{se,i}} \right) \
\cdot \left( {u''}_{\text{fu,i}} + {u'}_{\text{fu,i+1}} \right)^{2} \cdot .25
The optional formulation, which can be obtained using :math:`\text{IMOMEN} = 1` is:
(16.4-173)
.. _eq-16.4-173:
.. math::
\left( {\rho'} u^{2} \right)_{\text{fu,i}} = \left( {\rho'}_{\text{fu,i}} + {\rho'}_{\text{se,i}} \right) \cdot u_{\text{Mo,fu,i}}^{2}
where
(16.4-174a-d)
.. _eq-16.4-174a-d:
.. math::
u_{\text{Mo,Mi,i}} = \begin{cases}
{u''}_{\text{fu,i}} & \text{if } {u''}_{\text{fu,i}} \geq 0 \text{ and } {u'}_{\text{fu,i+1}} \geq 0 \\
\sqrt{{u''}_{\text{fu,i}}^{2} + {u'}_{\text{fu,i+1}}^{2}} & \text{if } {u''}_{\text{fu,i}} \geq 0 \text{ and } {u'}_{\text{fu,i+1}} \leq 0 \\
0 & \text{if } {u''}_{\text{fu,i}} < 0 \text{ and } {u'}_{\text{fu,i+1}} \geq 0 \\
{u'}_{\text{fu,i+1}} & \text{if } {u'}_{\text{fu,i}} < 0 \text{ and } {u'}_{\text{fu,i+1}} < 0 \\
\end{cases}
The quantity :math:`\theta_{\text{fu,i-1/2}}` is defined by Eqs. :ref:`16.4-142<eq-16.4-142a-d>` and
16.4-143, where the :math:`\theta_{\text{Mi,i}}` is replaced by :math:`\left( \theta_{\text{fu,i}} + \theta_{\text{se,i}} \right)`.
The source/sink terms in
:math:`\sum_{\text{m}}{{\Gamma'}_{\text{Mo,fu,i}}^{m}}` are defined
in detail below:
The condensation term:
(16.4-175)
.. _eq-16.4-175:
.. math::
{\Gamma'}_{\text{Mo,fu,i}}^{\text{cond}} = \frac{\Delta{\rho'}_{\text{fv,i}}^{\text{cond}}}{\Delta t} \cdot u_{\text{Mi,i}}
The vaporization term:
(16.4-176)
.. _eq-16.4-176:
.. math::
{\Gamma'}_{\text{Mo,fu,i}}^{\text{vap}} = - \frac{\Delta{\rho'}_{\text{fv,i}}}{\Delta t} \cdot u_{\text{fu,i}}
The wall friction term is dependent on the flow regime:
For the annular or bubbly fuel flow regimes:
(16.4-177)
.. _eq-16.4-177:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,fu,i}}^{\text{friction wall}} = - f_{\text{se,i}} \cdot \frac{\rho_{\text{se,i}} \cdot \left( \theta_{\text{fu,i}} + \theta_{\text{se,i}} \right)}{2 \cdot D_{\text{H,se,i}}} \cdot C_{\text{AREA,se,i}} \cdot u_{\text{fu,i}}^{n + 1} \cdot \left| u_{\text{fu,i}}^{n} \right| \\
- f_{\text{se,i}}\frac{\rho_{\text{se,i}} \left( \theta_{\text{fu,i}} + \theta_{\text{se,i}} \right)}{2 D_{\text{H,se,i}}} \cdot C_{\text{AREA,se,i}} \cdot \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} \right) \cdot \left| u_{\text{fu,i}} \right| \\
\end{matrix}
For the annular or bubbly fuel flow regimes:
(16.4-178)
.. _eq-16.4-178:
.. math::
{\Gamma'}_{\text{Mo,fu,i}}^{\text{friction wall}} = - f_{\text{fu,i}} \cdot \frac{\rho_{\text{fu,i}} \left( \theta_{\text{fu,i}} \
+ \theta_{\text{se,i}} \right)}{2 D_{\text{H,fu,i}}} \cdot C_{\text{AREA,fu,i}} \cdot \left( u_{\text{fu,i}} \
+ \Delta u_{\text{fu,i}} \right) \cdot \left| u_{\text{fu,i}} \right|
where
the :math:`D_{\text{H,se,i}}` and :math:`D_{\text{H,fu,i}}` are the hydraulic diameters
for steel and fuel, respectively, and are described in :numref:`section-16.4.3.9`.
the contact coefficients :math:`C_{\text{AREA,se,i}}` and :math:`C_{\text{AREA,fu,i}}`
account for the fact that only a fraction of the steel or fuel perimeter
is in contact with the stationary walls; they are described in :numref:`section-16.4.3.10`; and the friction coefficients :math:`f_{\text{se,i}}` and
:math:`f_{\text{fu,i}}` are defined below:
(16.4-179a-b)
.. _eq-16.4-179a-b:
.. math::
f_{\text{se,i}} = \begin{cases}
\text{CIFRFU} & \text{if } \text{Re}_{\text{se,i}} \geq \text{CIREFU} \\
\frac{80}{Re_{\text{se,i}}} & \text{if } \text{Re}_{\text{se,i}} < \text{CIREFU} \\
\end{cases}
(16.4-179c-d)
.. _eq-16.4-179c-d:
.. math::
f_{\text{fu,i}} = \begin{cases}
\text{CIFRFU} & \text{if } \text{Re}_{\text{fu,i}} \geq \text{CIREFU} \\
\frac{80}{\text{Re}_{\text{fu,i}}} & \text{if } \text{Re}_{\text{fu,i}} < \text{CIREFU} \\
\end{cases}
where :math:`\text{CIFRFU}` and :math:`\text{CIREFU}` are input constants. Currently, :math:`\text{CIFRFU} = .03` and
:math:`\text{CIREFU} = 2100`.
The fuel/steel-gas mixture drag source is similar, but of opposite sign,
to the term already presented for the gas mixture.
(16.4-180)
.. _eq-16.4-180:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,fu,i}}^{\text{drag mixture}} = - {C'}_{\text{D,Mi,fu,i}} \cdot \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} - u_{\text{Mi,i}} - \Delta u_{\text{Mi,i}} \right) \\
\cdot \left| u_{\text{fu,i}} - u_{\text{Mi,i}} \right| \\
\end{matrix}
The drag coefficient :math:`{C'}_{\text{D,Mi,fu,i}}` is defined by Eqs. :ref:`16.4-152<eq-16.4-152a>`
and :ref:`16.4-153<eq-16.4-153a>`.
The fuel/steel - chunk drag source is defined as follows:
(16.4-181)
.. _eq-16.4-181:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,fu,i}}^{\text{drag chunk}} = {C'}_{\text{D,fu,} l \text{u,i}} \cdot \left( u_{l \text{u,i}}^{n + 1} - u_{\text{fu,i}}^{n + 1} \right) \cdot \left| u_{l \text{u,i}}^{n} - u_{\text{fu,i}}^{n} \right| \\
= {C'}_{\text{D,fu,} l \text{u,i}} \cdot \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} - u_{\text{fu,i}} - \Delta u_{\text{fu,i}} \right) \cdot \left| u_{l \text{u,i}} - u_{\text{fu,i}} \right| \\
\end{matrix}
The generalized drag coefficient :math:`{C'}_{\text{D,fu,}l \text{u,i}}` is flow regime
dependent and is defined below:
For the steel annular and bubbly flow regimes:
(16.4-182)
.. _eq-16.4-182:
.. math::
{C'}_{\text{D,fu,} l \text{u,i}} = \theta_{l \text{u,i}} \cdot \rho_{\text{se,i}} \cdot \text{CIA4} \
\cdot \frac{1}{R_{l \text{u,i}}} \cdot \left\lbrack \frac{\theta_{\text{Mi,i}}}{\theta_{\text{Mi,i}} \
+ \theta_{l \text{u,i}}} \right\rbrack^{\text{CIA5}} \cdot C_{\text{DRAG}} \cdot C_{\text{AREA,se,} l \text{u,i}}
For the fuel annular and bubbly flow regimes:
(16.4-183)
.. _eq-16.4-183:
.. math::
{C'}_{\text{D,fu,} l \text{u,i}} = \theta_{l \text{u,i}} \cdot \rho_{\text{fu,i}} \cdot \text{CIA4} \
\cdot \frac{1}{R_{l \text{u,i}}} \cdot \left\lbrack \frac{\theta_{\text{Mi,i}}}{\theta_{\text{Mi,i}} \
+ \theta_{l \text{u,i}}} \right\rbrack^{\text{CIA5}} \cdot C_{\text{DRAG}} \cdot C_{\text{AREA,fu,} l \text{u,i}}
In the above equations, the drag coefficient :math:`C_{\text{DRAG}}` is defined
by 16.4-156. The area coefficients, which account for the fact that only
a fraction of the chunk lateral area is in contact with the molten fuel
or steel, are defined below:
(16.4-184)
.. _eq-16.4-184:
.. math::
C_{\text{AREA,se,} l \text{u,i}} = \begin{cases}
\left( 1 - C_{\text{AREA,Mi,} l \text{u,i}} \right) \cdot C_{\text{AREA,se,c} l \text{,i}} & \text{for annular steel flow} \\
1 & \text{for bubbly steel flow} \\
\end{cases}
where :math:`C_{\text{AREA,Mi,}l \text{u,i}}` indicates the fraction of the lateral chunk
area in contact with the gas mixture. The quantity :math:`1 - C_{\text{AREA,Mi,} l \text{u,i}}`
thus represents the fraction of the chunk area in
contact with the molten fuel/steel and cladding/hexcan wall. The
coefficient :math:`C_{\text{AREA,se,cl}}` represents the fraction of the
cladding/hexcan wall area covered by molten steel/fuel. Both these area
coefficients will be explained in more detail in :numref:`section-16.4.3.9`.
The momentum source due to fuel injection from the pin
(16.4-185)
.. _eq-16.4-185:
.. math::
{\Gamma'}_{\text{Mo,fu,i}}^{\text{injection}} = \frac{\Delta{\rho'}_{\text{fu,injection,i}}}{\Delta t} \
\cdot u_{\text{fu,ca,i}} \cdot C_{\text{Mo,injection,i}}
The quantity :math:`\Delta {\rho'}_{\text{fu,injection,i}}` represents the change in the
fuel generalized density due to injection via the pin rip or via the end
of the pin stubs, as explained in :numref:`section-16.3`. The velocity of the
injected material is :math:`u_{\text{fu,ca,i}}`, which is also explained in
:numref:`section-16.3`. The coefficient :math:`C_{\text{Mo,injection,i}}` accounts for the
axial momentum loss due to lateral acceleration and mixing during the
injection process, and is defined as follows:
(16.4-186a-b)
.. _eq-16.4-186a-b:
.. math::
C_{\text{Mo,injection,i}} = \begin{cases}
1, & \text{if injection is taking place via the pin stubs} \\
\text{CIFUMO}, & \text{if injection is taking place via the pin rip} \\
\end{cases}
where :math:`\text{CIFUMO}` is an input constant, with values between 0 and 1.
To summarize, the term
:math:`\sum_{\text{m}}{{\Gamma'}_{\text{Mo,fu,i}}^{m}}`, which appears
in Eq. :ref:`16.4-171<eq-16.4-171>`, has the form,
(16.4-187)
.. _eq-16.4-187:
.. math::
\begin{matrix}
\sum_{\text{m}}{{\Gamma'}_{\text{Mo,fu,i}}^{m}} = {\Gamma'}_{\text{Mo,fu,i}}^{\text{cond}} + {\Gamma'}_{\text{Mo,fu,i}}^{\text{vap}} + {\Gamma'}_{\text{Mo,fu,i}}^{\text{friction wall}} + {\Gamma'}_{\text{Mo,fu,i}}^{\text{drag mixture}} \\
+ {\Gamma'}_{\text{Mo,fu,i}}^{\text{drag chunk}} + {\Gamma'}_{\text{Mo,fu,i}}^{\text{injection}} \\
\end{matrix}
The term :math:`\sum_{\text{m}}{{\Gamma'}_{\text{Mo,fu,i-1}}^{m}}`
has a similar composition, and its terms are presented below:
(16.4-188)
.. _eq-16.4-188:
.. math::
{\Gamma'}_{\text{Mo,fu,i-1}}^{\text{cond}} = \frac{\Delta{\rho'}_{\text{fv,i-1}}^{\text{cond}}}{\Delta t} \
u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}}
(16.4-189)
.. _eq-16.4-189:
.. math::
{\Gamma'}_{\text{Mo,fu,i-1}}^{\text{vap}} = - \frac{\Delta{\rho'}_{\text{fv,i-1}}}{\Delta t} \cdot u_{\text{fu,i}} \cdot C_{\text{Mo,fu,i}}
For the steel flow regimes,
(16.4-190a)
.. _eq-16.4-190a:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,fu,i-1}}^{\text{friction}} = - f_{\text{se,i-1}} \cdot \frac{\rho_{\text{se,i-1}} \cdot \left( \theta_{\text{fu,i-1}} + \theta_{\text{se,i-1}} \right)}{2 D_{\text{H,se,i-1}}} \cdot C_{\text{AREA,se,i-1}} \\
\cdot C_{\text{Mo,fu,i}}^{2} \cdot \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} \right) \left| u_{\text{fu,i}} \right| \\
\end{matrix}
For the fuel flow regimes:
(16.4-190b)
.. _eq-16.4-190b:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,fu,i-1}}^{\text{friction}} = - f_{\text{fu,i-1}} \cdot \frac{\rho_{\text{fu,i-1}} \cdot \left( \theta_{\text{fu,i-1}} + \theta_{\text{se,i-1}} \right)}{2 D_{\text{H,fu,i-1}}} \cdot C_{\text{AREA,fu,i-1}} \\
\cdot C_{\text{Mo,fu,i}}^{2} \cdot \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} \right) \left| u_{\text{fu,i}} \right| \\
\end{matrix}
(16.4-191)
.. _eq-16.4-191:
.. math::
{\Gamma'}_{\text{Mo,fu,i-1}}^{\text{drag mixture}} = - {C'}_{\text{D,Mi,fu,i-1}} \cdot \left\lbrack \left( u_{\text{fu,i}} \
+ u_{\text{fu,i}} \right) \cdot C_{\text{Mo,fu,i}}^{n + 1} - \left( u_{\text{Mi,i}} + u_{\text{Mi,i}} \right) \
\cdot C_{\text{Mo,Mi,i}}^{n + 1} \right\rbrack \cdot \left| u_{\text{fu,i}} \cdot C_{\text{Mo,fu,i}} \
- u_{\text{Mi,i}} C_{\text{Mo,fu,i}} \right|
(16.4-192)
.. _eq-16.4-192:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,fu,i-1}}^{\text{drag chunk}} = {C'}_{\text{D,fu,} l \text{u,i-1}} \left\lbrack \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} \right) - \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} \right) \\
\cdot C_{\text{Mo,fu,i}}^{n + 1} \right\rbrack \cdot \left| u_{l \text{u,i}} - u_{\text{fu,i}} \cdot C_{\text{Mo,fu,i}} \right| \\
\end{matrix}
(16.4-193)
.. _eq-16.4-193:
.. math::
{\Gamma'}_{\text{Mo,fu,i-1}}^{\text{injection}} = \frac{\Delta{\rho'}_{\text{fu,injection,i-1}}}{\Delta t} \
\cdot u_{\text{fu,ca,i-1}} \cdot C_{\text{Mo,injection,i-1}}
The source terms, as given by Eqs. :ref:`16.4-175<eq-16.4-175>` through :ref:`16.4-193<eq-16.4-193>` are
substituted in Eq. :ref:`16.4-171<eq-16.4-171>`, and, after rearranging, the fuel momentum
equation is written in the form
(16.4-194)
.. _eq-16.4-194:
.. math::
\text{DFX} \cdot \Delta u_{\text{fu}} = \text{AFX} + \text{BFX} \cdot \Delta u_{\text{Mi}} + \text{CFX} \cdot \Delta u_{\text{fu}}
.. _section-16.4.3.8.3:
The Momentum Conservation Equation for the Fuel/Steel Chunks
''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
Using an integration procedure similar to that used for the gas mixture,
we obtain the following equation in finite difference form:
(16.4-195)
.. _eq-16.4-195:
.. math::
\begin{matrix}
\Delta u_{l \text{u,i}}\left\lbrack \left( {\rho'}_{\text{f} l \text{,i-1}}^{n + 1} + {\rho'}_{\text{s} l \text{,i-1}}^{n + 1} \right) \cdot \Delta z_{\text{i}} + \left( {\rho'}_{\text{f} l \text{,i-1}}^{n + 1} + {\rho'}_{\text{s} l \text{,i-1}}^{n + 1} \right) \cdot \Delta z_{2} \right\rbrack \cdot \frac{1}{\Delta t} \\
= - \left\lbrack \left( {\rho'} u^{2} \right)_{l \text{u,i}} - \left( {\rho'} u^{2} \right)_{l \text{u,i-1}} \right\rbrack - \theta_{l \text{u,i-1/2}} \cdot \left( P_{\text{i}} - P_{\text{i-1}} \right) \\
- u_{l \text{u,i}} \cdot \Delta z_{1} \cdot \left( \frac{\Delta{\rho'}_{\text{f} l \text{,i-1}}}{\Delta t} + \frac{\Delta{\rho'}_{\text{s} l \text{,i-1}}}{\Delta t} \right) - u_{l \text{u,i}} \cdot \Delta z_{2} \cdot \left( \frac{\Delta{\rho'}_{\text{f} l \text{,i}}}{\Delta t} + \frac{\Delta{\rho'}_{\text{s} l \text{,i}}}{\Delta t} \right) \\
+ \sum_{\text{n}}{ \left( {\Gamma'}_{\text{Mo,}l \text{u,i}}^{n} \cdot \Delta z_{\text{i}} + {\Gamma'}_{\text{Mo,} l \text{u,i}}^{n} \cdot \Delta z_{2} \right)} - \left( {\rho'}_{\text{f} l \text{,i-1}} + {\rho'}_{\text{s} l \text{,i-1}} \right) \cdot \Delta z_{1} \cdot g \\
- \left( {\rho'}_{\text{f} l \text{,i}} + {\rho'}_{\text{s} l \text{,i}} \right) \cdot \Delta z_{2} \cdot g \\
\end{matrix}
where
(16.4-196)
.. _eq-16.4-196:
.. math::
\left( {\rho'} u^{2} \right)_{l \text{u,i}} = \left( {\rho'}_{\text{f} l \text{,i}} \
+ {\rho'}_{\text{s} l \text{,i}} \right) \cdot \left( u_{l \text{u,i}} + u_{l \text{u,i+1}} \right)^{2} \cdot 0.25
The optimal formulation of the convective terms can be obtained by using
the input variable :math:`\text{IMOMEN} = 1` and is similar to Eq. :ref:`16.4-174<eq-16.4-174a-d>`, where
:math:`{u''}_{\text{fu,i}}` and :math:`{u'}_{\text{fu,i}}` are replaced
by :math:`u_{l \text{u,i}}` and :math:`u_{l \text{u,i}}` and :math:`u_{l \text{u,i+1}}`, respectively.
The quantity :math:`\theta_{l \text{u,i-1/2}}` is defined by Eqs. :ref:`16.4-143<eq-16.4-143a-b>` and
16.4-144, where :math:`\theta_{\text{Mi,i}}` is replaced by :math:`\theta_{l \text{u,i}}`. The source
terms are defined below:
(16.4-197)
.. _eq-16.4-197:
.. math::
\sum_{\text{n}}{ {\Gamma'}_{\text{Mo,} l \text{u,i}}^{n} = {\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{drag mixture}}} \
+ {\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{drag fuel}} + {\Gamma'}_{\text{Mo,lu,i}}^{\text{friction wall}} \
+ {\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{contraction}}
No momentum sources due to chunk formation or remelting are present in
Eq. :ref:`16.4-197<eq-16.4-197>`, because the routines modeling these processes are called
after the LEMOCO routine, and thus these effects will be considered
later. The individual sources are presented below:
The term due to the gas mixture/chunk drag:
(16.4-198)
.. _eq-16.4-198:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{drag mixture}} = - {C'}_{\text{D,Mi,lu,i}} \cdot \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} - u_{\text{Mi,i}} - \Delta u_{\text{Mi,i}} \right) \\
\cdot \left| u_{l \text{u,i}} + u_{\text{Mi,i}} \right| \\
\end{matrix}
The generalized drag coefficient
:math:`{C'}_{\text{D,Mi,}l \text{u,i}}` has been presented in Eq.
:ref:`16.4-181<eq-16.4-181>`.
The term due to the molten fuel/steel-chunk drag is as follows:
(16.4-199)
.. _eq-16.4-199:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{drag fuel}} = - {C'}_{\text{D,fu,} l \text{u,i}} \cdot \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} - u_{\text{fu,i}} - \Delta u_{\text{fu,i}} \right) \\
\cdot \left| u_{l \text{u,i}} + u_{\text{fu,i}} \right| \\
\end{matrix}
The generalized drag coefficient
:math:`{C'}_{\text{D,fu,} l \text{u,i}}` has been presented in Eq.
:ref:`16.4-181<eq-16.4-181>`.
The momentum sink due to wall friction is:
.. math::
{\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{friction wall}} = - 0.5 \cdot 10^{-3} \cdot P_{\text{i}} \cdot {A'}_{l \text{u,cl,i}} \cdot \text{SIGN} \left( u_{l \text{u,i}} \right)
where the generalized area of contact between chunks and clad/hexcan
wall, :math:`{A'}_{l \text{u,cl,i}}` is defined as:
(16.4-200)
.. _eq-16.4-200:
.. math::
{A'}_{l \text{u,cl,i}} = \frac{A_{l \text{u,cl,i}}}{\Delta z_{\text{i}} \cdot \text{AXMX}}
and is presented in :numref:`section-16.4.3.9`. Equation :ref:`16.4-199<eq-16.4-199>` was obtained
assuming that the frictional force between chunks and wall is due to the
normal force generated by the pressure :math:`P_{\text{i}}` and that the friction
coefficient between the two solid surfaces is :math:`0.5 \cdot 10^{-3}`.
The momentum sink due to jumbling at the contraction at the boundary :math:`i`
is due to chunks arriving at an abrupt contraction, where they lose
momentum upon hitting the wall normal to the flow path:
If :math:`u_{l \text{u,i}} < 0` and :math:`A_{\text{i}} > A_{\text{i-1}}`,
(16.4-201a)
.. _eq-16.4-201a:
.. math::
{\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{jumbling}} = \left( {\rho'}_{\text{f} l \text{,i}} \
+ {\rho'}_{\text{s} l \text{,i}} \right) \cdot u_{l \text{u,i}}^{2} \cdot \frac{A_{\text{i}} - A_{\text{i-1}}}{A_{\text{i}}}
Otherwise,
(16.4-201b)
.. _eq-16.4-201b:
.. math::
{\Gamma'}_{\text{Mo,} l \text{u,i}}^{\text{jumbling}} = 0
The sources
:math:`\sum_{\text{n}}{\Gamma'}_{\text{Mo,} l \text{u,i-1}}^{n}` are
defined in a similar way, as shown below:
(16.4-202)
.. _eq-16.4-202:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,} l \text{u,i-1}}^{\text{drag mixture}} = - {C'}_{\text{D,Mi,} l \text{u,i-1}} \cdot \left\lbrack \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} \right) - \left( u_{\text{Mi,i}} + \Delta u_{\text{Mi,i}} \right) \\
\cdot C_{\text{Mo,Mi,i}}^{n + 1} \right\rbrack \cdot \left| u_{l \text{u,i}} - u_{\text{Mi,i}} \cdot C_{\text{Mo,Mi,i}} \right| \\
\end{matrix}
(16.4-203)
.. _eq-16.4-203:
.. math::
\begin{matrix}
{\Gamma'}_{\text{Mo,} l \text{u,i-1}}^{\text{drag fuel}} = - {C'}_{\text{D,fu,} l \text{u,i-1}} \cdot \left\lbrack \left( u_{l \text{u,i}} + \Delta u_{l \text{u,i}} \right) - \left( u_{\text{fu,i}} + \Delta u_{\text{fu,i}} \right) \\
\cdot C_{\text{Mo,fu,i}}^{n + 1} \right\rbrack \cdot \left| u_{l \text{u,i}} - u_{\text{fu,i}} \cdot C_{\text{Mo,fu,i}} \right| \\
\end{matrix}
(16.4-204)
.. _eq-16.4-204:
.. math::
{\Gamma'}_{\text{Mo,} l \text{u,i-1}}^{\text{friction wall}} = - 0.5 \cdot 10^{- 3} \
\cdot P_{\text{i-1}} \cdot {A'}_{l \text{u,cl,i-1}} \cdot \text{SIGN} \left( u_{l \text{u,i}} \right)
If :math:`u_{l \text{u,i}} \geq 0` and :math:`A_{\text{i-1}} > A_{\text{i}}`,
(16.4-205a)
.. _eq-16.4-205a:
.. math::
{\Gamma'}_{\text{Mo,} l \text{u,i-1}}^{\text{contraction}} = \left( {\rho'}_{\text{f} l \text{,i-1}} \
+ {\rho'}_{\text{s} l \text{,i-1}} \right) \cdot u_{l \text{u,i}}^{2} \cdot \frac{A_{\text{i-1}} - A_{\text{i}}}{A_{\text{i-1}}}
Otherwise,
(16.4-205b)
.. _eq-16.4-205b:
.. math::
{\Gamma'}_{\text{Mo,} l \text{u,i-1}}^{\text{contraction}} = 0
The source terms given by Eqs. :ref:`16.4-198<eq-16.4-198>` through :ref:`16.4-205<eq-16.4-205a>` are substituted
in Eq. :ref:`16.4-195<eq-16.4-195>` and after rearranging, we obtain the chunk momentum
equation
(16.4-206)
.. _eq-16.4-206:
.. math::
\text{DLX} \cdot \Delta u_{l \text{u,i}} = \text{ALX} + \text{BLX} \cdot \Delta u_{\text{fu,i}} + \text{CLX} \cdot \Delta u_{\text{Mi,i}}
.. _section-16.4.3.8.4:
The Simultaneous Solution of the Momentum Conservation Equations
''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
The calculation of the new velocities in momentum cell i begins by
calculating the coefficients AMX and DMX for the gas-mixture Eq.
:ref:`16.4-170<eq-16.4-170>`. The terms due to the mixture interaction with the molten
fuel/steel and chunks are not included in AMX and DMX at this time. A
check is then performed to verify if the momentum cell I contains only
the gas mixture. If this is the case, the only equations solved is
(16.4-207)
.. _eq-16.4-207:
.. math::
\text{DMX} \cdot \Delta u_{\text{Mi,i}} = \text{AMX}
and only the gas-mixture velocity is updated:
(16.4-208)
.. _eq-16.4-208:
.. math::
u_{\text{Mi,i}}^{n + 1} = u_{\text{Mi,i}}^{n} + \Delta u_{\text{Mi,i}}
If either half of the momentum cell :math:`i` contains molten fuel and/or steel,
the coefficients are calculated for the momentum equations:
(16.4-209a)
.. _eq-16.4-209a:
.. math::
\text{DMX} \cdot \Delta u_{\text{Mi,i}} = \text{AMX} + \text{BMX} \cdot \Delta u_{\text{fu,i}}
(16.4-209b)
.. _eq-16.4-209b:
.. math::
\text{DFX} \cdot \Delta u_{\text{fu,i}} = \text{AFX} + \text{BFX} \cdot \Delta u_{\text{Mi,i}}
If the momentum cell does not contain fuel/steel chunks, these equations
are solved simultaneously for :math:`\Delta u_{\text{Mi,i}}` and :math:`\Delta u_{\text{fu,i}}` and
the new velocities are calculated:
(16.4-210a)
.. _eq-16.4-210a:
.. math::
u_{\text{Mi,i}}^{n + 1} = u_{\text{Mi,i}}^{n} + \Delta u_{\text{Mi,i}}
(16.4-210b)
.. _eq-16.4-210b:
.. math::
u_{\text{fu,i}}^{n + 1} = u_{\text{fu,i}}^{n} + \Delta u_{\text{fu,i}}
It is noted that the coefficients AMX and DMX in Eqs. :ref:`16.4-209<eq-16.4-209a>` are
obtained by adding to the values calculated for Eq. :ref:`16.4-207<eq-16.4-207>` the
additional terms due to the gas mixture-fuel interaction. If either half
of the momentum cell contains fuel/steel chunks, the coefficients are
calculated for the momentum equations in the form below:
(16.4-211a)
.. _eq-16.4-211a:
.. math::
\text{DMX} \cdot \Delta u_{\text{Mi,i}} = \text{AMX} + \text{BMX} \cdot \Delta u_{\text{fu,i}} + \text{CMX} \cdot \Delta u_{l \text{u,i}}
(16.4-211b)
.. _eq-16.4-211b:
.. math::
\text{DFX} \cdot \Delta u_{\text{fu,i}} = \text{AFX} + \text{BFX} \cdot \Delta u_{\text{Mi,i}} + \text{CFX} \cdot \Delta u_{l \text{u,i}}
(16.4-211c)
.. _eq-16.4-211c:
.. math::
\text{DLX} \cdot \Delta u_{l \text{u,i}} = \text{ALX} + \text{BLX} \cdot \Delta u_{\text{fu,i}} + \text{CLX} \cdot \Delta u_{\text{Mi,i}}
The coefficients that have been calculated using for Eqs. :ref:`16.4-210<eq-16.4-210a>` are
updated by adding the terms due to the presence of chunks. Equations
:ref:`16.4-211<eq-16.4-211a>` are solved simultaneously for :math:`\Delta u_{\text{Mi,i}}, \Delta_{\text{fu,i}}, \Delta u_{l \text{u,i}}`,
by using a substitution method and the new velocities
are calculated:
(16.4-212a)
.. _eq-16.4-212a:
.. math::
u_{\text{Mi,i}}^{n + 1} = u_{\text{Mi,i}}^{n} + \Delta u_{\text{Mi,i}}
(16.4-212b)
.. _eq-16.4-212b:
.. math::
u_{\text{fu,i}}^{n + 1} = u_{\text{fu,i}}^{n} + \Delta u_{\text{fu,i}}
(16.4-212c)
.. _eq-16.4-212c:
.. math::
u_{l \text{u,i}}^{n + 1} = u_{l \text{u,i}}^{n} + \Delta u_{l \text{u,i}}
.. _section-16.4.3.9:
Description of the Local Geometry
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In order to fully define the energy and momentum source terms used in
the conservation equations, we must supply the areas of contact between
various components. These areas are obtained in LEVITATE by defining the
local geometry in the routine LEGEOM. The definition of the geometry is
based on the local flow regime and the configuration of the stationary
elements, i.e., presence of fuel pins, presence of cladding, presence of
frozen fuel crusts, etc.
.. _section-16.4.3.9.1:
Local Flow Regime Definition
''''''''''''''''''''''''''''
Because the flow regimes are important in defining the local geometry,
the decision about the local flow regime is made before the geometry
definition in the LEVOFR routine. The physical models for the flow
regimes are described in :numref:`section-16.4.1.4`, and this section will
describe only the decision process used to select the appropriate flow
regime. The total flow regime is dependent on the local volumetric
fraction of various components in the cell and on the previously
established flow regime. The volumetric fraction for component :math:`i`,
:math:`\alpha_{\text{i}}`, is defined as follows:
(16.4-213)
.. _eq-16.4-213:
.. math::
\alpha_{\text{i}} = \frac{A_{\text{i}} \cdot \Delta z_{\text{i}}}{A_{\text{ch,op,i}} \
\cdot \Delta z_{\text{i}}} = \frac{\theta_{\text{i}}}{\theta_{\text{ch,op}}}
where :math:`A_{\text{ch,op}}` is the local cross sectional area of the open flow
channel. The flow regime in each cell is determined in the following
manner:
(I) If the previously established flow regime is bubbly fuel flow and
(a) if
:math:`\frac{\theta_{\text{fu,i}} + \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}} - \theta_{l \text{u,i}}} > \text{CIBBLY} \cdot 0.7` and
1. - if :math:`\theta_{\text{fu,i}} \geq \theta_{\text{se,i}} \cdot 0.9` → bubbly fuel flow regime
2. - if :math:`\theta_{\text{fu,i}} < \theta_{\text{se,i}} \cdot 0.9` → bubbly steel flow regime
(b) if
:math:`\frac{\theta_{\text{fu,i}} + \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}} - \theta_{l \text{u,i}}} \leq \text{CIBBLY} \cdot 0.7` and
1. - if :math:`\theta_{\text{fu,i}} \geq \theta_{\text{se,i}}` → annular fuel flow regime
2. - if :math:`\theta_{\text{fu,i}} < \theta_{\text{se,i}}` → annular steel flow regime
(II) If the previously established flow regime is bubbly steel flow and
(a) if
:math:`\frac{\theta_{\text{fu,i}} + \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}} - \theta_{l \text{u,i}}} > \text{CIBBLY} \cdot 0.7` and
1. - if :math:`\theta_{\text{se,i}} > \theta_{\text{fu,i}} \cdot 0.9` → bubbly steel flow regime
2. - if :math:`\theta_{\text{se,i}} \leq \theta_{\text{fu,i}} \cdot 0.9` → bubbly fuel flow regime
(b) if
:math:`\frac{\theta_{\text{fu,i}} + \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}} - \theta_{l \text{u,i}}} \leq \text{CIBBLY} \cdot 0.7` and
1. - if :math:`\theta_{\text{se,i}} \geq \theta_{\text{fu,i}}` → annular steel flow regime
2. - if :math:`\theta_{\text{se,i}} < \theta_{\text{fu,i}}` → annular fuel flow regime
(III) If the previously established flow regime is annular fuel or steel
flow and
(a) if
:math:`\frac{\theta_{\text{fu,i}} + \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}} - \theta_{l \text{u,i}}} < \text{CIBBLY}` and
1. - if :math:`\theta_{\text{fu,i}} \geq \theta_{\text{se,i}}` → annular fuel flow regime
2. - if :math:`\theta_{\text{fu,i}} < \theta_{\text{se,i}}` → annular steel flow regime
(b) if
:math:`\frac{\theta_{\text{fu,i}} + \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}} - \theta_{l \text{u,i}}} \geq \text{CIBBLY}` and
1. - if :math:`\theta_{\text{fu,i}} \geq \theta_{\text{se,i}}` → bubbly fuel flow regime
2. - if :math:`\theta_{\text{fu,i}} < \theta_{\text{se,i}}` → bubbly steel fuel flow regime
The input constant CIBBLY defines the threshold for the transition from
the annular to the bubbly flow regime. In the axial cells where the pins
have not yet been disrupted the value of CIBBLY is given by the input
constant CIBBIN. In disrupted nodes the value of CIBBLY is given the
input constant CIBBDI. The recommended value for CIBBIN is 0.7, and for
CIBBDI the recommended value is 0.2. Once a bubbly flow regime has been
established, a hysteresis effect is assumed to exist and the transition
threshold back to annular flow is CIBBLY :math:`\cdot` 0.7. Similar hysteresis
effects are used for the transitions from steel to fuel and vice versa
within the bubbly flow regime. Finally, it is noted that the volume
fractions used in the flow regime decision are based on the newly
calculated densities and thus are consistent with the conditions at the
end of the current time step.
.. _section-16.4.3.9.2:
Description of the Local Geometry of the Stationary and Moving Components
'''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''''
The definition of the local geometry of both stationary and moving
components is performed by the LEGEOM routine. The physical boundaries
of the rod bundle channel modeled in LEVITATE are the pin cladding and
the hexcan wall. Because of the different behavior of these boundaries,
the channel is visualized as being divided into two separate channels,
one associated with the fuel pins and another associated with the hexcan
wall, as shown in :numref:`figure-16.4-16`. The flow area is partitioned between
these two channels in proportion to the wetted perimeter of the
boundaries:
(16.4-214)
.. _eq-16.4-214:
.. math::
A_{\text{ch,cl,i}} = \frac{A_{\text{ch,i}}}{L_{\text{cl,i}} + L_{\text{sr,i}}} \cdot L_{\text{cl,i}}
(16.4-215)
.. _eq-16.4-215:
.. math::
A_{\text{ch,sr,i}} = \frac{A_{\text{ch,i}}}{L_{\text{cl,i}} + L_{\text{sr,i}}} \cdot L_{\text{sr,i}}
where :math:`L_{\text{cl,i}}` and :math:`L_{\text{sr,i}}` represent the perimeter of
the pins and hexcan wall in cell :math:`i`, respectively. It is
emphasized that LEVITATE models the whole subassembly, and not one pin
representing the subassembly. Thus, :math:`L_{\text{cl,i}}` represents the
perimeter of all the pins in the bundle. In the special situation when
no pins are present at a certain axial location due to pin disruption,
the perimeter :math:`L_{\text{cl,i}} = 0`, and the area of the channel
associated with the cladding, :math:`A_{\text{ch,cl,i}}` becomes zero. Each of
the two channel is treated, for the purpose of defining the geometry, as
a rectangular channel, with one dimension being the perimeter :math:`L`
and the other a characteristic length, :math:`l`. This characteristic
length is defined as follows:
(16.4-216)
.. _eq-16.4-216:
.. math::
l_{sr,i} = \frac{A_{\text{ch,sr,i}}}{L_{\text{sr,i}}}
.. _figure-16.4-16:
.. figure:: media/image24.png
:align: center
:figclass: align-center
Partition of Subassembly Flow Area into Two Distinct Channels
The characteristics length :math:`l` characterizes the width of the
channel and serves as an indicator for the maximum thickness of the fuel
crusts that can form on the pins and/or the hexcan wall. The conceptual
representation of these channels is presented in :numref:`figure-16.4-17`. The fuel
crust, when present, is characterized by length :math:`L_{\text{ff,cl,i}}` (and
:math:`L_{\text{ff,sv,i}}`) and thickness :math:`l_{\text{ff,cl,i}}` (and
:math:`l_{\text{ff,sv,i}}`). However, instead of storing the crust
:math:`L_{\text{ff}}`, two area coefficient are used:
(16.4-217)
.. _eq-16.4-217:
.. math::
C_{\text{ff,cl,i}} = \frac{L_{\text{ff,cl,i}}}{L_{\text{cl,i}}}
and
(16.4-218)
.. _eq-16.4-218:
.. math::
C_{\text{ff,sr,i}} = \frac{L_{\text{ff,sr,i}}}{L_{\text{sr,i}}}
As explained in :numref:`section-16.5`, the fuel crust can occasionally contain
molten or frozen steel, which is taken into account when the crust size
is calculated. The crust growth is dependent on the initial conditions,
the amount of new frozen fuel, and the local flow regime. This process
will be presented later in this chapter.
.. _figure-16.4-17:
.. figure:: media/image25.png
:align: center
:figclass: align-center
Conceptual Representation of the Pin and Hexcan Wall Channels
The configuration of the molten material, fuel and/or steel, is
characterized by the length :math:`L_{\text{fu,cl,i}} \left( L_{\text{fu,sr,i}} \right)`
and thickness :math:`l_{\text{fu,cl,i}} \left( l_{\text{fu,sr,i}} \right)`. However,
instead of storing the length :math:`L_{\text{fu}}`, two are
coefficients are used:
(16.4-219)
.. _eq-16.4-219:
.. math::
C_{\text{fu,cl,i}} = \frac{L_{\text{fu,cl,i}}}{L_{\text{cl,i}}}
(16.4-220)
.. _eq-16.4-220:
.. math::
C_{\text{ff,sr,i}} = \frac{L_{\text{ff,sr,i}}}{L_{\text{sr,i}}}
Note that the coefficient :math:`C_{\text{fu,cl,i}}` represents the fraction of
clading area covered by molten fuel and fuel crust. The same applies to
:math:`C_{\text{fu,sr,i}}`. All the contact areas required for the energy and
momentum equations are defined by using the lateral area of the clad
:math:`A_{\text{cl,i}}^{L}` and hexcan wall
:math:`A_{\text{sr,i}}^{L}` and the appropriate area coefficients.
Other area conditions used are:
:math:`C_{\text{fu,f} l \text{,cl,i}}` - fraction of cladding crust area covered by molten
fuel in cell :math:`i`
:math:`C_{\text{fu,ff,sr,i}}` - fraction of structure crust area covered by
molten fuel in cell :math:`i`
:math:`C_{\text{se,cl,i}}` - fraction of cladding area covered by molten steel
and fuel crust in cell :math:`i`
:math:`C_{\text{se,sr,i}}` - fraction of hexcan wall area covered by molten
steel and fuel crust in cell :math:`i`
:math:`C_{\text{se,ff,cl,i}}` - fraction of cladding crust area covered by molten
steel in cell :math:`i`
:math:`C_{\text{se,ff,sr,i}}` - fraction of hexcan wall crust area covered by
molten steel in cell :math:`i`
To each of these coefficients is attributed a specific value in the
routine LEGEOM, depending on the flow regime and the initial conditions.
Most of these values are self-explanatory and can be understood by
looking a :numref:`figure-16.4-1`, which illustrates the material configuration in
each LEVITATE flow regime. Some additional comments are required for the
partial annual fuel flow regime. To fully describe this flow regime, the
following assumptions were made:
1. | The molten fuel film maintains a constant ratio
:math:`\frac{l_{\text{fu,i}}}{L_{\text{fu,i}}}` both for the clad and the hexcan
wall. This ratio is defined by:
(16.4-221)
.. _eq-16.4-221:
.. math::
\frac{l_{\text{fu,i}}}{L_{\text{fu,i}}} = \frac{A_{\text{ch,op,i}} \cdot \text{CIANLR}}{\left( L_{\text{cl,i}} + L_{\text{sr,i}} \right)^{2}}
when no fuel crusts are present. It is built into the Eq. :ref:`16.4-223<eq-16.4-223a-b>`
and the user can affect it value only by changing the input
constant CIANLR. This assumption leads to a gradual increase of the
film-covered perimeter, together with a film thickness increase,
whenever the amount of molten fuel increases.
1. | The partial annual flow becomes fully annular when the volume
fraction of the molten fuel/steel reaches a certain input value,
i.e., when
(16.4-222)
.. _eq-16.4-222:
.. math::
\frac{\theta_{\text{fu,i}} + \theta_{\text{sc,i}}}{\theta_{\text{ch,op,i}}} = \text{CIANLR}
2. The existing fuel crust should be taken into account when calculating
the coefficients :math:`C_{\text{fu,cl,i}}` and :math:`C_{\text{fu,sr,i}}`.
These assumptions lead to the following definition for the area
coefficient :math:`C_{\text{fu,cl,i}}` in the annular fuel flow regime
(16.4-223a-b)
.. _eq-16.4-223a-b:
.. math::
C_{\text{fu,cl,i}} = \begin{cases}
\frac{\theta_{\text{fu,i}} + \theta_{\text{sc,i}}}{\theta_{\text{ch,op}}} + \frac{A_{\text{ff,cl,i}}}{A_{\text{ch,cl,i}}} \big/ \text{CIANLR} \\
1, & \text{if the above expression} > 1 \\
\end{cases}
For nodes where the fuel pins are undisrupted, CIANLR is set equal to
the input value CIANIN, currently 0.5. For disrupted nodes, CIANLR is
equal CIANDI. A similar formula applies to :math:`C_{\text{fu,sr,i}}`.
Another aspect treated in LEGEOM, which will be discussed in this
section, is the change in the fuel crust geometry due to additional fuel
freezing. It is noted that other aspects such as steel freezing, crust
breakup and remelting are treated in the routine LEFREZ and will be
discussed in :numref:`section-16.5`. Also, other aspects related to the chunk
geometry are treated in the LETRAN routine and will be introduced in
:numref:`section-16.4.3.10`.
The amounts of fuel/steel which have to be added to the fuel crusts due
to freezing are calculated in the LEFREZ routine in the previous time
step. In the routine LEGEOM, these amounts are converted to
corresponding are changes :math:`\Delta A_{\text{ff,cl,i}}` and :math:`\Delta A_{\text{ff,se,i}}`. The
decision on how to modify the crust parameters :math:`L_{\text{ff,cli,i}}`,
:math:`l_{\text{ff,cl,i}}`, :math:`L_{\text{ff,sr,i}}` and :math:`l_{\text{ff,sr,i}}` is made
in the following way (we will refer to the crust on the cladding only):
If no previous fuel crest is present and if the local fuel flow regime
is partially annular flow, the original crust has the same thickness as
the molten fuel film; however, if the local fuel flow regime is bubbly,
the original crust is rather thin and the area coefficient
:math:`C_{\text{ff,cl,i}}` is obtained as follows:
(16.4-224)
.. _eq-16.4-224:
.. math::
C_{\text{ff,cl,i}} = \begin{cases}
\sqrt{\frac{A_{\text{ff,cl,i}}}{A_{\text{ch,cl,i}}} \big/ \text{CIBBFZ}} \\
1, & \text{if above expression} > 1 \\
\end{cases}
where CIBBFZ is a constant, which is currently set equal to the input
constant CIBBDI. If a previously formed crust is present the thickness
of the crust is first increased until it reaches the thickness of he
fuel film or the channel characteristic size :math:`l_{\text{cl,i}}`, whichever
comes first. Then the length of the crust :math:`L_{\text{ff,cl,i}}` is
increased until :math:`C_{\text{ff,cl,i}} = 1`.
.. _section-16.4.3.9.3:
Description of the Fuel/Steel Chunk Geometry
''''''''''''''''''''''''''''''''''''''''''''
The chunks modeled in LEVITATE have a cylindrical geometry, as shown in
:numref:`figure-16.4-18`. They are characterized by the radius :math:`R`, which is different
in each axial cell. The length is related to the radius by the input
constant ASRALU, which defines the ratio :math:`L/2R` and has currently the
recommended value 1. Each chunk can contain either fuel, or steel or
both. The density of the steel chunks is constant and determined by the
input variable RHSESO. The fuel chunks, however, have a variable
density, which accounts for the possible porosity of the frozen fuel.
.. _figure-16.4-18:
.. figure:: media/image26.png
:align: center
:figclass: align-center
Fuel/Steel Chunk Geometry
The original geometry of the chunks depends on their origin. Several
mechanisms have been identified which lead to the formation of solid
fuel/steel chunks:
**A.** Disruption of the original fuel pins, which can occur when the
cladding is locally molten, leading to the formation of relatively large
fuel chunks. In this case the characteristic radius of the chunks is
determined from:
(16.4-224-1)
.. _eq-16.4-224-1:
.. math::
R_{\text{fu,i}}^{\text{NEW}} = \sqrt{\Delta R_{\text{pin,i}} \cdot \Delta L_{\text{perimeter,pin,i}}}
where
:math:`\Delta R_{\text{pin,i}}` - is the thickness of the solid fuel pin wall which
separate the pin cavity from the coolant channel
:math:`\Delta L_{\text{perimeter,pin,i}}` - is the characteristic size of the chunks
which is obtained by dividing the pin perimeter by the number of radial
cracks present at any axial location. The number of radial crack
currently used is 8.
If the input variable RALUDI is zero, the code will use the above
formula to determine the radius of the new chunks. However, this
calculation can be overridden by setting RALUDI to a non-zero value. The
radius of the newly formed chunks will be set equal to RALUDI.
**B.** Breakup of the frozen fuel crust that was formed previously on
the cladding and hexcan wall. The characteristic radius of these chunks
is determined from:
(16.4-224-2)
.. _eq-16.4-224-2:
.. math::
R_{l \text{u,i}}^{\text{NEW}} = \sqrt{\Delta A_{\text{ffc,i}}}
or
(16.4-224-3)
.. _eq-16.4-224-3:
.. math::
R_{l \text{u,i}}^{\text{NEW}} = \sqrt{\Delta A_{\text{ffs,i}}}
where
:math:`\Delta A_{\text{ffc,i}}` - is the cross sectional area of the frozen fuel crust
which breaks up during the current time step, in the axial cell :math:`i`
:math:`\Delta A_{\text{ffs,I}}` - is similar to :math:`\Delta A_{\text{ffc,i}}`, but applies to the
fuel crust associated with the hexcan wall.
**C.** Local bulk freezing of the fuel when no solid support for crust
formation is present. The characteristic radius of the chunks is
obtained from:
(16.4-224-4)
.. _eq-16.4-224-4:
.. math::
R_{\text{fu,i}}^{\text{NEW}} = \sqrt{\Delta A_{\text{fu,i}}}
where
:math:`\Delta A_{\text{fu,I}}` - is the change in the cross sectional area of the
molten fuel component due to removal of frozen fuel.
The volume of all the newly formed chunks in the axial cell :math:`i` is
calculated as follows:
(16.4-224-5)
.. _eq-16.4-224-5:
.. math::
\Delta V_{l \text{u,i}} = \left( \frac{\Delta{\rho'}_{\text{f} l \text{,i}}}{\rho_{\text{fl,i}}} \
+ \frac{\Delta{\rho'}_{\text{s}l \text{,i}}}{\rho_{\text{s} l \text{,i}}} \right) \cdot \Delta z_{\text{i}} \cdot \text{AXMX}
where
:math:`\Delta {\rho'}_{\text{fl,i}}` - is the generalized density of the fuel component in
the newly formed chunks
:math:`\Delta {\rho'}_{\text{sl,i}}` - is the generalized density of the fuel component in
the newly formed chunks
The number of new chunks is obtained by dividing the total volume
:math:`\Delta V_{l \text{u,i}}` by the volume of a single chunk:
(16.4-224-6)
.. _eq-16.4-224-6:
.. math::
\Delta N_{\text{fu,i}} = \frac{\Delta V_{\text{fu,i}} \cdot \text{ASRALU}}{2 \cdot \pi \cdot \left( R_{\text{lu,i}}^{\text{NEW}} \right)^{3}}
The new chunks will generally have different characteristics from the
chunks already present in a cell. Because only one type of chunks can
exist in one cell, the two categories are merged in one chunk population
with properties reflecting the characteristics of both original
components. This process is outlined below. Note that the same problem
can arise in any cell even without the formation of new chunks due to
the flux of chunk from neighboring cells. The first step in the merging
of the two chunk categories is to define the radius of the resulting
chunks.
(16.4-224-7)
.. _eq-16.4-224-7:
.. math::
R_{l \text{u,i}}^{n + 1} = \frac{R_{l \text{u,i}} \cdot \left\lbrack N_{l \text{u,i}} \
\cdot R_{\text{lu,i}}^{\text{ILUMER}} \right\rbrack + R_{l \text{u,i}}^{\text{NEW}} \
\cdot \left\lbrack \Delta N_{l \text{u,i}} \cdot \left( R_{l \
\text{u,i}}^{\text{NEW}} \right)^{\text{ILUMBER}} \right\rbrack}{N_{l \text{u,i}} \cdot R_{l \text{u,i}}^{\text{ILUMER}} \
+ \Delta N_{l \text{u,i}} \cdot \left( R_{l \text{u,i}}^{\text{NEW}} \right)^{\text{ILUMBER}}}
The new chunk radius is a weighted average of the radii of the two chunk
categories. The weights can be selected b the user, by changing the
input integer :math:`\text{ILUMER}`, which can have the values 0, 1, 2, or 3. For
:math:`\text{ILUMER} = 0`, the weights are equal to the number of chunks in the two
populations. The recommended value is 3, for which the weights are
proportional to the total volume. The number of resulting chunks is
given by:
(16.4-224-8)
.. _eq-16.4-224-8:
.. math::
N_{l \text{u,i}} = \frac{V_{l \text{u,i}} + \Delta V_{l \text{u,i}}}{2 \pi \left( R_{l \text{u,i}}^{n + 1} \right)^{3}} \text{ASRALU}
where the radius :math:`R_{l \text{,u,i}}^{n + 1}` is defined by
16.4-224-7. Finally the fraction of steel in the chunks is given by:
(16.4-224-9)
.. _eq-16.4-224-9:
.. math::
\begin{matrix}
\text{FR}_{\text{s} l, l \text{u,i}}^{n + 1} = \frac{L_{\text{STEEL,} l \text{u,i}}}{L_{\text{STEEL,} l \text{u,i}} + L_{\text{FUEL,} l \text{u,i}}} \\
= \frac{\left( {\rho'}_{\text{s} l \text{,i}} + \Delta{\rho'}_{\text{s} l \text{,i}} \right) \text{AXMX} \Delta z_{\text{i}}}{2 \pi \text{ASRALU} N_{l \text{u,i}} R_{l \text{u,i}}^{n + 1} \rho_{\text{s} l}} \\
\end{matrix}
The quantities :math:`L_{\text{FUEL}}` and :math:`L_{\text{STEEL}}` are introduced in :numref:`figure-16.4-18`.
If the input variable RALUFZ is zero, then the code will use the above
formulas to determine the radius of the new chunks formed by crust
break-up and/or bulk freezing. However, if RALUFZ is not zero, the
radius of the newly formed chunks will be set equal to RALUFZ.
.. _section-16.4.3.10:
Source Terms in the Energy and Momentum Equations
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This section completes the description of the channel hydrodynamic
model. It presents in detail the formulation of the source terms used in
the energy equations, as introduced previously. It also presents some
components of the source terms used in the momentum equations. The
description of these source terms has been delayed until now for two
reasons: first, the geometry elements introduced in :numref:`section-16.4.3.9` are
needed for the definition of the source terms; second, all the elements
described below are calculated in the routine LETRAN, and thus it was
deemed preferable to describe them separately from the energy and
momentum equations which are solved elsewhere.'
.. _section-16.4.3.10.1:
Source Terms in the Energy Equations
''''''''''''''''''''''''''''''''''''
In the energy conservation equations, we introduced the source terms in
the following form:
(1) For the fuel energy Eq. :ref:`16.4-14<eq-16.4-14>`:
(16.4-225)
.. _eq-16.4-225:
.. math::
\begin{matrix}
\sum_{\text{k}}{{H'}_{\text{fu,k,i}} \cdot \Delta T_{\text{fu,k,i}} = {H'}_{\text{fu,Na,i}} \cdot \Delta T_{\text{fu,Na,i}} + {H'}_{\text{fu,cl,i}} \cdot \Delta T_{\text{fu,cl,i}}} \\
+ {H'}_{\text{fu,f} l \text{,i}} \cdot \Delta T_{\text{fu,f} l \text{,i}} + {H'}_{\text{fu,s} l \text{,i}} \cdot \Delta T_{\text{fu,s} l \text{,i}} \\
+ {H'}_{\text{fu,sr,i}} \cdot \Delta T_{\text{fu,sr,i}} + {H'}_{\text{se,fu,i}} \cdot \Delta T_{\text{fu,se,i}} \\
+ {H'}_{\text{fu,ffc,i}} \cdot \Delta T_{\text{fu,ffc,i}} + {H'}_{\text{fu,ffs,i}} \cdot \Delta T_{\text{fu,ffs,i}} \\
\end{matrix}
(1) For the steel energy eq. 16.4-26:
(16.4-226)
.. _eq-16.4-226:
.. math::
\begin{matrix}
\sum_{\text{k}}{{H'}_{\text{se,k,i}} \cdot \Delta T_{\text{se,k,i}} = {H'}_{\text{se,fu,i}} \cdot \Delta T_{\text{fu,se,i}} + {H'}_{\text{se,cl,i}} \cdot \Delta T_{\text{se,cl,i}}} \\
+ {H'}_{\text{se,sr,i}} \cdot \Delta T_{\text{se,sr,i}} + {H'}_{\text{se,f} l \text{,i}} \cdot \Delta T_{\text{se,f} l \text{,i}} \\
+ {H'}_{\text{se,s} l \text{,i}} \cdot \Delta T_{\text{se,s} l \text{,i}} + {H'}_{\text{se,Na,i}} \cdot \Delta T_{\text{se,Na,i}} \\
+ {H'}_{\text{se,ffc,i}} \cdot \Delta T_{\text{se,ffc,i}} + {H'}_{\text{se,ffs,i}} \cdot \Delta T_{\text{se,ffs,i}} \\
\end{matrix}
(1) For the fuel chunk energy
(16.4-227)
.. _eq-16.4-227:
.. math::
\begin{matrix}
\sum_{\text{k}}{{H'}_{\text{f} l \text{,k,i}} \cdot \Delta T_{\text{f} l \text{,k,i}} = {H'}_{\text{fu,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,fu,i}} + {H'}_{\text{se,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,se,i}}} \\
+ {H'}_{\text{Na,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,Na,i}} + {H'}_{\text{c} l \text{,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,cl,i}} \\
+ {H'}_{\text{sr,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,sr,i}} + {H'}_{\text{ffc,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,ffc,i}} \\
+ {H'}_{\text{ffs,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,ffs,i}} + {H'}_{\text{f} l \text{,s} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,s} l \text{,i}} \\
\end{matrix}
(1) For the steel chunk energy equation:
(16.4-228)
.. _eq-16.4-228:
.. math::
\begin{matrix}
\sum_{\text{k}}{{H'}_{\text{s} l \text{,k,i}} \cdot \Delta T_{\text{s} l \text{,k,i}} = {H'}_{\text{fu,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,fu,i}} + {H'}_{\text{se,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,se,i}}} \\
+ {H'}_{\text{Na,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,Na,i}} + {H'}_{\text{cl,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,cl,i}} \\
+ {H'}_{\text{sr,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,sr,i}} + {H'}_{\text{ffc,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,ffc,i}} \\
+ {H'}_{\text{ffs,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,ffs,i}} + {H'}_{\text{f} l \text{,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,f} l \text{,i}} \\
\end{matrix}
(1) For the gas-mixture equation
(16.4-229)
.. _eq-16.4-229:
.. math::
\begin{matrix}
\sum_{\text{k}}{{H'}_{\text{k,Mi,i}} \cdot \Delta T_{\text{k,Mi,i}} = {H'}_{\text{Na,c} l \text{,i}} \cdot \Delta T_{\text{cl,Na,i}}} + {H'}_{\text{se,Na,i}} \cdot \Delta T_{\text{se,Na,i}} \\
+ {H'}_{\text{Na,sr,i}} \cdot \Delta T_{\text{sr,Na,i}} + {H'}_{\text{fu,Na,i}} \cdot \Delta T_{\text{fu,Na,i}} \\
+ {H'}_{\text{Na,f} l \text{,i}} \cdot \Delta T_{\text{f} l \text{,Na,i}} + {H'}_{\text{Na,s} l \text{,i}} \cdot \Delta T_{\text{s} l \text{,Na,i}} \\
+ {H'}_{\text{ffc,Na,i}} \cdot \Delta T_{\text{ffc,Na,i}} + {H'}_{\text{ffs,Na,i}} \cdot \Delta T_{\text{ffs,Na,i}} \\
\end{matrix}
where
(16.4-230)
.. _eq-16.4-230:
.. math::
{H'}_{\text{j,k,i}} = H_{\text{j,k,i}} \cdot \frac{A_{\text{j,k,i}}^{L}}{\Delta z_{\text{i}} \cdot \text{AXMX}}
We will now proceed to define all generalized heat-transfer coefficients
that appear in Eqs. :ref:`16.4-225<eq-16.4-225>` through :ref:`16.4-230<eq-16.4-230>`. As shown in Eq. :ref:`16.4-230<eq-16.4-230>`
both :math:`H_{\text{j,k,i}}` and :math:`A_{\text{j,k,i}}` will have to be defined.
Special reference will be made to each flow regime, and a short
description of the physical model will be included.
.. _section-16.4.3.10.1.1:
Heat-transfer Coefficient between Fuel and Sodium :math:`{H'}_{\text{fu,Na,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the annular and bubbly steel flow regimes, there is no contact
between the fuel and sodium:
(16.4-231)
.. _eq-16.4-231:
.. math::
{H'}_{\text{fu,Na,i}} = 0
For the annular fuel flow regime, the heat-transfer coefficient is
defined as follows:
(16.4-232)
.. _eq-16.4-232:
.. math::
H_{\text{fu,Na,i}} = \frac{1}{\frac{1}{H_{\text{fu,i}}} + \frac{1}{H_{\text{Na,i}}}}
where :math:`H_{\text{fu,i}}` is a convection-type transfer coefficient on the
fuel side:
(16.4-233)
.. _eq-16.4-233:
.. math::
H_{\text{fu,i}} = \frac{1}{D_{\text{H,fu,i}}} \cdot \text{CIA3} \cdot \mu_{\text{fu}} \
\cdot c_{\text{p,fu}} \cdot \text{Re}_{\text{fu,i}}^{0.8}
and the sodium heat-transfer coefficient in the two-phase region is
defined as:
(16.4-234)
.. _eq-16.4-234:
.. math::
H_{\text{Na,i}} = \begin{cases}
10^{5}, & \text{if } \alpha_{\text{Na,i}} < 0.5 \\
\left\lbrack 10^{5} \cdot \left( 1 - \alpha_{\text{na,i}} \right) + H_{\text{Na,i}}^{\text{Single Phase}} \cdot \left( \alpha_{\text{Na,i}} - 0.5 \right) \right\rbrack \cdot 2, & \text{ if } 0.5 \leq \alpha_{\text{Na,i}} < 1 \\
\end{cases}
where :math:`H_{\text{Na,i}}^{\text{Single,Phase}}` is defined
in Eq. :ref:`16.4-235<eq-16.4-235>`.
The convective sodium heat-transfer coefficient in the single-phase
region is defined below:
(16.4-235)
.. _eq-16.4-235:
.. math::
H_{\text{Na,i}}^{\text{Single Phase}} = \left\{ \text{C1} \cdot \left\lbrack D_{\text{H,Mi,i}} \
\cdot \rho_{\text{Mi,i}} \cdot \left| u_{\text{Mi,i}} - u_{\text{fu,i}} \right| \cdot C_{\text{p,Mi,i}} \
\cdot \frac{1}{k_{\text{Mi,i}}} \right\rbrack^{\text{C2}} \\
+ \text{C3} \right\} \cdot \frac{k_{\text{Mi,i}}}{D_{\text{H,Mi,i}}} \text{ for } \alpha_{\text{Na,i}} > 1
The area of contact between sodium and molten fuel is defined as
follows:
(16.4-236)
.. _eq-16.4-236:
.. math::
\begin{matrix}
A_{\text{fu,Na,i}}^{L} = A_{\text{cl,i}}^{L} \cdot \left\lbrack \left( C_{\text{fu,cl,i}} - C_{\text{ff,cli,i}} \right) + C_{\text{ff,cl,i}} \cdot C_{\text{fu,ff,cl,i}} \right\rbrack \\
+ A_{\text{sr,i}}^{L} \cdot \left\lbrack \left( C_{\text{fu,sr,i}} - C_{\text{ff,sr,i}} \right) + C_{\text{ff,sr,i}} \cdot C_{\text{fu,ff,sr,i}} \right\rbrack \\
\end{matrix}
For the bubbly fuel flow regime, the heat-transfer between sodium and
fuel is described as taking place between a continuous fuel component
and spherical sodium bubbles. The number of sodium bubbles (and thus
their radius) varies from one bubble at the transition boundary between
annular and bubbly flow to a maximum number determined by a maximum
number determined by a minimum radius of the bubble:
(16.4-237)
.. _eq-16.4-237:
.. math::
R_{\text{min,bubble,i}} = 0.05 R_{\text{max,bubble,i}}
(16.4-238)
.. _eq-16.4-238:
.. math::
R_{\text{max,bubble,i}} = \begin{cases}
R_{1} = \left( \frac{3 \cdot V_{\text{Na,i}}}{4 \cdot \pi} \right)^{1/3} & \text{if } R_{1} < D_{\text{H,Mi,i}} \cdot 0.5 \\
D_{\text{H,Mi,i}} \cdot 0.5 & \text{if } R_{1} \geq D_{\text{H,Mi,i}} \cdot 0.5 \\
\end{cases}
The actual radius of the bubbles is obtained by assuming an exponential
variation between :math:`R_{\text{max}}` and :math:`R_{\text{min}}`, dependent on how far
from the flow regime transition we are in the cell :math:`i`:
(16.4-239)
.. _eq-16.4-239:
.. math::
\begin{matrix}
R_{\text{bubble,i}} = R_{\text{min,bubble,i}} + \left( R_{\text{max,bubble,i}} - R_{\text{min,bubble,i}} \right) \\
\cdot \exp\left\lbrack \frac{- \left( \theta_{\text{Mi,bubbly}} - \theta_{\text{Mi,i}} \right)}{\theta_{\text{Mi,i}}} \right\rbrack \\
\end{matrix}
where :math:`\theta_{\text{Mi,bubbly}}` is the mixture volume fraction required for
the transition from the annular to bubbly flow.
The number of bubbles is defined by:
(16.4-240)
.. _eq-16.4-240:
.. math::
N_{\text{bubble,i}} = \frac{3 \cdot V_{\text{Na,i}}}{4 \cdot \pi \cdot R_{\text{bubbles,i}}^{3}}
and the heat-transfer area becomes:
(16.4-241)
.. _eq-16.4-241:
.. math::
A_{\text{fu,Na,i}}^{L} = N_{\text{bubble,i}} \cdot 4\pi \cdot R_{\text{bubble,i}}^{2}
The heat-transfer coefficient is determined by the heat-transfer
resistance on the fuel side and the resistance of the bubble:
(16.4-242)
.. _eq-16.4-242:
.. math::
H_{\text{fu,Na,i}} = \frac{1}{\frac{1}{H_{\text{fu,i}}} + \frac{1}{H_{\text{Na,i}}}}
where :math:`H_{\text{fu,i}} = 3 \cdot 10^{4}` J/m\ :sup:`2` K-s and wad determined
by analyzing the transient solution describing the propagation of a step
change in temperature in a semi-infinite medium. For a typical LEVITATE
time step :math:`\Delta t` and a temperature change :math:`\Delta T, H_{\text{fu,i}}` was determined
from:
(16.4-243)
.. _eq-16.4-243:
.. math::
H_{\text{fu,i}} = \frac{1}{\Delta T \cdot \Delta t} \int_{0}^{\Delta t}{Q \left( x = 0 \right) \text{dt}}
Where :math:`Q \left( x = 0 \right)` is the heat flux at the boundary of the semi-infinite
medium, in J/m\ :sup:`2` · s.
The heat-transfer coefficient in the bubble, :math:`H_{\text{Na,i}}` is defined
by Eq. :ref:`16.4-234<eq-16.4-234>` for the two-phase sodium region. For single-phase sodium:
(16.4-244)
.. _eq-16.4-244:
.. math::
\begin{align}
H_{\text{Na,i}} = \frac{k_{\text{Mi,i}}}{R_{\text{bubble,i}}} && \text{if } \alpha_{\text{Na,i}} = 1
\end{align}
.. _section-16.4.3.10.1.2:
Heat-transfer Coefficient between Fuel and Cladding or Structure :math:`{H'}_{\text{fu,cl,i}}` and :math:`{H'}_{\text{fu,sr,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the annular and bubbly steel flow regimes the fuel is in contact
with neither the cladding nor the structure.
(16.4-245)
.. _eq-16.4-245:
.. math::
{H'}_{\text{fu,cl,i}} = {H'}_{\text{fu,sr,i}} = 0
For the annular fuel flow regime, the heat-transfer coefficient from
fuel to cladding and structure is defined as:
(16.4-246)
.. _eq-16.4-246:
.. math::
H_{\text{fu,cl,i}} = H_{\text{fu,sr,i}} = \frac{1}{D_{\text{H,fu,i}}} \cdot \text{CIA3} \
\cdot \mu_{\text{fu,i}} \cdot C_{\text{p,fu}} \cdot \text{Re}_{\text{fu,i}}^{0.8}
and the respective heat-transfer areas are:
(16.4-247)
.. _eq-16.4-247:
.. math::
A_{\text{fu,cl,i}}^{L} = A_{\text{cl,i}}^{L} \cdot \left( C_{\text{fu,cl,i}} - C_{\text{ff,cl,i}} \right) \\
\\
A_{\text{fu,sr,i}}^{L} = A_{\text{sr,i}}^{L} \cdot \left( C_{\text{fu,sr,i}} - C_{\text{ff,sr,i}} \right) \\
For the bubbly fuel flow regime, the heat-transfer coefficients are
defined as follows:
(16.4-248)
.. _eq-16.4-248:
.. math::
H_{\text{fu,cl,i}} = H_{\text{fu,sr,i}} = \frac{1}{D_{\text{H,fu,i}}} \cdot \text{CIA3} \cdot \mu_{\text{fu}} \
\cdot C_{\text{p,fu}} \cdot \text{Re}_{\text{fu,i}}^{0.8} + \frac{4 \cdot k_{\text{fu}}}{D_{\text{H,fu,i}}}
and the respective heat-transfer areas are:
(16.4-249)
.. _eq-16.4-249:
.. math::
\begin{align}
A_{\text{fu,cl,i}}^{L} = A_{\text{cl,i}}^{L} ; && A_{\text{fu,sr,i}}^{L} = A_{\text{sr,i}}^{L}
\end{align}
.. _section-16.4.3.10.1.3:
Heat Transfer between Fuel and Frozen Fuel Crusts on Cladding and Structure :math:`{H'}_{\text{fu,ffc,i}}` and :math:`{H'}_{\text{fu,ffs,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These coefficients are zero in the annular and bubbly steel flow
regimes.
(16.4-250)
.. _eq-16.4-250:
.. math::
{H'}_{\text{fu,ffc,i}} = {H'}_{\text{fu,ffs,i}} = 0
For the annual fuel flow regime, the heat-transfer coefficients are
defined as:
(16.4-251)
.. _eq-16.4-251:
.. math::
H_{\text{fu,ffc,i}} = \frac{1}{\frac{1}{H_{\text{fu,i}}} + \frac{0.5 \cdot l_{\text{ff,cl,i}}}{k_{\text{fu}}}}
(16.4-252)
.. _eq-16.4-252:
.. math::
H_{\text{fu,ffs,i}} = \frac{1}{\frac{1}{H_{\text{fu,i}}} + \frac{0.5 \cdot l_{\text{ff,sr,i}}}{k_{\text{fu}}}}
where :math:`H_{\text{fu,i}}` is given by Eq. :ref:`16.4-246<eq-16.4-246>` and :math:`l_{\text{ff,cl,i}}`
and :math:`l_{\text{ff,sr,i}}` have been defined in :numref:`section-16.4.3.9`, which
describes the local geometry. The respective heat-transfer areas are:
(16.4-253)
.. _eq-16.4-253:
.. math::
A_{\text{fu,ffc,i}} = A_{\text{cl,i}}^{L} \cdot C_{\text{ff,cl,i}} \cdot C_{\text{fu,ff,cl,i}}
and
(16.4-254)
.. _eq-16.4-254:
.. math::
A_{\text{fu,ffs,i}} = A_{\text{sr,i}}^{L} \cdot C_{\text{ff,sr,i}} \cdot C_{\text{fu,ff,sr,i}}
For the bubbly fuel flow regime, the heat-transfer coefficients are
given by Eqs. :ref:`16.4-251<eq-16.4-251>` and :ref:`16.4-252<eq-16.4-252>`, but :math:`H_{\text{fu,i}}` in these is
given by Eq. :ref:`16.4-248<eq-16.4-248>` rather than Eq. :ref:`16.4-246<eq-16.4-246>`. The heat-transfer areas
are given by:
(16.4-255)
.. _eq-16.4-255:
.. math::
A_{\text{fu,ffc,i}} = A_{\text{cl,i}}^{L} \cdot C_{\text{ff,cl,i}}
(16.4-256)
.. _eq-16.4-256:
.. math::
A_{\text{fu,ffs,i}} = A_{\text{sr,i}}^{L} \cdot C_{\text{ff,sr,i}}
.. _section-16.4.3.10.1.4:
Heat-transfer Coefficient between Fuel and Steel :math:`{H'}_{\text{se,fu,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In the annular and bubbly steel flow regimes, the fuel exists in the
form of droplets imbedded in the continuous molten steel. The radius of
the fuel droplets varies between :math:`R_{\text{fu,droplet,min,i}} = 1 \cdot 10^{-4}` m
and :math:`R_{\text{fu,droplet,max,i}} = l_{\text{fu,cl,i}}`.
The actual radius is calculated as follows:
If :math:`1 \geq \frac{\theta_{\text{fu,i}}}{\theta_{\text{se,i}}} \geq 0.5`
(16.4-257a)
.. _eq-16.4-257a:
.. math::
R_{\text{fu,droplet,i}} = \left\lbrack R_{\text{fu,droplet,min,i}} \cdot \left( 1 - \frac{\theta_{\text{fu,i}}}{\theta_{\text{se,i}}} \right) \
+ R_{\text{fu,droplet,max,i}} \cdot \left( \frac{\theta_{\text{fu,i}}}{\theta_{\text{se,i}}} - 0.5 \right) \right\rbrack \cdot 2
If :math:`\frac{\theta_{\text{fu,i}}}{\theta_{\text{se,i}}} < 0.5`, then
(16.4-257b)
.. _eq-16.4-257b:
.. math::
R_{\text{fu,droplet,i}} = R_{\text{fu,droplet,min,i}}
The number of fuel droplets and heat-transfer area are calculated from:
(16.4-258)
.. _eq-16.4-258:
.. math::
N_{\text{fu,droplets,i}} = \frac{3 V_{\text{fu,i}}}{4 \pi \cdot R_{\text{fu,droplet,i}}^{3}}
and
(16.4-259)
.. _eq-16.4-259:
.. math::
A_{\text{se,fu,i}}^{L} = N_{\text{fu,droplets,i}} \cdot 4 \pi \cdot R_{\text{fu,droplet,i}}^{3}
The heat-transfer coefficient is obtained as follows:
(16.4-260)
.. _eq-16.4-260:
.. math::
H_{\text{se,fu,i}} = \frac{1}{\frac{R_{\text{fu,droplet,i}}}{k_{\text{fu}}} + \frac{1}{H_{\text{se,i}}}}
where :math:`H_{\text{se,i}} = 4 \cdot 10^{4}` and is obtained in a manner
similar to Eq. :ref:`16.4-243<eq-16.4-243>`.
For the annual and bubbly fuel flow regimes, the situation is reversed,
and the steel exists in the form of droplets imbedded in the continuous
molten fuel. The radius of the steel droplet is defined as follows:
For :math:`\frac{\theta_{\text{se,i}}}{\theta_{\text{fu,i}}} > 0.5`
(16.4-261a)
.. _eq-16.4-261a:
.. math::
\begin{matrix}
R_{\text{se,droplet,i}} = \left\lbrack R_{\text{se,droplet,min,i}} \cdot \left( 1 - \frac{\theta_{\text{se,i}}}{\theta_{\text{fu,i}}} \right) + R_{\text{se,droplet,max,i}} \\
\cdot \left( \frac{\theta_{\text{se,i}}}{\theta_{\text{fu,i}}} - 0.5 \right) \right\rbrack \cdot 2 \\
\end{matrix}
For :math:`\frac{\theta_{\text{se,i}}}{\theta_{\text{fu,i}}} \leq 0.5`
(16.4-261b)
.. _eq-16.4-261b:
.. math::
R_{\text{se,droplet,i}} = R_{\text{se,droplet,min,i}}
Note that the definitions 16.4-257 and 16.4-261 lead to a continuous
change in :math:`R_{\text{droplet,i}}` across the transition from a fuel to a
steel flow regime. The number of steel droplets and the heat-transfer
area are obtained from formulas similar to 16.4-258 and 16.4-259. The
heat-transfer coefficient is defined by:
(16.4-262)
.. _eq-16.4-262:
.. math::
H_{\text{se,fu,i}} = \frac{1}{\frac{R_{\text{se,droplet,i}}}{k_{\text{se}}} + \frac{1}{H_{\text{fu,i}}}}
where :math:`H_{\text{fu,i}} = 3 \cdot 10^{4}` and has already been explained
in Eq. :ref:`16.4-243<eq-16.4-243>`.
.. _section-16.4.3.10.1.5:
Heat-transfer Coefficients between Fuel and Fuel and Steel Chunks :math:`{H'}_{\text{fu,fl,i}}` and :math:`{H'}_{\text{fu,sl,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
As explained in 16.4.3.9.2, the fuel and steel chunks have a cylindrical
geometry and the aspect ratio R/L can be varied via the input. Thus, it
is necessary to distinguish between the heat transfer across the surface
normal to the flow :math:`A_{\text{f} l \text{,i}}^{N}` and that across the
surface parallel to the flow direction, :math:`A_{\text{f} l \text{,i}}^{L}`.
Note that the chunks are always assumed to be parallel to the flow
direction, and that the total surface of the fuel chunks is given by
:math:`A_{\text{f} l \text{,i}}^{L} + A_{\text{f} l \text{,i}}^{N}`. Thus, the
generalized heat-transfer coefficient between the fuel and the fuel and
steel chunks can be written as:
(16.4-263)
.. _eq-16.4-263:
.. math::
{H'}_{\text{fu,f} l \text{,i}} = H_{\text{fu,f} l \text{,i}}^{L} \cdot \
\frac{A_{\text{fu,f} l \text{,i}}^{L}}{\text{AXMX} \cdot \Delta z_{\text{i}}} \
+ H_{\text{fu,f} l \text{,i}}^{N} \cdot \frac{A_{\text{fu,f} l \text{,i}}^{N}}{\text{AXMX} \cdot \Delta z_{\text{i}}}
(16.4-264)
.. _eq-16.4-264:
.. math::
{H'}_{\text{fu,s} l \text{,i}} = H_{\text{fu,s} l \text{,i}}^{L} \cdot \frac{A_{\text{fu,s} l \text{,i}}^{L}}{\text{AXMX} \
\cdot \Delta z_{\text{i}}} + H_{\text{fu,s} l \text{,i}}^{N} \cdot \frac{A_{\text{fu,s} l \text{,i}}^{N}}{\text{AXMX}
\cdot \Delta z_{\text{i}}}
For the steel flow regimes, both bubbly and annular, no contact is
present between the molten fuel and chunks:
(16.4-265)
.. _eq-16.4-265:
.. math::
{H'}_{\text{fu,f} l \text{,i}} = {H'}_{\text{fu,s} l \text{,i}} = 0
For the bubbly fuel flow regime, the heat-transfer coefficients are
defined as follows:
(16.4-266)
.. _eq-16.4-266:
.. math::
\begin{align}
H_{\text{fu,f} l \text{,i}}^{L} = \frac{1}{\frac{1}{H_{\text{fu,i}}} \
+ \frac{R_{\text{f} l \text{,i}}}{k_{\text{fu}}}}; && H_{\text{fu,f} l \text{,i}}^{N} \
= \frac{1}{\frac{1}{H_{\text{fu,i}}} + \frac{L_{\text{f} l \text{,i}}}{2k_{\text{fu}}}}
\end{align}
and
(16.4-267)
.. _eq-16.4-267:
.. math::
\begin{align}
H_{\text{fu,s} l \text{,i}}^{L} = \frac{1}{\frac{1}{H_{\text{fu,i}}} \
+ \frac{R_{\text{s} l \text{,i}}}{k_{\text{se}}}}; && H_{\text{fu,s} l \text{,i}}^{N} \
= \frac{1}{\frac{1}{H_{\text{fu,i}}} + \frac{L_{\text{s} l \text{,i}}}{2k_{\text{se}}}}
\end{align}
where :math:`H_{\text{fu,i}}` has been defined in Eq. :ref:`16.4-243<eq-16.4-243>`. The corresponding
heat-transfer areas are:
(16.4-268)
.. _eq-16.4-268:
.. math::
\begin{align}
A_{\text{fu,f} l \text{,i}}^{L} = A_{\text{f} l \text{,i}}^{L} ; && A_{\text{fu,f} l \text{,i}}^{N} = 0.5 A_{\text{f} l \text{,i}}^{N}~.
\end{align}
(16.4-269)
.. _eq-16.4-269:
.. math::
\begin{align}
A_{\text{fu,s} l \text{,i}}^{L} = A_{\text{s} l \text{,i}}^{L} ; && A_{\text{fu,s} l \text{,i}}^{N} = 0.5 A_{\text{s} l \text{,i}}^{N}~.
\end{align}
For the annular fuel flow regime, the chunks can have velocities quite
different from those of the molten fuel, and convective effects must be
taken into account. The heat-transfer coefficients are still defined by
Eqs. :ref:`16.4-266<eq-16.4-266>` and :ref:`16.4-267<eq-16.4-267>`, but :math:`H_{\text{fu,i}}` is replaced by
:math:`H_{\text{fu,i}} + H_{\text{fu,convective,i}}`, where
:math:`H_{\text{fu,convective,i}}` is defined as:
(16.4-270)
.. _eq-16.4-270:
.. math::
H_{\text{fu,convective,i}} = \frac{1}{D_{\text{H,fu,i}}} \cdot \text{CIA3} \cdot \mu_{\text{fu}} \
\cdot C_{\text{p,fu}} \cdot \text{Re}_{\text{fu,i}}^{0.8}
where
(16.4-270-1)
.. _eq-16.4-270-1:
.. math::
\text{Re}_{l \text{u,i}} = \rho_{\text{fu}} \cdot \left| u_{\text{fu,i}} \
- u_{l \text{u,i}} \right| \cdot \frac{2 R_{l \text{u,i}}}{\mu_{\text{fu}}}
The heat-transfer areas for the annular flow are significantly different
from those used for the bubbly fuel flow. The normal surface is not in
contact with the molten fuel, thus:
(16.4-271)
.. _eq-16.4-271:
.. math::
A_{\text{fu,f} l \text{,i}}^{N} = A_{\text{fu,s} l \text{,i}}^{L} = 0
The lateral heat transfer areas are defined as follows:
(16.4-272)
.. _eq-16.4-272:
.. math::
\begin{matrix}
A_{\text{fu,f} l \text{,i}}^{L} = A_{\text{f} l \text{,i}}^{L} \cdot F_{\text{area,bd,} l \text{u,i}} \cdot C_{\text{fu,cl,i}} \cdot \frac{A_{\text{cl,i}}^{L}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \\
+ C_{\text{fu,sr,i}} \cdot \frac{A_{\text{sr,i}}^{L}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \\
\end{matrix}
where :math:`F_{\text{area,bd,} l \text{u,i}}` is the fraction of the area of the chunks
in contact with the channel boundaries. It is obtained by assuming that
chunks are uniformly distributed across the channel area and defining
the contact area for the chunks near the boundary (currently the contact
area is equal to :math:`0.5 \cdot A_{\text{fl,i}}^{L}`).
:math:`A_{\text{fu,s} l \text{,i}}^{L}` is defined similarly to Eq.
:ref:`16.4-272<eq-16.4-272>`, but :math:`A_{\text{f} l \text{,i}}^{L}` is replaced by
:math:`A_{\text{s} l \text{,i}}^{L}`.
.. _section-16.4.3.10.1.6:
Heat-transfer Coefficients between the Molten Steel and Fuel and Steel Chunks :math:`{H'}_{\text{se,fl,i}}` and :math:`{H'}_{\text{se,sl,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
For the annular and bubbly fuel flow regimes, these coefficients are
zero because the steel and the chunks are not in contact.
For the annular and bubbly steel flow regimes, the considerations made
in :numref:`section-16.4.3.10.1.5` still apply and the Eqs. :ref:`16.4-263<eq-16.4-263>` through
16.4-272 can be used, but :math:`H_{\text{fu,i}}` is replaced by :math:`H_{\text{se,i}}`,
defined similarly for steel, and :math:`H_{\text{fu,convective,i}}` is replaced
by :math:`H_{\text{se,convective,i}}` defined below:
(16.4-273)
.. _eq-16.4-273:
.. math::
H_{\text{se,convective,i}} = \frac{1}{D_{\text{H,se,i}}} \cdot \text{CIA3} \cdot \mu_{\text{se}} \
\cdot C_{\text{p,se}} \cdot \text{Re}_{l \text{u,i}}^{0.8}
.. _section-16.4.3.10.1.7:
Heat-transfer Coefficients between Molten Steel and Cladding and Structure :math:`{H'}_{\text{se,cl,i}}` and :math:`{H'}_{\text{se,sr,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The coefficients are zero for the fuel flow regimes, both annular and
bubbly:
(16.4-274)
.. _eq-16.4-274:
.. math::
{H'}_{\text{se,cl,i}} = {H'}_{\text{se,sr,i}} = 0
For the annular steel flow regime, heat-transfer coefficients are
defined as follows:
(16.4-275)
.. _eq-16.4-275:
.. math::
H_{\text{se,cl,i}} = H_{\text{se,sr,i}} = \frac{1}{D_{\text{H,se,i}}} \cdot \text{CIA3} \
\cdot \mu_{\text{se}} \cdot C_{\text{p,se}} \cdot \text{Re}_{\text{se,i}}^{0.8}
and the respective areas, :math:`A_{\text{se,c} l \text{,i}}` and :math:`A_{\text{se,sr,i}}` are
identical to Eqs. :ref:`16.4-247<eq-16.4-247>`.
For the bubbly steel regime, the heat-transfer is given by:
(16.4-276)
.. _eq-16.4-276:
.. math::
H_{\text{se,cl,i}} = H_{\text{se,sr,i}} = \frac{1}{D_{\text{H,se,i}}} \cdot \text{CIA3} \
\cdot \mu_{\text{se}} \cdot C_{\text{p,se}} \cdot \text{Re}_{\text{se,i}}^{0.8} + \frac{4k_{\text{se}}}{D_{\text{H,se,i}}}
and the heat-transfer areas
:math:`A_{\text{se,cl,i}}^{L}` and :math:`A_{\text{se,sr,i}}^{L}`
are defined by Eq. :ref:`16.4-249<eq-16.4-249>`.
.. _section-16.4.3.10.1.8:
Heat-transfer Coefficients between Steel and the Frozen Fuel on Cladding Structure :math:`{H'}_{\text{se,ffc,i}}` and :math:`{H'}_{\text{se,ffs,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These coefficients are zero for the fuel annular and bubbly flow
regimes:
(16.4-277)
.. _eq-16.4-277:
.. math::
{H'}_{\text{se,ffc,i}} = {H'}_{\text{se,ffs,i}} = 0
For the steel regimes, the considerations made in :numref:`section-16.4.3.10.1.3`
for the fuel regimes apply. Thus, Eqs. :ref:`16.4-250<eq-16.4-250>` through :ref:`16.4-256<eq-16.4-256>` can be
used to define the corresponding heat-transfer coefficients and transfer
areas if one replaces :math:`H_{\text{fu,i}}` by :math:`H_{\text{se,i}}`, defined as
follows:
(16.4-278)
.. _eq-16.4-278:
.. math::
H_{\text{se,i}} = \frac{1}{D_{\text{H,se,i}}} \cdot \text{CIA3} \cdot \mu_{\text{se}} \cdot C_{\text{p,se}} \cdot \text{Re}_{\text{se,i}}^{0.8}
.. _section-16.4.3.10.1.9:
Heat Transfer between Steel and Sodium/Fission Gas :math:`{H'}_{\text{se,Na,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The coefficient is zero for the fuel flow regimes:
(16.4-279)
.. _eq-16.4-279:
.. math::
{H'}_{\text{se},\text{Na},i} = 0
All considerations made for the fuel flow regimes in :numref:`section-16.4.3.10.1.1` apply here and Eqs. :ref:`16.4-232<eq-16.4-232>` through :ref:`16.4-244<eq-16.4-244>` should be
used, if :math:`h_{\text{fu,i}}` is replaced by :math:`H_{\text{se,i}}`, as appropriate.
Thus, the heat-transfer coefficient for the annual steel regime becomes
from Eq. :ref:`16.4-232<eq-16.4-232>`:
(16.4-280)
.. _eq-16.4-280:
.. math::
H_{\text{se,Na,i}} = \frac{1}{\frac{1}{H_{\text{se,i}}} + \frac{1}{H_{\text{Na,i}}}}
where :math:`H_{\text{se,i}}` is defined by Eq. :ref:`16.4-278<eq-16.4-278>`. The same formula
applies for the bubbly steel regime from Eq. :ref:`16.4-242<eq-16.4-242>`, but
:math:`H_{\text{se,i}}` is defined in a manner similar to Eq. :ref:`16.4-243<eq-16.4-243>`.
.. _section-16.4.3.10.1.10:
Heat-transfer Coefficients between Fuel Steel Chunks and Cladding and Structure :math:`{H'}_{\text{cl,f} l \text{,i}}, {H'}_{\text{cl,s} l \text{,i}}, {H'}_{\text{sr,f} l \text{,i}}, {H'}_{\text{sr,s} l \text{,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The chunks can be in contact with the cladding and structure only in the
partial annular fuel flow regime (this is also the default when neither
fuel not steel is present in the channel). For all other regimes, the
above coefficients are equal to zero.
For the partial annular fuel flow regime, the heat-transfer coefficients
are defined below:
(16.4-281)
.. _eq-16.4-281:
.. math::
H_{\text{cl,f} l \text{,i}} = H_{\text{sr,f} l \text{,i}} = \frac{k_{\text{fu}}}{R_{l \text{u,i}}}
(16.4-282)
.. _eq-16.4-282:
.. math::
H_{\text{cl,s} l \text{,i}} = H_{\text{sr,s} l \text{,i}} = \frac{k_{\text{se}}}{R_{l \text{u,i}}}
The definition of the heat-transfer areas accounts for the fact that
only the chunks close to the boundaries not covered by molten fuel
exchange energy directly with the cladding and structure.
(16.4-283)
.. _eq-16.4-283:
.. math::
A_{\text{cl,f} l \text{,i}} = A_{\text{f} l \text{,i}}^{L} \cdot F_{\text{area,bd,} l \text{u,i}} \
\cdot \frac{A_{\text{c} l \text{,i}}^{L}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \cdot \left( 1 - C_{\text{fu,cl,i}} \right)
(16.4-284)
.. _eq-16.4-284:
.. math::
A_{\text{sr,f} l \text{,i}} = A_{\text{f} l \text{,i}}^{L} \cdot F_{\text{area,bd,} l \text{u,i}} \
\cdot \frac{A_{\text{sr,i}}^{L}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \cdot \left( 1 - C_{\text{fu,sr,i}} \right)
(16.4-285)
.. _eq-16.4-285:
.. math::
A_{\text{cl,s} l \text{,i}} = A_{\text{s} l \text{,i}}^{L} \cdot F_{\text{area,bd,} l \text{u,i}} \
\cdot \frac{A_{\text{c} l \text{,i}}^{L}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \cdot \left( 1 - C_{\text{fu,cl,i}} \right)
(16.4-286)
.. _eq-16.4-286:
.. math::
A_{\text{sr,s} l \text{,i}} = A_{\text{s} l \text{,i}}^{L} \cdot F_{\text{area,bd,} l \text{u,i}} \
\cdot \frac{A_{\text{sr,i}}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \cdot \left( 1 - C_{\text{fu,sr,i}} \right)
.. _section-16.4.3.10.1.11:
Heat-transfer between Sodium/Fission Gas Mixture and the Fuel/Steel Chunks :math:`{H'}_{\text{Na,f} l \text{,i}}` and :math:`{H'}_{\text{Na,s} l \text{,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These coefficients are zero in the bubbly flow regimes when there is no
contact between sodium and chunks.
(16.4-287)
.. _eq-16.4-287:
.. math::
{H'}_{\text{Na,f} l \text{,i}} = {H'}_{\text{Na,s} l \text{,i}} = 0
For the annular fuel and steel regimes the generalized coefficients are
defined as follows:
(16.4-288)
.. _eq-16.4-288:
.. math::
{H'}_{\text{Na,f} l \text{,i}} = H_{\text{Na,f} l \text{,i}}^{L} \
\cdot \frac{A_{\text{Na,f} l \text{,i}}^{L}}{\Delta z_{\text{i}} \cdot \text{AXMX}} \
+ H_{\text{Na,f} l \text{,i}}^{N} \cdot \frac{A_{\text{Na,f} l \text{,i}}^{N}}{\Delta z_{\text{i}} \cdot \text{AXMX}}
(16.4-289)
.. _eq-16.4-289:
.. math::
{H'}_{\text{Na,s} l \text{,i}} = H_{\text{Na,s} l \text{,i}}^{L} \
\cdot \frac{A_{\text{Na,s} l \text{,i}}^{L}}{\Delta z_{\text{i}} \cdot \text{AXMX}} \
+ H_{\text{Na,s} l \text{,i}}^{N} \cdot \frac{A_{\text{Na,s} l \text{,i}}^{N}}{\Delta z_{\text{i}} \cdot \text{AXMX}}
The heat-transfer coefficients in Eqs. :ref:`16.4-288<eq-16.4-288>` and :ref:`16.4-290<eq-16.4-290>` are defined
below:
(16.4-290)
.. _eq-16.4-290:
.. math::
\begin{align}
H_{\text{Na,f} l \text{,i}}^{L} = \frac{1}{\frac{1}{H_{\text{Na,i}}} + \frac{R_{l \text{u,i}}}{k_{\text{fu}}}}; && H_{\text{Na,f} l \text{,i}}^{N} = \frac{1}{\frac{1}{H_{\text{Na,i}}} + \frac{L_{\text{f} l \text{,i}}}{2k_{\text{fu}}}}~ ;
\end{align}
(16.4-291)
.. _eq-16.4-291:
.. math::
\begin{align}
H_{\text{Na,s} l \text{,i}}^{L} = \frac{1}{\frac{1}{H_{\text{Na,i}}} + \frac{R_{l \text{u,i}}}{k_{\text{se}}}} ; && H_{\text{Na,s} l \text{,i}}^{N} = \frac{1}{\frac{1}{H_{\text{Na,i}}} + \frac{L_{\text{s} l \text{,i}}}{2k_{\text{se}}}}~ ;
\end{align}
The respective heat-transfer areas are defined as follow:
(16.4-292)
.. _eq-16.4-292:
.. math::
\begin{align}
A_{\text{Na,f} l \text{,i}}^{L} = A_{\text{f} l \text{,i}}^{L} \cdot \left( 1 - F_{\text{area,bd,} l \text{u,i}} \right) ; && A_{\text{Na,f} l \text{,i}}^{N} = 0.5 \cdot A_{\text{f} l \text{,i}}^{N}
\end{align}
(16.4-293)
.. _eq-16.4-293:
.. math::
\begin{align}
A_{\text{Na,s} l \text{,i}}^{L} = A_{\text{s} l \text{,i}}^{L} \cdot \left( 1 - F_{\text{area,bd,} l \text{u,i}} \right) ; && A_{\text{Na,s} l \text{,i}}^{N} = 0.5 \cdot A_{\text{s} l \text{,i}}^{N}
\end{align}
.. _section-16.4.3.10.1.12:
Heat-transfer between the Fuel and Steel Solid Chunks :math:`H_{fl,sl,i}^{'}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This coefficient is different from zero only when both fuel and steel
chunks are present in a cell. Given the geometry of the chunks presented
in :numref:`section-16.4.3.9`, the heat-transfer coefficient is defined by:
(16.4-294)
.. _eq-16.4-294:
.. math::
H_{\text{f} l \text{,s} l \text{,i}} = \frac{1}{\frac{L_{\text{f} l \text{,i}}}{2k_{\text{fu}}} \
+ \frac{L_{\text{s} l \text{,i}}}{2k_{\text{se}}}}
and the heat-transfer area by:
(16.4-295)
.. _eq-16.4-295:
.. math::
A_{\text{f} l \text{,s} l \text{,i}} = 0.5 \cdot A_{\text{f} l \text{,i}}^{N}
.. _section-16.4.3.10.1.13:
Heat Transfer between Fuel/Steel Chunks and the Frozen Fuel on Cladding and Structure, :math:`{H'}_{\text{f} l \text{,ffc,i}}, {H'}_{\text{f} l \text{,ffs,i}}, {H'}_{\text{s} l \text{,ffc,i}}, {H'}_{\text{s} l \text{,ffs,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These coefficients are non-zero only if there is little molten fuel
and/or steel present, i.e.:
(16.4-296)
.. _eq-16.4-296:
.. math::
\frac{\theta_{\text{fu,i}} + \theta_{\text{se,i}}}{\theta_{\text{ch,op,i}} - \theta_{l \text{u,i}}} \leq 0.1
and the frozen fuel crust covers a significant fraction of the channel
perimeter, i.e.:
(16.4-297)
.. _eq-16.4-297:
.. math::
C_{\text{ff,cl,i}} \geq 0.1 \text{ or } C_{\text{ff,sr,i}} \geq 0.1
If the above conditions are satisfied, the heat-transfer coefficients
are defined as follows:
(16.4-298)
.. _eq-16.4-298:
.. math::
\begin{align}
H_{\text{f} l \text{,ffc,i}} = \frac{1}{\frac{l_{\text{ff,cl,i}}}{2k_{\text{fu}}} \
+ \frac{R_{l \text{u,i}}}{k_{\text{fu}}}} ; && H_{\text{s} l \text{,ffc,i}} \
= \frac{1}{\frac{l_{\text{ff,cl,i}}}{2k_{\text{fu}}} + \frac{R_{l \text{u,i}}}{k_{\text{se}}}}
\end{align}
(16.4-299)
.. _eq-16.4-299:
.. math::
\begin{align}
H_{\text{f} l \text{,ffs,i}} = \frac{1}{\frac{l_{\text{ff,sr,i}}}{2k_{\text{fu}}} \
+ \frac{R_{l \text{u,i}}}{k_{\text{fu}}}}; && H_{\text{s} l \text{,ffs,i}} \
= \frac{1}{\frac{l_{\text{ff,sr,i}}}{2k_{\text{fu}}} + \frac{R_{l \text{u,i}}}{k_{\text{se}}}}
\end{align}
The respective heat-transfer areas are defined below:
(16.4-300)
.. _eq-16.4-300:
.. math::
A_{\text{f} l \text{,ffc,i}} = A_{\text{f} l \text{,i}}^{L} \cdot F_{\text{area,bd,} l \text{u,i}} \
\cdot \frac{A_{\text{cl,i}}^{L}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \cdot C_{\text{ff,cl,i}}
(16.4-301)
.. _eq-16.4-301:
.. math::
A_{\text{f} l \text{,ffs,i}} = A_{\text{f} l \text{,i}}^{L} \cdot F_{\text{area,bd,} l \text{u,i}} \
\cdot \frac{A_{\text{sr,i}}^{L}}{A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L}} \cdot C_{\text{ff,sr,i}}
The areas for the steel chunks, :math:`A_{\text{s} l \text{,ffc,i}}` and :math:`A_{\text{s} l \text{,ffs,i}}`
can be obtained from the above definitions by replacing
:math:`A_{\text{f} l \text{,i}}^{L}` with :math:`A_{\text{s} l \text{,i}}^{L}.`
.. _section-16.4.3.10.1.14:
Heat-transfer Coefficients between Sodium/Fission Gas and Cladding or Structure :math:`{H'}_{\text{Na,cl,i}}` and :math:`{H'}_{\text{Na,sr,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These coefficients are non-zero only for the partial annular fuel flow
regime. In all other regimes, the sodium/fission gas mixture is not in
direct contact with the cladding and structure. The heat-transfer
coefficient is calculated as follows:
If :math:`\alpha_{\text{Na,i}} \leq 0.15`,
(16.4-302)
.. _eq-16.4-302:
.. math::
H_{\text{Na,cl,i}} = H_{\text{Na,sr,i}} = \left\lbrack \text{C1} \left( D_{\text{H,Mi,i}} \
\cdot \rho_{\text{N1}} \cdot \left| u_{\text{Mi,i}} + u_{\text{Mi,i+1}} \cdot C_{\text{Mo,Mi,i+1}} \right| \
\cdot 0.5 \cdot \frac{C_{\text{p,N1}}}{k_{\text{N1}}} \right)^{\text{C2}} \
+ \text{C3} \right\rbrack \cdot \frac{k_{\text{N1}}}{D_{\text{H,Mi,i}}}
The value of :math:`k_{\text{N1}}` is given by the input constant :math:`\text{CDNL}`. The
constants :math:`C1`, :math:`C2` and :math:`C3` are the source input constants used in the
preboiling model.
If :math:`\alpha_{\text{Na,i}} > 0.15` and
a) :math:`T_{\text{Na,i}} \leq T_{\text{cl,i}}` (boiling situation)
(16.4-303a-c)
.. _eq-16.4-303a-c:
.. math::
H_{\text{Na,cl,i}} = \begin{cases}
H_{\text{Na,boiling}} = \text{CFNAEV} & \text{for } 1.5 < \alpha < 0.5 \\
\left\lbrack H_{\text{Na,boiling}} \cdot \left( 1 - \alpha_{\text{Na,i}} \right) + H_{\text{Nv,i}} \\
\cdot \left( \alpha_{\text{Na,i}} - 0.5 \right) \right\rbrack \cdot 2 & \text{for } 0.5 \leq \alpha < 1 \\
H_{\text{NV,i}} = \left\lbrack \text{C1} \cdot \left( D_{\text{H,Mi,i}} \cdot \rho_{\text{Mi,i}} \cdot \left| u_{\text{Mi,i}} \right| \cdot \frac{C_{\text{p,Mi}}}{k_{\text{Mi}}} \right)^{\text{C2}} + \text{C3} \right\rbrack \cdot \frac{k_{\text{Mi}}}{D_{\text{H,Mi,i}}} \text{ for } \alpha = 1 \\
\end{cases}
The value of :math:`k_{\text{Mi}}` is given by the input constant CDVG.
b) :math:`T_{\text{Na,i}} > T_{\text{cl,i}}` (condensation situation)
(16.4-304a-d)
.. _eq-16.4-304a-d:
.. math::
H_{\text{Na,cl,i}} = \begin{cases}
H_{\text{Nv,i}} \text{ as defined in Eq.16.4-303c} & \text{for } \alpha_{\text{Na,i}} = 1 \\
& \text{and } {T}_{\text{cl,i}} < T_{\text{sat,Na}} \left( P_{\text{Na,i}} \right) \\
\text{CFNACN} \cdot \frac{{\rho'}_{\text{Na}}}{{\rho'}_{\text{Na}} + {\rho'}_{\text{fi}}} + H_{\text{Nv}} & \text{for } \alpha_{\text{Na,i}} = 1 \\
\cdot \frac{{\rho'}_{\text{fi}}}{{\rho'}_{\text{Na}} + {\rho'}_{\text{fi}}} & \text{and } T_{\text{cl,i}} \geq T_{\text{sat,Na}} \left( P_{\text{Na,i}} \right) \\
H_{\text{Na,cond}} = \text{CFNACN} & \text{for } 0.5 \leq \alpha_{\text{Na,i}} < 1 \\
\left\lbrack H_{\text{Na,cond}} \cdot \alpha_{\text{Na,i}} + H_{\text{Na,cond}} \cdot 10^{- 2} \\
\cdot \left( 0.5 - \alpha_{\text{Na,i}} \right) \right\rbrack \cdot 2 & \text{for } 0.15 < \alpha < 0.5 \\
\end{cases}
It is noted that the branch Eq. :ref:`16.4-304b<eq-16.4-304a-d>` has not been implemented in the
initial release version but exists in the chunk version.
For :math:`\alpha_{\text{Na,i}} > .15`, the heat-transfer coefficient between sodium
and structure is obtained from Eq. :ref:`16.4-303<eq-16.4-303a-c>` for :math:`T_{\text{Na,i}} \leq T_{\text{sr,i}}`
and from Eq. :ref:`16.4-304<eq-16.4-304a-d>` for :math:`T_{\text{Na,i}} > T_{\text{sr,i}}`.
The respective heat-transfer areas are defined as follows:
(16.4-305)
.. _eq-16.4-305:
.. math::
A_{\text{Na,cl,i}}^{L} = A_{\text{cl,i}}^{L} \cdot \left( 1 - C_{\text{fu,cl,i}} \right) \cdot \left( 1 - C_{l \text{u,cl,i}} \right)
and
(16.4-306)
.. _eq-16.4-306:
.. math::
A_{\text{Na,sr,i}}^{L} = A_{\text{sr,i}}^{L} \cdot \left( 1 - C_{\text{fu,sr,i}} \right) \cdot \left( 1 - C_{l \text{u,sr,i}} \right)
.. _section-16.4.3.10.1.15:
Heat-transfer coefficients between sodium and Frozen Fuel on Cladding and Structure :math:`{H'}_{\text{ffc,Na,i}}` and :math:`{H'}_{\text{ffs,Na,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These coefficients are non-zero only in the partial annular flow regime.
For this regime, the heat transfer coefficients are defined below:
(16.4-307)
.. _eq-16.4-307:
.. math::
H_{\text{ffc,Na,i}} = \frac{1}{\frac{1}{H_{\text{Na,i}}} + \frac{0.5 \cdot l_{\text{ff,cl,i}}}{k_{\text{fu}}}}
(16.4-308)
.. _eq-16.4-308:
.. math::
H_{\text{ffs,Na,i}} = \frac{1}{\frac{1}{H_{\text{Na,i}}} + \frac{0.5 \cdot l_{\text{ff,sr,i}}}{k_{\text{fu}}}}
where :math:`H_{\text{Na,i}}` is defined by Eq. :ref:`16.4-234<eq-16.4-234>`.
The respective heat-transfer areas are defined as follows:
(16.4-309)
.. _eq-16.4-309:
.. math::
A_{\text{ffc,Na,i}}^{L} = A_{\text{cl,i}}^{L} \cdot C_{\text{ff,cl,i}} \cdot \left( 1 - C_{\text{fu,ff,cl,i}} \right)
(16.4-310)
.. _eq-16.4-310:
.. math::
A_{\text{ffs,Na,i}}^{L} = A_{\text{sr,i}}^{L} \cdot C_{\text{ff,sr,i}} \cdot \left( 1 - C_{\text{fu,ff,sr,i}} \right)
.. _section-16.4.3.10.2:
Friction Coefficients Used in the Momentum Equations
''''''''''''''''''''''''''''''''''''''''''''''''''''
In :numref:`section-16.4.3.8`, which describes the momentum equations, several
friction coefficients used in the momentum source terms were left
undefined. These coefficients are calculated in the routine LETRAN and
are presented in this section.
.. _section-16.4.3.10.2.1:
Friction Factor Between Mixture and Cladding/Structure :math:`f_{\text{Mi,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
This factor was referenced in Eq. :ref:`16.4-150<eq-16.4-150>`. It is zero for all flow
regimes except the partial annual fuel flow regime. For this flow
regime, it is defined as follows:
(16.4-311)
.. _eq-16.4-311:
.. math::
f_{\text{Mi,i}} = \text{AFRV} \cdot \text{Re}_{\text{Mi,i}}^{\text{BFRV}} \cdot C_{\text{Mi,bd,i}}
where AFRV, BFRV are input constants and :math:`C_{\text{Mi,bd,i}}` is the
fraction of the channel boundary (cladding + structure + crust) which is
in direct contact with the gas mixture in cell :math:`i`. It is defined as:
(16.4-312)
.. _eq-16.4-312:
.. math::
\begin{matrix}
C_{\text{Mi,bd,i}} = \left\lbrack A_{\text{cl,i}}^{L} \cdot \left( 1 - C_{\text{mfu,cl,i}} - Cf_{\text{ff,cl,i}} \cdot C_{\text{fu,ff,cl,i}} \right) \\
+ A_{\text{sr,i}}^{L} \cdot \left( 1 - C_{\text{mfu,sr,i}} - C_{\text{ff,sr,i}} \cdot C_{\text{fu,ff,sr,i}} \right) \right\rbrack \\
\big/ \left( A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L} \right) \\
\end{matrix}
where
(16.4-313)
.. _eq-16.4-313:
.. math::
C_{\text{mfu,cl,i}} = C_{\text{fu,cl,i}} - C_{\text{ff,cl,i}}
and
(16.4-314)
.. _eq-16.4-314:
.. math::
C_{\text{mfu,sr,i}} = C_{\text{fu,sr,i}} - C_{\text{ff,sr,i}}
.. _section-16.4.3.10.2.2:
Friction Factors Between Mixture and Fuel :math:`f_{\text{Mi,fu,i}}` and :math:`f_{\text{Mi,se,i}}`
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
These factors were referenced in Eq. :ref:`16.4-152<eq-16.4-152a>`. The factor
:math:`f_{\text{Mi,fu,i}}` is used only in the partial annular fuel flow regime
and is defined below:
(16.4-315)
.. _eq-16.4-315:
.. math::
f_{\text{Mi,fu,i}} = \text{AFRV} \cdot \text{Re}_{\text{Mi,fu,i}}^{\text{BFRV}} \cdot C_{\text{Mi,fu,i}}
:math:`\text{Re}_{\text{Mi,fu,i}}` is defined as:
(16.4-316)
.. _eq-16.4-316:
.. math::
\text{Re}_{\text{Mi,fu,i}} = D_{\text{H,Mi,i}} \cdot \left| u_{\text{MI,i}} \
- u_{\text{fu,i}} \right| \cdot \rho_{\text{Mi,i}} \cdot \frac{1}{\mu_{\text{Mi,i}}}
and :math:`C_{\text{Mi,fu,i}}`, the fraction of the mixture perimeter in contact
with the molten fuel is given by:
(16.4-317)
.. _eq-16.4-317:
.. math::
\begin{matrix}
C_{\text{Mi,fu,i}} = \left\lbrack A_{\text{cl}}^{L} \cdot \left( C_{\text{mfu,cl,i}} + C_{\text{ff,cl,i}} \cdot C_{\text{fu,ff,cl,i}} \right) \\
+ A_{\text{sr}}^{L} \cdot \left( C_{\text{mfu,sr,i}} + C_{\text{ff,sr,i}} \cdot C_{\text{fu,ff,sr,i}} \right) \right\rbrack \big/ \left( A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L} \right) \\
\end{matrix}
The factor :math:`f_{\text{Mi,se,i}}` is used in the annular steel flow regime
and takes into account the possibility of flooding. It is defined as
follows:
(16.4-318)
.. _eq-16.4-318:
.. math::
f_{\text{Mi,se,i}} = \text{AFRV} \cdot \text{Re}_{\text{Mi,fu,i}}^{\text{BRFV}} \cdot C_{\text{FLOOD,i}}
Where the flooding coefficient depends on the velocity ratio
:math:`V_{\text{RATIO}}`:
(16.4-319)
.. _eq-16.4-319:
.. math::
V_{\text{RATIO,i}} = \frac{\left( U_{\text{Mi,i}} + u_{\text{Mi,i+1}} \cdot C_{\text{Mo,Mi,i+1}} \right) \cdot 0.5}{u_{\text{FLOOD,i}}}
(16.4-320)
.. _eq-16.4-320:
.. math::
u_{\text{FLOOD,i}} - 2.3\sqrt{\frac{\rho_{\text{se,i}}}{\rho_{\text{Mi,i}}} \cdot \frac{2g \
\cdot l_{\text{se,ch,i}}}{\text{AFRV} \cdot \text{Re}_{\text{Mi,fu,i}}^{\text{BFRV}} \
\cdot F_{\text{FLOOD}} \left( \alpha_{\text{Na,i}} \right)}}
(16.4-321)
.. _eq-16.4-321:
.. math::
F_{\text{FLOOD}}\left( \alpha_{\text{Na,i}} \right) = \begin{cases}
1 + 75 \cdot \left( 1 - \alpha_{\text{Na,i}} \right) & 0.5 < \alpha \leq 1 \\
38.5 & \alpha \leq 0.5 \\
\end{cases}
The value of :math:`C_{\text{FLOOD,i}}` is also dependent on the previous
flooding history as shown below:
If :math:`C_{\text{FLODD,i}}^{n} \leq 1:`
(16.4-322)
.. _eq-16.4-322:
.. math::
C_{\text{FLODD,i}}^{n} = \begin{cases}
F_{\text{FLOOD}} \left( \alpha_{\text{Na,i}} \right) & \text{if } V_{\text{RATIO,i}} \geq 1.1 \\
C_{\text{FLODD,i}}^{n} & \text{otherwise} \\
\end{cases}
If :math:`C_{\text{FLODD,i}}^{n} > 1`
(16.4-323)
.. _eq-16.4-323:
.. math::
C_{\text{FLODD,i}}^{n + 1} = \begin{cases}
1 & \text{if } V_{\text{RATIO}} < 0.9 \\
C_{\text{FLODD,i}}^{n} & \text{otherwise} \\
\end{cases}
.. _section-16.4.3.10.2.3:
Several Area Coefficients Used in the Momentum Source Terms
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
The area coefficient :math:`C_{\text{AREA,Mi,fu,i}}` has been introduced in Eq.
:ref:`16.4-155<eq-16.4-155>`. It defines the fraction of the lateral area of chunks in
contract with the mixture and is zero for the fuel and steel bubbly flow
regimes. For the annular flow regimes, it is calculated as follows:
(16.4-324)
.. _eq-16.4-324:
.. math::
C_{\text{AREA,Mi,fu,i}} = \left( 1 - F_{\text{area,bd,fu,i}} \right)
The area coefficients :math:`C_{\text{AREA,se,i}}` and :math:`C_{\text{AREA,fu,i}}` have
been introduced in Eqs. :ref:`16.4-177<eq-16.4-177>` and :ref:`16.4-178<eq-16.4-178>`. They account for the fact
that only a fraction of the fuel steel perimeter is in contact with the
stationary walls. They are defined below:
For the fuel bubbly flow regime:
(16.4-325)
.. _eq-16.4-325:
.. math::
\begin{align}
C_{\text{AREA,fu,i}} = 1 ; && C_{\text{AREA,se,i}} = 0
\end{align}
For the steel bubbly flow regime:
(16.4-326)
.. _eq-16.4-326:
.. math::
\begin{align}
C_{\text{AREA,fu,i}} = 0 ; && C_{\text{AREA,se,i}} = 1
\end{align}
For the annular steel flow regime:
(16.4-327)
.. _eq-16.4-327:
.. math::
\begin{align}
C_{\text{AREA,fu,i}} = 0 ; && C_{\text{AREA,se,i}} = 1
\end{align}
For the partial annular fuel flow regime:
(16.4-328)
.. _eq-16.4-328:
.. math::
\begin{matrix}
C_{\text{AREA,fu,i}} = \left\lbrack A_{\text{cl,i}}^{L} \cdot \left( C_{\text{mfu,cl,i}} + C_{\text{ff,cl,i}} \cdot C_{\text{fu,ff,cl,i}} \right) \\
+ A_{\text{sr,i}}^{L} \cdot \left( C_{\text{mfu,sr,i}} + C_{\text{ff,sr,i}} \cdot C_{\text{fu,ff,sr,i}} \right) \right\rbrack \big/ \left( A_{\text{cl,i}}^{L} + A_{\text{sr,i}}^{L} \right) \\
\end{matrix}
.. math::
C_{\text{AREA,se,i}} = 0
.. _section-16.4.3.11:
Time Step Determination for the Channel Hydrodynamic Model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
This section describes the calculation of the maximum time step for the
coolant channel hydrodynamic model. This time step will be used in the
determination of the LEVITATE time step as outlined in :numref:`section-16.7.3`.
The maximum channel time step is calculated using the sonic Courant
condition for the multicomponent channel flow. Only a fraction of the
Courant calculated time is used, to account for inaccuracies in the
sonic velocity calculation. Thus:
(16.4-329)
.. _eq-16.4-329:
.. math::
\Delta t_{\text{LE,ch}} = 0.4 \cdot \min\left\lbrack \frac{\Delta z_{\text{i}}}{\left( u_{\text{sonic,i}} \
+ \left| u_{\text{Mi,i}} \right| \right)} \right\rbrack_{\text{i=IFMIBT,IFMITP}}
The minimum in Eq. :ref:`16.4-329<eq-16.4-329>` is evaluated over all axial cells of the
interaction region. The sonic velocity in each cell is calculated using
an expression [16-14] for an adiabatic homogeneous two-phase mixture of
liquid sodium and fission gas/sodium vapor. The compressibility of
liquid fuel is much smaller than that of liquid sodium and thus the fuel
is assumed to be incompressible in the calculation of the sonic velocity
in the channel. The effect of fuel and steel vapor is not included.
(16.4-330)
.. _eq-16.4-330:
.. math::
\begin{matrix}
u_{\text{sonic,i}}^{2} = \gamma_{\text{vg}} \cdot \left( P_{\text{fi,i}} + P_{\text{Na,i}} \right) \big/ \left\{ \alpha_{\text{vg,i}}^{2} \left( \rho_{\text{fi}} + \rho_{\text{Nv}} \right) + \alpha_{\text{vg,i}} \cdot \left( 1 - \alpha_{\text{vg,i}} \right) \\
\cdot \rho_{\text{N1}} + \left\lbrack \alpha_{\text{vg,i}} \cdot \left( 1 - \alpha_{\text{vg,i}} \right) \cdot \left( \rho_{\text{fi}} + \rho_{\text{Nv}} \right) + \left( 1 - \alpha_{\text{vg,i}} \right)^{2} \\
\cdot \rho_{\text{N1}} \right\rbrack \cdot \gamma_{\text{vg}} \cdot \left( P_{\text{fi}} + P_{\text{Na}} \right) \cdot \text{CMNL} \right\} \\
\end{matrix}
where:
(16.4-331)
.. _eq-16.4-331:
.. math::
\alpha_{\text{vg,i}} = \frac{\theta_{\text{vg,i}}}{\theta_{\text{ch,op,i}} \
- \theta_{\text{fu,i}} - \theta_{\text{se,i}} - \theta_{l \text{u,i}}}
:math:`\gamma_{\text{vg}}` = ratio of specific heat at constant pressure to that at
constant volume for the fission gas/sodium vapor mixture. A value of 1.4
is used in the LEVITATE code
:math:`\text{CMNL}` = compressibility of liquid sodium, an input constant
The fission-gas pressure :math:`P_{\text{fi}}` and the sodium pressure
:math:`P_{\text{Na}}` are obtained as explained in :numref:`section-16.4.3.5`.