.. _section-14.3:
Fuel and Fission-gas Ejection from the Pins
-------------------------------------------
.. _section-14.3.1:
Physical Model and Assumptions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In the previous section, the sinks for the in-pin motion due to fuel and
fission-gas ejection appeared in the fuel, free fission gas, and
dissolved fission-gas mass conservation equations (see Eqs. :ref:`14.2-9<eq-14.2-9>`,
14.2-15, and 14.2-20). These sink terms are, of course, source terms in
the channel thermal-hydraulics treatment.
The approach taken for the calculation of the fuel and fission-gas
ejection from the pins is based on the assumption that there will always
be pressure equilibrium established between a cavity node and the
adjacent channel node if fuel and fission-gas ejection occurs. This
assumption is justified for cladding ruptures with dimensions larger
than one-pin diameter because of the very short distance between cavity
nodes and corresponding coolant channel nodes. For pin-hole type
failures leading to a pure gas release, which is not currently treated
in PLUTO2, an orifice equation for calculating this gas release may be
more appropriate. In the older PLUTO code [14-1, 14-3, 14-4], both an
asymptotic orifice equation and an ejection calculation, based on
pressure equilibration, were computed for the fuel and gas ejections,
and the smaller of the two predictions was used. The asymptotic orifice
equation usually predicted the higher ejection rates. These were causing
a higher pressure in the channel node than in the adjacent cavity node
which was considered non-physical.
To achieve the pressure equilibrium between the cavity and channel nodes
in PLUTO2, an estimated amount of fuel and gas (with the same volume
ratio as present in the cavity node) is ejected into the channel and the
resulting cavity and channel pressures are calculated. This calculation
is repeated several times with updated estimates for the ejected fuel
and fission-gas masses until a pressure equilibrium is achieved. This
approach does not consider (or require) the cladding rupture area, which
is an advantage because the evolution of the cladding rupture is not
known. The cladding rupture sizes found in the post-test examinations of
TREAT tests H5 [14-24] and J1 [14-25], which were terminated at a time
when the pins were still largely intact, show large enough rupture sizes
to justify the assumption of pressure equilibration between the cavity
node behind the rupture and the adjacent channel node. Moreover, these
rupture areas are considerably larger than the cross section of the
molten cavity (50% areal melt fraction in an FFTF type pin means a
molten cross sectional area of about 0.1 cm\ :sup:`2`; cladding rupture
areas in H5 and J1 are about 1 cm\ :sup:`2`). This means that the
controlling aspect of the fuel and gas ejection (at least in mild TOP
accidents) is really the cross sectional area of the cavity, which is
taken into account in the in-pin motion calculation in PLUTO2. In the
SAS3D model SAS/FCI, which does not consider the in-pin motion
mechanistically, the cross sectional area for the orifice equation, used
in the calculation of the SAS/FCI fuel and gas ejection, has to be set
to about twice the cross sectional area of the cavity near the failure
location [14-3]. Under LOF-driven-TOP conditions, when axial cladding
rupture propagation is likely (see :numref:`section-14.3.3`), the in-pin motion is
not dominant in controlling the ejection because several contiguous
cavity nodes may eject fuel and gas simultaneously. In addition, the
time scales are very short under these conditions, leading to less axial
fuel and gas convection into an ejection node than can actually be
ejected.
Fuel and fission gas are being ejected from the fuel pins with the same
volume ratio as exists in the ejecting cavity node. For conditions
involving high void fractions in the ejecting cavity node, it is
possible that fission gas or fuel vapor will be ejected preferentially
because the remaining fuel may be in an annular flow configuration.
Another assumption about the fuel and gas ejection is that no backflow
of materials from the channel to the pin is allowed when the pressure in
the coolant channel node adjacent to the rupture becomes higher than the
pressure in the ejection node. To model this backflow would be difficult
and is not warranted because only a small amount of liquid sodium
flowing back into the pin would lead to enough vaporization and
pressurization to inhibit further backflow of materials. However, the
potential pressure increase in the cavity node may diminish the in-pin
fuel motion. This could be beneficial if midplane failures are assumed.
However, most channel pressurizations that have been observed
experimentally and that may be due to FCIs are not sufficient in
magnitude or duration to cause concern about this assumption.
.. _section-14.3.2:
Numerical Solution of the Fuel and Gas Ejection Calculation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The numerical evaluation of the amount of fuel and fission gas ejected
during a PLUTO2 time step is performed as a second step after the fuel
mass conservation Eq. :ref:`14.2-9<eq-14.2-9>` has been solved and advanced to the end of
the time step without accounting for the ejection term
:math:`S'_{\text{fuca,ej}}`. This section describes how the fuel smear
density value obtained at the end of the time step without accounting
for ejection :math:`{\rho'}_{\text{fuca}}^{\text{old}}` is corrected for
fuel ejection, thus obtaining the actual fuel smear density at the end
of time step the :math:`{\rho'}_{\text{fuca}}^{\text{new}}` and the
resulting equilibrated pressure of the ejecting cavity cell.
If :math:`\text{FN}_{\text{k}}` is the fraction of the fuel in a cavity node that is
ejected during a time step, the new generalized fuel smear density in
the cavity node will be:
(14.3‑1)
.. _eq-14.3-1:
.. math::
{\rho'}_{\text{fuca}}^{\text{new}} = {\rho'}_{\text{fuca}}^{\text{old}} - {\rho'}_{\text{fuca}}^{\text{old}} \cdot \text{FN}_{\text{k}} = {\rho'}_{\text{fuca}}^{\text{old}} - \text{FF}
where
:math:`\text{FF} = {\rho'}_{\text{fuca}}^{\text{old}} \cdot \text{FN}_{\text{K}}`
and is actually the same as the sink term for fuel ejection,
:math:`{S'}_{\text{fuca, ej}} \cdot \Delta t_{\text{PL}}` (see Eq. :ref:`14.2-13<eq-14.2-13>`).
Since the fission gas and fuel are assumed to be ejected with the same
volume ratio as present in the ejecting cell, the new generalized
free-fission-gas smear density is
(14.3‑2)
.. _eq-14.3-2:
.. math::
{\rho'}_{\text{fica}}^{\text{new}} = {\rho'}_{\text{fica}}^{\text{old}} - {\rho'}_{\text{fica}}^{\text{old}} \cdot \text{FN}_{\text{k}} = {\rho'}_{\text{fica}}^{\text{old}} - \text{FF} \cdot \frac{{\rho'}_{\text{fica}}^{\text{old}}}{{\rho'}_{\text{fuca}}^{\text{old}}}
The new cavity fuel fraction will be
(14.3‑3)
.. _eq-14.3-3:
.. math::
\theta_{\text{fica}}^{\text{new}} = \theta_{\text{fica}}^{\text{old}} - \frac{\text{FF}}{\rho_{\text{fuca}}}
where :math:`\rho_{\text{fuca}} = \rho_{\text{fuca}} \left( T_{\text{fuca}} \right)` is the
theoretical fuel density.
By using Eqs. :ref:`14.3-1<eq-14.3-1>`, :ref:`14.3-2<eq-14.3-2>`, and :ref:`14.3-3<eq-14.3-3>` in the cavity pressure
calculation of Eqs. :ref:`14.2-32<eq-14.2-32>` and :ref:`14.2-33<eq-14.2-33>` and by dropping the superscripts
"old", one obtains
(14.3‑4)
.. _eq-14.3-4:
.. math::
\begin{matrix}
P_{\text{ca}}^{\text{new}} = P_{\text{fvca}} + R_{\text{fi}} \cdot T_{\text{fuca}} \cdot \left( {\rho'}_{\text{fica}} - \text{FF} \cdot \frac{{\rho'}_{\text{fica}}}{{\rho'}_{\text{fuca}}} \right) \\
\big/ \left\{ \theta_{\text{ca}} - \theta_{\text{fuca}} + \frac{\text{FF}}{\rho_{\text{fuca}}\left( T \right)} + \left\lbrack \theta_{\text{fuca}} - \frac{\text{FF}}{\rho_{\text{fuca}}\left( T \right)} \right\rbrack K_{\text{fu}} \cdot \left( P_{\text{ca}}^{\text{new}} - P_{\text{fuca}} \right) \right\} \\
\end{matrix}
where
:math:`\theta_{\text{ca}} - \theta_{\text{fuca}} = \theta_{\text{fica}}`
By collecting the terms with FF on the left-hand side one obtains
(14.3‑5)
.. _eq-14.3-5:
.. math::
\begin{matrix}
\text{FF} \cdot \left\lbrack \frac{\left( P_{\text{ca}}^{\text{new}} - P_{\text{fvca}} \right)}{\rho_{\text{fuca}}} + \frac{{\rho'}_{\text{fica}}}{{\rho'}_{\text{fuca}}} \cdot R_{\text{fi}} \cdot T_{\text{fuca}} \\
- \frac{K_{\text{fu}}}{\rho_{\text{fu}}} \cdot \left( P_{\text{ca}}^{\text{new}} - P_{\text{fvca}} \right)^{2} \right\rbrack = R_{\text{fi}} \cdot T_{\text{fuca}} \cdot {\rho'}_{\text{fica}} \\
- \theta_{\text{fica}} \left( P_{\text{ca}}^{\text{new}} - P_{\text{fvca}} \right) - \theta_{\text{fica}} \cdot K_{\text{fu}} \cdot \left( P_{\text{ca}}^{\text{new}} - P_{\text{fvca}} \right)^{2} \\
\end{matrix}
This equation is used to calculate the fuel ejection FF for an
estimated :math:`P_{\text{ca}}^{\text{new}}`. FF is used, in turn, to
update the fission-gas and fuel densities in the channel node adjacent
to the ejecting cavity node. A new channel pressure is then calculated
based on the updated fuel and gas densities. The method of calculating
the channel pressures is discussed in :numref:`section-14.4.5`. If the new channel
pressure is not within 1% of the estimated new cavity pressure, a better
estimate for the new cavity pressure is made and the calculational
sequence for the fuel and gas ejection is repeated.
Because the conditions in the coolant channel node can vary from normal
coolant flow to fully voided, the procedure of estimating the new cavity
pressure is somewhat complex. It has evolved through trial and error and
it usually leads to convergence in a few iterations. If the iteration
does not converge, it usually indicates that a non-physical condition
has developed in the pin or channel. When the axial extent of the
cladding rupture includes more numerical nodes as a result of axial pin
failure propagation (which is likely under LOF′d′TOP conditions), the
ejection calculation will be performed for each of the axial failure
nodes.
The ejection calculation from the pins was originally formulated to
allow a simultaneous ejection from the three pin failure groups at any
given axial location. As mentioned earlier, the second and third pin
failure groups are not yet operational. Because of the complexity of the
current ejection calculation from only one pin failure group, it is
expected that a simultaneous ejection from three pin failure groups into
the same coolant channel node may become too complicated. Therefore, a
simpler approach, which would only allow the ejection from the cavity
node with the highest pressure during a PLUTO2 time step, may have to be
adopted. In this approach it would, of course, still be possible for the
different pin failure groups to eject fuel simultaneously at different
axial locations.
The fuel and gas ejection calculation is done at the end of subroutine
PL1PIN which was discussed in :numref:`section-14.1.2`. The ejection calculation
is done in the calculational sequence after all in-pin as well as
channel mass and energy conservation have been solved and before the
momentum equations for both the in-pin and channel flows are solved. The
ejection calculation thus provides updated pressures for the momentum
equations.
.. _section-14.3.3:
Axial Pin Failure Propagation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Rapid axial pin failure propagation due to the stress concentrations at
the edges of the initial rupture (zipper-type failure propagation) has
not been accepted as a mechanism for rapid axial failure propagation
(i.e., on a millisecond or submillisecond time scale) [14-26]. However,
if the conditions in a fuel-pin node are such that failure of the
cladding should occur irrespective of cladding stress concentrations due
to an adjacent cladding rupture and whether the potential new failure
node location is adjacent to the initial failure or not, there is no
reason to disallow this type of failure propagation.
The failure of nodes other than the initial one is determined by PLUTO2
because the DEFORM calculation is not done in a calculational channel
once PLUTO2 (or LEVITATE) has been initiated. There are two options for
pin failure propagation in PLUTO2. The first option is a mechanistic one
based on the burst strength of the cladding. The second option allows
the pin failure propagation to proceed according to a combination of
input failure criteria based on fuel melt fraction, cladding
temperature, and pressure differential between cavity and channel.
The mechanistic option (KFAILP=0) assumes that the solid fuel
surrounding the molten cavity is completely cracked and that the radial
stress at the fuel-cladding interface can therefore be calculated by
reducing the cavity pressure by a geometrical reduction factor.
(14.3‑6)
.. _eq-14.3-6:
.. math::
\sigma_{\text{radial,fc,K}} = \frac{P_{\text{ca,K}} \cdot D_{\text{ca,K}}}{\left( 2R_{\text{cl,fc,K}} \right)}
where
:math:`R_{\text{cl,fc,K}}` is the inner radius of the cladding at axial node
:math:`K`.
:math:`D_{\text{ca,K}}` is the diameter of the molten cavity at axial node :math:`K`.
The hoop stress in the cladding can be calculated from
(14.3‑7)
.. _eq-14.3-7:
.. math::
\sigma_{\text{hoop,K}} = \left( \sigma_{\text{radial,fc,K}} - P_{\text{ch,i}} \right) \frac{R_{\text{cl,fc,K}}}{\Delta R_{\text{cl,K}}}
where :math:`\Delta R_{\text{cl,K}}` is the cladding thickness at axial node :math:`K`,
and subscript :math:`i` refers to the channel node which is adjacent to cavity
node :math:`K`; :math:`i = K + \text{IDIFF}` where IDIFF is the number of channel
nodes below cavity node :math:`K = 1`.
By substituting Eq. :ref:`14.3-6<eq-14.3-6>` for the radial stress in Eq. :ref:`14.3-7<eq-14.3-7>`,
(14.3‑8)
.. _eq-14.3-8:
.. math::
\sigma_{\text{hoop,K}} = \frac{\left( P_{\text{ca,K}} \cdot D_{\text{ca,K}} - 2 \cdot P_{\text{ch,i}} \cdot R_{\text{cl,fc,K}} \right)}{\left( 2\Delta R_{\text{cl,K}} \right)}
Failure is assumed to occur when the following two conditions are
satisfied:
(14.3‑9)
.. _eq-14.3-9:
.. math::
\sigma_{\text{hoop,K}} > \sigma_{\text{UTS}} \left( T_{\text{cl,in,K}} \right)
(14.3‑10)
.. _eq-14.3-10:
.. math::
\text{AREA}_{\text{ca,K}} > \text{FNARME} \cdot \pi \cdot R_{\text{cl,fc,K}}^{2}
where
FNARME = an input melt fraction (or more precisely cavity cross
sectional area divided by cross sectional area of the fuel) which should
be set to :math:`\geq 0.2` because the PLUTO2 in-point motion model requires a
moderately sized cavity diameter to run well.
:math:`\sigma_{\text{UTS}}` = the burst strength of the cladding that is a function
of cladding temperature. This is calculated from the middle cladding
node temperature in function subroutine UTS, which is described in
:numref:`Chapter %s<section-8>` on fuel behavior.
A problem with this burst failure criterion is that it is not compatible
with a melt fraction failure criterion, cladding strain failure
criterion or any other criteria that are not directly related to the
cladding strength. Since these latter criteria may be used in DEFORM to
predict the initial failure, care has to be taken that the cladding will
not rip open along sizable length as soon as PLUTO2 takes over the
calculation. For example, this could happen if the initial pin failure
was assumed to occur due to a melt fraction criterion at a time when the
cavity pressure was already high enough to burst the cladding at several
axial elevations. If a melt fraction criterion was chosen for the
initial failure prediction, the problem that several nodes will
instantaneously rupture once PLUTO2 initiates, can be avoided by setting
the input parameter FNARME (see Eq. :ref:`14.3-10<eq-14.3-10>`) equal to or greater than
the input melt fraction criterion for the initial failure.
The non-mechanistic input failure propagation criterion (KFAILP=1)
involves several separate criteria. Failure occurs when the cladding
middle node and outer surface temperatures, the cross sectional area of
the cavity, and the pressure difference between cavity and channel
exceed input criteria.
(14.3‑11)
.. _eq-14.3-11:
.. math::
\begin{align}
T_{\text{cl,in,K}} && \text{and} && T_{\text{cl,os,K}} > \text{TEFAIL}
\end{align}
and
(14.3‑12)
.. _eq-14.3-12:
.. math::
\text{AREA}_{\text{ca,K}} > \text{FNARME} \cdot \pi \cdot R_{\text{cl,fc,K}}^{2}
and
(14.3‑13)
.. _eq-14.3-13:
.. math::
P_{\text{ca,K}} > P_{\text{ch,i}} + \text{PRFAIL}
where
TEFAIL, FNARME, and PRFAIL are input parameters.
Subscripts **in** and **os** refer to the middle and outer cladding nodes,
respectively. This combined criterion can of course be simplified by
setting some of the input values to extreme numbers. For example, if
TEFAIL is set to zero and PRFAIL to :math:`-10^{20}`, this criterion
will reduce to a pure melt fraction criterion. This input failure
propagation criterion may be useful for comparison calculations with
codes that allow failure propagation only according to one of the above
non-mechanistic criteria. Moreover, if the initial failure criterion is
a melt fraction criterion, one may also want to do the failure
propagation based on a melt fraction criterion or on a melt fraction
criterion combined with the condition that some overpressure has to
exist in the molten cavity nodes.
.. _section-14.3.4:
Complete Pin Disruption and Switch to LEVITATE
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Once the cladding has melted at a certain axial location and extensive
fuel melting has also occurred at the same location, complete fuel-pin
disruption will eventually occur. This cannot be treated with PLUTO2,
and therefore, a switch to LEVITATE is made under these conditions. For
each pin node, PLUTO2 checks whether the middle and outer cladding
temperatures have exceeded the liquidus and whether the fuel-surface
temperature has exceeded the solidus.
:math:`T_{\text{cl, in, K}} > T_{\text{cl, liq}}`
and
:math:`T_{\text{cl, os, K}} > T_{\text{cl, liq}}`
and
:math:`T_{\text{fu, os, K}} > T_{\text{fu,sol}}`
where
**cl**, **in** refers to the middle cladding node,
**cl**, **os** refers to the outer cladding node,
**fu**, **os** refers to the outermost fuel node.
If the above conditions are met, control will be transferred from PLUTO2
to LEVITATE. These checks are performed at the end of the PLUTO2 driver
rouÂtine. At that location in the code, it is also checked whether
cladding meltÂing has occurred in more than NCPLEV nodes (NCPLEV is
input). If this criteÂrion is met, a switch to LEVITATE will also occur.
Moreover, a switch to LEVITATE is also made if the fuel temperature in
the cavity or in the channel exceeds 4000 K because significant fuel
vapor pressures will develop beyond this temperature.