.. _section-8.4:
Fuel-pin Failure Criteria
-------------------------
DEFORM‑4 contains a number of methods for the determination of fuel‑pin
failure and failed pin model initiation. These can be divided into two
major areas: (1) failure based on specified conditions, and (2)
calculated failure correlations. The first method is currently used to
produce failure initiation because of the limited amount of integral
code validation with the failure correlations. Since these use
parameters such as stress and temperature to produce the predictions, it
is essential that these be checked against experimental tests to assure
the accuracy and appropriateness of their use under various transient
conditions. The first method involves parameters used previously with
the SAS3D code system, so some familiarity with their use and
limitations is assumed. Although these are simple, they can be used in a
manner consistent with the SAS3D implementation and available
experimental data.
All criteria are cast into the form of a failure fraction, which is the
current value for the parameter being used as a failure indicator, i.e.,
melt fraction, time, etc., divided by the value which would produce
failure. If this failure fraction becomes 1, then failure is assumed to
occur and the failed pin modeling is initiated. In addition to
satisfying the parameter under consideration, the fuel melt fraction
must be above a specified level, FMELTM, before PLUTO2 or LEVITATE will
be initiated. These calculations are performed in the subroutine CLDFAL.
.. _section-8.4.1:
Input Specified Failure Criteria
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Through the use of the input parameters MFAIL, IFAIL, JFAIL, FSPEC, and
FMELTM the code can be controlled to initiate failure with the
conditions discussed below. MFAIL selects the meaning of FSPEC, IFAIL
and JFAIL specify the radial and axial location of the test for failure,
and FMELTM is the minimum melt fraction necessary at the location of
failure for PLUTO2 or LEVITATE to be initiated. :numref:`table-8.4-1` gives the
selections currently available in conjunction with DEFORM-4. If IFAIL
and JFAIL are not specified, the location of the parameter maximum is
selected.
.. _table-8.4-1:
.. list-table:: Failure Initiation Options
:header-rows: 1
:align: center
:widths: auto
* - MFAIL
- FSPEC
- IFAIL
- JFAIL
* - 1
- Time
- \-
- Nec\*
* - 2
- Temperature
- Nec
- Opt\*\*
* - 3
- Melf fraction
- \-
- Opt
* - 4
- Cavity pressure
- \-
- Opt
* - 5
- Hoop stress
- \-
- Opt
* - 7
- Rip propagation
- \-
- Opt
* - \*Nec = Necessary parameter that must be specified.
\*\*Opt = Optional parameter. If not specified, the axial level with the maximum value will be chosen.
-
-
-
The use of MFAIL with a value of 7 initiates the failure with a criteria
that is consistent with the LEVITATE and PLUTO2 rip propagation
considerations, see :numref:`section-16.3.1` and :numref:`section-14.3.3`, respectively. This
criterion checks the cladding hoop stress introduced by a molten cavity
pressure, for fully cracked fuel, or fuel‑cladding mechanical
interaction, when solid fuel is present, against the ultimate tensile
strength of the cladding to determine if failure has occurred. If the
cladding does fail, then the routine checks to see if the molten fuel
has reached the crack boundary. Both conditions are necessary for the
initiation of the failed fuel routines.
When other failure criteria are used, there may be extensive rip
propagation after initiation of the failed fuel model because of the
change in criterion.
.. _section-8.4.2:
Calculated Failure Correlations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In DEFORM‑4, there exist a number of failure correlations that may be
used to determine the time and location of cladding failure. Care should
be taken when drawing any conclusions based on these correlations, since
the validation effort is in its initial stages with these correlations.
In addition, the method of determining cladding stress is not totally
consistent between DEFORM‑4, a more mechanistic approach, and the
correlations, which were based on initial cold dimensions and fill‑gas
pressures. DEFORM‑4 also considers fuel‑cladding mechanical interaction
rather than just gas pressure.
The use of these failure correlations currently involves an iterative
approach, using the failure fractions printed by SAS4A and one of the
failure conditions described in :numref:`section-8.4.1`. First, the case is run
with no failure conditions specified and restart files produced at
selected intervals. The output is then studied to determine when the
failure fraction of interest reaches one, and at which axial segment.
One of the conditions discussed above is then selected to initiate
failure at the appropriate time and axial segment. A restart case is
then run from a restart file produced before the correlation failure
condition was reached. In this case, one of the simple failure criteria
will be used to initiate failure. The values used in the simple criteria
should be chosen to produce failure initiation at the same time and
place as was predicted by the correlation criterion during the initial
run.
The various correlations are given below. Although the units are
inconsistent in many cases, the code has taken this into account and
made the necessary adjustments.
.. _section-8.4.2.1:
Biaxial Stress Rupture
^^^^^^^^^^^^^^^^^^^^^^
This correlation is from the Nuclear Systems Material Handbook [8-17]
and is based on unirradiated, 20% cold‑worked type 316 SS developmental
cladding.
.. math::
:label: 8.4-1
\log \theta = - 15.22 + 9.5342 \log \left\lbrack \log \left( \frac{\sigma^{*}}{\sigma} \right) \right\rbrack
where
:math:`\theta` = Dorn parameter = :math:`t_{\text{r}} \exp\left(\frac{-Q}{RT} \right)`
:math:`t_{\text{r}}` = Rupture time, hr
:math:`T` = Temperature, K
:math:`Q` = 83,508 cal/mole
:math:`R` = Universal gas constant
:math:`\sigma` = Hoop stress =
:math:`\frac{r_{\text{o}}^{2} + r_{\text{i}}^{2}}{r_{\text{o}}^{2} - r_{\text{i}}^{2}} P` , ksi
:math:`\sigma^{*}` = 135 ksi
:math:`r_{\text{o}}` = Clad outer radius
:math:`r_{\text{i}}` = Clad inner radius
:math:`P` = Internal gas pressure (or interface pressure), ksi
.. _section-8.4.2.2:
Burst Pressure
^^^^^^^^^^^^^^
This correlation is from the NSMH [8‑17] for unirradiated FTR cladding.
.. math::
:label: 8.4-2
P = \sum_{\text{i} = 0}^{10} a_{\text{i}} \sigma^{i}
where
:math:`P` = Burst pressure, ksi
:math:`\sigma` = Hoop stress, as defined above
:math:`a_{0} = 1.799988 \times 10^{1}`
:math:`a_{1} = 2.866442 \times 10^{-2}`
:math:`a_{2} = -3.986012 \times 10^{-4}`
:math:`a_{3} = 2.408207 \times 10^{-6}`
:math:`a_{4} = -8.090292 \times 10^{-9}`
:math:`a_{5} = 1.607218 \times 10^{-11}`
:math:`a_{6} = -1.962158 \times 10^{-14}`
:math:`a_{7} = 1.487159 \times 10^{-17}`
:math:`a_{8} = -6.821934 \times 10^{-21}`
:math:`a_{9} = 1.735220 \times 10^{-24}`
:math:`a_{10} = 1.879417 \times 10^{-28}`
.. _section-8.4.2.3:
Transient Burst Temperature
^^^^^^^^^^^^^^^^^^^^^^^^^^^
This correlation is from the NSMH [8‑17] for unirradiated cladding. It
involves two temperature ramp rates, 5.56 and 111.1 K/s. If the rate is
between these, a linear interpolation is performed.
.. math::
:label: 8.4-3
T_{\text{f}} = 2358.4 - 36.41 \sigma + 0.5649 \sigma^{2} - 3.455 \times 10^{- 3} \sigma^{3}
for :math:`\dot{T} \leq 5.56` K/s
.. math::
:label: 8.4-4
T_{\text{f}} = 2484.8 - 37.80 \sigma + 0.5827 \sigma^{2} - 3.585 \times 10^{- 3} \sigma^{3}
for :math:`\dot{T} \geq 111` K/s
where
:math:`T_{\text{f}}` = Failure temperature, °F
:math:`\Sigma` = Hoop stress, as defined above
.. _section-8.4.2.4:
Larson-Miller Life Fraction
^^^^^^^^^^^^^^^^^^^^^^^^^^^
This correlation is the LMP life fraction incorporated in the TEMECH
computer code [8‑30].
.. math::
:label: 8.4-5
\log \left( \frac{t_{\text{r}}}{3600} \right) = \frac{\text{LMP}}{1.8 T} - C
where
:math:`t_{\text{r}}` = Time to rupture, in seconds
:math:`C` = Material-dependent constant = 20
LMP = Experimentally determined Larson‑Miller parameter
:math:`T` = Cladding temperature, K
For 20% CW 316 SS cladding, the following correlation is used.
.. math::
:label: 8.4-6
\text{LMP}' = 4.6402 - 5.1218 \times 10^{-2} \sigma_{\text{m}} + 7.0417 \times 10^{-4} \sigma_{\text{m}}^{2} - 4.1349 \times 10^{-6} \sigma_{\text{m}}^{3}
for :math:`0 \leq` fluence :math:`\leq 1.9 \times 10^{22}`
.. math::
:label: 8.4-7
\text{LMP}' = 4.2281 - 2.0469 \times 10^{-2} \sigma_{\text{m}}
for :math:`1.0 \times 10^{22} < \text{ fluence } \leq 3 \times 10^{22}`
.. math::
:label: 8.4-8
\text{LMP}' = 7.488 - 0.138 \sigma_{\text{m}}
for :math:`5.56` K/s temperature ramp and fluences between :math:`3.0 - 4.0 \times 10^{22}`
.. math::
:label: 8.4-9
\text{LMP}' = 5.285 - 7.778 \times 10^{-2} \sigma_{\text{m}} + 6.027 \times 10^{-4} \sigma_{\text{m}}^{2}
for :math:`111.1` K/s temperature ramp and fluences between :math:`3.0 - 4.0 \times 10^{22}`
where
:math:`\text{LMP} = \text{LMP}' \times 10^{4}`
:math:`\sigma_{\text{m}}` = Modulus modified hoop stress in ksi = :math:`\sigma_{\text{c}} \frac{E \left( 1033C \right)}{E \left( T \right)}`
These rupture times are then used in the life fraction form as
.. math::
:label: 8.4-10
\text{life fraction} = \sum_{\text{i}}\frac{\Delta t_{\text{i}}}{t_{\text{r,i}}}
where
:math:`i` = Time-step number
:math:`\Delta t_{\text{i}}` = Time-step length
:math:`t_{\text{r,i}}` = Time for cladding rupture
.. _section-8.4.3:
Preliminary Metal Fuel Failure Criteria
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The consideration of a new fuel type in the context of DEFORM and SAS4A
raises the question of the type of failure these pins will experience.
Because metal fuel does not produce the same magnitude of fuel-cladding
mechanical interaction as the oxide fuel, it is reasonable to assume
that the cladding failure will result from a different phenomenon.
Metal uranium forms a low melting point eutectic alloy with both iron
and nickel, constituent materials in the cladding alloys used. As the
steady state irradiation proceeds, the iron and nickel can be leached
out of the cladding, leading to the formation of a low melting point
alloy layer adjacent to the cladding surface. Under transient conditions
the temperature in this outer layer, at a certain axial location, may
become high enough to melt this material. This in turn can produce a
failure at this location by accelerating the further thinning of the
load bearing cladding. A number of tests with UFs rods were performed
and analyzed to develop a correlation for the time to cladding failure
[8-31]. This correlation has been incorporated into the FAILUR routine
for preliminary use until a more mechanistic model is developed which
handles the cladding thinning and subsequent failure from the local
pressure.
.. math::
:label: 8.4-11
T_{\text{f}} = 9.142 \times 10^{4} \left\lbrack \frac{T_{\text{a}}}{T_{\text{cut}}} \right\rbrack^{-28.495} \left( 1 + B \right)^{- 0.54669}
where
:math:`T_{\text{f}}` = Time to failure at the current conditions, s
:math:`T_{\text{a}}` = Temperature at the fuel-cladding interface, °C
:math:`T_{\text{eut}}` = Input eutectic temperature, °C
:math:`B` = Burnup, atom percent
Once this time has been determined, a life fraction approach is used by
dividing the current time step by this failure time and adding it to the
previously calculated fractions. When the fraction reaches 1, failure is
assured.
.. math::
:label: 8.4-12
F_{\text{f,n}} = F_{\text{F,O}} + \frac{\Delta t}{T_{\text{f}}}
where
:math:`F_{\text{f,n}}` = New life fraction value
:math:`F_{\text{f,o}}` = Sum of previous life faction values
:math:`\Delta t` = Time step, s
This correlation was developed over a narrow range of temperatures so
its use should be considered preliminary until a more mechanistic
approach is developed.
.. _section-8.4.4:
Failure Modeling Coupling to Fuel Motion Models
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
In order to correctly assess the accident scenario in a transient that
leads to fuel pin failure and subsequent fuel motion, there should be a
high degree of consistency between the methods for predicting the
initial failure and that used to predict the axial propagation of this
failure. This consistency has now been incorporated in the FAILUR
subroutine for the option with MFAIL equal to 6. While the actual
criterion used is preliminary in nature, the fact that consistency
exists between failure and rip propagation is important in order to
study how this affects the accident scenario. It is noted that his
consistent failure model has been implemented only for an oxide-fuel
type failure. A similar consistent failure model for the metal fuel pins
will be added in the future.
PLUTO2 and LEVITATE use a rip propagation model that compares the
circumferential cladding stress, induced by a pressurized molten cavity
acting on cracked fuel, to the ultimate tensile strength of the
cladding. A stress greater than the ultimate tensile strength produces
failure at the axial location. This same approach has been followed in
this failure prediction.
First, the SAS4A/DEFORM cladding node structure is converted to one
compatible with to the failed fuel modeling routines. PLUTO2 and
LEVITATE divide the cladding into an inner cladding node containing
three-fourths of the cladding thickness, and an outer node containing
the rest.
.. math::
:label: 8.4-13
r_{\text{c}1} = r_{\text{ci}} + 0.25 \left( r_{\text{co}} - r_{\text{ci}} \right)
.. math::
:label: 8.4-14
r_{\text{c}2} = r_{\text{ci}} + 0.75 \left( r_{\text{co}} - r_{\text{ci}} \right)
where
:math:`r_{\text{c}1}` = Outer boundary of old inner cladding cell, m
:math:`r_{\text{c}2}` = Outer boundary of old central cladding cell, m
:math:`r_{\text{co}}` = Outer radius of the cladding, m
:math:`r_{\text{ci}}` = Inner radius of the cladding, m
The new inner cladding cell energy is then determined form weighting
factors based on the old cell radii
.. math::
:label: 8.4-15
E_{\text{c,a}} = \frac{\left\lbrack E_{\text{c}} \left( T_{\text{ci}} \right) \cdot W_{1} + E_{\text{c}} \left( T_{\text{cc}} \right) \cdot W_{2} \right\rbrack}{\left\lbrack W_{1} + W_{2} \right\rbrack}
where
:math:`E_{\text{c,a}}` = Average energy content of the new inner cladding node, J/kg
:math:`\left( E_{\text{c}} T \right)` = Cladding energy as a function of temperature, J/kg
:math:`T_{\text{ci}}` = Inner temperature of the cladding, K
:math:`T_{\text{cc}}` = Central temperature of the cladding, K
:math:`W = r_{\text{c}1}^{2} - r_{\text{ci}}^{2}`
:math:`W_{2} = r_{\text{c}2}^{2} - r_{\text{c}1}^{2}`
The temperature of the new inner cladding node is then determined.
.. math::
:label: 8.4-16
T_{\text{c}1} = T_{\text{c}} \left( E_{\text{c,a}} \right)
where
:math:`T_{\text{c}1}` = Temperature of modified inner cladding node, K
:math:`T_{\text{c}} \left( E \right)` = Cladding temperature as a function of energy, K
The ultimate tensile strength is then determined on the basis of the new
node structure.
.. math::
:label: 8.4-17
U_{\text{ts}} = \frac{\left\lbrack 3 U_{\text{t}} \left( T_{\text{c}1} \right) + U_{\text{t}} \left( T_{\text{co}} \right) \right\rbrack}{4}
where
:math:`U_{\text{ts}}` = Average ultimate tensile strength of the cladding, Pa
:math:`U_{\text{t}} \left( T \right)` = Ultimate tensile strength as a function of
temperature, Pa
:math:`T_{\text{co}}` = Cladding outer surface temperature, K
This is the value that determines if failure occurs. If the calculated
circumferential stress exclude :math:`U_{\text{ts}}`, failure is assumed to
occur.
The calculated stress is determined in two ways, depending on the
relationship between the melt boundary and cracked region of the fuel.
If a solid annulus of fuel exists between the melted and cracked
regions, the calculated fuel-cladding interface pressure is used
.. math::
:label: 8.4-18
\sigma_{\text{c}} = \frac{\left\lbrack P_{\text{fci}} r_{\text{ci}} - P_{\text{ext}} r_{\text{co}} \right\rbrack}{\left\lbrack r_{\text{co}} - r_{\text{ci}} \right\rbrack}
where
:math:`\sigma_{\text{c}}` = Calculated circumferential stress based on a force
balance, Pa
:math:`P_{\text{fci}}` = Pressure that exists at the fuel-cladding interface, Pa
:math:`P_{\text{ext}}` = Pressure that exists on the outer cladding surface,
If the melting has proceeded to the cracked boundary, then the cladding
stress is determined from a force balance on the cladding assuming an
inverse :math:`r` drop-off in pressure through the solid, cracked fuel.
.. math::
:label: 8.4-19
\sigma_{\text{c}} = \frac{\left\lbrack P_{\text{cav}} r_{\text{cav}} - P_{\text{ext}} r_{\text{co}} \right\rbrack}{\left\lbrack r_{\text{co}} - r_{\text{ci}} \right\rbrack}
where
:math:`P_{\text{cav}}` = Central molten cavity pressure, PA
:math:`r_{\text{cav}}` = Radius of the central molten region, m
The failure fraction is then calculated from the cladding stress and
ultimate tensile strength.
.. math::
:label: 8.4-20
F_{\text{f}} = \frac{\sigma_{\text{c}}}{U_{\text{ts}}}
where
:math:`F_{\text{f}}` = Cladding failure fraction at current time
When this fraction reaches 1, failure of the cladding at the axial
segment is assume to occur. However, this does not mean the ejection of
fuel into the coolant channel will take place. Besides cladding failure,
the molten fuel radius must have reached the cracked fuel radius.
Therefore, there are tow conditions to satisfy prior to initiation of
the post failure fuel motion modeling; (1) cladding failure, and (2)
complete solid fuel cracking.
Once the post failure fuel motion modeling has been initiated, DEFORM is
not used. Control is transferred to PLUTO2 or LEVITATE and the same
procedure given above is employed by these modules to determine if the
cladding failure propagates to other axial segments.
.. _section-8.4.5:
Time-step Control on Approach to Failure
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
As the conditions necessary for the failure are approached, the maximum
time step is reduced to specified levels. This is controlled through the
input parameters FIRLIM, SECLIM, and THRLIM, and the associated time
steps DTFALl, DTFAL2, and DTFAL3. When the failure fraction reaches
FIRLIM, the main time step will be reduced to DTFAL1 or the current time
step, whichever is smaller. This same procedure continues through SECLIM
and THRLIM. If
.. math::
:label: 8.4-21
0 \leq F_{\text{f}} < \text{FIRLIM}
where
:math:`F_{\text{f}}` = Failure fraction for the failure condition being used
then the maximum allowable time step is that defined by the initial
transient time‑step input parameter DT0.
.. math::
:label: 8.4-22
\Delta t = \text{DT}0
where
:math:`\Delta t_{\text{m}}` = Maximum allowable time-step length, s
If
.. math::
:label: 8.4-23
\text{FIRLM} \leq F_{\text{f}} < \text{SECLIM}
then
.. math::
:label: 8.4-24
\Delta t_{\text{m}} = \text{DTFAL}1
Similarly, if
.. math::
:label: 8.4-25
\text{SECLIM} \leq F_{\text{f}} < \text{THRLIM}
then
.. math::
:label: 8.4-26
\Delta t_{\text{m}} = \text{DTFAL}2
or, if
.. math::
:label: 8.4-27
\text{THRLM} \leq F_{\text{f}} \leq 1
then
.. math::
:label: 8.4-28
\Delta t_{\text{m}} = \text{DTFAL}3
This procedure serves two functions. First, the reduction in main time
step produces a corresponding reduction in the PRIMAR time step, thereby
allowing the loop model to continue a stable calculation on the
initiation of pin failure.Second, the reduced time step on initiation of
failure avoids excessive fuel motion in PLUTO2 and LEVITATE prior to the
time these failed pin models can initiate their own time‑step control
procedures.