.. _section-11.3:
Metal Fuel Element Models
-------------------------
Although the FPIN2 code was originally developed for the purpose of
analyzing oxide fuels, it has been modified for the analysis of metallic
fuels. The metal fuel version of the code includes numerous improvements
to the basic mechanical analysis models that reflect the experience
gained from the in-pile TREAT tests on EBR-II irradiated prototypic fuel
of the IFR concept. In addition, models for molten cavity formation, the
large gas plenum, molten fuel extrusion, and fuel-cladding eutectic
formation are provided to complement the fuel element mechanics
calculation.
There are many differences between metal and oxide fuels that affect the
transient response of the fuel elements. The existence of a large
fission gas plenum in metal-alloy fueled elements plays a major role in
determining the internal pin pressure during a transient. The sodium
bond and comparatively large radial fuel-cladding gap requires
consideration of expelling sodium into the plenum as the fuel expands
and the gap closes. Metal fuel thermal conductivity is an order of
magnitude larger than oxide fuel, and that leads to a flatter radial
temperature profile and a more rapid spreading of the region of molten
fuel. Since the metal fuel solidus temperature is well below the
cladding melting temperature, in many cases fuel can be expected to melt
entirely before the cladding fails. Due to the high fuel thermal
conductivity, melting usually begins at or near the top of the fuel
column with the axial profile of the fuel centerline temperature more
closely following the coolant temperature profile than that for oxide
fuel. This prevents large cavity pressurization effects in metallic
fuels. In addition, a low melting temperature eutectic alloy is expected
to form in metallic fuels between the fuel and cladding at temperatures
below the anticipated cladding failure. This eutectic formation allows
the fuel and cladding to slip freely in the axial direction and can lead
to an accelerated cladding failure at elevated temperatures. Cracking is
not a significant phenomenon in metallic fuel pins, but metal fuel
swelling can be important.
The current version of the FPIN2 coupled with SASSYS/SAS4A includes
various model modifications and additions that address these differences
between oxide and metal fuels. The following sections contain
discussions of these additions and modifications made to FPIN2 for the
analysis of metal-fueled elements. The most significant extension to the
code is the inclusion of a model for the plenum region above the fuel
column as discussed in :numref:`section-11.3.1`. Extrusion of molten fuel into the
plenum as an additional axial expansion mechanism is discussed in
:numref:`section-11.3.2`. Basic mechanical properties of metal-alloy fueled
elements are outlined in :numref:`section-11.3.3`. The constitutive equations for
fuel creep and a non-equilibrium approximation to fuel swelling are
presented in :numref:`section-11.3.4` and :numref:`section-11.3.5`, respectively. The
plastic flow behavior of three types of cladding materials commonly used
in metallic fuel elements of the IFR concept is discussed in :numref:`section-11.3.6`. Fuel-cladding eutectic formation and its impact on the
mechanical analysis of fuel elements are summarized in :numref:`section-11.3.7`.
And finally, the fuel element failure formulation is outlined in :numref:`section-11.3.8`.
.. _section-11.3.1:
Central Cavity and Plenum Models
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Internal pin pressure is calculated in FPIN2 using straightforward
volume accounting models of the central cavity and the gas plenum. The
radial boundary of the cavity for each axial segment is assumed to be at
the point where the fuel has reached the solidus temperature. Elements
inside the cavity boundary are assumed to be in a hydrostatic state of
stress equal to the cavity pressure and are dropped out of stress-strain
calculation. These elements occupy a volume in proportion to their
density and contribute to the volume and mass balance iteration that
determines the cavity pressure.
Metallic fuel elements lead to relatively low transient induced internal
pin pressure since (1) fuel porosity easily accommodates the fuel volume
expansion upon melting, (2) fuel melting begins at or near the top of
fuel column, and (3) metallic fuels have less restrictive flow path for
fission gas passage to the plenum. Thus, the large gas plenum is very
effective in mitigating any large pressure buildup inside the fuel
elements. The pin plenum model of FPIN2 is based on the following
assumptions:
1. The internal pin pressure, :math:`p_{\text{g}}`, is considered uniform
throughout the interior of a metallic fuel element.
2. The plenum region is assumed to be at the uniform temperature. [#4]_
3. The thermal expansion of the fill sodium and its expulsion from the
fuel-cladding gap as the gap closes is accounted for. However, sodium
compressibility is assumed to be negligible at the expected pressure
range.
The first level of iteration, or outer loop, in the mechanical analysis
section of the FPIN2 is the search for the internal pin pressure
:math:`p_{\text{g}}`. According to the calculation sequence, the temperatures
of the cavity and plenum materials are known at the beginning of a new
time step before the pin pressure is found. Using these temperatures,
the following can be calculated:
1. Cavity boundary changes due to fuel melting,
1. Mass of the gas in the cavity, including the gas released upon
melting,
2. Mass-average temperature of the gas in the cavity,
3. Volume available from porosity released after fuel melting,
4. The thermal expansion of the sodium in the plenum.
The algorithm for calculating the plenum and cavity pressures are
outlined below for the case where the two volumes are connected and
share a common pressure, :math:`p_{\text{g}}`. The pin pressure, :math:`p_{\text{g}}`,
is calculated by considering the two relationships between the gas
volume and pressure that must be satisfied: the ideal gas law, and the
interaction between pressure and cavity boundary displacements. Pin
molten-cavity and plenum gases are considered as separate entities with
individual compositions and temperatures. Since the masses of the gases
in the cavity, :math:`m_{\text{g}}^{C}`, and the plenum, :math:`m_{\text{g}}^{P}`, as
well as the temperatures of the cavity gas, :math:`T_{\text{g}}^{C}`, and the
plenum gas, :math:`T_{\text{g}}^{P}`, are known, the ideal gas law reduces to
an inverse relationship between the total gas volume,
:math:`V_{\text{g}}^{\text{IGL}}`, and the pressure as follows
(11.3‑1)
.. _eq-11.3-1:
.. math::
V_{\text{g}}^{\text{IGL}} = \frac{m_{\text{g}}^{C} R^{C} T_{\text{g}}^{C} + m_{\text{g}}^{P} R^{P} T_{\text{g}}^{P}}{p_{\text{g}}}
where :math:`R^{C}` and :math:`R^{P}` are the gas constants for cavity
and plenum gases, respectively. The second relationship involves the
geometrical calculation of the volume available to the gas that depends
on displacement of the solid fuel, cladding, and plenum tube. The liquid
fuel and sodium are assumed to be incompressible. Since the gas in the
cavity and plenum are considered separately, the total volume has two
components
(11.3‑2)
.. _eq-11.3-2:
.. math::
V_{\text{g}}^{\text{MECH}}\left( p_{\text{g}} \right) = V_{\text{g}}^{C}\left( p_{\text{g}} \right) + V_{\text{g}}^{P}\left( p_{\text{g}} \right)
where :math:`V_{\text{g}}^{C}` is the available cavity volume that is
calculated using the finite element analysis of solid fuel and cladding,
and :math:`V_{\text{g}}^{P}` is the volume available for the gas in the plenum.
A negative value of :math:`V_{\text{g}}^{C}` means that the molten fuel
material is extruded upward into the plenum.
The two values for the total volume, :math:`V_{\text{g}}^{\text{IGL}}` and
:math:`V_{\text{g}}^{\text{MECH}}`, are equal when the correct value of
:math:`p_{\text{g}}` is found. In FPIN2, Newton's method is used to fine the
value of :math:`p_{\text{g}}` such that
(11.3‑3)
.. _eq-11.3-3:
.. math::
f\left( p_{\text{g}} \right) = V_{\text{g}}^{\text{IGL}} - V_{\text{g}}^{\text{MECH}} = 0
The Newton iteration equation
(11.3‑4)
.. _eq-11.3-4:
.. math::
p_{\text{g}}^{i + 1} = p_{\text{g}}^{i} - \frac{f\left( p_{\text{g}}^{i} \right)}{f'\left( p_{\text{g}}^{i} \right)}
requires the derivative of :math:`V_{\text{g}}^{\text{IGL}}` and
:math:`V_{\text{g}}^{\text{MECH}}` with respect to :math:`p_{\text{g}}` (:math:`i` is the
iteration counter). The derivative of :math:`V_{\text{g}}^{\text{IGL}}` is
easily calculated from the ideal gas law and the derivative of
:math:`V_{\text{g}}^{\text{MECH}}` is found by forming a finite difference
quotient from two calculations of :math:`V_{\text{g}}^{\text{MECH}}` at two
neighboring values of :math:`p_{\text{g}}`.
The total mass of sodium, :math:`m_{\text{Na}}^{T}`, and mass of the gas
in the plenum, :math:`m_{\text{g}}^{P}`, are obtained from the initial
conditions. The inventory of sodium within the fuel element consists of
sodium in the fuel-cladding gap, :math:`m_{\text{Na}}^{G}`, and in the
plenum, :math:`m_{\text{Na}}^{P}` :
(11.3‑5)
.. _eq-11.3-5:
.. math::
m_{\text{Na}}^{T} = m_{\text{Na}}^{P} + m_{\text{Na}}^{G}
where
(11.3‑6)
.. _eq-11.3-6:
.. math::
m_{\text{Na}}^{P} = \rho_{\text{Na}}\left( T^{P} \right) \pi r_{\text{P}}^{2} h_{\text{Na}}^{P}
(11.3‑7)
.. _eq-11.3-7:
.. math::
m_{\text{Na}}^{G} = \sum_{\text{j}} \rho_{\text{Na}}\left( T_{\text{j}} \right) \pi \left( r_{\text{ci}}^{2} - r_{\text{fo}}^{2} \right)_{\text{j}} \Delta z_{\text{j}}
The symbols in Eqs. :ref:`11.3-6<eq-11.3-6>` and :ref:`11.3-7<eq-11.3-7>` are defined as follows:
:math:`\rho_{\text{Na}}` = sodium density,
:math:`T^{P}` = plenum temperature,
:math:`r_{\text{p}}` = cladding inner radius in plenum region,
:math:`h_{\text{NA}}^{P}` = height of sodium column in plenum,
:math:`r_{\text{ci}}` = cladding inner radius for axial segment *j*,
:math:`r_{\text{fo}}` = fuel outer radius for axial segment *j*,
:math:`\Delta z_{\text{j}}` = height of fuel axial segment *j*.
The mass of the plenum gas is determined by using the ideal gas law:
(11.3‑8)
.. _eq-11.3-8:
.. math::
m_{\text{g}}^{P} = \frac{P^{P} \pi r_{\text{P}}^{2}\left( h^{P} - h_{\text{Na}}^{P} \right)}{R^{P}T^{P}}
where :math:`P^{P}` is the plenum pressure and :math:`h^{P}` is initial
plenum height. The plenum pressure is obtained from a user specified
reference pressure at a reference temperature as follows
(11.3‑9)
.. _eq-11.3-9:
.. math::
P^{P} = P_{\text{ref}}\frac{T^{P}}{T_{\text{ref}}}
During the transient analysis, the cavity pressure algorithm requires
the volume in the plenum available to the plenum fission gas and to the
material extruded from the molten cavity. This value is calculated by
subtracting the volume of the sodium in the plenum from the total volume
of the deformed plenum tube. Thus,
(11.3‑10)
.. _eq-11.3-10:
.. math::
V_{\text{g}}^{P} = V^{P} - V_{\text{Na}}^{P}
The total plenum volume, :math:`V^{P}`, is found by assuming that the
plenum tube can be treated as a thermoelastic thin shell. The volume of
sodium in the plenum is found using the equation
(11.3‑11)
.. _eq-11.3-11:
.. math::
V_{\text{Na}}^{P} = \frac{m_{\text{Na}}^{T} - m_{\text{Na}}^{G}}{\rho_{\text{Na}}\left( T^{P} \right)}
.. _section-11.3.2:
Extrusion of Molten Fuel into the Plenum
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The procedure in FPIN2 for calculating the behavior of the molten cavity
materials uses a simple hydrostatic model based on volume accounting. As
discussed in previous section, a volume and mass balance iteration
determines the cavity pressure. For transients in which initial fuel
melting is observed below the fuel-plenum interface, plenum and cavity
pressure equations are solved separately, leading generally to a cavity
pressure larger than the plenum pressure. Once melting reaches the top
axial segment, the two pressures eventually equilibrate to give a common
pin pressure. The temporary imbalance between the cavity and plenum
pressures plays an important role in the dynamics of the molten fuel
extruded into the plenum.
The extrusion of the molten cavity material into the plenum is
calculated at the completion of the pressure iteration and consists of
calculating the excess volume of the cavity materials over the volume
available in the molten cavity. Both the radial and axial dimensions of
the cavity section for each axial segment are based on the deformed
geometry. The materials in the cavity include liquid fuel at a
temperature above the liquidus temperature, solid fuel at a temperature
between the solidus and liquidus, and the fission gas that is assumed to
obey the ideal gas law and is released to the cavity in proportion to
the melt fraction. Open porosity is available to the fission gas volume
when the fuel reaches the solidus temperature, but closed grain boundary
porosity is release d in proportion to the metal fraction.
For low-burnup fuel pins, the bond sodium in the fuel-cladding radial
gap affects the extrusion of molten fuel into the plenum. The metallic
fuel can melt completely across the fuel radial cross section in an
axial segment. Although the large gap in fresh metallic fuels is not
expected to close by the time of 100% fuel melting, the bond sodium in
the gap is expected to escape into the plenum as the molten fuel slumps
to fill up the gap due to the large density difference between fuel and
sodium. This local slumping lessens the fuel extrusion; however, metal
fuels at as low as 0.35 at.% burnup contain sufficient fission gas to
result in net upward axial fuel motion [11-12]. Two options are provided
in FPIN2 to model this behavior. The first option is to assume that the
fuel does not slump and expel the bond sodium. In this case, the fuel
outer boundary is assumed to remain at the position it was in at the
time of 100% melting. The second (default) option allows the molten fuel
to move out to the cladding, with the bond sodium being expelled into
the plenum and the gap volume being added to the molten cavity.
To handle the second option, a gap closure model has been added to move
the molten fuel out to the cladding over several time steps as the bond
sodium is expelled into the plenum and the gap volume becomes available
to the molten cavity materials. In order to maintain the stability of
the numerical calculation, a varying closure rate is coded in FPIN2 in
which the fraction of the gap closed per time step is specified.
Somewhat artificial time step dependence of this transition is necessary
because the temperature drop across the gap may be as large as 30K.
Since the radial temperature profile across the fuel element is
relatively flat, an excessively rapid reduction of the gap :math:`\Delta T` may cause
computational difficulties such as fuel refreezing during a power rise.
.. _section-11.3.3:
Basic Metal Fuel Properties
~~~~~~~~~~~~~~~~~~~~~~~~~~~
*Thermal Properties*: In the interfaced mode, the SAS-FPIN2 integrated
model uses SASSYS/SAS4A routines to calculate fuel and cladding thermal
material properties as discussed in :numref:`Chapter %s<section-10>`. In the stand-alone mode,
FPIN2 uses its own built-in correlations for metallic IFR fuel that have
been developed by the "Metallic Fuel Properties Working Group" [11-12].
Melting related properties for the metallic fuels such as solidus and
liquidus temperatures, heat of fusion, and volume change on melting are
calculated using identical algorithms in FPIN 2 and SASSYS/SAS4A.
*Elastic Properties*: Tensile test data at room temperature for uranium
and its alloys indicate yielding at very low stresses. The equation used
in FPIN2 code for the Young's modulus is as follows
(11.3‑12)
.. _eq-11.3-12:
.. math::
E = 0.12 \cdot 10^{6} \left( 1 - 1.2 p \right) \left( 1 - 0.754 \cdot 10^{-3} \left( T - 588 \right) \right)
where :math:`E` is Young's modulus (in Bars), :math:`p` is fractional porosity, and
:math:`T` is temperature :math:`\left( K \right)`. This equation gives a value of one-tenth the
handbook [11-7] value and should more properly be called a tangent
modulus. However, for the expected monotonic loading of most FPIN2
calculations, the fuel is treated as a pseudo-elastic-plastic material
with plastic straining handled as secondary creep and yielding is
included as part of the "elastic" behavior. In the code, Poisson's ratio
is calculated from
(11.3‑13)
.. _eq-11.3-13:
.. math::
\nu = 0.27 \left( 1 - 0.8 p \right) \left( 1 + 0.854 \cdot 10^{- 3} \left( T - 588 \right) \right)
where :math:`\nu` is Poisson's ratio, :math:`p` is fractional porosity, and
:math:`T` is temperature :math:`\left( K \right)`.
*Linear Thermal Expansion*: In the metal fuel version of FPIN2, the
thermal expansion is expressed in terms of handbook linear thermal
expansion data, :math:`\Delta L/L_{0}` [11-7]. Using the data rather than the
coefficient of thermal expansion, :math:`\alpha`, results in a more accurate
accounting of the total expansion from room temperature to melting since
it automatically includes the expansion at solid-to-solid phase
transitions. The thermal expansion data and phase transition expansions
have been approximated in the code as three straight line segments with
breaks at the two solid-to-solid phase change temperature associated
with the beginning and ending of the transformation to pure :math:`\gamma` phase
solid solution. The equations used for the binary fuel are
(11.3‑14)
.. _eq-11.3-14:
.. math::
\frac{\Delta L}{L_{0}} = \begin{cases}
1.695 \cdot 10^{- 5}\left( T - 293 \right) & T < 900 \\
0.0103 + 7 \cdot 10^{- 5}\left( T - 900 \right) & 900 < T < 1000 \\
0.0173 + 2.12 \cdot 10^{- 5}\left( T - 1000 \right) & T > 1000
\end{cases}
And, the equations for the ternary fuel are
(11.3‑15)
.. _eq-11.3-15:
.. math::
\frac{\Delta L}{L_{0}} = \begin{cases}
1.67 \cdot 10^{-5} \left( T - 293 \right) & T < 864 \\
0.0095 + 6.7 \cdot 10^{- 5}\left( T - 864 \right) & 864 < T < 950 \\
0.0153 + 2.12 \cdot 10^{- 5}\left( T - 950 \right) & T > 950 \\
\end{cases}
where and :math:`T` is the temperature :math:`\left( K \right)`.
.. _section-11.3.4:
Secondary Creep of Metallic Fuels
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Uranium alloys deform plastically under constant load at elevated
temperatures. The secondary, or minimum, creep rate is defined as the
steady-state rate that is attained under these conditions. The equations
added to FPIN2 for metal fuels are obtained from a review of the data
and theory for modeling secondary creep of U-Pu-Zr alloys that are of
interest to the IFR concept [11-13]. Application of creep data to fuel
pin analysis often requires correlating the data using mathematical
functions of the governing variables. Such correlations range from
purely empirical equations to theoretical equations involving
fundamental physical properties. In FPIN2, and intermediate approach is
preferred where theoretical models are used to obtain the form of the
equation, but the parameters are determined form the creep data rather
than the fundamental properties. Such an approach allows an
interpolation of the data and gives reasonable confidence in
often-required extrapolations beyond the database.
The particular form of the secondary creep equation used in FPIN2 is the
form used to represent creep of UO\ :sub:`2` [11-12]. The total plastic
strain rate, :math:`\dot{\varepsilon}`, is given by
(11.3‑16)
.. _eq-11.3-16:
.. math::
\dot{\varepsilon} = C_{1}\left( \dot{F},d \right)\sigma \exp{\left( - \frac{Q}{RT} \right)} + C_{2}\left( \dot{F} \right) \sigma^{n} \exp{\left( - \frac{Q}{RT} \right)} + C_{3} \left( T \right) \sigma \dot{F}
where
:math:`\dot{\varepsilon}` = secondary creep rate, s\ :sup:`-1`
:math:`\sigma` = equivalent stress, MPa
:math:`R` = universal gas constant, (1.987 cal/g-mole-K)
:math:`T` = temperature, K
:math:`Q` = activation energy, (cal/g-mole)
:math:`d` = grain size, µm
:math:`\dot{F}` = fission rate, (fissions/cm\ :sup:`3`-s)
:math:`C` = material functions.
In Eq. :ref:`11.3-16<eq-11.3-16>`, the first term represents diffusional creep, the second
term dislocation creep, and the third fission-induced creep. Here, the
low temperature deformation mechanisms are neglected such as dislocation
glide and twinning that may occur in :math:`\alpha` uranium at temperatures less
than 675K. On the other hand, the expected dependence of in-reactor
creep on the fission rate is considered.
Some of the parameters in Eq. :ref:`11.3-16<eq-11.3-16>` are estimated from knowledge of
the structure of material and the values that these parameters take for
similar materials. At low temperatures, it is reasonable to postulate
that the creep deformation of uranium alloys is dominated by the
deformation of the :math:`\alpha` uranium matrix. This is particularly true if the
other phases present are coarsely dispersed. When the :math:`\alpha` uranium matrix
deformation is dominant, the activation energy is expected to be the
self-diffusion energy in :math:`\alpha` uranium. The creep activation energy :math:`Q=52000`
cal/g-mole has been determined experimentally for :math:`\alpha` uranium. It is
shown that the temperature dependence of creep can be deduced using such
a single activation energy and that the value of n=4.5 is appropriate in
the high stress range [11-13]. Using these values for :math:`Q` and :math:`n`,
:math:`C_{1}` and :math:`C_{2}` in Eq. :ref:`11.3-16<eq-11.3-16>` are chosen to minimize
the least squares error between the calculated and measures values of
:math:`\log\left( z \right)` where :math:`z` is the Zener-Holloman parameter
(11.3‑17)
.. _eq-11.3-17:
.. math::
z = \dot{\varepsilon} e^{\frac{Q}{\text{RT}}}
The result is
(11.3‑18)
.. _eq-11.3-18:
.. math::
&C_{1} = 0.5 \cdot 10^{4} &\ & \text{MPa}^{- 1}\text{s}^{- 1} \\
&C_{2} = 6.0 &\ & \text{MPa}^{- 1}\text{s}^{- 1}
When the calculated time to 2% strain using these values is compared to
data from creep tests of U-Pu-Zr alloys, results are found to be in
acceptable agreement. Examination of experimental data suggests that the
grain-size dependence of :math:`C_{1}` can be included as
(11.3‑19)
.. _eq-11.3-19:
.. math::
\begin{align}
C_{1} = \left( \frac{160}{d} \right)^{2} 0.5 \cdot 10^{4} && \text{MPa}^{- 1}\text{s}^{- 1}
\end{align}
for grain sizes near 160 μm.
At temperatures above about 900K, the structure of the U-Pu-Zr alloys no
longer contains an :math:`\alpha` uranium matrix. The transformations between 900K
and 975K are complex and depend strongly on composition. Beyond 975K,
however, the solid phase consists of a solid solution of uranium,
plutonium, and zirconium over a wide range of compositions. In FPIN2, it
is assumed that the creep rate in the intermediate temperature range
falls between the creep rate of :math:`\alpha` uranium and solid solution :math:`\gamma`
phase. It is also assumed that creep of uranium alloys is governed by
dislocation glide in the :math:`\gamma` regime. The phenomenological creep equation
is of the form
(11.3‑20)
.. _eq-11.3-20:
.. math::
\dot{\varepsilon} = C_{4} \sigma^{3} e^{\left( - \frac{Q_{\gamma}}{RT} \right)}
where :math:`Q_{\gamma}` is the creep activation energy in the :math:`\gamma` phase, and the
parameter :math:`C_{4}` is, in general, a function of composition and
fission rate. Based on tracer diffusion data and/or additional creep
data, :math:`Q_{\gamma}` is approximated as the activation energy for creep of
the pure :math:`\gamma` uranium solvent
(11.3‑21)
.. _eq-11.3-21:
.. math::
\begin{align}
Q_{\gamma} = 28500 && \text{cal/g-mole}
\end{align}
The constant :math:`C_{4}` is chosen to fit data at 973K as
(11.3‑22)
.. _eq-11.3-22:
.. math::
\begin{align}
C_{4} = 8.0 \cdot 10^{- 2} && \text{MPa}^{- 3} \text{s}^{- 1}
\end{align}
Since the data is somewhat limited and the Eqs. :ref:`11.3-16<eq-11.3-16>` and :ref:`11.3-20<eq-11.3-20>` are
consistent with the creep of alloys with a high percentage of uranium,
FPIN2 uses these expressions for all metal fuels.
Experience with earlier versions of the FPIN2 reveals that the
evaluation of Eqs. :ref:`11.3-16<eq-11.3-16>` and :ref:`11.3-20<eq-11.3-20>` uses a substantial fraction of
the computing time. Since the terms involving the exponential function
depend only on the temperature, in the current version such terms are
evaluated only at the beginning of the mechanics calculation. Therefore,
Eq. :ref:`11.3-16<eq-11.3-16>` is coded in the following form
(11.3‑23)
.. _eq-11.3-23:
.. math::
\dot{\varepsilon} = \sigma \left( A + B \sigma^{n - 1} \right)
where :math:`A` and :math:`B` are previously evaluated constants.
The creep equations given above are the same as those given in the
Metallic Fuels Handbook [11-7]. However, fission-induced creep term is
not included in the FPIN2 because it does not contribute significantly
to the fuel stains on the time scale of the transients that the code has
been designed to analyze.
.. _section-11.3.5:
Fission Induced Gas Swelling of Metal Fuels
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Irradiated metal fuels experience rapid, large-scale swelling when
subjected to overheating. From the safety perspective, this transient
swelling could provide an inherent, self-limiting, negative reactivity
feedback mechanism. In FPIN2, a transient swelling model has been
incorporated to get an estimate for the magnitude of this
fission-gas-induced swelling. This non-equilibrium model is based on
diffusive growth of grain boundary bubbles. [#5]_
The distribution of fission gas retained in the fuel matrix is specified
as input in FPIN2, and the remainder of the gas is assumed to be in
solution or in small bubbles within the fuel grains. Fission gas that is
retained in the fuel during steady-state irradiation provides a source
for expansion of both solid and liquid fuel during overheating. The
amount of gas in the pin plenum is also important since the plenum
pressure is a major contributor to cladding loading and, therefore,
cladding failure.
The following assumptions are the basis of the transient swelling model
of FPIN2 for the analysis of the metallic fuels:
1. All "retained gas" (as measured from experiments on small samples of
irradiated fuel) is in solution within the fuel or in the form of
small bubbles within the fuel grains or on the grain boundaries.
1. A certain fraction of this gas (input parameter) is contained in
grain boundary bubbles, fixed in number and all of the same initial
radius (0.1 microns).
2. The open porosity contains fission gas in equilibrium with the plenum
pressure.
3. Only the grain boundary bubbles contribute to the swelling of solid
fuel.
In order to use this model, a relationship between the swelling strain,
:math:`\varepsilon^{s}`, the mean stress, :math:`\sigma_{m}`, and the fuel temperature,
:math:`T`, is required as explained in :numref:`section-11.2.1.2.3`. Studies on
gas-bubbles growth mechanisms in metallic fuels indicate [11-14] that
the transient fuel swelling is dominated by diffusive growth of grain
boundary bubbles that is given by
(11.3‑24)
.. _eq-11.3-24:
.. math::
\frac{\text{dr}}{\text{dt}} = \frac{\Omega D_{\text{gb}}w \sin^{3}{\theta}}{2k T r^{2} L \eta} \phi
where
:math:`r` = bubble radius in the grain boundary,
:math:`D_{\text{gb}}` = grain boundary diffusion coefficient,
:math:`w` = the boundary thickness,
:math:`\Omega` = the atomic (or molecular) volume, and
:math:`kT` = thermal energy.
The overpressure, :math:`\phi`, in Eq. :ref:`11.3-24<eq-11.3-24>` is defined as
(11.3‑25)
.. _eq-11.3-25:
.. math::
\phi = p_{\text{g}} + \sigma_{\text{m}} - \frac{2\gamma}{\rho}
and it represents the excess of gas pressure, :math:`p_{\text{g}}`, over the
sum of the mean stress, :math:`\sigma_{\text{m}}`, and the surface tension
restraint. This restraint is defined in terms of the specific surface
free energy, :math:`\gamma`, and the radius of curvature, :math:`\rho`. For the
grain-boundary bubbles,
(11.3‑26)
.. _eq-11.3-26:
.. math::
\rho = \frac{r}{\sin{\theta}}
where :math:`\theta` is the intersection angle between the bubble surface and the
grain boundary. In FPIN2, the fuel swelling rate is determined from Eq.
:ref:`11.3-24<eq-11.3-24>` for the bubble growth rate and from the assumed bubble density.
The remaining constants in the above equations are geometric constants
related to the bubble volume and the bubble spacing [11-14]. The gas in
the bubbles is assumed to satisfy the ideal gas law so that
(11.3‑27)
.. _eq-11.3-27:
.. math::
p_{\text{g}} = \frac{mRT}{V_{\text{b}}}
where :math:`m` is the fixed mass of the gas per bubble and :math:`V_{\text{b}}` is
the volume of the bubbles. The mass of the gas in a bubble is determined
by using the equilibrium conditions at the pretransient time when the
bubble radius is assumed to be :math:`r_{\text{o}}`. Thus
(11.3‑28)
.. _eq-11.3-28:
.. math::
m = \frac{V_{\text{bo}}}{RT_{\text{o}}}\left( - \sigma_{\text{mo}} + \frac{2\gamma}{r_{\text{o}}} \right)
Since all of the grain boundary gas is assumed to reside in the closed
porosity bubbles, the number density of bubbles, :math:`N`, is given by
(11.3‑29)
.. _eq-11.3-29:
.. math::
N = \frac{F_{\text{g}}}{m}
where :math:`F_{\text{g}}` is the grain boundary fission gas density with units
of g per cm\ :sup:`3` of fuel. For this simple model, only the swelling
of the grain boundary bubbles is considered so that
(11.3‑30)
.. _eq-11.3-30:
.. math::
\varepsilon^{s} = \frac{1}{3}\left( \frac{\Delta V}{V_{\text{o}}} \right) = \frac{1}{3} N\left( V_{\text{b}} - V_{\text{bo}} \right)
These equations, when combined, give the complete specification of the
transient swelling model in the form
(11.3‑31)
.. _eq-11.3-31:
.. math::
g\left( \varepsilon^{s}, \sigma_{\text{m}}, T \right) = 0
The treatment of this constitutive equation for swelling in finite
element mechanics is discussed in :numref:`section-11.2.2`.
.. _section-11.3.6:
Generalized Plastic Flow Behavior of Cladding Materials
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The tensile properties of the cladding materials (HT9, D9, and Type 316
stainless steel) are not frequently used in analyses of fuel element
performance during normal operation, as designs are usually such that
stresses imposed do not challenge the points where yielding should
occur. However, under transient conditions at elevated temperatures the
applied stresses can result in general yielding. This behavior is
modeled in FPIN2 by using a unified plastic deformation model that
includes both rate-independent deformation (classical plasticity) and
rate-dependent deformation (creep). The equations describing the
generalized plastic flow behavior of Type 316 stainless steel, D9, and
HT9 cladding are presented in the following sections. The yield strength
and ultimate strength of the materials can be calculated from the
plastic flow equations by applying them to the geometry of a tensile
test.
.. _section-11.3.6.1:
Type 316 SS and D9 Cladding
^^^^^^^^^^^^^^^^^^^^^^^^^^^
The deformation behavior of Type 316 stainless steel and D9 cladding can
be described over a broad range of temperature and strain-rate with the
set of equations shown below. These equations are used in calculations
of the strength and deformation at temperatures ranging from room
temperature to 1675 K under a variety of loading conditions. In this
formalism, the true equivalent flow stress, :math:`\sigma_{\text{e}}`, is described
by the equation
(11.3‑32)
.. _eq-11.3-32:
.. math::
\sigma_{\text{e}} = \sigma_{\text{s}} - \left( \sigma_{\text{s}} - \sigma_{\text{l}} \right) \exp\left( - \frac{\hat{\varepsilon}}{\varepsilon^{*}} \right)
where :math:`\sigma_{\text{l}}` is yield stress of fully annealed unirradiated
material, and :math:`\sigma_{\text{s}}` is the saturation value of the flow stress
that is asymptotically approached at large values of plastic strain.
Ongoing true plastic strain is incorporated in the "hardness" parameter,
:math:`\hat{\varepsilon}` (input in the model), as this strain is
accumulated. The hardness parameter also contains contributions from
prior cold work, irradiation hardening, and softening caused by
annealing, all scaled as true plastic strain. Consequently,
:math:`\hat{\varepsilon} = 0` for the fully annealed unirradiated
material, and :math:`\hat{\varepsilon} = 0.223` for 20% cold-worked
material (default value in the code). Finally, the value of the
parameter :math:`\varepsilon^{*}` is determined by the initial
hardening rate :math:`\theta_{\text{l}}` of fully annealed material
:math:`\left( \hat{\varepsilon} = 0 \right)` which is given by
(11.3‑33)
.. _eq-11.3-33:
.. math::
\theta_{\text{l}} = \frac{\sigma_{\text{s}} - \sigma_{\text{l}}}{\varepsilon^{*}}
The initial hardening rate, :math:`\theta_{\text{l}}`, is found to be only
temperature dependent through the shear modulus; so, the rate and
temperature dependencies of :math:`\varepsilon^{*}` is obtained from the above
equation.
The yield stress and saturation flow stress are rate and temperature
dependent in accordance with the equations
(11.3‑34)
.. _eq-11.3-34:
.. math::
\frac{\sigma_{\text{s}}}{G} = \frac{\sigma_{\text{so}}}{G}\left\lbrack 1 - \exp\left( - \left( \frac{{\dot{\varepsilon}}_{\text{p}}}{{\dot{\varepsilon}}_{\text{os}}} \right)^{mK} \right) \right\rbrack^{1/K}
(11.3‑35)
.. _eq-11.3-35:
.. math::
\frac{\sigma_{\text{l}}}{G} = \frac{\sigma_{\text{lo}}}{G}\left\lbrack 1 - \exp\left( - \left( \frac{{\dot{\varepsilon}}_{\text{p}}}{{\dot{\varepsilon}}_{\text{ol}}} \right)^{mK} \right) \right\rbrack^{1/K}
where :math:`G` is the temperature dependent shear modulus,
:math:`{\dot{\varepsilon}}_{\text{p}}` is the equivalent plastic strain rate
(the :math:`\frac{\text{d}{\overline{\varepsilon}}^{p}}{\text{dt}}` term
in Eq. :ref:`11.3-26<eq-11.3-26>`), :math:`m` and :math:`K` are constants, and :math:`\sigma_{\text{so}}`,
:math:`\sigma_{\text{lo}}`, :math:`{\dot{\varepsilon}}_{\text{os}}` and
:math:`{\dot{\varepsilon}}_{\text{ol}}` are temperature dependent
functions. The constant :math:`m` is the rate sensitivity (the reciprocal of the
stress-exponent n in a power-law creep equation), and :math:`K` is a
non-physical fitting parameter that governs the sharpness of the
transition between rate-dependent flow and rate-independent flow. At a
given temperature, in the high strain-rate limit, the equation for the
saturation stress reduces to
(11.3‑36)
.. _eq-11.3-36:
.. math::
\frac{\sigma_{\text{s}}}{G} = \frac{\sigma_{\text{so}}}{G}
whereas in the low strain-rate limit, it becomes
(11.3‑37)
.. _eq-11.3-37:
.. math::
\frac{\sigma_{\text{s}}}{G} = \frac{\sigma_{\text{so}}}{G}\left( \frac{{\dot{\varepsilon}}_{\text{p}}}{{\dot{\varepsilon}}_{\text{os}}} \right)^{m}
which reflects the typical power-law creep behavior observed for these
materials at high temperatures and low strain rates. The equation for
the yield stress behaves similarly. The temperature dependencies of
:math:`\sigma_{\text{so}}` and :math:`\sigma_{\text{lo}}` are in large part eliminated by
dividing by :math:`G`. However, :math:`{\dot{\varepsilon}}_{\text{os}}` and
:math:`{\dot{\varepsilon}}_{\text{ol}}` reflect a temperature dependence
of high-temperature creep of the form:
(11.3‑38)
.. _eq-11.3-38:
.. math::
{\dot{\varepsilon}}_{\text{os}} = {\dot{\varepsilon}}_{\text{oos}} \exp\left( - Q/RT \right)
(11.3‑39)
.. _eq-11.3-39:
.. math::
{\dot{\varepsilon}}_{\text{ol}} = {\dot{\varepsilon}}_{\text{ool}} \exp\left( - \frac{Q}{RT} \right)
where :math:`{\dot{\varepsilon}}_{\text{oos}}` and
:math:`{\dot{\varepsilon}}_{\text{ool}}` are constants; Q, R, and T are
the creep activation energy, gas constant, and absolute temperature,
respectively. The values for the parameters in all of the above
equations are:
:math:`G = 92.0 - 4.02 10^{-2} T` (GPa)
:math:`\frac{\theta_{\text{l}}}{G} = 3.66 \times 10^{-2}`
:math:`\frac{\sigma_{\text{so}}}{G} = 2.00 \times 10^{-2} - 9.12 \times 10^{-6} T`
:math:`\frac{\sigma_{\text{lo}}}{G} = 2.06 times 10^{-3} + 7.12 \times 10^{-1} T`
:math:`m = \frac{1}{5.35} = 0.187`
:math:`\frac{Q}{R} = 38533` (K)
:math:`{\dot{\varepsilon}}_{\text{oos}} = 1.062 \times 10^{14}` (s\ :sup:`-1`)
:math:`{\dot{\varepsilon}}_{\text{ool}} = 3.794 \times 10^{12}` (s\ :sup:`-1`)
:math:`K = 2.0`
The combination of the Eq. :ref:`11.3-32<eq-11.3-32>` and the above strain-rate and
temperature laws allow one to generate the complete rate and temperature
dependent true-stress/true-strain curves. The yield stress,
:math:`\sigma_{\text{l}}`, in these equations can be compared to the 0.2% offset
yield stress. The ultimate strength, while not specifically denoted in
the equations, can be determined by differentiating the Eq. :ref:`11.3-32<eq-11.3-32>` in
accordance with the construction of true stress, :math:`\sigma_{\text{u}}`, that
corresponds to the engineering ultimate strength:
(11.3‑40)
.. _eq-11.3-40:
.. math::
\frac{\text{d}\sigma}{\text{d}\varepsilon} \bigg\rvert_{\varepsilon_{\text{pu}}} = \theta_{\text{l}} \exp\left( - \frac{\varepsilon_{\text{pu}}}{\varepsilon^{*}} \right) = \sigma_{\text{u}}
where :math:`\varepsilon_{\text{pu}}` is the true plastic strain at the
ultimate strength which corresponds to the uniform elongation. Solving
the above equation and Eq. :ref:`11.3-32<eq-11.3-32>` simultaneously gives :math:`\sigma_{\text{u}}`
and :math:`\varepsilon_{\text{pu}}`.
.. _section-11.3.6.2:
HT9 Cladding
^^^^^^^^^^^^
The deformation behavior of the martensitic-ferritic stainless steel HT9
cladding can be described over a broad range of temperature and
strain-rate with the set of equations shown below. These equations are
incorporated in FPIN2 and used in calculations of strength and ductility
of this alloy at temperatures ranging from room temperature to 1110 K
under a variety of loading conditions. The high temperature creep
behavior is adequately described by an equation of the Dorn power-law
form:
(11.3‑41)
.. _eq-11.3-41:
.. math::
\frac{{\dot{\varepsilon}}_{\text{p}}}{{\dot{\varepsilon}}_{\text{oos}}} = \left( \frac{E}{\sigma_{\text{so}}} \right)^{n} \left( \frac{\sigma_{\text{s}}}{E} \right)^{n} \exp\left( - \frac{Q_{\text{c}}}{\text{kT}} \right)
In this equation, :math:`{\dot{\varepsilon}}_{\text{p}}` is the steady-state
equivalent creep rate normalized by the constant
:math:`{\dot{\varepsilon}}_{\text{oos}}`, :math:`\sigma_{\text{s}}` is the
equivalent applied stress in the creep test normalized by the constant
:math:`\sigma_{\text{so}}`, :math:`E` is the temperature dependent Young's modulus,
:math:`Q_{\text{c}}` is the creep activation energy, :math:`k` is Boltzmann's
constant, and :math:`T` is absolute temperature. The temperature dependence of
:math:`E` is expressed as
(11.3‑42)
.. _eq-11.3-42:
.. math::
\begin{align}
E\left( T \right) = 2.12 \cdot 10^{11}\left\lbrack 1.144 - 4.856 \cdot 10^{- 4} T \right\rbrack && \left( \text{Pa} \right)
\end{align}
and the remaining parameters in the creep equation are given by
.. math::
\begin{align}
{\dot{\varepsilon}}_{\text{oos}} = 5.1966 \cdot 10^{10} && \left( \text{s}^{- 1} \right)
\end{align}
.. math::
\frac{\sigma_{\text{so}}}{E} = 3.956 \cdot 10^{- 3}
.. math::
n = 2.263
.. math::
\begin{align}
\frac{Q_{\text{c}}}{k} = 36739 && \left( \text{K} \right)
\end{align}
The flow-stress/strain-hardening behavior can be described by an
equation of the type
(11.3‑43)
.. _eq-11.3-43:
.. math::
\frac{\sigma_{\text{e}}}{E} = \frac{\sigma_{\text{s}}}{E} - \frac{\sigma_{\text{s}} - \sigma_{\text{l}}}{E} \exp\left( - \frac{{\overline{\varepsilon}}_{\text{p}}}{\varepsilon^{*}} \right)
where :math:`\sigma_{\text{e}}` is the equivalent flow stress at some value of true
equivalent plastic strain :math:`{\overline{\varepsilon}}^{p}` (defined in Eq.
:ref:`11.3-26<eq-11.3-26>`), measured from the reference state of as-heat-treated material
:math:`\left( {\overline{\varepsilon}}^{p} = 0 \right)`. As
described in :numref:`section-11.3.6.1`, :math:`\theta_{\text{l}}` is the yield stress of
as-heat-treated material, :math:`\varepsilon^{*}` is a temperature dependent
parameter extracted from the hardening rate at the ultimate strength,
and :math:`\sigma_{\text{s}}` is the saturation stress, or steady-state flow stress,
approached at a large plastic strain and assumed compatible with the
stress in the above steady-state creep equation. It has been found that
assuming that the yield stress :math:`\sigma_{\text{l}} = 0.8 \sigma_{\text{s}}`
throughout allows the generation of flow stress-strain curves that agree
well with data. The true stress at the ultimate strength (maximum load),
after differentiating Eq. :ref:`11.3-43<eq-11.3-43>`, is given by
(11.3‑44)
.. _eq-11.3-44:
.. math::
\frac{\sigma_{\text{u}}}{E} = \frac{\sigma_{\text{s}} - \sigma_{\text{l}}}{E} \varepsilon^{*} \exp\left( \frac{{- \varepsilon}_{\text{pu}}}{\varepsilon^{*}} \right)
where :math:`\varepsilon_{\text{pu}}` is the true strain at the ultimate strength
corresponding to the uniform elongation. Simultaneously solving this
equation in conjunction with the flow stress equation evaluated at
:math:`\varepsilon_{\text{pu}}`, the following expression is obtained for the quantity
:math:`\varepsilon^{*}`:
(11.3‑45)
.. _eq-11.3-45:
.. math::
\varepsilon^{*}\left( T \right) = 0.12733 - 3.5027 \cdot 10^{- 4} T + 2.9934 \cdot 10^{- 7}T^{2}
An equation to model the transition between high-rate, low-temperature,
rate independent flow behavior and creep behavior has the form
(11.3‑46)
.. _eq-11.3-46:
.. math::
\frac{\sigma_{\text{s}}}{E} = \frac{\sigma_{\text{so}}}{E}\left\{ 1 - exp\left\lbrack \left( \frac{{\overline{\varepsilon}}_{\text{p}} \exp\left( \frac{Q_{\text{t}}}{kT} \right)}{\varepsilon_{\text{oos}}} \right)^{K/n} \right\rbrack \right\}^{1/K}
Because of microstructural changes that occur during long-time creep
testing of HT9, and the effects they have on flow stress, there is not a
smooth transition between short term tensile behavior and creep
behavior. However, setting the non-physical fitting parameter as :math:`K =
0.2`, and with all the other parameters set as indicated above, Eq.
:ref:`11.3-46<eq-11.3-46>` follows the tensile data quite well, and reduces to the
power-law creep equation at very low strain rates.
For temperatures above the completion of the :math:`\gamma` transformation
temperature (1233 K), the flow stress model in FPIN2 assumes that the
deformation rate for HT9 cladding is the same as that given by the Type
316 SS and D9 cladding equations in :numref:`section-11.3.6.1`. A simple mixture
rule is used to calculate the deformation rates in the :math:`\alpha`-:math:`\gamma` transition
region (1110K-1233K).
.. _section-11.3.7:
Fuel-Cladding Eutectic Formation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
An additional complication for metallic fuels is the formation of a low
melting point eutectic alloy between the fuel and the cladding that can
contribute to fuel element failure during transient overheating events.
The eutectic alloy forms due to interdiffusion of fuel and cladding
constituents at the fuel-cladding interface. It melts at a temperature
lower than the fuel and cladding solidus temperatures.
.. _section-11.3.7.1:
Eutectic Penetration of the Cladding
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The primary effect of the liquid eutectic alloy formation on fuel
element failure is thinning of the cladding wall. The liquid does not
further damage the remaining cladding tendon by mechanisms such as
liquid metal embrittlement. Experimental data documenting this effect
comes from constant temperature time-to-failure experiments on
irradiated EBR-II fuel elements and form out-of-pile dipping tests of
penetration of iron by molten uranium-iron eutectic alloy. These data
have recently been reviewed and the following correlation has been
recommended [11-15]
(11.3‑47)
.. _eq-11.3-47:
.. math::
\dot{M} = \begin{cases}
0 && T < 1353 \text{ K} \\
922 + 2.93 \left( T - 1388 \right) - 0.215 \left( T - 1388 \right)^{2} + 0.001134 \left( T - 1388 \right)^{3} && 1355 \text{ K} \leq T \leq 1506 \text{ K} \\
\exp\left( 22.85 - \frac{27624}{T} \right) && T > 1506 \text{ K}
\end{cases}
where :math:`\dot{M}` is the melt rate in μm/s, and :math:`T` is the
absolute temperature in Kelvins. The major feature of the data and these
equations is that the melt rate is very rapid for temperatures above
1353 K. For typical cladding dimensions, this means that cladding will
completely melt-through is somewhat less than one second once molten
fuel reaches 1353K. On the other hand, for the transient rate of concern
for a typical FPIN2 application, negligible cladding melt-through occurs
until this "rapid eutectic attack" temperature is reached. [#6]_ The Eq.
:ref:`11.3-47<eq-11.3-47>` is used in cladding failure criterion subroutine of FPIN2 to
calculate wall thinning.
.. _section-11.3.7.2:
Eutectic Release of Cladding Axial Restraint
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
When fuel and cladding are in contact, FPIN2 has an option that allows
that fuel and cladding to either slip freely axially or to be locked
together. In the latter option, it is assumed that the fuel and cladding
remain locked, either by metallurgical bonding or by friction, at
temperatures below the assumed eutectic alloy melting threshold,
:math:`T_{\text{th}}` (an input in the code). At temperatures above
:math:`T_{\text{th}}`, the fuel and cladding are assumed to slip freely because
of the liquid phase that forms at the fuel cladding interface. This
discontinuity in the cladding axial constraint model of FPIN2 at the
assumed eutectic alloy melting temperature causes a sudden drop in
stress in both fuel and cladding elements as the fuel expands slightly
in the axial direction and contracts in the radial direction due to
elastic recovery.
When the fuel-cladding gap is open, the fuel and cladding axial
displacements are calculated independently. When the fuel and cladding
are locked together, the increments in axial strain in the fuel are
assumed to be equal to the increments in axial strain in the cladding.
This is handled in mechanics calculation by modifying the terms in the
stiffness matrix. Specifically, the finite-element algorithm is solved
for the resultant fuel and cladding axial forces by combining the fuel
and cladding stiffness matrices in a non-iterative procedure. In this
procedure, the two equations for the axial displacements are replaced by
a single equation for their joint displacement. The redundant equation
is then replaced with an identity so that the size of the stiffness
matrix and all the associated coding remains the same. When the sudden
release of cladding restraint occurs at eutectic alloy melting point,
fairly large changes take place in the axial displacements. However,
experience has shown that, due to the robust procedure used, this
substantial sudden change in the mechanics causes no computational
difficulties.
.. _section-11.3.8:
Fuel Element Failure
~~~~~~~~~~~~~~~~~~~~
The failure of the cladding due to temperature and pressure transients
is essentially independent of fuel type except that failure in metal
fueled elements is augmented by eutectic formation which penetrates and,
in effect, thins the cladding as discussed in the previous section.
Also, the loss in cladding stiffness is not considered in calculating
the continued deformation of the cladding in the oxide fuel version of
FPIN2. For conditions where simultaneous eutectic erosion and plastic
deformation are important, oxide fuel version under-predicts the
permanent cladding strains at the time of failure. Therefore, the FPIN2
analysis of cladding deformation has been modified to account for this
eutectic thinning and to determine the permanent cladding strain
remaining after a transient more accurately.
The technique that has been implemented in the FPIN2 cladding
deformation model to account for the eutectic attack uses an effective
ligament thickness ratio to decease the contribution that the material
stresses in a given element make to the overall balance between internal
and external forces. Elements that are fully liquid are assumed to have
a ligament thickness ratio of zero and those that are solid are assumed
to have ligament ratio of one. An element that has partially liquefied
then has a ratio equal to the fraction of the element that is still
solid. In other words, the effect of eutectic formation is included in
FPIN2 by considering only the thickness of unaffected cladding that is
available to carry the load. This model has the additional benefit that
it can easily be modified to account for other damage mechanisms such as
cavitation and void growth during tertiary creep. In the current version
of the FPIN2, however, these additional mechanisms are not considered.
For transients with relatively short time scale, the eutectic
penetration is a factor only if the cladding temperature exceeds the
"rapid eutectic attack" temperature (1353K) where the penetration rate
increases by three orders of magnitude. The change in cladding wall
thickness above 1353K is calculated using Eq. :ref:`11.3-47<eq-11.3-47>` for Type 316 SS,
D9, and HT9 cladding. At temperatures below 1353K, the rate of eutectic
penetration is generally insignificant and assumed to be zero.
Cladding rupture is predicted in the code by using the life fraction
criteria. The TCD-2 life fraction criterion [11-16] is used for the fuel
elements with D9 and Type 316 stainless steel cladding; and the HEDL
transient Dorn parameter correlation [11-17] is used for the fuel
elements with HT9-cladding. The fuel adjacency effect in the TDC-2
correlation has been neglected. The life fraction over a time step,
:math:`\text{dt}`, is calculated from the rupture time, :math:`t_{\text{r}}`, for the
instantaneous average cladding temperature, :math:`T_{\text{c}}`, and hoop
stress, :math:`\sigma_{\text{c}}`. Life fractions are summed in the usual way so
that cladding failure is predicted to occur at time :math:`t_{\text{f}}` when
(11.3‑48)
.. _eq-11.3-48:
.. math::
\int_{0}^{t_{\text{f}}}{\frac{1}{t_{\text{r}}\left( \sigma_{\text{c}}, T_{\text{c}} \right)}} \text{dt} = 1
The location of the failure is the location of first axial cladding
segment to reach a life fraction of 1.0. The hoop stress in the cladding
tendon is calculated by the thin-shell equations consistent with the
methodology used in developing the life fraction correlations. The
thickness of this tendon is reduced by the amount of eutectic attack.
.. rubric:: Footnotes
.. [#4]
In stand-alone FPIN2 calculation, this plenum temperature is assumed
to be equal to the coolant outlet temperature.
.. [#5]
FPIN2 also provides an equilibrium swelling model as an option.
.. [#6]
The rate at 1000K is about 10\ :sup:`-2` µm/s, which would require 10
hours to completely penetrate the full cladding wall thickness.