.. _section-9.8.2:

Model Formulation
-----------------

.. _section-9.8.2.1:

Fission Gas
~~~~~~~~~~~

Reference 9.8‑2 reviews the current understanding of fission gas
generation, release, and transport in metallic fuel. DEFORM‑5 does not
currently model steady-state fission gas generation and migration, but
assumes that an independent assessment has been made to determine the
fission gas content in the fuel element at the initial steady-state. The
fission gas inventory is specified as input in the form of a axially
uniform fuel element pressure at the |SAS| initial reference
temperature. The steady-state fission gas mass is then calculated from
plenum and available pore volumes using the ideal gas law. Transient
fuel element and internal fuel pressures are calculated using the
calculated fission gas mass and the transient temperatures in the ideal
gas formulation.

.. _section-9.8.2.2:

Fuel/Cladding Eutectic Alloy Formation
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Uranium‑ and uranium/plutonium‑based metallic fuels interact chemically
with iron-based cladding to form low-melting-point eutectic alloys. For
typical metallic fuel element geometry and for the transient sequences
analyzed with |SAS|, the impact of this phenomena is to form a
molten region at the fuel/cladding interface if the fuel is contacting
the cladding, if the contact temperature has been raised to a sufficient
level to cause eutectic alloy formation, and if the contact temperature
has been held at the elevated level long enough to form the eutectic
alloy, given the formation rate at that temperature. :numref:`figure-9.8.2-1` shows
a binary phase diagram for uranium and iron [9.8‑3]. This diagram shows
that in equilibrium, liquid material may be formed at a temperature as
low as 725°C for an alloy that is 89 wt. % uranium.

.. _figure-9.8.2-1:

..  figure:: media/image2.png
	:align: center
	:figclass: align-center
	:width: 6.57292in
	:height: 7.22500in

	Uranium-Iron Phase Diagram [9.8‑3].

In |SAS| analyses, where the fuel/cladding behavior model is
concerned mainly with predicting margin to cladding failure, cladding
failure time, and cladding failure location, the primary importance of
eutectic alloy formation at the fuel cladding interface is an effective
thinning of the cladding, and a lessening of its ability to contain the
internal, hydrostatic pressure due to released fission gas. To quantify
the cladding thinning in DEFORM‑5, the correlation developed by Bauer
[9.8‑4] for the eutectic penetration rate as a function of absolute
temperature is used. This correlation is

.. math::
    :label: 9.8.2-1

	\dot{r} = 1 \times 10^{-6} \exp\left( 22.847 - \frac{27624}{T} \right)

except in the range 1353 K (1080°C) to 1506 K (1233°C) where

.. math::
    :label: 9.8.2-2

	\dot{r} = 1 \times 10^{-6} \left\lbrack 922 + 2.9265 \left( T - 1388 \right) - 0.21522 \left( T - 1388 \right)^{2} + 0.0011338 \left( T - 1388 \right)^{3} \right\rbrack

Where :math:`\dot{r}` is the eutectic penetration rate in m/s, and
:math:`T` is the absolute temperature in Kelvins. This correlation is
based on 1) tests in which iron capsules were dipped into molten uranium
and uranium/iron alloy baths [9.8‑5], 2) tests of EBR-II Mark-II driver
fuel [9.8‑6, 9.8‑7, 9.8‑8], and 3) tests of ternary alloy fuel (U‑19Pu‑10Zr)
clad with stainless steel (D9) [9.8‑4]. Waltar and Kelman [9.8‑5] associated
the rate increase in the range from 1353 K to 1506 K with the formation
characteristics of the compound UFe\ :sub:`2`. :eq:`9.8.2-1` and
:eq:`9.8.2-2` are used in DEFORM‑5 to calculate the cladding
penetration as a function of time and the effective cladding thickness
at each axial location and in each |SAS| channel.

.. _section-9.8.2.3:

Clad Strain
~~~~~~~~~~~

In DEFORM‑5, a calculation of the cladding strain is performed to
provide input to the cladding failure assessment. The basic approach
adopted for calculating cladding strain is that developed by DiMelfi and
Kramer [9.8‑9, 9.8‑10] in their studies on plastic flow in steel cladding,
and implemented in the FPIN2 computer code (see :numref:`section-11.3.6`). The
hoop stress in the cladding, :math:`\sigma_{\theta}`, is determined for a thin shell
under internal pressure loading:

.. math::
    :label: 9.8.2-3

	\sigma_{\theta} = \left\lbrack P - P_{\text{ch}} \right\rbrack\frac{a}{h}

where :math:`P` is the internal pressure, :math:`P_{\text{ch}}` is the
coolant channel pressure, :math:`a` is the inner cladding radius, and
:math:`h` is the cladding thickness. Assuming the hydrostatic loading of
a thin, cylindrical shell, the hoop stress is converted to an equivalent
stress:

.. math::
    :label: 9.8.2-4

	\sigma = \frac{\sqrt{3}}{2}\sigma_{\theta}

and this stress is then used in the flow equation developed by DiMelfi
and Kramer:

.. math::
    :label: 9.8.2-5

	\dot{\epsilon} = {\dot{\epsilon}}_{\text{oos}}\left( \frac{\sigma}{\sigma_{\text{so}}} \right)^{m}\exp\left( \frac{- Q}{\text{RT}} \right)

where :math:`\dot{\epsilon}` is the equivalent strain rate,
:math:`{\dot{\epsilon}}_{\text{oos}}` is a material constant,
:math:`\sigma` is the equivalent stress, :math:`\sigma_{\text{so}}` is
the saturation stress, :math:`m` is the stress exponent,
:math:`\frac{Q}{R}` is the creep activation temperature, and :math:`T`
is the absolute steel temperature.

This formulation was chosen because it allows the use of the same model
for all cladding types of interest. Through the use of appropriate
cladding parameters, SS316, D9, and HT‑9 can all be modeled within the
DEFORM‑5 context. (In the absence of data, the following parameters for
SS 316 are used for D9).

For type 316 stainless steel, Kramer and coworkers [9.8‑11] derived the
following parameters:

:math:`m` = 5.35

:math:`\frac{Q}{R} = 38,533` K

:math:`{\dot{\epsilon}}_{\text{oos}} = 1.062 \times 10^{14}` s\ :sup:`-1`

:math:`\sigma_{\text{so}} = \left( 2000 - 9.12 T \right) \left( 92,000 - 40.2 T \right)` Pa

For type HT‑9 stainless steel, Kramer and coworkers [9.8‑12] derived the
following parameters:

:math:`m = 2.263`

:math:`\frac{Q}{R} = 36,739` K

:math:`{\dot{\epsilon}}_{\text{oos}} = 5.1966 \times 10^{10}` s\ :sup:`-1`

:math:`\sigma_{\text{so}} = 3.956 \times 10^{-3} \times 2.12 \times 10^{11} \left( 1.144 - 4.856 \times 10^{-4} T \right)` Pa

.. _section-9.8.2.4:

Cladding Failure
~~~~~~~~~~~~~~~~

The determination of cladding breach, or margin to cladding breach, is
based on the eutectic thinning of the cladding and the reduced ability
to contain the internal pressure. Besides the eutectic thinning, the
cladding wall thickness is reduced as plastic flow takes place.
Different accident scenarios and pin conditions lead to different modes
of breach, whether through cladding penetration or plastic strain or a
combination.

In the DEFORM‑5 calculation, the time‑dependent cladding damage fraction
is calculated based on the time remaining to breach, following Kramer
and DiMelfi's assessment [9.8‑13] of the TDC‑2 transient damage
correlation for cold‑worked 316 (and also for D9) stainless steel fuel
pin cladding [9.8‑14]:

.. math::
    :label: 9.8.2-6

	\ln t_{\text{r}} = A + B\ln{\ln{\left( \frac{\sigma^{*}}{\sigma} \right) + \frac{Q_{1}}{T} + \text{TRAMP} + C\tanh{\phi t}}}

where

:math:`t_{\text{r}}` = time to rupture, hours

:math:`A` = -43.06

:math:`B` = 7.312

:math:`C` = -1.73

:math:`Q_{1}` = 41339

:math:`D` = 1000

:math:`E` = 200

:math:`F` = 0.438

:math:`\sigma^{*} = 775 - \left\lbrack 387.5 - 387.5\tanh\left( \frac{D - TI}{E} \right)\right\rbrack\tanh\left( \frac{\phi t}{2.0} \right) \\ + 125\left\lbrack \tanh\left( \frac{\sigma}{550} \right)^{10} \right\rbrack\left\lbrack 1 - \tanh\left( \frac{\text{TI}}{583} - F \right)^{25} \right\rbrack\tanh\left( \frac{\phi t}{2.5} \right)`

:math:`\text{TRAMP} = -0.28 + 1.18 \tanh{\left(-0.5 \lbrack \ln{\dot{T}} - 1 \rbrack \right)}`

:math:`\sigma` = hoop stress, MPa (use thin wall formula with midwall tube diameter)

:math:`\phi t` = fluence n/cm\ :sup:`2`, E > 0.1 MeV, x10\ :sup:`22`

:math:`T` = transient temperature, K

:math:`\text{TI}` = steady-state irradiation temperature, K

:math:`\dot{T}` = transient temperature ramp rate, K/sec

In DEFORM‑5, the fast fluence is calculated from the input local linear
power, the channel flux-to-power ratio, and number of full-power days of
steady-state irradiation.

For HT‑9, the time remaining to breach is assumed to be given by [9.8‑15]:

.. math::
    :label: 9.8.2-7

	t_{\text{r}} = \theta\exp\left( \frac{Q}{RT} \right)

where

:math:`t_{\text{r}}` = time to rupture, seconds

:math:`\ln\theta = A + B\ln{\ln\left( \frac{\sigma^{*}}{\sigma} \right)}`

:math:`A = - 34.8 + \tanh{\left( \frac{\sigma - 200}{50} \right) + C}`

:math:`B = \frac{12}{1.5 + 0.5\tanh\left( \frac{\sigma - 200}{50} \right)}`

:math:`C = - 0.5\left\lbrack 1 + \tanh\left( \frac{\sigma - 200}{50} \right) \right\rbrack 0.75\left\lbrack 1 + \tanh\left( \frac{\dot{T} - 58}{17} \right) \right\rbrack`

:math:`\sigma^{*}` = 730 MPa

:math:`\sigma` = hoop stress, MPa

:math:`T` = transient temperature, K

:math:`\dot{T}` = heating rate, K/s

:math:`Q` = 70170 cal/mole

:math:`R` = 1.986 cal/mole-K