.. _section-16.3:

Fuel Ejection from the Pins and Fuel-pin Disruption
---------------------------------------------------

The hydrodynamic in-pin model is connected to the channel model via the
mechanisms of molten fuel ejection into the channel and fuel-pin
disruption. The models describing these processes are described below.

.. _section-16.3.1:

Fuel Ejection via a Cladding Rupture
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The molten fuel/fission-gas mixture can be ejected from the pin cavity
into the channel in any axial cell where a cladding rupture has already
occurred. The original cladding failure location is determined in the
routines FAILUR and FUINIT, just before LEVITATE is initiated. Once
control passes to LEVITATE, additional nodes can fail, leading to the
increase of the original failure and, occasionally, the initiation of
new failures. Thus, two or more disjoint failures can be present in a
channel under certain circumstances. The fuel-pin rip enlargement is
performed in the routine LEIF.

The stress in the cladding in the axial cell is calculated as follows:

(16.3-1)

.. _eq-16.3-1:

.. math::

	\sigma_{\text{cl,i}} = \frac{P_{\text{ca,k}} \cdot R_{\text{ca,k}} - P_{\text{cl,i}} \cdot R_{\text{cl,os,i}}}{\Delta R_{\text{cl,i}}}

The ultimate tensile strength for the cladding is calculated using the
function :math:`\sigma_{\text{U}} \left( T \right)` for each radial cladding node in the axial
cell :math:`i`:

(16.3-2)

.. _eq-16.3-2:

.. math::

	\sigma_{\text{cl,i}}^{U} = \frac{\sigma^{U}\left( T_{\text{cl,os,i}} \right) \cdot \left( R_{\text{cl,os,i}} \
	- R_{\text{cl,in,i}} \right) + \sigma^{U}\left( T_{\text{cl,in,i}} \right) \cdot \left( R_{\text{cl,in,i}} \
	- R_{\text{cl,is,i}} \right)}{\left( R_{\text{cl,os,i}} - R_{\text{cl,is,i}} \right)}

The rupture will be extended to cell :math:`i` if:

(16.3-3)

.. _eq-16.3-3:

.. math::

	\sigma_{\text{cl,i}} > \sigma_{\text{cl,i}}^{U}

during any time step. All the nodes in the LEVITATE region are scanned
every cycle. Once the rupture has occurred in one axial node, that node
will continuously be checked for fuel ejection in each cycle.

The ejection of the molten fuel from the pin cavity into the channel is
performed in the LE1PIN routine. Only the nodes where the cladding has
been ruptured can eject fuel. Ejection will occur in these nodes if:

(16.3-4)

.. _eq-16.3-4:

.. math::

	P_{\text{ca,K}} > P_{\text{ch,i}}

and

(16.3-5)

.. _eq-16.3-5:

.. math::

	\alpha_{\text{fu,ca,k}} = \frac{\theta_{\text{fu,ca,k}}}{\theta_{\text{ca,k}}} > 0.1

An additional constraint, designed to avoid numerical problems related
to a fully incompressible channel configuration, is:

(16.3-6)

.. _eq-16.3-6:

.. math::

	\frac{\left( \theta_{\text{ch,i}} - \theta_{\text{ch,op,i}} \right) + \theta_{\text{fu,i}} \
	+ \theta_{\text{se,i}} + \theta_{\text{fu,i}}}{\theta_{\text{ch,i}}} \leq 0.95

i.e., the volume fraction of the incompressible components (crust, fuel,
steel, and chunks) should not be more than 95% of the original channel
volume. If all the conditions 16.3-4 through 16.3-6 are satisfied, the
injection calculation begins by calculating the in-pin pressure after
injection. An estimate of the amounts of fuel that can be ejected during
a typical LEVITATE time step indicated that at the time of failure, the
limiting factor in the ejection process is the inertia for the molten
fuel. For the time steps usually used in LEVITATE (approximately
2⨉10\ :sup:`-5` s) and the typical conditions for a loss-of-flow
situation, where the coolant channel is voided, it was found that the
pressure in the pin cavity will decrease initially by about 0.5% of the
pressure difference between the cavity and the channel. However, the
fuel in the cavity can be accelerated laterally quite rapidly, and the
inertial constraint becomes insignificant within a few milliseconds of
the failure time. Afterwards, the ejection process is rapid enough to
equilibrate the pressures in the cavity and channel within one or two
typical LEVITATE time steps.

(16.3-7)

.. _eq-16.3-7:

.. math::

	P_{\text{ca,k}}^{n + 1} = P_{\text{ca,k}}^{n} - \left( P_{\text{ca,k}} \
	- P_{\text{ch,i}} \right) \cdot \text{CIPINJ} \cdot \text{DTPLU}

The input constant CIPINJ has the recommended value :math:`2.5 \times 10^{4}`.
For this value of CIPINJ, the product CIPINJ ⨉ DTPLU has the value 0.5
and the pressure difference between the cavity and the coolant channel
will be equilibrated very rapidly. This model does not account for the
inertial effects mentioned above, which are present for a short time
after the failure has occurred. A more mechanistic model accounting for
these effects is currently being developed. With the assumption that the
fuel, fission gas and fuel vapor in the axial cell :math:`k` are homogeneously
mixed, we can now calculate the amount of fuel which has to be ejected
from the cavity in order to establish the pressure
:math:`P_{\text{ca,k}}^{n + 1}`:

(16.3-8)

.. _eq-16.3-8:

.. math::

	\begin{matrix}
	\Delta {\rho'}_{\text{fu,k}} = \left\{ \left\lbrack \theta_{\text{fi,ca,k}} - \theta_{\text{fu,ca,k}} \cdot \text{CMFU} \cdot \left( P_{\text{ca,k}} - P_{\text{fv,ca,k}} \right) \right\rbrack \cdot P_{\text{ca,k}} \\
	+ \left( R_{\text{fi}} \cdot {\rho'}_{\text{fi,ca}} + R_{\text{fv}} \cdot {\rho'}_{\text{fv,ca,k}} \right) \cdot T_{\text{fu,ca,k}} \right\} \cdot \rho_{\text{fu,ca,k}} \\
	\div \left( P_{\text{ca,k}} + f_{\text{fi,fu,k}} \cdot R_{\text{fi}} \cdot T_{\text{fu,ca,k}} \cdot \rho_{\text{fu,ca,k}} \\
	+ f_{\text{fv,fu,k}} \cdot R_{\text{fv}} \cdot T_{\text{fu,ca,k}} \cdot \rho_{\text{fu,ca,k}} \right) \\
	\end{matrix}

where the input variable CMFU represents the compressibility of the
molten fuel and

(16.3-9)

.. _eq-16.3-9:

.. math::

	f_{\text{fi,fu,k}} = \frac{{\rho'}_{\text{fi,ca,k}}}{{\rho'}_{\text{fu,ca,k}}}

(16.3-10)

.. _eq-16.3-10:

.. math::

	f_{\text{fi,fu,k}} = \frac{{\rho'}_{\text{fv,ca,k}}}{{\rho'}_{\text{fu,ca,k}}}

In the derivation of Eq. :ref:`16.3-8<eq-16.3-8>`, we used the assumption that, for
injection purposes, the fuel vapor behaves as a gas. Further fuel
vaporization can take place in the next cycle. It should also be noted
that :math:`\Delta {\rho'}_{\text{fu,k}}` represents the change in the generalized
density of the in-pin fuel. This change refers to the fuel in all the
pins in the subassembly, as explained in the corresponding PLUTO2
section.

The amount of fission gas and fuel vapor ejected is then calculated as

(16.3-11a)

.. _eq-16.3-11a:

.. math::

	\Delta {\rho'}_{\text{fi,k}} = f_{\text{fi,fu,k}} \cdot \Delta {\rho'}_{\text{fu,ca,k}}

(16.3-11b)

.. _eq-16.3-11b:

.. math::

	\Delta {\rho'}_{\text{fv,k}} = f_{\text{fv,fu,k}} \cdot \Delta {\rho'}_{\text{fu,ca,k}}

The fuel vapor mass ejected is then added to the mass of the liquid fuel
ejected. It is noteworthy that the fuel vapor pressure is used only for
the momentum calculation in the pin and for the fuel ejection. No fuel
vapor conservation equations are solved in the pin cavity.

The new channel partial pressures are then recalculated:

(16.3-12)

.. _eq-16.3-12:

.. math::

	P_{\text{fv,i}} = \frac{R_{\text{fv}} \cdot {\rho'}_{\text{fv,i}} \cdot T_{\text{fv,i}}}{\theta_{\text{vg,i}}^{n + 1}}

(16.3-13)

.. _eq-16.3-13:

.. math::

	P_{\text{sv,i}} = \frac{R_{\text{sv}} \cdot {\rho'}_{\text{sv,i}} \cdot T_{\text{sv,i}}}{\theta_{\text{vg,i}}^{n + 1}}

If :math:`\frac{\theta_{\text{Na,lq,i}}}{\theta_{\text{ch,op,i}}} \leq 0.3`

(16.3-14a)

.. _eq-16.3-14a:

.. math::

	P_{\text{fi,i}} = \frac{R_{\text{fi}} \cdot \left( {\rho'}_{\text{fi,i}} + \Delta {\rho'}_{\text{fi,k}} \right) \
	\cdot T_{\text{fi,i}}}{\theta_{\text{vg,i}}^{n + 1}}

If :math:`\frac{\theta_{\text{Na,lq,i}}}{\theta_{\text{ch,op,i}}} > 0.3`, then

(16.3-14b)

.. _eq-16.3-14b:

.. math::

	P_{\text{fi,i}} = \frac{- \theta_{\text{vg,i}}^{n + 1} + \sqrt{\left( \theta_{\text{vg,i}}^{n + 1} \right)^{2} \
	+ 4 \theta_{\text{Na,}l \text{q,i}} \cdot \text{CMNL} \cdot \left( {\rho'}_{\text{fi,i}} \
	+ \Delta {\rho'}_{\text{fi,ca,k}} \right) \cdot R_{\text{fi}} \cdot T_{\text{fi,i}} }}{2 \theta_{\text{Na,} l \text{q,i}} \cdot \text{CMNL}}

If :math:`\theta_{\text{Na},l \text{q}} = 0`, i.e., if the sodium present is in
the form of single-phase vapor:

(16.3-15a)

.. _eq-16.3-15a:

.. math::

	P_{\text{Na,i}} = \frac{R_{\text{Na}} \cdot {\rho'}_{\text{Na,i}} \cdot T_{\text{Na,i}}}{\theta_{\text{vg,i}}^{n + 1}}

If :math:`0 < \theta_{\text{Na}, l \text{q}}`, i.e., two-phase sodium is
present:

(16.3-15b)

.. _eq-16.3-15b:

.. math::

	P_{\text{Na,i}} = P_{\text{Na}}\left( T_{\text{Na,i}} \right)

where CMNL is the compressibility coefficient of the liquid sodium and
:math:`\theta_{\text{vg,i}}^{n + 1}` is:

(16.3-16)

.. _eq-16.3-16:

.. math::

	\theta_{\text{vg,i}}^{n + 1} = \theta_{\text{vg,i}} - \frac{\Delta {\rho'}_{\text{fu,ca,k}}}{\rho_{\text{fu,k}}}

The new tentative channel pressure is:

(16.3-17)

.. _eq-16.3-17:

.. math::

	P_{\text{ch,i}}^{n + 1} = P_{\text{Na,i}} + P_{\text{fi,i}} + P_{\text{sv,i}} + P_{\text{fv,i}}

If this pressure is below the pressure in the cavity after injection,
i.e.,

(16.3-18)

.. _eq-16.3-18:

.. math::

	P_{\text{ch,i}}^{n + 1} < P_{\text{ca,k}}^{n + 1}

then no further iterations are necessary and
:math:`P_{\text{ch,i}}^{n + 1}` remains the new pressure in the channel.
If Eq. :ref:`16.3-18<eq-16.3-18>` is not satisfied, too much material was ejected from the
cavity into the channel. A new pressure in the pin cavity is selected:

(16.3-19)

.. _eq-16.3-19:

.. math::

	P_{\text{ca,k}}^{n + 1} = \left( P_{\text{ca,k}}^{n} + P_{\text{ca,k}}^{n + 1} \right) \cdot 0.5

and the procedure outlined in Eqs. :ref:`16.3-7<eq-16.3-7>` through :ref:`16.3-17<eq-16.3-17>` is repeated
until the condition in 16.3-18 is satisfied.

It should be noted that in Eq. :ref:`16.3-14<eq-16.3-14a>`, the fission-gas pressure is
calculated by assuming that the fission gas injected has the same
temperature as the fission gas in the channel. This is necessary because
the fission gas and sodium have to be at the same temperature, which has
already been calculated in the routine LENAEN. The additional energy
carried by the injected fission gas, which in fact has the temperature
:math:`T_{\text{fu,ca,k}}` will be added to the fission
gas/sodium mixture in the next cycle, when solving the energy equation
for these components. Finally, the fuel and fission-gas densities in the
pin and channel are updated.

(16.3-20)

.. _eq-16.3-20:

.. math::

	{\rho'}_{\text{fu,ca,k}} = {\rho'}_{\text{fu,ca,k}} - \Delta{\rho'}_{\text{fu,ca,k}}

(16.3-21)

.. _eq-16.3-21:

.. math::

	{\rho'}_{\text{fi,ca,k}} = {\rho'}_{\text{fi,ca,k}} - \Delta{\rho'}_{\text{fi,ca,k}}

(16.3-22)

.. _eq-16.3-22:

.. math::

	{\rho'}_{\text{fu,i}} = {\rho'}_{\text{fu,i}} + \Delta{\rho'}_{\text{fu,ca,k}}

(16.3-23)

.. _eq-16.3-23:

.. math::

	{\rho'}_{\text{fu,i}} = {\rho'}_{\text{fi,i}} + \Delta{\rho'}_{\text{fi,ca,k}}

In the Release 1.1 version of LEVITATE a mechanistic calculation of the
in-pin fuel ejection has been added. This model can be used by setting
the input variable :sasinp:`INRAEJ`\ =1. If INRAEJ=0 the
ejection of the fuel into the channel is calculated as described above.

The mechanistic ejection model calculates the radial velocity of the
in-pin molten fuel/gas mixture, :math:`u_{\text{ca-ch,k}}`, at each axial
location where the cladding has been ruptured. This velocity is
recalculated each time step. The amounts of fuel and fission gas ejected
each time step are calculated as follows:

(16.3-23-1)

.. _eq-16.3-23-1:

.. math::

	\Delta{\rho'}_{\text{fu,k}} = {\rho'}_{\text{fu,k}} \cdot \frac{V_{\text{fuel ejected}}}{V_{\text{fuel cavity}}} \
	= {\rho'}_{\text{fu,k}} \cdot \frac{u_{\text{fu,ca-ch,k}} \cdot \Delta t}{\pi \cdot R_{\text{ca,k}}}

(16.3-23-2)

.. _eq-16.3-23-2:

.. math::

	\Delta{\rho'}_{\text{fi,k}} = {\rho'}_{\text{fi,k}} \cdot \frac{u_{\text{fu,ca-ch,k}} \cdot \Delta t}{\pi \cdot R_{\text{ca,k}}}

(16.3-23-3)

.. _eq-16.3-23-3:

.. math::

	\Delta {\rho'}_{\text{fv,k}} = {\rho'}_{\text{fv,k}} \cdot \frac{u_{\text{fu,ca-ch,k}} \cdot \Delta t}{\pi \cdot R_{\text{ca,k}}}

The pressure in the cavity is then calculated solving equation 16.3-8
for :math:`P_{\text{ca,k}}` and the pressure in the coolant channel is calculated
using Eqs. :ref:`16.3-12<eq-16.3-12>` through :ref:`16.3-17<eq-16.3-17>`. If

(16.3-23-4)

.. _eq-16.3-23-4:

.. math::

	P_{\text{ch,i}}^{n + 1} < P_{\text{ca,k}}^{n + 1}

then no further iterations are necessary. If 16.3-23-4 is not satisfied
then too much material was ejected from the cavity into the channel. The
radial velocity is reduced:

(16.3-23-5)

.. _eq-16.3-23-5:

.. math::

	u_{\text{fu,ca-ch,k}}^{n + 1} = u_{\text{fu,ca-ch,k}}^{n + 1} \cdot 0.5

and the procedure for pressure calculation is repeated.

The key element in the calculation described above is the radial fuel
velocity :math:`u_{\text{fu,ca-ch,k}}`, which is calculated as described below.

The radial velocity calculation first updates all radial velocities
considering only the radial acceleration :math:`a_{\text{fu,ca-ch,k}}`, and
ignoring temporarily the axial transport of radial momentum:

(16.3-23-6)

.. _eq-16.3-23-6:

.. math::

	u_{\text{fu,ca-ch,k}}^{n + 1} = u_{\text{fu,ca-ch,k}} + a_{\text{fu,ca-ch,k}} \cdot \Delta t

where :math:`a_{\text{fu,ca-ch,k}}` is the radial acceleration of the molten fuel
in the cavity in the axial cell :math:`k`. This acceleration is calculated using
the simplified geometry illustrated in :numref:`figure-16.3-1`. The pressure
difference between the pin cavity and the coolant channel is assumed to
act on the fuel contained in the shaded volume. The rip size is
considered to be of the same order of magnitude as the cavity radius.
The acceleration is calculated as follows:

(16.3-23-7)

.. _eq-16.3-23-7:

.. math::

	a_{\text{fu,ca-ch,k}} = \frac{\sum\text{Forces}}{\text{Mass}} = \frac{R_{\text{ca,k}} \
	\cdot z_{\text{k}}}{0.9 \cdot R_{\text{ca,k}}^{z} \cdot z_{\text{k}} \cdot \
	\rho_{\text{fu,k}}}\left\{ P_{\text{ca-ch,k}} - 0.5 \cdot \rho_{\text{fu,k}} \
	\cdot u_{\text{fu,ca-ch,k}} \\
	\cdot \left| u_{\text{fu,ca-ch,k}} \right| \cdot C_{\text{ORIFICE}} \
	- C_{\text{SHEAR}} \cdot \eta_{\text{fu,k}} \cdot \frac{u_{\text{fu,ca-ch,k}}}{R_{\text{ca,k}}} \right\}

where: :math:`\Delta P_{\text{ca-ch,k}} = P_{\text{ca,k}} - P_{\text{ch,i}}`

:math:`C_{\text{ORFICE}}` is an orifice coefficient used in the calculation of
the flow pressure drop across the rip. This coefficient is currently
zero in the code.

:math:`C_{\text{SHEAR}}` is a coefficient associated with the shear forces
exerted on the accelerating fuel. It can be determined from geometrical
considerations. The current value is 6.88. :math:`\eta_{\text{fu,k}}` is the
viscosity of the molten fuel in cell :math:`k`

As the fuel from the radial control volume is ejected into the channel,
it is replaced with new fuel from outside the control volume, assumed to
carry no radial momentum. Thus, the radial velocity of the fuel has to
be recalculated. into the channel, it is replaced with new fuel from
outside the control volume, assumed to carry no radial momentum. Thus,
the radial velocity of the fuel has to be recalculated.

(16.3-23-8)

.. _eq-16.3-23-8:

.. math::

	\begin{matrix}
	u_{\text{fu,ca-ch,k}} = u_{\text{fu,ca-ch,k}} \cdot \frac{V_{\text{control}} - V_{\text{ejected}}}{V_{\text{control}}} \\
	= u_{\text{fu,ca-ch,k}} \cdot \frac{0.9 R_{\text{ca,k}} - u_{\text{fu,ca-ch,k}} \cdot \Delta t}{0.9 R_{\text{ca,k}}} \\
	\end{matrix}

.. _figure-16.3-1:

..  figure:: media/image7.png
	:align: center
	:figclass: align-center

	Geometry for the Radial Acceleration Calculation

Finally, the radial velocity is recalculated to account for the axial
convection of radial momentum:

(16.3-23-9)

.. _eq-16.3-23-9:

.. math::

	u_{\text{fu,ca-ch,k}}^{n + 1} = \left\{ u_{\text{fu,ca-ch,k}}^{n + 1} \cdot {\rho'}_{\text{fu,ca,k}} \
	- \left( \left\lbrack {\rho'} \cdot u \cdot u_{\text{ca-ch}} \right\rbrack_{\text{fu,k+1}} \
	- \left\lbrack {\rho'} \cdot u \cdot u_{\text{ca-ch}} \right\rbrack_{\text{fu,k}} \right) \cdot \
	\frac{t}{z_{\text{k}}} \right\} \cdot \frac{1}{{\rho'}_{\text{fu,ca,k}}}

where the momentum convective terms are defined as follows:

(16.3-23-10)

.. _eq-16.3-23-10:

.. math::

	\left\lbrack {\rho'} \cdot u \cdot u_{\text{ca-ch}} \right\rbrack_{\text{fu,k}} = \begin{cases}
	{\rho'}_{\text{fu,ca,k-1}} \cdot u_{\text{fu,ca,k}} \cdot u_{\text{fu,ca-ch,k-1}} & \text{if } u_{\text{fu,ca,k}} \geq 0 \\
	{\rho'}_{\text{fu,ca,k}} \cdot u_{\text{fu,ca,k}} \cdot u_{\text{fu,ca-ch,k}} & \text{if } u_{\text{fu,ca,k}} < 0 \\
	\end{cases}

The mechanistic ejection model has been compared with the pressure
equilibration ejection model (:math:`\text{INEAEJ} = 0`) in several calculations. It
appears that the pressure equilibration ejection model provides
generally a good approximation to the results calculated by the
mechanistic model. The larger differences appear during high power
excursions and during the time immediately following the pin failure,
when the inertial forces accounted for in the mechanistic model delay
the early fuel ejection into the channel. It is expected that after
validation in several experiment analyses the mechanistic ejection model
will become the standard model to be used in SAS4A calculations.

.. _section-16.3.2:

Fuel-pin Disruption
~~~~~~~~~~~~~~~~~~~

When certain conditions are satisfied, the fuel pins can be totally
disrupted at certain axial locations. The decision process which
triggers the pin disruption is performed in the LE1PIN routine and has
been outlined in :numref:`section-16.2`. The nodes to be disrupted are flagged in
LE1PIN by setting :math:`\text{IFLAG} \left( I \right) = 9`. The disruption process itself is
performed in the LEDISR routine and is described below.

The LEDISR routine first checks whether a node that is to be disrupted
is within the physical region for fuel, fission gas, steel and fuel
chunks, since all these components are likely to be created via the
disruption process. If the node to be disrupted lies outside the fuel
physical region, for example, the fuel region will be extended so that
the molten fuel generated via the disruption process will be modeled
appropriately. Then, the generalized densities and the temperatures of
the components present in the channel are reset to account for the
addition of mass due to the pin disruption:

(16.3-24)

.. _eq-16.3-24:

.. math::

	\Delta{\rho'}_{\text{fu,i}} = {\rho'}_{\text{fu,ca,k}}

(16.3-25)

.. _eq-16.3-25:

.. math::

	\Delta{\rho'}_{\text{fi,i}} = {\rho'}_{\text{fi,ca,k}}

(16.3-26)

.. _eq-16.3-26:

.. math::

	\Delta{\rho'}_{\text{f} l \text{,i}} = {\rho'}_{\text{ffc,i}} + \frac{M_{\text{fu,i}} \
	\cdot N_{\text{PIN}}}{\Delta z_{\text{i}} \cdot \text{AXMX}}

(16.3-27)

.. _eq-16.3-27:

.. math::

	\Delta{\rho'}_{\text{s} l \text{,i}} = {\rho'}_{\text{sic,i}} + \frac{M_{\text{se,i}} \
	\cdot N_{\text{PIN}}}{\Delta z_{\text{i}} \cdot \text{AXMX}}

where :math:`M_{\text{fu,i}}` and :math:`M_{\text{se,i}}` represent the mass of solid
stationary fuel and steel, respectively, present in the axial node :math:`i`,
per pin. The temperatures are reset using an energy balance. For
example, the new fuel enthalpy is given by:

(16.3-28)

.. _eq-16.3-28:

.. math::

	h_{\text{fu,i}} = \frac{h_{\text{fu,i}} \cdot {\rho'}_{\text{fu,i}} + h_{\text{fu,ca,k}} \
	\cdot \Delta{\rho'}_{\text{fu,i}}}{{\rho'}_{\text{fu,i}} + \Delta{\rho'}_{\text{fu,i}}}

The new fuel temperature is obtained from :math:`h_{\text{fu,i}}`, using the
function :math:`T_{\text{fu}} \left( h \right)`. The velocities are reset using a momentum
conservation equation. For example, the new fuel velocity at boundary :math:`i`,
when node :math:`i` is disrupted, is calculated as follows:

(16.3-29)

.. _eq-16.3-29:

.. math::

	\begin{matrix}
	{u''}_{\text{fu,i}} = \left\lbrack u_{\text{fu,i}} \cdot \left( {\rho'}_{\text{fu,i-1}} + {\rho'}_{\text{s} l \text{,i-1}} \right) \cdot 0.5 \cdot \Delta z_{\text{i-1}} \\
	+ {u''}_{\text{fu,i}} \cdot \left( {\rho'}_{\text{fu,i-1}} + {\rho'}_{\text{s} l \text{,i-1}} \right) \cdot 0.5 \cdot \Delta z_{\text{i-1}} \\
	+ u_{\text{fu,ca,k}} \cdot 0.5 \cdot \Delta{\rho'}_{\text{fu,i}} \cdot \Delta z_{\text{i}} \right\rbrack \\
	\div \left\lbrack \left( {\rho'}_{\text{fu,i-1}} + {\rho'}_{\text{s} l \text{,i-1}} \right) \cdot 0.5 \cdot \Delta z_{\text{i-1}} \\
	+ \left( {\rho'}_{\text{fu,i}} + {\rho'}_{\text{s} l \text{,i}} \right) \cdot 0.5 \cdot \Delta z_{\text{i}} \\
	+ \Delta{\rho'}_{\text{fu,i}} \cdot 0.5 \cdot \Delta z_{\text{i}} \right\rbrack \\
	\end{matrix}

Disruption of node :math:`i` also affects the fuel velocity at the boundary :math:`i+1`:

(16.3-30)

.. _eq-16.3-30:

.. math::

	\begin{matrix}
	{u''}_{\text{fu,i+1}} = \left\lbrack {u'}_{\text{fu,i+1}} \cdot \left( {\rho'}_{\text{fu,i}} + {\rho'}_{\text{s} l \text{,i}} \right) \cdot 0.5 \cdot \Delta z_{\text{i}} \\
	+ {u''}_{\text{fu,i+1}} \left( {\rho'}_{\text{fu,i+1}} + {\rho'}_{\text{s} l \text{,i+1}} \right) \cdot 0.5 \cdot \Delta z_{\text{i+1}} \\
	+ u_{\text{fu,ca,k}} \cdot 0.5 \cdot \Delta{\rho'}_{\text{fu,i}} \cdot \Delta z_{\text{i}} \right\rbrack \\
	\div \left\lbrack \left( {\rho'}_{\text{fu,i}} + {\rho'}_{\text{s} l \text{,i}} \right) \cdot 0.5 \cdot \Delta z_{\text{i}} \\
	+ \left( {\rho'}_{\text{fu,i+1}} + {\rho'}_{\text{s} l \text{,i+1}} \right) \cdot 0.5 \cdot \Delta z_{\text{i+1}} \\
	+ \Delta{\rho'}_{\text{fu,i}} \cdot 0.5 \cdot \Delta z_{\text{i}} \right\rbrack \\
	\end{matrix}

The velocities :math:`{u'}_{\text{fu,i}}` and :math:`{u''}_{\text{fu,i}}` represent the molten
fuel velocities at the channel boundary :math:`i` and are described in more
detail in :numref:`section-16.4.2.1`.

Finally, the densities are reset as follows:

(16.3-31)

.. _eq-16.3-31:

.. math::

	{\rho'}_{\text{fu,i}} = {\rho'}_{\text{fu,i}} + \Delta{\rho'}_{\text{fu,i}}

(16.3-32)

.. _eq-16.3-32:

.. math::

	{\rho'}_{\text{fi,i}} = {\rho'}_{\text{fi,i}} + \Delta{\rho'}_{\text{fi,i}}

(16.3-33)

.. _eq-16.3-33:

.. math::

	{\rho'}_{\text{f} l \text{,i}} = {\rho'}_{\text{f} l \text{,i}} + \Delta{\rho'}_{\text{f} l \text{,i}}

(16.3-34)

.. _eq-16.3-34:

.. math::

	{\rho'}_{\text{s} l \text{,i}} = {\rho'}_{\text{s} l \text{,i}} + \Delta{\rho'}_{\text{s} l \text{,i}}

The partial pressures are then reset to account for the new gas volume
and gas densities in the disrupted node. A number of quantities related
to the cladding are set to zero before the end of the routine. These
quantities include the perimeter and thickness of cladding and the
amount of fuel and steel in the cladding crust.

.. _section-16.3.3:

Fuel Ejection via the Open Pin Stubs
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Once the fuel-pin disruption has occurred in a subassembly, the
remaining pin stubs can eject fuel and fission gas axially into the
disrupted region, in addition to the regular radial gas ejection. Fuel
and fission gas can also reenter the fuel-pin stubs when the pressure in
the disrupted region exceeds the pressure in the pin cavity. The
ejection (or reentry) of the molten fuel via the open pin stubs is
calculated by solving the momentum conservation equation for the cavity
half-cells adjacent to the disrupted region. This equation is solved in
the routine LE1PIN. It is derived in the same manner as described in the
PLUTO2 chapter for all regular in-pin momentum cells, but using the cell
geometry illustrated in :numref:`figure-16.3-2`. The momentum cell for the bottom
pin stub extends from the center of the cell :math:`\text{IDISBT-1}` to the end of the
stub. The pressure difference used in the momentum equation is
:math:`P_{\text{IDISBT}} - P_{\text{ca,IDISBT-1}}`, where :math:`P_{\text{IDISBT}}` is the
pressure in the disrupted region the convective flux coming *from* the
disrupted region is defined using the generalized density
:math:`{\rho'}_{\text{fu,IDISBT}}`, i.e., the fuel generalized density in the
disrupted region.

.. _figure-16.3-2:

..  figure:: media/image8.png
	:align: center
	:figclass: align-center

	Geometry Used in the Derivation of the Momentum Equations for Ejection via the Pin Stubs