.. _section-14.1:
Introduction and Overview
-------------------------
.. _section-14.1.1:
Historical Background and Description of the Physical Model
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The PLUTO2 module calculates the post‑failure fuel motion and sodium
voiding in subassemblies experiencing an overpower condition leading to
significant fuel melting, fuel‑pin failures and fuel ejection into
unvoided or partially voided coolant channels. The degree of fuel‑pin
disruption is limited to cladding ruptures through which molten fuel can
be ejected into the coolant channels. Thus, this model will be
appropriate for treating the early post‑pin‑failure phase of higher ramp
rate transient overpower (TOP) accidents and the entire post‑failure
phase of lower ramp rate TOPs if no complete pin disintegration or
extensive cladding melting occurs. Of additional importance is the
module's application to the early post‑pin‑failure fuel motion and
sodium voiding in unvoided or partially voided subassemblies
experiencing a Loss‑of‑Flow driven TOP (LOF'd'TOP). Once complete pin
disintegration, high fuel vapor pressures, or molten cladding motion
have to be considered, a switch to the LEVITATE module, which is
designed to treat these phenomena, is made. This compatibility between
PLUTO2 and LEVITATE is important for a consistent continuation of the
analysis of the transient. This compatibility did not exist between the
corresponding SAS3D modules SAS/FCI [14‑1] and SLUMPY [14-2] which
could never be used sequentially to treat the same fuel pin.
The PLUTO2 module can be used in all situations for which the SAS/FCI
module of SAS3D was used. Moreover, the PLUTO2 fuel pins can fail into
partially voided coolant channels, which was not possible with SAS/FCI.
The models in PLUTO2 are considerably more mechanistic than those used
in SAS/FCI. For example, fuel motion in SAS/FCI was largely treated with
a lumped parameter approach, whereas PLUTO2 has one‑dimensional models
of the fuel motion inside the pins and in the coolant channels.
Moreover, PLUTO2 treats phenomena that were not addressed in SAS/FCI
such as continuous molten fuel flow regimes and the plateout of freezing
fuel.
The PLUTO2 module is an outgrowth of the earlier PLUTO code [14‑3, 14‑4]
which has been successfully used to simulate the early sodium voiding
and fuel motion in TOP experiments [14‑5, 14‑6]. Therefore, the basic
PLUTO concept has been retained in PLUTO2. However, as mentioned above,
PLUTO2 is also designed to analyze the longer‑term phenomena in mild
TOPs and the early phases of LOF'd'TOPs. Moreover, PLUTO2 has an
Eulerian hydrodynamic treatment which eliminates most of the cumbersome
rezoning that was necessary in the Lagrangian PLUTO code. The Eulerian
treatment has allowed the incorporation of axial cladding rupture
propagation, fuel plateout, and variable cross section flow.
Furthermore, PLUTO2 is considerably more economical to run than PLUTO;
this is of importance for a module of a whole‑core analysis code.
The PLUTO2 (and LEVITATE) coolant‑channel treatment considers the entire
subassembly cross section with all associated cladding and structure
although the treatment is still one‑dimensional. This is somewhat
different from the concept that is used in the pre‑fuel motion phase in
SAS4A. The latter considers the coolant flow area, cladding and
structure associated with only one pin.
Schematics of the PLUTO2 model when used for a mild TOP calculation and
for a LOF'd'TOP condition are shown in :numref:`figure-14.1-1` and :numref:`figure-14.1-2`,
respectively. These schematics are for a single pin with a coolant
channel that can belong to one or more pins. Although the PLUTO2 channel
treatment is one‑dimensional (i.e., there is a common treatment of all
subchannels), not all of the fuel pins in a subassembly have to fail
simultaneously. A number of the pins, as specified by input, can remain
intact. A treatment of the delayed failure of these pins is not yet
operational. In cases that assume that only some of the pins in a
subassembly fail, the fuel and gas ejection from these failed pins will
be added to all coolant subchannels since all subchannels in a
subassembly are treated alike. This has prompted the use of the
above‑mentioned concept in PLUTO2 in which the basic channel cell has a
cross section equal to the entire coolant flow area in a subassembly.
The PLUTO2 model addresses three major modeling areas pertinent to the
post‑pin‑failure behavior. The first area is the in‑pin fuel motion
toward a cladding rupture, the second is the fuel and gas ejection
through the cladding rupture, and the third is the multi‑component,
multi‑phase hydrodynamics treatment in the coolant channel.
The basic assumption for the in‑pin motion is the treatment of the
molten fuel region inside the pin (see :numref:`figure-14.1-1` and :numref:`figure-14.1-2`) as a
pressurized cavity containing fuel and fission gas, which flow toward a
cladding rupture. This general concept was already used in SAS/FCI,
PLUTO, and EPIC [14‑7, 14‑8]. The key PLUTO2 features in the modeling of
the in‑pin flow are:
A1) Treatment of the two‑phase, two‑component flow as a compressible
homogeneous flow with variable flow cross section and strong mass sinks
(due to fuel ejection) and mass sources (due to the addition of melting
fuel).
A2) Modeling of the radial and axial growth of the molten fuel cavity
due to additional fuel melt‑in determined by a heat‑transfer calculation
in the solid fuel annulus which is performed by subroutine PLHTR.
A3) The treatment of two types of fission‑gas bubbles in the molten
cavity. These are the large fission‑gas bubbles on the grain boundaries
which are assumed to act like free gas and exert pressure in the molten
cavity and very small intragranular bubbles which do not affect the
cavity pressure directly because they are assumed to be totally
constrained by surface tension. However, these small bubbles can
coalesce into large bubbles and thus contribute to the cavity
pressurization in a delayed fashion. In the code, the rate of
coalescence is controlled by an input time constant. Up to the time of
pin failure, when PLUTO2 is initiated, the relative fractions of these
two types of gas bubbles in the solid fuel and molten fuel cavity are
calculated by DEFORM. DEFORM has the same two‑bubble fission‑gas
treatment and uses the same input grain‑boundary gas fraction and time
constant for small bubble coalescence. Once PLUTO2 is active, additional
fuel can melt into the cavity which brings fission gas with it. The
total amount of fission gas in the melting fuel node is known from the
steady‑state DEFORM calculation. In the solid fuel, the DEFORM
calculation tracks grain boundary gas (large bubbles) and intragranular
gas (small bubbles) separately. The input fraction (FIFNGB) should
roughly correspond to the fraction of gas on the grain boundaries in the
solid fuel as determined from post‑irradiation examinations of fuel
pins.
A4) A simple fuel vapor pressure calculation which is based on the
radially averaged temperature in a cavity node is performed. The fuel
vapor pressure and the fission‑gas pressure are assumed to be additive.
A better fuel vapor pressure calculation would be based on the maximum
fuel temperature in a node. However, computing the latter in a moving
fluid is difficult. Also, it is not clear that any significant radial
temperature gradients remain present after the onset of fuel motion.
This problem will be addressed in future SAS4A work.
The fuel ejection from the pins is based on the assumption that the
cavity pressure in the node behind the cladding rupture of the failed
pins equilibrates instantaneously with the coolant channel pressure at
the same elevation by ejecting appropriate amounts of fuel and gas. The
latter are ejected with the same volume ratio as present in the cavity
nodes. If the pressure in the coolant channel becomes temporarily higher
than the pressure in the failed cavity nodes, the fuel and gas ejection
is halted. Backflow of fuel, gas or sodium into the pins is not allowed.
Two items concerning pin failure coherency and the axial pin‑failure
propagation are of importance:
B1) There is an option to fail only some of the pins in a subassembly.
The other pins remain intact. This option is relevant for mild TOPs.
Because of the common one‑dimensional treatment of all subchannels, this
requires that the failed pins are reasonably well distributed over the
subassembly cross section (see :numref:`figure-14.1-3`). A problem with this option
is that LEVITATE cannot be switched on after PLUTO2 because LEVITATE
assumes that all pins in a subassembly are failed.
B2) The cladding failures can enlarge axially; this is also referred to
as axial pin‑failure propagation. There is a nonmechanistic pin‑failure
propagation option available which is keyed on input fuel melt fraction,
cladding temperature, and required pressure difference between cavity
and channel. The mechanistic option compares the calculated cladding
hoop stress with an ultimate tensile strength function. Moreover, an
input fuel melt fraction has also to be exceeded.
The axial pin‑failure propagation is of key importance for lower
power channels experiencing a high overpower condition due to an LOF
accident (LOF'd'TOP). In this case, the cladding can be relatively
soft along a considerable length of the channel at the time of the
initial failure. This may lead to rapid axial failure propagation.
For milder overpower conditions, the axial failure propagation would
be slower and mostly caused by the overheating of the cladding by
molten fuel that has been ejected into the coolant channels.
.. _figure-14.1-1:
.. figure:: media/image2.png
:align: center
:figclass: align-center
PLUTO2 Schematic for Transient Overpower Conditions
.. _figure-14.1-2:
.. figure:: media/image3.png
:align: center
:figclass: align-center
PLUTO2 Schematic for Loss of Flow Driven Transient Overpower Conditions
.. _figure-14.1-3:
.. figure:: media/image4.png
:align: center
:figclass: align-center
Illustration of Particulate Fuel Flow Regions (Upper Figure) and Partial Annular Fuel Flow Regime (Lower Figure) for a Seven-Pin Bundle in Which Only Four Pins Are Assumed to Have Failed
In the coolant channels, PLUTO2 treats the flow of materials as a
one-dimensional, compressible two‑fluid flow with variable flow cross
section. One component of this two‑fluid treatment is the mobile fuel
and the fission gas that is dissolved in the fuel. The other component
is the mixture of liquid sodium, sodium vapor, fuel vapor, and free
fission gas. The salient features of the channel modeling can be seen in
:numref:`figure-14.1-1` and :numref:`figure-14.1-2` and are briefly discussed below.
C1) Coolant channel boiling is modeled by treating a static sodium film
that is left behind by the expelled coolant slugs (see upper voided
region in :numref:`figure-14.1-1` and :numref:`figure-14.1-2`). The sodium film can be entrained by
vapor streaming and by the action of fuel particles. Once the fuel flow
regime becomes continuous (see below), the sodium film is
instantaneously entrained as droplets in the gaseous phase. This sodium
film is of importance because it provides a significant sodium vapor
source and it cools the cladding surface for some time after sodium
voiding has occurred.
C2) The liquid sodium slugs above and below the interaction region
(which encompasses all the two‑phase sodium, fuel, and fission gas) are
treated as incompressible slugs with variable flow cross section.
However, during the first few milliseconds after failure, an acoustic
approximation is made to determine the slug interface velocities.
C3) Three different fuel flow regimes are treated in PLUTO2:
particulate, partially or fully annular, and bubbly. The flow regime
selection is mainly keyed to the liquid sodium fraction and to the
channel fuel volume fraction.
The fuel motion in TOP accidents has been traditionally modeled as a
particulate fuel suspension in a two‑phase sodium/fission‑gas mixture.
However, the breakup of fuel into droplets or particles is likely only
when the liquid sodium fraction is fairly high. From TREAT experiments,
it can be concluded that continuous molten fuel flow regimes exist in
voided regions. In PLUTO2, continuous fuel flow regimes are, therefore,
considered in addition to the particulate flow regime. The treatment of
a partially annular fuel flow regime is not a common approach but has
been prompted by the notion that a relatively small amount of molten
fuel in a voided channel will not cover the entire cladding and
structure perimeter, but will rather behave like a single or multiple
rivulet flow. For higher fuel fractions, a complete annular fuel flow
regime is assumed and for an even higher fuel volume fraction, a bubbly
fuel flow is modeled. :numref:`figure-14.1-3` illustrates the particulate and
partially annular flow regime models for a seven‑pin TREAT test bundle
for the case in which only a certain fraction of the pins has failed.
For the particulate or bubbly fuel flows, the fuel is simply uniformly
distributed in all subchannels. For the partially annular flow, the fuel
mass is assumed to be distributed between the pin cladding and structure
in proportion to the cladding‑to‑structure surface area ratio. The
cladding of all failed and unfailed pins is assumed to be covered by
equal amounts of fuel film with equal film thickness. The fraction of
the cladding perimeters covered by the fuel films is dependent on the
fuel volume fraction and the input constant CIANIN. Where this fuel is
exactly located on the pin perimeters is not relevant since there is no
azimuthal cladding temperature distribution calculated. The fraction of
the fuel covering the structure is of considerable importance in small
bundles. Once frozen fuel crusts are generated (see below), they are
also distributed in the same manner. Once the fuel fraction is high
enough to lead to a fully annular flow, all cladding and structure in a
given node is covered by the fuel films. The fuel flow regimes will be
discussed in more detail in :numref:`section-14.4.2`. :numref:`figure-14.4-1`, which is
shown later in :numref:`section-14.4`, illustrates the fuel flow regimes in more
detail for an equivalent cylindrical geometry.
Modeling these different fuel flow regimes explicitly has the advan‑tage
that one has all interaction areas for heat, mass, and momentum transfer
readily available.
C4) Frozen fuel plateout is treated in PLUTO2 because there is
over‑whelming evidence from in‑pile and out‑of‑pile experiments that
fuel freezing and plateout are key phenomena. In PLUTO2, only fuel in
the continuous flow regime can plate out on cladding and structure upon
fuel freezing. The fuel particles are not allowed to stick to cladding
and structure because the fuel particles are assumed to have a solid
shell due to their interaction with liquid sodium.
The modeling of the fuel plateout in PLUTO2 can either be of the
bulk‑freezing type or conduction‑limited type. This is controlled by
input parameter CIFUFZ, which also allows intermediate modes. Frozen
fuel crusts can also become mobile after remelting. In addition,
frozen fuel crusts are released into the moving fuel stream if the
underlying clad or structure becomes significantly molten.
C5) The Fuel‑Coolant Interaction (FCI) treatment depends on the fuel
flow regime. In the particulate flow regime, the FCI treatment is
largely based on the Cho‑Wright approach [14‑9] which considers the heat
flow resistance in the fuel particles and ignores the resistance of
liquid sodium. Moreover, the resistance due to vapor blanketing is
treated in a parametric fashion in PLUTO2. In the code, separate FCI
calculations are done for every numerical cell, whereas the Cho‑Wright
model is a lumped‑parameter approach. Of importance for the FCI
treatment in PLUTO2 is also the treatment of the slip between fuel and
liquid sodium which mitigates the strength of the FCI's.
In the annular fuel flow regimes in PLUTO2, the convective heat
transfer between the hot fuel film and/or fuel crusts and the
two‑phase sodium/fission‑gas mixture is considered. Since the fuel
surface area for the annular flow is significantly smaller than that
for the particulate flow, this type of FCI is much milder.
In the bubbly fuel flow regime high heat‑transfer rates between fuel
and liquid sodium are possible, but the bubbly fuel flow regime is
usually generated at an axial elevation where no liquid sodium is
present. The penetration of liquid sodium into a bubbly fuel flow
regime is also unlikely in the one‑dimensional PLUTO2 because the
cladding near a node with bubbly flow is usually too hot to allow
reentry of liquid sodium.
The numerical grids on which the hydrodynamics equations are solved are
shown in :numref:`figure-14.1-4`. The stationary Eulerian grid in the molten pin
cavity is aligned with the stationary Eulerian grid in the coolant
channel. The cavity grid can expand continuously in the radial direction
due to fuel melt‑in and stepwise (by whole cells) in the axial
direction. The grid on which the reactivity calculations are done
covers the pin and blankets and the adjacent coolant channel cells from
K = 1 to K = MZ. Outside the molten cavity region, this grid is also
subdivided radially for the heat‑transfer calculation in the solid fuel
and cladding. The interaction region in the coolant channel can expand
or contract continuously in the axial direction, which requires a
partially Lagrangian treatment for the edge cells of the interaction
region. The channel fuel region and the fission‑gas region also expand
or contract continuously in the axial direction. However, when they have
moved into a new cell, the fuel or fission gas is assumed to be
homogeneously distributed in that cell for calculating the pressure,
heat transfer, momentum change, and reactivity. On the mesh grids above
and below the interaction region, only the liquid sodium temperatures
are calculated. The momentum change of the liquid sodium slugs is
calculated in an integral fashion.
The current status of the PLUTO2 validation and its future validation
needs are discussed in the SAS4A Validation and Verification Plan
[14‑10]. Here only an enumeration of the integral validation efforts
already performed will be given. As mentioned earlier, the PLUTO code
has been successfully used to simulate the early sodium voiding and fuel
motion in two in‑pile TOP experiments [14‑5, 14‑6]. PLUTO2 comparison
calculations with PLUTO showed good agreement for the early fuel motion
and sodium voiding [14‑11]. An intercode comparison with the EPIC code
for LOF'd'TOP conditions showed that the two codes compare rather well
when several of the advanced features in PLUTO2, such as fuel flow
regimes and fuel plateout, are switched off [14‑12]. PLUTO2 was also
used in the EEC‑WAC TOP comparative exercise [14‑20]. Because its
advanced features were active in this comparison exercise, the PLUTO2
results differed considerably from those calculated with simpler models.
A reasonably good post‑test simulation of the major flow event in the H6
50 ¢/s TOP TREAT test [14‑13] was achieved through input parameter
adjustments [14‑6]. However, some uncertainty with regard to the mode of
the FCI observed in this test could not be resolved. A good post‑test
simulation of the L8 LOF'd'TOP TREAT Test [14‑14, 14‑15, 14‑12] was also
achieved after introduction of a model for frozen fuel crust release
from molten cladding. This was prompted by a pre‑test analysis which
underpredicted the fuel dispersal [14-16]. Other pretest predictions
with PLUTO2 were made for the AX1 $3/s TOP test using carbide fuel
[14‑17], the W2 10 ¢/s TOP test in the ETR [14‑18], and the 37‑pin
bundle P4 pin‑failure propagation test in ETR [14‑19]. The prediction
for the AX1 test was quite reasonable, whereas the prediction for W2
suffered from the assumption of too coherent pin failures. The PLUTO2 P4
pretest analysis predicted a complete sweepout of the fuel ejected from
the three fuel canisters used in this test, whereas the experiment led
to very little sweepout and a sizable frozen fuel blockage. Possible
explanations of the observed behavior include sodium bypass effects in a
large bundle or the fuel canister ejecting the fuel more violently than
the fuel pins in TOP tests.
.. _figure-14.1-4:
.. figure:: media/image5.png
:align: center
:figclass: align-center
Axial Mesh Cells Used in the Pin and the Coolant Channel for the Numerical Solution of Conservation Equations. Also Shown Are the Different Component Regions in the Channel
.. _section-14.1.2:
Overview of the Program Flow
~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Either PLUTO2 or LEVITATE is initiated once the subroutine FAILUR has
predicted pin failure and a minimum fuel melt fraction equal to the
input parameter FMELTM exists at the failure node. Subroutine FAILUR is
usually called from the driver of the DEFORM pin‑behavior module.
However, in the case when cladding motion has already begun, it is
directly called from the transient SAS4A driver routine TSTHRM. The
decision as to which one of the two modules, PLUTO2 or LEVITATE, is to
be initiated depends on the existence and size of a boiling region at
the time of pin failure and whether the pins are predicted to fail into
a voided or unvoided region of the coolant channel. If there is no
boiling, as in an intermediate ramp rate TOP accident or if the pins
fail into the liquid region of a partially voided channel, PLUTO2 will
be initiated because of its capability of treating FCI's and sweepout of
particulate fuel. For the case of fuel failing into a multibubble
boiling region, PLUTO2 will only be initiated if the average void
fraction in this region is less than 70%. If the average void fraction
is larger or if cladding motion has begun, LEVITATE will be initiated.
The PLUTO2 initialization is described in more detail in :numref:`section-14.7.1`.
The subroutines involved in this procedure are PLINPT, PLSET, PLSETl,
and PLSAIN. In these routines all permanent PLUTO2 variables are either
set to input values or values calculated by the single‑phase hydraulics,
the pin heat‑transfer routines, the pin behavior module DEFORM, or the
boiling module. It should also be mentioned here that the flag ICALC,
which designates which major module is currently active, will be set to
3 if PLUTO2 has been initialized and to 2 if LEVITATE has been
initialized.
Once the PLUTO2 initialization routines have been executed and the flag
ICALC has been set to 3, the transient SAS4A driver TSTHRM (see flow
diagram in :numref:`Chapter %s<section-2>`) will call PLUDRV (*PLU*\ TO2 *DR*\ IVER
SUBROUTINE). PLUDRV will retain control and advance the solution using
PLUTO2 time steps until the end of the primary‑loop time step is
reached. If another SAS4A channel is using PLUTO2 at this time, its
solution will also be advanced until the end of the current primary‑loop
time step.
The flowcharts in :numref:`figure-14.1-5` and :numref:`figure-14.1-6` show the logic of the PLUTO2
driver. They are complete except for an option for switching off all
subroutines performing the material velocity calculations. This option
is discussed later in this section. If the flag MODEPL, which controls
this option, is zero, the flow diagrams in :numref:`figure-14.1-5` and :numref:`figure-14.1-6` are
appropriate.
If the flag ILEPLI has been set to 1 in the PLUTO2 initialization (see
:numref:`figure-14.7-2` in :numref:`section-14.7`), LEVITATE will have to be initiated via
PLUTO2. In this case, the PLUTO2 driver calls LEPLIN which is the
interface routine between PLUTO2 and LEVITATE. The flag ICALC is then
set to 2 and control is returned to TSTHRM. The latter will call the
LEVITATE driver at the beginning of the next coolant time step upon
encountering ICALC = 2 for the channel under consideration (see flow
diagram of TSTHRM in :numref:`Chapter %s<section-2>`).
If ILEPLI is not equal to 1, subroutine PLSET2 will be called (see :numref:`figure-14.1-5`). In this subroutine, all temporary arrays (i.e., arrays which
can be overwritten once control is returned to TSTHRM) are initiated.
Moreover, temporary integers are set.
.. _figure-14.1-5:
.. figure:: media/image6.png
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:figclass: align-center
First Part of the Flow Chart of the PLUTO2 Driver Subroutine, PLUDRV, Showing the Functions of the Major Subroutines
.. _figure-14.1-6:
.. figure:: media/image7.png
:align: center
:figclass: align-center
Second Part of the Flow Chart of the PLUTO2 Driver Subroutine, PLUDRV, Showing the Functions of the Major Subroutines
Next, the PLUTO2 time is advanced by adding the PLUTO2 time‑step size to
the time at the beginning of the PLUTO2 cycle. The time‑step increment
for the very first PLUTO2 cycle is an input value which is also the
minimum PLUTO2 time‑step size. Later, in the logic flow, the time‑step
size is calculated.
Next, the inlet and outlet pressures at the end of the PLUTO2 time step
are determined from the PRIMAR calculated inlet and outlet pressures and
rates of inlet and outlet pressure changes.
Next, the power level at the end of the PLUTO2 time step is calculated
from an exponential fit of the power‑time history that takes the power
level at the beginning of the previous and the current main time steps
and the precalculated power level at the end of the current main time
step into account. By using the calculated power level and the axial
input power distribution, the specific power for each axial pin node is
calculated. The same specific power is set for the corresponding channel
nodes.
In subroutine PSHAPE, which is not a PLUTO2 subroutine, the total power
for all axial fuel‑pin nodes is calculated assuming that the pin is
still intact. This is needed for the heat‑transfer calculation in the
solid fuel annulus in the pin.
Next, cladding and structure temperatures in the nodes adjacent to the
interaction region are initialized in order to make sure that they will
be available if the interaction zone expands into one of these nodes.
In subroutine PLIF (*PL*\ UTO2 *I*\ NTER\ *FA*\ CES), the axial
displacements of the sodium slug interfaces, the interfaces of the
fission‑gas region and the interfaces of the regions containing fuel
are reset. The actual calculations of the velocities needed for this
resetting are performed later in subroutine PLMOCO. The initial sodium
velocities come from the single‑phase hydraulics or the boiling model
and are set in PLSAIN. Fission gas and mobile sodium velocities are
always the same in PLUTO2. Subroutine PLIF also calculates the axial
pin‑failure propagation and resets the pointer array IDISR(I) which
indicates which pin nodes have failed.
In subroutine PLREZO (*PL*\ UTO2 *REZ*\ ONE), mesh cells are added to an
expanding interaction region or deleted from a shrinking interaction
region. Moreover, it cuts off short columns of liquid sodium slugs
which are attempting to reenter into a cell of the interaction zone
containing frozen fuel or ruptured cladding. The liquid sodium is added
homogeneously to such cells. If fuel pins fail into the lower sodium
slug, PLREZO will enlarge the interaction region downwards.
In subroutine PLFREZ (*PL*\ UTO2 *FRE*\ E\ *Z*\ ING ROUTINE), the amount
of fuel plating out per time step and per node is calculated. Moreover,
PLFREZ calculates the amount of crust released because of remelting or
because the underlying cladding has become molten. The released fuel
crusts, which have an axial velocity of zero, are mixed with the moving
fuel and an updated velocity is calculated by momentum averaging.
In PLMACO (*PL*\ UTO2 *MA*\ SS *CO*\ NSERVATION), the mass conservation
equations for the moving components in the channels are solved. This
includes a combined mass conservation for solid or liquid fuel and fuel
vapor, and mass conservations for the sodium, free fission gas and
dissolved fission gas. There is a special treatment for the top and
bottom cells of the channel. Fuel, sodium and fission‑gas will be taken
out and stored in a reservoir if the interaction region has extended
into the lowermost or uppermost channel cell.
In subroutine PLVOFR (*PL*\ UTO2 *VO*\ LUME *FR*\ ACTIONS), the
entrainment of the static sodium film by the flow of the two‑phase
sodium/fission‑gas mixture and of the fuel particles as well as the
de‑entrainment of liquid droplets onto the film, is calculated. PLVOFR
also sets the volume fractions of the various components based on the
results of the plateout and crust release calculation in PLFREZ and the
results of the mass conservation equations. The final section of
subroutine PLVOFR is devoted to the selection of the fuel flow regime.
In subroutine PLMISC (*PL*\ UTO2 *MISC*\ ELLANEOUS), several important
items are calculated. First, the molten‑ and frozen‑fuel configurations
in the flow channel are determined (see :numref:`figure-14.4-1`). Second, most
energy and some momentum exchange terms between the various flow
components, cladding and structure are calculated. Because many of these
interaction terms depend on the fuel flow regimes, three different
exchange coefficients are needed for several of the components. Third,
the mobile fuel energy equation is solved.
In subroutine PLTECS (*PL*\ UTO2 *TE*\ MPERATURE CALCULATION OF
*C*\ LADDING AND *S*\ TRUCTURE}, the cladding, reflector, and structure
temperatures within the interaction zone are calculated. This
calculation is preceded by the determination of the energy exchange
coefficients for the frozen fuel crust and the solution of the frozen
crust energy equation. Since the energy exchange between fuel crust and
moving fuel was not considered in subroutine PLMISC, an updating of the
moving fuel energies is also performed here.
In subroutine PLNAEN (*PL*\ UTO2 *NA* *EN*\ ERGY EQUATION), the
two‑phase and single‑phase enthalpy equations for the mixture of sodium
and fission gas are solved. From the resulting temperatures and the
previously calculated volume fractions (see PLVOFR), the liquid‑phase
sodium pressure or the sodium saturation or superheated sodium vapor
pressure, as well as the free fission gas pressure, are calculated. In
addition, the total end‑of‑time‑step pressure, which includes the fuel
vapor pressure, is calculated.
In subroutine PLlPIN (*PL*\ UTO2 NO.\ *1* *PIN* EQUATIONS}, the mass and
energy equations for the in‑pin fuel motion are solved. At first, the
fuel and free and dissolved fission‑gas mass sources due to fuel melt‑in
are calculated and the in‑pin cavity enlargement is determined.
Moreover, the free fission‑gas mass sources due to dissolved gas
coalescence and the corresponding sinks for the dissolved gas are
evaluated. Following these, the fuel mass and energy conservation
equations and the free fission‑gas mass conservation equation are
solved. Then, a preliminary end‑of‑time‑step pressure is calculated for
all cavity cells. This pressure calculation takes the prior‑calculated
end‑of-time‑step densities and temperatures into account. Only the
pressures in the ejecting nodes will be further updated; in the other
cells, the "preliminary end‑of‑time‑step pressure" is the pressure that
will actually be used in the momentum equation. The last major item in
PLlPIN is the calculation of the mass of fuel and fission gas ejected
from all cavity cells that have failed cladding and a pressure higher
than that in the corresponding channel cell. The masses of fuel and free
and dissolved fission gases in these cavity cells are then reduced and
the pressure is updated. The fuel and free and dissolved fission‑gas
masses in the channel cells are correspondingly increased and the
end‑of‑time‑step pressures in the channel nodes receiving fuel and/or
fission gas are updated. The fuel energy in the channel nodes receiving
fuel is also updated due to the addition of fuel with a higher
temperature.
In subroutine PL2PIN (*PL*\ UTO2 NO. *2* *PIN* EQUATIONS), the momentum
equation for the homogeneous fuel/fission gas mixture in the molten pin
cavity is solved. Next, the mass conservation equation for the dissolved
fission gas is solved. This could have been solved earlier in PLlPIN but
that routine had become too crowded. The calculation of the time‑step
size for the in‑pin calculation during the next time step is also
performed in PL2PIN.
In subroutine PLMOCO (*PL*\ UTO2 *MO*\ MENTUM *CO*\ NSERVATION), many
quantities which were previously needed only at the cell centers have to
be defined at the cell edges. Moreover, most of the momentum exchange
terms and, in particular, the drag between the fuel and the two‑phase
sodium/fission‑gas mixture are evaluated for the particulate and bubbly
fuel flow regimes. The main section of this routine deals with the
simultaneous solution of the two momentum equations at all cell edges
in the interaction region. Moreover, momentum equations will also be
solved for fuel particles at either end of the fuel region if the end
nodes of the fuel region are in a particulate flow regime. Another
section of PLMOCO deals with the calculation of the interaction zone
interface velocities. This includes an acoustic approach in the liquid
slugs during the first few milliseconds after pin failure and later an
incompressible variable‑cross‑section treatment of the upper and lower
coolant slugs. Also calculated in PLMOCO are instantaneous coolant slug
flow rates for the entire calculational channel, as well as the
integrated channel flow rates over a PRIMAR time step. These quantities
are needed by the PRIMAR4 module for recalculating the inlet and outlet
pressures.
The next task in the PLUTO2 driver routine is the time‑step size
determination in the coolant channel. This is compared with the
time‑step size calculated for the in‑pin motion and the smaller of the
two will be the PLUTO2 time‑step size for the next calculational step.
The next task of PLUDRV is to calculate the transient mass distributions
for the fuel and voiding reactivities. This calculation will be
described in the next section on the interaction with other SAS4A
modules.
If the time at the end of a PLUTO2 time step coincides with the end of
the heat‑transfer time step, several heat‑transfer routines are called.
This "coincidence" is forced to occur whenever the calculated PLUTO2
time step would overshoot the end of the heat‑transfer time step. In
this case, the PLUTO2 time step is set to coincide with the end of the
heat‑transfer time step. This is actually done right after the time‑step
size calculation described above.
The heat‑transfer time step is determined next. Its maximum value is
based on a characteristic heat‑transfer time of the cladding and can be
further limited by small primary‑loop time steps which will be small in
high power situations because the main (or point kinetics) time step
becomes small. Since the heat‑transfer time step is also used for the
liquid slug temperature calculations, it can be further limited by a
Courant condition based on the slug velocities.
In subroutine PLCOOL (*PL*\ UTO2 *COO*\ LANT SLUG TEMPERATURE), the
temperature in all numerical nodes in the liquid sodium slugs is
calculated. This involves the calculation of heat‑transfer exchange
terms between liquid sodium and cladding, plenum cladding, reflectors,
and structure, and the solution of an energy equation. Moreover, PLCOOL
checks whether a node in the lower sodium slug has started to boil.
In subroutine PLSTR (*PL*\ UTO2 *STR*\ UCTURE TEMPERATURE CALCULATION),
the temperatures of the structure, the plenum cladding, and the
reflectors outside of the interaction region are calculated.
In subroutine PLHTR (*PL*\ UTO2 *H*\ EAT *TR*\ ANSFER), calculations are
made of the temperature fields in the solid fuel annulus surrounding the
molten fuel cavity, in the unmelted fuel and blanket cells, and in the
fuel and blanket cladding outside the interaction region. The
heat‑transfer boundary condition at the interface between the molten
cavity and the solid fuel annulus is treated by applying a
time‑integrated heat flow rate term whose contributions were calculated
and summed up in subroutine PLlPIN. The boundary condition between the
inner cladding and the outer fuel surface is also based on such an
integrated heat flow rate term. The latter is calculated and summed up
in subroutine PLTECS, in which the cladding temperature field in the
interaction region is calculated using the PLUTO2 time step.
If it is time to produce output, subroutine PLOUT (*PL*\ UTO2
*OUT*\ PUT) will be called. If the end of the PRIMAR time step has not
yet been reached, the logic flow will return to point B (see :numref:`figure-14.1-5`).
If the end of the PLUTO2 time step coincides with the end of the
primary loop time step, a check is made whether the conditions require
a switch to the LEVITATE module. This will be necessary if extensive
cladding melting has occurred or if complete fuel‑pin disruption is
imminent or if the fuel vapor pressures have become quite high. In this
case, the LEVITATE‑PLUTO2 interface routine LEPLIN is called and the
integer flag ICALC is set to 2 which will assure the calling of the
LEVITATE driver routine LEVDRV at the beginning of the next PRIMAR step.
Whether a switch to LEVITATE is made or not, control will now be
returned to the transient driver TSTHRM.
An option, which is not shown in the flow charts in :numref:`figure-14.1-5` and
14.1‑6, allows the user to set all material velocities to small values
and to shut off all the subroutines or subroutine sections that are
calculating the motion of materials in PLUTO2. This option, which allows
an economical but very simplistic continuation of a PLUTO2 calculation
for several tens of seconds, is useful for the later treatment of a lead
channel failing long before other channels in a low‑ramp‑rate TOP
calculation. This situation is most likely to occur when the negative
reactivity introduced due to fuel sweepout from the lead channel is not
enough to insure permanent subcriticality. This option may be
reasonable for subassemblies that are completely blocked by frozen fuel
as long as they do not lose much heat to neighboring subassemblies (this
subassembly‑to‑subassembly heat loss should eventually be modeled in
SAS4A). Moreover, if the fuel in the disrupted assembly heats up too
much, it could also become mobile again; this is not currently treated.
The above‑mentioned PLUTO2 option will be activated if the input time
TIPLMX is exceeded by the PLUTO2 time.
.. _section-14.1.3:
Interaction with other SAS4A Modules
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
During the PLUTO2 initialization, data from several other modules are
transferred to PLUTO2 (see :numref:`figure-14.1-7`). When PLUTO2 is used for a TOP
calculation, it requires data from the single‑phase hydraulics, the
DEFORM pin behavior module, and the pin heat‑transfer module for
non‑boiling conditions (whose driver routine is TSHTRN). When PLUTO2 is
used in a SAS4A channel experiencing a LOF'd'TOP condition, data may
also be needed from the boiling module and the heat‑transfer module for
boiling conditions (whose driver routine is TSHTRV). The PLUTO2
initialization has already been briefly discussed at the beginning of
:numref:`section-14.1.2` and will be described in more detail in :numref:`section-14.7.1`.
Once PLUTO2 is active, there will only be an interaction with the point
kinetics module and also with the primary loop module if the PRIMAR‑4
module has been selected (see :numref:`figure-14.1-7`). Moreover, there is an
interaction with the PLHTR pin heat‑transfer calculation which is
virtually a PLUTO2 routine but uses a different time step than PLUTO2
(see :numref:`section-14.1.2`). There is no interaction with other modules because
PLUTO2 calculates the motion of the liquid sodium slugs (in PLMOCO), has
the capability to treat sodium boiling in regions which are not yet
occupied by fuel, and also has a simplified calculation of the axial
cladding failure propagation (in PLIF).
When PLUTO2 is active, it uses the user input axial fission power
distribution with the magnitude calculated by the point kinetics
module. PLUTO2 provides the point kinetics module with the sodium and
fuel axial mass distributions for all channels in which PLUTO2 is
active. Moreover, the Doppler reactivity calculation is based on PLUTO2
calculated average fuel temperatures. The details of these calculations
are given in :numref:`section-14.6.1`.
If the PRIMAR-1 option is chosen PLUTO2 will use a constant outlet
coolant plenum pressure which is input and an inlet coolant plenum
pressure which is calculated by PRIMAR-1 at the beginning of the
transient and later modified by an input table PLUTO2 will not feed back
and information to the primary-loop module if the PRIMAR‑1 option has
been chosen.
.. _figure-14.1-7:
.. figure:: media/image8.png
:align: center
:figclass: align-center
Data Transfer Between the PLUTO2 Module and the Other SAS4A Module
If the PRIMAR‑4 option has been chosen, PLUTO2 uses the time‑dependent
inlet and outlet plenum pressures and temperatures which are calculated
by PRIMAR‑4. PLUTO2 feeds back to the PRIMAR‑4 module the sodium masses
ejected into or received from the inlet and outlet plena during a PRIMAR
time step. PLUTO2 also provides the PRIMAR-4 module with the energy of
the sodium ejected into the plena and with the liquid sodium flowrates
at the end of the PRIMAR time step. This is described in more detail in
:numref:`section-14.6.2`.
The PLUTO2 calculation will be taken over by the LEVITATE module if
extensive cladding melting has occurred or if complete pin disruption
is imminent or if the fuel vapor pressure becomes quite high. This
transition, which can be controlled by input, is necessary because
PLUTO2 is not designed to treat these situations. However, the
transition to LEVITATE will not occur if only some of the pins have
failed in PLUTO2. This PLUTO2 option, which is useful for mild TOP
conditions, causes problems for LEVITATE because the latter assumes that
all pins have failed when it is initiated.