.. _section-4.5:
Net Reactivity
--------------
For applications other than EBR-II, the net reactivity in :eq:`4.2-4` is
the sum of nine reactivity components:
.. math::
:label: 4.5-1
\delta k \left( t \right) = \delta k_{\text{p}}\left( t \right) + \delta k_{\text{cs}}\left( t \right) + \delta k_{\text{D}}\left( t \right) + \delta k_{\text{d}}\left( t \right) + \delta k_{{\text{Na}}}\left( t \right) + \delta k_{\text{re}}\left( t \right) + \delta k_{\text{cr}}\left( t \right) + \delta k_{\text{fu}}\left( t \right) + \delta k_{\text{cl}}\left( t \right)
where
:math:`\delta k_{\text{p}}` = User-programmed reactivity,
:math:`\delta k_{\text{cs}}` = Control system reactivity,
:math:`\delta k_{\text{D}}` = Fuel Doppler feedback reactivity,
:math:`\delta k_{\text{d}}` = Fuel, cladding, and structure axial expansion
feedback reactivity,
:math:`\delta k_{\text{Na}}` = Coolant density or voiding
feedback reactivity,
:math:`\delta k_{\text{re}}` = Core radial expansion feedback
reactivity,
:math:`\delta k_{\text{cr}}` = Control rod drive expansion feedback
reactivity,
:math:`\delta k_{\text{fu}}` = Fuel relocation reactivity feedback,
:math:`\delta k_{\text{cl}}` = Cladding relocation reactivity feedback.
The reactivity feedback models available for these components are
covered in the sections that follow. For EBR-II applications, the net
reactivity is calculated as formulated in :numref:`section-4.5.9`.
.. _section-4.5.1:
User-Programmed Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~~
The user-programmed reactivity is specified by the user at execution
time as either an input table or as subprogram FUNCTION PREA. It is
intended to be used as a means of specifying any reactivity effect not
explicitly modeled as a feedback. An example might be the simulation of
a control rod withdrawal or insertion, or the dropping of a fuel
subassembly during reloading.
The user-programmed reactivity option is triggered by setting :sasinp:`IPOWER` to 0. For :sasinp:`NPREAT` equal to
0, the value returned by the user-supplied subroutine function PREA is
used as the programmed reactivity. For :sasinp:`NPREAT` > 0, :sasinp:`NPREAT` gives the
number of pairs of values of programmed reactivity and time input on the
standard input file in :sasinp:`PREATB` and :sasinp:`PREATM`. A maximum of twenty pairs of programmed reactivity and
time may be entered in :sasinp:`PREATB` and :sasinp:`PREATM`. These data are then either
used directly or fit to curves according to the user's specification of
input data :sasinp:`IFIT`.
.. _section-4.5.2:
Control System Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~
The control system reactivity is the value supplied by the reactivity
control signal (JTYPE = -1) generated by the control system model (see
:numref:`Chapter %s<section-6>` and Input Block 5-INCONT).
.. _section-4.5.3:
Fuel Doppler Feedback Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The fuel Doppler reactivity effect at any axial location in a
subassembly is estimated from
.. math::
:label: 4.5-2
T\frac{\text{d}\left( \delta k_{\text{D}} \right)}{\text{dT}_{\text{f}}} = \alpha_{\text{D}}
where :math:`T_{\text{f}}` is the radial mass-averaged fuel temperature for that axial location, and
:math:`\alpha_{\text{D}}` is the local fuel Doppler coefficient, an input
quantity. To obtain the Doppler reactivity feedback at time :math:`t`,
:eq:`4.5-2` is integrated from steady-state conditions to conditions at
time :math:`t` to obtain
.. math::
:label: 4.5-3
\delta k_{\text{D}}\left( t \right) = \alpha_{\text{D}}\ln\left\lbrack \frac{T_{\text{f}}\left( t \right)}{T_{\text{f}}\left( 0 \right)} \right\rbrack
:eq:`4.5-3` is used at each axial location where a fuel temperature is
calculated. The local fuel Doppler coefficient, :math:`\alpha_{\text{D}}`, is
adjusted linearly between the input coolant-in (flooded) and coolant-out
(voided) values to correct for the effect of coolant voiding on neutron
leakage.
The coolant-in and coolant-out Doppler coefficients are entered as :sasinp:`ADOP`
and :sasinp:`BDOP`, and the axial weighting
of the Doppler coefficients is input as :sasinp:`WDOPA`.
.. _section-4.5.4:
Fuel, Cladding, and Structure Axial Expansion Feedback Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-4.5.4.1:
Simple Axial Expansion Reactivity Model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
A simple model for the reactivity effects of thermal expansion of fuel
and cladding is included in |SAS|. The simple thermal expansion
feedback model is based on a few assumptions. It is assumed that before
the start of the transient, a combination of fuel cracking, fuel
re-structuring, and stress relaxation cause the gap between the fuel and
the cladding to close, but there is little contact force between the
fuel and the cladding. During the transient, if the cladding expands
faster than the fuel, then the fuel-cladding gap opens, and the fuel can
expand freely in the axial direction. If the fuel expands faster than
the cladding, then the fuel binds with the cladding, and the axial
expansion is determined by balancing the axial forces between the fuel
and the cladding. Slip between fuel and cladding is ignored in this
case. The axial expansion feedback is calculated separately for each
channel. The formulation for each channel follows.
For each axial node, :math:`j`, in the core, the fuel and cladding
expansion fractions are calculated as
.. math::
:label: 4.5-4
\epsilon_{\text{f}}\left( j,t \right) = \alpha_{\text{f}}\left\lbrack T_{\text{f}}\left( j,t \right) - T_{\text{f}}\left( j,0 \right) \right\rbrack
and
.. math::
:label: 4.5-5
\epsilon_{\text{e}}\left( j,t \right) = \alpha_{\text{e}}\left\lbrack T_{\text{e}}\left( j,t \right) - T_{\text{e}}\left( j,0 \right) \right\rbrack
where
:math:`\alpha_{\text{f}}` = fuel thermal expansion coefficient
:math:`\alpha_{\text{e}}` = cladding thermal expansion coefficient
:math:`T_{\text{f}}` = mass-averaged fuel temperature for axial node j
and
:math:`T_{\text{e}}` = cladding mid-point temperature
If :math:`\epsilon_{\text{e}}\left( j \right)` is greater than
:math:`\epsilon_{\text{f}}\left( j \right)`, then the fuel axial expansion for
node :math:`j` is
.. math::
:label: 4.5-6
\delta_{\text{f}}\left( j,t \right) = \epsilon_{\text{f}}\left( j,t \right)D_{\text{z}}\left( j \right)
and the cladding expansion is
.. math::
:label: 4.5-7
\delta_{\text{e}}\left( j,t \right) = \epsilon_{\text{e}}\left( j,t \right)D_{\text{z}}\left( j \right)
where :math:`D_{\text{z}}` is the nominal axial height of the node. If
:math:`\epsilon_{\text{f}}\left( j \right)` is greater than or equal to
:math:`\epsilon_{\text{e}}\left( j \right)`, then a simple balance between the
axial forces of the fuel and cladding gives
.. math::
:label: 4.5-8
\delta_{\text{f}}\left( j,t \right) = \frac{\epsilon_{\text{f}}\left( j,t \right)Y_{\text{f}}A_{\text{f}} + \epsilon_{\text{e}}\left( j,t \right)Y_{\text{e}}A_{\text{e}}}{Y_{\text{f}}A_{\text{f}} + Y_{\text{e}}A_{\text{e}}}D_{\text{z}}\left( j \right)
and
.. math::
:label: 4.5-9
\delta_{\text{e}}\left( j,t \right) = \delta_{\text{f}}\left( j,t \right)
where
:math:`Y_{\text{f}}` = fuel Young's modulus
:math:`Y_{\text{e}}` = cladding Young's modulus
:math:`A_{\text{f}}` = nominal cross-sectional area of the fuel
and
:math:`A_{\text{e}}` = nominal cross-sectional area of the cladding
The reactivity calculation is based on the fuel and cladding worth
tables used by the code. First an unexpanded axial mesh,
:math:`z_{0}\left( j \right)`, is calculated using
.. math::
:label: 4.5-10
z_{0}\left( 1 \right) = 0
and
.. math::
:label: 4.5-11
z_{0}\left( j + 1 \right) = z_{0}\left( j \right) + D_{\text{z}}\left( j \right)
Note that :math:`z_{0}` is zero at the bottom of the lower axial
blanket.
New, expanded axial meshes for the fuel and cladding,
:math:`z_{\text{nf}}\left( j \right)` and
:math:`z_{\text{ne}}\left( j \right)`, are calculated using
.. math::
:label: 4.5-12
z_{\text{nf}}\left( 1 \right) = 0
.. math::
:label: 4.5-13
z_{\text{ne}}\left( 1 \right) = 0
.. math::
:label: 4.5-14
z_{\text{nf}}\left( j + 1 \right) = z_{\text{nf}}\left( j \right) + f_{\text{f}}\left( j \right)D_{\text{z}}\left( j \right)
and
.. math::
:label: 4.5-15
z_{\text{ne}}\left( j + 1 \right) = z_{\text{ne}}\left( j \right) + f_{\text{e}}\left( j \right)D_{\text{z}}\left( j \right)
where
.. math::
:label: 4.5-16
f_{\text{f}}\left( j \right) = 1 + \frac{\delta_{\text{f}}\left( j,t \right)}{D_{\text{z}}\left( j \right)}
and
.. math::
:label: 4.5-17
f_{\text{e}}\left( j \right) = 1 + \frac{\delta_{\text{e}}\left( j,t \right)}{D_{\text{z}}\left( j \right)}
|SAS| uses a fuel worth per unit mass,
:math:`R_{\text{f}}\left( j \right)`, and a cladding worth per unit mass,
:math:`R_{\text{e}}\left( j \right)`, defined on the original mesh,
:math:`z_{0}\left( j \right)`. If node :math:`j` has been shifted and
expanded so that
.. math::
:label: 4.5-18
z_{0}\left( j \right) \leq z_{\text{nf}}\left( j \right) \leq z_{0}\left( j + 1 \right)
and
.. math::
:label: 4.5-19
z_{0}\left( j + 1 \right) \leq z_{\text{nf}}\left( j + 1 \right) \leq z_{0}\left( j + 2 \right)
as in :numref:`figure-4.5-1`, then :math:`\Delta\rho_{\text{f}}\left( j \right)`, the
fuel contribution to axial expansion feedback from node :math:`j` is
calculated as
.. math::
:label: 4.5-20
\Delta\rho_{\text{f}}\left( j \right) = m_{\text{f}}\left( j \right)R_{\text{f}}\left( j \right)\frac{z_{0}\left( j + 1 \right) - z_{\text{nf}}\left( j \right)}{z_{\text{nf}}\left( j + 1 \right) - z_{\text{nf}}\left( j \right)} + m_{\text{f}}\left( j \right)R_{\text{f}}\left( j + 1 \right)\frac{z_{\text{nf}}\left( j + 1 \right) - z_{0}\left( j + 1 \right)}{z_{\text{nf}}\left( j + 1 \right) - z_{\text{nf}}\left( j \right)} - m_{\text{f}}\left( j \right)R_{\text{f}}\left( j \right)
A similar expression is used to calculate
:math:`\Delta\rho_{\text{e}}\left( j \right)`, the cladding contribution to
axial expansion feedback. If :math:`z_{\text{nf}}\left( j + 1 \right)`
has expanded past :math:`z_{0}\left( j + 2 \right)`, then a summation
over the appropriate nodes is used instead of :eq:`4.5-20`.
.. _figure-4.5-1:
.. figure:: media/image1.png
:align: center
:figclass: align-center
Original and Expanded Fuel Axial Meshes
By default, |SAS| calculates the axial expansion of fuel and
cladding as described above. In addition to the gap-dependent model
described above, the user can choose cladding-controlled expansion,
independent free expansion, or force balance controlled expansion at all
times. Cladding-controlled expansion may be appropriate for cases where
there is no gap between fuel and cladding and the fuel Young's modulus
is much less than the cladding Young's modulus. In this case, the
cladding expansion is calculated by :eq:`4.5-7`, and the fuel expansion is
set equal to the cladding expansion. Independent free expansion may be
appropriate for cases where the gap is expected to be maintained
throughout the transient; the expansion is then determined by :eq:`4.5-6`
and :eq:`4.5-7`, without the need for force balance. The continual force
balance option may be appropriate for cases where the fuel and cladding
are expected to be in contact at the initial condition, and remain in
contact throughout the transient. In which case, expansion is determined
by :eq:`4.5-8` and :eq:`4.5-9`, regardless of the relative expansion rates of
fuel and cladding.
A feature has been added to include the reactivity effect associated
with the axial thermal expansion of structure during the transient. This
feature may be used, for instance, to treat the thermal expansion of the
subassembly duct walls. Structure is considered to be free, and
expansion is calculated in an analogous way as the independent free
expansion model for fuel and cladding. That is, equations for the
structure expansion are written similar to :eq:`4.5-5` and :eq:`4.5-7`, with no
need for force balance. An expression for the reactivity change in node
:math:`j`, :math:`\Delta \rho_{\text{s}} \left( j \right)`, is written similar to
:eq:`4.5-20`, based on changes in the grid calculated in an analogous
fashion as :eq:`4.5-15`.
The total reactivity change, :math:`\delta k_{\text{d}}`, is
.. math::
:label: 4.5-21
\delta k_{\text{d}} = \epsilon_{\text{ex}}\sum_{\text{j}}\left\lbrack \Delta\rho_{\text{f}}\left( j \right) + \Delta\rho_{\text{e}}\left( j \right) + \Delta\rho_{\text{s}}\left( j \right) \right\rbrack
where :math:`\epsilon_{\text{ex}}` is an effective axial expansion
multiplier.
The summation in :eq:`4.5-21` is only over the core nodes. The axial
blankets are ignored in the fuel expansion feedback calculations. In
order to obtain an accurate value for the axial expansion reactivity
feedback, the fuel worth input for the upper axial blanket nodes must be
the worth of core fuel in the blanket region.
The cladding, fuel, and structure worth used in :eq:`4.5-20` are input in
arrays :sasinp:`CLADRA`, :sasinp:`FUELRA`, and :sasinp:`STRCRA`. The effective axial expansion multiplier in :eq:`4.5-21` is
input as :sasinp:`EXPCFF`. Thermal expansion
coefficients for fuel, cladding, and structure are input in :sasinp:`FUELEX`,
:sasinp:`CLADEX`, and :sasinp:`StructEX`. The
initial mass of cladding and structure in each channel is computed using
the density :sasinp:`DENSS`. The simple axial
expansion reactivity model is invoked by :sasinp:`IAXEXP` with the axial expansion mode option selection in :sasinp:`MODEEX`.
.. _section-4.5.4.2:
DEFORM-4 Axial Expansion Reactivity Model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
For any channel in which the DEFORM-4 module (see :numref:`Chapter %s<section-8>`) has been
specified, the axial expansion reactivity feedback will be calculated as
described in :eq:`4.5-10` through :eq:`4.5-21` but using fuel and cladding
expanded axial mesh heights as calculated by DEFORM-4. The DEFORM-4
effective axial expansion multiplier is entered as :sasinp:`EXPCOF`.
.. _section-4.5.5:
Coolant Density Feedback Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
Reactivity feedback effects from either single-phase coolant density
changes or two-phase coolant boiling are calculated using the input
coolant void reactivity worth table :sasinp:`VOIDRA`. The reactivity feedback from coolant density and voiding changes
is calculated from
.. math::
:label: 4.5-22
\delta k_{{\text{Na}}} = \epsilon_{\text{d}}\sum_{i}{\sum_{j}{\left( \rho_{c} \right)_{ij}\alpha_{ij}}}
where
:math:`\epsilon_{\text{d}}` = sodium density multiplier
:math:`\left( \rho_{c} \right)_{ij}` = coolant void worth in
axial segment :math:`j` of channel :math:`i`
and
:math:`\alpha_{ij}` = average coolant void fraction in segment
:math:`j` of channel :math:`i`.
The local coolant void fraction is calculated based on either a) liquid
coolant density changes from the initial steady-state condition, or b)
combined liquid density and boiling-induced voiding density changes from
the steady-state condition. The sodium density multiplier is entered as :sasinp:`COLFAC`.
.. _section-4.5.6:
Radial Expansion Feedback Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-4.5.6.1:
Simple Radial Expansion Reactivity Model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Two radial expansion feedback models are available in |SAS|: a
simple model described here and a more detailed model described in
:numref:`section-4.5.6.2`.
The simple radial expansion feedback model in |SAS| is based on
a model by Huebotter [4-4]. The radial growth of the core is determined
by the expansion of the lower grid support structure and by the
expansion of the duct walls at the above core load pads. The expansion
of the lower grid support structure is assumed to be proportional to the
rise in the subassembly inlet temperature above its initial steady-state
value. The expansion at the location of the above core load pads is
assumed to be proportional to the change in the average structure
temperature at this location.
The equations actually used in |SAS| are
.. math::
:label: 4.5-23
\delta k_{\text{re}} = C_{\text{re}}\left\lbrack \Delta T_{{\text{in}}} + \frac{{\text{XMC}}}{{\text{XAC}}}\left( \Delta{\overline{T}}_{\text{SLP}} - \Delta T_{{\text{in}}} \right) \right\rbrack
where
:math:`t` = time, s
:math:`\delta k_{\text{re}}` = reactivity change due to radial
expansion, $
:math:`C_{\text{re}}` = radial expansion coefficient, $/K
:math:`\Delta T_{{\text{in}}}` =
:math:`T_{{\text{in}}}\left( t \right) - T_{{\text{in}}}\left( t_{1} \right)`,
K
:math:`T_{{\text{in}}}` = coolant inlet temperature, K
:math:`t_{1}` = time at the end of the first main time step, s
:math:`\text{XMC}` = distance from nozzle support point to core midplane, m
:math:`\text{XAC}` = distance from nozzle support point to above core load pad, m
.. math::
:label: 4.5-24
\Delta{\overline{T}}_{\text{SLP}} = {\overline{T}}_{\text{SLP}}\left( t \right) - {\overline{T}}_{\text{SLP}}\left( t_{1} \right) \quad K
.. math::
:label: 4.5-25
{\overline{T}}_{\text{SLP}}\left( t \right) = \frac{\sum_{\text{i}}{N_{\text{s}}\left( i \right)T_{\text{SLP}}\left( i,t \right)f_{\text{i}}}}{\sum_{\text{i}}{N_{\text{s}}\left( i \right)f_{\text{i}}}}
:math:`{\overline{T}}_{\text{SLP}}` = average structure
temperature at the above core load pads
:math:`T_{\text{SLP}}\left( i,t \right)` = structure temperature (outer
structural radial node) in channel *i* at the axial node corresponding
to the above core load pad
:math:`i` = channel number
:math:`N_{\text{s}}\left( i \right)` = number of subassemblies represented by
channel :math:`i`
:math:`f_{\text{i}} = \begin{cases}
1 & \text{if channel is to be included in the average} \\
0 & \text{if channel is not to be included} \\
\end{cases}`
An option has been included in the code to use the inlet plenum wall
temperature instead of the coolant inlet temperature for
:math:`T_{{\text{in}}}`. This option can be used to account for
time delays in the heating of the lower grid support structure. Note
that for coding simplicity the temperatures at the end of the first main
time step are used as the reference values, rather than using
steady-state temperatures for the reference. This makes very little
difference in the results, since the changes in the inlet temperature
and the outer structure node temperature above the core are normally
extremely small during the first time step. Also, the user can choose
which channels will be included in the averaging of :eq:`4.5-25` by
specifying :math:`f_{\text{i}}` for each channel using :sasinp:`IRDEXP`.
This model was not explicitly set up to account for subassembly bowing
or flowering of the core, but the user can set arbitrary values for
:math:`C_{\text{re}}` and
:math:`\left\lbrack \frac{{\text{XMC}}}{{\text{XAC}}} \right\rbrack`
in :eq:`4.5-23`. Therefore, if the bowing reactivity effect is
proportional to :math:`\Delta{\overline{T}}_{{\text{SLP}}}` or to
:math:`\Delta{\overline{T}}_{{\text{SLP}}} - \Delta{\overline{T}}_{{\text{in}}}`,
then bowing reactivity can be accounted for by adjusting
:math:`C_{\text{re}}` and
:math:`\frac{{\text{XMC}}}{{\text{XAC}}}`, which are
entered as input variables :sasinp:`RDEXPC` and :sasinp:`XMCXAC`. The simple radial expansion model is invoked by specifying
:sasinp:`IRADEX`.
.. _section-4.5.6.2:
Detailed Radial Expansion Reactivity Model
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The basic radial core expansion reactivity feedback model described in
:numref:`section-4.5.6.1` incorporated several major assumptions that restricted
the ability of the model to calculate the reactivity feedback
accurately, particularly in extended transients. These assumptions
include the following:
1. The reactivity feedback is determined solely by thermal expansions of
the grid support plate and load pad region, with all regions having
the same thermal expansion coefficient.
2. The displacement of the core midplane is sufficient to estimate the
reactivity feedback from radial core expansion.
3. All of the subassembly load pads are in contact throughout the
transients.
With these assumptions, the model does not explicitly account for
subassembly bowing, and is not capable of calculating changes in core
loading conditions during the course of a transient. This deficiency
becomes especially important for the extended transients typically
encountered for unprotected accidents.
In order to provide a more mechanistic approach to the calculation of
the radial core expansion reactivity feedback, and to provide a
framework for more detailed modeling as required, the following detailed
model was developed. It is intended to overcome the restrictions
associated with the assumptions listed above, and to allow a more
appropriate use of results from detailed computer code simulations of
core behavior, such as those obtained with NUBOW-3D [4-5].
Model Description
'''''''''''''''''
The approach taken in the development of the detailed model is to relate
the reactivity feedback from radial core expansion to a change in the
size of the core, in the same manner as a uniform dilation of the core
is used to calculate the reactivity effect of a change in effective core
radius. However, rather than maintain the cylindrical shape associated
with a uniform core dilation, an axial profile of core radius is
calculated. During a transient, the changes in the axial profile are
used in conjunction with a worth curve for radial core expansion to
yield the reactivity feedback.
The axial profile of the core radius is obtained from the behavior of an
average subassembly in the outer row of the core. The shape of this
subassembly is determined by the relative location of the grid plate,
the core, the load pads, and any core restraint rings. The shape is also
affected by the thermal gradient across the subassembly, as this
introduces additional bending of the subassembly. The subassembly is
treated as a continuous beam subject to these conditions and restraints.
The basic equation that describes these deflections is the differential
equation of the elastic curve of the beam,
.. math::
:label: 4.5-26
\text{EI}\frac{\text{d}^{2}\text{y}}{\text{dx}^{2}} = M_{\text{x}}
where
:math:`E` = modulus of elasticity, N/m\ :sup:`2`
:math:`I` = moment of inertia of the beam cross-sectional area,
m\ :sup:`4`
:math:`M` = bending moment, N-m
:math:`x` = distance along the beam, m
:math:`y` = distance perpendicular to the beam, m
This equation is solved subject to various loads and moments, depending
on the state of the core. However, since only the displacement is
needed, and the forces and moments are never evaluated, the solution is
not dependent on the value of EI.
At present, the model is only applicable to the "limited free bow" type
of restraint. For this type of restraint, there are load pads just above
the top of the core (ACLP) and at the top of the subassembly (TLP).
There is also a restraint ring (RR), or core former, around the core at
the top load pad elevation. This restraint ring limits the outward
motion of the top of the subassembly. With this type of restraint, the
shape of the subassembly is determined by one of the following
possibilities:
A. Grid Plate/Subassembly Nozzle Clearances Not Exceeded
A.1. No contact at ACLP, RR, or TLP
.. math::
:label: 4.5-27a
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} \quad \text{for } x_{1} \leq x \leq a
A.2. No contact at ACLP or TLP; contact at RR
.. math::
:label: 4.5-27b
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GR}}x \quad \text{for } x_{1} \leq x \leq a
.. math::
S_{\text{GR}} = \frac{R_{3} - \frac{M_{1}}{\text{EI}}\left\lbrack \frac{\left( a - x_{1} \right)\left( L - a \right)}{2} + \frac{\left( a - x_{1} \right)^{2}}{6} \right\rbrack - \frac{M_{2}}{\text{EI}}\frac{\left( L - a \right)^{2}}{2}}{L}
A.3. No contact at ACLP or RR; contact at TLP
.. math::
:label: 4.5-27c
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GR}}x \quad \text{for } x_{1} \leq x \leq a
.. math::
S_{\text{GR}} = \frac{R_{2} - \frac{M_{1}}{\text{EI}}\left\lbrack \frac{\left( a - x_{1} \right)\left( L - a \right)}{2} + \frac{\left( a - x_{1} \right)^{2}}{6} \right\rbrack - \frac{M_{2}}{\text{EI}}\frac{\left( L - a \right)^{2}}{2}}{L}
A.4. Contact at ACLP; no contact at TLP or RR
.. math::
:label: 4.5-27d
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GR}}x \quad \text{for } x_{1} \leq x \leq a
.. math::
S_{\text{GR}} = \frac{R_{1} - \frac{M_{1}}{\text{EI}}\left\lbrack \frac{\left( a - x_{1} \right)^{2}}{6} \right\rbrack}{a}
A.5. Contact at ACLP and RR; no contact at TLP
.. math::
:label: 4.5-27e
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} - \frac{V_{\text{GR}}}{\text{EI}}\frac{x^{3}}{6} - \frac{C_{1}}{\text{EI}}x \quad \text{for } x_{1} \leq x \leq a
.. math::
\frac{V_{\text{GR}}}{\text{EI}} &= \frac{P}{\text{EI}}\left\lbrack 1 - \frac{a}{L} \right\rbrack \\
\frac{P}{\text{EI}} &= \frac{\frac{R_{1}L}{a} - R_{3} + \frac{M_{1}}{\text{6EI}}\left( 2a^{2} - ax_{1} - x_{1}^{2} \right)\left(\frac{L}{a} - 1 \right) + \frac{M_{2}}{\text{EI}}\frac{\left( L - a \right)^{2}}{2}}{\frac{a}{3}\left( L - a \right)^{2}} \\
\frac{C_{1}}{\text{EI}} &= \frac{C_{3}}{\text{EI}} + \frac{M_{1}}{\text{EI}}\frac{x_{1}^{2}}{2\left( a - x_{1} \right)} \\
\frac{C_{3}}{\text{EI}} &= - \frac{R_{1}}{a} + \frac{M_{1}}{\text{EI}}\frac{1}{a - x_{1}}\left\lbrack \frac{a^{2}}{6} - \frac{x_{1}a}{2} + \frac{x_{1}^{3}}{6a} \right\rbrack - \frac{V_{\text{GR}}}{\text{EI}}\frac{a^{2}}{6}
B. Grid Plate/Subassembly Nozzle Clearances Exceeded
B.1. No contact at ACLP; contact at RR
.. math::
:label: 4.5-27f
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GRMAX}}x - \frac{V_{\text{GR}}}{\text{EI}}\frac{x^{3}}{6} - \frac{M_{\text{GR}}}{\text{EI}}\frac{x^{2}}{2} \quad \text{for } x_{1} \leq x \leq a
.. math::
\frac{M_{\text{GR}}}{\text{EI}} &= - \frac{V_{\text{GR}}}{\text{EI}}L \\
\frac{V_{\text{GR}}}{\text{EI}} &= 3\frac{R_{3}}{L^{3}} - \frac{M_{1}}{\text{EI}}\left\lbrack \frac{3\left( a - x_{1} \right)\left( L - a \right) + \left( a - x_{1} \right)^{2}}{2L^{3}} \right\rbrack - \frac{M_{2}}{\text{EI}}\left\lbrack \frac{3\left( L - a \right)^{2}}{2L^{3}} \right\rbrack - 3\frac{S_{\text{GRMAX}}}{L^{2}}
B.2. No Contact at ACLP or RR; contact at TLP
.. math::
:label: 4.5-27g
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GRMAX}}x - \frac{V_{\text{GR}}}{\text{EI}}\frac{x^{3}}{6} - \frac{M_{\text{GR}}}{\text{EI}}\frac{x^{2}}{2} \quad \text{for } x_{1} \leq x \leq a
.. math::
\frac{M_{\text{GR}}}{\text{EI}} &= - \frac{V_{\text{GR}}}{\text{EI}}L \\
\frac{V_{\text{GR}}}{\text{EI}} &= 3\frac{R_{2}}{L^{3}} - \frac{M_{1}}{\text{EI}}\left\lbrack \frac{3\left( a - x_{1} \right)\left( L - a \right) + \left( a - x_{1} \right)^{2}}{2L^{3}} \right\rbrack - \frac{M_{2}}{\text{EI}}\left\lbrack \frac{3\left( L - a \right)^{2}}{2L^{3}} \right\rbrack - 3\frac{S_{\text{GRMAX}}}{L^{2}}
B.3. Contact at ACLP; no contact at TLP or RR
.. math::
:label: 4.5-27h
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GRMAX}}x - \frac{V_{\text{GR}}}{\text{EI}}\frac{x^{3}}{6} - \frac{M_{\text{GR}}}{\text{EI}}\frac{x^{2}}{2} \quad \text{for } x_{1} \leq x \leq a
.. math::
\frac{M_{\text{GR}}}{\text{EI}} &= - \frac{P}{\text{EI}}a \\
\frac{V_{\text{GR}}}{\text{EI}} &= \frac{P}{\text{EI}} \\
\frac{P}{\text{EI}} &= 3\frac{R_{1}}{a^{3}} - \frac{M_{1}}{\text{EI}}\frac{\left( a - x_{1} \right)^{2}}{2a^{3}} - 3\frac{S_{\text{GRMAX}}}{a^{2}}
B.4. Contact at ACLP and RR; no contact at TLP
.. math::
:label: 4.5-27i
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GRMAX}}x - \frac{V_{\text{GR}}}{\text{EI}}\frac{x^{3}}{6} - \frac{M_{\text{GR}}}{\text{EI}}\frac{x^{2}}{2} \quad \text{for } x_{1} \leq x \leq a
.. math::
\frac{M_{\text{GR}}}{\text{EI}} &= \frac{P}{\text{EI}} \left( L-a \right) - \frac{V_{\text{GR}}}{\text{EI}}L
\frac{V_{\text{GR}}}{\text{EI}} &= \frac{P}{\text{EI}} \left\lbrack 1 + \frac{a^3}{2L^3} - \frac{3a^2}{2L^2} \right\rbrack + \frac{3R_3}{L^3} - \frac{M_1}{\text{EI}} \frac{\left\lbrack 3 \left( a-x_1 \right) \left( L-a \right) + \left( a-x_1 \right)^2 \right\rbrack}{2L^3} - \frac{M_2}{\text{EI}} \frac{\left\lbrack 3 \left( L-a \right)^2 \right\rbrack}{2L^3} - \frac{3S_{\text{GRMAX}}}{L^2}
.. math::
\frac{P}{\text{EI}} = \left\lbrack R_3 \frac{\left( \frac{3a^2}{L^2} - \frac{a^3}{L^3} \right)}{2} - R_1 - \frac{M_1}{\text{EI}} \left\lbrack \left( \frac{3a^2}{L^2} - \frac{a^3}{L^3} \right) \frac{ \left( L-a \right)}{4} + \left( \frac{a^2}{4L^2} - \frac{a^3}{12L^3} - \frac{1}{6} \right) \left( a-x_1 \right) \right\rbrack \left( a-x_1 \right) \right. \\
\left. - \frac{M_2}{\text{EI}} \left( \frac{3a^2}{L^2} - \frac{a^3}{L^3} \right) \frac{ \left( L-a \right)^2}{4} + S_{\text{GRMAX}} \left(a + \frac{a^3}{L^2} - \frac{3a^2}{2L} \right) \right\rbrack \bigg/ a^3 \left( \frac{a^3}{12L^3} - \frac{a^2}{2L^2} + \frac{3a}{4L} - \frac{1}{3} \right)
B.5. Contact at ACLP and TLP; no contact at RR
.. math::
:label: 4.5-27j
y\left( x \right) = \frac{M_{1}}{6\text{EI}\left( a - x_{1} \right)}\left( x - x_{1} \right)^{3} + S_{\text{GRMAX}}x - \frac{V_{\text{GR}}}{\text{EI}}\frac{x^{3}}{6} - \frac{M_{\text{GR}}}{\text{EI}}\frac{x^{2}}{2} \quad \text{for } x_{1} \leq x \leq a \\
.. math::
\frac{M_{\text{GR}}}{\text{EI}} &= \frac{P}{\text{EI}} \left( L-a \right) - \frac{V_{\text{GR}}}{\text{EI}}L
\frac{V_{\text{GR}}}{\text{EI}} &= \frac{P}{\text{EI}} \left\lbrack 1 + \frac{a^3}{2L^3} - \frac{3a^2}{2L^2} \right\rbrack + \frac{3R_2}{L^3} - \frac{M_1}{\text{EI}} \frac{\left\lbrack 3 \left( a-x_1 \right) \left( L-a \right) + \left( a-x_1 \right)^2 \right\rbrack}{2L^3} - \frac{M_2}{\text{EI}} \frac{\left\lbrack 3 \left( L-a \right)^2 \right\rbrack}{2L^3} - \frac{3S_{\text{GRMAX}}}{L^2}
.. math::
\frac{P}{\text{EI}} = \left\lbrack R_2 \frac{\left( \frac{3a^2}{L^2} - \frac{a^3}{L^3} \right)}{2} - R_1 - \frac{M_1}{\text{EI}} \left\lbrack \left( \frac{3a^2}{L^2} - \frac{a^3}{L^3} \right) \frac{ \left( L-a \right)}{4} + \left( \frac{a^2}{4L^2} - \frac{a^3}{12L^3} - \frac{1}{6} \right) \left( a-x_1 \right) \right\rbrack \left( a-x_1 \right) \right. \\
\left. - \frac{M_2}{\text{EI}} \left( \frac{3a^2}{L^2} - \frac{a^3}{L^3} \right) \frac{ \left( L-a \right)^2}{4} + S_{\text{GRMAX}} \left(a + \frac{a^3}{L^2} - \frac{3a^2}{2L} \right) \right\rbrack \bigg/ a^3 \left( \frac{a^3}{12L^3} - \frac{a^2}{2L^2} + \frac{3a}{4L} - \frac{1}{3} \right)
where
:math:`y` = radial displacement with respect to the core radius at the
grid plate
:math:`R_{1}` = minimum core radius at the above-core load pad with
respect to the core radius at the grid plate
:math:`R_{2}` = minimum core radius at the top load pad with respect to
the core radius at the grid plate
:math:`R_{3}` = maximum core radius at the restraint ring with respect
to the core radius at the grid plate
:math:`x` = axial elevation
:math:`x_{1}` = elevation of the lower axial blanket/lower reflector
interface
:math:`a` = elevation of the above core load pad
:math:`L` = elevation of the top load pad
:math:`S_{\text{GR}}` = subassembly slope with respect to vertical at
the grid plate
:math:`S_{\text{GRMAX}}` = maximum subassembly slope with respect to
vertical at the grid plate
:math:`\frac{M_{1}}{\text{EI}}` = thermally induced bending moment in
the core region
:math:`\frac{M_{2}}{\text{EI}}` = thermally induced bending moment in
the above core region
:math:`V_{\text{GR}}` = radial reaction at the grid plate
:math:`M_{\text{GR}}` = applied moment at the grid plate
:math:`P` = radial force at the above core load pad
:math:`\text{EI}` = modulus of elasticity times the moment of inertia of
the subassembly cross-sectional area
The use of the word contact in this context implies that either the
outward motion is sufficient for the restraint ring to apply a force
preventing further outward motion, or there is sufficient inward motion
such that all of the intra-subassembly gaps in the load pad region(s)
are eliminated, thus generating a force preventing further inward
motion. A grid plate/subassembly nozzle clearance is required for the
replacement of subassemblies, and results in a corresponding maximum
possible deviation of the subassembly from vertical at the grid plate.
When this clearance is exceeded, a moment is applied to the subassembly
at the nozzle.
The subassembly is also subjected to a bending moment related to the
temperature difference of opposite hex can walls within the subassembly.
This temperature difference is converted into an equivalent bending
moment, as described in the following section. The temperature
difference increases linearly through the core region, from the lower
axial blanket to the upper axial blanket. In the upper subassembly
region, the temperature difference is assumed to be constant from the
upper axial blanket to the top of the subassembly, with the value
varying with time as the transient progresses.
The major assumptions incorporated in this model at present include the
uniform distribution of core material in the radial direction at every
axial elevation and the completely rigid subassembly load pads.
Distributing the material uniformly at each axial elevation as the core
radius changes is the same assumption used for the uniform core dilation
calculation to obtain the radial core expansion reactivity feedback
coefficient, and implies that all of the subassemblies are moving in
proportion to their distance from the center of the core. The radial
expansion worth gradient and the intra-subassembly temperature gradients
tend to be greatest at the edge of the active core, with the result that
most of the reactivity feedback effect comes from movement of the outer
row of subassemblies. Any movement in the central region of the core
that is not proportional to the distance from the center of the core is
expected to cause a minor effect. The accuracy of this assumption for
any specific reactor core can be evaluated by comparison with results
from NUBOW-3D [4-5].
The use of completely rigid subassembly load pads provides slightly
greater expansion of the core during certain events as in an unprotected
loss-of-flow, and less for other accidents, such as an unprotected
loss-of-heat-sink transient. The error introduced by using this
assumption is on the order of 15% to 20%, and can be design dependent.
The incorporation of deformable load pads and "bridging" of
subassemblies may be desirable, and is being considered for a future
version of this model. The accuracy of this assumption can also be
checked by comparison with NUBOW-3D.
Since the expression given by :eq:`4.5-26` is solved for the various core
loading possibilities listed above, the resulting algebraic formulas are
incorporated in the code. This avoids the need for a finite difference
solution of :eq:`4.5-26`, simplifying the computer coding and providing a
rapid calculation of the core shape.
Code Description and Input Requirements
'''''''''''''''''''''''''''''''''''''''
This section contains a description of the algorithm used for
calculating the appropriate core shape. This calculation is performed at
the start of the transient to establish the steady-state core
configuration, and for every step during the transient. The use of the
detailed radial core expansion reactivity feedback model requires some
of the same input as the simple model plus several other variables.
These will be discussed as their use occurs.
The optional model is activated by setting
:math:`|` :sasinp:`IRADEX` :math:`| =` 4 or 5, where a value of ±5
gives a much more detailed printout, while ±4
only gives results in the PSHORT printout. The first step is to
calculate the average temperature of the above-core load pad (ACLP) and
top load pad (TLP) regions. This is done using :eq:`4.5-25`, as in the
basic model. The model sets the location of the ACLP, so that :sasinp:`JSTRDX` does not need to be input. The temperature of the
grid plate can be given by either the inlet coolant temperature, or by
the wall temperature of the compressible volume used to represent the
inlet plenum in PRIMAR-4. This option is discussed in detail for the
basic model, and is activated by setting :sasinp:`IRADEX` to the
appropriate negative value.
The next step is to calculate the equivalent core radius at the grid
plate, the ACLP and the TLP. For this calculation, the following input
is needed:
.. list-table::
:header-rows: 1
:align: center
:widths: 1,4
* - Input
- Description
* - :sasinp:`NSUBTC`
- total number of subassemblies in the active core region, including internal blankets and control subassemblies
* - :sasinp:`MTGRD`
- material used for the grid plate, where
1 = 316 SS
2 = HT-9
* - :sasinp:`MTACLP`
:sasinp:`MTTLP`
- material used for the ACLP and TLP, where
1 = 316 SS
2 = HT-9
* - :sasinp:`PITCHG`
- subassembly pitch at the grid plate at reference temperature :sasinp:`TR`
* - :sasinp:`PITCHA`
- flat-to-flat dimension across the ACLP at reference temperature :sasinp:`TR`
* - :sasinp:`PITCHT`
- flat-to-flat dimension across the TLP at reference temperature :sasinp:`TR`
Using the pitch at the grid plate along with the steady-state inlet
temperature, the equivalent radius of the subassemblies in active core
is calculated. As part of the calculation, there is a call to subroutine
THRMEX that gives the material thermal expansion as a function of
temperature for either 316 SS or HT-9. For the load pad regions, a
minimum allowable core radius is calculated based on the size of the
load pad region when all of the load pads are pushed together and there
are no intra-subassembly gaps. This is possible since the model assumes
the load pads all have the same temperature, as described above.
In addition to these dimensions, there are two other geometric
constraints, as follows:
.. list-table::
:header-rows: 1
:align: center
:widths: 1,4
* - Input
- Description
* - :sasinp:`SLLMAX`
- maximum allowable slope of the subassembly at the grid plate with respect to vertical, based on subassembly nozzle/grid plate clearances and dimensions
* - :sasinp:`TLPRRC`
- clearance between the top load pads and the restraint ring
The value for :sasinp:`SLLMAX` is calculated from the radial clearances of the
subassembly nozzle/grid plate socket connection and the length of the
connection. The maximum tilt of the subassembly occurs when the maximum
radial motion of the nozzle is used, usually inward at the bottom of the
nozzle and outward at the top. This number is design-dependent and can
vary greatly, even when the subassembly sizes are comparable. The
clearance between the top load pad region and the restraint ring is
determined by the maximum clearance that would occur between the
subassemblies in the outer row of active core and the first row of
radial blankets when all of the core subassembly load pads are pushed
inward together and all of the radial blanket load pads are pushed
outward against the restraint ring. The top load pad/restraint ring
clearance is kept constant throughout the transient, i.e. the restraint
ring expands as the top load pads expand thermally. This approximation
tends to be conservative. Default values for these two input variables
have been provided for cases where such detailed design information is
not available. Design information should be used wherever possible, as
the results can be especially sensitive to the value for :sasinp:`SLLMAX`.
The only other input variable required for determining the core shape
are those related to the thermally-induced bending moment:
.. list-table::
:header-rows: 1
:align: center
:widths: 1,4
* - Input
- Description
* - :sasinp:`BNDMM1`
- applied bending moment at the top of the core region, representing the flat-to-flat temperature difference in the radial direction for the subassemblies at the outer edge of the active core
* - :sasinp:`BNDMM2`
- applied bending moment in the region above the core, representing the flat-to-flat temperature difference in the radial direction in this region for the subassemblies at the outer edge of active core
The data on the temperature difference must be obtained from a code
which performs detailed calculations of the steady-state subassembly
temperatures with intersubassembly heat transfer, such as SUPERENERGY-2
[4-6]. Default values are included if such information is not available.
The input variables can then be calculated using :eq:`4.5-28`.
.. math::
:label: 4.5-28
\text{BNDMM}1 = \frac{\alpha \Delta T}{D}
where
:math:`\alpha` = mean thermal expansion coefficient of the subassembly
hexcan, :math:`1/K`
:math:`\Delta T` = flat-to-flat temperature difference, K
:math:`D` = hexcan flat-to-flat dimension, m
The model uses a linear variation in bending moment through the core
region, from zero at the bottom of the core to :sasinp:`BNDMM1` at the top. The
bending moment :sasinp:`BNDMM2` is applied uniformly from the top of the core to
the top of the subassembly. In the transient, the bending moments are
modified in proportion to the power-to-flow ratio changes.
Once all of these conditions have been calculated, the algorithm goes
through a series of logic to determine the correct combination of forces
and moments, as given above. The subassemblies are assumed to be
vertical at the grid plate unless there are forces at the ACLP or TLP,
or both, which would cause the subassembly to tilt. With the appropriate
choice, the algebraic equation corresponding to that loading condition
is evaluated for every axial node in the core region. In the printout,
the algebraic equation selected is indicated by "CORE SHAPE MODEL =",
where the value printed corresponds to the particular case as listed:
**CORE SHAPE MODEL**
.. list-table::
:header-rows: 1
:align: center
:widths: auto
* -
- Steady-State
- Transient
* - Grid plate/subassembly nozzle clearances not exceeded
-
-
* - No contact at ACLP, RR or TLP
- 1.0
- 21.0
* - No contact at ACLP; contact at RR
- 2.0
- 22.0
* - No contact at ACLP; contact at TLP
- 3.0
- 23.0
* - Contact at ACLP; no contact at TLP or RR
- 4.0
- 24.0
* - Contact at ACLP and RR
- 5.0
- 25.0
* - Grid plate/subassembly nozzle clearances exceeded
-
-
* - No contact at ACLP; contact at RR
- 8.0
- 28.0
* - No contact at ACLP; contact at TLP
- 9.0
- 29.0
* - Contact at ACLP; no contact at TLP or RR
- 10.0
- 30.0
* - Contact at ACLP and RR
- 11.0
- 31.0
* - Contact at ACLP and TLP
- 12.0
- 32.0
In the steady-state, the axial profile of core radius is stored for
comparison during the transient. For each step during the transient, the
process is repeated and the difference in core radius at each elevation
is calculated.
The reactivity worth curve is based on the radial expansion coefficient
for a uniform core dilation,
.. list-table::
:header-rows: 1
:align: center
:widths: 1,4
* - Input
- Description
* - :sasinp:`RDEXCF`
- radial expansion coefficient for a uniform core dilation, $/m
This coefficient is then proportioned among the axial fuel nodes
according to the axial power shape. The resulting worth curve provides
the radial displacement worth for each axial node in the core. When used
in combination with the deflections from steady-state described above,
the reactivity feedback from each axial node is determined and the total
reactivity feedback from radial core expansion is calculated by summing
over the axial fuel nodes. As stated above, this is a very rapid
calculation due to the use of algebraic expressions for the subassembly
shape, which are the solutions given in :eq:`4.5-27a` for the various
combinations of force and moments.
.. _section-4.5.7:
Control Rod Drive Expansion Feedback Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
For the control rod drive feedback model, it is assumed that the control
rod drives are washed by the outlet coolant from the core. Thermal
expansion of the drives due to a rise in core outlet temperature will
cause the control rods to be inserted further into the core, providing a
negative reactivity component. On the other hand, if the control rod
drives are supported on the vessel head, and if the core is supported by
the vessel walls, then heating the vessel walls will either lower the
core or raise the control rod drive supports, leading to a positive
reactivity component. Both the control drive expansion and the vessel
wall expansion are accounted for in |SAS|. This model is
invoked with input variable :sasinp:`ICREXP`.
A simple one-node treatment is used for calculating the temperature of
the control rod drives. The equation used is
.. math::
:label: 4.5-29
M_{\text{cr}}C_{\text{cr}}\frac{\text{dT}_{\text{cr}}}{\text{dt}} = h_{\text{cr}}A_{\text{cr}}\left( T_{\text{ui}} - T_{\text{cr}} \right)
where
:math:`M_{\text{cr}}` = mass of the control rod drivelines, kg
:math:`C_{\text{cr}}` = specific heat of the rod drivelines, J/kg-K
:math:`T_{\text{cr}}` = control rod drive temperature, K
:math:`T_{\text{ui}}` = coolant temperature in the upper internal
structure region, K
:math:`h_{\text{cr}}` = heat transfer coefficient between the coolant
and the control rod drive, W/m\ :sup:`2`-K
:math:`A_{\text{cr}}` = heat transfer area between the coolant and the
control rod drive, :math:`\text{m}^2`
:math:`t` = time, s
The product of the control rod driveline mass and specific heat is
entered as input variable :sasinp:`CRDMC`, and the
product of the driveline heat transfer coefficient and area is entered
as :sasinp:`CRDHA`.
The coolant temperature in the upper internal structure region is
calculated using
.. math::
:label: 4.5-30
\frac{\text{dT}_{\text{ui}}}{\text{dt}} = w_{\text{c}}\frac{T_{\text{mm}} - T_{\text{ui}}}{\rho_{\text{u}}V_{\text{ui}}}
where
:math:`w_{\text{c}}` = core outlet flow rate
:math:`T_{\text{mm}}` = mixed mean coolant outlet temperature
:math:`\rho_{\text{u}}` = sodium density in the outlet plenum
:math:`V_{\text{ui}}` = coolant volume in the upper internal structure
region
Initially both :math:`T_{\text{ui}}` and :math:`T_{\text{cr}}` are set
equal to the steady-state mixed mean outlet temperature. The UIS volume,
:math:`V_{\text{ui}}` is entered as input variable :sasinp:`UIVOL`.
During the transient calculation, :eq:`4.5-30` is approximated with
.. math::
:label: 4.5-31
T_{\text{ui}}\left( t + \Delta t \right) = \frac{T_{\text{ui}}\left( t \right) + x{T}_{\text{mm}}\left( t + \Delta t \right)}{1 + x}
where
.. math::
:label: 4.5-32
x = \frac{w_{\text{c}}\left( t + \Delta t \right)\Delta t}{\rho_{\text{u}}V_{\text{ui}}}
For this calculation, only channels with positive outlet flow rates
contribute to :math:`w_{\text{c}}` and :math:`T_{\text{mm}}`. :eq:`4.5-29` is
approximated as
.. math::
:label: 4.5-33
M_{\text{cr}}C_{\text{cr}}\frac{T_{\text{cr}}\left( t + \Delta t \right) - T_{\text{cr}}\left( t \right)}{\Delta t} = h_{\text{cr}}A_{\text{cr}}\left\lbrack T_{\text{ui}}\left( t + \Delta t \right) - T_{\text{cr}}\left( t + \Delta t \right) \right\rbrack
or
.. math::
:label: 4.5-34
T_{\text{cr}}\left( t + \Delta t \right) = \frac{T_{\text{cr}}\left( t \right) + d\ T_{\text{ui}}\left( t + \Delta t \right)}{1 + d}
where
.. math::
:label: 4.5-35
d = \frac{h_{\text{cr}}A_{\text{cr}}}{M_{\text{cr}}C_{\text{cr}}}\Delta t
The axial expansion of the control rod drive,
:math:`\Delta z_{\text{cr}}`, is calculated as
.. math::
:label: 4.5-36
\Delta z_{\text{cr}}\left( t \right) = L_{\text{cr}}\alpha_{\text{cr}}\left\lbrack T_{\text{cr}}\left( t \right) - T_{\text{cr}}\left( 0 \right) \right\rbrack
where :math:`L_{\text{cr}}` is the length of the control rod drive
washed by the outlet sodium, and :math:`\alpha_{\text{cr}}` is the
thermal expansion coefficient. These data are entered as :sasinp:`CRDLEN` and
:sasinp:`CRDEXP`.
The vessel wall expansion is calculated on the basis of the temperatures
calculated by PRIMAR-4 for the walls of the liquid elements or
compressible volumes that represent the vessel wall. For a typical pool
reactor, the vessel wall would be the wall of the cold pool; but for
some reactor designs a number of compressible volumes and liquid
elements would be used to represent the vessel wall. The expansion of
the vessel wall, :math:`\Delta z_{\text{v}}`, is calculated as
.. math::
:label: 4.5-37
\Delta z_{\text{v}} = \sum_{\text{k}}{\left\lbrack {\overline{T}}_{\text{k}}\left( t \right) - {\overline{T}}_{\text{k}}\left( 0 \right) \right\rbrack L_{\text{k}}\alpha_{\text{k}}}
where
:math:`{\overline{T}}_{\text{k}}` = average wall temperature of the
k-th compressible volume or liquid element in the vessel wall
:math:`L_{\text{k}}` = length of the vessel wall represented by the k-th
compressible volume or element
:math:`\alpha_{\text{k}}` = thermal expansion coefficient of the vessel wall
The net movement, :math:`\Delta z_{\text{n}}`, is calculated as
.. math::
:label: 4.5-38
\Delta z_{\text{n}} = \Delta z_{\text{cr}} - \Delta z_{\text{v}}
and the reactivity feedback, :math:`\delta k_{\text{cr}}`, is calculated
as
.. math::
:label: 4.5-39
\delta k_{\text{cr}} = a_{\text{cr}}\Delta z_{\text{n}} + b_{\text{cr}}\left( \Delta z_{\text{n}} \right)^{2}
where :math:`a_{\text{cr}}` and :math:`b_{\text{cr}}` are user-supplied
coefficients entered as :sasinp:`ACRDEX` and :sasinp:`BCRDEX`.
A multiple-node version of this model is in development, but has not
been verified for production use.
.. _section-4.5.8:
Fuel and Cladding Relocation Feedback Reactivity
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The fuel and cladding relocation reactivity feedbacks are calculated as
the product of the input material reactivity worth (:sasinp:`CLADRA` and :sasinp:`FUELRA`) and the change in the axial
material mass distribution since the initial steady-state condition.
Symbolically this is represented by
.. math::
\delta k \left( t \right) = \sum_{i}{\sum_{j}{\left( \frac{\Delta k}{\Delta m} \right)_{ij}\left\lbrack m_{ij}\left( t \right) - m_{ij}\left( 0 \right) \right\rbrack}}
where :math:`\left( \frac{\Delta k}{\Delta m} \right)_{ij}` is
the material reactivity worth in axial node :math:`j` of channel
:math:`i`, :math:`m_{ij}\left( t \right)` is the material mass at
axial node :math:`j` in channel :math:`i` at time :math:`t`, and
:math:`m_{ij}\left( 0 \right)` is the initial steady-state node
material mass. The input worth curves may be input on the fuel (MZ)
mesh, or the coolant (MZC) mesh according to the input value of :sasinp:`IREACZ`.
The initial and transient axial mass
distributions are computed internally from input design geometry and
density data, and the fuel and cladding relocation models and solutions.
.. _section-4.5.9:
EBR-II Reactivity Feedback model
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A reactor-specific set of reactivity feedback correlations has been
implemented in |SAS| for analysis of the EBR-II reactor and
plant. The formulation of these correlations is based on the reactivity
feedback model used in the NATDEMO computer program [4-7, 4-8], which is
based on analysis of reactivity temperature coefficients in EBR-II Run
93 [4-9]. This documentation of the |SAS| EBR-II reactivity
feedback model is taken from notes provided by White [4-10] and Herzog
[4-11].
In |SAS|, the EBR-II reactivity feedback is assumed to be
composed of nine components. These components are: 1) fuel expansion, 2)
coolant expansion, 3) stainless steel expansion, 4) axial reflector
sodium expansion, 5) radial reflector sodium expansion, 6) fuel Doppler
effect, 7) control rod bank expansion, 8) upper grid plate expansion,
and 9) core subassembly bowing.
Each of these effects will be considered independently and the method in
which the feedback magnitude is determined will be given. The source and
nature (linear or nonlinear) of the term will also be discussed.
The reactivity in a steady state critical reactor is defined as zero. In
the EBR-II feedback calculation, a parameter, which shall be named
:math:`\zeta\left( 0 \right)`, is defined at time zero (this treatment
is not used in the other reactivity calculations performed by
|SAS|). At later times, :math:`\zeta\left( t \right)` is
calculated. The difference,
:math:`\delta k \left( t \right) = \zeta\left( t \right) - \zeta\left( 0 \right)`,
is the reactivity introduced from feedback effects at time :math:`t`.
.. _section-4.5.9.1:
Fuel Expansion
^^^^^^^^^^^^^^
Both radial and axial expansion of the fuel are considered to be linear
terms.
.. _section-4.5.9.1.1:
Axial Expansion
'''''''''''''''
The method used is derived from Ref. 4-12. It has been modified to
include the possibility that contact can occur between the fuel and
cladding, thus altering the expression for the amount of expansion.
Correlations from the Metallic Fuels Handbook [4-13] are used to
evaluate the linear expansion coefficient and Young's modulus for the
fuel and cladding types used in EBR-II fuel elements. First, the
following correlation is used to determine the coefficient of linear
expansion for U-5FS fuel:
.. math::
\alpha_{\text{LF}} = \begin{cases}
1.264 \times 10^{- 5} - 1.7964 \times 10^{- 9}\overline{T} + 2.0532 \times {10^{- 11}\overline{T}}^{2} & \overline{T} < 941 \\
1.73 \times 10^{- 5} & 941 < \overline{T} < 1048 \\
1.775 \times 10^{- 5} + 8.761 \times 10^{- 9}\overline{T} - 3.717 \times 10^{- 12}{\overline{T}}^{2} & 1048 < \overline{T} < 1480 \\
2.55 \times 10^{- 5} & 1480 < \overline{T} \\
\end{cases}
and for U-10Zr fuel, the following correlation is used:
.. math::
\alpha_{\text{LF}} = \begin{cases}
1.658 \times 10^{- 5} - 2.104 \times 10^{- 8}\overline{T} + 3.345 \times 10^{- 11}{\overline{T}}^{2} & \overline{T} < 900 \\
2.25 \times 10^{- 5} & 900 < \overline{T} \\
\end{cases}
and for U-10Zr-20Pu fuel, the following correlation is used:
.. math::
\alpha_{\text{LF}} = \begin{cases}
1.73 \times 10^{- 5} & \overline{T} < 868 \\
1.98 \times 10^{- 5} & 868 < \overline{T} \\
\end{cases}
The :math:`\overline{T}` used here is the mass-average temperature for
the fuel in a particular channel for a particular axial layer.
For SS316 or D-9 cladding, the steel linear thermal expansion
coefficient is calculated from:
.. math:: \alpha_{\text{Lss}} = 5.189 \times 10^{- 5} - 6.4375 \times 10^{- 4}{\overline{T}}^{- \frac{1}{2}} - 1.00862 \times 10^{- 8}\overline{T}
and for HT-9 cladding the linear thermal expansion coefficient is
calculated from:
.. math:: \alpha_{\text{Lss}} = 1.62307 \times 10^{- 6} + 2.84714 \times 10^{- 8}\overline{T} - 1.65103 \times 10^{- 11}{\overline{T}}^{2}
where :math:`\overline{T}` is the mass-average cladding
temperature at a particular axial location in a particular channel.
Correlations [4-13] for the Young's modulus of metal fuels are functions
of the fuel temperature and porosity :sasinp:`PRSTY`. The following correlation is used to determine the Young's
modulus of U-5Fs fuel:
.. math::
Y_{\text{f}} = \begin{cases}
1.5123 \times 10^{11}\left( 1 - 1.2P \right)\left( 1 - 1.06\frac{\overline{T} - 588}{1405} \right) & \overline{T} < 923 \\
1.5123 \times 10^{- 11}\left( 1 - 1.2P \right)\left( 1 - 1.06\frac{\overline{T} - 588}{1405} \right) - 0.3Y_{\text{f}}\left( 923 \right) & 923 < \overline{T} \\
\end{cases}
For U-10Zr fuel, the following correlation is used:
.. math::
Y_{\text{f}} = \begin{cases}
1.4349 \times 10^{11}\left( 1 - 1.2P \right)\left( 1 - 1.06\frac{\overline{T} - 588}{1405} \right) & \overline{T} < 923 \\
1.4349 \times 10^{- 11}\left( 1 - 1.2P \right)\left( 1 - 1.06\frac{\overline{T} - 588}{1405} \right) - 0.3Y_{f}\left( 923 \right) & 923 < \overline{T} \\
\end{cases}
And for U-10Zr-20 Pu fuel, the following correlation is used:
.. math::
Y_{\text{f}} = \begin{cases}
1.1149 \times 10^{11}\left( 1 - 1.2P \right)\left( 1 - 1.06\frac{\overline{T} - 588}{1405} \right) & \overline{T} < 923 \\
1.1149 \times 10^{- 11}\left( 1 - 1.2P \right)\left( 1 - 1.06\frac{\overline{T} - 588}{1405} \right) - 0.3Y_{\text{f}}\left( 923 \right) & 923 < \overline{T} \\
\end{cases}
For SS316 or D-9 cladding, the steel Young's modulus is calculated from:
.. math::
Y_{\text{ss}} = 2.01 \times 10^{11} - 7.929 \times 10^{7}\overline{T}
and for HT-9 cladding the Young's modulus is calculated from:
.. math::
Y_{\text{ss}} = 2.137 \times 10^{11} - 1.0274 \times 10^{8}\overline{T}
The expression for the expansion in the fuel is
.. math::
\delta_{\text{f}} = \alpha_{\text{Lf}}\Delta T_{\text{f}}
for no contact (fuel burnup < 2.9%), and
.. math::
\delta_{\text{f}} = \frac{\alpha_{\text{Lss}}\Delta T_{\text{ss}}Y_{\text{ss}}A_{\text{ss}} + \alpha_{\text{Lf}}\Delta T_{\text{f}}Y_{\text{f}}A_{\text{f}}}{Y_{\text{ss}}A_{\text{ss}} + Y_{\text{f}}A_{\text{f}}}
for contact (fuel burnup > 2.9%), where :math:`\alpha_{\text{Lf}}` and
:math:`\alpha_{\text{Lss}}` are the thermal expansion coefficients for
the fuel and stainless steel, :math:`\Delta T_{\text{f}}` and
:math:`\Delta T_{\text{ss}}` are the temperature changes for the fuel
and stainless steel, :math:`Y_{\text{f}}` and :math:`Y_{\text{ss}}` are Young's
modulus for the fuel and stainless steel, and :math:`A_{\text{f}}` and
:math:`A_{\text{ss}}` are the cross sectional areas of the fuel and
stainless steel.
As described above, the change in temperature is not calculated directly
by the code. Instead, the code expands the expression into two terms:
one involving steady state conditions, :math:`\zeta\left( 0 \right)`,
and the second involving the conditions at some time :math:`t`,
:math:`\zeta\left( t \right)`. The reactivity feedback is determined
using the expression
:math:`\delta k \left( t \right) = \zeta\left( t \right) - \zeta\left( 0 \right)`.
The parameter :math:`\zeta` due to the expansion can be expressed as:
.. math::
\zeta_{\text{fa}} = - \left\lbrack \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{\text{f}} - \left( \frac{\partial \text{k}}{\frac{\partial \text{H}}{H}} \right)_{\text{f}} \right\rbrack\frac{\alpha_{\text{Lss}}{\overline{T}}_{\text{ss}}Y_{\text{ss}}A_{\text{ss}} + \alpha_{\text{Lf}}{\overline{T}}_{\text{f}}Y_{\text{f}}A_{\text{f}}}{\left( Y_{\text{ss}}A_{\text{ss}} + Y_{\text{f}}A_{\text{f}} \right)\beta}
where :math:`\beta` is the delayed neutron fraction,
:math:`\zeta_{\text{fa}}` is the reactivity due to fuel axial expansion
in dollars,
:math:`\left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{\text{f}}` is
the change in :math:`k` with a relative change in the number density of the
fuel, and
:math:`\left( \frac{\partial \text{k}}{\frac{\partial \text{H}}{H}} \right)_{\text{f}}` is
the change in :math:`k` with a relative change in the height of the fuel, for
the case when the clad and fuel are in contact and
.. math::
\zeta_{\text{fa}} = - \left\lbrack \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{\text{f}} - \left( \frac{\partial \text{k}}{\frac{\partial \text{H}}{H}} \right)_{\text{f}} \right\rbrack\frac{\alpha_{\text{Lf}}{\overline{T}}_{\text{f}}}{\beta}
for the case where there is no contact. The two partial derivatives in
both of these expressions are entered as input to |SAS| as
input variables :sasinp:`YKNF` and :sasinp:`YKHF`.
Because some channels will have both assemblies in which the cladding
and fuel do contact and assemblies in which they do not contact, the
code uses the equation:
.. math::
\zeta_{\text{fa}} = - \left\lbrack \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{\text{f}} - \left( \frac{\partial \text{k}}{\frac{\partial \text{H}}{H}} \right)_{\text{f}} \right\rbrack\frac{\alpha_{\text{Lss}}{\overline{T}}_{\text{ss}}Y_{\text{ss}}A_{\text{ss}} + \alpha_{\text{Lf}}{\overline{T}}_{\text{f}}Y_{\text{f}}A_{\text{f}}}{\left( Y_{\text{ss}}A_{\text{ss}} + Y_{\text{f}}A_{\text{f}} \right)\beta}\left( 1 - F_{{\text{LowBU}}} \right) \\
- \left\lbrack \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{\text{f}} - \left( \frac{\partial \text{k}}{\frac{\partial \text{H}}{H}} \right)_{\text{f}} \right\rbrack \frac{\alpha_{\text{Lf}}{\overline{T}}_{\text{f}}}{\beta}F_{{\text{LowBU}}} \\
where :math:`F_{\text{LowBU}}` is the fraction of pins with < 2.9%
burnup. This fraction is entered to |SAS| as variable :sasinp:`FLOWBU`.
The temperature of the stainless steel (cladding) is determined using a
weighted mass-average of the cladding temperature. The cladding is
composed of three radial nodes. In the weighting, the middle node is
given twice the weight of the two other nodes.
.. _section-4.5.9.1.2:
Radial Expansion
''''''''''''''''
The radial expansion of the fuel contributes to the reactivity
principally by displacing the bond gap sodium from the core. This
displacement is of the sodium between the cladding and the fuel. Once
the fuel reaches a burnup of 2.9% the cladding is in contact with the
fuel and there is no additional displacement, and therefore, no
additional reactivity change. The amount by which the fuel volume
increases can be determined using a two dimensional isotropic
approximation, i.e.: fuel volume increase is
:math:`V_{\text{f}}{2\alpha}_{\text{Lf}}`. The fractional decrease in the
sodium volume is then
:math:`\frac{V_{\text{f}}2\alpha_{\text{Lf}}}{V_{{\text{Na}}}}`. The
:math:`\zeta` value associated with this sodium expulsion is:
.. math::
\zeta_{\text{fr}} = \left( - \frac{2V_{\text{f}}\ \alpha_{\text{Lf}}}{V_{{\text{Na}}}} \right)\left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{{\text{Na}}}\frac{{\overline{T}}_{\text{f}}}{\beta}
when there is no contact and :math:`\zeta_{\text{fr}} = 0` when there is
contact. In these expressions,
:math:`\left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{{\text{Na}}}` is
the change in :math:`k` with respect to a relative change in the number
density of the sodium, which is entered as input variable :sasinp:`YKNNA`.
Therefore, the value of :math:`\zeta` for the
radial expansion of the fuel can be expressed as:
.. math::
\zeta_{\text{fr}} = \left( - \frac{2V_{\text{f}}\alpha_{\text{Lf}}}{V_{{\text{Na}}}} \right)\left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{{\text{Na}}}\frac{{\overline{T}}_{\text{f}}}{\beta}F_{{\text{LowBU}}}
The same type of summation as given above is used to volume weight the
reactivities (and temperatures, :math:`\zeta` values) from the different
axial layers and channels.
.. _section-4.5.9.2:
Coolant Expansion
^^^^^^^^^^^^^^^^^
When the sodium coolant expands, it increases the leakage from the
reactor, decreases sodium capture and results in spectral shifts. Sodium
expansion is assumed to be a linear effect. The sodium coolant expansion
is treated in a manner developed in Ref. 4.9. The equation used is:
.. math::
\zeta_{\text{c}} = \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)\alpha_{\text{V}_{{\text{Na}}}}\frac{{\overline{T}}_{{\text{Na}}}}{\beta}
where :math:`{\alpha_{\text{V}}}_{{\text{Na}}}` is the thermal
volumetric expansion coefficient,
:math:`{\overline{T}}_{{\text{Na}}}` is the sodium
temperature, and the sodium temperature is the value assigned by the
code for a particular channel and axial layer. The sodium volumetric
thermal expansion coefficient is calculated from the local sodium
temperature in the correlation for the sodium volumetric thermal
expansion coefficient given in :eq:`NaLiqThermalExpansion`.
The reactivity worths are summed in the manner described in the fuel
axial expansion section. The sodium number density reactivity
coefficient is entered as :sasinp:`YKNNA`, the
radial sodium reactivity worth factor is entered as :sasinp:`XRNSHP`, and the axial sodium reactivity worth weighting is
taken as the normalized axial shape of :sasinp:`VOIDRA`.
.. _section-4.5.9.3:
Stainless Steel Expansion
^^^^^^^^^^^^^^^^^^^^^^^^^
Both axial and radial steel expansion coefficients are considered to be
linear effects.
.. _section-4.5.9.3.1:
Axial Expansion
'''''''''''''''
The axial expansion of the stainless steel cladding results in a
decrease in the number density. The amount of expansion, as in the case
of the fuel, is different for conditions where there is contact or no
contact. In the event of no contact (burnup < 2.9%), the expression for
the reactivity is:
.. math::
\zeta_{\text{ssa}} = {- \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)}_{\text{ss}}\frac{\alpha_{\text{Lss}}{\overline{T}}_{\text{ss}}}{\beta}
where :math:`\alpha_{\text{Lss}}` is the linear thermal expansion coefficient for
the cladding, and
:math:`\left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{{\text{ss}}}`
is the change in :math:`k` with respect to the relative number density in the
stainless steel.
The partial derivative above is an input parameter, :sasinp:`YKNSS`.
If there is contact, the following expression applies:
.. math::
\zeta_{\text{ssa}} = {- \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)}_{\text{ss}}\frac{\alpha_{\text{Lss}}{\overline{T}}_{\text{ss}}Y_{\text{ss}}A_{\text{ss}} + \alpha_{\text{Lf}}{\overline{T}}_{\text{f}}Y_{\text{f}}A_{\text{f}}}{\left( Y_{\text{ss}}A_{\text{ss}} + Y_{\text{f}}A_{\text{f}} \right)\beta}
The code therefore contains the expression:
.. math::
\zeta_{\text{ssa}} = {- \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)}_{\text{ss}}\frac{\alpha_{\text{Lss}}{\overline{T}}_{\text{ss}}}{\beta}F_{{\text{LowBU}}} - \left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{\text{ss}}\frac{\alpha_{\text{Lss}}{\overline{T}}_{\text{ss}}Y_{\text{ss}}A_{\text{ss}} + \alpha_{\text{Lf}}{\overline{T}}_{\text{f}}Y_{\text{f}}A_{\text{f}}}{\left( Y_{\text{ss}}A_{\text{ss}} + Y_{\text{f}}A_{\text{f}} \right)\beta}\left( 1 - F_{{\text{LowBU}}} \right)
The radial and axial weighting of the steel expansion reactivity is
carried out in the same manner as for the fuel expansion reactivity. The
radial steel reactivity worth shape factor is entered as :sasinp:`XRSSHP`,
and the axial shape is taken as the normalized
axial shape of input array :sasinp:`CLADRA`.
.. _section-4.5.9.3.2:
Radial Expansion
''''''''''''''''
The reactivity feedback due to radial expansion of the cladding results
from the displacement of sodium from the core. This removal is
independent of the burnup because the expansion is directed toward the
coolant channel. The expression for the radial expansion of the
stainless steel structure is:
.. math::
\zeta_{\text{ssr}} = - 2\frac{V_{\text{ss}}}{V_{{\text{Na}}}}\left( \frac{\partial \text{k}}{\frac{\partial \text{N}}{N}} \right)_{{\text{Na}}}\frac{\alpha_{\text{Lss}}{\overline{T}}_{\text{ss}}}{\beta}
where :math:`V_{\text{ss}}` is the volume of stainless steel in the
cladding, and :math:`\zeta_{\text{ssr}}` is the reactivity feedback due
to radial expansion of the clad.
The same type of summation as given above (:numref:`section-4.5.9.3.1`) is used to
volume weight the reactivities.
.. _section-4.5.9.4:
Axial Reflector Sodium Expansion
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In the axial reflector the reactivity feedbacks stem mostly from the
change in leakage. The upper and lower axial reflector sodium expansion
are treated as linear effects.
Upper Reflector
'''''''''''''''
The expression for the upper reflector sodium parameter :math:`\zeta`
is:
.. math:: \zeta_{\text{ur}} = \left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{ar}}\frac{{\overline{T}}_{\text{ur}}\ }{\beta}
where :math:`{\overline{T}}_{\text{ur}}` is the average
temperature in the upper reflector, and
:math:`\left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{ar}}` is the
change in :math:`k` with respect to a change in axial reflector
temperature.
The input value for the above partial derivative is entered as variable
:sasinp:`YRCUR`.
To determine the average temperature of the coolant in the upper
reflector, the following equation is used:
.. math:: {\overline{T}}_{\text{uf}} = \frac{\sum_{\text{i} = 1}^{N}{V_{\text{i}}T_{\text{i}}} + \sum_{\text{j} = 1}^{M}{\sum_{\text{k} = 1}^{L\left( j \right)}{V_{\text{j,k}}T_{\text{j,k}}}}}{\sum_{\text{i} = 1}^{N}V_{\text{i}} + \sum_{\text{j} = 1}^{M}{\sum_{\text{k} = 1}^{L\left( j \right)}V_{\text{j,k}}}}
where :math:`N` is the number of axial nodes in the gas plenum,
:math:`M` is the number of zones in the upper reflector,
:math:`L\left( j \right)` is the number of axial nodes in zone
:math:`j`. Also, :math:`V_{\text{i}}` is the volume of coolant node :math:`i`
in the plenum space, :math:`T_{\text{i}}` is the temperature of the node
:math:`i` coolant in the plenum space, :math:`V_{\text{j,k}}` is the volume of
coolant node :math:`k` in zone :math:`j` in the reflector area, and
:math:`T_{\text{j,k}}` is the temperature of the coolant in node :math:`k` of
zone :math:`j` in the reflector area.
In this analysis, only the sodium coolant is considered to determine the
reactivity feedback. This is done for two reasons: the difference
between the sodium coolant and stainless steel temperature is small and
the relative contributions of the stainless steel and coolant expansions
are unknown. The volumes used are that of the flow, not of the structure
and flow.
Lower Reflector
'''''''''''''''
The expression for the lower reflector sodium feedback is:
.. math::
\zeta_{\text{lr}} = \left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{ar}}\frac{{\overline{T}}_{\text{lr}}}{\beta}
where the partial derivative is entered as input variable :sasinp:`YLCLR` and
:math:`{\overline{T}}_{\text{lr}}` is the average
temperature in the lower reflector.
The same temperature volume-weighting scheme used in the upper reflector
is also used in the lower reflector. Again, the temperatures are that of
the coolant and the volumes are that in which flow occurs.
.. _section-4.5.9.5:
Radial Reflector Expansion
^^^^^^^^^^^^^^^^^^^^^^^^^^
The reactivity change that results from a temperature change in the
radial reflector is mainly due to the change in density of the sodium in
the reflector, which results in a change in the leakage. The radial
reflector expansion is treated as a linear effect. The expression for
the radial reflector sodium feedback is:
.. math::
\zeta_{\text{rr}} = \left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{rr}}\frac{{\overline{T}}_{\text{rr}}}{\beta}
where :math:`\left( \frac{\delta \text{k}}{\delta \text{T}} \right)_{\text{rr}}` is
the change in :math:`k` with respect to the radial reflector
temperature, and :math:`{\overline{T}}_{\text{rr}}` is the average
temperature of the radial reflector.
The partial derivative is entered as input variable :sasinp:`YRCRR`.
The nodes from which the average temperature is
determined are those in which :sasinp:`LCHTYP` is set to 2, indicating a stainless
steel reflector subassembly, and those of the bypass region at the core
level. For the channel subassemblies, only zone 5 (currently the core)
is used in the volume weighting. For the bypass channel, the
temperatures are also volume weighted. The coolant temperature is volume
weighted in all axial nodes of all channels to obtain the average
temperature. The equation used in the code is:
.. math::
{\overline{T}}_{\text{rr}} = \frac{\sum_{\text{i}}{V_{\text{by,i}}T_{\text{by,i}}} + \sum_{\text{j}}{V_{\text{ch,j}}T_{\text{ch,j}}}}{V_{\text{by}} + V_{\text{ch}}}
where :math:`V_{\text{by,i}}` is the volume of bypass region :math:`i`,
:math:`T_{\text{by,i}}` is the temperature of the bypass region :math:`i`,
:math:`V_{\text{by}}` is the total volume of the bypass region
corresponding to the reflector, :math:`V_{\text{ch,j}}` is the volume of axial
layer :math:`j` of the stainless steel containing channel,
:math:`T_{\text{ch,j}}` is the temperature of axial layer :math:`j` of the
stainless steel containing channel, :math:`V_{\text{ch}}` is the total
volume of the stainless steel channel.
.. _section-4.5.9.6:
Doppler Effect
^^^^^^^^^^^^^^
The Doppler effect is a nonlinear phenomenon that results from the
change in effective cross sections at different temperatures. The
temperature coefficient for the Doppler effect
:math:`\left( \frac{\delta \text{k}}{\delta \text{T}} \right)_{\text{D}}` is a function of
fuel temperature:
.. math::
\left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{D}} = \frac{k_{\text{D}}}{{\overline{T}}_{\text{f}}}
where :math:`k_{\text{D}}` is an input variable :sasinp:`YRCDOP` and :math:`{\overline{T}}_{\text{f}}` is the channel
average fuel temperature. Since the Doppler temperature coefficient is
temperature dependent, an average value is used.
.. math::
\left\langle \left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{D}} \right\rangle = \frac{\int_{{\overline{T}}_{\text{f}}\left( 0 \right)}^{{\overline{T}}_{\text{f}}\left( t \right)}{\left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{D}}\text{d}{\overline{T}}_{\text{f}}}}{\int_{{\overline{T}}_{\text{f}}\left( 0 \right)}^{{\overline{T}}_{\text{f}}\left( t \right)}{\text{d}{\overline{T}}_{\text{f}}}} = \frac{k_{\text{D}}}{{\overline{T}}_{\text{f}}\left( t \right) - {\overline{T}}_{\text{f}}\left( 0 \right)}\ln\left\lbrack \frac{{\overline{T}}_{\text{f}}\left( t \right)}{{\overline{T}}_{\text{f}}\left( 0 \right)} \right\rbrack
where :math:`{\overline{T}}_{\text{f}}\left( t \right)` is the channel average
fuel temperature at time :math:`t`, and
:math:`{\overline{T}}_{\text{f}}\left( 0 \right)` is the steady-state channel
average fuel temperature.
The total feedback :math:`\delta k_{\text{D}}` due to the Doppler effect is the
sum of the Doppler feedbacks for each fueled channel. This is
implemented in the code as:
.. math::
\delta k_{\text{D}} = \sum_{\text{i} = 1}^{N}{\left\langle \left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{D}} \right\rangle_{\text{i}}\frac{V_{\text{i}}}{V_{{\text{tot}}}}\left\lbrack {\overline{T}}_{\text{f,i}}\left( t \right) - {\overline{T}}_{\text{f,i}}\left( 0 \right) \right\rbrack} = \frac{k_{\text{D}}}{V_{{\text{tot}}}}\sum_{\text{i} = 1}^{N}{V_{\text{i}}\ln\left\lbrack \frac{{\overline{T}}_{\text{f,i}}\left( t \right)}{{\overline{T}}_{\text{f,i}}\left( 0 \right)} \right\rbrack}
where the temperatures given above are currently weighted by the axial
and radial reactivity worths.
.. _section-4.5.9.7:
Control Rod Bank Expansions
^^^^^^^^^^^^^^^^^^^^^^^^^^^
The change in reactivity that results from control rod bank temperature
change is a result of the support of these rods. These are suspended
from the top of the reactor. When a temperature change occurs, the rods
expand or contract, forcing some of the fuel into or out of the core.
This results in a reactivity change. The control rod expansion is
treated as a linear effect. Although there are several portions of the
control rod that undergo expansion when the temperature is increased,
the current code only models two. The first of these is the control rod
fuel and the second is the driveline expansion. The expression found in
the code is:
.. math::
\zeta_{\text{cr}} = \frac{\left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{cr}}}{\beta}\left\lbrack F_{\text{cr}}\frac{\sum_{\text{i} = 1}^{N}{{\overline{T}}_{\text{f,i}}V_{\text{i}}}}{V_{{\text{tot}}}} + \left( 1 - F_{\text{cr}} \right)T_{\text{CV}2} \right\rbrack
where :math:`T_{\text{CV}2}` is the temperature in the upper plenum (CV#2 in
|SAS|),
:math:`\left( \frac{\delta \text{k}}{\delta \text{T}} \right)_{\text{cr}}` is the
change is the change in :math:`k` with respect to control rod
temperature, and :math:`F_{\text{cr}}` is the fraction of the response
due to the channel temperature change.
The partial derivative above is entered as input variable :sasinp:`YRCCR` and
:math:`F_{\text{cr}}` is entered as :sasinp:`FCR`.
The model given above differs from previous models [Ref. 4-12] because
it does not include different time constants which characterize the
different portions of the stainless steel control rod structure;
instead, they are treated as the two terms given above; one proportional
to fuel temperature and the other proportional to the exit plenum
temperature. For steady state or a series of steady states this will not
result in any significant error.
.. _section-4.5.9.8:
Upper Grid Plate Expansion
^^^^^^^^^^^^^^^^^^^^^^^^^^
The upper grid plate is located below the core and serves as a support
for the hex can. Grid plate expansion impacts the reactivity by
displacing the assembly rows in the reactor. The upper grid plate
expansion is treated as a linear effect. Reference 4-9 gives the
following expression for the feedback:
.. math::
\zeta_{\text{gp}} = \frac{T_{\text{CV}1}\left( t \right)}{\beta}\left( \frac{\partial \text{k}}{\partial \text{T}} \right)_{\text{gp}}
where :math:`\zeta_{\text{gp}}` is the reactivity due to the upper grid
plate expansion, :math:`T_{\mathrm{CV1}}\left( t \right)` is the
temperature of the inlet plenum (CV#1 in |SAS|), and
:math:`\left( \frac{\delta \text{k}}{\delta \text{T}} \right)_{\text{gp}}` is the
feedback coefficient associated with the grid plate.
The partial derivative above is an input parameter entered as :sasinp:`YRCGP`.
.. _section-4.5.9.9:
Bowing and Unspecified Parameters
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
In the EBR-II reactivity feedback model, "bowing" is a "catch all" for
terms which can not specifically be identified and therefore may contain
terms other than that which result from the bowing of the assembly. True
assembly bowing results when one side of a fuel assembly is at a higher
temperature than the other. This results in a different expansion of the
two sides moving the assembly radially. Bowing contributes to the
reactivity by increasing or decreasing the amount of fuel in relatively
high worth portions of the core.
The overall "bowing" term in the EBR-II model is treated as a linear
effect that is modeled by one line below a threshold and by another line
above the threshold. The effect is a function of the normalized
temperature rise across the core
:math:`\Delta T_{{\text{norm}}}`. For normalized temperatures
above the threshold, which is given by the input variable :sasinp:`YTCUT`, the bowing reactivity
:math:`\delta k_{\text{bw}}` is given by:
.. math::
\delta k_{\text{bw}} = A + B\Delta T_{{\text{norm}}}
For normalized temperatures below the threshold, the bowing reactivity
:math:`\delta k_{\text{bw}}` is given by:
.. math::
\delta k_{\text{bw}} = \left( \frac{A}{{\text{YTCUT}}} + B \right)\Delta T_{{\text{norm}}}
The values for the input coefficients :math:`A` and :math:`B` are
entered as :sasinp:`YABOW` and :sasinp:`YBBOW`.
The normalized temperature rise is the ratio of the temperature rise
across the core at time t to the temperature rise across the core at
full power and flow conditions. The temperature rise is modeled as the
difference between the upper and lower plena temperatures (found in
YTLCV2 and YTLCV1) or as the difference between the average sodium
temperature for the fueled channels
:math:`{\overline{T}}_{{\text{Na}}}` and the
temperature of the lower plenum YTLCV1. Which model is used for the
normalized temperature rise is determined by an input parameter :sasinp:`IBOWTP`.
For :sasinp:`IBOWTP` equal to zero, the normalized
temperature rise is based on the outlet and inlet plena temperatures. In
the case, the normalized factor :sasinp:`YDELT0`
gives the difference between the outlet and inlet plena temperatures at
full power and flow conditions. For :sasinp:`IBOWTP` not equal to zero, the
normalized temperature rise is based on the average sodium temperature
in fueled channels and the temperature of the inlet plenum. In this
case, the normalized factor :sasinp:`YDELT0` gives the difference between the
average sodium temperature in the fueled channels and the temperature of
the inlet plenum at full power and flow conditions.