.. _section-A3:
Appendices
----------
.. _section-A3.1:
APPENDIX 3.1: Degree of Implicitness for Flow and Temperature Calculations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
A typical calculation in SASSYS-1 involves finding a temperature or flow
rate at the end of a time step given the values of relevant parameters
at the beginning of the step, as well as the values of driving functions
at the beginning and end of the step. Linearized finite difference
approximations to the relevant differential equations are usually used,
and the accuracy of the finite differencing in time will depend both on
the size of the time step and on the degree of implicitness in the
solution. By picking appropriate values for the degree of implicitness,
it is possible to improve the accuracy of the solution.
The differential equations used for temperatures and flow rates are of
the form
.. math::
:label: A3.1-1
\frac{\text{dy}}{\text{dt}} = f\left( y,t \right)
Where :math:`y` is the flow rate or temperature being calculated, and :math:`f` is a
function of y and time. The finite difference approximation used for
:eq:`A3.1-1` is
.. math::
:label: A3.1-2
\frac{y_{2} - y_{1}}{ \Delta t } = \theta_{1}f\left( y_{1},t_{1} \right) + \theta_{2}f\left( y_{2},t_{2} \right)
where
:math:`t_1` = time at the beginning of the time step
:math:`t_2` = time at the end of the time step
:math:`y_1` = :math:`y` at :math:`t_1`
:math:`y_2` = :math:`y` at :math:`t_2`
.. math::
:label: A3.1-3
\Delta t = t_{2} - t_{1}
.. math::
:label: A3.1-4
\theta_{1} + \theta_{2} = 1.0
The parameters :math:`\theta_1` and :math:`\theta_2` determines the degree of
implicitness of the solution: for a fully explicit solution
:math:`\theta_1` = 1.0 and :math:`\theta_2` = 0.0, whereas for a fully
implicit solution :math:`\theta_1` = 0.0 and :math:`\theta_2` = 1.0.
After linearizing, :eq:`A3.1-1` can be put in the form
.. math::
:label: A3.1-5
\frac{\text{dy}}{\text{dt}} = f\left( y_{1},t_{1} \right) + \left( t - t_{1} \right)\frac{\text{df}}{\text{dt}} + \left( y - y_{1} \right)\frac{\text{df}}{\text{dy}}
or
.. math::
:label: A3.1-6
\frac{\text{dy}}{\text{dt}} = A + B \left( t - t_{1} \right) + C\left( y - y_{1} \right)
where
.. math::
:label: A3.1-7
A = f\left( y_{1},t_{1} \right)
.. math::
:label: A3.1-8
B = \frac{\text{df}}{\text{dt}}
.. math::
:label: A3.1-9
C = \frac{\text{df}}{\text{dy}}
For instance :eq:`eq-3.9-5` for the coolant flow rate in a channel is
.. math::
:label: A3.1-10
I_{1}\frac{\text{dw}}{\text{dt}} = p_{\text{b}} - p_{\text{t}} - w^{2}I_{2} - A'_{\text{fL}}wI_{\text{3L}} - A'_{\text{fr}}w \left| w \right|^{1 + b_{\text{fr}}}I_{\text{3T}} - w\left| w \right|I_{4} - gI_{5}
Linearizing the :math:`w^2`,
:math:`w \left| w \right|^{1 + b_{\text{fr}}}`, and
:math:`w\left| w \right|` terms gives
.. math::
:label: A3.1-11
w^{2} \simeq w_{1}^{2} + 2w_{1}\left( w - w_{1} \right)
.. math::
:label: A3.1-12
w \left| w \right|^{1 + b_{\text{fr}}} \simeq w_{1}\left| w_{1} \right|^{1 + b_{\text{fr}}} + \left( 2 + b_{\text{fr}} \right) \left| w_{1} \right|^{1 + b_{\text{fr}}}\left( w - w_{1} \right)
and
.. math::
:label: A3.1-13
w \left| w \right| \simeq w_{1}\left| w_{1} \right| + \left| w_{1} \right|\left( w - w_{1} \right)
Also, :math:`p_{\text{b}}` and :math:`p_{\text{t}}` are approximated as
.. math::
:label: A3.1-14
p_{\text{b}}\left( t \right) = p_{\text{b}}\left( t_{1} \right) + \left( t - t_{1} \right)\frac{\text{dp}_{\text{b}}}{\text{dt}}
.. math::
:label: A3.1-15
p_{\text{t}}\left( t \right) = p_{\text{t}}\left( t_{1} \right) + \left( t - t_{1} \right)\frac{\text{dp}_{\text{t}}}{\text{dt}}
where
.. math::
:label: A3.1-16
\frac{\text{dp}_{\text{b}}}{\text{dt}} = \frac{p_{\text{b}}\left( t_{2} \right) - p_{\text{b}}\left( t_{1} \right)}{t_{2} - t_{1}}
and
.. math::
:label: A3.1-17
\frac{\text{dp}_{\text{t}}}{\text{dt}} = \frac{p_{\text{t}}\left( t_{2} \right) - p_{\text{t}}\left( t_{1} \right)}{t_{2} - t_{1}}
Then :eq:`A3.1-10` has the form of :eq:`A3.1-6` if
.. math::
:label: A3.1-18
A = \frac{\left\{ p_{\text{b}}\left( t_{1} \right) - p_{\text{t}}\left( t_{1} \right) - w_{1}^{2}I_{2} - A'_{\text{fL}}w_{1}I_{\text{3L}} - A'_{\text{fr}} w_{1}\left| w_{1} \right|^{1 + b_{\text{fr}}} I_{\text{3T}} - w_{1} \left| w_{1} \right|\ I_{4} - gI_{5} \right\}}{I_{1}}
.. math::
:label: A3.1-19
B = \frac{\frac{\text{dp}_{\text{b}}}{\text{dt}} - \frac{\text{dp}_{\text{t}}}{\text{dt}}}{I_{1}}
and
.. math::
:label: A3.1-20
C = - \frac{\left\lbrack 2w_{1} I_{2} - A'_{\text{fL}}I_{\text{3L}} - \left( 2 + b_{\text{fr}} \right)A'_{\text{fr}}\left| w_{1} \right|^{1 + b_{\text{fr}}}I_{\text{3T}} - \left| w_{1} \right|I_{4} \right\rbrack}{I_{1}}
The exact solution for :eq:`A3.1-6` is
.. math::
:label: A3.1-21
y - y_{1} = \left( A\tau - B\tau^{2} \right) \left( 1 - e^{- \left( t - t_{1} \right)/\tau} \right) + B\tau\left( t - t_{1} \right)
where
.. math::
:label: A3.1-22
\tau = - \frac{1}{C}
The finite difference approximation used for :eq:`A3.1-6` is
.. math::
:label: A3.1-23
\frac{y_{2} - y_{1}}{ \Delta t } = A + \theta_{2}B\Delta t + \theta_{2} C \left( y_{2} - y_{1} \right)
where
.. math::
:label: A3.1-24
\Delta t = t_{2} - t_{1}
The solution of :eq:`A3.1-23` for :math:`y_2` is
.. math::
:label: A3.1-25
y_{2} - y_{1} = \frac{A \Delta t + \theta_{2} B\Delta t^{2}}{1 + \theta_{2} \frac{\Delta t}{\tau}}
:numref:`figure-A3.1-1` shows :math:`y_2` - :math:`y_1`, as given by
:eq:`A3.1-25`, as a function of :math:`\theta_2` for the case where :math:`A` = 1, :math:`B`
= .5, :math:`\Delta t` = 2, and :math:`\tau` = 1. Also shown is the exact solution from
:eq:`A3.1-21`. Note that a value :math:`\theta_2` can be found such that the
finite difference solution of :eq:`A3.1-25` exactly matches the
differential equation solution, given by :eq:`A3.1-21`, for this case.
Also note that neither a fully explicit solution, :math:`\theta_2` = 0, nor
a fully implicit solution, :math:`\theta_2` = 1.0, is very accurate for
particular case. An explicit solution would be numerically unstable for
:math:`\Delta t` greater than :math:`\tau`; and values considerably smaller than the time
constant :math:`\tau`.
For any values of the parameters in :eq:`A3.1-21` and :eq:`A3.1-25`, a value
of :math:`\theta_2` can be chosen such that the finite difference solution
matches the solution of the differential equation by setting :math:`t =
t_2` in :eq:`A3.1-21` and combining this equation with :eq:`A3.1-25`
to give
.. math::
:label: A3.1-26
\left( A\tau - B\tau^{2} \right) \left( 1 - e^{- \Delta t/\tau} \right) + B \tau \Delta t = \frac{A \Delta t + \theta_{2} B\Delta t^{2}}{1 + \theta_{2} \frac{\Delta t}{\tau}}
solving for :math:`\theta_2` gives
.. math::
:label: A3.1-27
\theta_{2} = \frac{\frac{ \Delta t }{\tau} - \left( 1 - e^{- \Delta t/\tau} \right)}{\frac{ \Delta t }{\tau}\left( 1 - e^{- \Delta t/\tau} \right)}
Note that A and B cancelled out of :eq:`A3.1-27`, and :math:`\theta_2` is a
function only of :math:`\Delta t/t`.
:numref:`figure-A3.1-2` shows :math:`\theta_2`, given by :eq:`A3.1-27`, as a function
of :math:`\Delta t/\tau`. This figure also shows some approximations to this
function, as discussed below. For small values of :math:`\Delta t/\tau`,
:math:`\theta_2` approaches 0.5, where as for large values of :math:`\Delta t/\tau`,
:math:`\theta_2` approaches 1.0.
.. _figure-A3.1-1:
.. figure:: media/image35.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
Finite Difference Solution as a Function of the Degree of Implicitness
.. _figure-A3.1-2:
.. figure:: media/image36.png
:align: center
:figclass: align-center
:width: 7.29236in
:height: 5.88472in
Degree of Implicitness as a Function of Normalized Time Step Size
For any equation that can be put in the form of :eq:`A3.1-5` or :eq:`A3.1-6`,
:eq:`A3.1-22` can be used to find :math:`\tau`, and then :eq:`A3.1-27` can be used to
find the appropriate value of :math:`\theta_2` for any value of :math:`\Delta t`.
Note that if driving pressures and coolant flows are being solved for
simultaneously, as is done in PRIMAR-4, then the value of :math:`B` in
:eq:`A3.1-6` is not known until the pressures have been solved for. In such
cases, a direct analytic solution of the differential equation, as in
:eq:`A3.1-22`, could not be made except by iterating between :eq:`A3.1-22`
and the pressure solution. On the other hand, the calculation of
:math:`\theta_2`, as in :eq:`A3.1-27`, requires only :math:`\Delta t` and :math:`\tau`; and the
calculation of :math:`\tau` requires only information that is available at the
beginning of a time step; so an appropriate value for :math:`\theta_2` can
be found for use in finite difference approximations without iteration,
even if driving pressures and flow rates are being solved for
simultaneously.
Even if finite differencing in time does not introduce any error into a
solution, there are usually other sources of error, such as the
linearization approximations of :eq:`A3.1-11` through :eq:`A3.1-17`. Also,
the term represented by :math:`B \left(t-t_1 \right)` in :eq:`A3.1-6` may in fact
not be linear in time. Therefore, simpler and easier to compute
approximations to the expression on the right-hand side of :eq:`A3.1-27`
might sometimes be used without losing much overall accuracy. A simple
approximation, :math:`{\theta'}_{2}` that approaches the correct limits
for very small and very large values of :math:`\Delta t/\tau` is
.. math::
:label: A3.1-28
{\theta'}_{2} = \frac{a + \frac{ \Delta t }{\tau}}{2a + \frac{ \Delta t }{\tau}}
where the parameter :math:`a` can be chosen to give some best overall fit.
:numref:`figure-A3.1-2` shows this function for various values of the parameter
:math:`a`. The curve for :math:`a` = 1.65 gives a fairly good fit, although no value
of :math:`a` will give a good fit over the whole range.
A better fit to :math:`\theta_2` can be obtained by an expression of the
form
.. math::
:label: A3.1-29
{\theta'}_{2} = \frac{a + bx + x^{2}}{2a + cx + x^{2}}
where
.. math::
:label: A3.1-30
x = \frac{ \Delta t }{\tau}
with :math:`a` = 6.12992, :math:`b` = 2.66054, and :math:`c` = 3.56284, this expression
fits the exact value of :eq:`A3.1-27` to within 1% over the whole range of
:math:`\Delta t/\tau`, as shown in :numref:`figure-A3.1-3`.
In general, when linear approximations are valid, and when a single
known time constant dominates the behavior, :eq:`A3.1-27` or :eq:`A3.1-29`
will give a value of :math:`\theta_2` that will provide accurate finite
difference solutions. For cases where a number of different time
constants are important, none of these expressions for :math:`\theta_2`
will give a precise finite difference solution for all time-step sizes.
Even in this case, though, :math:`\theta_2` should approach 0.5 for small
time steps, and it should approach 1.0 for large time steps. Therefore,
an expression of the form given by :eq:`A3.1-28` may be appropriate in
such cases.
.. _figure-A3.1-3:
.. figure:: media/image37.png
:align: center
:figclass: align-center
:width: 6.47222in
:height: 5.54444in
Approximate Correlation for the Degree of Implicitness
.. _section-A3.2:
APPENDIX 3.2: Input Preprocessor
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-A3.2-1:
Introduction
^^^^^^^^^^^^
A preliminary version of an input preprocessor has been written for the
three-dimensional core thermal hydraulics model described in :numref:`section-3.14`. This preprocessor reads a relatively small amount of information
from the user and produces a large fraction of the input for the new
model. This appendix is split into two main sections. The first section
details the calculations carried out by the preprocessor, while the
second section is a brief user guide, which details the necessary input
and its structure.
.. _section-A3.2-2:
Calculation Details
^^^^^^^^^^^^^^^^^^^
The preprocessor calculates and formats many of the variables necessary
for the detailed subchannel model. Some of these variables are
calculated by the preprocessor, while others are input directly into the
preprocessor. The preprocessor functions can be broken down into three
main components, as seen in :numref:`figure-A3.2-1`. First, it calculates
geometrical parameters such as coolant flow areas and hydraulic
diameters. Second, it uses correlations to compute the parameters used
for subchannel-to-subchannel heat flow. Lastly, the parameters used in
subassembly-to-subassembly heat transfer are determined. This section
details each of these calculations
.. _figure-A3.2-1:
.. figure:: media/Flowchart_A3.2-1.*
:align: center
:figclass: align-center
Preprocessor Coding Overview
.. _section-A3.2-2.1:
Geometric Properties
''''''''''''''''''''
This first step to calculating the necessary geometric input for
|SAS| is to determine the number of rows [1]_ of pins in each
subassembly. This value is determined from the user-specified number of
pins per subassembly :math:`n_{\text{SA}}` (preprocessor variable
NPINSB), as shown in :numref:`table-A3.2-1`. Note that if the user input for the
number of pins per subassembly :math:`n_{\text{SA}}` (NPINSB) is not
listed in :numref:`table-A3.2-1`, the preprocessor will return an error.
.. _table-A3.2-1:
.. list-table:: Number of Rows of Pins
:header-rows: 1
:align: center
:widths: auto
* - Number of Pins per Subassembly
:math:`n_{\text{SA}}`
- Number of Rows of Pins
:math:`n_{\text{rows}}`
* - 1
- 1
* - 7
- 2
* - 19
- 3
* - 37
- 4
* - 61
- 5
* - 91
- 6
* - 127
- 7
* - 169
- 8
* - 217
- 9
* - 271
- 10
After :math:`n_{\text{rows}}` is calculated, the remaining geometric
properties depend on the type of pin treatment specified by the user. As
described in :numref:`section-3.15.3.1`, the subchannel model has several pin
options, which are designated using the |SAS| input
:sasinp:`JJMLTP`. These different options are selected using the
preprocessor input ISC, as shown in :numref:`table-A3.2-2`.
.. _table-A3.2-2:
.. list-table:: Preprocessor Pin Options ISC
:header-rows: 1
:align: center
:widths: auto
* - ISC Option
- Description
* - 0
- JJMLTP = 0, No multi-pin treatment
* - 1
- JJMLTP = 1, 1 pin treatment
* - 2
- Multi-pin, subchannels grouped by row and sector
* - 3
- Multi-pin, same as 2, but bypass flow channel on outside
* - 4
- Multi-pin, individual subchannels
* - 5
- Multi-pin, individual subchannels, bypass flow channel on the outside
* - 6
- JJMLTP = 1, 1 pin treatment, bypass flow channel
The choice of ISC also affects the number of channels used to represent
different areas and regions of the subassembly. :numref:`table-A3.2-3` documents
these results, where the values in columns A-E are the number of
channels used to represent that region, where :numref:`figure-A3.2-2` shows the
layout of the pin regions. As shown in :numref:`table-A3.2-3`, many of the values
for the number of channels in each region depend on
:math:`n_{\text{rows}}`, or :math:`n_{\text{c}_{\text{ISC}}}`, which is the
number of channels for the inner subchannels and can take on value 0 or
6, depending on the input for IPDOPT.
.. _table-A3.2-3:
.. list-table:: Number of Channels in the Preprocessor Pin Channel Calculation [2]_
:header-rows: 2
:align: center
:widths: 1,1,1,1,1,1
* -
- A
- B
- C
- D
- E
* - ISC
- Edge SC
- Corner SC
- Thimble Flow Region
- Represent SA Type
- Inner SC
* - 0
-
-
-
- 1
- 1
* - 1
-
-
-
- 1
- 1
* - 2
- 6
-
-
- 6\ :math:`n_{\text{rows}}`
- D-6-\ :math:`n_{c_{\text{ISC}}}`
* - 3
- 6
-
- 6
- 6(\ :math:`n_{\text{rows}}`\ +1)
- D-12-\ :math:`n_{c_{\text{ISC}}}`
* - 4
- 6(\ :math:`n_{\text{rows}}`-1)
- 6
-
- 2\ :math:`n_{\text{rows}}`\ +4
- D-A-\ :math:`n_{c_{\text{ISC}}}`\ 1-B
* - 5
- 6(\ :math:`n_{\text{rows}}`-1)
- 6
- A-B
- 4+8\ :math:`n_{\text{rows}}`
- D-A-\ :math:`n_{c_{\text{ISC}}}`-1-C-B
* - 6
-
-
- 1
- 2
- 1
.. _figure-A3.2-2:
.. figure:: media/image39.png
:align: center
:figclass: align-center
:width: 4.19236in
:height: 2.70764in
Pin Region Layout [3]_
There are several geometric calculation options available using the
input IPDOPT, as seen in :numref:`table-A3.2-4`. These options determine how the
distance from the duct wall to the center of the first row
:math:`d_{\text{DC}}`, the pitch :math:`p`, and the flat-to-flat inner
diameter :math:`D_{\text{ff}}` are calculated. :numref:`figure-A3.2-3` presents a
diagram of these variables.
.. _table-A3.2-4:
.. list-table:: IPDOPT Options
:header-rows: 1
:align: center
:widths: 1,1,1,3
* - IPDOPT
- Input
- Calculated
- Description
* - 0
- :math:`p,D_{\text{ff}}`
- :math:`d_{\text{DC}}`
- Use input :math:`p` everywhere, calculate :math:`d_{\text{DC}}` based on remainder
* - 1
- :math:`D_{\text{ff}}`
- :math:`p`, :math:`d_{\text{DC}}`
- Use :math:`D_{\text{ff}}`, calculate uniform :math:`p`, :math:`d_{\text{DC}}`
* - 2
- :math:`p`
- :math:`d_{\text{DC}},D_{\text{ff}}`
- Use input :math:`p`, calculate :math:`d_{\text{DC}}` with same slop
* - 3
- :math:`D_{\text{ff}},p`
- :math:`d_{\text{DC}}`
- Hot channel option: use: :math:`p = D_{\text{ww}} + D_{\text{pin}}`\ [*]_ for 1 inner row of subchannels, use :math:`p` for other inner subchannels, calculate :math:`d_{\text{DC}}` based on remainder
* - 4
- :math:`p,d_{\text{DC}}`
- :math:`D_{\text{ff}}`
- Use input :math:`p` and :math:`d_{\text{DC}}`, calculate :math:`D_{\text{ff}}`
.. [*] :math:`D_{\text{ww}}` = Wire-wrap diameter, :math:`D_{\text{pin}}` = Pin diameter
.. _figure-A3.2-3:
.. figure:: media/image40.png
:align: center
:figclass: align-center
:width: 5.00764in
:height: 3.30764in
Assembly Layout Diagram
.. _section-A3.2-2.1.1:
Basic Dimensions Calculation
""""""""""""""""""""""""""""
Depending on the choice of preprocessor inputs IPDOPT and ISC, the
|SAS| input for the channel geometry will be calculated
differently. In any case, the basic geometric inputs listed in :numref:`table-A3.2-5` are then used to calculate additional geometric information, such
as those variables listed in :numref:`table-A3.2-6`. Some of these variables are
needed for direct |SAS| input, such as :sasinp:`ACCZ`, while
others are used to calculate other variables in the following sections,
such as the flow splits or subassembly-to-subassembly heat transfer
coefficients.
.. _table-A3.2-5:
.. list-table:: Preprocessor Input Geometric Variables
:header-rows: 1
:align: center
:widths: auto
* - Symbol
- Input
- Description
* - :math:`D_{\text{ff}}`
- FFID
- Flat-to-flat inner diameter
* - :math:`d_{\text{DC}}`
- DEDGE
- Distance from duct wall to center of 1\ :sup:`st` row of pins
* - :math:`D_{\text{pin}}`
- DPIN
- Diameter of pin
* - :math:`D_{\text{WW}}`
- DWW
- Diameter of wrapper wire
* - :math:`p`
- PITCH
- Pin centerline to centerline distance
* - :math:`l_{\text{WW}}`
- HWW
- Wrapper wire lead length
* - :math:`t_{C}`
- DRE
- Thickness of clad
* - :math:`t_{\text{DW}}`
- TDW
- Thickness of duct wall
* - :math:`t_{\text{GAP}}`
- TGAP
- Thickness of gap between subassemblies
.. _table-A3.2-6:
.. list-table:: Calculated Geometric Variables
:header-rows: 1
:align: center
:widths: auto
* - Symbol
- Variable
- Description
* - :math:`A_{\text{pin}}`
-
- Area of pin
* - :math:`A_{\text{WW}}`
-
- Area of wrapper wire
* - :math:`P_{\text{WW}}`
-
- Perimeter of wrapper wire
* - :math:`A_{\text{hex}}`
-
- Area of subassembly hex
* - :math:`P_{\text{hex}}`
-
- Perimeter of subassembly hex
* - :math:`A_{\text{CC}}`
- :sasinp:`ACCZ`
- Cross-sectional flow area per pin
* - :math:`D_{h}`
- :sasinp:`DHZ`
- Hydraulic diameter
.. _section-A3.2-2.1.2:
Flow Splits Calculation
"""""""""""""""""""""""
The flow split calculation starts with establishing the exponential
variable :math:`\alpha`, which is calculated using the preprocessor
input BFR. BFR is also used in |SAS| for the calculation of the
turbulent friction factor.
.. math::
:label: A3.2-1
\alpha = \frac{1 - \text{BFR}}{2 + \text{BFR}}
From there, the flow split per region :math:`f_{\text{XX}}` can be
calculated using the following equation, where XX is the region
abbreviation seen in :numref:`table-A3.2-1`.
.. math::
:label: A3.2-2
f_{\text{XX}} = A_{\text{CC}_{\text{XX}}}D_{\text{h}_{\text{XX}}}^{\alpha}
.. _table-A3.2-7:
.. list-table:: Region Abbreviations
:header-rows: 1
:align: center
:widths: auto
* - Abbreviation - XX
- Region
* - INR
- Inner Row of Subchannels
* - INB
- Inner Subchannels (except inner row if IPDOPT = 3)
* - RN
- Corner Subchannels
* - ED
- Edge Subchannels
The total flow area can be found from :numref:`table-A3.2-8`, where each value in
the right hand column is added to the total flow area.
.. _table-A3.2-8:
.. list-table:: Total Flow Area Contributions
:header-rows: 1
:align: center
:widths: auto
* - Region
-
- Amount Added to Total Flow Area
* - INR
-
- :math:`3f_{\text{INR}}`
* - INB
- If :math:`n_{\text{rows}} = 1`,
If :math:`n_{\text{rows}} \neq 1`,
- :math:`f_{\text{INB}}`
:math:`(3 - 6n_{\text{rows}} + 3n_{\text{rows}}^{2} - 3^{*})f_{\text{INB}}`
* - RN
-
- :math:`f_{\text{RN}}`
* - ED
- If :math:`n_{C_{\text{RN}}} > 0`,
If :math:`n_{C_{\text{RN}}} = 0`,
- :math:`\left( 3\left( n_{\text{rows}} - 1 \right) \right)f_{\text{ED}}`
:math:`\left( 1 + 3\left( n_{\text{rows}} - 1 \right) \right)f_{\text{ED}}`
\* Subtract 3 only if the number of inner subchannels > 0
Then the normalized initial coolant flow split :math:`w_{\text{N}_{\text{XX}}}`
is the value of the flow split for region XX divided by the total flow
area, determined by :numref:`table-A3.2-8`, times the total core flow
:math:`w_{\text{core}}` , which is defined by the preprocessor input
WTOTL.
.. math::
:label: A3.2-3
w_{\text{N}_{\text{XX}}} = \left( \frac{f_{\text{XX}}}{\text{Total}} \right)w_{\text{core}}
If bypass subchannels (thimble flow corners) are present, the same
process holds, where XX is the region from :numref:`table-A3.2-9`.
.. math::
:label: A3.2-4
f_{\text{XX}} = A_{\text{CC}_{\text{XX}}}D_{\text{h}_{\text{XX}}}^{\alpha}
.. _table-A3.2-9:
.. list-table:: Bypass Region Abbreviations
:header-rows: 1
:align: center
:widths: auto
* - Abbreviation
- Region
* - BYE
- Bypass Edge Subchannels
* - BYC
- Bypass Corner Subchannels
The total bypass flow area can be found from :numref:`table-A3.2-10`
.. _table-A3.2-10:
.. list-table:: Total Bypass Flow Area Contributions
:header-rows: 1
:align: center
:widths: auto
* - Region
-
- Amount Added to Total Bypass Flow Area
* - BY
- If ISC = 3
If ISC = 5
If ISC = 6
- :math:`{6f}_{\text{BYE}}`
:math:`{6\left( n_{\text{rows}} - 1 \right)f}_{\text{BYE}}`
:math:`f_{\text{BYE}}`
* - NR
- If ISC = 3
If ISC = 5
If ISC = 6
- :math:`0`
:math:`6f_{\text{NR}}`
:math:`0`
The bypass normalized initial coolant flow split is then just the
regional flow split divided by the total bypass flow area, multiplied by
the total bypass flow :math:`w_{\text{bypass}}`, which is defined by the
preprocessor input WBYFL.
.. math::
:label: A3.2-5
w_{\text{N}_{\text{XX}}} = \left( \frac{f_{\text{XX}}}{\text{Total}_{\text{bypass}}} \right)w_{\text{bypass}}
.. _section-A3.2-2.2:
Subchannel-to-Subchannel Heat Flow Coefficients
'''''''''''''''''''''''''''''''''''''''''''''''
For the multiple pin options (ISC = 2, 3, 4, or 5), the
subchannel-to-subchannel heat flow coefficients must be determined (
:eq:`3.14-7` in :numref:`section-3.14.2`). The |SAS| input :sasinp:`UACHM1`\, which
is used in the coolant subchannel-to-subchannel heat flow per pin unit
height calculation, can be split into coolant and clad contributions, as
explained in :numref:`section-3.14.2`,
.. math::
:label: A3.2-6
\text{UACHM}1_{\text{total}} = \text{UACHM}1_{\text{XXXX}} + \text{UACHM}1_{\text{Clad}}
where :math:`\text{UACHM}1_{\text{XXXX}}` is the coolant contribution
from the XXXX region, and :math:`\text{UACHM}1_{\text{Clad}}` is the
contribution from the cladding. The regions are described in :numref:`table-A3.2-11`.
.. _table-A3.2-11:
.. list-table:: Region Abbreviations
:header-rows: 1
:align: center
:widths: auto
* - Abbreviation - XXXX
- Region
* - ININ
- Inner row to inner row
* - IAIA
- Inner average row to inner average row
* - IAE
- Inner average row to edge
* - INIA
- Inner row to inner average row
* - EDED
- Edge to edge
* - EDCN
- Edge to corner
The clad contribution is independent of the region, and is defined
below, where :math:`k_{\text{ww}}` is the thermal conductivity of the
wrapper wire (preprocessor input XKSS), :math:`M` is a multiplier
(preprocessor input FU1E), and :math:`k_{\text{gap}}` is the thermal
conductivity of the coolant in the gap (preprocessor input XKGAP).
.. math::
:label: A3.2-7
\text{UACHM}1_{\text{Clad}} = \frac{12k_{\text{ww}}t_{\text{c}}M}{\pi k_{\text{gap}}D_{\text{pin}}}
The coolant contribution to :sasinp:`UACHM1` depends on the region, and
uses the following equations for each of the regions defined in :numref:`table-A3.2-11`,
.. math::
:label: A3.2-8
\text{UACHM}1_{\text{ININ}} = \left( \frac{0.774D_{\text{ww}}\left( D_{\text{pin}} + D_{\text{ww}} \right)}{\left( \frac{D_{\text{pin}} + D_{\text{ww}}}{\sqrt{3}} \right)D_{\text{pin}}} \right)\left( \frac{D_{\text{ww}}}{D_{\text{pin}}} \right)^{- 0.2627}
.. math::
:label: A3.2-9
\text{UACHM}1_{\text{IAIA}} = \left( \frac{0.774\left( p - D_{\text{pin}} \right)p}{\frac{p}{\sqrt{3}}D_{\text{pin}}} \right)\left( \frac{p - D_{\text{pin}}}{D_{\text{pin}}} \right)^{- 0.2627}
.. math::
:label: A3.2-10
\text{UACHM}1_{\text{IAE}} = \text{UACHM}1_{\text{IAIA}}
.. math::
:label: A3.2-11
\text{UACHM}1_{\text{INIA}} = \left( \frac{0.774D_{\text{ww}}\left( D_{\text{pin}} + D_{\text{ww}} \right)}{\left( \frac{\frac{p}{\sqrt{3}} + \frac{D_{\text{pin}} + D_{\text{ww}}}{\sqrt{3}}}{2} \right)D_{\text{pin}}} \right)\left( \frac{D_{\text{ww}}}{D_{\text{pin}}} \right)^{- 0.2627}
.. math::
:label: A3.2-12
\text{UACHM}1_{\text{EDED}} = \frac{1}{2}\left( \frac{0.774\left( 2\left( p - D_{\text{pin}} \right) \right)p}{pD_{\text{pin}}} \right)\left( \frac{2\left( p - D_{\text{pin}} \right)}{D_{\text{pin}}} \right)^{- 0.2627}
.. math::
:label: A3.2-13
\text{UACHM}1_{\text{EDCN}} = \frac{1}{2}\left( \frac{0.774\left( 2\left( p - D_{\text{pin}} \right) \right)p}{{\left( \frac{p}{2} + \frac{d_{\text{DC}}}{\sqrt{3}} \right)D}_{\text{pin}}} \right)\left( \frac{2\left( p - D_{\text{pin}} \right)}{D_{\text{pin}}} \right)^{- 0.2627}
The |SAS| input :sasinp:`UACHM2`\, which is used in the coolant
subchannel-to-subchannel heat flow per pin unit height calculation, can
be found by the following (if IPODPT = 3) for inner row to inner row
heat flow,
.. math::
:label: A3.2-14
\text{UACHM}2_{\text{ININ}} = \frac{\text{CXTC}*D_{\text{ww}}}{2A_{\text{CC}_{\text{INR}}}}
and for the inner row to the inner average,
(:math:`\text{UACHM}2_{\text{ININ}}` and
:math:`\text{UACHM}2_{\text{INIA}}` are zero for IPODPT :math:`\neq 3`)
.. math::
:label: A3.2-15
\text{UACHM}2_{\text{INIA}} = \frac{\text{CXTC}*D_{\text{ww}}}{A_{\text{CC}_{\text{INR}}} + A_{\text{CC}_{\text{INB}}}}
For all IPODPT options, the inner average to inner average value
follows,
.. math::
:label: A3.2-16
\text{UACHM}2_{\text{IAIA}} = \frac{\text{CX}*(p - D_{\text{pin}})}{2A_{\text{CC}_{\text{INB}}}}
For the inner average to the edge,
.. math::
:label: A3.2-17
\text{UACHM}2_{\text{IAE}} = \frac{\text{CX}*(p - D_{\text{pin}})}{A_{\text{CC}_{\text{INB}}} + A_{\text{CC}_{\text{ED}}}}
The value for UACHM2 for both edge to edge, and edge to corner heat flow
is zero. The placeholder variables are defined below,
.. math::
:label: A3.2-18
\text{CXTC} = \text{CMMTC}\left( \text{TAN}\text{SQR}*\frac{\text{AR}1}{A1\text{PTC}} \right)^{\frac{1}{2}}
.. math::
:label: A3.2-19
\text{CX} = \text{CMT}\left( \text{TANSQR}*\frac{\text{AR}1}{A1\text{PTC}} \right)^{\frac{1}{2}}
.. math::
:label: A3.2-20
\begin{aligned}
\text{CMMTC} = 0.1 \frac{D_{\text{ww}}}{D_{\text{pin}}}^{- \frac{1}{2}} && \text{If } n_{\text{SA}} < 7 \\
\text{CMMTC} = 0.14\frac{D_{\text{ww}}}{D_{\text{pin}}}^{- \frac{1}{2}} && \text{If } n_{\text{SA}} \geq 7 \\
\end{aligned}
.. math::
:label: A3.2-21
\begin{aligned}
\text{CMT} = 0.1 \left( \frac{p - D_{\text{pin}}}{D_{\text{pin}}} \right)^{- \frac{1}{2}} && \text{If } n_{\text{SA}} < 7 \\
\text{CMT} = 0.14\left( \frac{p - D_{\text{pin}}}{D_{\text{pin}}} \right)^{- \frac{1}{2}} && \text{If } n_{\text{SA}} \geq 7 \\
\end{aligned}
.. math::
:label: A3.2-22
\text{TANSQR} = \frac{1}{\left( \frac{h_{\text{ww}}}{\sqrt{h_{\text{ww}}^{2} + \left( \pi\left( D_{\text{pin}} + D_{\text{ww}} \right) \right)^{2}}} \right)^{2}}
.. math::
:label: A3.2-23
\text{AR}1 = \frac{\pi\left( D_{\text{pin}} + D_{\text{ww}} \right)D_{\text{ww}}}{6}
.. math::
:label: A3.2-24
\text{A}1\text{PTC} = \frac{1}{4}p^{2}\sqrt{3} - \frac{\pi D_{\text{pin}}^{2}}{8}
For swirl flow, the |SAS| input :sasinp:`XKSWRL`\, which is used to
calculate swirl flow between subchannels, can be found using the
following for the edge to edge calculation,
.. math::
:label: A3.2-25
\text{XKSWR}L_{\text{EDED}} = \frac{C1L(p - D_{\text{pin}})}{A_{\text{CC}_{\text{ED}}}\beta}
where,
.. math::
:label: A3.2-26
\begin{aligned}
\beta = \frac{1}{2} && \text{If } \text{ISC} > 3 \\
\beta = \frac{1}{6} + \frac{1}{2}\left( n_{\text{rows}} - 1 \right) && \text{If } \text{ISC} \leq 3 \\
\end{aligned}
For XKSWRL from the edge to corner,
.. math::
:label: A3.2-27
\text{XKSWRL}_{\text{EDCN}} = \text{XKSWRL}_{\text{EDED}}
Lastly, the value for XKSWRL from the corner to edge depends on the
selection for ISC,
.. math::
:label: A3.2-28
\begin{aligned}
\text{XKSQRL}_{\text{CNED}} = \frac{6(\text{C}1\text{L})(p - D_{\text{pin}})}{A_{\text{CC}_{\text{CN}}}} && \text{If } \text{ISC} > 3 \\
\beta = \text{XKSWRL}_{\text{CNED}} = \text{XKSWRL}_{\text{EDED}} && \text{If } \text{ISC} \leq 3 \\
\end{aligned}
Where the placeholder variables are defined below,
.. math::
:label: A3.2-29
\text{C}1\text{L} = \text{CST}\left( \text{TANSQR}\left( \frac{\left( \frac{3}{2} \right)\pi\left( D_{\text{pin}} + D_{\text{ww}} \right)\left( \frac{D_{\text{ww}}}{6} \right)\ }{pd_{\text{DC}} - \frac{\pi D_{\text{pin}}^{2}}{8}} \right) \right)^{\frac{1}{2}}
.. math::
:label: A3.2-30
\begin{aligned}
\text{CST} = 0.1 \left( \frac{h_{\text{ww}}}{D_{\text{pin}}} \right)^{0.3} && \text{If } n_{\text{SA}} < 7 \\
\text{CST} = 0.75\left( \frac{h_{\text{ww}}}{D_{\text{pin}}} \right)^{0.3} && \text{If } n_{\text{SA}} \geq 7 \\
\end{aligned}
The calculation of the |SAS| input :sasinp:`ALATRL`\, which is the
coolant lateral flow area per pin, depends greatly on the pin options
chosen by ISC and IPDOPT, with too many possible combinations to review
in this document.
.. _section-A3.2-2.3:
Subassembly-to-Subassembly Heat Transfer
''''''''''''''''''''''''''''''''''''''''
Before the subassembly-to-subassembly heat transfer calculations can be
conducted, a series of more detailed geometric variables must be found
that includes many |SAS| inputs. These variables, shown in
:numref:`table-A3.2-12`, are calculated using the basic geometric information
found in :numref:`section-A3.2-2.1`.
.. _table-A3.2-12:
.. list-table:: Detailed Geometric Variables
:header-rows: 1
:align: center
:widths: auto
* - Block
- Variable
- Description
* - GEOMIN
- RER0
- Nominal cladding outer radius
* -
- RBR0
- Nominal cladding inner radius
* -
- DRF0
- Thickness of the outer reflector node
* -
- DRFI
- Thickness of the inner reflector node
* -
- SER
- Reflector perimeter per pin wetted by coolant
* -
- RBRPL
- Cladding inner radius in gas plenum
* -
- RERPL
- Cladding outer radius in gas plenum
* -
- AREAPC
- Coolant plus pin area per pin
* -
- SRFSTZ
- Structure perimeter
* -
- DSTIZ
- Thickness of inner structure node
* -
- DSTOZ
- Thickness of outer structure node
* - POWINC
- PRSHAP
- Ratio of power per subassembly
* - PMATCH
- XKSTIZ
- Inner structure thermal conductivity
* -
- XKSTOZ
- Outer structure thermal conductivity
* -
- RHOCSI
- Density\*heat capacity for the inner structure
* -
- RHOCSO
- Density\*heat capacity for the outer structure
* -
- RHOCR
- Density\*heat capacity for the reflector
* -
- XKRF
- Thermal conductivity of reflector/cladding (in gas plenum)
* - COOLIN
- W0
- Steady-state coolant flow rate per fuel-pin
.. _section-A3.2-2.3.1:
Single-Pin Options
""""""""""""""""""
For the single pin treatment with no bypass channel (ISC = 0 or 1), the
preprocessor will only calculate subassembly-to-subassembly heat
transfer if the channel represents one subassembly. If the channel
represents more than one subassembly, this input must be calculated
manually. The first step for calculating subassembly-to-subassembly heat
transfer is to determine which subassemblies are in contact. Using the
core symmetry option ISYM and the number of rows NROW, the preprocessor
finds the six adjacent assemblies. The preprocessor resolves whether a
subassembly is at an adiabatic boundary.
Once the adjacent subassemblies are known, the |SAS| input
related to subassembly heat transfer, shown in :numref:`table-A3.2-13`, is
calculated for each flat of the subassembly. This information is then
printed to the preprocessor output in the correct format for insertion
into a |SAS| input deck.
.. _table-A3.2-13:
.. list-table:: Subassembly Heat Transfer SAS Inputs
:header-rows: 1
:align: center
:widths: auto
* - Block
- Variable
- Description
* - INPCHN
- :sasinp:`NCHCH`
- Number of other channels in contact
* - INPCHN
- :sasinp:`ICHCH`
- Channel number that is in contact
* - INPCHN
- :sasinp:`IOPCHC`
- Subassembly to subassembly heat transfer option
* - PMATCH
- :sasinp:`HACHCH`
- Heat-transfer coefficient x area per unit height
For the single pin treatment with bypass channel (ISC = 6), the
additional geometric parameters relating to the bypass channel are
found, and then the adjacent subassemblies are determined. From there,
the process follows a similar path as the single pin options with no
bypass channel, where the variables in :numref:`table-A3.2-13` are found (but
taking into account the bypass) and the output is printed.
.. _section-A3.2-2.3.2:
Multiple-Pin Options
""""""""""""""""""""
The calculation process is more complicated for the multi-pin options
(ISC = 2, 3, 4, or 5). For each row of pins, the detailed geometric
variables in :numref:`table-A3.2-11` are calculated. The power gradient is found
first using the preprocessor input IDRGRD. This information, along with
the preprocessor input PRSHP0, is necessary to calculate the
|SAS| input POWINC:PRSHAP. The formulas used to calculate the
variables in :numref:`table-A3.2-11` differ depending on the choice of IPDOPT and
ISC (whether grouped by sector and row or individual subchannels). Where
the channel is located, whether it is an inner, edge, or corner channel,
also affects the calculation. The location is determined using the
number of rows NROW and the choices for IPDOPT and ISC. This is also
used to calculate the |SAS| input :sasinp:`JCHMPN`, which denotes
what channels are in contact.
The heat flow parameters found in :numref:`section-A3.2-2.2` are then adjusted
(normalized) for the different pin nodes. The calculation of the
subassembly heat transfer variables in :numref:`table-A3.2-12` is also more
intricate than the single pin models, and depends on whether individual
subchannels are being used (ISC = 4, 5) or subchannels grouped by row
and sector (ISC = 2, 3).
.. _section-A3.2-3:
User Guide
^^^^^^^^^^
.. _section-A3.2-3.1:
Overview
''''''''
The current preprocessor is useful for a limited subset of the cases
that |SAS| can handle, as shown in :numref:`table-A3.2-14`. The current
preprocessor was written only for hexagonal subassemblies with solid
duct walls and wire wrapped pins. A modified preprocessor would be
needed to handle square arrays, pins with grid spacers instead of wire
wraps, or ductless subassemblies. Also, the preprocessor does not handle
subassembly-to-subassembly heat transfer involving a channel that
represents more than one subassembly.
.. _table-A3.2-14:
.. list-table:: Scenarios Covered by Preprocessor
:header-rows: 1
:align: center
:widths: auto
* - Covered
- Not Covered
* - - Hexagonal Subassemblies
- Solid Duct Walls
- Wire Wrapped Pins
- - Square Subassemblies
- Ductless Subassemblies
- Pins with Grid Spacers
The current preprocessor provides input only for the pin section of a
subassembly. Input for the reflector zones above and below the pin
section must be input into |SAS| manually, as shown in :numref:`figure-A3.2-4`, but such information only needs to be input once for each
subassembly type, as explained in :numref:`section-3.15.3.1`.
The preprocessor numbers the channels in a consistent manner, and
determines subchannel connectivity. In the pin zone the preprocessor
calculates three types of parameters. As described in :numref:`section-A3.2-2`, it
first calculates geometrical parameters, such as coolant flow areas and
hydraulic diameters. Second, it uses correlations to calculate the
parameters used for subchannel-to-subchannel heat flow. Third, it
calculates the parameters used in subassembly-to-subassembly heat
transfer.
The preprocessor can also print additional user-specified information
directly to the created |SAS| input in order to help reduce the
time needed to create the |SAS| input. This allows a block of
text to be input once into the preprocessor, which then duplicates that
text for all channels of that type. This is done through the ICARD
option, and is explained in more detail in :numref:`section-A3.2-2.3` and :numref:`section-A3.2-3.5`.
.. _figure-A3.2-4:
.. figure:: media/image41.png
:align: center
:figclass: align-center
:width: 3.76944in
:height: 3.96944in
Preprocessor Coverage
The preprocessor could be of some use even if the three dimensional core
thermal hydraulics model is not being used. For a single average channel
treatment, the preprocessor can calculate the geometrical parameters
such as coolant flow areas and hydraulic diameters for each subassembly.
Also, if a separate channel represents each subassembly, then input for
subassembly-to-subassembly heat transfer can be calculated by the
preprocessor.
.. _section-A3.2-3.2:
Input/Output Format
'''''''''''''''''''
The layout of the input file can be broken into three main sections, as
shown in :numref:`figure-A3.2-5`. The general input is first, then the subassembly
type(s) information, which can also include the optional visualization
input, followed by the individual subassembly information. This last
section can also include the ICARD information if the preprocessor
options ICD or ICDBFR are being used, which allow additional text to be
added to the preprocessor output.
:numref:`table-A3.2-15` provides more detail about the input structure, beginning
with the general input. Each subassembly type must have its ITYPE value
in consecutive order. Then, ITYPE = 0 is used to signal the end of the
subassembly type input. Also, the optional visualization input is
entered independently for each subassembly type. A similar structure is
used for the subassembly input, where the value for ISUBAS must be in
consecutive order, and ISUBAS = 0 signals the end of the subassembly
input. The optional ICARD data is entered independently for each
subassembly. The format of the input is described in :numref:`table-A3.2-16`.
.. _figure-A3.2-5:
.. figure:: media/image42.png
:align: center
:figclass: align-center
:width: 3.25417in
:height: 3.55417in
Preprocessor Input Layout
.. _table-A3.2-15:
.. list-table:: Preprocessor Input Structure
:header-rows: 1
:align: center
:widths: auto
* - General Input
-
* - 1st Subassembly Type Input (ITYPE = 1)
-
* -
- Visualization Input (optional)
* - 2nd Subassembly Type Input (ITYPE = 2)
-
* -
- Visualization Input (optional)
* - …
-
* - Last Subassembly Type Input
-
* -
- Visualization Input (optional)
* - ITYPE 0
-
* - 1st Subassembly Input (ISUBAS = 1)
-
* -
- ICARD Input (optional)
* - 2nd Subassembly Input (ISUBAS = 2)
-
* -
- ICARD Input (optional)
* - …
-
* - Last Subassembly Input
-
* -
- ICARD Input (optional)
* - ISUBAS 0
-
|
.. _table-A3.2-16:
.. list-table:: Preprocessor Input Formatting
:header-rows: 1
:align: center
:widths: auto
* - **Section/Variable**
- **Fortran Format**
* - **General Input**
-
* - ITITLE
- (A80)
* - NROW, ISYM, IFSTCH, IHEXPL, IHEXP2, IDBUG, IFSTSA
- 7(7X,I5)
* - **Subassembly Type Input**
-
* - ITYPTL
- (A80)
* - ITYPE, ISC(ITYPE), NPINSB(ITYPE), IPDOPT(ITYPE), KZPIN(ITYPE), IBYOPT(ITYPE), ISWDIR(ITYPE)
- 7(7X,I5)
* - DPIN(ITYPE), DWW(ITYPE), PITCH(ITYPE), DEDGE(ITYPE)
- 4(10X,F10.5)
* - FFID(ITYPE), HWW(ITYPE), TBY(ITYPE), TDWBY(ITYPE)
- 4(10X,F10.5)
* - AFR(ITYPE), BFR(ITYPE), AFLAM(ITYPE), RELAM(ITYPE)
- 4(10X,F10.5)
* - DRE(ITYPE), FU1E(ITYPE), HCBY(ITYPE)
- 3(10X,F10.5)
* - TDW(ITYPE), TGAP(ITYPE), RHOCSS(ITYPE), RHOCNA(ITYPE)
- 4(10X,E10.3)
* - XKSS(ITYPE), XKGAP(ITYPE), FRSTIN(ITYPE), FRSTED(ITYPE)
- 4(10X,E10.3)
* - **Visualization Input (optional)**
-
* - LDETL(ITYPE), MZ(ITYPE), NT(ITYPE), IEQMAS(ITYPE)
- 6(7X,I5)
* - (ZMZ(IZ,ITYPE),IZ=1,MZ)
- (6E12.5)
* - RINFP, ROUTFP
- (2E12.5)
* - **Subassembly Input**
-
* - ISUBAS, ITYPEV(ISUBAS), KMAX, IDRGRD(ISUBAS), NSUB(ISUBAS), ICD(ISUBAS)
- 6(7X,I5)
* - SATITL
- (A80)
* - ICDBFR, IHTLAT(ISUBAS)
- 2(7X,I5)
* - WTOTL(ISUBAS), PRSHP0(ISUBAS), PGRAD(ISUBAS), ADOPPN(ISUBAS)
- 4(10X,F10.6)
* - BDOPPN(ISUBAS), WBYFL(ISUBAS), HALATP(ISUBAS)
- 3(10X,F10.6)
* - (IHXPSS(K), IHXPSE(K),K=1,KMAX)
- (20X,12I5)
* - NPRINT, (ICHPRT(I,ISUBAS),I=1,NPRINT)
- (20x,12I5)
.. _section-A3.2-3.3:
Input Description
'''''''''''''''''
As stated in :numref:`section-A3.2-3.2` and shown in :numref:`table-A3.2-17`, the input for
the preprocessor can be split into three major sections with the
visualization input under the subassembly type section. The following
tables outline the input for each section. The input needed for the
visualization tool, described in :numref:`section-A3.3`, is also provided.
.. _table-A3.2-17:
.. list-table:: Preprocessor Input Sections
:header-rows: 1
:align: center
:widths: auto
* - **Section**
- **Description**
* - General Input
- General Input
* - Subassembly Type Input
- Specifies pin treatment and geometric options for each subassembly type
* - Visualization Input (optional)
- Specifies details for creation of visualization mesh, described in :numref:`section-A3.3`
* - Subassembly Input
- Subassembly specific data
|
.. list-table:: General Input
:header-rows: 1
:align: center
:widths: 1,4
* - Symbol
- Description
* - ITITLE
- One line of title information. (80 character limit)
* - NROW
- Number of rows of subassemblies in the model.
* - ISYM
- Symmetry information
= 1, Full core
= 2, Half core
= 3, Third core
= 4, Quarter core
= 6, Sixth core
= 12, 1/12 core
* - IFSTCH
- First channel number to use.
* - IHEXPL
- Plot information
= 0, Skip plot of subassembly hexes numbered with hex positions for the symmetry and number of rows being used
= 1, Plot to file HEX_NUM.ps and continue
= 2, Plot to file HEX_NUM.ps and quit
* - IHEXP2
- Plot information
= 0, Skip plot of subassembly hexes numbered with subassembly numbers
> 0, Plot subassembly hexes numbered with subassembly numbers to file SUB_NUM.ps.
= 2, Generate visualization geometry file for use with VisIt,
additional input must be provided as described in :numref:`section-A3.3`
* - IDBUG
- Debugging output
> 0, Debugging output
* - IFSTSA
- First subassembly number to use.
|
.. list-table:: Subassembly Type Input
:header-rows: 1
:align: center
:widths: 1,4
* - Symbol
- Description
* - ITYPTL
- One line of type title information. (80 character limt)
* - ITYPE
- Subassembly type. If ITYPE=0, this is the end of the type input.
* - ISC
(ITYPE)
- Pin treatment option
= 0, JJMLTP = 0, No multi-pin treatment
= 1, JJMLTP = 1, 1 pin treatment
= 2, Multi-pin, subchannels grouped by row and sector
= 3, Multi-pin, same as 2, but bypass flow channel on outside
= 4, Multi-pin, individual subchannels
= 5, Multi-pin, individual subchannels, bypass flow channel on the outside
= 6, JJMLTP = 1, 1 pin treatment, bypass flow channel
* - NPINSB
(ITYPE)
- Number of pins per subassembly.
* - IPDOPT
(ITYPE)
- Geometry Calculation Options
= 0, Use input PITCH everywhere, calculate DEDGE based on what is left over, use input FFID
= 1, Use FFID, calculate uniform PITCH, DEDGE
= 2, Use input PITCH, calculate DEDGE with same slop, calculate FFID
= 3, Hot channel option: use FFID, use:
P = DPIN + DWW for 1 inner row of subchannels, use PITCH for other inner subchannels, calculate DEDGE based on what is left over
= 4, Use PITCH and DEDGE, calculate FFID
* - KZPIN
(ITYPE)
- Zone KZ
* - IBYOPT
(ITYPE)
- Bypass information
= 0, Bypass flow the whole length of the subassembly
= 1, Bypass flow only in the pin section
Note: IBYOPT is only relevant if ISC = 3, 5, or 6. IBYOPT = 1 is not currently implemented
* - ISWDIR
(ITYPE)
- Swirl information
= 1, Swirl flow is counter-clockwise, when viewed from above
= 2, Swirl flow is clockwise
* - DPIN
(ITYPE)
- Pin Diameter.
* - DWW
(ITYPE)
- Wrapper wire diameter.
* - PITCH
(ITYPE)
- Pin Centerline to Centerline Distance.
* - DEDGE
(ITYPE)
- Distance from duct wall to center of first row of pins.
* - FFID
(ITYPE)
- Flat-to-flat ID.
* - HWW
(ITYPE)
- Wire wrap lead length.
* - TBY
(ITYPE)
- Thickness of bypass flow gap.
* - TDWBY
(ITYPE)
- Thickness of the outer duct wall beyond the bypass gap.
* - AFR
(ITYPE)
- Turbulent Friction Factor = AFR\*Re\*\*BFR
* - BFR
(ITYPE)
- Turbulent Friction Factor = AFR\*Re\*\*BFR
* - AFLAM
(ITYPE)
- Laminar Friction Factor = AFLAM/Re
* - RELAM
(ITYPE)
- Reynolds number for transition from turbulent to laminar. If RELAM = 0, then it is calculated so that turbulent friction factor equals laminar friction factor at RELAM.
* - DRE
(ITYPE)
- Clad thickness.
* - FU1E
(ITYPE)
- Geometry multiplier for subchannel-to-subchannel heat transfer by conduction through cladding. Typical value near 2.0.
* - HCBY
(ITYPE)
- Coolant heat transfer coefficient from outer surface of duct wall to thimble flow coolant. For laminar flow, HCBY = 7.6\*k/Dh to 8.23\*k/Dh.
* - TDW
- Thickness of duct wall.
* - TGAP
- Thickness of gap between subassemblies.
* - RHOCSS
- Density × heat capacity of the wrapper wire and duct wall.
* - RHOCNA
- Density × heat capacity of the coolant in the gap.
* - XKSS
- Thermal conductivity of the wrapper wire and duct wall.
* - XKGAP
- Thermal conductivity of the coolant in the gap.
* - FRSTIN
- Fraction of structure thickness in the inner node, inner subchannel.
* - FRSTED
- Fraction of structure thickness in the inner node, edge subchannel.
* - Visualization Input (if IHEXP2 = 2) [4]_
-
* - **Symbol**
- **Description**
* - LDETL
- Level of detail in the visualization mesh geometry file.
= 0, Do not generate geometry for this assembly type.
= -1, Only simplified coolant subchannels are defined.
= -2, Simplified coolant subchannels and duct wall structure.
= 1, Detailed coolant subchannels are defined.
= 2, Detailed coolant subchannels and duct wall structure.
= 3, Detailed subchannels, duct wall, and simple fuel pin.
= 4, Detailed subchannels, duct wall, and detailed fule pin.
* - MZ
- Number of axial segments in zone KZPIN (e.g., the MZ mesh).
* - NT
- Number of radial temperature nodes within the fuel (only relevant for LDETL = 4).
* - IEQMAS
- Radial fuel mesh size assumptions (relevant for LDETL=4).
= 0, Equal radial distance.
> 0, Equal cylindrical area.
* - ZMZ
- Axial segment boundaries for zone KZPIN, where the lowest boundary is assumed to be zero. Note that these and the above values are only used to generate the geometry file, and are not used to define NZNODE, AXHI, or ZONEL.
* - RINFP
- Fuel inner radius.
* - ROUTFP
- Fuel outer radius (default = clad inner radius)
|
.. list-table:: Subassembly Input
:header-rows: 1
:align: center
:widths: 1,4
* - Symbol
- Description
* - ISUBAS
- Subassembly number. Max = NSBMAX = 1000. If ISUBAS = 0, then this is the end of the subassembly input.
* - ITYPEV
(ISUBAS)
- ITYPE = subassembly type.
* - KMAX
- Highest pin axial node, as shown in :numref:`figure-3.15-2`
* - IDRGRD
(ISUBAS)
- Direction of power gradient.
= 1, N(high) - S(low)
= 2, NNE - SSW
= 3, NE - SW
= 4, E - W
= 5, SE - NW
= 6, SSE - NNW
* - NSUB
(ISUBAS)
- Number of real subassemblies represented by this ISUBAS
* - ICD
- If > 0, Read in cards, ICARD, to be put at the end of the first channel in ISUBAS
* - SATITL
(ISUBAS)
- Subassembly title information.
* - ICDBFR
- If > 0, Read in cards to be put before the first channel in ISUBAS.
* - IHTLAT
- If > 0, Use HALATP(ISUBAS) for lateral heat transfer across pins. Currently, only implemented for the detailed subchannel treatment (ISC = 4 or 5). Not implemented for the one pin per subassembly case.
* - WTOTL
(ISUBAS)
- Total subassembly coolant flow, excluding thimble flow.
* - PRSHP0
(ISUBAS)
- Average PRSHAP for the subassembly.
PRSHAP = Ratio of power per subassembly averaged over this channel to the power per subassembly averaged over all channels.
* - PGRAD
(ISUBAS)
- Power gradient.
:math:`\text{PRSHAP}\ = \ \frac{\text{PRSHP}0\left( 1 + \text{PGRAD} \right)\text{XPIN}}{\text{NPINSB}}` at SA [5]_ Edge
:math:`\text{PRSHAP}\ = \ \frac{\text{PRSHP}0\ \times \ \text{XNPIN}}{\text{NPINSB}}` at SA Center
:math:`\text{PRSHAP}\ = \ \frac{\text{PRSHP}0\left( 1 - \text{PGRAD} \right)\text{XPIN}}{\text{NPINSB}}` at SA Other Edge
* - ADOPPN
- ADOP per pin.
ADOP = Doppler coefficient for this channel when part of the core represented by this channel is not voided.
* - BDOPPN
- BDOP per pin.
BDOP = Doppler coefficient for this channel when part of the core represented by this channel is fully voided.
* - WBYFL
- Total thimble flow. Used only if ISC = 3, 5, or 6.
* - HALATP
- Heat transfer coefficient × area/height for lateral heat transfer across pins.
* - IHXPSS(K)
- Subassembly ISUBAS represents subassemblies with hex position numbers from IHXPSS(K) to IHXPSE(K)
* - IHXPSE(K)
- See above
* - NPRINT
- Number of channels in this subassembly which will be printed out in normal prints (set IPRSKP = 0, in all other channels IPRSKP will be set to 1).
* - ICHPRT
- Relative channel number (1 for the first channel in the subassembly).
* - ICARD
- Information to be read and directly printed to fort.1 output. See ICD and ICDBFR.
.. _section-A3.2-3.4:
Input Example
'''''''''''''
An example input is provided in :numref:`figure-A3.2-6`, where "#" is used as a
placeholder for user specified values. This example has two subassembly
types, and two subassemblies. The visualization tool is also used for
both subassembly types. The two subassemblies use the ICARD option, with
the first subassembly (ISUBAS = 1) using ICDBFR, and the second
subassembly (ISUBAS = 2) using ICD.
.. _figure-A3.2-6:
.. figure:: media/image43.png
:align: center
:figclass: align-center
Preprocessor Example Input
.. _section-A3.2-3.5:
Output Description
''''''''''''''''''
Lastly, several output files are created by the preprocessor, as seen in
:numref:`table-A3.2-18`. The first output file is the direct output from the
preprocessor, which documents notifications printed during its
execution. The second file, fort.1 is the |SAS| input created
by the preprocessor. If the ICARD option is being used (ICD or ICDBFR >
0), then ICARD information will be printed directly to fort.1. The
location of the ICARD information will be before the first channel in
ISUBAS if ICDBFR is being used, or at the end of the first channel in
ISUBAS if ICD is used.
The preprocessor can optionally produce postscript plots to HEX_NUM.ps and
SUB_NUM.ps. HEX_NUM.ps plots the subassembly hexes numbered
with the position numbers used internally by the preprocessor.
SUB_NUM.ps plots the subassembly hexes numbered with the
subassembly numbers assigned by the user. The two plots could be
identical if the user picks the same numbering scheme that the
preprocessor uses internally. Lastly, the optional visualization geometry file, SAS.sasgeom,
can be created. More details on this file can be found in :numref:`section-A3.3`.
.. _table-A3.2-18:
.. list-table:: Output File Description
:header-rows: 1
:align: center
:widths: 1,4
* - Output
- Description
* - \*.out
- Log
* - fort.1
- |SAS| input
* - HEX_NUM.ps
- Plot of the subassembly hexes numbered with the position numbers used internally by the preprocessor (postscript)
* - SUB_NUM.ps
- Plot of the subassembly hexes numbered with the subassembly numbers assigned by the user (postscript)
* - SAS.sasgeom
- Visualization mesh file (optional)
.. _section-A3.3:
APPENDIX 3.3: Visualization
~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-A3.3-1:
Introduction
^^^^^^^^^^^^
An advanced visualization capability has been developed for the
|SAS| code [1] that allows for the generation, display, and
animation of large datasets that are calculated during a transient
simulation. The key features of the new capability include its support
for multiple-pin, subchannel modeling, integration with the subchannel
input preprocessor, geometry visualization, fully-detailed temperature
field support with unlimited dataset sizes, and interactive
visualization and animation generation on a desktop PC using the
visualization tool VisIt [2].
The motivation for developing an advanced visualization capability
results from the ever-increasing fidelity with which fast reactor
transient simulations are being carried out and the enormous amounts of
data generated during those simulations. For example, the EBR-II SHRT-17
test has recently been modeled [3] with subchannel-level detail and a
coarse axial mesh for the experimental assembly XX09 and its six nearest
neighbors. For a simulation that lasts 250 seconds with data saved every
one-tenth of a second, the full temperature field in the active core
region for these seven assemblies is estimated to be approximately
15 GB. [6]_ While it is trivial to identify maxima and minima within
this large dataset, identifying other features is much more difficult.
Items of interest might include the identification of power, flow, or
temperature fluctuations caused by instabilities in the numerical
solution, regions of temperature near (but not at) the peak, flow
reversals at the subchannel level, unusual results caused by input
error, time-derivatives of solution variables, and so on.
The following sections describe changes made to the subchannel input
preprocessor to support creation of visualization mesh geometry and
preliminary changes made to the |SAS| code to support large
dataset generation. Finally, examples of using the visualization tool
VisIt to render three-dimensional images from the SHRT-17 simulation are
provided.
.. _section-A3.3-2:
Mesh Generation
^^^^^^^^^^^^^^^
Channel modeling in the |SAS| code is fundamentally based on a
single-pin representation that includes associated coolant and
structure. Geometric input to the |SAS| code is represented by
generalized parameters such as flow area, hydraulic diameter,
heat-transfer area, etc, and details of the problem-specific geometry
are not fully known. In order to project computed transient data into
three-dimensional space, a visualization mesh representing the actual
geometry of the problem needs to be defined.
Detailed core models using the multiple-pin subchannel capability in the
|SAS| safety analysis code are usually created using the
subchannel input preprocessor, detailed in :numref:`section-A3.2`. The input
preprocessor takes subassembly design information and creates detailed
|SAS| input descriptions of the multiple-pin subchannels in
terms of geometric, thermal, and hydraulic parameters. Because the input
preprocessor contains subassembly design information that is not
available to the |SAS| code, it was modified to act as the tool
to be used in creating the visualization mesh geometry. A visualization
mesh is defined so that computed data can be mapped into
three-dimensional space. Furthermore, the mesh can be viewed independent
of computed data to identify subchannel locations in the global
geometry. Details on the format of the visualization mesh geometry file
are available at the end of this appendix.
An example of the visualization mesh created for the SHRT-17 simulation
is shown in :numref:`figure-A3.3-1`, where the mesh is colored according to the
type of geometry present. In this case, gray represents cladding and
structure, blue represents sodium coolant, orange is an average fuel
region, and red is a detailed fuel region. The bypass flow area of the
experimental assembly XX09 is visible in the assembly located in the
upper-right portion of the figure.
.. _figure-A3.3-1:
.. figure:: media/image44.png
:align: center
:figclass: align-center
:width: 6.48472in
:height: 4.86181in
Pseudo Color Plot of Steady-State Coolant, Structure, and Fuel Temperatures (°C) for XX09 and Surrounding Assemblies.
Because of the potential for the enormous amounts of data that need to
be generated in order to fill a visualization mesh, an option is
provided to control the level of detail represented in the visualization
mesh geometry file on an assembly-by-assembly basis. :numref:`figure-A3.3-1`
displays two of the many detail options available. In :numref:`figure-A3.3-1`,
experimental assembly XX09 is represented by a detailed radial pin mesh
that includes nine radial temperature zones in the fuel region. The
surrounding assemblies, however, are represented by an average fuel
region that includes only a single radial zone for the bulk average fuel
temperature. This reduces the data file for the 250 second simulation
from the 15 GB estimate described earlier down to approximately 3.5 GB.
The different options for the level of detail in the visualization mesh
geometry file are listed in :numref:`table-A3.3-1` and examples of the mesh for
each option are shown in :numref:`figure-A3.3-2`.
The detail options shown in :numref:`figure-A3.3-2` include two very simplified
subchannel representations that use a single cell volume for each
subchannel (LDETL = -1 or -2). For the interior subchannels, triangular
cells are defined; and for the edge and corner subchannels,
quadrilateral cells are defined. For LDETL = -1, these cells
(artificially) extend over the fuel pin and structure regions in order
to occupy the full assembly volume. For LDETL = -2, the cells only
extend over the fuel pin regions, and additional cells are defined to
represent the duct wall structure regions.
Positive values of LDETL define multiple coolant cells to approximate
the actual shape of the coolant subchannel. LDETL = 1 and LDETL = 2 have
void regions in the geometry since neither define regions for fuel pins
and the former does not define regions for the duct wall structure. The
advantage of LDETL = 1 and LDETL = 2 is that they require the same data
as with the corresponding negative values for LDETL but they more
accurately represent the physical geometry. LDETL options 3 and 4
introduce regions defined for the fuel pin, including fuel and cladding.
The difference between the two options is that LDETL = 3 represents the
fuel region by a single radial zone, while LDETL = 4 represents the fuel
region by multiple radial zones corresponding to the parameter NT. (See
below.)
.. _table-A3.3-1:
.. list-table:: Geometry Detail Options for Creating Visualization Mesh Geometry Files.
:header-rows: 1
:align: center
:widths: 1,4
* - LDETL
- Geometric Detail
* - 0
- No Geometry Defined
* - -1
- Simplified Coolant Subchannels
* - -2
- Simplified Coolant Subchannels and Duct Wall Structure
* - 1
- Detailed Coolant Subchannels
* - 2
- Detailed Coolant Subchannels and Duct Wall Structure
* - 3
- Detailed Coolant Subchannels, Duct Wall Structure, and Simplified Fuel Pin
* - 4
- Detailed Coolant Subchannels, Duct Wall Structure, and Detailed Fuel Pin
|
.. _figure-A3.3-2:
.. list-table:: Examples of Geometry Detail Options Applied to the XX09 Experimental Assembly.
:header-rows: 1
:align: center
:widths: auto
* - |LDETL-1|
LDETL = -1
- |LDETL-2|
LDETL = -2
* - |LDETL+1|
LDETL = 1
- |LDETL+2|
LDETL = 2
* - |LDETL+3|
LDETL = 3
- |LDETL+4|
LDETL = 4
It should be noted that even though the coolant subchannels are defined
by multiple cells for positive values of LDETL (four in the case of the
interior subchannels) |SAS| still only calculates a single
temperature for each axial position along a subchannel and that
temperature is applied to all cells that belong to a given subchannel.
Therefore, it is important to recognize the distinction between the
*visualization* mesh described here and the *computation* mesh (i.e.
channel model) used by the |SAS| code.
In the multiple-pin subchannel preprocessor, all new parameters
associated with the generation of a visualization mesh geometry file are
confined to the assembly type definition input. This includes the
geometric detail option parameter LDETL. Therefore, even if two
assemblies are of the same *type* (both driver assemblies, for example)
different types will need to be defined if a different level of
geometric detail is desired for each. In the modified preprocessor, the
additional parameters are expected only if the existing parameter IHEXP2
= 2. This assures compatibility with existing input files created for
earlier versions of the preprocessor. When IHEXP2 = 2, the additional
visualization input, shown in :numref:`section-A3.2-3.3`, must be added to the end
of each assembly type definition.
Using the geometry information contained in the preprocessor input file,
along with the additional visualization input shown in :numref:`section-A3.2-3.3`,
the multiple-pin subchannel preprocessor can create a |SAS|
visualization mesh geometry file. By default, the generated file is
named "SAS.sasgeom". Although the basename of the file is not critical,
the file extension (\*.sasgeom) is used by the VisIt visualization
software to identify the correct file reader with which to parse the
mesh.
The VisIt visualization software can be used to view the visualization
mesh geometry file created by the preprocessor without needing to carry
out a simulation to generate data. VisIt has the capability to plot
numerous qualities of the visualization mesh, including determining
individual cell volumes. In addition to the visualization mesh geometry,
the file contains two parameters that can be plotted: channel ID and
channel type. Channel ID directly corresponds to the |SAS|
channel number, and can be used to identify specific channels in the
global geometry. Channel type is somewhat of a misnomer, [7]_ in that it
does not define a type for a given channel but rather a region to which
a cell in the channel's geometry belongs. (:numref:`figure-A3.3-1` was created by
plotting the channel type parameter.) The currently defined types
include
- 0: Coolant (always present)
- 1-3: Cladding Radial Components (Inner, Middle, Outer; present when LDETL ≥ 3)
- 4-5: Structure Components (Inner, Outer; present when \|LDETL\| ≥ 2)
- 10: Bulk Fuel (present when LDETL = 3)
- 11-(10+NT): Radial Fuel Components (present when LDETL = 4)
Data generated during a simulation is associated with a cell in the
visualization mesh by matching three parameters: channel ID, axial
position, and channel type. This data can then further vary as a
function of time. The creation of a |SAS| visualization *data*
file is described in the following section.
.. _section-A3.3-3:
Data Generation
^^^^^^^^^^^^^^^
The visualization capabilities introduced into |SAS| assume
that geometry is invariant. This significantly simplifies implementation
since the |SAS| code has limited knowledge of the specific
geometric details needed to project computed simulation data into
three-dimensional space. As described in the previous section, a
visualization mesh geometry file is created by the multiple-pin
subchannel preprocessor. Modifications have been made to the
|SAS| code to produce a separate visualization data file that
contains time-dependent simulation data that can be associated with the
visualization mesh geometry. Details on the format of the visualization
data file are contained in :numref:`section-A3.3-7`.
Two parameters have been added to the |SAS| input description
to control the creation of the visualization data file. These two
parameters are defined in :numref:`table-A3.3-2`. The first parameter, IVIS3D,
indicates how often to write simulation data to the visualization data
file. By default, the file created will be named "SAS.sasdata". Like
with the visualization mesh geometry file, the basename of the file is
not critical, but the file extension (\*.sasdata) is used by the VisIt
visualization software to identify the correct file reader with which to
parse the data file.
The second parameter, LDETL, is a channel-dependent parameter that
controls the extent of data written for each channel. It corresponds to
the LDETL parameter defined for the multiple-pin subchannel input
preprocessor. When creating channel-dependent input using the input
preprocessor, the LDETL parameter will automatically be written to the
generated input description. Therefore, a user does not normally need to
set this input parameter.
.. _table-A3.3-2:
.. list-table:: |SAS| Input Parameters for Generation of Visualization Data
:header-rows: 1
:align: center
:widths: 1,4
* - Parameter
- Description
* - :sasinp:`IVIS3D`
- Visualization Data Generation Flag:
= 0, Do not write visualization data
> 0, Write visualization data every IVIS3D time steps.
* - :sasinp:`LDETL`
- Visualization Data Detail Flag:
= 0, Do not write visualization data for this channel
= 1, Write coolant temperatures only
= 2, Coolant and duct wall temperatures
= 3, Coolant, structure, and average fuel temperatures
= 4, Coolant, structure, average fuel, and detailed radial fuel temperatures.
There are two minor differences in the treatment of the LDETL parameter
between the multiple-pin subchannel input preprocessor and the
|SAS| code. The input preprocessor allows negative values for
LDETL, which correspond to coolant subchannel cells that completely
occupy the assembly dimensions. However, the data needed for these two
options is identical to the data needed for positive values of LDETL
that have the same magnitude. Therefore, |SAS| only defines
positive values for LDETL. The second difference is that for LDETL = 4,
|SAS| also writes out the average fuel temperatures
(corresponding to LDETL = 3) in addition to the detailed radial fuel
temperatures. This is not done for the visualization mesh because it
would define multiple cells for the same physical region. By writing
this additional data, a visualization data file created as part of a
simulation with LDETL = 4 can be used with a visualization mesh geometry
file created with LDETL = 3 without having to re-run the simulation. In
fact, once a visualization data file is created, it can be combined with
a visualization mesh geometry file with a lower-magnitude LDETL
parameter.
.. _section-A3.3-4:
Data Visualization
^^^^^^^^^^^^^^^^^^
The preceding sections describe the two-step process needed to create a
visualization mesh geometry file and a visualization data file
respectively. Interactively viewing the results of a simulation
contained in these files is accomplished using the VisIt visualization
software package freely available from Lawrence Livermore National
Laboratory. VisIt is available for multiple computing platforms,
including Linux, Windows, Mac OS X, Solaris, and others. In
collaboration with LLNL, a Windows-based file reader plug-in for VisIt
has been written that can parse the two files described here and import
data into VisIt.
Each of the two files requires a unique file extension (\*.sasgeom and
\*.sasdata) that VisIt uses to identify the correct file-reader plug-in
to use when reading the data files. While the geometry file can be
viewed independently of the data file, the data file requires a matching
geometry file so that data can be projected into three-dimensional
space. The matching of the geometry and data files is done by comparing
a file's basename. Therefore, SAS.sasdata will be associated with
SAS.sasgeom (and vice versa). This process is handled automatically by
the file reader.
VisIt supports extensive data visualization capabilities, including
pseudo color plots, slices, contours, and animations as well as data
picks (queries), filters, time derivatives, and the construction of
arbitrary expressions based on existing data. The use of the VisIt
visualization tool is beyond the scope of this memo. Details on
downloading and using VisIt are currently found at
http://www.llnl.gov/visit/.
To illustrate some of the visualization capabilities, two (static)
examples from the SHRT-17 simulation are shown in :numref:`figure-A3.3-3` and
:numref:`figure-A3.3-4`. :numref:`figure-A3.3-3` shows a plot of the XX09 assembly and its
neighbors at *t* = 0 (steady state). The plot has been cropped through
the center of XX09 to show the interior detail of fuel-pin temperatures.
:numref:`figure-A3.3-4` shows a contour plot of a slice at the top of the active
core region at *t* = 90 seconds into the transient. Animated versions of
both of these plots are presently archived at
http://internal.ne.anl.gov/~fanning/SHRT17.html.
.. _figure-A3.3-3:
.. figure:: media/image51.png
:align: center
:figclass: align-center
:width: 5.33056in
:height: 3.76944in
Pseudo Color Plot of Steady-State Coolant, Structure, and Fuel Temperatures (°C) for XX09 and Surrounding Assemblies.
.. _figure-A3.3-4:
.. figure:: media/image52.png
:align: center
:figclass: align-center
:width: 5.86944in
:height: 3.19236in
Contour Plot of Coolant, Structure, and Fuel Temperatures (°K) for XX09 and Adjacent Assemblies. (Top of Core at :math:`t = 90` seconds)
.. _section-A3.3-5:
Summary
^^^^^^^
An advanced visualization capability has been developed for the
|SAS| code that allows for the generation, display, and
animation of large datasets that are calculated during a transient
simulation. The key features of the new capability include its support
for multiple-pin, subchannel modeling, integration with the subchannel
input preprocessor, geometry visualization, fully-detailed temperature
field support with unlimited dataset sizes, and interactive
visualization and animation generation on a desktop PC using the
visualization tool VisIt.
.. _section-A3.3-6:
References
^^^^^^^^^^
1. J. E. Cahalan, et al., "Advanced LMR Safety Analysis Capabilities in
the SASSYS-1 and SAS4A Computer Codes," *Proc. Int'l Topical
Meeting on Advanced Reactor Safety*, American Nuclear Society,
Pittsburgh, PA, April 17-21, 1994.
2. "VisIt User's Manual", UCRL-SM-220449, Lawrence Livermore National
Laboratory, October 2005.
3. F. E. Dunn, et al., "Whole Core Subchannel Analysis Verification with
the EBR-II SHRT-17 Test," *Proc. ICAPP 2006*, Reno NV, June 4-8,
2006.
.. _section-A3.3-7:
File Formats
^^^^^^^^^^^^
.. _section-A3.3-7.1:
Visualization Mesh Geometry File
''''''''''''''''''''''''''''''''
The visualization mesh geometry file is an unformatted (binary) Fortran
file that contains a file header followed by two data sections. The
first data section defines assembly types in a local coordinate system.
The second data section defines actual assemblies in the problem
geometry in terms of previously-defined types and offsets into a global
coordinate system. This arrangement very much mimics the definition of
assembly types and positions in the multiple-pin subchannel
preprocessor. All numeric data is represented by four byte integers and
double-precision (eight-byte) floating point values. Byte order does not
matter, since the file reader for VisIt was designed to detect and
support both conventions. However byte order must be consistent within a
single file.
.. _section-A3.3-7.1.1:
File Header
"""""""""""
The file header consists of two records that identify file attributes
and a problem title. The first record defines the following file
attributes:
FILEID File ID. Must be the four characters ``SAS_``. (The fourth
character is a space.)
FILETYPE File Type. Must be the four characters ``GEOM``.
GEOMVER1 Integer File Version. Currently, GEOMVER1 = 1.
GEOMVER2 Integer File Sub-version. Currently, GEOMVER2 = 0.
CHRDATE File Creation Date. (Eight-byte character string.)
CHRTIME File Creation Time. (Eight-byte character string.)
The first four attributes are used by the VisIt file reader to ensure
that the correct reader is being used to parse the data file. If major
or minor changes to the file format are introduced, the GEOMVER1 and
GEOMVER2 attributes will be changed to identify the correct file format
for the reader to parse.
The second record in the file header is an 80-character problem title.
The multiple-pin subchannel preprocessor uses the problem title provided
as input for this record.
.. _section-A3.3-7.1.2:
Assembly Types
""""""""""""""
The first data section following the file header defines assembly types
in a local coordinate system. Since most assemblies (for example, driver
assemblies) will have the same geometry, this simplifies (and shortens)
the overall visualization mesh geometry file.
The structure of the assembly type definitions is represented by the
following pseudo code::
NTYPES
DO ITYPE = 1, NTYPES
TYPE\_TITLE(ITYPE)
ITYPEID(ITYPE), NCHAN(ITYPE), NZ(ITYPE),
(Z(K,ITYPE),K=1,NZ(ITYPE))
DO J = 1, NCHAN(ITYPE)
IRELCH(J,ITYPE), NP(J,ITYPE),
(X(I,J,ITYPE),Y(I,J,ITYPE),I=1,NP(J,ITYPE))
ENDDO
ENDDO
The assembly type definitions begin with a record containing a single
integer parameter, NTYPES, defining the number of assembly types
contained in this data section. Each of the assembly type definitions
that follow contains a title record (TYPE\_TITLE, 80 characters) and an
attributes record containing the following parameters:
ITYPEID Unique identification number for this assembly type.
NCHAN Number of "data channels" defined for this assembly type.
NZ Number of axial segments boundaries. (Typically, NZ = MZ + 1.)
Z Z coordinates of the axial segment boundaries in the local coordinate
system.
Following the assembly type attributes record is one record for each of
the NCHAN data channels defined for an assembly type. The data channel
record defines the attributes and geometric shape (in the x-y plane) of
the data channel:
IRELCH Relative data channel identification number. (See ICHOFF below.)
NP Number of vertices in the polygon defining this data channel.
(Presently, NP must be 3 or 4.)
X, Y (X,Y) coordinates of the vertices defining the polygon for this
data channel in local coordinates.
.. _section-A3.3-7.1.3:
Assembly Definitions
""""""""""""""""""""
The second data section defines "actual" assemblies in the problem
geometry by placing assembly types defined earlier into a global
coordinate system. The structure of the assembly definitions is
represented by the following pseudo code::
NSUBS
DO ISUB = 1, NSUBS
SUB\_TITLE(ISUB)
ISUBID(ISUB), ISUBTYPEID(ISUB), ICHOFF(ISUB),
XOFF(ISUB),YOFF(ISUB),ZOFF(ISUB)
ENDDO
The subassembly definitions begin with a record containing a single
integer parameter, NSUBS, defining the number of assemblies being
specified in the visualization mesh geometry file. This need not
represent all the assemblies in the problem geometry, only those for
which a visualization mesh is needed. Each of the assembly definitions
that follow contains a title record (SUB\_TITLE, 80 characters) and an
assembly attributes record containing the following parameters:
ISUBID Identification number for this subassembly.
ISUBTYPEID Identification number of the assembly type being used to
define this assembly.
ICHOFF Integer offset to translate each relative data channel
identification number into an absolute (global) data channel
identification number.
XOFF, YOFF, ZOFF Coordinate offsets to translate local coordinates of
the type definition into global coordinates of the problem geometry.
Absolute data channel identification numbers are constructed from the
sum of the relative data channel identification number (IRELCH) and the
data channel offset (ICHOFF) for each of the data channels that define
an assembly type. For data channels that represent coolant, IRELCH
typically starts at 1 and increases by 1 for each coolant subchannel in
the assembly type definition. When added to ICHOFF, the resulting data
channel identification corresponds to the channel number (ICH) in the
|SAS| simulation. Note that because multiple data channels may
be used to represent the complex shape of a particular region (the
coolant region of a subchannel, for example) absolute data channel
identification numbers need not be unique in the visualization mesh
geometry file.
For data channels that represent regions other than coolant, an
additional term is introduced into the definition of IRELCH. If ICHTYPE
represents the channel types as defined by the bulleted list on
Page 3-184, then the additional term added to IRELCH is equal to
ICHTYPE×10\ :sup:`6`. This procedure is also used by |SAS| to
create data channel identification numbers based on the channel number,
ICH, and guarantees correct association between the visualization mesh
and the simulation data. That is, simulation data is correctly
associated with the visualization mesh when (IRELCH + ICHOFF) =
(ICHTYPE×10\ :sup:`6` + ICH).
.. _section-A3.3-7.2:
Visualization Data File
'''''''''''''''''''''''
The visualization data file is an unformatted (binary) Fortran file that
contains a file header followed by one data section for each time step
saved during a simulation. All numeric data is represented by four byte
integers and double-precision (eight-byte) floating point values. Byte
order does not matter, since the file reader for VisIt was designed to
detect and support both conventions. However byte order must be
consistent within a single file.
.. _section-A3.3-7.2.1:
File Header
"""""""""""
The simulation data file header consists of a record that identifies
file attributes and two problem title records. The first record defines
the following file attributes:
FILEID File ID. Must be the four characters ``SAS_``. (The fourth
character is a space.)
FILETYPE File Type. Must be the four characters ``DATA``.
DATAVER1 Integer File Version. Currently, DATAVER1 = 1.
DATAVER2 Integer File Sub-version. Currently, DATAVER2 = 0.
CHRDATE File Creation Date. (Eight-byte character string.)
CHRTIME File Creation Time. (Eight-byte character string.)
The first four attributes are used by the VisIt file reader to ensure
that the correct reader is being used to parse the data file. If major
or minor changes to the file format are introduced, the DATAVER1 and
DATAVER2 attributes will be changed to identify the correct file format
for the reader to parse.
The second and third records in the file header represent two
80-character problem titles. The problem titles are determined by the
two title records provided as part of the |SAS| input.
.. _section-A3.3-7.2.2:
Simulation Data
"""""""""""""""
The remainder of the visualization data file consists of repeated blocks
of simulation data. Data blocks contain simulation data for each time
step that is recorded as determined by the value of the :sasinp:`IVIS3D` input
parameter. The details of the simulation data
that is recorded are determined by the channel-dependent input
parameter, :sasinp:`LDETL`. The arrangement of data in
the simulation data block is described by the following pseudo code::
TIME, NDATA
DO J = 1, NDATA
ICHID(J), ND(J), (DATA(J,K),K=1,ND)
ENDDO
The first record in the data block contains the current simulation time
(TIME) and the number of data channels that are to follow (NDATA).
Although all data blocks from a simulation will have the same length and
the same number of data channels, the NDATA parameter is provided to
facilitate parsing and verification by file readers.
Following the initial record is a series of NDATA records containing
simulation data represented by the following parameters:
ICHID Unique ID for this data channel.
ND Number of axial data values for this data channel. (Typically, ND =
MZ.)
DATA Simulation data, with one value for each axial position.
Data channel IDs must be unique within each data block and are
associated with zero or more data channels in the visualization mesh
geometry in order to project the data into three-dimensional space. The
construction of the data channel ID is described in the previous section
as :math:`\text{ICHID} = \left( \text{ICHTYPE} \times 10^{6} + \text{ICH} \right)` and ensures a unique value based
on |SAS| channel numbers (ICH) and the type of data being
stored.
.. [1]
A row of pins is the same as a ring of pins
.. [2]
Equations reference other variable columns using the assigned letter.
SC=Subchannel, SA=Subassembly.
.. [3]
Thimble flow region not shown
.. [4]
These parameters are only used to generate the visualization mesh
geometry and, with the exception of LDETL, are not incorporated into
the |SAS| input that is generated for the subchannel
definitions.
.. [5]
SA = Subassembly
.. [6]
Animations of the SHRT-17 simulation can be viewed at
http://internal.ne.anl.gov/~fanning/SHRT17.html
.. [7]
The misnomer arises because the channel ID and channel type
parameters actually exist in the geometry file as a single integer
parameter that represents a unique "data channel" that is used to
associate simulation data with the visualization mesh.
.. |LDETL-1| image:: media/image45.png
:width: 2.49236in
:height: 2.49236in
.. |LDETL-2| image:: media/image46.png
:width: 2.49236in
:height: 2.49236in
.. |LDETL+1| image:: media/image47.png
:width: 2.49236in
:height: 2.49236in
.. |LDETL+2| image:: media/image48.png
:width: 2.49236in
:height: 2.49236in
.. |LDETL+3| image:: media/image49.png
:width: 2.49236in
:height: 2.49236in
.. |LDETL+4| image:: media/image50.png
:width: 2.49236in
:height: 2.49236in
.. _section-A3.4:
APPENDIX 3.4: Coolant Physical Properties
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
.. _section-A3.4-1:
Introduction
^^^^^^^^^^^^
The expressions used in |SAS| to model coolant physical
properties as functions of temperature are derived from data
correlations generated by Padilla :ref:`[12.1] <section-12.references>`
and by Fink and Leibowitz :ref:`[12-12] <section-12.references>`.
These correlations were developed using sodium coolant properties that match the most recent measurements available,
namely those of Bhise and Bonilla :ref:`[12-13] <section-12.references>`
and Da Gupta and Bonilla :ref:`[12-14] <section-12.references>`. The mathematical forms of
these properties are maintained for coolants independent of their composition.
The mathematical expressions for coolant physical properties are provided in the sections
below, including brief descriptions of their origins. In all cases, the temperature :math:`T` is in Kelvin.
.. _section-A3.4-2:
Sodium Coolant Physical Properties
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
Details for sodium coolant properties and the associated polynomial coefficients are
described in :numref:`section-12.13`. This information is not repeated here.
.. _section-A3.4-3:
Lead Coolant Physical Properties
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The lead coolant properties are derived from the data and correlations available in
:ref:`[A3.4-1] <section-A3.4-7>`. The mathematical expressions in :ref:`[A3.4-1] <section-A3.4-7>` are not directly implemented, but rather a least-squares approximation to these correlations was generated for lead coolant properties to match the existing sodium-derived forms implemented within |SAS|. The error introduced by this approach, which is largely considered negligible, has been quantified and documented in the sections below.
.. _section-A3.4-3.1:
Lead Critical Temperature
'''''''''''''''''''''''''
The critical temperature (T\ :sub:`c`\) is an important parameter for the development of an equation of state (EOS) and the extension of information on the available thermophysical data to higher temperatures and pressures.
A very large uncertainty exists in the critical temperature for lead as indicated in :ref:`[A3.4-1] <section-A3.4-7>`.
The literature review performed in :ref:`[A3.4-1] <section-A3.4-7>` takes into account the large uncertainty in the estimations of the critical temperature.
The value recommended is:
.. math::
:label: A3.4-1
\text{T}_{Crit} = 5000.0 \pm 200 \text{ K}
.. _section-A3.4-3.2:
Lead Heat of Vaporization
'''''''''''''''''''''''''
The heat of vaporization at the normal boiling point is also referred to the latent heat (enthalpy) of vaporization in the literature.
It quantifies the energy required to break the atomic binding in lead.
The literature review in :ref:`[A3.4-1] <section-A3.4-7>` outlines a relatively small standard deviation.
The recommended value in :ref:`[A3.4-1] <section-A3.4-7>` for the latent heat of vaporization for pure lead at the normal boiling point is:
.. math::
:label: A3.4-2
\text{Q} = 858.6 \pm 1.9 \text{ kJ/kg}
The heat of vaporization correlation implemented in |SAS| uses a temperature-dependent polynomial to compute this quantity:
.. math::
:label: A3.4-3
\lambda = A_1 + A_2 T + A_3 T^{2} + A_4 T^{3}
Lead’s latent heat is a constant and, therefore, only :math:`{A}_{1}` is specified. The corresponding values for the coefficients of :eq:`A3.4-3` are provided in :numref:`section-A3.4-6`.
.. _section-A3.4-3.3:
Lead Saturation Pressure
''''''''''''''''''''''''
The lead saturation pressure as a function of temperature is an important property of coolants.
It determines the dependence of the boiling temperature on pressure and heat-mass transfer in the cover gas space.
It is directly related to the latent heat (enthalpy) of vaporization. :ref:`[A3.4-1] <Section-A3.4-7>` provides two correlations for the saturation pressure.
The expression chosen for the latent heat of vaporization for pure lead at the normal boiling point is:
.. math::
:label: A3.4-4
P_{S} = 1.88 × 10^{13} T^{-0.985} e^{\left ( \frac{-23325}{T} \right )}
The saturated vapor pressure is defined within a temperature range of :math:`[\text{T}_{M,0(Pb)}-\text{T}_{B,0(Pb)} ]` where :math:`\text{T}_{M,0(Pb)}` represents the normal melting point :math:`(600.6\text{ K})` and :math:`\text{T}_{B,0(Pb)}` is the normal boiling point :math:`(2021 \pm 3 \text{ K})`.
Per :ref:`[A3.4-1] <Section-A3.4-7>`, the standard deviation with respect to the data is:
- 15% for temperatures within the range :math:`[\text{T}_{M,0(Pb)} - 1100\text{ K}]`
- 5% for temperatures within the range :math:`[1100\text{ K} - \text{T}_{B,0(Pb)}]`
The calculation of the saturation pressure in |SAS| correlates the natural logarithm of the saturation pressure to a temperature-dependent polynomial.
.. math::
:label: A3.4-5
ln(P_{S}) = A_5 - \frac{A_6}{T} - \frac{A_7}{T^2}
The constant coefficients in the |SAS| second order correlation were evaluated using a least squares equal weighted residual minimization fit to the OECD compressibility correlation. The corresponding values for the coefficients of :eq:`A3.4-5` are provided in :numref:`section-A3.4-6`.
The temperature range provided in :ref:`[A3.4-1] <Section-A3.4-7>` was maintained when performing the least square fit to adapt to the form of :eq:`A3.4-5`. Uncertainties for this correlation are not determined. However, absolute and relative errors with respect to the OECD correlation were calculated. The relative error is provided in the plot below.
.. figure:: media/LeadSaturationPressureComparison.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-3.4:
Lead Saturation Temperature
'''''''''''''''''''''''''''
|SAS| requires an expression for temperature as a function of pressure. This expression is obtained by inverting :eq:`A3.4-5`. Direct inversion yields:
.. math::
:label: A3.4-6
T_{S} = \frac{A_8}{A_9+\sqrt{A_{10}+A_{11}ln \left( P_S\right)}}
The corresponding values for the coefficients of :eq:`A3.4-6` are provided in :numref:`section-A3.4-6`.
.. _section-A3.4-3.5:
Lead Liquid Density
''''''''''''''''''''
The lead density depends on temperature and pressure.
The expression recommended in :ref:`[A3.4-1] <Section-A3.4-7>` for the lead density defines a linear temperature dependence of the form:
.. math::
:label: A3.4-7
\rho_{l} = 11441 - 1.2795 T
The accuracy with which this correlation reproduces selected literature sources is discussed in :ref:`[A3.4-1] <Section-A3.4-7>`.
The maximum difference does not exceed 1% within the temperature range from the normal melting point :math:`\text{T}_{M,0(Pb)}` to :math:`1500 \text{ K}`.
This difference becomes about 2% when temperature increases up to the normal boiling point (defined as :math:`\text{T}_{B,0(Pb)} = 2021 \pm 3 \text{ K}`).
The |SAS| mathematical expression correlating the coolant density to the temperature is a second order polynomial:
.. math::
:label: A3.4-8
\rho_{l} = A_{12} + A_{13} T + A_{14} T^2
The corresponding values for the coefficients of :eq:`A3.4-8` are provided in :numref:`section-A3.4-6`. The correlation accuracy and temperature range of :ref:`[A3.4-1] <Section-A3.4-7>` is consistent with :eq:`A3.4-8` since the correlation mathematical form is maintained.
.. _section-A3.4-3.6:
Lead Vapor Density
''''''''''''''''''
Discussion of lead vapor density is not available in :ref:`[A3.4-1] <Section-A3.4-7>`, and alternative literature/experimental sources are not widely available.
The calculation of the vapor density in |SAS| uses a temperature-dependent polynomial which is multiplied by the saturation temperature:
.. math::
:label: A3.4-9
\rho_{v} = P_{S} \left ( \frac{A_{15}}{T} + A_{16} + A_{17} T + A_{18} T^2 + A_{19} T^3 + A_{20} T^4 \right )
The corresponding values for the coefficients of :eq:`A3.4-9` are provided in :numref:`section-A3.4-6`. For the current implementation, a value for :math:`A_{15} = 0.01` is assumed, and all other coefficients are assumed to be zero. The temperature range over which this correlation is applicable is unknown, but it is assumed that it applies to temperatures above :math:`\text{T}_{B,0(Pb)} = 2021 \pm 3 \text{ K}` where lead vapor is anticipated to be present.
.. _section-A3.4-3.7:
Lead Liquid Specific Heat
'''''''''''''''''''''''''
Calculation of the coolant enthalpy change as a function of temperature requires knowledge of the isobaric heat capacity. To account for large uncertainties in the data, :ref:`[A3.4-1] <Section-A3.4-7>` proposes the following expression for liquid lead specific heat:
.. math::
:label: A3.4-10
C_{l} = 176.2 - 4.923 \times 10^{-2} T + 1.544 \times 10^{-5} T^2 - \frac{1.524 \times 10^{6}}{T^2}
:ref:`[A3.4-1] <Section-A3.4-7>` identifies a 1991 correlation, which was later refined, as a reasonable choice for the isobaric heat capacity of pure liquid lead in the temperature range of :math:`[\text{T}_{B,0(Pb)} - 2000 \text{ K}]`. The accuracy with which this refined correlation reproduces selected literature sources is discussed in :ref:`[A3.4-1] <Section-A3.4-7>`. The estimated uncertainty is 5-7% within :math:`[\text{T}_{B,0(Pb)} - 1500 \text{ K}]` and 7-15% at higher temperatures where no experimental data is available in literature.
In |SAS|, the liquid specific heat is approximated by a polynomial that considers the difference between coolant temperature and critical temperature :eq:`A3.4-1`:
.. math::
:label: A3.4-11
C_{l} = \frac{A_{28}}{\left( T_{C} - T \right)^2} + \frac{A_{29}}{\left( T_{C} - T \right)} + A_{30} + A_{31} (T_{C} - T) + A_{32} \left ( T_{C} - T \right )^{2}
The mathematical expressions in :ref:`[A3.4-1]<Section-A3.4-6>` and that of |SAS| have different forms. The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to the :ref:`[A3.4-1] <Section-A3.4-7>` specific heat correlation. The temperature range over which this correlation was determined in :ref:`[A3.4-1] <Section-A3.4-7>` is maintained as the least square fit to :eq:`A3.4-11` was applied over the range outlined above. The corresponding values for the coefficients of :eq:`A3.4-11` are provided in :numref:`section-A3.4-6`.
Absolute and relative errors with respect to the OECD correlation were calculated, with relative error provided in the plot below.
.. figure:: media/LeadLiquidHeatCapacityComparison.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-3.8:
Lead Vapor Specific Heat
''''''''''''''''''''''''
Reference :ref:`[A3.4-1] <Section-A3.4-7>` does not discuss vapor specific heat. No experimental data related to the specific heat of lead vapor could be found.
Lead boiling is not expected to occur within the temperature ranges typically considered for lead fast reactors in the design basis analysis envelope. Moreover, in the unlikely event that lead boiling were to occur, this phenomenon will likely be localized for these particular scenarios. Due to the lack of literature on this subject, its limited relevance the typical |SAS| analysis scope, and unavailability of a two-phase lead coolant model in |SAS|, vapor specific heat is ignored. The corresponding values for the related coefficients are provided in :numref:`section-A3.4-6`.
It should be noted that this assumption is not applicable for severe accidents where coolant boiling occurs at a larger scale.
.. _section-A3.4-3.9:
Lead Adiabatic Compressibility
''''''''''''''''''''''''''''''
Reference :ref:`[A3.4-1] <Section-A3.4-7>` assumes compressibility is a reversible adiabatic process. As such, the isentropic compressibility of the coolant at normal pressure is calculated as the inverse of the bulk elastic modulus.
The bulk elastic modulus discussed in :ref:`[A3.4-1] <Section-A3.4-7>` uses a polynomial as a function of temperature.
As such, the expression for the compressibility becomes:
.. math::
:label: A3.4-12
\beta_{s} = \left( \left( 43.50 - 1.552 \times 10^{-2}T + 1.622 \times 10^{-6}T^2 \right) \times 10^{9} \right)^{-1}
The accuracy with which this correlation predicts compressibility is not directly discussed in :ref:`[A3.4-1] <Section-A3.4-7>`.
The report outlines how both compressibility and sound velocity are basic thermodynamic properties of molten lead.
The Dreyfus thermodynamic relationship is used to infer the sound celerity from the bulk elastic modulus, and compressibility.
Per :ref:`[A3.4-1]<Section-A3.4-6>`, the correlation is applicable for sound celerity of the longitudinal sound waves in molten lead for temperatures up to 1400K (i.e. :math:`[\text{T}_{M,0(Pb)} - 1400 \text{ K}]`).
This range is assumed to be the same for compressibility.
The compressibility coefficient is calculated in |SAS| by a polynomial expression based on the inverse of the coolant temperature subtracted from the critical temperature :eq:`A3.4-1`.
.. math::
:label: A3.4-13
\beta_{s} = A_{40} + \frac{A_{41}}{\left( T_{C} - T \right)}
The mathematical expressions in :ref:`[A3.4-1] <Section-A3.4-7>` and |SAS| have different forms.
The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to :eq:`A3.4-12`. The corresponding values for the coefficients of :eq:`A3.4-13` are provided in :numref:`section-A3.4-6`.
Uncertainties for this correlation were not determined. However, absolute and relative errors with respect to :eq:`A3.4-12` were calculated. The relative error is provided in the plot below.
.. figure:: media/LeadLiquidCompressibilityComparison.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-3.10:
Lead Thermal Expansion
''''''''''''''''''''''
The density of liquid metals changes with temperature due to thermal expansion related to anharmonicity of interatomic forces. As outlined in :ref:`[A3.4-1] <Section-A3.4-7>`, the relationship between the density and the coefficient of volumetric thermal expansion (CVTE) at constant pressure :math:`p` is defined by the following formula:
.. math::
:label: A3.4-14
\alpha_{p} = -\frac{1}{\rho} \left ( \frac{\partial \rho}{\partial T} \right )_{p}
Substituting the correlations for density recommended by :ref:`[A3.4-1] <Section-A3.4-7>` yields:
.. math::
:label: A3.4-15
\alpha_{p} = \frac{1}{8942-T}
Since the thermal expansion correlation is defined from that of the density, the same temperature range of :math:`[\text{T}_{M,0(Pb)} - 1400 \text{ K}]`) applies.
The uncertainty of this quantity can be derived from the thermal expansion expression and the coolant density uncertainty.
The thermal expansion coefficient is calculated in |SAS| by a polynomial expression based on the inverse of the coolant temperature subtracted to the critical temperature:
.. math::
:label: A3.4-16
\alpha_{p} = A_{42} + \frac{A_{43}}{\left( T_{C} - T \right)} + \frac{A_{44}}{\left( T_{C} - T \right)^2} + \frac{A_{45}}{\left( T_{C} - T \right)^3} + \frac{A_{46}}{\left( T_{C} - T \right)^4} + \frac{A_{47}}{\left( T_{C} - T \right)^5}
A simple polynomial form matching between equations in :eq:`A3.4-15` and :eq:`A3.4-16` does not allow the |SAS| expression to match that of :ref:`[A3.4-1] <Section-A3.4-7>` with a reasonably small variation.
Therefore, additional terms had to be included in the |SAS| correlation. The correlation implemented includes terms from :eq:`A3.4-16` up to the third order.
The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to the OECD thermal expansion correlation. The corresponding values for the coefficients of :eq:`A3.4-16` are provided in :numref:`section-A3.4-6`.
Uncertainties for this correlation were not determined. However, absolute and relative errors with respect to :eq:`A3.4-15` were calculated. The relative error is provided in the plot below.
.. figure:: media/LeadLiquidThermalExpasionComparison.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-3.11:
Lead Thermal Conductivity
'''''''''''''''''''''''''
Thermal conductivity is the fundamental thermophysical parameter of liquid metal coolants, which is used in all heat transfer calculations.
The accuracy of its determination for liquid metals is poor and usually large discrepancies exist among different sets of data.
Per :ref:`[A3.4-1] <Section-A3.4-7>`, the following linear correlation was used as the thermal conductivity of the technically pure liquid lead at normal pressure:
.. math::
:label: A3.4-17
k_{l} = 9.2 + 0.011 T
The accuracy with which this correlation reproduces selected literature sources is discussed in :ref:`[A3.4-1] <Section-A3.4-7>`.
The correlation is applicable from the lead melting temperature at normal atmospheric pressure up to 1300K (i.e. :math:`[\text{T}_{M,0(Pb)} - 1300 \text{ K}]`) with an estimated standard deviation of 15%.
The |SAS| mathematical expression correlating the coolant density to the temperature is a third order polynomial as defined below:
.. math::
:label: A3.4-18
k_{l} = A_{48} + A_{49} T + A_{50} T^2 + A_{51} T^3
The corresponding values for the coefficients of :eq:`A3.4-18` are provided in :numref:`section-A3.4-6`. The correlation accuracy and temperature ranges of :eq:`A3.4-17` and :eq:`A3.4-18` are consistent since the mathematical correlation form is maintained.
.. _section-A3.4-3.12:
Lead Viscosity
''''''''''''''
Viscosity is a measure of friction among atoms of liquids. Because there is very little data on the impact of impurities on viscosity, the correlations below are for pure lead.
Lead is believed to be a Newtonian liquid. Therefore, the temperature dependence of their dynamic viscosity is usually described by an Arrhenius-type equation as described in :ref:`[A3.4-1] <Section-A3.4-7>`.
.. math::
:label: A3.4-19
\mu_{l} = 4.55 \times 10^{-4}e^{\left ( \frac{1069}{T}\right )}
This correlation is valid in the temperature range from :math:`T_{\text{M,0(Pb)}}` to :math:`1473 \text{K}` with an estimated uncertainty of :math:`\pm 5\%`.
Coolant dynamic viscosity is calculated in |SAS| by a polynomial expression based on the inverse of the coolant temperature.
.. math::
:label: A3.4-20
\mu_{l} = A_{52} + \frac{A_{53}}{T} + \frac{A_{54}}{T^2} + \frac{A_{55}}{T^3}
The mathematical expressions in :eq:`A3.4-19` and :eq:`A3.4-20` have different forms.
The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to the OECD viscosity correlation. The corresponding values for the coefficients of :eq:`A3.4-20` are provided in :numref:`section-A3.4-6`.
The temperature range over which this correlation was determined in :ref:`[A3.4-1] <Section-A3.4-7>` is maintained because the least square fit to :eq:`A3.4-19` was performed over the applicability range outlined above. Uncertainties for this correlation were not determined. However, absolute and relative errors with respect to :eq:`A3.4-19` were calculated. The relative error is provided in the plot below.
.. figure:: media/LeadViscosityComparison.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-4:
LBE Coolant Physical Properties
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The LBE coolant properties are derived from the data and correlations available in
:ref:`[A3.4-1] <section-A3.4-7>`. The mathematical expressions in :ref:`[A3.4-1] <section-A3.4-7>` are not directly implemented for some properties, but rather a least-squares approximation to these correlations was generated for coolant properties to match the existing sodium-derived forms implemented within |SAS|. The error introduced by this approach, which is largely considered negligible, has been quantified and documented in the sections below.
.. _section-A3.4-4.1:
LBE Critical Temperature
''''''''''''''''''''''''
The critical temperature (T\ :sub:`c`\) is an important parameter for the development of an equation of state (EOS) and the extension of information on the available thermophysical data to higher temperatures and pressures.
A very large uncertainty still exists in the critical temperature for LBE as indicated in :ref:`[A3.4-1] <section-A3.4-7>`, as no measurement data is available in public literature. Critical parameters are only available through theoretical estimations. The literature review performed in :ref:`[A3.4-1] <section-A3.4-7>` takes into account the three available estimations of the critical temperature. The value recommended by :ref:`[A3.4-1] <section-A3.4-7>`, bounded by the uncertainties of lead and bismuth, is:
.. math::
:label: A3.4-21
\text{T}_{Crit} = 4800.0 \pm 500 \text{ K}
.. _section-A3.4-4.2:
LBE Heat of Vaporization
''''''''''''''''''''''''
The heat of vaporization at the normal boiling point is also referred to the latent heat (enthalpy) of vaporization in the literature. It quantifies the energy required to break the atomic binding in LBE. The literature review in :ref:`[A3.4-1] <section-A3.4-7>` indicates limited availability of information. The recommended value in :ref:`[A3.4-1] <section-A3.4-7>` for the latent heat of vaporization of pure LBE at the normal boiling point is:
.. math::
:label: A3.4-22
\text{Q} = 856 \pm 5 \text{ kJ/kg}
The heat of vaporization correlation implemented in |SAS| uses a temperature-dependent polynomial to compute this quantity:
.. math::
:label: A3.4-23
\lambda = A_1 + A_2 T + A_3 T^{2} + A_4 T^{3}
The latent heat of LBE is a constant and, therefore, only :math:`{A}_{1}` is specified. The corresponding values for the coefficients of :eq:`A3.4-23` are provided in :numref:`section-A3.4-6`.
.. _section-A3.4-4.3:
LBE Saturation Pressure
'''''''''''''''''''''''
Vapor pressure as a function of temperature is an important property of coolants. It determines the dependence of the boiling temperature on pressure and heat-mass transfer in the cover gas space. It is directly related to the latent heat (enthalpy) of vaporization. The expression recommended by :ref:`[A3.4-1] <section-A3.4-7>` for the saturation pressure for pure LBE is:
.. math::
:label: A3.4-24
P_{S} = 1.22 \times 10^{10} e^{\left ( \frac{-22552}{T} \right )}
The literature review in :ref:`[A3.4-1] <section-A3.4-7>` indicates very large uncertainties exist at low temperatures :math:`\left( < ~773 \text{K} \right)`. Experimental data used to develop the above correlation is defined within a temperature range of :math:`[500 - 1943 \text{K}]`. The OECD correlation standard deviation with respect to the data is, per :ref:`[A3.4-1] <section-A3.4-7>`:
- :math:`\pm 60\%` for temperatures within the range :math:`[673 - 1927 \text{K}]`, and
- No agreement below 673 K.
The calculation of the saturation pressure in |SAS| correlates the natural logarithm of the saturation pressure to a temperature-dependent polynomial.
.. math::
:label: A3.4-25
ln(P_{S}) = A_5 - \frac{A_6}{T} - \frac{A_7}{T^2}
The constant coefficients in the |SAS| second order correlation were evaluated using a least squares equal weighted residual minimization fit to the OECD correlation. The corresponding values for the coefficients for saturation pressure are provided in :numref:`section-A3.4-6`.
The temperature range provided in :ref:`[A3.4-1] <Section-A3.4-7>` was maintained when performing the least square fit to adapt to the form of :eq:`A3.4-25`. Uncertainties for this correlation are not determined. However, absolute and relative errors with respect to the OECD correlation were calculated. The relative error is provided in the plot below.
.. figure:: media/LBESaturationPressure.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-4.4:
LBE Saturation Temperature
''''''''''''''''''''''''''
|SAS| requires an expression for temperature as a function of pressure. This expression is obtained by inverting :eq:`A3.4-25`. Direct inversion yields:
.. math::
:label: A3.4-26
T_{S} = \frac{A_8}{A_9+\sqrt{A_{10}+A_{11}ln \left( P_S\right)}}
The corresponding values for the coefficients of :eq:`A3.4-26` are provided in :numref:`section-A3.4-6`.
.. _section-A3.4-4.5:
LBE Liquid Density
'''''''''''''''''''
The LBE density depends on temperature and pressure.
The expression recommended in :ref:`[A3.4-1] <Section-A3.4-7>` for the lead density defines a linear temperature dependence of the form:
The coolant density depends on temperature and pressure. This quantity is essential to the development of an EOS. The expression recommended for the LBE density defines a linear temperature dependence of the form :ref:`[A3.4-1] <Section-A3.4-7>`:
.. math::
:label: A3.4-27
\rho_{l} = 11065 - 1.293 T
The accuracy with which this correlation reproduces selected literature sources is discussed in :ref:`[A3.4-1] <Section-A3.4-7>`. The maximum difference does not exceed 0.8% within the temperature range of :math:`[673 - 1927 \text{K}]`.
The |SAS| mathematical expression correlating the coolant density to the temperature is a second order polynomial:
.. math::
:label: A3.4-28
\rho_{l} = A_{12} + A_{13} T + A_{14} T^2
The corresponding values for the coefficients of :eq:`A3.4-28` are provided in :numref:`section-A3.4-6`. The correlation accuracy and temperature range of :ref:`[A3.4-1] <Section-A3.4-7>` is consistent with :eq:`A3.4-28` since the correlation mathematical form is maintained.
.. _section-A3.4-4.6:
LBE Vapor Density
'''''''''''''''''
Reference :ref:`[A3.4-1] <section-A3.4-7>` does not discuss vapor density. No experimental data related to this property could be found. Boiling is not expected to occur within the temperature range anticipated for design basis accidents in LBE-cooled fast reactors. However, in the unlikely event that coolant boiling were to occur, this phenomenon will likely be localized. Because of degraded heat transfer to vapor form coolant, coolant vapor temperature is not expected to raise significantly above saturation temperature.
The calculation of the vapor density in |SAS| uses a temperature-dependent polynomial which is multiplied by the saturation pressure:
.. math::
:label: A3.4-29
\rho_{v} = P_{S} \left ( \frac{A_{15}}{T} + A_{16} + A_{17} T + A_{18} T^2 + A_{19} T^3 + A_{20} T^4 \right )
The corresponding values for the coefficients of :eq:`A3.4-29` are provided in :numref:`section-A3.4-6`. For the current implementation, a value for :math:`A_{15} = 0.01` is assumed, and all other coefficients are assumed to be zero. The temperature range over which this correlation is applicable is unknown, but it is assumed that it applies to temperatures above :math:`\text{T}_{B,0(LBE)} = 1927 \pm 16 \text{ K}` where LBE vapor is anticipated to be present.
.. _section-A3.4-4.7:
LBE Liquid Specific Heat
''''''''''''''''''''''''
Heat capacity is one of the most important properties of a coolant. Calculation of the coolant enthalpy change as a function of temperature requires knowledge of the isobaric heat capacity.
Discussion in :ref:`[A3.4-1] <section-A3.4-7>` indicates a limited availability of independent data on liquid specific heat. Leveraging the similarity of Pb-Bi to its components, available experimental data, and Kopp’s law (typically used to derive heat capacities of binary systems), a single polynomial was developed in :ref:`[A3.4-1] <section-A3.4-7>`. As such, the expression for the molten coolant specific heat becomes:
.. math::
:label: A3.4-30
C_{l} = 164.8 - 3.94 \times 10^{-2} T + 1.25 \times 10^{-5} T^2 - \frac{4.56 \times 10^{5}}{T^2}
The OECD correlation uncertainty with respect to the data is, per :ref:`[A3.4-1] <section-A3.4-7>`:
- < 5% for temperatures within the range :math:`[400 – 1100 \text{K}]`, and
- 1 – 2% within the range :math:`[1100 - \text{T}_{B,0(LBE)} = 1927 \pm 16 \text{K}`, where the validity of Kopp’s law is most effective.
In |SAS|, the liquid specific heat is approximated by a polynomial that considers the difference between coolant temperature and critical temperature :eq:`A3.4-1`:
.. math::
:label: A3.4-31
C_{l} = \frac{A_{28}}{\left( T_{C} - T \right)^2} + \frac{A_{29}}{\left( T_{C} - T \right)} + A_{30} + A_{31} (T_{C} - T) + A_{32} \left ( T_{C} - T \right )^{2}
The mathematical expressions in :ref:`[A3.4-1]<Section-A3.4-7>` and that of |SAS| have different forms. The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to the :ref:`[A3.4-1] <Section-A3.4-7>` specific heat correlation. The temperature range over which this correlation was determined in :ref:`[A3.4-1] <Section-A3.4-7>` is maintained as the least square fit to :eq:`A3.4-31` was applied over the range outlined above. The corresponding values for the coefficients of :eq:`A3.4-31` are provided in :numref:`section-A3.4-6`.
Absolute and relative errors with respect to the OECD correlation were calculated, with relative error provided in the plot below.
.. figure:: media/LBELiquidHeatCapacity.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-4.8:
LBE Vapor Specific Heat
'''''''''''''''''''''''
Reference :ref:`[A3.4-1] <section-A3.4-7>` does not discuss vapor specific heat. No experimental data related to this property could be found. Boiling is not expected to occur within the temperature range anticipated for design basis accidents in LBE-cooled fast reactors. However, in the unlikely event that coolant boiling were to occur, this phenomenon will likely be localized. Because of degraded heat transfer to vapor form coolant, coolant vapor temperature is not expected to raise significantly above saturation temperature.
Specific heat is ignored until more information becomes available. This is not applicable for severe accidents where coolant boiling occurs at a larger scale and for higher temperatures.
.. _section-A3.4-4.9:
LBE Adiabatic Compressibility
'''''''''''''''''''''''''''''
No direct measurement of compressibility are available in literature, as per the discussion in :ref:`[A3.4-1] <section-A3.4-7>`. Reference :ref:`[A3.4-1] <section-A3.4-7>` assumes compressibility in this case as a reversible adiabatic process. As such, the isentropic compressibility of the coolant at normal pressure is calculated as the inverse of the bulk elastic modulus. Compressibility is then found from direct measurements of sound velocity, :math:`u_{s}`, and density:
.. math::
\beta_{s} = \left( \rho u_{s}^{2} \right)^{-1}
The bulk elastic modulus discussed in :ref:`[A3.4-1] <section-A3.4-7>` uses a polynomial as a function of temperature. As such, the expression for the compressibility becomes:
.. math::
:label: A3.4-32
\beta_{s} = \left( 3.8075 \times 10^{10} - 1.3152 \times 10^{7}T + 1.5143 \times 10^{3}T^2 - 5.8113 \times 10^{-2}T^3 \right)^{-1}
Reference :ref:`[A3.4-1] <section-A3.4-7>` indicates limited availability of sound velocity measurements in molten LBE, with tests conducted for temperatures :math:`[430 – 1100 \text{K}]`. Hence, applicability of the above correlation is considered to be :math:`[430 – 1100 \text{K}]`, where the deviation from measured data is less than 1% in this temperature range as per :ref:`[A3.4-1] <section-A3.4-7>`.
The compressibility coefficient is calculated in |SAS| by a polynomial expression based on the inverse of the coolant temperature subtracted from the critical temperature :eq:`A3.4-1`.
.. math::
:label: A3.4-33
\beta_{s} = A_{40} + \frac{A_{41}}{\left( T_{C} - T \right)}
The mathematical expressions in :ref:`[A3.4-1] <Section-A3.4-7>` and |SAS| have different forms. The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to :eq:`A3.4-32`. The corresponding values for the coefficients of :eq:`A3.4-33` are provided in :numref:`section-A3.4-6`.
Uncertainties for this correlation were not determined. However, absolute and relative errors with respect to :eq:`A3.4-32` were calculated. The relative error is provided in the plot below.
.. figure:: media/LBELiquidCompressibility.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-4.10:
LBE Thermal Expansion
'''''''''''''''''''''
The density of liquid metals changes with temperature due to thermal expansion related to anharmonicity of interatomic forces. As outlined in :ref:`[A3.4-1] <Section-A3.4-7>`, the relationship between the density and the coefficient of volumetric thermal expansion (CVTE) at constant pressure :math:`p` is defined by the following formula:
.. math::
:label: A3.4-34
\alpha_{p} = -\frac{1}{\rho} \left ( \frac{\partial \rho}{\partial T} \right )_{p}
Substituting the correlations for density recommended by :ref:`[A3.4-1] <Section-A3.4-7>` yields:
.. math::
:label: A3.4-35
\alpha_{p} = \frac{1}{8558-T}
Since the thermal expansion correlation is defined from that of the density, the same temperature range of :math:`[400 - 1273 \text{K}]` applies. The uncertainty of this quantity can be derived from the thermal expansion expression and the coolant density uncertainty.
The thermal expansion coefficient is calculated in |SAS| by a polynomial expression based on the inverse of the coolant temperature subtracted to the critical temperature:
.. math::
:label: A3.4-36
\alpha_{p} = A_{42} + \frac{A_{43}}{\left( T_{C} - T \right)} + \frac{A_{44}}{\left( T_{C} - T \right)^2} + \frac{A_{45}}{\left( T_{C} - T \right)^3} + \frac{A_{46}}{\left( T_{C} - T \right)^4} + \frac{A_{47}}{\left( T_{C} - T \right)^5}
A simple polynomial form matching between equations in :eq:`A3.4-35` and :eq:`A3.4-36` does not allow the |SAS| expression to match that of :ref:`[A3.4-1] <Section-A3.4-7>` with a reasonably small variation.
Therefore, additional terms had to be included in the |SAS| correlation. The correlation implemented includes terms from :eq:`A3.4-36` up to the third order.
The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to the OECD thermal expansion correlation. The corresponding values for the coefficients of :eq:`A3.4-36` are provided in :numref:`section-A3.4-6`.
Uncertainties for this correlation were not determined. However, absolute and relative errors with respect to :eq:`A3.4-35` were calculated. The relative error is provided in the plot below.
.. figure:: media/LBELiquidThermalExpansion.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-4.11:
LBE Thermal Conductivity
''''''''''''''''''''''''
Thermal conductivity is the fundamental thermophysical parameter of liquid metal coolants, which is used in all heat transfer calculations. The accuracy of its determination for liquid metals is poor and usually large discrepancies exist among different sets of data.
Per :ref:`[A3.4-1] <section-A3.4-7>`, the following linear correlation was used as the thermal conductivity of the technically pure liquid LBE at normal pressure:
.. math::
:label: A3.4-37
k_{l} = 3.284 + 1.617 \times 10^{-2}T - 2.305 \times 10^{-6}T^2
The correlation is applicable over the range [400 – 1100 K] with a maximum deviation of 10-15%.
The |SAS| mathematical expression correlating the thermal conductivity to the temperature is a third order polynomial as defined below:
.. math::
:label: A3.4-38
k_{l} = A_{48} + A_{49} T + A_{50} T^2 + A_{51} T^3
The corresponding values for the coefficients of :eq:`A3.4-38` are provided in :numref:`section-A3.4-6`. The correlation accuracy and temperature ranges of :eq:`A3.4-37` and :eq:`A3.4-38` are consistent since the mathematical correlation form is maintained.
.. _section-A3.4-4.12:
LBE Viscosity
'''''''''''''
Viscosity is a measure of friction among atoms of liquids. Because there is very little data on the impact of impurities on viscosity, the correlations below are for pure LBE.
LBE is believed to be a Newtonian liquid. Therefore, the temperature dependence of their dynamic viscosity is usually described by an Arrhenius-type equation as described in :ref:`[A3.4-1] <section-A3.4-7>`.
.. math::
:label: A3.4-39
\mu_{l} = 4.94 \times 10^{-4}e^{\left ( \frac{754.1}{T}\right )}
This correlation is valid in the temperature range of :math:`[400 – 1200 \text{K}]` and an estimated deviation of 6-8%.
Coolant dynamic viscosity is calculated in |SAS| by a polynomial expression based on the inverse of the coolant temperature.
.. math::
:label: A3.4-40
\mu_{l} = A_{52} + \frac{A_{53}}{T} + \frac{A_{54}}{T^2} + \frac{A_{55}}{T^3}
The mathematical expressions in :eq:`A3.4-39` and :eq:`A3.4-40` have different forms.
The constant coefficients in the |SAS| correlation were evaluated using a least squares equal weighted residual minimization fit to the OECD viscosity correlation. The corresponding values for the coefficients of :eq:`A3.4-40` are provided in :numref:`section-A3.4-6`.
The temperature range over which this correlation was determined in :ref:`[A3.4-1] <Section-A3.4-7>` is maintained because the least square fit to :eq:`A3.4-39` was performed over the applicability range outlined above. Uncertainties for this correlation were not determined. However, absolute and relative errors with respect to :eq:`A3.4-39` were calculated. The relative error is provided in the plot below.
.. figure:: media/LBEViscosity.png
:align: center
:figclass: align-center
:width: 7.70764in
:height: 6.21528in
.. _section-A3.4-5:
Physical Properties of Other Coolants
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
The mathematical expressions utilized for the physical properties of sodium-potassium and heavy water coolants are identical to those described in :ref:`Lead Properties <section-A3.4-3>` and :ref:`Sodium Properties <section-A3.4-3>`.
The corresponding coefficients utilized for all available built-in coolant properties are provided in :numref:`section-A3.4-6`. Additional details on formulation of these coefficients is not provided here.
.. _section-A3.4-6:
|SAS| Coolant Physical Properties
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
.. csv-table:: Coolant Thermophysical Property Coefficients for |SAS|
:file: media/CoolantProperties.csv
:header-rows: 1
:widths: auto
:align: center
.. [8] Coefficient A8, used in :ref:`[A3.4-2.4] <section-A3.4-3.4>`, is independent of the coolant type. It is defined as a function of the saturation pressure coefficients as: :math:`A8 = 2 \times A7`.
.. [9] Coefficient A9, used in :ref:`[A3.4-2.4] <section-A3.4-3.4>`, is independent of the coolant type. It is defined as a function of the saturation pressure coefficients as: :math:`A9 = -A6`.
.. [10] Coefficient A10, used in :ref:`[A3.4-2.4] <section-A3.4-3.4>`, is independent of the coolant type. It is defined as a function of the saturation pressure coefficients as: :math:`A10 =A6^2+4 \times A5 \times A7`.
.. [11] Coefficient A11, used in :ref:`[A3.4-2.4] <section-A3.4-3.4>`, is independent of the coolant type. It is defined as a function of the saturation pressure coefficients as: :math:`A11 = -4 \times A7`.
.. _section-A3.4-7:
References
^^^^^^^^^^
1. "Handbook on Lead-bismuth Eutectic Alloy and Lead Properties, Materials Compatibility, Thermal-hydraulics and Technologies", Nuclear Science,
OECD/NEA, 2015.