.. _section-5.4.1:
.. _Pipe Temperatures:
Pipe Temperatures
~~~~~~~~~~~~~~~~~
The pipe temperature model is a slug flow model with heat transfer to
the pipe walls, as indicated in :numref:`figure-pipetemp`. The coolant in a pipe is
divided into a number of moving nodes or slugs. The node boundaries move
with the coolant flow. All nodes in a pipe have equal volumes except for
the first and last nodes. The inlet node size starts at zero and grows
as the flow continues until it reaches the size of the other nodes. At
that point a new node is started at the inlet. Similarly, the outlet
node shrinks and eventually is removed when its volume reaches zero. The
temperature in a coolant node changes due to internal heating (see :ref:`Direct Heating`)
or heat transfer with the pipe wall, while the temperature in a wall node may change due to
internal heating, heat transfer with the coolant, or heat transfer from the outside of the pipe wall.
There is one wall node for each coolant node. One radial node
is used in the pipe wall. Heat transfer from the outside of the pipe
wall is described in :numref:`section-5.4.6` on component-to-component heat
transfer. Wall nodes do not move, so the wall node in contact with a
given coolant node changes periodically as the coolant node boundaries
pass wall nodes.
All of the elements in a pipe temperature group are handled at the same
time as if they made a single long pipe. The use of equal coolant
volumes for each node determines the locations of the wall nodes. If the
region represented by a wall node spans the boundary between two
elements, then weighted averages are used to obtain the coolant flow
area, :math:`A_{\text{c}}`, wall perimeter, :math:`P_{\text{er}}`, wall mass,
:math:`M_{\text{w}}`, wall heat capacity, :math:`c_{\text{w}}`, and wall heat-transfer
coefficient, :math:`h_{\text{w}}`, for the node. The averaging is done so as to
conserve coolant volume and wall mass times heat capacity.
.. _figure-pipetemp:
.. figure:: media/image6.png
:align: center
:figclass: align-center
Pipe Temperature Calculations.
The primary loop time step is divided into sub-intervals for the pipe
temperatures calculations. The coolant slug is ejected from the end, a
new slug is formed at the inlet, and the node indexes for intermediate
slugs are increased by one. In subsequent sub-intervals, the coolant
moves exactly one node per sub-interval until the end of the primary
loop time step is approached. Usually, the coolant does not move exactly
an integral number of nodes in a primary loop step, so in the last
sub-interval the coolant usually moves only a fraction of a node.
For any node except the inlet node, the heat-transfer equation used for
the coolant is:
.. math::
:label: eq-5.4-1
\rho_{\text{c}}c_{\text{c}}A_{\text{c}}\frac{\partial \text{T}_{\text{c}}}{\partial \text{t}} = P_{\text{er}} h_{\text{wc}} \left( T_{\text{w}} - T_{\text{c}} \right) + q_{\text{c}}'
and that for the wall is
.. math::
:label: eq-5.4-2
M_{\text{w}}c_{\text{w}}\frac{\partial \text{T}_{\text{w}}}{\partial \text{t}} = P_{\text{er}} h_{\text{wc}} \left( T_{\text{c}} - T_{\text{w}} \right) + \left( \text{hA} \right)_{\text{snk}} \left( T_{\text{snk}} - T_{\text{w}} \right) + q_{\text{w}}'
where
:math:`T_{\text{c}}` = coolant temperature
:math:`T_{\text{w}}` = wall temperature
:math:`\rho_{\text{c}}` = coolant density
:math:`c_{\text{c}}` = coolant specific heat
:math:`h_{\text{wc}}` = effective heat-transfer coefficient between the wall and the
coolant
:math:`M_{\text{w}}` = wall mass per unit length
:math:`c_{\text{w}}` = specific heat of the wall
:math:`P_{\text{er}}` = wall perimeter, :math:`4A/D_h`
:math:`T_{\text{snk}}` = temperature of a heat sink outside the wall
:math:`\left( hA \right)_{\text{snk}}` = heat transfer coefficient times area per unit
length for heat transfer to air or liquid sodium outside the pipe wall
:math:`q_{\text{c}}'` = linear heat generation in the coolant
:math:`q_{\text{w}}'` = linear heat generation in the wall
The effective heat-transfer coefficient :math:`h_{\text{wc}}` contains a coolant heat-transfer
coefficient, :math:`h_{\text{c}}`, in series with a wall heat-transfer
coefficient, :math:`h_{\text{w}}`:
.. math::
:label: eq-5.4-3
\frac{1}{h_{\text{wc}}} = \frac{1}{h_{\text{c}}} + \frac{1}{h_{\text{w}}}
or
.. math::
:label: eq-5.4-4
h_{\text{wc}} = \frac{h_{\text{c}}h_{\text{w}}}{h_{\text{c}} + h_{\text{w}}}
The coolant heat-transfer coefficient is calculated as
.. math::
:label: eq-5.4-5
h_{\text{c}} = \frac{k_{\text{c}}}{D_{\text{h}}} \left\lbrack C_{1}\left( \frac{D_{\text{h}} \left| w \right| c_{\text{c}}}{A_{\text{c}}k_{\text{c}}} \right)^{c_{2}} + C_{3} \right\rbrack
:math:`C_1`, :math:`C_2` and :math:`C_3` = user-supplied
correlation coefficients
:math:`D_{\text{h}}` = pipe hydraulic diameter
:math:`w` = coolant flow rate
:math:`k_{\text{c}}` = thermal conductivity of the coolant
:math:`c_{\text{c}}` = specific heat of the coolant
The wall heat-transfer coefficient represents heat transfer from the
interior of the wall to the surface in contact with the coolant.
Assuming :math:`q_{\text{c}}'` and :math:`q_{\text{w}}'` to be constant over a timestep, finite differencing of :eq:`eq-5.4-1` and :eq:`eq-5.4-2` gives
.. math::
:label: eq-5.4-6
\rho_{\text{c}} c_{\text{c}} A_{\text{c}} \frac{\left( T_{\text{c}6} - T_{\text{c}5} \right)}{\delta t} = \frac{P_{\text{er}}h_{\text{wc}}}{2} \left( T_{\text{w}6} - T_{\text{c}6} + T_{\text{w}5} - T_{\text{c}5} \right) + q_{\text{c}}'
and
.. math::
:label: eq-5.4-7
M_{\text{w}}c_{\text{w}} \frac{\left( T_{\text{w}6} - T_{\text{w}5} \right)}{\delta t} = \frac{P_{\text{er}}h_{\text{wc}}}{2} \left( T_{\text{c}6} - T_{\text{w}6} + T_{\text{c}5} - T_{\text{w}5} \right) \\
+ \left( hA \right)_{\text{snk}} \left\lbrack T_{\text{snk}} - \frac{\left( T_{\text{w}6} + T_{\text{w}5} \right)}{2} \right\rbrack + q_{\text{w}}'
where
:math:`\delta t` = sub-interval time-step size
:math:`T_{\text{c}5}` = coolant temperature at beginning of the sub-interval
:math:`T_{\text{c}6}` = coolant temperature at end of the sub-interval
:math:`T_{\text{w}5}`, :math:`T_{\text{w}6}` = wall temperatures at
the beginning and end of the sub-interval
Simultaneous solution of these equations gives
.. math::
:label: eq-5.4-8
T_{\text{c}6} = \frac{\left\lbrack d_{2} - d_{1} d_{5} \left( M_{\text{w}} c_{\text{w}} + d_{3} \right) \right\rbrack T_{\text{c}5} + 2M_{\text{w}} c_{\text{w}} d_{1} d_{5} T_{\text{w}5} + 2d_{1} d_{3} d_{5} T_{\text{snk}} + \delta t q_{\text{c}}' + d_1 d_5 \delta t q_{\text{w}}'} {d_{2} + d_{1} d_{5} \left( M_{\text{w}} c_{\text{w}} + d_{3} \right)}
and
.. math::
:label: eq-5.4-9
T_{\text{w}6} = d_{5} \left( M_{\text{w}} c_{\text{w}} - d_{1} - d_{3} \right) T_{\text{w}5} + d_{1} d_{5} \left( T_{\text{c}6} + T_{\text{c}5} \right) + 2d_{3} d_{5} T_{\text{snk}} + d_5 \delta t q_{\text{w}}'
where
.. math::
:label: eq-5.4-10
d_{1} = \frac{\delta t}{2} h_{\text{wc}} P_{\text{er}}
.. math::
:label: eq-5.4-11
d_{2} = \rho_{\text{c}}c_{\text{c}}A_{\text{c}}
.. math::
:label: eq-5.4-11b
d_{3} = \frac{\delta t}{2} \left( hA \right)_{\text{snk}}
and
.. math::
:label: eq-5.4-12
d_{5} = \frac{1}{d_{1} + M_{\text{w}} c_{\text{w}} + d_{3}}
For the inlet node, the wall temperature calculation is the same as that
used for the other nodes, but the coolant temperature calculation is
different, since new coolant is being added to the node. The basic
equation used for the coolant temperature is
.. math::
:label: eq-5.4-13
L_{\text{n}} \rho_{\text{c}} c_{\text{c}} A_{\text{c}} \frac{\partial}{\partial \text{t}} \left( f_{\text{r}} T_{\text{c}} \right)
= L_{\text{n}} \rho_{\text{c}} c_{\text{c}} A_{\text{c}} {\overline{T}}_{\text{in}} \frac{\partial f_{\text{r}}}{\partial \text{t}} + L_{\text{n}} h_{\text{wc}} P_{\text{er}} f_{\text{r}} \left( T_{\text{w}} - T_{\text{c}} \right) + f_{\text{r}} q_{\text{c}}'
where
:math:`L_{\text{n}}` = length of a full node at the inlet
:math:`f_{\text{r}}` = fraction of a full node at the inlet
:math:`f_{\text{r}} L_{\text{n}}` = current length of the inlet node
:math:`{\overline{T}}_{\text{in}}` = pipe inlet temperature
After finite differencing, this equation becomes
.. math::
:label: eq-5.4-14
\rho_{\text{c}} c_{\text{c}} A_{\text{c}} f_{\text{r}6} T_{\text{c}6} - \rho_{\text{c}} c_{\text{c}} A_{\text{c}} f_{\text{r}5} T_{\text{c}5} = \rho_{\text{c}} c_{\text{c}} A_{\text{c}} {\overline{T}}_{\text{in}} \left( f_{\text{r}6} - f_{\text{r}5} \right) \\
+ \frac{\delta t}{2} h_{\text{wc}} P_{\text{er}} \left\lbrack f_{\text{r}5} \left( T_{\text{w}5} - T_{\text{c}5} \right) + f_{\text{r}6} \left( T_{\text{w}6} - T_{\text{c}6} \right) \right\rbrack + f_{\text{r6}} q_{\text{c}}'
where
:math:`f_{\text{r}5}` = :math:`f_{\text{r}}` at beginning of step
:math:`f_{\text{r}6}` = :math:`f_{\text{r}}` at end of step.
The simultaneous solution of :eq:`eq-5.4-14` and :eq:`eq-5.4-7` gives
.. math::
:label: eq-5.4-15
T_{\text{c}6} = \left\{ \left( d_{2} f_{\text{r}5} - d_{1} f_{\text{r}5} + d_{1}^{2} d_{5} f_{\text{r}6} \right) T_{\text{c}5} + d_{1} \left\lbrack f_{\text{r}5} + f_{\text{r}6} d_{5} \left( M_{\text{w}} c_{\text{w}} - d_{1} - d_{3} \right) \right\rbrack T_{\text{w}5} \right. \\ \left.
+ d_{2} \left( f_{\text{r}6} - f_{\text{r}5} \right) {\overline{T}}_{\text{in}} + 2 f_{\text{r}6} d_{1} d_{3} d_{5} T_{\text{snk}} + f_{\text{r6}} \delta t q_{\text{c}}' + f_{\text{r6}} d_1 d_5 \delta t q_{\text{w}}' \right\} \big/ \\
f_{\text{r}6} \left\lbrack d_{2} + d_{1} d_{5} \left( M_{\text{w}} c_{\text{w}} + d_{3} \right) \right\rbrack
and :eq:`eq-5.4-9` is again used for the wall temperature.
If flow in the pipe reverses direction, then the temperature
calculations are the same, except that the outlet node becomes the inlet
node and the inlet node becomes the outlet node.
.. _section-5.4.1.1:
Eulerian Calculations
^^^^^^^^^^^^^^^^^^^^^
The slug flow pipe temperature model described above is a LaGrangian
treatment that avoids the spurious numerical diffusion that results from
typical Eulerian treatments. On the other hand, there are situations in
which this treatment requires significantly more computing time than a
Eulerian treatment would; and in many of these situations the effects of
numerical diffusion would be small, so there is little gain from the
time consuming LaGrangian treatment. The Eulerian computation time per
subinterval is comparable to the LaGrangian computation time per
subinterval, but the LaGrangian time step subinterval size is limited by
the restriction that the coolant can not be allowed to move more than
one node per subinterval, whereas no such restriction applies in the
Eulerian case. Therefore, there is no advantage to a Eulerian treatment
if the coolant flow rates are small and the time step sizes are small;
but a Eulerian treatment is much faster than a LaGrangian treatment if
the coolant flow rate is high and time steps are large enough that the
coolant moves many nodes per time step. Therefore, a Eulerian speed-up
option has been added to the pipe temperature calculations in the code.
The Eulerian speed-up option can be especially useful in the null
transient used for steady-state initialization when
component-to-component heat transfer is used. In this case, the
temperature time constants are often large, requiring a long null
transient to obtain converged temperatures. The temperature solution is
numerically stable for large time steps, so one would use a large time
step size in the null transient to reduce computing time; but much of
the benefit from a large time step size is nullified if a LaGrangian
pipe temperature calculation limits the subinterval size to a small
value. Also, the numerical diffusion from a Eulerian solution is small
or non-existent in a steady-state pipe temperature result, so there is
no reason not to use the Eulerian treatment in the null transient.
There are three options for using the Eulerian speed-up. The default
option is to always us only the LaGrangian treatment. The second option
is to use the Eulerian speed-up in the steady-state null transient but
not in the regular transient. The third option is to use the Eulerian
speed-up both in the null transient and in the regular transient. In any
case, the LaGrangian calculation is used for small time steps in which
the coolant will move less than two nodes. If the Eulerian speed-up is
being used for a large time step, then first a LaGrangian subinterval is
used to move the coolant to the next node boundary. Next, a Eulerian
calculation is used to move the coolant the maximum whole number of
nodes that will fit within the time step. Finally, a LaGrangian
subinterval is used to finish the time step and move the coolant many
fraction of a node remaining. The Eulerian part of the calculation is
described in :numref:`section-A5.6`.
.. _section-5.4.1.2:
Annular Element Temperatures
^^^^^^^^^^^^^^^^^^^^^^^^^^^^
An annular element is treated the same as a pipe except that an annular
element has two walls in contact with the coolant instead of one. The
annular element was added to |SAS| in order to model the
coolant flow in an RVACS/RACS system in which a relatively thin annulus
of sodium flows between the vessel wall and an inner liner. Significant
heat transfer occurs between the sodium in the annulus and both the
vessel wall and the inner liner.
For the annular element, the heat transfer equation used for the coolant
is
.. math::
:label: eq-5.4-15a
\rho_{\text{c}} c_{\text{c}} A_{\text{c}} \frac{\partial \text{T}_{\text{c}}}{\partial \text{t}} = P_{\text{era}} h_{\text{wca}} \left( T_{\text{wa}} - T_{\text{c}} \right) + P_{\text{erb}} h_{\text{wcb}} \left( T_{\text{wb}} - T_{\text{c}} \right) + q_{\text{c}}'
where :math:`P_{\text{era}}`, :math:`h_{\text{wca}}`, and :math:`T_{\text{wa}}` refer to wall
:math:`a`, and :math:`P_{\text{erb}}`, :math:`h_{\text{wcb}}`, and :math:`T_{\text{wb}}` refer to
wall :math:`b`. :eq:`eq-5.4-2` and :eq:`eq-5.4-7` are still applicable to each wall.
Finite differencing of :eq:`eq-5.4-15a` gives
.. math::
:label: eq-5.4-15b
\rho_{\text{c}} c_{\text{c}} A_{\text{c}} \frac{\left( T_{\text{c}6} - T_{\text{c}5} \right)}{\delta \text{t}} = \frac{P_{\text{era}} h_{\text{wca}}}{2} \left( T_{\text{wa}6} - T_{\text{c}6} + T_{\text{wa}5} - T_{\text{c}5} \right) \\
+ \frac{P_{\text{erb}} h_{\text{wcb}}}{2} \left( T_{\text{wb}6} - T_{\text{c}6} + T_{\text{wb}5} - T_{\text{c}5} \right) + q_{\text{c}}'
Simultaneous solution of the finite difference equations for
:math:`T_{\text{c}}`, :math:`T_{\text{wa}}`, and :math:`T_{\text{wb}}` gives
.. math::
:label: eq-5.4-15c
T_{\text{c}6} = \left\{\left\lbrack d_{2} - d_{1\text{a}} d_{5\text{a}} \left( M_{\text{wa}} c_{\text{wa}} + d_{3\text{a}} \right) - d_{1\text{b}} d_{5\text{b}} \left( M_{\text{wb}} c_{\text{wb}} + d_{3\text{b}} \right) \right\rbrack T_{\text{c}5} \right. \\ \left.
+ 2d_{1\text{a}} d_{5\text{a}} \left( M_{\text{wa}} c_{\text{wa}} T_{\text{wa}5} + d_{3\text{a}} T_{\text{snka}} + \frac{\delta t}{2} q_{\text{wa}}' \right) \right. \\ \left.
+ 2d_{1\text{b}} d_{5\text{b}} \left( M_{\text{wb}} c_{\text{wb}} T_{\text{wb}5} + d_{3\text{b}} T_{\text{snkb}} + \frac{\delta t}{2} q_{\text{wb}}' \right) + \delta t q_{\text{c}}' \right\} \\
\big/ \left\lbrack d_{2} + d_{1\text{a}} d_{5\text{a}} \left( M_{\text{wa}} c_{\text{wa}} + d_{3\text{a}} \right) + d_{1\text{b}} d_{5\text{b}} \left( M_{\text{wb}} c_{\text{wb}} + d_{3\text{b}} \right) \right\rbrack
where :math:`q_{\text{wa}}'` and :math:`q_{\text{wb}}'` refer to linear heat sources for walls :math:`a` and :math:`b`, respectively.
The solutions for :math:`T_{\text{wa}}` and :math:`T_{\text{wb}}` are the same as
:eq:`eq-5.4-9`. For the inlet node, simultaneous solution of the finite
difference equations gives
.. math::
:label: eq-5.4-15d
T_{\text{c}6} = \left\{ \left\lbrack d_{2} f_{\text{r5}} - d_{1\text{a}} \left( f_{\text{r5}} - f_{\text{r6}} d_{1\text{a}} d_{5\text{a}} \right) - d_{1\text{b}} \left( f_{\text{r5}} - f_{\text{r6}} d_{1\text{b}} d_{5\text{b}} \right) \right\rbrack T_{\text{c}5} \right. \\ \left.
+ d_{1\text{a}} \left\lbrack f_{\text{r5}} + f_{\text{r6}} d_{5\text{a}} \left( M_{\text{wa}} c_{\text{wa}} - d_{1\text{a}} - d_{3\text{a}} \right) \right\rbrack T_{\text{wa}5} \right. \\ \left.
+ d_{1\text{b}} \left\lbrack f_{\text{r5}} + f_{\text{r6}} d_{5\text{b}} \left( M_{\text{wb}} c_{\text{wb}} - d_{1\text{b}} - d_{3\text{b}} \right) \right\rbrack T_{\text{wb}5} \right. \\ \left.
+ d_{2} {\overline{T}}_{\text{in}} \left( f_{\text{r6}} - f_{\text{r5}} \right) + 2f_{\text{r6}} \left\lbrack d_{1\text{a}} d_{3\text{a}} d_{5\text{a}} T_{\text{snka}} + d_{1\text{b}} d_{3\text{b}} d_{5\text{b}} T_{\text{snkb}} \right\rbrack \right. \\ \left.
+ f_{\text{r6}} \left\lbrack \delta t q_{\text{c}}' + d_{\text{1a}} d_{\text{5a}} \delta t q_{\text{wa}}' + d_{\text{1b}} d_{\text{5b}} \delta t q_{\text{wb}}' \right\rbrack \right\} \\
\big/ f_{\text{r6}} \left\lbrack d_{2} + d_{1\text{a}} d_{5\text{a}} \left( M_{\text{wa}} c_{\text{wa}} + d_{3\text{a}} \right) + d_{1\text{b}} d_{5\text{b}} \left( M_{\text{wb}} c_{\text{wb}} + d_{3\text{b}} \right) \right\rbrack
.. _Direct Heating:
Direct Element Heating
^^^^^^^^^^^^^^^^^^^^^^
During operation, heat may be generated and/or removed in a variety of different components.
For example, during pump operation, heat may be generated in both the working fluid and in the pump structure itself.
This phenomenon may strongly impact system temperatures under certain conditions, and therefore SAS provides users with flexibility when modeling it.
Direct element heating is currently supported in all pipe-like elements. Pipe-like elements include: pipes, pumps, check valves, valves, annular pipes, and annular pumps.
Two mechanisms are in place for defining direct element heat, time-dependent functions and built-in pump heat models.
Time-Dependent Element Heating
''''''''''''''''''''''''''''''
Time-dependent functions for direct fluid and direct structure heat can be defined using :sasinp:`CoolHeatTableID` and :sasinp:`WallHeatTableID`, respectively.
.. math::
:label: direct-nonannular-fluid
q_{\text{c}}' = \frac{f_{\text{c,k}}(t)}{L_{\text{k}}}
where
:math:`f_{\text{c,k}}` = user-defined, time-dependent function for element :math:`k` coolant (W).
:math:`L_{\text{k}}` = length of element :math:`k`
and
.. math::
:label: direct-nonannular-wall
q_{\text{w}}' = \frac{f_{\text{w,k}}(t)}{L_{\text{k}}}
where
:math:`f_{\text{w,k}}` = user-defined, time-dependent function for element :math:`k` wall (W).
A description of the CoolantHeat Table and the WallHeat Table, identified in :sasinp:`CoolHeatTableID` and :sasinp:`WallHeatTableID`,
as well as example tables are presented below.
.. list-table:: Input in a Direct Coolant Heat Table
:header-rows: 1
:align: center
:widths: 1,3,1
* - Column Label
- Description
- Range
* - ``iEll``
- Element ID.
- 1 :math:`\leq` ``iEll`` :math:`\leq` `NELEMT`
* - ``fID``
- ID of function defining direct coolant heat for element ``iELL``.
The output of this function should be in Watts.
-
.. list-table:: Input in a Direct Wall Heat Table
:header-rows: 1
:align: center
:widths: 1,3,1
* - Column Label
- Description
- Range
* - ``iEll``
- Element ID.
If ``iEll`` is greater than 1000, the second wall of ``iEll``\ -1000 is identifeid.
- 1 :math:`\leq` ``iEll`` :math:`\leq` `NELEMT`
* - `fID`
- ID of function defining direct wall heat for element ``iELL``.
The output of this function should be in Watts.
-
.. _DirectHeatInput:
Example Direct Coolant Heat Table input::
TABLE <id> Coolant
iEll fID
3 1
4 100
END
Example Direct Wall Heat Table input::
TABLE <id> Structure
iEll fID
1003 200
3 100
4 1
END
.. _Pump Heating:
Pump Heat
'''''''''
Pump heating is implemented by calculating the contributions to linear heat rates :math:`q_{\text{c}}'`, :math:`q_{\text{w}}'`, :math:`q_{\text{wa}}'`, and :math:`q_{\text{wb}}'` in the equations above.
Currently this treatment is limited only to :ref:`equivalent circuit pump models<Equivalent Circuit EM pump>`, where the scaling factor :ref:`fH<common inputs>` is used to control the fraction of power dissipated in various resistors representing fluid and wall components.
**Detailed Equivalent Circuit Model**
For reference to the variables cited below, see the :ref:`governing circuit<detailed circuit diagram>`.
Power dissipated by :math:`R_1` and :math:`R_D` in each phase is deposited into the element wall while power disspated in :math:`R_J` is deposited in the fluid.
For nodes of element :math:`k` that corresponds to an equivalent circuit pump, this is expressed as
.. math::
:label: detailed-nonannular-fluid
q_{\text{c}}' = \frac{3 f_H I_3^2 R_J}{L_{\text{k}}}
where
:math:`L_{\text{k}}` = length of element :math:`k`
:math:`I_3` = current through resistor :math:`R_J`
and
.. math::
:label: detailed-nonannular-wall
q_{\text{w}}' = \frac{3 f_H \left(I^2 R_1 + I_2^2 R_D \right)}{L_{\text{k}}}
If the pump element is annular (see `ITYPEL`), users can control how much of the wall heat is put into each of :math:`q_{wa}'` and :math:`q_{wb}'` as
.. math::
:label: detailed-annular-wall
q_{\text{wa}}' &= f_{OD} q_{\text{w}}' \\
q_{\text{wb}}' &= (1 - f_{OD}) q_{\text{w}}'
where :math:`f_{OD}` is :ref:`fOD<common inputs>`.
**Simple Equivalent Circuit Model**
For reference to the variables cited below, see the :ref:`governing circuit<simple circuit diagram>`.
Power dissipated by :math:`R_1` in each phase is deposited into the element wall as
.. math::
:label: simple-nonannular-wall
q_{\text{w}}' = \frac{3 f_H I^2 R_1}{L_{\text{k}}}
No power is deposited into the coolant in the simple circuit model.
If the pump element is annular (see `ITYPEL`), users can control how much of the wall heat is put into each of :math:`q_{wa}'` and :math:`q_{wb}'` as
.. math::
:label: simple-annular-wall
q_{\text{wa}}' &= f_{OD} q_{\text{w}}' \\
q_{\text{wb}}' &= (1 - f_{OD}) q_{\text{w}}'
where :math:`f_{OD}` is :ref:`fOD<common inputs>`.