.. _section-3.12:
Interaction with Other Models
-----------------------------
As mentioned in :numref:`section-3.1`, the heat-transfer routines interact with a
number of other models. These interactions are indicated in :numref:`figure-3.1-1`.
If PRIMAR-4 is being used, then the subassembly coolant outlet
temperatures calculated in the heat-transfer routines are used by
PRIMAR-4 to calculate the outlet plenum temperature. Also, if flow
reversal occurs in a channel, then the temperature calculated in the
heat-transfer routines for the coolant leaving the bottom of the
subassembly is used by PRIMAR-4 in the calculation of the inlet plenum
temperature. :numref:`section-3.3.6` describes how the inlet and outlet plenum
temperatures are used in the calculations of the subassembly coolant
inlet and reentry temperatures.
Before the onset of voiding, TSHTRN calculates the coolant temperatures
used in the coolant channel hydraulics calculations, and the hydraulics
routines calculate the coolant flow rates used by TSHTRN.
After the onset of voiding, coolant temperatures are calculate din
TSBOIL. This module supplies the heat flux at the cladding outer
surface, as defined in :eq:`3.5-2` and used in :eq:`3.5-9` to TSHTRV. TSBOIL
uses the calculated cladding temperatures form TSHTRV to obtain
extrapolated cladding temperatures, as in :eq:`3.5-3`, for use in its
coolant temperature calculations.
The point kinetics model supplies the power level used in the
heat-transfer routines. In return, the heat-transfer routines supply the
temperatures used to calculate reactivity feedback.
.. _section-3.12.1:
Reactivity Feedback
~~~~~~~~~~~~~~~~~~~
The temperatures calculated in the core thermal hydraulics routines are
used to calculate various components of reactivity feedback. These
reactivity components include Doppler feedback, axial expansion of the
fuel and cladding, density changes in the sodium, core radial expansion
and control rod drive expansion. These reactivity feedbacks are
described in :numref:`Chapter %s<section-4>`.
.. _section-3.12.2:
Coupling Between Core Channels and PRIMAR-4
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
As described in :numref:`section-5.2.2`, the PRIMAR-4 calculations for a PRIMAR time
step are carried out before the core channel coolant calculations.
PRIMAR-4 must make estimates of the core flows for a new time step, and
it also makes corrections for the differences between its estimates for
the previous step and values computed by the core channel coolant
routines. The coolant routines supply information to PRIMAR-4 for use in
these estimates and corrections. Also, PRIMAR-4 supplies inlet and
outlet coolant plenum pressures and temperatures for the use in the core
channel calculations.
.. _section-3.12.2.1:
Supplied By PRIMAR-4
^^^^^^^^^^^^^^^^^^^^
The information supplied by PRIMAR-4 is
:math:`p_{\text{in}}\left( t_{\text{p}1} \right)` = inlet plenum pressure at
the beginning of the PRIMAR time step
:math:`p_{\text{x}}\left( t_{\text{p}1} \right)` = outlet plenum pressure at the
beginning of the PRIMAR time step
:math:`\frac{\text{dp}_{\text{in}}}{\text{dt}}` =time derivative of the inlet
plenum pressure
:math:`\frac{\text{dp}_{\text{x}}}{\text{dt}}` =time derivative of the outlet plenum
pressure
:math:`\rho_{\text{cin}}` =coolant density in the inlet plenum
:math:`\rho_{\text{cout}}` =coolant density in the outlet plenum
:math:`T_{\text{cin}}` =coolant temperature in the inlet plenum
and
:math:`T_{\text{cout}}` = coolant temperature in the outlet plenum
The pressure :math:`p_{\text{in}}` is at an elevation
:math:`z_{\text{pll}}` and :math:`p_{\text{x}}` is at :math:`z_{\text{plu}}`.
At any time, :math:`t`, during the PRIMAR time step, the inlet plenum pressure
is
.. math::
:label: 3.12-1
p_{\text{in}}\left( t \right) = p_{\text{in}}\left( t_{\text{p}1} \right) + \left( t - t_{\text{p}1} \right)\frac{\text{dp}_{\text{in}}}{\text{dt}}
and the exit plenum pressure is
.. math::
:label: 3.12-2
p_{\text{x}}\left( t \right) = p_{\text{x}}\left( t_{\text{p}1} \right) + \left( t - t_{\text{p}1} \right)\frac{\text{dp}_{\text{x}}}{\text{dt}}
.. _section-3.12.2.2:
Supplied To PRIMAR-4
^^^^^^^^^^^^^^^^^^^^
The information supplied to PRIMAR-4 by the core coolant routines for
channel :math:`ic` at the subassembly inlet :math:`\left( L=1 \right)` or outlet
:math:`\left( L=2 \right)` is the following:
.. math::
:label: 3.12-3
\Delta m_{\text{c}}\left( L \right) = \sum_{\text{ic}}{N_{\text{ps}}\left( \text{ic} \right)}\int_{t_{\text{p}1}}^{t_{\text{p}2}}{w \left( L,ic \right) }\text{dt}~ ,
.. math::
:label: 3.12-4
\Delta m_{\text{c}}T_{\text{c}}\left( L \right) = \sum_{\text{ic}}{\ N_{\text{ps}}\left( \text{ic} \right) }\int_{t_{\text{p}1}}^{t_{\text{p}2}}{w\left( L,ic \right)\ T_{\text{ex}}\left( L,ic \right) }\text{dt}~ ,
where
:math:`w \left(L,ic,t=t_{\text{p}2} \right)` = computed flow rate at
:math:`t_{\text{p}2}`
:math:`T_{\text{ex}} \left( L,ic \right)` = coolant temperature at the subassembly inlet or
outlet
.. :math:`\Delta E_{\text{v}} \left( L,ic \right)` = heat added to the inlet or outlet plenum by
.. condensing sodium vapor. (This term is zero before the onset of
.. boiling.)
Additionally, the core coolant routines supply the coefficients :math:`C_{\text{0}}' \left( L,ic \right)`, :math:`C_{\text{1}} \left( L,ic \right)`,
:math:`C_{2} \left( L,ic \right)`, and :math:`C_{3}' \left( L,ic \right)` used by PRIMAR-4 to
estimate the core channel flow. PRIMAR-4 estimates the flow into or out
of each subassembly using
.. math::
:label: 3.12-5
\frac{\Delta w_{\text{e}}\left( L,ic \right)}{\Delta \text{t}} = C_{0}'\left( L,ic \right)
+ C_{1} \left( L,ic \right) \left( p \left( \text{JIN} \right) + \theta_{2c} \Delta p\left( \text{JIN} \right) \right) \\
+ C_{2} \left( L,ic \right) \left( p \left( \text{JX} \right) + \theta_{2c} \Delta p\left( \text{JX} \right) \right) \\
+ 2 \theta_{2c} C_{3}' \left( L,ic \right) w_{\text{e}}\left( L,ic \right) \Delta w_{\text{e}}\left( L,ic \right)
The core channel calculations use :math:`w` as the flow rate per pin, whereas
PRIMAR-4 estimates the total flow represented by a channel, so
:math:`N_{\text{ps}} \left( ic \right)`, the number of pins per subassembly times the
number of subassemblies represented by the channel, comes into
:eq:`3.12-3`, :eq:`3.12-4` and the calculations of the coefficients :math:`C_0'`,
:math:`C_1`, :math:`C_2`, and :math:`C_3'`. In the pre-voiding module the
coefficients are calculated as
.. math::
:label: 3.12-6
C_{0}' = -\frac{N_{ps}}{I_1} \left( w_{2}^{2} I_{2} + A'_{\text{fL}} w_{2}I_{\text{3L}} + A'_{\text{fr}}w_{2}\left| w_{2} \right|^{1 + b_{\text{fr}}} I_{\text{3T}} + w_{2}\left| w_{2} \right|I_{4} \right. \\ \left.
+ g\left\lbrack I_{5} + \rho_{\text{cin}}\left( z_{\text{c}}\left( 1 \right) - z_{\text{pll}} \right) + \rho_{\text{cout}} \left( z_{\text{plu}} - z \left( \text{MZC} \right) \right) \right\rbrack\right)
.. math::
:label: 3.12-7
C_{1} = \frac{N_{\text{ps}}}{I_{1}}
.. math::
:label: 3.12-8
C_{2} = - C_{1}
and
.. math::
:label: 3.12-9
C_{3}' = - \frac{\left( I_{2}w_{2} + 0.5A'_{\text{fL}}I_{\text{3L}} + \left(1+0.5b_{\text{fr}}\right)A'_{\text{fr}}I_{\text{3T}}\left| w_{2} \right|^{1 + b_{\text{fr}}} + I_{4}\left| w_{2} \right| \right)}{\left| w_{2} \right|I_{1}N_{\text{ps}}}
In this case, the coefficients for :math:`L=2` are equal to those for
:math:`L=1`.