.. _section-3.4:

Steady-State Thermal Hydraulics
-------------------------------

The steady-state thermal hydraulics calculations for a channel using the
single pin per subassembly option consist of direct solutions of the
relevant steady-state equations, rather than running the transient
calculations until they converge to a steady-state solution.

For the steady-sate calculations, the user specifies the coolant flow
rate for each channel, the coolant inlet temperature and exit pressure,
and the power in each node of each channel. The code then calculates the
remaining coolant temperatures and pressures, as well as the
temperatures in the fuel, cladding, structure, and reflectors. First,
the coolant temperatures in a channel are calculated, starting at the
inlet and working upward. The steady-state coolant temperature
calculation requires only the coolant flow rate, the total power in each
axial node, and the coolant heat capacity; so coolant temperatures can
be calculated before the fuel and cladding temperatures are known. The
second step is to calculate the coolant pressures, starting at the
subassembly outlet and working down. Inlet orifice coefficients are
adjusted so that all channels have the same total pressure drop. The
pressure calculations are described in :numref:`section-3.9`. The third step is to
set the structure and reflector temperatures equal to the coolant
temperatures everywhere except in the core and axial blankets. The gas
plenum temperatures are also set equal to the coolant temperature in
this region, and the cladding temperature in the gas plenum region is
also set equal to the coolant temperature. Fourth, the fuel-pin
temperatures are calculated for each axial node in the core and axial
blankets, starting at the cladding outer surface and working inward.
Last, the structure temperatures in the core and axial blankets are
calculated.

.. _section-3.4.1:

Basic Equations
~~~~~~~~~~~~~~~

The basic heat-transfer equations used in the steady-state calculations
are the same as those used for the transient solution, except that all
of the time derivatives are dropped in the steady-state solution. These
equations include :eq:`3.3-1` and :eq:`3.4-4` to :eq:`3.3-10`. Also, the spatial
finite differencing used in the steady-state is the same as that used in
the transient.

For the steady-state calculations, :eq:`3.3-5` becomes

.. math::
    :label: 3.4-1

    \frac{\text{d}}{\text{dz}} \left( wc_{\text{c}}T \right) = Q_{\text{ct}}A_{\text{c}}

where the total heat source per unit volume, :math:`Q_{\text{ct}}`, at node
:math:`jc` is

.. math::
    :label: 3.4-2

    Q_{\text{ct}} \left( jc \right) = Q_{\text{c}} \left( jc \right) + Q_{\text{ec}} \left( jc \right) + Q_{\text{sc}} \left( jc \right) = \frac{ \overline{P} \left( jc \right)}{A_{\text{c}}} \Delta z \left( jc \right)

and :math:`\overline{P}\left( jc \right)` is the
total steady-state power (watts) in the node. For this equation, it is
assumed that all heat generated in the fuel, cladding, and structure
ends up in the coolant. Note that outside the core and axial blankets
:math:`\overline{P}\left( jc \right)` and
:math:`Q_{\text{ct}}\left( jc \right)` are zero.

For the steady-state fuel and cladding calculations, :eq:`3.3-1` is
multiplied by :math:`2 \pi r` and integrated from the fuel inner
surface, :math:`r_{\text{if}}`, to give

.. math::
    :label: 3.4-3

    2\pi kr \frac{\text{dT}}{\text{dr}} = - 2 \pi \int_{r_{\text{if}}}^{r}{r' Q \left( r' \right)} dr'

where the adiabatic boundary condition at :math:`r_{\text{if}}` has been used.

.. _section-3.4.2:

Coolant Temperatures
~~~~~~~~~~~~~~~~~~~~

The finite difference form for :eq:`3.4-1` may be written as

.. math::
    :label: 3.4-4

    w{\overline{c}}_{\text{c}}\left( \text{j} \right) \frac{\left\lbrack T_{c}\left( jc + 1 \right) - T_{c}\left( jc \right) \right\rbrack}{\Delta z \left( jc \right)} = \frac{\overline{P}\left( jc \right)}{ \Delta z \left( jc \right)}

or

.. math::
    :label: 3.4-5

    T_{\text{c}}\left( jc + 1 \right) = T_{\text{c}}\left( jc \right) + \frac{\overline{P}\left( jc \right)}{w{\overline{c}}_{\text{c}}\left( jc \right)}

where :math:`{\overline{c}}_{\text{c}}\left( jc \right)` is
the specific heat evaluated at the average temperature,
:math:`{\overline{T}}_{\text{c}}\left( jc \right)`, given by

.. math::
    :label: 3.4-6

    {\overline{T}}_{\text{c}}\left( jc \right) = \frac{T_{\text{c}}\left( jc \right) + T_{\text{c}}\left( jc + 1 \right)}{2}

Also, :math:`T_{\text{c}}\left( 1 \right)` is equal to the inlet temperature:

.. math::
    :label: 3.4-7

    T_{\text{c}}\left( 1 \right) = T_{\text{in}}

Starting from *jc* = 1, :eq:`3.4-5` is used to match up the channel. An
iteration is used to obtain consistency between
:math:`{\overline{c}}_{\text{c}}\left( jc \right)` and
:math:`{\overline{T}}_{\text{c}}\left( jc \right)`.

.. _section-3.4.3:

Fuel and Cladding Temperatures in the Core and Axial Blankets
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

At each axial node, the radial node powers, :math:`Q(i)`, are calculated
using :eq:`3.3-22` to :eq:`3.3-25`. Note that the :math:`Q` in :eq:`3.4-3` is a power
per unit volume, whereas :math:`Q(i)` is an integral value for a node:

.. math::
    :label: 3.4-8

    Q \left( i \right) = \int_{z \left( j \right)}^{z \left( j + 1 \right)}\int_{r_{\text{i}}}^{r_{i + 1}}{\ \ 2\pi r\ \ Qdr\ \ \text{dz}}

The sums, :math:`Q_{\text{sm}} (i)`, are calculated as

.. math::
    :label: 3.4-9

    Q_{\text{sm}}\left( i \right) = \sum_{ii = 1}^{i}{\text{Q}\left( ii \right)}

:eq:`3.4-3` becomes

.. math::
    :label: 3.4-10

    2\pi\ {\overline{k}}_{\text{i,i} + 1}\ \frac{r \left( i + 1 \right) \left \lbrack T \left( i + 1 \right) -  T\left( i \right) \right\rbrack}{\Delta r_{\text{i,i} + 1}}\  = \ \frac{Q_{\text{sm}}\left( i \right)}{ \Delta z }

or

.. math::
    :label: 3.4-11

    T\left( i \right) = T\left( i + 1 \right) +  \frac{\Delta r_{\text{i,i} + 1} Q_{\text{sm}}\left( i \right)}{2\pi k_{\text{i,i} + 1} r \left( i + 1 \right) \Delta z }

where :math:`\Delta r_{i,i + 1}` and
:math:`{\overline{k}}_{i,i + 1}` are given by :eq:`3.3-21`
and :eq:`3.3-26`.

The calculations for an axial node start with the coolant temperature
that has already been calculated, as in the section above:

.. math::
    :label: 3.4-12

    T \left( \text{NC} ,j \right) = \overline{T} \left( jc \right)

Then the cladding surface temperature is given by

.. math::
    :label: 3.4-13

    T\left( \text{NE}' \right) = T\left( \text{NC} \right) + \frac{Q_{\text{sm}}\left( \text{NE}' \right)}{2\pi r\left( \text{NE}' \right) \Delta z~ h_{\text{c}}}

Cladding temperatures at nodes :math:`\text{NE}` and :math:`\text{NE}''` are calculated using
:eq:`3.4-11`. Since :math:`{\overline{k}}_{\text{ii} + 1}` can be a
function of :math:`T_{\text{i}}`, a simple iteration between :eq:`3.4-11` and
:eq:`3.3-26` is used.

The equation used for the fuel surface temperature is

.. math::
    :label: 3.4-14

    2\pi r\left( \text{NR} \right) \left\{ h_{\text{b}}\left\lbrack T \left( \text{NT} \right) - T\left( \text{NE}'' \right)\  \right\rbrack + \varepsilon\sigma\left\lbrack T\left( \text{NT} \right)^{4} - T\left( \text{NE}'' \right)^{4} \right\rbrack  \right\} = \frac{Q_{\text{sm}}\left( \text{NT} \right)}{\Delta z}

or

.. math::
    :label: 3.4-15

    T\left( \text{NT} \right) = d_{1} - d_{2} T \left( \text{NT} \right)^{4}

where

.. math::
    :label: 3.4-16

    d_{1} = T\left( \text{NE}'' \right) + \frac{Q_{\text{sm}}\left( \text{NT} \right)}{2\pi r\left( \text{NR} \right)\Delta z~ h_{\text{b}}} + \frac{\epsilon\sigma T\left( \text{NE}'' \right)^{4}}{h_{\text{b}}}

and

.. math::
    :label: 3.4-17

    d_{2} = \frac{\epsilon \sigma}{h_{\text{b}}}

:eq:`3.4-15` is solved by iteration.

After the fuel surface temperature has been calculated, the inner fuel
node temperatures are calculated one at a time, starting at the outside
and working inward, by iterating between :eq:`3.4-11` and :eq:`3.3-26`. In this
procedure, *T*\ (*i*) is to be found after *T*\ (*i*\ +1) is known.
First, *T*\ (*i*) is set equal to :math:`T(i + 1)`. Second, *k*\ (*i*)
is to be found after *T*\ (*i*\ +1) is known. First, *T*\ (*i*) is set
equal to :math:`T(i + 1)`. Second, *k*\ (*i*) is calculated as a
function of the temperature, *T*\ (*i*). Third,
:math:`{\overline{k}}_{\text{i,i} + 1}`\ is calculated using
:eq:`3.3-26`. Fourth, a new value for *T*\ (*i*) is calculated, using
:eq:`3.4-11`. Fifth, the new *T*\ (*i*) from the fourth step is compared with
old value used in the second step. If the two values differ by less than
a user-specified convergence criterion, then the iteration is finished,
and the code goes on to the next node. Otherwise, the code goes back to
the second step, using the new value of *T*\ (*i*), and repeats the
process.

.. _section-3.4.4:

Structure Temperatures in the Core Axial Blankets
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The inner structure node temperature is calculated using

.. math::
    :label: 3.4-18

    \Delta z~S_{\text{pr}}\ H_{\text{sic}}\ \left\lbrack T \left( \text{NSI} \right) - T \left( \text{NC} \right) \right\rbrack = \gamma_{\text{s}}\overline{P}\left( j \right)

or

.. math::
    :label: 3.4-19

    T\left( \text{NSI} \right) = T\left( \text{NC} \right) + \frac{\gamma_{\text{s}}\overline{P}\left( j \right)}{ \Delta z ~S_{\text{pr}}\ H_{\text{sic}}}

The outer structure node is then calculated using

.. math::
    :label: 3.4-20

    T\left( \text{NSO} \right) = T\left( \text{NSI} \right) + \frac{\gamma_{\text{s}}\overline{P}\left( j \right)\ d_{\text{sto}}}{ \Delta z ~S_{\text{pr}}\ H_{\text{stio}}\ \left( d_{\text{sti}} + d_{\text{sto}} \right)}

.. _section-3.4.5:

Reflector, Structure, Cladding, and Gas Plenum Temperature Outside the Core and Axial Blankets
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

Outside the core and axial blankets no power sources are considered, so
the reflector and structure temperatures at an axial node are the same
as the coolant temperature for the steady-state. The coolant
temperatures are the same at all axial nodes in the gas plenum region,
and the cladding and gas temperatures in this region are equal to the
coolant temperatures.