Merge branch 'updates-for-mr-1786' into 'main'

down again after 4s. Note that the timestep is not adjusted to sync with

transient BCs.

.. image:: ./images/transient-bc.png

.. figure:: ./images/transient-bc.png

   :height: 4in

   :align: center

   :alt: Plot of boundary conditions versus time. Fluid velocity increases linearly from 0 to 15 cm/s over the first 3 seconds and then remains constant. Temperature stays at 300 K from 0 to 3 seconds, rises abruptly to 500 K between 3 and 4 seconds, and then returns to 300 K after 4 seconds.

   Linear velocity ramp to 15 cm/s (0–3 s), constant thereafter; temperature step from 300 K to 500 K (3–4 s) then return to 300 K

Thermal boundary conditions

---------------------------

|                      | A valid is required for ``constant`` EB temperature model.            |             |           |

+----------------------+-----------------------------------------------------------------------+-------------+-----------+

The conjugate heat transfer model approximates heat transfer through the embedded boundary using one-dimensional,

steady-state conduction. The external environment is treated as a thermal reservoir with a constant temperature

in both space and time. This approach is illustrated in :numref:`fig_conjugate_heat_transfer_bc`, where heat flows

through a series of thermal resistances: from the fluid to the inner wall surface, through the wall, and finally

to the environment. The direction of heat transfer---into or out of the system---is governed by the temperature

gradient between the fluid and the surroundings.

The conjugate heat transfer model approximates heat transfer through the embedded boundary using a one-dimensional thermal

resistance representation. By default, the wall temperature is computed from an algebraic heat balance; optionally, a

lumped-capacity transient wall model can be enabled.

The external environment is treated as a thermal reservoir with constant temperature :math:`T_{\infty}`. Heat is

transferred from the fluid to the inner wall surface, through the wall, and then from the outer wall surface to

the environment, as illustrated in :numref:`fig_conjugate_heat_transfer_bc`.

.. _fig_conjugate_heat_transfer_bc:

   Schematic of the conjugate heat transfer boundary condition.

An effective resistance is calculated based on the convective heat transfer coefficient between the fluid

and wall, :math:`h_{int}`, conduction through the wall with thickness :math:`L_w` and thermal conductivity

:math:`\kappa_w`, and convective heat transfer from the exterior wall to the environment :math:`h_{ext}`.

The effective outward thermal conductance per unit area is

.. math::

   R_{eff} = \left( \frac{1}{h_{int}} + \frac{L_w}{\kappa_w} + \frac{1}{h_{ext}} \right)^{-1}

   G_{out} = \left( \frac{L_w}{\kappa_w} + \frac{1}{h_{ext}} \right)^{-1}

where :math:`L_w` is the wall thickness, :math:`\kappa_w` is the wall thermal conductivity,

and :math:`h_{ext}` is the external convective heat transfer coefficient. This expression

represents the combined resistance of conduction through the wall and convection from the

outer wall surface to the environment.

The interior wall temperature is determined by balancing the heat flux contributions from internal

convection, wall conduction, and external convection, resulting in a closed-form expression based

on the effective thermal resistance of the system.

For the zero-capacity wall model, the inner wall temperature is determined from an algebraic

heat balance between internal convection, radiation, and heat transfer to the environment:

.. math::

   T_{w,int} = T_f - \frac{1}{h_{int}}\left[ \frac{T_f - T_{\infty}}{R_{eff}} \right]

   T_{w,int} = \frac{ h_{int} T_f - q_{rad} + G_{out} T_{\infty}}{h_{int} + G_{out}}

where :math:`h_{int}` is the internal convective heat transfer coefficient. Positive :math:`q_{rad}`

denotes radiative heat loss from the wall. By default, the CHT-1D wall has no thermal storage,

so the wall temperature responds instantaneously to changes in the local heat balance.

If a wall heat capacity :math:`C_A` [J/(m\ :sup:`2`\ K)] is specified, the wall is modeled as a lumped thermal

mass with a single spatially uniform temperature. In that case, the wall temperature evolves according to

.. math::

   C_A \frac{dT_{w,int}}{dt} = h_{int} \left( T_f - T_{w,int} \right) - q_{rad} - G_{out} \left( T_{w,int} - T_{\infty} \right)

In this lumped-capacity approximation, the inner and outer wall temperatures are not resolved separately; instead,

the wall is represented by a single transient temperature. When :math:`C_A = 0` or is not specified, the algebraic

wall temperature model is used.

The model settings are defined using the prefix ``bc.[region_name].eb.temperature.CHT-1D``:

+----------------------------+------------------------------------------------------------------------+-------------+-----------+

| wall.thickness             | Wall thickness.                                                        | Real        | None      |

+----------------------------+------------------------------------------------------------------------+-------------+-----------+

| wall.heat_capacity         | Lumped wall heat capacity per area [J/(m\ :sup:`2`\ K)]                | Real        |  0.       |

+----------------------------+------------------------------------------------------------------------+-------------+-----------+

| phase_averaged_temperature | Use a phase averaged temperature for simulations containing particles. | Bool        | false     |

|                            | The phase averaged temperature is computed as the volume fraction      |             |           |

|                            | weighted sum of the fluid and averaged particle temperatures.          |             |           |

+---------------------+-----------------------------------------------------------------------+-------------+-----------+

| plt_X_g             | Plot fluid species mass fractions.                                    |    Int      | 0         |

+---------------------+-----------------------------------------------------------------------+-------------+-----------+

| plt_T_eb            | Plot EB wall temperature. Only applicable for CHT-1D BCs.             |    Int      | 0         |

+---------------------+-----------------------------------------------------------------------+-------------+-----------+

The following inputs must be preceded by the prefix ``mfix.solids`` and allow to write additional plotfiles which

contain only solids variables in specific regions at fixed timesteps or approximated simulation times.