__1. Variables stored and solved__

2. Media properties

3. Conjugate heat-transfer for the enthalpy equation

4. How to set boundary conditions for IMMERSOL:

4.1 Apertures in the Domain Boundaries

4.2 Solid surfaces at the Domain Boundaries

4.2.2 Shell wall with a given external heat flux

4.2.1 Shell wall with fixed temperature.

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**1. Variables stored and solved**

The IMMERSOL radiation model requires solution of the energy equation which involves the energy transport in solids by conduction and that within the space between solids by radiation. This equation is formulated in terms of T3, which is the temperature of the solid phase in solids and the radiation temperature in gaps separating solids. The solution of this equation is activated in PHOENICS by the PIL command,

SOLVE(T3).

The T3 equation includes only diffusion terms, thus TERMS(T3,N,N,Y,P,P,P).

The solution of the T3 equation requires the storage of the PIL variable WGAP to calculate the heat transport coefficient within gaps by radiation. WGAP stands for the distance between adjacent solids computed from the solution for LTLS. The later is activated by the PIL commands,

DISWAL; STORE(WGAP).

The diffusivity for the radiation flux, i.e. T3, in the gaps is set by EARTH internally. The heat conduction coefficient in solids can either be set to a constant, PRANDTL(T3)= -CONST, or defined in the same way as for the conjugate heat-transfer problems, i.e. by the PIL commands,

STORE(PRPS); PRNDTL(T3)= -GRND10.

The 3D array PRPS stores material flags.

The emissivity coefficient of solid surfaces by default is taken to be 0.8. The user can change the default value by the following PIL command

SPEDAT(SET,EMISSIVITY,IMAT,R,EPS)

Where IMAT is the flag of the material which will have the surface emissivity set to EPS, e.g. SPEDAT(SET,EMISSIVITY,111,R,0.1) sets the emissivity coefficient of material 111 to 0.1.

The heat transfer by conduction and convection in the gaps between solids filled by a participating media is defined by the energy equation formulated in terms of either the gas temperature TEM1 or enthalpy H1; the solution of this equation being activated by the PIL command,

SOLVE(TEM1) or SOLVE(H1).

The heat conduction coefficient for TEM1 or H1 is normally defined by the PIL command, PRNDTL(TEM1)= -GRND10 or PRNDTL(H1)= -GRND10. Note, that PROPS file stores heat conduction coefficients, which are retrieved as for TEM1, as for H1. The flow field problem is set and solved using the standard PHOENICS approach.

The contribution of radiation heat transfer to the energy equation in participating media is described by a source term 4*A*(R - E)

The physical meaning of this term is that it represents the net rate of loss or gain of energy per unit volume. The term A*E represents the local rate of emission, while the term A*R represents the local rate of absorption of radiation per unit volume.

By default the absorption coefficient A is constant and defined by the PIL variable RADIA. Setting of RADIA to one of the GRND numbers causes EARTH to call function GXABSR for the value of absorption coefficient in a fluid-cell (function GXABSR resides in GXIMSL.FOR file). RADIA=GRND1 enables to set absorption coefficients for each non-solid material. Value of A is retrieved from the PIL command

SPEDAT(SET,ABSORPTION,IMAT,R,Val)

Here IMAT is a material flag; and Val is the value of A set for this material.

__3. Conjugate heat-transfer for the enthalpy equation__

In IMMERSOL, the thermal-energy equation for the gaps between solids can be formulated in terms of either temperature TEM1, or enthalpy H1.

If all solid objects in the domain are defined using special IMSW-patches (see 1.2 below), i.e. if there are no solid regions participating in the conjugate heat transfer, the user can specify either the laminar Prandtl number, or the diffusivity as for TEM1, as for H1 using the PIL variable PRNDTL (see PRNDTL entry to PHENC).

To activate the conjugate heat transfer the user should set either PRNDTL(TEM1)=-GRND10, or PRNDTL(H1)=-GRND10 if the enthalpy is solved.

This will force EARTH to retrieve the thermal conductivity for each involved material from the PROPS file using the material flag stored in PRPS.

For the enthalpy EARTH divides the thermal conductivity over the specific heat. Values of the specific heat are normally set for each individual material using the entry to PROPS file according to the material flag; this is activated by the PIL command CP1=GRND10.

When solving conjugate-heat-transfer problems for enthalpy, EARTH visits subroutine GXIMSL to calculate the enthalpy of a gas near the solid surface. This subroutine is visited for each fluid cell at the solid-fluid interfaces (see comments in GXIMSL which resides in the GXIMSL.FOR file).

By default, GXIMSL provides for two enthalpy-temperature relations:

4. How to set boundary conditions for IMMERSOL:

Solid-fluid interfaces within the computational domain are treated by IMMERSOL automatically, thus no special boundary conditions are normally necessary. T3 is a standard solved variable, thus the user can introduce any additional sources or sinks of heat by defining appropriate patches and COVAL's commands for this variable.

There are special types of boundary conditions (normally set at the domain boundaries) which are accounted for by the IMMERSOL. The way to set these special boundary conditions is described below.

4.1 Apertures in the Domain Boundaries

An aperture, such as a fluid inflow or outflow boundary, should be represented as a radiative link with a prediscribed radiating surface temperature (that of the external environment). This can be achieved by means of *RAD patches defined for T3. Thus

PATCH(*RAD1,EAST,NX,NX,IYF,IYL,IZF,IZL,1,LSTEP) COVAL(*RAD1,T3,COEF,TEXT)

introduces into the T3 energy equation an additional source term of the following form Area*COEF*(TEXT**4 - T3p**4)

where Area is the aperture area; T3p is the value of T3 in cells adjacent to an aperture; and COEF is an appropriate coefficient.

4.2 Solid surfaces at the Domain Boundaries

Special attention needs to be given to the setting of boundary conditions at the domain edges which represent external surfaces of radiating solid objects. There are two types of boundary conditions provided by the IMMERSOL and triggered by the special IMSW patches of a wall-type (WWALL, EWALL, SWALL, NWALL, LWALL or HWALL).

NOTE, these patches can be set either at the domain boundaries or at the external surfaces of non-participating solids. IMSW patches are standard wall-type patches, i.e. it is also possible to use them to introduce wall functions for turbulent flows (see WALL functions entry to PHENC).

4.2.1 Shell wall with fixed temperature.

To define a radiating solid surface at the edge of the domain which has fixed temperature Twal, the user should define a wall-type patch IMSW... and set the following COVAL commands: PATCH(IMSW...,WWALL,IXF,IXF,IYF,IYL,IZF,IZL,1,LSTEP)

for T3

COVAL(IMSW...,T3,GRND4,TWALL)

for TEM1:

COVAL(IMSW...,TEM1,WALLCO,TWALL)

here WALLCO defines the wall-function to be applied to calculate the heat-transfer coefficient, e.g. WALLCO=GRND2 activates the equilibrium wall-function (see the WALL-function entry to PHENC).

if H1 is solved instead of TEM1, then:

COVAL(IMSW...,H1,WALLCO,GRND4)

value set to GRND4 causes EARTH to visit GXIMSL subroutine to derive the enthalpy of a gas at Twal temperature.

By default the value of the emissivity of the solid surface defined by the IMSW patch is taken to be 1.0. However, the user can change this value by inserting the following PIL command SPEDAT(SET,EMISSIVITY,OF NAMPAT,R,VAL)

Here NAMPAT is the full name of an IMSW patch; while VAL is the value of emissivity.

4.2.2 Shell wall with a given external heat flux

To define a radiating solid surface at the edge of the domain which is subject to the external heat flux Qwal, given by the formula

Qwal = A*(Te^{4} - Tw^{4}) + B*(Te - Tw)^{D} + C,

where Tw is the temperature of the solid wall calculated by EARTH from the heat balance; Te is fixed external temperature; A,B,C and D are constants; the user should define a wall-type patch IMSW... and set the following COVAL commands:

PATCH(IMSW...,WWALL,IXF,IXF,IYF,IYL,IZF,IZL,1,LSTEP)

for T3:

COVAL(IMSW...,T3,GRND4,GRND5)

for TEM1 or H1:

COVAL(IMSW...,TEM1,WALLCO,GRND5)

By default A= B= C= 0.0, i.e. the solid surface defined by the IMSW patch is adiabatic; D= 1.0 and Te= 300. The user can change either of these values by the following PIL commands:

for Te:

SPEDAT(SET,EXTRN_TEMP,OF NAMPAT,R,VAL)

for A :

SPEDAT(SET,EXTRN_CO1 ,OF NAMPAT,R,VAL)

for B :

SPEDAT(SET,EXTRN_CO2 ,OF NAMPAT,R,VAL)

for D :

SPEDAT(SET,EXTRN_PWR ,OF NAMPAT,R,VAL)

for C :

SPEDAT(SET,EXTRN_QEXT,OF NAMPAT,R,VAL)

Here NAMPAT is the full name of an IMSW patch; while VAL is the new value of an appropriate coefficient.