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**Solution for the variable**from a heat-conduction-type equation with a local conductivity dependent on:*T3*,- the nature of the medium,
- on its temperature, and on
- the distance between nearby solid walls.

is defined, for media within which thermal radiation is active, as the fourth root of the radiosity, divided by the Stefan-Boltzmann constant,**T3**.*sigma*For solid materials which may be immersed in the medium,

is defined as being equal to the local solid temperature.*T3*-
**Solution for the temperature of the fluid phase(s)**by means of the conventional energy equation(s), having either temperature or enthalpy as the dependent variable(s).The energy equation(s) contain radiation-source terms proportional to:

*sigma * ( T3 ** 4 - T12 ** 4)*

where T12 stands for the temperature of the fluid phase to which the equation relates.Equal and opposite sources appear in the

*T3*equation. -
**Modifications of the source terms**, in cells adjacent to solids, which account for the departures from unity of the surface emissivity from unity.

- Origins
- Character
- Mathematical formulation
- Implementation in PHOENICS
- An early example
- More recent examples
- Concluding remarks regarding IMMERSOL

IMMERSOL can be regarded as a development of the ideas which were partly implemented in PHOENICS version 2.2 as the "radiosity model" in the 'advanced-radiation' option.

However, the implemented version omitted the proposal, which is a central feature of IMMERSOL, to introduce the WGAP term into the "radiation-conductivity term".

Also, the present proposal goes beyond the first in the smoothness of the connexion which it makes between the radiation across the intervening fluid spaces and the conduction within the immersed solids.

IMMERSOL provides an economically-realisable approximation to the precise mathematical representation of radiative transfer.

The claims that can be made for it are that:-

- its predictions agree exactly with the precise representation in simple (specifically, one-dimensional) circumstances; and
- in more complex circumstances, its predictions are always plausible, and of the right order of magnitude.

In these respects, IMMERSOL is like the LVEL model, with which it shares the WDIS and WGAP calculations based on the LTLS equation.

Its use can be justified in part by reference to the facts that:

(a) it is rare for the absorption and scattering coefficients to be known with great precision, and

(b) the neglect of wave-length dependencies, which is commonly regarded as acceptable, is probably no less serious a cause of error.

[Moreover, its economy is such that it would allow refinement in respect of wavelength, the single variable E3 (i.e. sigma*T3**4) being replaced by a set of variables, one for each wavelength band; but this possibility has not yet been exploited.]

Its practical realisability is its greatest asset; but it should NOT be represented or defended as providing more than plausible predictions.

Like LVEL, and the radiosity model, IMMERSOL is based upon intuition as much as on rigorous analysis. It should therefore be treated with both respect and caution.

The temperature of the solid phase, T3, is governed by the differential equation which describes how:

- the variation of T3 with time is balanced by
- conduction within the solid and by
- such sources as are present within it.

This equation can be written as:

s.dT3/dt - div( lamda * grad T3) = qdot (1)

where:

s is the specific heat capacity of the solid,

lamda is its thermal conductivity, and

qdot is the heat source per unit volume.

Within the space between solids, the distribution of radiosity, i.e. sigma*T3**4, can be represented as obeying the equation:

- div( gamma_rad * grad E3 ) = (a + s) * (E12 - E3) (2)

where:

- E3 stands for radiosity,
- E12 stands for the phase-surface-average of sigma * T ** 4
- of the first and second fluid phases,
- a stands for the absorptivity of the fluid (possibly 2-phase) medium,
- s stands for the scattering coefficient of that medium,
- gamma_rad is defined as the reciprocal of 0.75*( a + s + 1/WGAP ), and
- WGAP stands for the distance between adjacent wall computed
- from the solution for LTLS.

How T3 can represent both the solids temperature and (E3/sigma)**.25

For the between-solids spaces, because it has no other significance where immersed solids are absent, T3 can be defined by:

T3 = ( E3 / sigma)**0.25, so that E3 = sigma * T3**4 , and dE3 = 4 * (E3 / T3) * dT3 , i.e. = 4 * sigma * T3**3 * dT3

Hence, equation (2) can be expressed as a differential equation in terms of T3 as:

div( lamda_rad * grad T3 ) = (a + s) * (E12 - sigma * T3**4) (3)

where: lamda_rad = (16/3) * sigma * T3**3 / ( a + s + 1/WGAP )

This equation has no dT3/dt term, because radiosity is not a quantity which is stored (in the physical rather than computational sense).

The inclusion of WGAP in the radiative-conductivity expression is the most significant innovative feature of IMMERSOL; for it allows the conductivity to remain finite even when a and s and zero.

Moreover it gives it the CORRECT value for the only case which is easy to test, namely that of radiative transfer between wide parallel surfaces.

A plausible justification in physical terms is as follows:

- 1/(a+s) is the mean free path (or average ray-stopping distance) associated with absorption and scattering
- WGAP, the distance between the bounding walls of the space, is very definitely the "stopping distance" effected by those walls.

Taken together, equations (1) and (3) imply that:

A SINGLE HEAT-CONDUCTION-TYPE EQUATION

describes the influences of:

BOTH CONDUCTION IN SOLIDS AND RADIATION BETWEEN THEM

on the distribution of immersed-solids temperature, T3, throughout the domain, with however a

POSITION- AND T3-DEPENDENT CONDUCTIVITY and

SOURCE TERMS WHICH VARY SIMILARLY.

Equations of this type can be easily solved, in an iterative manner, by PHOENICS.

//|<- surface absorptivity aa |// //| |// / a<---------------------- WGAP-------------------- >b / //| |// //| fluid-phase absorptivity a, scattering s |// //| surface absorptivity bb ->|//

Between two parallel plates a and b, with emissivities aa and ab, in the absence of absorption or scattering within the fluid, the net heat flux is well known to be given by:

(Ea - Eb) / [ (1/aa -1 ) + (1/ab - 1) + 1 ]

The terms (1/aa-1) and (1/ab-1) have the significances of

EXTRA RESISTANCES TO HEAT TRANSFER

occasioned by the departure of the solid surface from black-body conditions.

<near-wall cell> Were it not for this extra / | | resistance near the wall, solid / | fluid | the resistance from a to y x / a y | would be computed as * / | * | WDIS / WGAP . / |<- WDIS > | / | | To account for the wall-emissivity / |<------ WGAP ----->>>| effect, it is necessary simply to add to this the quantity

1/aa -1 , for the a boundary.

The addition of resistances at phase boundaries is a common practice in PHOENICS, presenting no difficulty.

The phase boundary is also the location at which energy exchanges occur between the solid on one side and the fluid on the other.

The driving forces are the temperature differences:

T1 - T3 and T2 - T3 , where T1 and T2 are fluid-phase temperatures.

The resistances to these energy exchange are the sums of: the conductive resistance on the solid side and the conductive + convective resistance on the fluid side.

When the fluid-phase temperatures are solved for indirectly, by way of the enthalpies, some additional algebraic equations are needed; but their inclusion and simultaneous solution present no difficulty.

Absorption of radiation by the fluid materials between the immersed solids, and emission of radiation by those materials, are represented by the term: (a + s) * (E12 - E3) of equation (2).

This term can be spelled out in more detail as:

r1 * (a1 + s1) * (E1 - E3) + r2 * (a2 + s2) * (E2 - E3)

where r1 and r2 represent respectively the volume fractions of the first and second phases, and a1, s1 etc are the phase absorption and scattering coefficients.

Each of the above two terms appears, with opposite sign, also in the enthalpy equation of the phase in question. employed for the radiosity method.

Few actions are needed in order to activate IMMERSOL. They comprise:

SOLVE(T3) to ensure that T3 is solved

DISWAL to activate the WDIS and WGAP calculation

STORE(WGAP) to enable this quantity to be accessed

together with such information as provides material properties and initial and boundary conditions.

Examples may be found in the IMMERSOL-related cases, in the advanced-radiation-option library.

From PHOENICS-3.4 onwards, however, the setting of radiation-related properties has been made more similar to the setting of other properties, such as density and specific heat.

Thus:

- there now exist two new PIL variables with meaningful names, namely:
- EMISS,
- the emission (and absorption) coefficient of a transparent medium, with the dimensions of 1/length; and
- SCATT,
- the scattering coefficient of the transparent medium, also with the dimensions of 1/length

- If these quantities are set to non-negative values, those values
are used for all regions which are not occupied by solids, unless
further instructions are given.
- Negative values are treated as zero, being without either physical
or formula-choosing significance[
**unlike**the case of RHO1, etc]. - If the commands:

STORE(EMIS) and/or STORE(SCAT)

are placed in the Q1, for print-out or other purposes, the fields of values are printed in the result file, and stored in the phi file, in the usual manner.Moreover

**different**values can then be placed in different parts of the domain by the conventional use of:

FIINIT( ),

PATCH(name,INIVAL, ),

INIT(name, EMIS, ) and

INIT(name, SCAT, ) - If InForm-style property formulae are the provided in the Q1,
it is the values which the formulae provide which are used in the
radiation calculations.
- PLANT-generated or user-generated Fortran coding can also be used to provide non-uniform emissivity and scattering coefficients.

The **differences** from the treatment of other properties are:

- no entries have been provided for emissivity and scattering
coefficients in the PROPS file, because it is usually small contaminants
ot the pure materials which affect the radiation, for example water
droplets (mist) in air, or soot particles in combustion gases; and
- no GX-style formulae, accessed via GRNDx settings, are provided, because the availability of InForm.

Examples will be found in the input-library cases:

r201.htm,
r202.htm,
r203.htm and
r204.htm.
However, it should be noted that, when EMIS is set for a cell containing a
non-transparent solid, it becomes dimensionless; and it then represents the
emissivity/absorptivity of the **surface** of the solid.

For example, it may be desired to represent an aperture in the bounding surface, though which radiation can enter and leave; or a domain boundary may be required to simulate a solid wall of prescribed temperature, for which the emissivity must also be defined.

To meet this requirement, PHOENICS has been caused to react
appropriately to PATCH and COVAL statements, the which the patch-name
begins with the characters 'IM' **and** the dependent variable in
question is T3.

How such patches are used is explained in the section of the lecture on IMMERSOL which may be viewed by clicking here,

- The flow is steady and laminar.
- The Reynolds number is 10
- The pressure is atmospheric.
- The dimensions are small and the air is unpolluted, so that it does not participate at all in the radiative process.
- The emissivities of the surfaces are:-
- bottom surface... 0.9
- top surface.... 0.9
- first solid...... 0.1
- second solid... 0.2

The practically interesting question is: What temperatures will be attained in the solid?

First some images showing the geometry and some results:

The grid and velocity vectors

The distributions of WGAP and of PRPS

The true (TEM1) and radiation (T3) temperatures

Since it is not easily seen from the above contour plot that TEM1 and T3 are very close within the solid, the following plot is provided of the difference between them. This is easily calculated by placing the following In-Form line in the Q1 file.

(stored tdif is tem1-t3)

True temperature minus radiation temperature.

The upward (y) and rightward (x) radiative heat fluxes

IMMERSOL has allowed this problem to be set up and solved with great ease, the advantage of which can be best appreciated by those who have attempted to solve such problems by the methods advocated in many text-books, involving use of 'view-factor' calculations.

- Comparisons with text-book solutions (A.F.Mills):
- Other examples
- Use of IMMERSOL in connexion with the calculation of stresses in solids
- Application to an electronics-cooling problem.

- IMMERSOL was conceived and introduced into PHOENICS during
1996.
- Wherever tested, it has performed well, in respect of:
- accuracy, where exact solutions are known;
- plausibility, where they are not;
- economy, in all circumstances.

- Since it is the ONLY model which can be economically used in complex circumstances, including multi-phase, chemically reactive and body-fitted coordinates, it seems likely to become the most widely used of all the radiation models in PHOENICS.

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