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3. The IMMERSOL model of Radiative Heat Transfer

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Summary: The IMMERSOL method involves three main elements:

  1. Solution for the variable T3, from a heat-conduction-type equation with a local conductivity dependent on:

    T3 is defined, for media within which thermal radiation is active, as the fourth root of the radiosity, divided by the Stefan-Boltzmann constant, sigma.

    For solid materials which may be immersed in the medium, T3 is defined as being equal to the local solid temperature.

  2. 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.

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

Contents

  1. Origins
  2. Character
  3. Mathematical formulation
  4. Implementation in PHOENICS
    1. Activation via Q1
    2. Setting emissivities/absorptivities and scattering coefficients
    3. Domain-boundary conditions
  5. An early example
  6. More recent examples
  7. Concluding remarks regarding IMMERSOL

3.1. Origins

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.

3.2. Character

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

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

  1. its predictions agree exactly with the precise representation in simple (specifically, one-dimensional) circumstances; and

  2. 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.

Justification

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.

3.3. Mathematical formulation

The differential equation for T3 within the immersed solids

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

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.

The differential equation for T3 between the immersed solids

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

where:


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).

Justification of the inclusion of WGAP

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:

Combining the equations

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.

The value of lamda_rad in the fluid-phase cell close to the immersed-solid boundary


   //|<- 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.

Other phase-boundary effects

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.

Radiative interchanges with the fluid(s) between the solids

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.

3.4. Implementation in PHOENICS

(a) Activation via Q1

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.

Warning

It is also desirable to set varmax(tem1) and varmax(t3) to the lowest reasonable value, lest unrealistic temperatures generated at the start of iteration fail, because of the inherent non-linearity of the equation system, to die away.

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

(b) Setting emissivities/absorptivities and scattering coefficients

When it was first created, IMMERSOL was provided with means of supplying relevant property data by way of the PIL variables RADIA, RADIB, and of special SPEDAT statements,

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:

The differences from the treatment of other properties are:

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.

(c) Domain-boundary conditions

Although the settings of EMIS within the transparent and non-transparent media suffice to enable internal-to-the-domain boundaries to be correctly represented, there is also sometimes a need to ascribe radiation-influencing conditions at the boundaries of the domain.

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,

3.5 An example: Input-file library case r209

The application of IMMERSOL to a steady-state heating-and-cooling problem will be described, by reference to the following sketch:

Problem-specifying input data

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

Evidently both plates are heated, by radiation, to temperatures significantly above that of te cooling air which surrounds them.

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

The y-direction fluxes are the greater, because it is that direction that the temperature variations are largestand both negative (downward directed) and positive (upward-directed) fluxes are in evidence.

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.

3.6 More recent examples of IMMERSOL

3.7 Concluding remarks regarding IMMERSOL

End of the IMMERSOL section

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