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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.
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.
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:-
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:
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:
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).
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:
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:
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, )
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,
The application of IMMERSOL to a steady-state heating-and-cooling problem will be described, by reference to the following sketch.
///////////// radiating surface at 400 K //////////////
--------------------------------------------------------
----->
-----> -----------------------------------
cooling |/////// first solid ////////////|
air -> -----------------------------------
at -->
300 K> -----------------------------------
-----> |/////// second solid ////////////|
-----> -----------------------------------
----->
--------------------------------------------------------
//////////// radiating surface at 500 K ////////////////
The practically interesting question is:
What temperatures will be attained in the solid?
The upward-direction (y) radiative heat flux
The rightward-direction (x) radiative heat flux
From the following plot, it can be seen that:
Strictly speaking, TEM1 has no meaning within the blocks; but it is plotted as equal to T3 to ensure their equality at the phase interface.
Profiles 1. 1.1.1+....+....+....+....+....+....+....+....+....+
. 1 1 .
of: .9 + +
. 2 2 2 .
1. T3 .8 + 1 1 1 1 1 2 2 2 +
2. TEM1 2 .
.7 + +
. .
hotter .6 + 2 2 2 2 +
wall--->. 1 1 1 .
.5 + <-solid-> 1 1
. <-solid-> .
.4 + 2 cooler wall->+
. 2
.3 + 2 +
. 2 2 2 .
.2 + 2 +
. 2 2 2 .
.1 + +
. .
0. +....+....+....+....+....+....+....+....+....+....+
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