Modeling of Radiative Heat Transfer
In a thermal fluid system, the solid surfaces, fluid flow, or both the solid surfaces and fluid flow can be subjected to heating or cooling due to radiation. In CFA models, radiative heat transfer is explained by solving the Radiative Transport Equation (RTE) and then obtaining the radiative source term for the total energy conservation equation. A widely-used modeling approach, the Surface-to-Surface (S2S) radiation model, is the chosen model in Creo Flow Analysis.
Radiative Transfer Equation
As a ray of radiation traverses a layer of an absorbing, emitting, and scattering medium in a particular direction, the ray loses energy through both absorption and scattering away from the ray. The ray also gains energy from light sources in the medium through emission and the scattering directed towards the ray. The overall energy balance of the ray over an infinite layer of the medium results in a differential equation, known as the radiative transport equation (RTE).
To derive the radiative transport equation, consider that an incoming ray of radiation with the intensity of I travels across a medium, for example, a gas, with the incremental thickness ds in the direction of 
, as shown in the figure. Through the medium layer, the incidence at the location 
⃗ and direction 
change in four ways, acting to either increase (energy gained +) or decrease (energy lost -) the radiation intensity 
:
, as shown in the figure. Through the medium layer, the incidence at the location ⃗ and direction change in four ways, acting to either increase (energy gained +) or decrease (energy lost -) the radiation intensity :• Absorption—A medium, for example, gas, absorbs a fraction of the radiation traversing it. With the absorption coefficient 
, the radiation energy lost through absorption is:
, the radiation energy lost through absorption is:
equation 2.282
• Scattering—A medium, for example, gas, scatters a fraction of the radiation energy to another direction (away from direction ) when the ray traverses the medium. With the scattering coefficient 
, the radiation energy lost through scattering is:
) when the ray traverses the medium. With the scattering coefficient , the radiation energy lost through scattering is:
Equation 2.283
• Emission—A medium emits radiation energy to the ray as a gray-body according to its local temperature (
) and emission characteristics to the ray. From Stefan-Boltzmann’s law and the reciprocity between emission and absorption, equation 2.274 and equation 2.279, the radiance emitted by the medium is: 
) and emission characteristics to the ray. From Stefan-Boltzmann’s law and the reciprocity between emission and absorption, equation 2.274 and equation 2.279, the radiance emitted by the medium is: Further, assuming that n is the refractive index of the medium (defined as the ratio of the velocity of light in vacuum to its velocity in the specified medium), the actual energy gained by the ray of radiation is:
Equation 2.284
• Scattering of Other Radiation—A fraction of other radiation sources in the layer of the medium is scattered into the ray of radiation dependent on the position and direction vectors 
and 
. Introducing 
to represent the direction and solid angle of the radiation beam, and 
to be the phase function, you have the fraction of intensity of a ray traveling in all directions to be scattered to the direction of 
⃗ as:
and . Introducing to represent the direction and solid angle of the radiation beam, and to be the phase function, you have the fraction of intensity of a ray traveling in all directions to be scattered to the direction of ⃗ as:
equation 2.285
Note that in equation 2.285, the scattering processes are ignored.
With the incoming radiation 
, and the outgoing radiation 
, the radiative energy balance in the direction 
⃗ has the form:
, and the outgoing radiation , the radiative energy balance in the direction ⃗ has the form:
equation 2.286
Substituting equation 2.282 to equation 2.285 into equation 2.286, and dividing it by 
, gives the radiative transport equation (RTE) as follows:
, gives the radiative transport equation (RTE) as follows:
equation 2.287
The RTE is a first order integro-differential equation for the radiation intensity 
in a fixed direction 
. To solve this equation within a domain, you need the temperature field within the domain, and you also require boundary conditions for 
on both the internal and external surfaces, and also the interfaces between two different media.
in a fixed direction . To solve this equation within a domain, you need the temperature field within the domain, and you also require boundary conditions for on both the internal and external surfaces, and also the interfaces between two different media.The local medium temperature is obtained by solving the total energy (including the radiative sources) conservation equation, described in the Heat module. For the thermal radiation, however, the boundary treatment is complex and depends on the radiation models. In general, a boundary can be an opaque medium that emits, reflects, and absorbs, or a semitransparent medium that also transmits. Reflection and transmission can be diffuse or specular or both diffuse and specular. For example, on an emitting and reflecting opaque boundary with gray radiation, and depending on the type of reflection, the intensity of a ray can be expressed as follows:
◦ Opaque boundary with diffuse emission and reflection:
Equation 2.288
◦ Opaque boundary with diffuse emission and specular reflection:
Equation 2.289
where,
 ⃗ | normal-to-surface unit vector at position  |
 | direction and solid angle of a diffusely reflected ray (uniform reflection in all directions) |
 | direction of the specularly reflected ray (perfect reflection depending on the incidence) |
 | surface reflectivity, diffuse reflectivity and specular reflectivity, respectively which have the following relationship:  equation 2.290 |
With the given boundary conditions, equation 2.287 governs the transport of the radiation intensity in a specified direction. For gray radiations, equation 2.287 should be solved in all the different directions within a sphere. For non-gray radiations, the intensity also depends on the wavelengths. Therefore, it needs to be solved in all directions over the entire spectrum of the wavelengths. Clearly, the direct solution of the radiative transfer equation is very time consuming. In many engineering simulations, therefore, it is desirable to use somewhat simplified yet approximate models to account for the directional and spectral dependencies. In CFD simulations, the following radiation models are routinely adopted, in which the detailed description may be found in references.
References: R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer”, Hemisphere Publishing Corporation, Washington DC, 1992.
▪ Rosseland Radiation Model
References: R, Siegel and J. R. Howell, Thermal Radiation Heat Transfer”, Hemisphere Publishing Corporation, Washington DC, 1992.
▪ P-1 Radiation Model
References: R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer, Hemisphere Publishing Corporation, Washington DC, 1992.
▪ Discrete Transfer Radiation Model
References: N. G. Shah, A New Method of Computation of Radiant Heat Transfer in Combustion Chambers”, PhD thesis, Imperial College of Science and Technology, London, England, 1979.
References: M. G. Carvalho, T. Farias, and P. Fontes, Predicting Radiative Heat Transfer in Absorbing, Emitting, and Scattering Media Using the Discrete Transfer Method”, In W. A. Fiveland et al., editor, Fundamentals of Radiation Heat Transfer, volume 160, pages 17-26. ASME HTD, 1991.
▪ Surface-to-Surface (S2S) Radiation Model
References: R. Siegel and J. R. Howell, Thermal Radiation Heat Transfer”, Hemisphere Publishing Corporation, Washington DC, 1992.
▪ Discrete Ordinates (DO) Radiation Model
References: G. D. Raithby and E. H. Chui, A Finite-Volume Method for Predicting a Radiant Heat Transfer in Enclosures with Participating Media”, J. Heat Transfer, 112:415-423, 1990.
References: E. H. Chui and G. D. Raithby, Computation of Radiant Heat Transfer on a Non-Orthogonal Mesh Using the Finite-Volume Method”, Numerical Heat Transfer, Part B, 23:269-288, 1993.
Each model has its own advantages and limitations regarding accuracy and cost. For example, while the Rosseland model does not solve a transport equation for the incident radiation. it is the fastest radiation model and requires the least extra memory. Rosseland can only be used for optically thick (optical thickness is the natural logarithm of the ratio of incident to transmitted radiant power in a medium) media due to its overly simplification of the radiative transport equation.
The discrete ordinates (DO) radiation model transforms equation 2.287 into a transport equation for radiation intensity in the spatial coordinates 
and solves it over a finite number of discrete solid angles associated with the vector direction 
. The number of the solid angles selected directly determines the accuracy and the computational cost. The DO modeling approach is also identical to the approach used for the fluid flow and energy equations. At present, it is the most general radiation model that spans the entire range of optical thicknesses and can be applied to problems ranging from surface-to-surface radiation to participating radiation such as a combustion system. However, the computational cost of the DO model is high for non-gray radiations.
and solves it over a finite number of discrete solid angles associated with the vector direction . The number of the solid angles selected directly determines the accuracy and the computational cost. The DO modeling approach is also identical to the approach used for the fluid flow and energy equations. At present, it is the most general radiation model that spans the entire range of optical thicknesses and can be applied to problems ranging from surface-to-surface radiation to participating radiation such as a combustion system. However, the computational cost of the DO model is high for non-gray radiations.Among the above-mentioned radiation models, the surface-to-surface (S2S) radiation model is particularly good for modeling the enclosure radiative transfer without the consideration of the participating media. The typical examples are the radiative space heaters, and automotive underhood and underbody systems. In those situations, the radiation models for participating radiation are sometimes not efficient. As compared to the DO radiation model, the S2S model is faster per iteration, though the view factor calculation itself can be CPU-intensive. In Creo Flow Analysis, the current choice of model for radiative heat transfer is the S2S radiation model.
Surface-to-Surface (S2S) Radiation Model
The surface-to-surface radiation model accounts for the radiation exchange in an enclosure of gray-diffuse surfaces without participating media. The surface-to-surface radiative energy exchange depends on two main factors: the radiative characteristics of the involved surfaces, and the geometrical parameters including the surface areas and shapes, and the relative position to reach other (separation distance and orientation). In the S2S radiation model, the surface radiative heat transfer is considered by the gray-diffuse radiation model, while the geometrical parameters are accounted for by a geometric function referred to as view factor.
• Gray-Diffuse Radiation
The S2S radiation model assumes that the surfaces are gray and diffuse (gray radiation). For a gray surface, both emissivity 
and absorptivity 
of the surfaces are independent of the wavelength of the outgoing and incoming rays. According to Kirchhoff’s law of thermal radiation in equation 2.274, emissivity equals the absorptivity:
and absorptivity of the surfaces are independent of the wavelength of the outgoing and incoming rays. According to Kirchhoff’s law of thermal radiation in equation 2.274, emissivity equals the absorptivity:
Equation 2.291
Also, with the assumption of a diffuse surface, no specular reflection occurs on the surface and the reflectivity (
) of incident radiation at the surface is isotropic with respect to the solid angle. From equation 2.290, surface reflectivity is determined as:
) of incident radiation at the surface is isotropic with respect to the solid angle. From equation 2.290, surface reflectivity is determined as:
Equation 2.292
where,
 | surface specular |
 | diffuse reflectivity |
For a nonopaque or a semitransparent surface, transmissivity 
is also independent of the wavelengths:
is also independent of the wavelengths:
Equation 2.293
The gray-diffuse surface-to-surface model assumes that the exchange of radiative energy between surfaces is virtually unaffected by the medium that separates them. Thus, if a certain amount of radiant energy 
is incident on a surface per unit area (irradiance), the portions of the radiative energy reflected, absorbed, and transmitted are 
, 
and 
respectively. Since for most applications, the surfaces are opaque to thermal radiation in the infrared spectrum, the radiative surfaces can be further considered as opaque. The transmissivity, therefore, can be neglected 
. From equation 2.273 and equation 2.274, the surface reflectivity 
is expressed as:
is incident on a surface per unit area (irradiance), the portions of the radiative energy reflected, absorbed, and transmitted are , and respectively. Since for most applications, the surfaces are opaque to thermal radiation in the infrared spectrum, the radiative surfaces can be further considered as opaque. The transmissivity, therefore, can be neglected . From equation 2.273 and equation 2.274, the surface reflectivity is expressed as:
Equation 2.294
with the assumptions of the surface gray-diffuse radiation, the S2S modeling equation is constructed based on energy conservation on each surface.
• S2S Modeling Equation
The main assumption of the S2S model is that in an enclosed system, the radiative heat transfer only occurs between gray-diffuse surfaces (gray radiation). You can ignore the absorption, emission, or scattering of radiation in the medium separating the surfaces. Therefore, only consider surface-to-surface radiation for numerical analysis.
The radiative energy flux leaving a given surface consists of directly emitted and reflected energy. The reflected energy flux is dependent on the incident energy flux from the surroundings, which then can be expressed in terms of the energy flux leaving all other surfaces. To compute the net radiative energy flow in a surface, it is convenient to define the radiosity 
, which is the sum of the emissive power per unit area (emittance) 
, and the reflected part of the radiation power received by the surface per unit area (irradiance) 
:
, which is the sum of the emissive power per unit area (emittance) , and the reflected part of the radiation power received by the surface per unit area (irradiance) :
Equation 2.295
For an opaque surface, 
, you have the radiosity:
, you have the radiosity:
equation 2.296
with the assumptions in S2S model, therefore, the following system of linear equations can be formulated to calculate the radiosity on each surface in an enclosed system. Assuming that 
represents the radiosity on an arbitrary surface 
, 
is the surface temperature, and 
is the view factor between surface 
and 
, you have the radiosity in the surface 
:
represents the radiosity on an arbitrary surface , is the surface temperature, and is the view factor between surface and , you have the radiosity in the surface :
equation 2.297
where 
is the number of the surfaces participating in the radiative heat transfer. Introducing the Kronecker symbol 
and applying Stefan-Boltzmann’s law for gray radiation, equation 2.278, you can rearrange equation 2.297 and derive the S2S modeling equation:
is the number of the surfaces participating in the radiative heat transfer. Introducing the Kronecker symbol and applying Stefan-Boltzmann’s law for gray radiation, equation 2.278, you can rearrange equation 2.297 and derive the S2S modeling equation:
equation 2.298
With the pre-calculated view factor 
, the system of linear equation 2.298 is solved to obtain 
for the participating surfaces. Then the radiation net heat flows on each surface is computed easily. For the surface 
, the net radiative heat flux 
is the difference between the per-unit-area outgoing (
) and incoming (
) radiation. From equation 2.278 and equation 2.296, you can derive the following flux formulation:
, the system of linear equation 2.298 is solved to obtain for the participating surfaces. Then the radiation net heat flows on each surface is computed easily. For the surface , the net radiative heat flux is the difference between the per-unit-area outgoing () and incoming () radiation. From equation 2.278 and equation 2.296, you can derive the following flux formulation:
Equation 2.299
For a given surface area 
, the net radiation heat flows leaving the surface 
is computed as:
, the net radiation heat flows leaving the surface is computed as:
Equation 2.300
The S2S model is composed of a system of linear equations in the form of equation 2.298. The advantage in the application of the model is that for given view factors and temperatures, you calculate the net heat flows by solving a system of linear equations, which are computed by applying numerical algorithms. However, the main difficulty in applying the proposed surface-to-surface model is computation of the 
view factors, for n-number of participating surfaces. This can be time consuming, in particular, with an increase in the number of surfaces.
view factors, for n-number of participating surfaces. This can be time consuming, in particular, with an increase in the number of surfaces.Calculation of View Factor
In the S2S modeling equation 2.298, the view factor 
is the proportion of the radiation which leaves surface 
and strikes surface 
. As shown in figure 2.37, assuming that 
is the differential area on surface 
, and 
is the differential area on surface 
, and the distance between and is 
, you express the view factor 
from 
to 
at a distance 
as follows:
is the proportion of the radiation which leaves surface and strikes surface . As shown in figure 2.37, assuming that is the differential area on surface , and is the differential area on surface , and the distance between and is , you express the view factor from to at a distance as follows:
Equation 2.301
where 
and 
are the angle between the surface normal directions and a ray between the two differential areas.
and are the angle between the surface normal directions and a ray between the two differential areas.
Figure 2.37
If 
and 
are the given areas of surface 
and 
, respectively, the view factor from surface 
to surface 
is the area averaged integral of equation 2.301 over the surfaces 
and 
:
and are the given areas of surface and , respectively, the view factor from surface to surface is the area averaged integral of equation 2.301 over the surfaces and :
Equation 2.302
Note that surface-to-surface radiation only occurs when the two surfaces are visible to each other, or the view factor is non-zero. Introducing the Kronecker symbol 
with respect to the visibility between 
and 
:
with respect to the visibility between and :
Equation 2.303
you can rewrite equation 2.302 as:
Equation 2.304
For any two surfaces visible to each other, a given surface 
only radiates a portion of the outgoing radiative energy to the surface 
, as shown in figure 2.378. Therefore, the dimensionless view factor 
represents the portion of energy that leaves surface 
and reaches surface 
. The characteristics are in the list that follows:
only radiates a portion of the outgoing radiative energy to the surface , as shown in figure 2.378. Therefore, the dimensionless view factor represents the portion of energy that leaves surface and reaches surface . The characteristics are in the list that follows:
Figure 2.378
• Summation of View Factors—Since radiation leaving a surface is conserved, the sum of all view factors from a given surface 
is unity. For 
surface enclosed system, you have
is unity. For surface enclosed system, you have
Equation 2.305
• Self-Viewing Surfaces—Since radiation travels in straight lines, no radiation ray from a convex surface can leave the surface and then hit the same surface later. Hence, the convex surfaces cannot be self-viewed:
Equation 2.306
For concave surfaces, the outgoing ray from one position on the surface can hit the same surface later at a different position. Therefore, the concave surface can be visible to itself:
Equation 2.307
• Superposition—For 
surface system, if a given surface 
radiates to 
number of surfaces (
), then the view factor between the surface 
and the 
-number of surfaces equals the sum of the view factors between surface 
and each one of the 
-number of surfaces:
surface system, if a given surface radiates to number of surfaces (), then the view factor between the surface and the -number of surfaces equals the sum of the view factors between surface and each one of the -number of surfaces:
Equation 2.308
The superposition rule or summation rule is useful when a geometry is not available with given charts or graphs. With the superposition rule you can express the geometry that is being sought using the sum or difference of geometries that are known.
• Reciprocity—Equation 2.304 defines the view factor 
as the fraction of the radiative energy that leaves surface 
and reaches surface 
. Similarly, the view factor 
which is the portion of energy that leaves the surface 
and reaches surface 
is expressed as:
as the fraction of the radiative energy that leaves surface and reaches surface . Similarly, the view factor which is the portion of energy that leaves the surface and reaches surface is expressed as:
Equation 2.309
If you compare equation 2.309 with equation 2.304, you obtain the following relationship:
Equation 2.310
Equation 2.310 is referred to as the reciprocity of the view factors. With the reciprocity theorem you can directly calculate only one of the pair of view factors.
Clustering
The S2S radiation model is computationally expensive when the number of radiating surfaces is large. To reduce the computational time and the storage requirement, you can reduce the number of radiating surfaces by grouping a certain number of the neighboring boundary-cell faces to create surface clusters. The radiosity (
) is then calculated for surface clusters. These values are then distributed to the boundary-cell faces within each cluster to calculate the wall temperatures. Since the radiation source terms are highly nonlinear (proportional to the fourth power of temperature), make sure you calculate the average temperature of the surface clusters and distribute the flux and source terms appropriately among the boundary faces that form the clusters.
) is then calculated for surface clusters. These values are then distributed to the boundary-cell faces within each cluster to calculate the wall temperatures. Since the radiation source terms are highly nonlinear (proportional to the fourth power of temperature), make sure you calculate the average temperature of the surface clusters and distribute the flux and source terms appropriately among the boundary faces that form the clusters.The surface cluster temperature is obtained by area-averaging of boundary face temperature as shown in the following equation:
Equation 2.311
where 
is the temperature of the surface cluster, and 
and 
are the face area and temperature of the boundary cell in CFA simulations. The summation is carried over all the faces within a surface cluster.
is the temperature of the surface cluster, and and are the face area and temperature of the boundary cell in CFA simulations. The summation is carried over all the faces within a surface cluster.