How to interpret radiation flux report at external semi-transparent boundary of solid cell zone in Fluent 6.1
In applications which involve modeling solar radiation through windows using the non-gray DO model it is often desirable to model the window as a solid cell zone that participates in radiation. The reason is that if a semi-transparent thin wall is used and there is significant absorption of radiation in the window, unphysical temperatures may be predicted elsewhere in the domain. When the window is modeled as a solid zone, users often would like to relate the value returned for the external boundary of the solid by Report > Fluxes > Radiation Heat Transfer Rate to the inputs for beam irradiation in the boundary conditions panel. This process can be very confusing because there are several different quantities involved, some of which are found in different reporting panels and some that must be calculated by hand.
A second example of radiation accounting at an external semi-transparent boundary of a solid cell zone in Fluent 6.1
The case is the same as in resolution #1 but the angle of incidence and the index of refraction were changed.
Angle of incidence for external beam irradiation = 70.5 degrees
index of refraction for glass = 1.3
Specular reflectivity = .130 (from Equation 11.3-53)
Diffuse reflectivity = .0611 (from Equations 11.3-57 & 11.3-58)
External beam irradiation = 100 W/m^2 * 1 m^2 = 100 W
Radiation from thermal b.c. = 5.669e8 W/(m^2 * K^4) * 0.8 * (300 K)^4 * 1 m^2
= 367.35 W
Total incoming radiation = 467.35 W
Radiation transmitted from inside enclosure to outside environment:
Transmitted Radiation Flux (band-0) = 9.37 W
Transmitted Radiation Flux (band-1) = 431.22 W
Total transmitted radiation from inside enclosure to outside environment = 440.59 W
Reflected Incoming Radiation
Beam Irradiation = .130 * 100 W = 13.0 W
Diffuse radiation from thermal bc = .0611 * 367.35 W = 22.45 W
Total reflected incoming radiation = 35.45 W
Net Radiation = Total incoming - total reflected incoming - total transmitted from interior
= 467.35 W - 35.45 W - 440.59 W
= -8.69 W
Value reported by Report > Fluxes > Radiation Heat Transfer Rate
= -8.73 W
The difference of 0.04 W from the value in Report > Fluxes > Radiation Heat Transfer Rate is less than 0.01% of the total incoming radiation and can easily be attributed to not carrying the same number of significant figures in the hand calculations that FLUENT uses internally. The following example will help to illustrate how to interpret the radiation heat transfer rate report at the external boundary of the solid cell zone used for the window. The problem involves a single enclosure with one window. The non-gray DO model is used. Two wavelength bands are defined. One band represents short wavelength solar radiation and the other represents long wavelength radiation emitted by surfaces in the enclosure. The absorption coefficient for the solid material of the window is such that all radiation in the long wavelength band will be absorbed. At the external boundary of the window, an external irradiation flux of 100 w/m^2 is prescribed. Additionally, the thermal boundary condition is of type "Mixed" with an external radiation temperature of 300K and an external emissivity of 0.8.
After the solution has converged, the value returned for the window by Report > Fluxes > Radiation Heat Transfer is -8.50 W. A negative value indicates net radiation from inside the window to the external surroundings. This is due to radiation emitted within the solid cells representing the windows.
To understand where this value comes from, we simply need to evaluate the incoming and outgoing radiation fluxes at the window and sum them.
Incoming fluxes include the external beam irradiation and radiation from the thermal boundary condition. These must be obtained through hand calculations. In this example, these are
Beam irradiation = 100 W/m^2 * 1 m^2 = 100 W
Radiation from thermal b.c.= 5.669e8 W/(m^2 * K^4) * 0.8 * (300 K)^4 * 1 m^2
= 367.35 W
Total incoming radiation: 467.35 W
A portion of the incoming radiation will be reflected at the semi-transparent interface. Hand calculations must be performed to obtain the external reflectivity for both the beam irradiation (specular reflectivity) and the radiation from the thermal b.c (diffuse reflectivity). To do this, the angle of incidence of the beam irradiation and the index of refraction of the solid material are required. In this example the values are:
Angle of incidence: 54.7 degrees
Index of refraction 1.4
From these values, it is possible to calculate the specular reflectivity at the boundary using equation 11.3-53 and the diffuse reflectivity using equations 11.3-57 and 11.3-58. The results are:
Specular reflectivity 0.0532
Diffuse reflectivity 0.0768
Therefore the reflected external radiation values are
Beam Irradiation = .0532 * 100 W = 5.32 W
Diffuse radiation from thermal bc = .0768 * 367.35 W = 28.21 W
Total reflected incoming radiation = 33.53 W
Outgoing fluxes include radiation transmitted from inside the computational domain to the external environment through the external boundary of the window. The radiation transmitted from inside the computational domain must be obtained through the Surface Integrals panel. Select Integral for Report Type and select Wall Fluxes > Transmitted Radiation Flux (band-n)
Transmitted Radiation Flux (band-0) = 13.56 W
Transmitted Radiation Flux (band-1) = 428.74 W
Total Transmitted Radiation = 442.30 W
Now it is possible to evaluate the net radiation transfer at the external semi-transparent boundary and compare it with the value of -8.50 W returned by Report > Fluxes > Radiation Heat Transfer
Net Radiation = Total incoming radiation - total reflected incoming radiation - total transmitted radiation
= 467.35 W - 442.30 W - 33.53 W
= -8.48 W
The difference of 0.02 W from the value in Report > Fluxes > Radiation Heat Transfer Rate is less than 0.01% of the total incoming radiation and can easily be attributed to not carrying the same number of significant figures in the hand calculations that FLUENT uses internally.
Please note that this calculation procedure is valid for FLUENT v6.1.22 but may not apply to future software releases.