My understanding of foil radiant barriers is that their effective heat transmittance is not the same if the radiant energy is moving in opposite directions. For example, if a foil faced insulation is installed below the sheathing of a flat roof with an air gap between the foil face and the sheathing, the foil will reflect heat away from the air space, however the foil will not reflect heat from the insulation below because they are in direct contact and conduction is occurring.
Is there any way to model this behavior of a radiant barrier? I understand that a user defined material is needed to model the impact of the foil radiant barrier on vapor diffusion, however I am concerned that temperature results may not be correct.
Thank you for any thoughts you have on this.
Foil Radiant Barriers
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Andrew Bishop
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Re: Foil Radiant Barriers
Dear Mr. Bishop,
I would not expect a difference in the exchanged net amounts of radiation for different exchange directions.
Consider an air gap, bounded by a surface with temperature T1 and emissivity eps1 on one side, and a surface with temperature T2 and emissivity eps2 on the other side. If one of the surfaces is highly reflective (low emissivity), then the standard argument is that the reflective surface reflects the radiation emitted by the other surface and thus reduces the amount of radiation transported across the gap.
However, radiation transport in the other direction is reduced by the same amount, because the highly reflective surface has low emissivity and thus emits only a small amount of thermal radiation to cross the gap.
If E1 is the radiant energy leaving surface 1, then E1 is the sum of M1, the thermal emission of surface 1, and the radiation which came from surface 2 and is reflected back by surface 1 towards surface 2:
E1 = M1 + r1*E2
The thermal emission M1 can be described by the thermal emission B1 of a blackbody at the same temperature T1, multiplied by the emissivity eps1 of the surface: M1 = eps1*B1.
The reflectivity r1 is one minus the emissivity: r1 = 1 - eps1. So we have the equations
E1 = eps1*B1 + (1-eps1)*E2, for surface 1 and, correspondingly,
E2 = eps2*B2 + (1-eps2)*E1, for surface 2
Solving for E1 and E2, and taking the difference E1 - E2 yields the net radiation exchange
E1 - E2 = (B1 - B2)/( 1/eps1 + 1/eps2 - 1 )
This result shows that for given surface emissivities eps1 and eps2 the net radiation exchange remains the same if eps1 and eps 2 are exchanged (mathematically speaking: the formula is symmetrical with respect to eps1 and eps2).
In other words: the net radiation exchange E1 - E2 depends on the involved surface temperatures T1 and T2 (which determine B1 and B2, respectively), and it depends on the involved surface emissivities eps1 and eps2, but it does _not_ matter whether the surface with the higher reflectivity is the warmer or the colder one.
In still other words: the net radiation exchange E1 - E2 is equal to the net radiation exchange B1 - B2 of blackbodies with the respective temperatures, reduced by the "radiation exchange factor" 1/( 1/eps1 + 1/eps2 - 1 ). The radiation exchange factor depends on the emissivities eps1 and eps2, but it does not depend on which of them is on the warmer side. The radiation exchange factor is thus a property of the air gap (more correctly: of the surfaces bounding the air gap) and independent of the temperatures and the direction of the temperature gradient. Note also that it is independent of the width of the air gap (as long as the width is small enough so that radiation exchange with the edges of the air gap can be neglected).
Now consider the air gap between the sheathing and the foil face. One surface has high emissivity (the sheathing) and one surface has low emissivity (the foil), resulting in a reduced radiation exchange factor - independent of whether the radiation exchange occurs in this or in that direction.
On the insulation side of the foil, it is true that the contact beween the foil and the insulation will lead to some heat conduction, but that is a different subject, and it should be independent of the direction of the heat flow as well. As far as radiation exchange is concerned, the foil will be in some radiation exchange with the insulation (the air gaps between the insulation fibres allowing the radiation to penetrate to some depth into the insulation layer), and the above considerations should apply here as well, making the size of this radiative heat flux dependent on the radiative properties of the foil and the insulation fibres, but independent of the direction of the radiation exchange.
Due to the presence of the reflective foil and its influence on the radiative heat exchange, the heat transmittances of the air gap and of the insulation layer should thus be adjusted accordingly. For the air layer, the relevant formula is given above. For the insulation layer, I don't know how large the effect is, it may be more or less negligible. There is no dependence on the direction of the radiative heat flow. The convective heat flow which occurs simultaneously depends on the heat flow direction (up vs. down vs. sideways).
Kind regards,
Thomas
I would not expect a difference in the exchanged net amounts of radiation for different exchange directions.
Consider an air gap, bounded by a surface with temperature T1 and emissivity eps1 on one side, and a surface with temperature T2 and emissivity eps2 on the other side. If one of the surfaces is highly reflective (low emissivity), then the standard argument is that the reflective surface reflects the radiation emitted by the other surface and thus reduces the amount of radiation transported across the gap.
However, radiation transport in the other direction is reduced by the same amount, because the highly reflective surface has low emissivity and thus emits only a small amount of thermal radiation to cross the gap.
If E1 is the radiant energy leaving surface 1, then E1 is the sum of M1, the thermal emission of surface 1, and the radiation which came from surface 2 and is reflected back by surface 1 towards surface 2:
E1 = M1 + r1*E2
The thermal emission M1 can be described by the thermal emission B1 of a blackbody at the same temperature T1, multiplied by the emissivity eps1 of the surface: M1 = eps1*B1.
The reflectivity r1 is one minus the emissivity: r1 = 1 - eps1. So we have the equations
E1 = eps1*B1 + (1-eps1)*E2, for surface 1 and, correspondingly,
E2 = eps2*B2 + (1-eps2)*E1, for surface 2
Solving for E1 and E2, and taking the difference E1 - E2 yields the net radiation exchange
E1 - E2 = (B1 - B2)/( 1/eps1 + 1/eps2 - 1 )
This result shows that for given surface emissivities eps1 and eps2 the net radiation exchange remains the same if eps1 and eps 2 are exchanged (mathematically speaking: the formula is symmetrical with respect to eps1 and eps2).
In other words: the net radiation exchange E1 - E2 depends on the involved surface temperatures T1 and T2 (which determine B1 and B2, respectively), and it depends on the involved surface emissivities eps1 and eps2, but it does _not_ matter whether the surface with the higher reflectivity is the warmer or the colder one.
In still other words: the net radiation exchange E1 - E2 is equal to the net radiation exchange B1 - B2 of blackbodies with the respective temperatures, reduced by the "radiation exchange factor" 1/( 1/eps1 + 1/eps2 - 1 ). The radiation exchange factor depends on the emissivities eps1 and eps2, but it does not depend on which of them is on the warmer side. The radiation exchange factor is thus a property of the air gap (more correctly: of the surfaces bounding the air gap) and independent of the temperatures and the direction of the temperature gradient. Note also that it is independent of the width of the air gap (as long as the width is small enough so that radiation exchange with the edges of the air gap can be neglected).
Now consider the air gap between the sheathing and the foil face. One surface has high emissivity (the sheathing) and one surface has low emissivity (the foil), resulting in a reduced radiation exchange factor - independent of whether the radiation exchange occurs in this or in that direction.
On the insulation side of the foil, it is true that the contact beween the foil and the insulation will lead to some heat conduction, but that is a different subject, and it should be independent of the direction of the heat flow as well. As far as radiation exchange is concerned, the foil will be in some radiation exchange with the insulation (the air gaps between the insulation fibres allowing the radiation to penetrate to some depth into the insulation layer), and the above considerations should apply here as well, making the size of this radiative heat flux dependent on the radiative properties of the foil and the insulation fibres, but independent of the direction of the radiation exchange.
Due to the presence of the reflective foil and its influence on the radiative heat exchange, the heat transmittances of the air gap and of the insulation layer should thus be adjusted accordingly. For the air layer, the relevant formula is given above. For the insulation layer, I don't know how large the effect is, it may be more or less negligible. There is no dependence on the direction of the radiative heat flow. The convective heat flow which occurs simultaneously depends on the heat flow direction (up vs. down vs. sideways).
Kind regards,
Thomas