Radiative heat flux

[mgri]

Hi there everyone. I have what I think should be a straight forward question. I'm trying to model the radiative heat flux leaving a surface. The following is the standard solution to the problem.

P(out) = ??A(Tsurface)^4 (1)

P(in) = ??A(Tenvironment)^4 (2)

So, Pnet = ??A(Tenvironment^4 ?Tsurface^4) (3)

My question is that this solution only seems to take the emissivity of the surface in question into account i.e. it isn't concerned with the emissivity of the surface from which the environment radiates... walls in my case. I've come across a paper which deals with this in the following way:

Pnet = ??(s)?(e)A(Tenvironment^4 ?Tsurface^4)

... where ?(s) and ?(e) are the emissivities of the surface and environment respectively. I'm not convinced that this is correct so I thought I'd ask.

Any help is much appreciated.

Brian

 

[25362]

The total radiation leaving a surface is the sum of emitted energy and reflected radiation. So the radiation heat transfer between two surfaces A₁ at T₁, and A₂ at T₂, where A₁ is completely enclosed by surface A₂, would be:

Q_1,2 = C_bA₁(T₁⁴-T₂⁴)[÷][(1/e₁)+(A₁/A₂)(1/e₂-1)]​

where

C_b =black body radiation coefficient, 5.7 [×]10^-8 W/(m²K⁴)
e₁, e₂ = emissivities of each surface (dimensionless)
T₁, T₂= absolute temperatures of the emitting surfaces, K

Although this equation was derived for spheres and long cylinders it gives a fairly good estimate for other cases where one body is completely surrounded by another.

The above was taken from Fluid Flow and Heat Transfer by professor Lydersen, John Wiley & Sons.

 

[corus]

The environment or ambient is assumed to have infinite area and geometric factor of 1. '25362' has omitted the inverse power in his expression and has used a + instead of a *. For infinite A2 you arrive at the expression you have.

corus

 

[25362]

The formula I quoted is considered correct. It was derived by Prof. Lydersen using the reasoning by Ernst Schmidt: Thermodynamics, Principles and Applications to Engineering, Dover, New York, 1966, for a cylindrical surface surrounded by a larger cylinder or for a spherical surface surrounded by a larger sphere.

C_b÷[(1/e₁)+(A₁/A₂)(1/e₂-1)]= C_1-2​

is termed the radiation exchange coefficient, and the formula for the net heat transfer becomes shortened to:

Q_1-2 = C_1-2A₁(T₁⁴-T₂⁴)​

BTW, C_b(T₁⁴-T₂⁴) is considered the "overall potential difference" which is to be divided by the "surface and space resistances" to obtain the neat heat transfer by radiation, following J.P. Holman in his using the network method. See Heat Transfer by J.P. Holman (McGraw-Hill).

When the shape factor = 1, the formula (contested by corus) is in fact the same as that given by Holman for a two-body problem, namely for two surfaces that exchange heat with each other and nothing else.