Radiation mean beam length

TABLE 5-7 Mean Beam Lengths for Volume Radiation... 

Surface absorptivity or absorptance subscript 1 refers to the surface temperature while subscript 2 refers to the radiation source Gas absorptivity, emissivity, and transmissivity Dimensionless constant in mean beam length equation, LM = (5-LM0... 

Eq. (5-169) reduces to simply SiG/Ai = Eg < 1.0, and it is evident that the magnitude of the radiation contribution never exceeds unity. At high temperatures, radiative effects can easily dominate other modes ofheat transfer by an order of magnitude or more. When mean beam length calculations are employed, use LM/D = 0.94 for a cylindrical cross section of diameter D, and... 

The mean beam length for a cylinder of equal diameter and height for radiation emitted to all surfaces is, from Table 13-4,... 

Discussion This is the average emissivity for radiation emitted to ali surfaces of the cylindrical enclosure. For radiation emitted towards the center of the base, the mean beam length is 0.7ID jnstead of 0.60D, and the emissivity value WQuld e different. 

As we will show in section 5.6.4, the complicated determination of the emis-sivities g and q °f any shape of gas space can be traced back to the standard case of the gas hemisphere we have just dealt with. A mean beam length sm is determined for the gas space under consideration from the following condition a gas hemisphere with the radius R = sm should give rise to the same spectral irradiance on a surface element at its centre as that for the radiation from any shaped gas volume on a certain element of its surface. As follows from (5.188) and (5.191)... 

The emission of gas radiation depends on the size and shape of the gas space it is described quantitatively by the irradiance, which the gas radiation generates at the surface of the gas space. The decisive equations (5.188) and (5.189) include the spectral emissivity e))(. and the emissivity integrated over all wavelengths q, which, according to (5.193) and (5.194), can be replaced by the emissivity of a gas hemisphere with a radius the same as the mean beam length. sni of any shaped gas space. 

The mean beam length sm of a gas space of any shape that radiates on an element dA = dA2 of its surface, is defined by the fact that the spectral irradiance I a,g of dA 2 has exactly the same magnitude as the spectral irradiance of a surface element in the centre of a gas hemisphere of radius R = sm. According to section 5.6.2, the spectral irradiance of this surface element is... 

In the determination of a constant mean beam length sm we first will consider the limiting case of an optically thin gas with kq = kGL0 —> 0. The spectral radiation flow emitted from a volume element of the gas in all directions is, according to (5.181),... 

With this mean beam length, the mean irradiance EG of the total surface A of the radiating gas space of volume V is found to be... 

The foregoing charts are based oe gases in a hemispherical container of radius L radiating to differential area at the center of the base (Rg. 8.3). In terms of an equivalent mean beam length, however, the use of these charts may be extended to other gas shapes. It can be shown for irregular gas shapes that the mean length is... 

Table 10.1 Mean beam length for gas radiation (from Rohsenow and Hartnett [24]).

Mean beam lengths for gas radiation to entire surface area of enclosnret... 

Thickness of material, radius of hemisphere or mean beam length in radiating gas, m or ft Absorption length, m or ft... 

For the selected firebox dimensions, calculate the equivalent cold plane surface, the effective refractory surface and the mean beam length, L. Read the partial pressure of carbon dioxide plus water from Figure 1-7 and calculate the PL product (P is the partial pressure of the radiating components as a function of excess air for the usual hydrocarbon fuels). 

For a black differential receiving surface area dA located in the center of the base of a hemisphere of radius L containing a radiating gas, the mean beam length is L. The mean beam length has been evaluated for various geometries and is given in Table 4.11-1. For other shapes, L can be approximated by... 

Table 4.11-1. Mean Beam Length for Gas Radiation to Entire Enclosure Surface Ml, R2, P3)... 

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