Stefan-Boltzmann equations

The models proposed to represent radiation transport process can be grouped into two classes. The first and simpler approach is to use some form of the Stefan-Boltzmann equation for radiant exchange between opaque gray bodies,... 

Both constants o and o are expressions of and can be calculated from the natural constants k, h, and c, which were discussed earlier. Equation 17.5, which relates the energy flux to the absolute temperature, is known as the Stefan-Boltzmann equation. The coefficient o is known as the Stefan-Boltzmann constant. The average photon energy can be calculated from... 

Heat transport by radiation is described by the Stefan-Boltzmann equation (for two parallel planes) ... 

Equation 7.12 is known as the Stefan-Boltzmann equation, and C is the Stefan-Boltzmann constant, which has the value... 

Observe the relationship to the Stefan - Boltzmann equation. It can be easily verified that... 

Solids radiate heat, even at low temperatures. The net radiant heat actually transferred to a receiver is the difference between radiant heat received from a souree and the radiant heat re-emitted from the receiver to the source. The net radiant heat flux between a hot body (heat source) and a cooler body (heat receiver) can be ealculated by any of the following Stefan-Boltzmann equations. 

The Stefan-Boltzmann equations (2.6, 2.7, 2.8, and 2.9) show that heat transfer rate to most black or gray bodies varies as the difference in the 4th power of their absolute temperatures, which accentuates the difference between furnace temperature or furnace wall temperature and poc gas temperature. Case A In Figure 6.3, at 1000 F furnace temperature and 20 fps gas velocity, the temperature of the exiting poc gas is on the order of 1800 F. With a combustion air temperature of 600 F, if someone erroneously took the %available heat (from fig. 5.1) af 1000 F he would read 78%. He should have taken the %available heat at 1800 F, where he would read 57%. Therefore, if the required available heat were 100 kk Btu/hr (105.5 kJ/h), the gross heat required will be 100/0.57 = 175 kk Btu/hr (185 kJ/h), NOT 100/0.78 = 128 kk Btu/hr (135 kJ/h) as with the erroneous method. Case B At 2500 F furnace temperature, with the same 20 fps, the poc gas temperature would be 2560 F. Corresponding figures are in table 8.16. 

When radiation heat transfer occurs from a surface it is usually accompanied by convective heat transfer unless the surface is in a vacuum. When the radiating surface is at a uniform temperature, we can calculate the heat transfer for natural or forced convection using the methods described in the previous sections of this chapter. The radiation heat transfer is calculated by the Stefan-Boltzmann equation (4.10-6). Then the total rate of heat transfer is the sum of convection plus radiation. 

EMISSIVITY OF ALUMINUM SHIELDS AND APPARATUS SURFACES. Tests on the small samples were conducted with the same single-guarded cold plate thermal conductivity apparatus as was used in previous investigations [2]. This apparatus was found useful in establishing the mean emissivities of foils and films. To obtain measurements, the foil was glued to the cold and warm surfaces of the apparatus, keeping one surface at 46 F and the other at -320 F. The space between the surfaces, 1/4 in., was evacuated to 3.10 mm Hg, so that radiation was the major factor in the heat transferred. The heat flux between the surfaces was calculated from the observed boil-off rate on the assunaption that the emissivities of the warm and cold surfaces were equal. The mean emissivity was established by means of the Stefan-Boltzmann equation ... 

Thermal radiation is usually calculated using surface emitted flux, E, rather than the Stefan-Boltzmann equation, as the latter requires the flame temperature. Typical energy fluxes for BLEVEs (200-350 kW/m ) are much higher than in pool fires as the flame is not smoky. Roberts (1981) and Hymes (1983) provide a means to estimate surface heat flux based on the radiative fraction of the total heat of combustion. 

The surface emitted power or radiated heat flux may be computed from the Stefan-Boltzmann equation. This is very sensitive to the assvuned flame temperature, as radiation varies with temperature to the fourth power (Perry and Green, 1984). Further, the obscuring effect of smoke substantially reduces the total emitted radiation integrated over the whole flame surface. 

Burning rate, flame height, flame tilt, surface emissive power, and atmospheric transmissivity are all empirical, but well established, factors. The geometric view factor is soundly based in theory, but simpler equations or summary tables are often employed. The Stefan-Boltzmann equation is frequently used to estimate the flame surface flux and is soundly based in theory. However, it is not easily used, as the flame temperature is rarely known. 

This concept of alternating reflective layers with nonconducting fibrous spacers has been known since 1951. This arrangement results in heat impedance excelling by far the performance of the best dewar vessels by reducing the apparent k factor to below 0.5 xw/cm- C. In spite of this spectacular reduction of the heat flux, the reduction in heat flow is considerably lower than what one would expect from the Stefan-Boltzmann equation for the applied number of shields and their emissivity. The ratio of observed heat flow to the theoretically computed flux can be referred to as the efficiency of the radiation shields (see Table 1). 

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