View-factor

The calculation of radiative exchange between two surfaces requires a quantity that describes the influence of their position and orientation. This is the view factor, which is also known by the terms configuration factor or angle factor. The view factor indicates to what extent one surface can be seen by another, or more exactly, what proportion of the radiation from surface 1 falls on surface 2. 

The first step in the calculation of the view factor is to determine the radiation flow d2 f 12, emitted from surface element dj4j that strikes surface element dA2, Fig. 5.50. With Ll as the intensity of the radiation emitted from dA from (5.11), we get 

duj2 is the solid angle at which the surface element d.42 appears to dA,  

This relationship is also known as the photometric fundamental law. According to this, the radiation that reaches dA2 decreases with the square of the distance r between radiation source and receiver. In addition to this, the orientation of the surface elements to the straight line between them is of importance. This is expressed in terms of a cosine function of the two polar angles /3y and / 2. 

We will now calculate the radiation that is emitted by the finite surface 1 that strikes surface 2, Fig. 5.50. This involves the assumption that the intensity Ly is constant over the entire surface 1. Integration of (5.128) over both surfaces yields 

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F = direct view factor Ey, fraction of isotropic radiation from Aj intercepted directly by Aj. 

Example 3 Calculation of View Factor Evaluate the view factor between two parallel circular tubes long enough compared with their diameter D or their axis-to-axis separating distance C to make the problem two-dimensional. With reference to Fig. 5-18, the crossed-strings method yields, per unit of axial length,... 

The view factor F may often be evaluated from that for simpler configurations by the application of three principles that of reciprocity, AjFij = AjFp that of conservation, XF = 1 and that due to Yamauti [Res. Electrotech. Lab. (Tokyo), 148, 1924 194, 1927 250, 1929], showing that the exchange areas AF between two pairs of surfaces are equal when there is a one-to-one correspondence for all sets of symmetrically placed pairs of elements in the two surface combinations. 

FIG. 5-16 View factors for a system of two concentric coaxial c to inner cylinder, (h) Inner surface of outer cylinder to itself. 

FIG. 5-17 Distribution of radiation to rows of tubes irradiated from one side. Dashed lines direct view factor F from plane to tubes. Solid lines total view factor F for black tubes backed by a refractory surface. 

Only one direct-view factor F o of direct-exchange area 12 is needed because Fn equals 1 — Fig and Foo equals 1 — Fo equals 1 — FioA /Ao. Then 11 equals A] — 12 and 22 equals A2 — 21, With these substitutions, Eq, (5-131) becomes... 

Here, Qr is the energy loss per second by a surface at temperature to its suiToundings at temperature T, , the emissivity of the subsU ate being e, the view factor F being the fraction of tire emitted radiation which is absorbed by the cool sunoundings, and a being tire Stefan-Boltzmairn radiation constant (5.67 X 10 Jm s In the present case, tire emissivity will have a value of about 0.2-0.3 for the metallic subsU ates, but nearly unity for the non-metals. The view factor can be assumed to have a value of unity in the normal situation where the hot subsU ate is enclosed in a cooled container. 

In all of these systems, the rate of generation at the gas-solid interface is so rapid that only a small fraction is canied away from the particle surface by convective heat uansfer. The major source of heat loss from the particles is radiation loss to tire suiTounding atmosphere, and the loss per particle may be estimated using unity for both the view factor and the emissivity as an upper limit from tlris source. The practical observation is that the solids in all of these methods of roasting reach temperatures of about 1200-1800 K. 

The thermal radiation received from the fireball on a target is given by equation 9.1-31, where Q is the radiation received by a black body target (kW/m ) r is the atmospheric transmissivity (dimensionless), E = surface emitted flux in kW/m", and f is a dimensionless view factor. 

The heat flux, E, from BLEVEs is in the range 200 to 350 kW/m is much higher than in pool fires because the flame is not smoky. Roberts (1981) and Hymes (1983) estimate the surface heat flux as the radiative fraction of the total heat of combustion according to equation 9.1-32, where E is the surface emitted flux (kW/m ), M is the mass of LPG in the BLEVE (kg) h, is the heat of combustion (kJ/kg), is the maximum fireball diameter (m) f is the radiation fraction, (typically 0.25-0.4). t is the fireball duration (s). The view factor is approximated by equation 9.1-34. where D is the fireball diameter (m), and x is the distance from the sphere center to the target (m). At this point the radiation flux may be calculated (equation 9.1-30). 

Tj is the surface temperature of the panel, Tj, the thermometer bulb temperature, Tj the air temperature, and T the temperature of the walls of the building. F is the view factor from the bulb to the heating panel, e is the emissivity of the thermometer bulb at temperature T cr is the Stefan-Boltzmann constant (5.67 X 10 W m K " ), and is the convective heat transfer coefficient from bulb to air. 

If the thermometer is situated symmetrically relative to the heating panel, then the view factor will be four times higher than calculated from Eq. (8.31). 

In the case of a given surface temperature, the amount of energy released is determined by the parameters for the convective and radiative heat exchange. As far as convection is concerned, these are the temperatures ol the heat source surface and room air, respectively, and the heat transfer coefficient. The radiative heat exchange is determined by the view factors and the temperatures of the surrounding surfaces. 

In the second approach, the energy release is split by a predefined (mostly constant) factor between convection and radiation. The convective part is directly transferred as energy gain to the room air, while the radiative part is distributed to the surrounding walls by the area-weighted method or the view-factor method. 

Form view factor A factor which describes the effects of the relative area of two surfaces, the geometry of the surfaces in relation to each other, and the two emissivities on radiation heat exchange between the surfaces. 

The view factor is the fraction of the radiation falling directly on the receiving target. The view factor depends on the shapes of the fire and receiving target, and on the distance between them. 

Fn = view factor or geometric configuration factor E = emissive power of emitting surface 2 = incident radiation-receiving surface... 

The view factor depends on the shape of the emitting surface (plane, cylindrical, spherical, or hemispherical), the distance between emitting and receiving surfaces, and the orientation of these surfaces with respect to each other. In general, the view factor from a differential plane dAj) to a flame front (area A,) on a distance L is determined (Figure 3.10) by ... 

A fireball is represented as a solid sphere with a center height H and a diameter D. Let the radius of the sphere be / (/ = DU). (See Figure 3.11.) Distance x is measured from a point on the ground directly beneath the center of the fireball to the receptor at ground level. When this distance is greater than the radius of the fireball, the view factor can be calculated. 

For a vertical surface beneath the fireball (jc < D/2), the view factor is given by... 

For a flash fire, the flame can be represented as a plane surface. Appendix A contains equations and tables of view factors for a variety of configurations, including spherical, cylindrical, and planar geometries. 

In order to compute the thermal radiation effects produced by a burning vapor cloud, it is necessary to know the flame s temperature, size, and dynamics during its propagation through the cloud. Thermal radiation intercepted by an object in the vicinity is determined by the emissive power of the flame (determined by the flame temperature), the flame s emissivity, the view factor, and an atmospheric-attenuation factor. The fundamentals of heat-radiation modeling are described in Section 3.5. 

The only model ever published in the literature is poor. The fact, for instance, that burning speed is taken as proportional to wind speed implies that, under calm atmospheric conditions, burning velocities become improbably small, and flash-fire duration proportionately long. The effect of view factors, which change continuously during flame propagation, requires a numerical approach. 

The total radiation received by an object also depends on the fireball s position relative to the object (i.e., the view factor) and radiation adsorption by the atmosphere. 

The solid-flame model, presented in Section 3.5.2, is more realistic than the point-source model. It addresses the fireball s dimensions, its surface-emissive power, atmospheric attenuation, and view factor. The latter factor includes the object s orientation relative to the fireball and its distance from the fireball s center. This section provides information on emissive power for use in calculations beyond that presented in Section 3.5.2. Furthermore, view factors applicable to fireballs are discussed in more detail. 

View Factor. The view factor of a point on a plane surface located at a distance L from the center of a sphere (fireball) with radius r depends not only on L and r, but also on the orientation of the surface with respect to the fireball. If 2 is the view angle, and 0 is the angle between the normal vector to the surface and the line connecting the target point and the center of the sphere (see Figure 6.9), the view factor (F) is given by... 

When the distance (X) is greater than the radius of the fireball, the view factor for... 

Radiation effects from a fireball of the size calculated above, and assumed to be in contact with the ground, have been calculated by Pietersen (1985). A fireball duration of 22 s was calculated from the formula suggested by Jaggers et al. (1986). An emissive power of 350 kW/m was used for propane, based on large-scale tests by British Gas (Johnson et al. 1990). The view factor proposed in Section 6.2.5. 

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