View factors, calculation

The view factors F, and can easily be found by summing the view factors calculated from the surfaces I and II. Values of the view factor F , as function of X, are given in Table A-2 and Figure A-6. 

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 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). 

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). 

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. 

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. 

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. 

The problem focuses on the determination of the geometric view factor. For example, the view factor after 5 seconds of flame propagation can be calculated as follows ... 

Calculate the view factor using the equation given in Appendix A for a vertical plane surface emitter, or else read the view factor from Table A-2 of Appendix A for the appropriate X and fi. This results in F = 0.062 for each portion of the flame surface, and implies a total view factor of... 

In the case of a vertical plane surface, it is assumed that the emitter and receiver are parallel to each other. The view factor is calculated from the sum of view factors from surface I and surface II (see Figure A-5). Surfaces I and II are defined as those to the left and the right of a plane through the center of the receiver and perpendicular to the intersections of the receiver with the ground. 

The amount of calculation involved here can be very considerable and use of a computer is usually required. A simpler approach is to make use of the many expressions, graphs and tables available in the heat transfer literature. Typical data, presented by Incropera and DE Witt(45) and by Howell(47), are shown in Figures 9.38-9.40, where it will be seen that in many cases, the values of the view factors approach unity. This means that nearly all the radiation leaving one surface arrives at the second surface as, for example, when a sphere is contained within a second larger sphere. Wherever a view factor approaches zero, only a negligible part of one surface can be seen by the other surface. 

In a series of papers, Derby and Brown (144, 149-152) developed a detailed TCM that included the calculation of the temperature field in the melt, crystal, and crucible the location of the melt-crystal and melt-ambient surfaces and the crystal shape. The analysis is based on a finite-ele-ment-Newton method, which has been described in detail (152). The heat-transfer model included conduction in each of the phases and an idealized model for radiation from the crystal, melt, and crucible surfaces without a systematic calculation of view factors and difiuse-gray radiative exchange (153). 

Dudokovic and co-workers (154, 155) extended the analysis of a QSSM to include difluse-gray radiation. They computed view factors by approximating the shapes of the crystal and melt by a few standard geometrical elements and incorporating analytical approximations to the view factors. Atherton et al. (153) developed a scheme for a self-consistent calculation of view factors and radiative fluxes within the finite-element framework and implemented this scheme in the QSSM. 

In the light of this, it is necessary to calculate the view factor between two facing squares. Values for such view factors are tabulated in the literature [20] as a function of L/d, where L is the side of the square and d is the distance between the two sur-... 

Radiation heat transfer is assumed to be significant between each portion of the cells and those portions of other cells that are in the same plane, in the upper plane and in the lower plane. The error associated with this assumption increases when small vertical portions are considered. Figure 7.26 shows the view factors, for a cavity constituted of four cells, as the function of the length of the cell slice (vertical discretization). The four view factors have the following meaning (1) Fap - view factor between a cell and the two adjacent cells in the same horizontal plane (2) Fop - view factor between opposite cells in the same horizontal plane (3) Fad - view factor between a cell and the two adjacent cells in the upper and lower plane (4) Fod -view factor between opposite cells in the same horizontal plane. When the element length is larger than 0.04 m, the summation of these coefficients is about 0.97, which means that the error is less than 3%. The calculation of these coefficients is shown in Section 7.4.2.2. 

When the gas emissivity is sufficiently small, the calculation is quite simple. The procedure can be presented by introducing an equivalent electric network, as shown in many textbooks (see for example Incropera and DeWitt, 2002). To calculate the heat transfer fluxes it is necessary to calculate the equivalent resistances due to the non-black surfaces (R, ) and those associated with the view factors (RfY-... 

View factor of a cylinder of length Z and diameter D (surface 1) and a tangent plane (surface 2). This view factor is used for calculating the radiative heat transfer between the cell and the in-stack reformer or the stack walls (see Figure 7.27). 

Study the geometry in the figure above. It is symmetric in both vertical and horizontal direction. The view factor can be computed by calculating the upper and lower paths respectively and adding them. Because of the symmetry, only the upper path need to be considered. 

The objective of this problem is to calculate the view factor of a solar field, taking into account all of the above-mentioned facts. 

Calculate the view factor using Hottel string theorem assuming the clouds to be opaque. (The view factor should be calculated assuming no radiation passes through clouds then calculate the view factor assuming open air gaps are opaque and clouds are translucent, and add them. The atmosphere is also translucent.)... 

The Hottel crossed-string method allows us to calculate the view factor of a surface that is very long in the direction perpendicular to the cross section of the objects. This problem stretches this assumption to use the method for a space heater in a room that has a shape of a cylinder and a person at some distance away. 

Study the value obtained by taking the sum of the crossed strings And subtracting away the sum of the uncrossed strings Half of this calculated value is the required product Of the length of one area and its view factor to the other. 

Many literature sources document closed-form algebraic expressions for view factors. Particularly comprehensive references include the compendia by Modest (op. cit., App. D) and Siegel and Howell (op. cit., App. C). The appendices for both of these textbooks also provide a wealth of resource information for radiative transfer. Appendix F of Modest, e.g., references an extensive listing of Fortan computer codes for a variety of radiation calculations which include view factors. These codes are archived in the dedicated Internet web site maintained by the ublisher. The textbook by Siegel and Howell also includes an extensive atabase of view factors archived on a CD-ROM and includes a reference to an author-maintained Internet web site. Other historical sources for view factors include Hottel and Sarofim (op. cit., Chap. 2) and Hamilton and Morgan (NACA-TN 2836, December 1952). 

Physically, a two-zone speckled enclosure is characterized by the fact that the view factor from any point on the enclosure surface to the sink zone is identical to that from any other point on the bounding surface. This is only possible when the two zones are intimately mixed. The seemingly simplistic concept of a speckled enclosure provides a surprisingly useful default option in engineering calculations when the actual enclosure geometries are quite complex. 

Solution Let zone 1 represent one tube and zone 2 represent the effective plane 2, that is, the unit cell for the tube bank. Thus Aj = tcD and A2 = C are the corresponding zone areas, respectively (per unit vertical dimension). This notation is consistent with Example 3. Also put i = 0.8 with 2 = 1.0 and define R = C/D = 12/5 = 2.4. The gray plane effective emissivity is then calculated as the total view factor for the effective plane to tubes, that is, J2-1s % For R = 2.4, Fig. 5-15, curve 5, yields the refractory augmented view factor F2ji = 0.81. Then is... 

Clearly when K = 0, the two direct exchange areas involving a gas zone g[ j and g gj vanish. Computationally it is never necessary to make resort to Eq. (5-155) for calculation of gg,. This is so because sj, gSj, and gg may all be calculated arithmetically from appropriate values of StSj by using associated conservation relations and view factor algebra. 

---