1、4732 Simple Approach to Evaluate the View Factors between Internal Heat Sources and Their Environment Qingpeng Wei, Ph.D. ABSTRACT It is important to determine the view factors between inter- nal heat source surfaces and their environment in order to better understand the thermal radiation in rooms.
2、 With reason- able approximation to the theoretical derivation, this paper presents a simple approach to evaluate the view factors between internal heut sources and their surroundings. The approach is based on the contour-integral representation and the superposition principle of view factors. The i
3、ntegral inter- mediate value theorem is applied to reduce the complexity of calculation. The simple approach is validated through comparison with a theoretical method for several basic conjg- urations. A comparison to numerical results obtained by the Monte Carlo method is also presented. In additio
4、n, four typical configurations between human postures and the surroundings are presented. Compared to Fangers experimental data, the simple approach produced satisfactory results. Based on this approach, thermal radiation heat exchange between internal heat sources and their surroundings can be easi
5、ly predicted with suficient accuracy, INTRODUCTION Thermal radiation between internal heat sources and their surroundings impacts indoor environment significantly. With certain assumptions, the process can be analyzed using radi- osity equations (Hottel 1954) or network representations (Oppenheim 19
6、56). However, one barrier on the way to apply- ing these approaches in practice is that it is difficult to deter- mine the view factors between internal heat source surfaces and their environment. A simple method applicable to various internal heat source shapes with adequate precision is not availa
7、ble yet. Yi Jiang, Ph.D., P.E. Member ASHRAE The view factors of human bodies to a surrounding enclo- sure are well studied. Photographic methods have been applied and the experimental results obtained by Fanger (1 982) have been widely used to date (ASHRAE 200 1). Hori- koshi et al. (1990) measured
8、 the view factors between standing and seated postures and rectangular planes close to the subjects using the photographic method. Jones et al. (1998) developed projected area data for the whole body and for indi- vidual body segments, using photographic methods in succes- sion. Based on the foregoi
9、ng experimental works, new algorithms have been developed. Rizzo et ai. (1991) developed algorithms to calculate the mean projected area factors of seated and standing persons. Cannistraro et al. (1992) devel- oped algorithms for calculating the view factors between a human body and rectangular surf
10、aces in parallelepiped envi- ronments. Kalisperis et al. (199 1) developedview factor tables for a variety of inclined surfaces. Nucara et al. (1999) proposed a simple algorithm for the automatic calculation of the view factors between people and composite plane surfaces based on Fangers data. Conve
11、ntional photographic methods have a limitation in practice of measuring various types of internal heat sources because they require much time and money. Numerical approaches have been well developed to calculate the view factors between a human body and its surroundings in recent years. Ozeki et ai.
12、 (2000) divided the human body surface into 4396 quadrilateral elements for both standing and seated postures. Based on that fine-meshed model, Ozeki et al. (2000) developed a numerical method that can predict the view factors of the whole body and surface parts of the body in any posture with enoug
13、h accuracy for prac- tical use. Miyanaga et al. (2000) developed a simplified human body model with 6 17 small elements for evaluating thermal Qingpeng Wei is an assistant professor and Yi Jiang is a professor and head of the Department of Building Science, School of Architecture, Tsinghua Universit
14、y, Beijing, P.R. China. 400 02004 ASHRAE. radiant environment in a radiant cooled space. These methods are based on the calculation of the effective radiation area of a human body with a given posture. The Monte Carlo (MC) method, which works well to solve radiant heat transfer prob- lems in complic
15、ated conditions (Howell 1968), has also been used to calculate view factors between humans and the surrounding environment in recent years. Omori et al. (1998) applied the MC method to simulate the radiant heat transfer of complicated shapes with unstructured grid systems. Murakami et al. (2000) car
16、ried out combined simulation of airflow, radiation, and moisture transport for heat release from a human body using the MC method. Li (2002) traced the rays emitted from a small surface in a cubic enclosure and validated that when the number of rays exceeds 30,000, the view factor calculated by the
17、MC method has an uncertainty of less than 1 Yo compared with theory. However, the numerical methods require large amounts of computer memory, which can make implementation impractical. A simple approach to evaluate the view factors between internal heat sources and their surroundings is presented in
18、 this paper. The approach is based on the contour-integral represen- tation and the superposition principle of view factors. The integral intermediate value theorem is applied to reduce the complexiy of calculation. This simple approach is validated through comparison with a theoretical method for s
19、ome basic configurations. Comparison to numerical results obtained by the Monte Carlo method is presented. In addition, four basic configurations of human postures and their surroundings are presented. Compared with Fangers experimental data, the simple approach produces satisfactory results. Using
20、this approach, the thermal radiation heat exchange between inter- nal heat sources and their surroundings can be easily predicted with sufficient accuracy. METHODOLOGY Representation of the View Factor between Infinitesimal and Finite Areas Fundamentals. By definition, the view factors for diffuse r
21、adiation interchange are represented in terms of area integrals when one or both of the participating surfaces are finite. An alternate form of the defining equations for view factors can be achieved by replacing the area integrals by contour integrais. Thus, the Stokes theorem is adopted (Sparrow 1
22、978), and one obtains (zI-zIYJ-o;-YI) FdAz-AJ = I I 2 2nr (1) c, (x - x,)dz - (z - z,) -Y,)dx, - (XI -XlMJ 2 i+nzJ 2nr 2 CJ 2nr +m,j CJ This is the contour-integral representation of the view factor for interchange between an infinitesimal and a finite surface. In particular, the absolute value of t
23、he first contour integral represents the view factor for an element (xl, y, 2,) having direction cosines 1, = *I, m, = n, = O. The choice of sign depends on whether the normal to the aforementioned element lies along the +x or -x axis as the element views Ai. In light of this interpretation, one def
24、ines uni Repeating the foregoing, one obtains Equation 3 can be recognized as stating a superposition principle; namely, that the view factor between an element dA, with arbitrary direction cosines (li, mi, ni) and a surface Ai is expressible as a linear sum of the basic view factors: weighting fact
25、ors in the superposition are the direction cosines of the element. This superposition principle serves to gener- alize results for basic configurations to more complex config- urations. Analytical Formulas of the Basic Configurations. Analytical formulas for view factors of several basic config- ura
26、tions, shown in Figure 1, are given in Equations 4 and 5. Fd?, -A, (* 190,o) 9 FdA, -A/( o,* 1, o), and FdA, - AJ (0 ,o, however, when thk area A, is much smaller than A, FdA -A(P) does not vary in magnitude. Hence, taking the centroid ofA, as Po is a reasonable choice. In addition, the numencal int
27、egral theory shows that zero- exponent approximation in Equation 9 has the same accuracy in magnitude to the one-exponent approximation. Simplifying Internal Heat Sources with Regular Shapes. Internal heat sources, such as people, light-emitting elements or lamps, and various kinds of appliances or
28、equip- ment, have all kinds of complex shapes; however, it is possible to simplie and catalog them into several regular shapes from the viewpoint of building energy analysis. Rectangular, cube, cuboid, column, hemisphere, or their combination are gener- ally adequate to approximate all kinds of inte
29、rnal heat sources; detailed study will determine the characteristic dimensions of the foregoing shapes in order to best approximate reality. Illustration of Representing View Factors for the Regular Configurations. To illustrate how the foregoing approach is applied, the case of diffuse radiation in
30、terchange between internal heat sources and a horizontal cooled ceiling is considered. The configuration and representation of view factors for the regular shapes, such as rectangle, cube or cuboid, column, and hemisphere, to the ceiling are shown schematically below. In light of the interpretation,
31、 one defines Deducing the rest by analogy, one obtains representation of view factors for regular shapes to the ceiling in a room, as shown in Figures 3a to 3d and Equations 12 to 15. 4SFS-C = abF+,(o) (12) RESULTS AND VALIDATION View Factor Distribution in Rooms Consider a room with the dimensions,
32、 A x B x H= 4.5 x 6.0 x 3.0 m3. A medium-sized computer monitor is represented as a cube of a = 0.4 m, which can be put anywhere in the room. The view factor between the cube and ceiling is calculated by Equation 13, and results referring to different positions in the room are shown in Figure 4. In
33、Figure 4a, it appears that the view factor between the internal heat source and the ceiling varies only slightly at the same altitude but increases rapidly in the vertical direction, closer to the ceiling. Analysis of the range of possible view factors, shown in Figure 4b, shows that the view factor
34、 does not exceed 0.35 in most configurations. 402 ASHRAE Transactions: Research /! Ceiling Figure 3 Illustrations of calculating the vim factor from (a) a horizontal rectangle upward to a ceiling, (b) a cube or cuboid to a ceiling, (c) a column to a ceiling, and (d) a hemisphere upward to a ceiling.
35、 Percent Possibility of View factor between cube internai heat source aiid ceiling - _. _ 30.0% , 25.0% b rl 20.0% Il n View factor range 0.113 0.170 0.226 0.283 0.340 0.396 0.453 0.509 Figure 4 Vlew factor between cube internal heat source and ceiling: (a) projles and (b) possibility. ASHRAE Transa
36、ctions: Research 403 Figure 5 Comparison with theoretical approach: (a) relative error for diyerent locations and similarity ratio and (b) permitted similarity ratio in room with an acceptable relative error of I %. Symbols Al A2 Table 1, Parameters of the Illustration for Parallel Rectangular Surfa
37、ces with Different Dimensions Representing Normal Direction internal heat sources fZ horizontal ceiling -Z I I I I I l 1 Characteristic Length 2a, 2b A,B Area Centers 4ab bo3 Ya, 20) AB (Al2, Bl2, H) Validation with Theoretical Approach It is possible to calculate the view factor between two finite
38、surfaces with a transformed contour-integral represen- tation by employing Stokes theorem, as shown in Equation 7. To illustrate this principle and validate the aforementioned approaches, consider a pair of parallel rectangular surfaces with different dimensions as shown schematically in Figure 3a.
39、The parameters of this configuration are shown in Table 1. Upon substituting these parameters in Equation 7, one obtains the theoretical representation as where Validation with Numerical Approach Consider the aforementioned room and computer monitor (simplified as a cube of a = 0.4 m and can be put
40、anywhere in the room). The view factor between the cube and ceiling is calculated by Equation 13 and by a Monte Carlo method (Li 2002). The results are shown in Figure 6a. The approximate method is consistent with the numerical method, with less than 5% of difference. In addition, it takes more than
41、 12 hours by the MC method to accomplish the computation of 729 different configurations, but less than 1 second by the approximate method. 404 ASHRAE Transactions: Research MC Method AsFs-c calculated by two methods, cube - rectangle 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00 O - -1. O0 O05
42、O 10 0.15 0.20 0.25 O30 O35 o 40 0.45 Approrimilte method MC 0.25 AsFs-c calculated by two methods, column - rectangle 0.20 0.15 0.10 0.05 0.00 0.00 0.05 0.10 0.15 0.20 0.2s Approximate Method Figure 6 Comparing the view factors by the two approaches: (a) a cube to a rectangle ceiling and (a) a colu
43、mn to a rectangle ceiling. The comparison between these two methods was also to calculate the view factor between a column (R carriedJ2t = a /n = 0.226 m , H= 0.4 m) and a rectangular ceiling, based on Equation 14. The results of 27 configurations are shown in Figure 6b. Validation with Experimental
44、 Approach A detailed validation of this simple approach was carried out by comparison with Fangers (1 982) experimental data. Four basic configurations of human posture and its surround- ings are presented in the following sections: seated or standing person to horizontal or vertical rectangle (Figu
45、res 7 and 8). A seated person is presented by a cuboid with dimensions 0.35 xO.3 x 1.2m3,and,forastandingperson,0.35 xO.3 x 1.75m3. Therefore, the view factors obtained by Equation 13 are comparing with the ones from Fangers charts. In summary, while the view factors are less than 0.04, the simple a
46、pproach has large relative errors to the experimental data; however, the absolute errors are less then 0.005. There- fore, it is concluded that the proposed simple approach to eval- uate the view factors between a body and its surroundings offers a satisfactory result compared with the experimental
47、data. SUMMARY AND DISCUSSION It is important to determine the view factors between internal heat sources surfaces and their environment in order to get a better understanding of the thermal radiation in rooms. With reasonable approximation to the theoretical derivation, a simple approach to evaluate
48、 the view factors between internal heat sources and their surroundings is presented in this paper. During conduct of this approach, the contour-integral representation and the superposition principle of view factors are employed. As a result, the view factor of element with arbi- trary direction can
49、 be represented through a summation of vectors. In addition, the integral intermediate value theorem is applied to reduce the complexity of integral calculation. When internal heat sources are simplified to regular shapes such as rectangle, cube or cuboid, column, or hemisphere, the view factors between internal heat sources and their surroundings can be obtained easily, as shown in Equations 12 to 15. The simple approach is validated through a comprehen- sive comparison with theoretical, numerical, and experimental methods. A basic configuration, a pair of parallel rectang
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