1、4696 Heat Transfer through a Composite Wall with a Rectangular Graybody Radiating Cavity: A Numerical Solution Mohamed A. Antar, Ph.D. ABSTRACT A numerical$nite-dzfference analysis is developed in this paperfor steady-state heat transfer in a composite wall with a two-dimensional rectangular graybod
2、y radiating cavity with and without natural convection circulation of air Theprimary objective of this paper is to provide a clear and systematic approach that can be used to accurately account for the effects of radiation in practical applications of this type and to estab- lish a basis for evaluat
3、ing the error of the first-order two- dimensional method. INTRODUCTION The importance of radiant heat transfer in composite walls with air gaps or evacuated spaces has long been recog- nized (ASHRAE 1993). Practical design information is avail- able that serves as a guide for estimating the effects
4、of graybody radiation on the R-value of thin one-dimensional air gaps that are used in building construction. Based on this information, the thermal resistance associated with a 40 mm plane air space increases by factors as large as 10 for a decrease in surface emissivity from 0.82 to 0.03. The corr
5、ela- tions that are available apply for ideal conditions associated with single air spaces of uniform thickness bounded by plane, smooth, parallel surfaces with no leakage. No information pertaining to the levei of accuracy provided by these early correlations is available. Furthermore, no practical
6、 approach is available for adapting these correlations to applications involving multiple spaces and multidimensional effects. Computer models have been developed by Kosny and Christian (1 995) and Soylemez (1999) that analyze conduc- tive and convective heat transfer in multidimensional compos- ite
7、 walls. In addition, numerical studies have been published Lindon C. Thomas, Ph.D. for graybody radiation and natural convection in rectangular and square cavities (Balaji and Venkateshan 1993; Mezrhab and Bcir 1998; Mohrpatra et al. 1999; Antar and Thomas 2001). However, the studies by Kosny and Ch
8、ristian and Soylemez do not include the effects of radiation, and the stud- ies by Balaji and Venkateshan, Mezrhab and Bcir, and Mohr- patra et al. do not include the effects of conduction in the surrounding structure. A practical first-order two-dimensional method for analyzing heat transfer in two
9、-dimensional composite walls has recently been presented by Antar and Thomas (2001) that accounts for graybody radiation across interior enclosures. This method is computationally efficient and quite general. However, a comprehensive evaluation of the accuracy of this practical analysis approach has
10、 not yet been presented. Because of the significance of graybody radiation in multidimensional composite building construction and the potential usefulness of approximate methods for the design and analysis of applications of this type, a formal numerical analysis is developed in this paper to deter
11、mine the extent to which practical approximate methods can be employed in dealing with the complicating issue of radiant heat transfer. The problem selected for study is a basic building block with a rectangular radiating space with and without natural convec- tion circulation of air under steady-st
12、ate and uniform property conditions. In addition to providing a means of evaluating the accuracy of practical methods for modeling the radiant heat transfer in such enclosures, this analysis is intended to provide the framework for accounting for the effects of conduction or convection in air and fo
13、r dealing with multiple spaces. One of the objectives of this paper is to provide a practical numerical approach to analyzing basic problems involving Mohamed Antar is an associate professor and Lindon Thomas is a professor in the Mechanical Engineering Department, King Fahd Univer- sity of Petroleu
14、m and Minerals, Dhahran, Saudi Arabia. 36 02004 ASHRAE. Surface A, with uniform or nonuniform temperature Tg L Fns4-ns1 and Fns4- - - NS J., = sEbns+ Ps JnlFns-n, (21) n, = 1 This equation must be satisfied for all N, subsurfaces. Closure. To obtain an accurate numerical solution for the heat transf
15、er characteristics associated with this problem, the nodal equations and subsurface radiosity equations must be solved simultaneously for a sufficiently fine finite-difference grid. The solution approach featured in this study involves a double iteration using the successive approximation method. Th
16、e solution steps are outlined as follows: 1. Set initial approximations for all nodal temperatures Zm,n. 2. Use Equations 17 and 19 to calculate approximate values for the radiosity J, at each subsurface by setting Gns = O. 3. a. b. 4. a. b. Using approximate values for the subsurface radiosi- ties
17、Jns from the previous step, solve the system of subsurface radiosity equations indicated by Equation 21 to obtain refined values for Jnsi, Jn,s2, . , . , JNs using successive iterations. Use Equations18 and 16 to calculate G, and Ag,. With the subsurface radiant heat-transfer rate specified in accor
18、dance with step 3b, solve the system of nodal equations indicated by Equation 15 to obtain refined values for Tm,n using successive iterations. With the subsurface temperatures specified in accor- dance with step 4a and with G, specified according to step 3b, calculate refined values for the subsurf
19、ace radiosities J,. 5. Continue with steps 3 and 4 until satisfactory convergence is achieved e., (Tm,(k+l)-T m.n (k)/T m,n (k) This solution scheme has been incorporated into a numer- ical finite-difference program. To enhance the computational efficiency, the symmetry of the particular problem und
20、er investigation has been accounted for in writing the nodal equa- tions. The total rates of heat transferred into and out of the block are expressed in terms of the nodal temperatures by writ- ing ASHRAE Transactions: Research In addition to requiring that the solution converges as the number of no
21、des increases, the difference between qi and qo must be very small in accordance with energy balance require- ments for steady-state conditions, i.e., q = qi = qo. R-value. The R-value of the block is expressed in terms of9 by which condition & is specified by the following convection correlation (J
22、acob 1946): Ti - To R-vaue = - q/A To provide a basis for testing the program, consideration is given to the cases for which Hlw approaches zero and unity. Relations for the R-value associated with these two limiting cases are given by for 1 1 0.5 and E, 0.5. However, the error associated with the f
23、irst-order two-dimensional model for small values of emissivity is significantly higher, with the error being as large as 36% for E, = O. 1. Although the effect of other variables (k, LI, L, wl, wz) needs to be assessed for this basic problem, these results clearly demonstrate the capability of the
24、first-order two-dimensional method for providing reasonable approximate solutions for graybody radiation in 40 E = 0.1 - E = 0.3 0.5 0.6 0.7 0.8 0.9 1 HNV Figure 11 Error in jrst-order two-dimensional calculation for R-value: evacuated cavity. composite walls with an evacuated rectangular space. How
25、ever, the approximate method must be used with caution since significantly larger errors are found to occur for smaller values of emissivity for practical arrangements for which Hlw 0.5. Clearly, the higher-order methodology employed in the present paper is required in the design of systems involvin
26、g highly reflective surfaces. In connection with the effect of emissivity E, on the accu- racy of the first-order method, it should be noted that the dependence of radiosity J, on irradiation G, increases with decrease in E, according to Equation 4. Because it is the irra- diation that accounts for
27、the contribution of radiation from the bottom and top walls in establishing the values of radiosity along the vertical surfaces, it is concluded that the significance of the two-dimensional radiation effects increase with decrease in E,. Therefore, it is concluded that the two-dimen- sional effect a
28、ssociated with emissivity is responsible for the decrease in the accuracy of the first-order method with decrease in E,. The fact that the variation in temperature along the verti- cal surfaces of the cavity is small, as shown in Table 1, provides justification for use of the practical approach to m
29、odeling convection presented in the section on “Effects of Natural Convection.” The effect of natural convection on the R-value for an air-filled cavity is shown in Figure 12. As expected, natural convection results in significantly lower R- values. The difference between the first-order two-dimen-
30、sional analysis (Antar and Thomas 200 1) and the higher-order numerical analysis for this case is shown in Figure 12. These calculations indicate that the differences between the first- order and numerical solutions with and without convection have similar characteristics. It follows that the practi
31、cal first- ASHRAE Transactions: Research 43 Table la. Temperatures at Vertical Sides of the Enclosure (E = 1) Right Side 32.42725 39.549 19 39.53678 39.51396 40 u1 o 2, m C .- - 30 i f! o .I- iF g 20 !g i 8 L .I- 10 8 n c 32.43076 32.44184 32.46234 39.47651 39.41531 39.30732 32.49623 32.55220 32.652
32、22 39.07922 38.43009 32.86652 33.4834 1 Left Side Right Side 41.707 15 41.68807 30.263 12 30.28165 41.52534 41.36589 30.43969 30.59449 40.77605 40.18670 3 1.16603 3 1.73432 Left Side T Right Side T 42.25349 42.18029 29.71608 29.78739 41.85636 41.57152 30.10269 30.37946 40.46698 39.0971 1 3 1.44689 3
33、2.75831 - With convection Without Convection 0.5 0.6 0.7 0.8 0.9 1 H/W Figure 12 Error in first-order two-dimensional calculation for R-value. T T order two-dimensional method can be expected to provide reasonable approximate solutions for cases in which E, is in the range 0.5 to 1. However, more ac
34、curate numerical solutions should be employed for highly reflective surfaces. Concerning future work, the present numerical analysis provides a basis for conducting a thorough parametric study on the basis of which criteria can be established for offering guidance concerning the range of conditions
35、associated with the basic problem considered in this paper for which low to moderate errors in the approximate first-order two-dimen- sional method can be expected. In addition, the analysis can be readily generalized to account for convection and radiation at the outer surfaces and the effects of c
36、onduction across the cavity for the case in which the Rayleigh number is less than 1000. I 41.62922 I 30.33881 I I 41.13062 I 30.82273 I I 38.96156 I 32.90730 I Table IC. Temperatures at Vertical Sides of the Enclosure (E = 0.1) ACKNOWLEDGMENT The authors are pleased to acknowledge the support provi
37、ded for this study by King Fahd University of Petroleum & Minerals. I 42.27728 I 29.69290 I NOMENCLATURE I 42.05 183 I 29.91249 I dF Ebs F = radiatin shape factor h = heat transfer coefficient = infinitesmal radiation shape factor = local blackbody heat flux = irradiation GS Js = radiosity k = therm
38、al conductivity I 41.14989 I 30.7881 1 I 1 * Case: L = 0.02, H = 0.20, HIL = 10, H/W = 0.5 44 ASHRAE Transactions: Research L = length LI, L2, L3= lengths indicated in Figure 1 Lab, La, Lb, L, L,=dimensions used for calculating the shape factor using crossed-strings method MI, M2, M3=nurnber of node
39、s in the x-direction of subsections 1,2, and 3, respectively (see Figure 3) Nl, N2 = number of nodes in the y-direction of subsections 1, 2, and 3, respectively (see Figure 3) m = increment in the x-direction Nsl, Ns2, Ns, Ns4=t0tal number of nodes for subsections in y- nSl, ns2, ns3, ns4=nodes of s
40、ubsections in y-direction (Figure 5) n 9” = heat flux R = thermal resistance R, = longitudinal radiation resistance R, = longitudinal convection resistance Ra = Rayleighnumber T = temperature q, T, = temperature at lefi and right surfaces, respectively W = width wl, w2 = widths indicated in Figure 1
41、 X = horizontal coordinate Y = vertical coordinate xsl , ys2 = node locations indexes (Figure 6) z = depth Greek Symbols a = thermal diffisivity ES = surface emissivity V = kinematic viscosity Ps = surface reflectivity (3 = Stefan Boltzrnan constant direction (Figure 5) = increment in the y-directio
42、n = coefficient of thermal expansion Su bscripts 1,3 2,4 = horizontal lower and upper surfaces of the cavity = horizontal lefi and right surfaces of the cavity S = surface n = node R = radiation C = convection f = fluid t = transverse REFERENCES Antar, M., and L.C. Thomas. 2001. Heat transfer throug
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45、gh enclosed plane gas layers. Trans. ASME: 189. Kosny, J., and J.E. Christian. 1995. Reducing the uncertain- ties associated with using the ASHRAE zone method for R-value calculations of metal frame walls. ASHRAE Transactions 1 lO(2): 779-788. Mahapatra, S.K., S. Sen, and A. Sarkar. 1999. Interactio
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