Definition and Evaluation of Mean Beam Lengths for Applications in Multidimensional Radiative Heat Transfer: A Mathematically Self-Consistent Approach

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2 The concept of emission MBL is introduced based on the volume-surface exchange process characterized by the two GEFs, g 1 s 2,pp and g 2 s 2,pd, with the emitting volume V 1 and absorbing area A 2 as shown in Fig. 1. As an extension to Eq. 4, the emission MBLs for the two configurations are defined by g 1 s 2,xx D 2 = F 12,xx n x,n y,n z 1 e a 1 L em,xx e a m L t,xx, xx = pp,pd 6 where a 1 is the absorption coefficient of the emitting medium in volume V 1. The two emission MBLs L em,pp and L em,pd are functions of the two absorption coefficients a 1 and a m, as well as the geometrical configuration represented by n x, n y, n z. Finally, the concept of absorption MBL is introduced based on the volume-volume exchange GEFs g 1 g 2,pp and g 1 g 2,pd, with the emitting volume V 1 and absorbing volume V 2 as shown in Fig. 1. Generalizing the mathematical expression introduced by Eq. 6, the absorption MBLs for the two components are given by Fig. 1 Geometry of the emitting volume area and receiving volume element used in the definition of the two components of the GEF and the associated coordinate system Mathematically, the two surface-surface GEFs s 1 s 2,pp and s 1 s 2,pd correspond to the actual exchange factors between the two square surfaces A 1 and A 2. For the four volume-surface and volume-volume GEFs g 1 s 2,pp, g 1 g 2,pd, g 1 g 2,pp and g 1 s 2,pd, the exchange factors include only the portion of the radiation emitted by V 1 through the surface A 1. For the two volume-volume GEFs g 1 g 2,pp and g 1 g 2,pd, the exchange factor is further limited by including only the radiation received by the volume V 2 through the surface A 2. Actual volume-surface and volume-volume exchange factors are generated from the six GEFs by superposition. The detail of the superposition is presented in Ref. 9 and will not be repeated here. It should be noted that by restricting the emission/absorption through specific areas in the definition of the three basic GEFs, the MACZM approach can simulate accurately the variation of optical properties in an inhomogeneous medium, particularly in a region of large discontinuity in optical properties e.g. a continuously changing 3D solid/liquid/gas phase boundary. The six GEFs are the basis for the definition of six corresponding MBLs in the present work. The Concepts of Transmission, Absorption, and Emission Mean Beam Length Mathematically, the two surface-surface GEFs s 1 s 2,pp and s 1 s 2,pd, normalized by the area D 2, depend only on the optical thickness, and the relative orientation between the two areas as specified by n x, n y, n z. The optical thickness 12 is evaluated along the line of sight between the center-point of the two areas. The two transmission MBLs are defined by s 1 s 2,xx D 2 = F 12,xx n x,n y,n z e a m L t,xx, xx = pp,pd 4 In the above expressions, F 12,xx n x,n y,n z, are the view factors between the two areas for the two configurations. The average absorption coefficient between the two surfaces, a m,is a m = 12 5 L c with L c being the distance between the center of the two surfaces. g 1 g 2,xx D 2 = F 12,xx n x,n y,n z 1 e a 1 L em,xx 1 e a 2 L a,xx e a m L t,xx, xx = pp,pd 7 where a 2 is the absorption coefficient of the medium in volume V 2. The two absorption MBLs L a,pp and L a,pd are functions of the three absorption coefficients a 1, a 2, and a m, as well as the geometrical configuration represented by n x, n y, n z. It is important to note that, together, the three MBLs absorption, emission, and transmission MBLs in its two-component are a set of path lengths, which reduce the six GEFs to equivalent one-dimensional expressions Eqs. 4, 6, and 7. Since these path lengths are introduced to account for the three distinct physical processes, their mathematical behaviors are influenced most strongly by the absorption coefficient associated with the specific process. The geometric effect is largely accounted for by the view factor and, therefore, the dependence of the MBLs on n x, n y, n z is generally weak. The one-dimensional form of the GEFs can potentially serve as a basis for further development of additional engineering correlation to account for the nongray effect of multidimensional radiative heat transfer. The Transmission Mean Beam Length, L t,pp and L t,pd Numerical data show that for two disjoint surfaces, the two transmission MBLs L t,pp and L t,pd are generally independent of optical thickness and can be approximated accurately by the corresponding center-to-center distances L c,pp and L c,pp. One exception is that the pd component with n x,n y,n z = 0,0,1, when A 1 and A 2 are two perpendicular square surfaces with a common edge. For this case, the transmission MBL can be correlated by L t,pd 2 = L 12 n x,n y,n z = 0,0,1 8 c,pd The Emission Mean Beam Length L em,pp and L em,pd Physically, the behavior of the emission MBL for the case with the absorbing surface in contact with the emitting volume the pp component with n x,n y,n z = 0,0,0 is expected to differ from cases in which the absorbing area and the emitting volume are disjoint. Different correlations are thus needed for the two possibilities. Specifically, the GEF g 1 s 2,pp with n x,n y,n z = 0,0,0 corresponds to the actual exchange factor between a cubical volume and one of its bounding surface. The transmission MBL is zero. The emission MBL is tabulated and shown in Fig. 2 a. The conventional MBL, given by Eq. 2 with C=0.9, is shown in the same figure for comparison. The corresponding prediction for the exchange factor is shown in Fig. 2 b. The conventional MBL is clearly not an accurate approximation of the emission MBL. The / Vol. 130, NOVEMBER 2008 Transactions of the ASME

3 Fig. 2 Comparison of the emission MBL and the conventional MBL and the corresponding exchange factor for the g 1 g 2,pp exchange factor with n x,n y,n z = 0,0,0 discrepancy in the exchange factor prediction is large in the limit of an optically thick emitting volume i.e., large a 1 D. The numerical data presented in Fig. 2 a can be correlated by L em,pp 0,0,0 = a 1 D a 1 D 2 9 D For cases in which the emitting volume and the absorbing area are disjoint, the emission MBLs are influenced strongly by the two optical thicknesses a 1 D and 12 and geometry n x, n y, n z. Since algebraic correlations are difficult to obtain, a neural network is created to provide a simple mathematical correlation. The capability of using neural networks to correlate experimental or numerical data accurately is well documented 10. Specifically, using a two-layer neural network as shown in Fig. 3, neural network correlations are developed for the two emission MBLs. For L em,pp in ranges of 0 n x 7, 0 n y 7, 1 n z 7, 0.01 a 1 D 4.6, and , the numerical data can be correlated with the following equation: 18 5 tanh W 1,ij p 10 where p is a normalized input vector with components n x, n y, n z, a 1 D, and 12 and z=l em,pp /L c,pp is the normalized output. All normalized variables are generated by a linear transformation from its range of value to the common range of 1 to 1. Numerical values for the network elements Ŵ 1, b 1, W 2, and b 2 are available upon request to the author. The accuracy of the neural network prediction is illustrated for some typical data with n z =1 in Fig. 3 Schematic of a two-layer neural network used in generating the correlation for the emission MBL Fig. 4. The conventional MBL, given by Eq. 2 with C=0.9, is shown in the same figure to demonstrate the general inaccuracy of the conventional MBL concept. Because of the more complex mathematical behavior of L em,pd, two separate neural network correlations are generated to simulate the numerical data. For 0 n x 5, 1 n z 7, 0.01 a 1 D 4.6, and with n y =0, the data are correlated by the following equation: with 13 4 F W 1,ij p 11 1 F x = 1+e x 12 The normalized vector p is a vector with components n x, n z, a 1 D, and 12 Note that z=l em,pd /L c,pd is normalized to be within the range 0 1, which is the range of the log-sigmoid transfer function as represented by Eq. 12. For cases with 1 n y 6, a second network is generated to correlate the numerical data as follows: 21 5 F W 1,ij p 13 with p= n x,n y,n z,a 1 D, 12. Detailed numerical data of the two networks are available upon request to the author. The accuracies of the two neural network correlations are illustrated for some selected geometry with n z =1 in Fig. 5. The conventional MBL, given by Eq. 2 with C=0.9, is again shown in the same figure to demonstrate the inaccuracy of the conventional MBL concept. It should be noted that the numerical data and the associated neural network correlation for the emission MBLs L em,pp and L em,pd are generated only for a finite range of the input variables a 1 D, 12, n x, n y, and n z. Since the emitting volume is expected to approach a blackbody in the limit of a large emission optical thickness a 1 D and GEF is generally small approaches zero in the limit of a large distance between the emitting volume and the absorbing surface, the accuracy of the emission MBL is less critical in those limits. For the optically thick limit a 1 D 4.6 and/or , it is sufficient to use the corresponding emission MBL evaluated at a 1 D=4.6 and/or 12 =4.6. For a geometric configura- Journal of Heat Transfer NOVEMBER 2008, Vol. 130 /

4 Fig. 4 Comparison between the neural network prediction of the emission MBL, L em,pp,for different geometric configurations with n z =1 Fig. 5 Comparison between the neural network prediction of the emission MBL, L em,pd,for different geometric configurations with n z =1 tion with n x, n y, n z outside of the specified range, the normalized emission MBL can be taken to be the value evaluated at a configuration with the closest distance. Absorption Meam Beam Length, L a,pp and L a,pd Numerical data show that, in general, the absorption MBL is effectively independent of the optical properties of the emitting volume and can be taken to be the same as the emission MBL evaluated at the optical property of the absorbing volume. Due to the selection of the x-direction as the orientation of A 1 in the definition of the pd component of the GEF, the equivalence between the pd component of the absorption MBL and the mission MBL would require an exchange in the value of n x and n z as follows: L a,pp n x,n y,n z,a 1 D,a 2 D, 12 = L em,pp n x,n y,n z,a 2 D, 12 14a L a,pd n x,n y,n z,a 1 D,a 2 D, 12 = L em,pd n x 1,n y,n x +1,a 2 D, 12 14b Nomenclature a 1 absorption coefficient of the emitting volume V 1 a 2 absorption coefficient of the absorbing volume V 2 a absorption coefficient at wavelength a m average absorption coefficient of the medium between surfaces A 1 and A 2 A surface area C constant used in the definition of the conventional mean beam length, Eq. 2 D characteristic dimension used in the definition of GEF e b blackbody emissive power / Vol. 130, NOVEMBER 2008 Transactions of the ASME

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