INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 Effect of surface radiation on Conjugate Convection from an electronic board with multiple embedded discrete heat sources Gururaja Rao.C, Narasimha Suri.T Department of Mechanical Engineering, National Institute of Technology, [Deemed University], Warangal ­ 506 004 (A. P.) cgr_gcr@yahoo.co.in ABSTRACT The present paper reports the important results of the problem of combined conduction­ convection­radiation from a vertical rectangular electronic board equipped with three identical discrete heat sources. The heat sources are located along the vertical central axis of the board. The heat generated in the three heat sources is conducted both along and across the board, before subsequently getting dissipated by convection and radiation. Air, a radiatively transparent medium, is considered to be the cooling agent. The governing partial differential equations for temperature distribution in the entire computational domain are obtained by appropriate energy balance between the heat generated, conducted, convected and radiated. These equations are solved using finite difference formulation coupled with Gauss­Seidel iterative solver. A computer code is written to solve the problem and exhaustive parametric studies have been carried out. Interactive effect of radiation on conjugate convection has been elucidated. Keywords: Surface radiation, Conduction, Convection, Electronic Board, Discrete Heat Sources, Interaction. 1. Introduction Literature on multi­mode heat transfer contains several analytical, numerical and experimental works involving interplay between two or more than two modes of heat transfer, viz., conduction, convection and radiation. One such initial work is credited to Zinnes (1970), who presented his analytical and numerical results of laminar natural convection from a vertical, heat­conducting, flat plate of finite thickness with an arbitrary heating distribution along its surface. Subsequently, several other researchers came out with their findings on problems of this kind. Specifically, if one looks into the last two decades, Lee and Yovanovich (1989) proposed a quasi­analytical conjugate heat transfer model that addresses a two dimensional vertical flat plate provided with arbitrarily sized discrete heat sources dissipating heat by free convection. Tewari and Jaluria (1990) performed an experimental study on fundamental aspects of conjugate mixed convection from two heat sources of finite width and negligible thickness with a uniform heat flux input at the surface. A numerical­cum­experimental study of laminar mixed convection from a vertical composite plate provided with isolated, discontinuous, surface heating elements was performed by Kishinami et al. (1995). Vynnycky and Kimura (1996) solved the problem of two­ dimensional conjugate free convection from a vertical plate in communication with a semi­ infinite fluid region both by analytical and numerical means. Mendez and Trevino (2000) dealt with numerical solution to the problem of conjugate free convection from a thin vertical strip with non­uniform internal heat generation. Gururaja Rao et al. (2001) obtained
561 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 numerical solution to the problem of conjugate laminar mixed convection with surface radiation from a vertical plate with a flush­mounted discrete heat source. They solved the governing fluid flow and heat transfer equations without boundary layer approximations using a finite volume method. Gururaja Rao et al. (2002) also tackled the geometry of a vertical channel with a discrete heat source in each wall involved in conjugate mixed convection with radiation. Subsequently, Gururaja Rao (2004) considered a traversable discrete heat source along a vertical electronic board involved in buoyancy­aided conjugate mixed convection with radiation. Kanna and Das (2005) gave an analytical solution for conjugate forced convection from a flat plate involved in laminar flow. Gururaja Rao (2007) solved, numerically, the problem of combined conduction­convection­radiation in a vertical channel with multiple discrete heat sources in the left wall. Interplay between conduction, convection and radiation from the geometry of a rectangular cavity with a discrete heat source in its bottom wall has been explored by Gururaja Rao et al. (2008). Conjugate mixed convection with radiation from a vertical electronic board with multiple identical discrete heat sources has been attempted by Sawant and Gururaja Rao (2009). Further, the geometry of an L ­ corner with a discrete heat source in each of its legs and involved in multi­mode heat transfer has been explored by Gururaja Rao et al. (2009). Very recently, Sawant and Gururaja Rao (2010) solved the problem of conjugate mixed convection with surface radiation from a uniformly heated vertical plate. A review of literature points out that an exhaustive probe into the problem of interaction of radiation with conjugate convection from a vertical two­dimensional electronic board with multiple discrete heat sources has not been reported. On account of this, the present paper aims at a numerical study of the effect of radiation on combined conduction and convection from a two dimensional rectangular plate with three identical discrete heat sources. 2. Problem Definition and Mathematical Formulation Figure 1 shows the problem geometry considered in the presented study. It consists of a rectangular slab shaped electronic board of dimensions L×W. There are three identical embedded discrete heat sources of dimensions Lh×Lh in this slab. The heat sources are provided along the vertical central axis of the slab, as shown. There is one heat source at the geometric center of the plate, while the other two heat sources are symmetrically located on either side of it. The bottom most heat source and top most heat source are at a distance L/8 from the corresponding horizontal boundaries of the rectangular slab. There is a uniform volumetric heat generation at the rate qv W/m 3 in each heat source. The thermal conductivity and surface emissivity of the rectangular slab are, respectively, k and ε. The heat generated in the three heat sources is initially conducted along and across the electronic board, before getting dissipated from the boundaries of the device by combined convection and radiation. The cooling medium is air at temperature T∞, while the convection heat transfer coefficient is h W/m 2 K. The governing equations for temperature distribution in the entire computational domain are obtained by making energy balance between the heat generated, conducted, convected and radiated. For example, the governing equation for all the interior nodes of each of the three discrete heat sources, upon energy balance, turns out to be:
562 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 ¶ 2 T ¶ 2 T q v +
+
= 0 k ¶x 2 ¶y 2
(1) Air [h, T∞] Lh ε k T(x, y)
ε qv
Air [h, T∞] Air [h, T∞] L qv
Lh ε q v x, i y, j W ε Air [h, T∞] Figure 1: Schematic of the problem geometry chosen for study. With regard to interior nodes in the non­heat source portions, the governing equation would be Eq. (1) with source term absent. For a typical interface element between the top boundary of the central heat source and the non­heat source portion, the energy balance turns out to be: ΔxΔy (2) q
+ q + q v = q + q x, cond, in y, cond, in x, cond, out y, cond, out 4 Assuming that Dx = Dy, and substituting for various terms in the above equation and simplifying, one gets: ¶ 2T ¶ 2 T q v +
+
= 0 (3) ¶x 2 ¶ y 2 2 k
563 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 For a typical corner element of any of the heat sources, the governing equation for temperature distribution is same as Eq. (3) with the third term on the left hand side replaced by (q v/4k). With regard to the boundaries of the board, making energy balance on an arbitrarily chosen element along the left boundary: q = q + q +q
+ q x, cond, in x, cond, out y, cond, out conv rad (4) Substitution of appropriate expressions for various terms in the above equation and subsequent simplification leads to the following governing equation for temperature distribution along the left boundary of the board: ¶ 2 T
2 ¶T q v 2h 2 es
(5) +
+ (T - T¥ ) (T 4 - T ¥ 4 ) = 0 2 ¶x
Dy2 ¶y k k Dy2
k D y2 With regard to the two corner elements of the left boundary, the governing equation for temperature distribution would get modified as: ¶T Dx2 ¶T h Dx
es
Dx + (1 + 2 )(T - T¥ ) +
(1 + 2 )(T 4 - T ¥ 4 ) = 0 ¶x Dy2 ¶y k
Dy2
k
D y2 (6) Similar treatment is used for obtaining the temperature of the remaining boundaries as well. 3. Solution Methodology and Ranges of Parameters The governing equations for temperature distribution in the entire computational domain derived as above are non­linear partial differential equations. They are converted into algebraic equations using finite difference formulation. The resulting equations are later solved using Gauss­Seidel iterative solver. Full relaxation has been used on temperature during iterations, while a convergent criterion of 10 ­6 has been imposed to terminate the iterations. A computer code in C ++ has been written to solve the problem. With regard to ranges of independent parameters concerning the present problem, the surface emissivity of the electronic board (ε) is varied between 0.05 and 0.85. The lower limit of ε corresponds to a poor emitter like polished aluminum, while the upper limit of ε signifies a good emitter like black paint. The thermal conductivity (k) of the electronic board is varied between 0.25 W/m K and 1 W/m K. This is done so in view of the fact that electronic devices are typically made of materials of thermal conductivity of the order of unity. An example is glass epoxy that has k = 0.26 W/m K. For convection heat transfer coefficient (h), the appropriate range is from 5 W/m 2 K to 100 W/m 2 K. The above limiting values of h, respectively, correspond to asymptotic free convection limit and asymptotic forced convection limit. Further, the height (L) and width (W) of the electronic board are fixed at 20 cm and 10 cm, respectively. The mean temperature (T∞) of the cooling agent (air) is taken to be 25 o C.
564 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 4. Results and Discussion 4.1. Grid independence study The grid sensitivity text is carried out in two phases for a standard input: qv = 10 5 W/m 3 , k = 0.25 W/m K, h = 25 W/m 2 K and ε = 0.45. In Phase – 1 of the study, the number of grids (N) in the horizontal direction is arbitrarily fixed at 113. The number of grids in the vertical direction (M) is varied. The maximum board temperature (Tmax) is found to be changing by 8.37% as M is increased from 171 to 191. A further increase in M brings down the percentage change in Tmax. It is noticed that Tmax changes by 5.09% as M increases from 231 to 251. A subsequent slight increase in M from 251 to 255 brings down the change in Tmax to just 0.92%. Simultaneous to these observations on T max, even the energy balance is checked and the best possible energy balance check occurs at M = 259. In view of these observations, the value of M is finalized to be 259. In Phase – 2 of the study, the value of M is taken to be 259 as finalized above, while the number of grids in horizontal direction (N) is varied. It is seen that though the values of N ³ 111 give a converging Tmax, the check for energy balance is at its best for N = 113. In view of this, the value of N is finalized to be 113. In both the phases above, the number of grids in each heat source is taken as 30×30. It is noticed that any other grid size in the heat sources does not give better result than the above. Thus, all the ensuing parametric studies are carried out taking M = 259, N = 113 and number of grids per heat source = 30×30. 4.2 Variation of local temperature in the electronic board with other parameters Figure 2 shows the local temperature distribution along the axial direction at mid­plane drawn for five different values of convection heat transfer coefficient (h). The data is obtained for a fixed input of q v = 10 5 W/m 3 , ε = 0.45 and k = 0.25 W/m K. For a given h, the temperature is increasing as one moves from the bottom of the board, reaching a local maximum a little head of top end of the first heat source. It is decreasing again, reaching a local minimum before rising again as one reaches the second heat source. The temperature now reaches a second maximum exactly at the geometric centre of the board before decreasing again. After reaching a second local minimum, the temperature is shooting up again to a third maximum a little ahead of the top end of the third heat source and is decreasing again. There is an expected symmetry in the local temperature profile with reference to the geometric center of the board. Further, the local maximum noticed at the geometric center of the board is the largest temperature in the board, while the other two local maxima on either side of the above are identical but smaller than the central temperature. The figure further shows that the temperature at any location along the board decreases as h increases from 5 to 100 W/m 2 K. This is expected because the flow regime transits from free convection dominance to forced convection dominance with h increasing between the above two limits. In the present example, the local temperature at the geometric center of the board decreases by 24.37% as h increases from 5 to 100 W/m 2 K.
565 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association Local temperature distribution along the axial direction, T(x), o C RESEARCH ARTICLE ISSN ­ 0976­4259 2 1 3 5 1. h = 5 W/m 2 K 4 2. h = 10 W/m 2 K 3. h = 25 W/m 2 K 4. h = 50 W/m 2 K qv = 10 5 W/m 3 , ε = 0.45, k = 0.25 W/m K 5. h = 100 W/m 2 K Axial distance along the board, x, m Local temperature distribution along the axial direction, T(x), o C Figure 2: Local board temperature profiles in various regimes of convection. 1 3 1.ε = 0.05 2.ε = 0.45 3.ε = 0.85 2 q v = 10 5 W/m 3 , h = 5 W/m 2 K, k = 0.25 W/m K
Axial distance along the board, x, m Figure 3: Local temperature profiles for various surface emissivities of the board. The effect of surface emissivity (ε) on local axial temperature distribution at mid­plane along the electronic board is as shown in Fig. 3. The trend followed by the temperature profile for a given ε looks similar to what has been noticed in Fig. 2. Further, at a given location, the temperature decreases with increasing ε owing to increased radiative heat dissipation from the board with other parameters held fixed. The figure further shows a comparatively larger 566 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 drop in local temperature between ε = 0.05 and ε = 0.45, while the drop between ε = 0.45 and ε = 0.85 is less pronounced. In the present example, the temperature at the geometric center of the board decreases by 11.93% as ε increases from 0.05 to 0.45, while the drop in temperature is only by 6.23% as ε subsequently increases from 0.45 to 0.85. Figure 4 describes the effect of thermal conductivity (k) on local temperature distribution along the axial direction at mid­plane of the electronic board. Six different values of thermal conductivity (k) are chosen as shown. As can be seen, the general trend of the temperature profile is again the same as that seen in Figs. 2 and 3. The local temperature in the board comes down with increasing k on account of increased percolation of heat along the board. In the present example, the temperature at the geometric center of the board decreases by 81.3% as k increases from 0.25 W/m K to 10 W/m K. However, the temperature profile gets progressively flatter with increasing thermal conductivity of the board. This means, towards larger values of k, there is hardly any temperature gradient noticed along the board and the board almost appears to be isothermal. In order to bring out the contrast between the local axial temperature profiles pertaining to cases, viz., (1) all three heat sources present, (2) bottom heat source only present, (3) middle heat source only present and (4) top heat source only present, Fig.5 has been drawn. As can be noticed, the curve ­ 1 resembles what has been obtained in the last three figures (Figs. 2­4). This is because here all the three heat sources are present. Curve ­ 2 corresponds to the case that has only the bottommost heat source in the board. Thus, the local temperature here, after reaching peak somewhere near the top end of the heat source, diminishes continuously and almost becomes asymptotic towards the trailing edge of the board. Curve ­ 3 belongs to the case where the board has only the central heat source. Thus, the temperature is maximum at the geometric center of the board and decreases symmetrically in both the directions along the vertical axis from here. This is attributed to identical rate of percolation of heat on either side of the heat source. Curve ­ 4 pertains to the case that has only the topmost heat source. Therefore, the temperature profile appears to be an exact mirror image of curve ­ 2. It can also be noticed that the maximum board temperature is the largest in case ­ 1 and the least in case ­ 3. This is expected because in case ­ 1 all the three heat sources are there. Though in cases 2 to 4 only one heat source is there in the board, cases 2 and 4 have the heat source either towards the bottom end or towards the top end of the board. In case – 3, the heat source is centrally located. In the present example, the values of Tmax in cases 1, 2, 3 and 4 are, respectively, 159.5 o C, 120.95 o C, 115.15 o C and 120.95 o C. 4.3 Variation of maximum board temperature with other parameters Figure 6 describes the variation of maximum board temperature (Tmax) with surface emissivity (ε) for various values of convection heat transfer coefficient (h). Results are obtained for five typical values of h starting from the asymptotic free convection limit of 5 W/m 2 K to the asymptotic forced convection limit of 100 W/m 2 K. There is a clear effect of ε noticed on Tmax for h = 5 W/m 2 K. Here, Tmax is decreasing very largely from ε = 0.05 to ε = 0.85.
567 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association Local temperature distribution along the axial direction, T(x), o C RESEARCH ARTICLE ISSN ­ 0976­4259 1. k = 0.25 W/m K , 2. k = 0.5 W/m K, 3. k = 0.75 W/m K qv = 10 5 W/m 3 4. k = 1 W/m K, 5. k = 5 W/m K, 6. k = 10 W/m K ε = 0.05 h = 5 W/m 2 K
1 2 3 5 4 6 Axial distance along the board, x, m Local temperature distribution along the axial direction, T(x), o C Figure 4: Local temperature profiles for various thermal conductivities of the board. 1. Three heat sources, 2. Bottom heat source 3. Middle heat source, 4. Top heat source qv = 10 5 W/m 3, ε = 0.45 h = 5 W/m 2 K k = 0.25 W/m K 1 2 3 4 Axial distance along the board, x, m Figure 5: Comparison of local temperature profiles with various heat source configurations. Though similar kind of decrement in Tmax with ε is noticed for all other values of h too, the degree of decrement is found to be coming down as one moves towards forced convection limit of h. This is attributed to increasing dominance of convection over radiation for larger values of h. The figure also shows that, for a given surface emissivity, the peak board temperature decreases with increasing h on account of reasons already explained. In the present example, Tmax is decreasing by 17.41% as ε increases from 0.05 to 0.85, while it decreases only by 0.15% between the same limits of ε for h = 100 W/m 2 K. Likewise, for ε = 568 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 0.05, Tmax is decreasing by a huge 33.34% as h increases from 5 to 100 W/m 2 K, while the decrease in T max is just by 19.4% for ε = 0.85 between the same limits of h. Maximum temperature, Tmax, o C The variation of maximum board temperature (Tmax) with convection heat transfer coefficient (h) for different thermal conductivities (k) of the board is shown in Fig. 7. Seven values of h and three values of k are considered for study as depicted in the figure. For a given k, Tmax decreases sharply as h increases from 5 W/m 2 K and 25 W/m 2 K, while the decrease in T max is less pronounced towards larger values of h. This implies that the optimum value of h lies somewhere around 25 W/m 2 K in the present problem. The figure further shows that Tmax, for a given h, decreases with increase in k. Here again the decrease is quite large between k = 0.25 and 0.5 W/m K, while a subsequent increase in k from 0.5 to 1 W/m K brings a lesser pronounced drop in Tmax. In the present example, for k = 0.25 W/m K, Tmax decreases by 27.4% as h increases from 5 to 25 W/m 2 K, while Tmax is decreasing by just 8.18% due to subsequent increase of h to 100 W/m 2 K. Likewise, for h = 5 W/m 2 K, for example, Tmax is coming down by 27.48% as k increases from 0.25 to 0.5 W/m K. In contrast, the decrease in Tmax is lesser between k = 0.5 and 1 W/m K and is by 21.04%. q v = 10 5 W/m 3 k = 0.25 W/m K 1. h = 5 W/m 2 K, 2. h = 10 W/m 2 K 3. h = 20 W/m 2 K, 4. h = 25 W/m 2 K 5. h = 100 W/m 2 K 1 2 3 4 5 0.05 0.25 0.45 0.65 0.85 Emissivity, ε Figure 6: Variation of maximum board temperature with emissivity in various convection regimes.
569 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 Maximum temperature, Tmax, o C 1. k = 0.25 W/m K 2. k = 0.5 W/m K 3. k = 1 W/m K qv = 10 5 W/m 3 , ε = 0.05 1 2 3 100 50 75 Convection heat transfer coefficient, h, W/m 2 K Figure 7: Variation of maximum board temperature with convection heat transfer coefficient for different thermal conductivities of the board. 5 10 20 25
Maximum temperature, Tmax, o C Variation of maximum board temperature (Tmax) with surface emissivity (ε) is probed for three different thermal conductivities of the board as shown in Fig. 8. There is a huge drop in Tmax, for a given ε, as k increases from 0.25 to 0.5 W/m K. However, for a subsequent increase in k to 1 W/m K, the drop in Tmax is not that pronounced. In this example, Tmax decreases by 30.29% as k increases from 0.25 to 0.5 W/m K for ε = 0.45. Against this, T max further drops only by 23.61% as k increases to 1 W/m K from 0.5 W/m K for the same ε. The figure also shows an expected progressive decrease in Tmax, for a given k, as ε increases from 0.05 to 0.85. There is more or less an identical drop in Tmax between the two limiting values of ε for all the three values of k chosen. 1. k = 0.25 W/m K, 2. k = 0.5 W/m K 3. k = 1 W/m K 1 5 3 2 q v = 10 W/m , h = 5 W/m K 2 3 0.0 5 0.6 0.4 0.8 5 5 5 Surface emissivity, ε Figure 8: Maximum board temperature with emissivity for different thermal conductivities. 0.2 5 570 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 4.4 Relative contributions of convection and radiation in heat dissipation Relative contributions of convection and radiation, (% age)
Figure 9 shows the nature of variation of the relative contributions of convection and radiation in heat dissipation from the board with emissivity (ε) for three typical values of convection heat transfer coefficient. For a given h, the percentage convective dissipation is progressively decreasing with increasing ε. The above is accompanied by a corresponding progressive increase in the percentage radiative dissipation. This kind of variation, though noticed with all values of h, is less pronounced for larger values of h. This is because a smaller value of h implies free convection, which naturally results in smaller convective heat dissipation. Thus, increasing ε here would obviously enhance the rate of radiation heat transfer by a larger extent. In the present example, for h = 5 W/m 2 K, an increase in ε from 0.05 to 0.85 is increasing the relative contribution of radiation from 6.71% to 52.2%. As can be seen, the curves belonging to convective and radiative dissipation for this particular value of h are crossing each other for ε @ 0.7745. Further, for a given ε, the role of convection is increasing with a corresponding decrease in the role of radiation as one transits from free convection to forced convection dominant regime, i.e., smaller h to larger h. 3, a 2, a q v = 10 5 W/m 3 , k = 0.25 W/m K 1, a 1. h = 5 W/m 2 K a. Convection 2. h = 10 W/m 2 K b. Radiation 3. h = 25 W/m 2 K 1, b 2, b 3, b 0.05 0.25 0.85 0.45 0.65 Surface emissivity, ε Figure 9: Relative contributions of convection and radiation with emissivity in various convection regimes. 5. Concluding Remarks The problem of combined conduction­convection­radiation from a vertical rectangular electronic board possessing three identical embedded discrete heat sources has been solved numerically. A computer code making use of finite difference method of solution in conjunction with Gauss­Seidel solver is written for the purpose. A grid sensitivity test has been carried out and the best possible grid system that gives the optimum solution vis­à­vis peak board temperature as well as energy balance has been evolved. A detailed probe into local temperature distribution in the board and maximum board temperature is made with 571 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 reference to pertinent independent parameters, viz., number of heat sources, thermal conductivity, surface emissivity and convection heat transfer coefficient. Further, the relative roles played by convection and radiation in carrying the mandated heat load have been studied with regard to surface emissivity of the board in different regimes of convection. 6. References 1. Gururaja Rao, C., Balaji, C., and Venkateshan, S. P., 2001, “Conjugate mixed convection with surface radiation from a vertical plate with a discrete heat source”, ASME Journal of Heat Transfer, 123, pp 698­702. 2. Gururaja Rao, C., Balaji, C., and Venkateshan, S. P., 2002, “Effect of surface radiation on conjugate mixed convection in a vertical channel with a discrete heat source in each wall”, International Journal of Heat and Mass Transfer, 45, pp 3331­3347. 3. Gururaja Rao, C., 2004, “Buoyancy­aided mixed convection with conduction and surface radiation from a vertical electronic board with a traversable discrete heat source”, Numerical Heat Transfer, Part A: Applications, 45, pp 935­956. 4. Gururaja Rao, C., 2007, “Interaction of surface radiation with conduction and convection from a vertical channel with multiple discrete heat sources in the left wall”, Numerical Heat Transfer, Part A: Applications, 52, pp 831­848. 5. Gururaja Rao, C., Nagabhushana Rao, V., and Krishna Das, C., 2008, “Simulation studies on multi mode heat transfer from an open cavity with a flush mounted discrete heat source”, Heat and Mass Transfer, 44, pp 727­737. 6. Gururaja Rao, C., Santhosh, D., and Vijay Chandra, P., 2009, “Multi­mode heat transfer studies on L ­ corner with multiple discrete heat sources”, Heat and Mass Transfer, 45, pp 1293­1302. 7. Kishinami, K., Saito, H., and Suzuki, J., 1995, “Combined forced and free laminar convective heat transfer from a vertical plate with coupling of discontinuous surface heating”, International Journal of Numerical Methods for Heat and Fluid flow, 5, pp 839­851. 8. Kanna, P. R., and Das, M. K., 2005, “Conjugate forced convection heat transfer from a flat plate by laminar plane wall jet flow”, International Journal of Heat and Mass Transfer, 48, pp 2896­2910. 9. Lee, S., and Yovanovich, M. M., 1989, “Conjugate heat transfer from a vertical plate with discrete heat sources under natural convection”, ASME Journal of Electronic Packaging, 111, pp 261­267. 10. Mendez, F., and Trevino, C., 2000, “The conjugate conduction­ natural convection heat transfer along a thin vertical plate with non­ uniform internal heat generation”, International Journal of Heat and Mass Transfer, 43, pp 2739­2748.
572 INTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 1, No 3, 2010 © Copyright 2010 All rights reserved Integrated Publishing Association RESEARCH ARTICLE ISSN ­ 0976­4259 11. Sawant, S. M., and Gururaja Rao, C., 2009, “Fluid flow and heat transfer studies and correlations for mixed convection with conduction and radiation from a discretely heated vertical plate”, International Journal of Fluid Mechanics Research, 36, pp 255­271. 12. Sawant, S. M., and Gururaja Rao, C., 2010, “Combined conduction­mixed convection­surface radiation from a uniformly heated vertical plate”, Chemical Engineering Communications, 197, pp 881­899. 13. Tewari, S. S., and Jaluria, Y., 1990, “Mixed convection heat transfer from thermal sources mounted on horizontal and vertical sources”, ASME Journal of Heat Transfer, 112, pp 975­987. 14. Vynnycky, M., and Kimura, S., 1996, “Conjugate free convection due to a heated vertical plate”, International Journal of Heat and Mass Transfer, 39, pp 1067­1080. 15. Zinnes, A. E, 1970, “The coupling of conduction with laminar natural convection from a vertical flat plate with arbitrary surface heating”, ASME Journal of Heat Transfer, 92, pp 528­534.
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