Heat transer and fluid flow charaectertics of vertical symmetrical

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Heat transer and fluid flow charaectertics of vertical symmetrical

  1. 1. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME271HEAT TRANSFER AND FLUID FLOW CHARACTERISTICS OFVERTICAL SYMMETRICAL TRIANGULAR FIN ARRAYSN.G.Narve1, N.K.Sane2LNBCIE & T, Satara, Kolhapur,IndiaJSCOE, Hadapsar, Pune,IndiaABSTRACTThis paper deals with study of heat transfer and fluid flow characteristics ofnatural convection heat flow through vertical symmetrical triangular fin arrays. It wasstudied numerically and its results were compared with equivalent rectangular fin arrays.In the numerical arrangement, spacing between fins was varied. Results were generatedfor S+= 0.5 & 0.105 and GrH =106to 108. Average, base Nusselt number and Grashofnumber were calculated. It was observed that with increase in Grashof number, averageand base Nusselt number increases. Similarly average Nusselt number increases withspacing whereas base Nusselt number increases to maximum value with spacing and thendecreases [11].Keywords: Fin arrays, Grashof number, Heat transfer, Natural convection, Spacing.I INTRODUCTIONMany proposed applications of electronic and thermo electric devices depend uponthe feasibility of rejecting waste heat by economical, trouble free methods. For theseapplications, better utilization of the available heat rejection area may be realized by theproper application of outstanding fins.Fins are extended surfaces used to improve the overall heat transfer rate when it islimited by low rate between a solid surface and surrounding fluids. Fins provide largersurface area for heat dissipation. Fins are casted or fabricated by pressing, soldering orwelding. Fins find application in variety of fields of which some are-INTERNATIONAL JOURNAL OF ADVANCED RESEARCH INENGINEERING AND TECHNOLOGY (IJARET)ISSN 0976 - 6480 (Print)ISSN 0976 - 6499 (Online)Volume 4, Issue 2 March – April 2013, pp. 271-281© IAEME: www.iaeme.com/ijaret.aspJournal Impact Factor (2013): 5.8376 (Calculated by GISI)www.jifactor.comIJARET© I A E M E
  2. 2. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME272(i)The heads and cylinders of the air cooled engines and compressors(ii)Electronic components such as power diodes, transistors etc.(iii)Tubes of various heat exchangers for example condenser tubes of domesticrefrigerators, radiators of automobiles.(iv)Outside surfaces of the cooling and dehumidifying coils in the air conditioningsystems.(v)Direct energy conversion devices.(vi)Nuclear fuel modules.(vii)Chemical and Cryogenic equipments.(viii)Conventional furnaces and gas turbineThere are different types and shapes of fins used in practice. Fins are used onplane surfaces or cylindrical surfaces. Fins may be of having different cross sections.Depending on cross section we may have rectangular, parabolic or triangular fins.The heat can be removed effectively if the fluid flow and the resulting flow patternare capable of removing the heat efficiently. The heat dissipation from fins under naturalconvection condition depends on the geometry and orientation of finned surface.The literature survey revealed that the problem of free convection heat transferfrom vertical fin arrays has been investigated by a few investigators. Elenbass [1] haddone extensive work on channels and parallel plates on experimental and semi-empiricalbasis. Starnner and Mcmanus [2] presented free convection data for four rectangulararrays in three positions including vertical position for the fin base. Similarlyexperimental work for vertical fin arrays are carried out[3-8].Theoretical work onrectangular fin arrays for natural convection is also reported[9,10].In the present work, the free convection heat transfer from isothermal verticalsymmetric triangular fin arrays was analyzed theoretically. It is proved by manyinvestigators that free convection heat transfer from vertical triangular fin array results inthe single chimney flow pattern.The object of this theoretical study is to determine the local, average and baseNusselt number for free convective heat transfer from vertical triangular fin arrays alongwith stream function and velocity distribution. In order to achieve this objective the set ofdifferential equations governing the fluid flow and heat transfer are to be used. These arederived from fundamental laws. Then its comparison was done with equivalentrectangular fins arrays. In both cases spacing was the variable.II FORMULATION OF THE PROBLEM2.1 Statement of the problemThe vertical symmetric triangular fin array to be analyzed is shown in Fig.1.Itconsists of large number of vertical triangular fins of height ‘H’ and length ‘L’. Thespacing between two adjacent fin flats is ‘S’. Each array has number of fin channels. The
  3. 3. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, Marchassumption of the entire fin array to be isothermal is supported by fin material havinghigh thermal conductivity, so that the fin flats and fin base are at same temperature Twhere Tb is the temperature of the base. The fin is surroundedtemperature and all fluid properties are considered as constant. Due to large number of finchannels, the end effects can be neglected.2.2 Domain of InterestFrom Fig.1, it is clear that symmetry is in the Z direction. Due to large number ofchannels, only a single fin channel needs to bethe vertical symmetric triangular fin channel ABCDEFinterest. It is surrounded by solid wall viz. fin flat and fin base and planes of opening i.eat the top, bottom and side of channel.Figure 1 Domain2.3 Governing equationsFollowing are the fundamental equations used for analyzing and solving theproblem.Continuity equation-International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766499(Online) Volume 4, Issue 2, March – April (2013), © IAEME273mption of the entire fin array to be isothermal is supported by fin material havinghigh thermal conductivity, so that the fin flats and fin base are at same temperature Tis the temperature of the base. The fin is surrounded by fluidtemperature and all fluid properties are considered as constant. Due to large number of finchannels, the end effects can be neglected.it is clear that symmetry is in the Z direction. Due to large number ofonly a single fin channel needs to be analyzed and due to symmetry, only halfthe vertical symmetric triangular fin channel ABCDEF is chosen and is the domain ofby solid wall viz. fin flat and fin base and planes of opening i.eat the top, bottom and side of channel.Domain-Half Symmetrical Triangular fin ductFollowing are the fundamental equations used for analyzing and solving the∂u ∂v ∂w∂x ∂y ∂zInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –April (2013), © IAEMEmption of the entire fin array to be isothermal is supported by fin material havinghigh thermal conductivity, so that the fin flats and fin base are at same temperature Tb,by fluid at ambienttemperature and all fluid properties are considered as constant. Due to large number of finit is clear that symmetry is in the Z direction. Due to large number ofue to symmetry, only halfthe domain ofby solid wall viz. fin flat and fin base and planes of opening i.e.Following are the fundamental equations used for analyzing and solving the-------[1]
  4. 4. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME274Momentum equation‫ݑ‬ப୳ப୶൅ ‫ݒ‬ப୳డ௬൅ ‫ݓ‬ப୳డ௭ൌ െଵఘడ௣డ௫൅ ‫ݒ‬ ቀப²୳ப୶²൅ப²୳ப୷²൅ப²୳ப୸²ቁ ൅ gβሺT െ Taሻ -----[2]‫ݑ‬ப୴ப୶൅ ‫ݒ‬ப୴డ௬൅ ‫ݓ‬ப୴డ௭ൌ െଵఘడ௣డ௬൅ ‫ݒ‬ ቀப²୴ப୶²൅ப²୴ப୷²൅ப²୴ப୸²ቁ -----[3]‫ݑ‬ப୵ப୶൅ ‫ݒ‬ப୵డ௬൅ ‫ݓ‬ப୵డ௭ൌ െଵఘడ௣డ௭൅ ‫ݒ‬ ቀப²୵ப୶²൅ப²୵ப୷²൅ப²୵ப୸²ቁ -----[4]Energy Equation‫ݑ‬ப୘ப୶൅ ‫ݒ‬ப୘డ௬൅ ‫ݓ‬ப୘డ௭ൌ௞ఘ஼௣ቀப²୘ப୶²൅ப²୘ப୷²൅ப²୘ப୸²ቁ ------[5]Substituting w=0, following equations are simplified,ப୳ப୶൅డ௨డ௬ൌ 0 ----[6]‫ݑ‬ப୳ப୶൅ ‫ݒ‬ப୳డ௬ൌ െଵఘడ௣డ௫൅ ‫ݒ‬ ቀப²୳ப୶²൅ப²୳ப୷²൅ப²୳ப୸²ቁ ൅ gβሺT െ Taሻ ----[7]‫ݑ‬ப୴ப୶൅ ‫ݒ‬ப୴డ௬ൌ െଵఘడ௣డ௬൅ ‫ݒ‬ ቀப²୴ப୶²൅ப²୴ப୷²൅ப²୴ப୸²ቁ -------[8]ப୮ப୸ൌ 0 -------[9]Pressure term is eliminated in above equation by cross differentiation and subtracting(7) from equation (8) and equations are non-dimensionlised by using characteristicdimension ‘H’X ൌ‫ݔ‬‫ܪ‬, ܻ ൌ‫ݕ‬‫ܪ‬, ܼ ൌ‫ݖ‬‫ܪ‬ߠ ൌܶ െ ܾܶܽܶ െ ܶܽ, U ൌuHԂ, V ൌvHԂ, Pr ൌCpµkDimensionless vorticity is defined by -ܷ ൌ߲߲߰‫ݒ‬, V ൌ െ߲߲߰‫ݔ‬ܽ݊݀ ߱ ൌ߲ܸ߲‫ݔ‬െ߲ܷ߲‫ݕ‬The basic governing equation can be represented in the following way-Energy equation:-ܷபθபଡ଼൅ ܸபθడ௒ൌ െଵ௉௥ቀப²ԕப୶²൅ப²ԕப୷²൅ப²ԕப୸²ቁ -----[10]Vorticity Transport Equation Equation
  5. 5. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME275ܷபωபଡ଼൅ ܸபωడ௒ൌப²ωப୶²൅ப²ωப୷²൅ப²ωப୸²– GrHபθడ௒------[11]Stream Function Equationെ߱ ൌப²ψப୶²൅ப²ψப୷²------[12]Above set of (10) to (12) governing equations are partial differential equationand their simultaneous analytical solution is not possible. So finite difference technique isused to solve these equations. The second order terms are replaced by central differenceswhile non-linear convective terms are replaced by upwind difference procedure. These formthree algebraic equations with three unknown at each nodal point in the grids2.4 Fixing of Boundary ConditionsThe array is having opening from the top, bottom and side, therefore no definiteboundary conditions can be assumed at these open surfaces. Therefore an attempt has beenmade to accommodate these open boundaries by extending them, a certain distance awayfrom channel. At these extended top, bottom and side boundary surfaces, ambient conditionscan be assumed. This approach has been made previously many investigators.In the bottom region, below the channel, incoming flow of air is assumed at ambienttemperatures. Above the temperature at the side entrance of channel has been assumed to beequal to ambient value.2.5 Calculation of local and average Nusselt numberThe expression for local Nusslet number can be obtained as follows:-The heat transfer coefficient at the fin base surface is given byhb = k.பθడ௒at Y=0The local Nusselt number for the fin base is definedNub = hb .H/k = H.பθడ௒at Y=0Similarly the local Nusslet number for the fin flat isNu = -பθడ௭at Z=0The temperature gradientபθడ௒at Y=0 and -பθడ௭at Z=0 are obtained by usinga five point numerical differentiation based on Taylor’s expansion series[8]. Then averageNusselt number for the entire fin array is obtained by numerically integrating the localNusselt number over the surface of the fin array.III RESULTS AND DISCUSSIONSThe numerical analysis using computational technique is applied to symmetricaltriangular and equivalent rectangular fine arrays. The solutions are obtained for differentvalues of dimensionless parameters of S+ and GrH for constant L+.The results are generatedfor following values of parameters-L+ - 0.5, and S+ - 0.105, 0.5GrH -106to 108and Pr- 0.7
  6. 6. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, MarchThe results are obtained inwithin the domain of interest.characteristic The heat transfer rate for the fin arrays is effectively studied in average andbase Nusselt number.3.1 FLUID FLOW CHARACTERFluid flow characteristicscontours and the movement of air in X and Y directi3.1.1 Stream FunctionStream lines describe the actual flow of the air inside the entireclosed contours satisfying the continuity equation. The stream lines shows the flow linesentering the fin channel from thecomponent of velocity as they approach in the base surface and then resulting in the outgoingmain flow from the top of the channelFig. 2 to Fig.5 shows the stream line contours in the entire fin domain at the vertsection of (z=50) for GrH at 106It is observed that GrH has a strong influence on the nature of flow in the domain ofinterest. Overall observation is that the flow lines tend to concentrate at the heated edges. Atlow Grashof number that is at 10towards vertex of fin flat in case of triangular fin arrays. At these Grashof number,recirculation is formed above the fin array. This negative loop is very weak in the rectangulfin arrays or almost absent. This is due to higher buoyancy towards the vertical base. In thiscase stream line due not tend to move away from the vertical base but are almost parallel to itgiving more uniform flow.It is observed that values of stretriangular fins than rectangular fins. This seems to be one of theeffectiveness of this geometry. It is also seen that when fins spacing is increased the negativeloop or recirculation weakens or becomes completelyFigure 2 Stream line contours for Figure 3 Stream line contours forTriangular fin array Rectangular fin arrayInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766499(Online) Volume 4, Issue 2, March – April (2013), © IAEME276he results are obtained in terms of distribution of stream function, U and V velocitiesThese gives the nature of fluid flow and heat transfercharacteristic The heat transfer rate for the fin arrays is effectively studied in average andFLUID FLOW CHARACTERISTICSFluid flow characteristics describe the flow pattern of air in terms of stream linecontours and the movement of air in X and Y direction expressed in U and V velocitydescribe the actual flow of the air inside the entire domain. Theysatisfying the continuity equation. The stream lines shows the flow linesentering the fin channel from the bottom and side of the array developing verticalof velocity as they approach in the base surface and then resulting in the outgoingtop of the channelshows the stream line contours in the entire fin domain at the vertto 107for symmetrical triangular and rectangular fin arrays.has a strong influence on the nature of flow in the domain ofobservation is that the flow lines tend to concentrate at the heated edges. At106to 107stream lines moves away from the vertical baseof fin flat in case of triangular fin arrays. At these Grashof number,recirculation is formed above the fin array. This negative loop is very weak in the rectangulfin arrays or almost absent. This is due to higher buoyancy towards the vertical base. In thism line due not tend to move away from the vertical base but are almost parallel to itvalues of stream functions are always higher for symmetricaltriangular fins than rectangular fins. This seems to be one of the reasons for the bettereffectiveness of this geometry. It is also seen that when fins spacing is increased the negativeeakens or becomes completely absent.Figure 2 Stream line contours for Figure 3 Stream line contours forTriangular fin array Rectangular fin arrayInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –April (2013), © IAEMEand V velocitiesand heat transfercharacteristic The heat transfer rate for the fin arrays is effectively studied in average andflow pattern of air in terms of stream lineon expressed in U and V velocity.domain. They formsatisfying the continuity equation. The stream lines shows the flow linesdeveloping verticalof velocity as they approach in the base surface and then resulting in the outgoingshows the stream line contours in the entire fin domain at the verticalfor symmetrical triangular and rectangular fin arrays.has a strong influence on the nature of flow in the domain ofobservation is that the flow lines tend to concentrate at the heated edges. Atstream lines moves away from the vertical baseof fin flat in case of triangular fin arrays. At these Grashof number,recirculation is formed above the fin array. This negative loop is very weak in the rectangularfin arrays or almost absent. This is due to higher buoyancy towards the vertical base. In thism line due not tend to move away from the vertical base but are almost parallel to itm functions are always higher for symmetricalfor the bettereffectiveness of this geometry. It is also seen that when fins spacing is increased the negativeFigure 2 Stream line contours for Figure 3 Stream line contours for
  7. 7. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, MarchFigure 4 Stream line contours forTriangular fin array3.1.2 Distribution of velocityVelocity distribution describes the type of fluid motion present in the X and Y directions.It also indicates the formation of the boundary layer near the heated surface.X- component of velocity (U)-Fig. 6 to Fig.7 shows the Urectangular fin arrays. It is seen that Ufin increased. In Y direction U- Velocitygrowth adjacent to fin base. It is also observed that as the distance from fin flat increases in Zdirection and is maximum at the center of the channel (Z=5). The trend for Usimilar for both arrangements.Y- Component of velocity (V)-Fig. 6 to Fig.7 shows the V velocity distribution for symmetrical triangular andrectangular fin arrays. It is seen that as one approaches the fin base in Y direction, the magnitudeof velocity decreases because some fluid movesboth the arrangements.Figure 6 U and V Components Figure 7 U and V Componentsof Triangular fin array of Rectangular fin arrayInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766499(Online) Volume 4, Issue 2, March – April (2013), © IAEME277Figure 4 Stream line contours for Figure 5 Stream contours forTriangular fin array Rectangular fin arrayVelocity distribution describes the type of fluid motion present in the X and Y directions.It also indicates the formation of the boundary layer near the heated surface.shows the U- velocity distribution for symmetrical triangular andrectangular fin arrays. It is seen that U- Velocity increases as the distance X from the bottom ofVelocity clearly indicates the formation of boundary layer and itsgrowth adjacent to fin base. It is also observed that as the distance from fin flat increases in Zdirection and is maximum at the center of the channel (Z=5). The trend for U- Velocity is almostshows the V velocity distribution for symmetrical triangular andrectangular fin arrays. It is seen that as one approaches the fin base in Y direction, the magnitudeof velocity decreases because some fluid moves upward due to heating. Same trend is observed inFigure 6 U and V Components Figure 7 U and V Componentsof Triangular fin array of Rectangular fin arrayInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –April (2013), © IAEMEFigure 5 Stream contours forRectangular fin arrayVelocity distribution describes the type of fluid motion present in the X and Y directions.triangular andVelocity increases as the distance X from the bottom ofboundary layer and itsgrowth adjacent to fin base. It is also observed that as the distance from fin flat increases in ZVelocity is almostshows the V velocity distribution for symmetrical triangular andrectangular fin arrays. It is seen that as one approaches the fin base in Y direction, the magnituderd due to heating. Same trend is observed inFigure 6 U and V Components Figure 7 U and V Componentsof Triangular fin array of Rectangular fin array
  8. 8. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March3.2 HEAT TRANSFER CHARACTERISTICSIn the following the variation of average and base Nnumber and fin spacing is discussed3.2.1 Average Nusselt numberThe heat transfer rate from the fin array can be calculated by knowing the values ofaverage Nusselt number. Average Nusselt(a) Effect of Grashof number:Fig 7.17 shows the variation ofthere is a marked increase in Nusymmetrical triangular fin arrays than rectangular fins.all the spacing.Fig. 8 Effect of Gr(b) Effect of Spacing:Fig 9 shows variation of Nuof GrH has been obtained durinbecause with increased spacing, fluid flow through the fin channel more freely without anyinterference. The values of Nua are higher for symmetrical triangular fins than rectangular finfor the given spacing.International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766499(Online) Volume 4, Issue 2, March – April (2013), © IAEME278TRANSFER CHARACTERISTICSvariation of average and base Nusselt number with tis discussed.The heat transfer rate from the fin array can be calculated by knowing the values ofaverage Nusselt number. Average Nusselt number is found to vary with S+ and GrGrashof number:Fig 7.17 shows the variation of Nua with GrH for different spacing. It is observed thatNua with increase in GrH. The value of Nuasymmetrical triangular fin arrays than rectangular fins. Similar trend has been observed8 Effect of GrH on average Nusselt numbershows variation of Nua with S+ for one particular Grashof number. This valuehas been obtained during experiment. As expected, Nua increases with spacingbecause with increased spacing, fluid flow through the fin channel more freely without anyare higher for symmetrical triangular fins than rectangular finInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –April (2013), © IAEMEusselt number with the GrashofThe heat transfer rate from the fin array can be calculated by knowing the values ofGrH.for different spacing. It is observed thatis higher forSimilar trend has been observed for+ for one particular Grashof number. This valueincreases with spacingbecause with increased spacing, fluid flow through the fin channel more freely without anyare higher for symmetrical triangular fins than rectangular fin
  9. 9. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, MarchFig. 9 Effect of Spacing on average Nusselt number3.2.2 Base Nusselt numberBase Nusselt number is found to v(a)Effect of spacing :Fig 10 shows variation of Nuvalue of curve indicates optimum fin spacing. This plot confirm the postulate that the peakvalue of Nub is obtained for the spacing at which transition just starflow pattern to some other disturbed flFig. 10 Effect of spacinInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 09766499(Online) Volume 4, Issue 2, March – April (2013), © IAEME2799 Effect of Spacing on average Nusselt numberBase Nusselt number is found to vary with Grashof number and fin spacing.shows variation of Nub with S+ for assumed value of Grvalue of curve indicates optimum fin spacing. This plot confirm the postulate that the peakis obtained for the spacing at which transition just starts from single chimneyflow pattern to some other disturbed flow.10 Effect of spacing on Base Nusselt numberInternational Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –April (2013), © IAEMEry with Grashof number and fin spacing.value of GrH. The peakvalue of curve indicates optimum fin spacing. This plot confirm the postulate that the peaks from single chimney
  10. 10. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME280NOMENCLATUREA Area,m2GrH Grashof numberg Acceleration due to gravity, m/s2ha Average heat transfer coefficient/ m2Khb Base heat transfer coefficient, W/ m2Kk Thermal conductivity of the air, W/ m KL Length of fins, mL+L/H, length to height ratioNua Average Nusselt numberNub Base Nusselt numberPr Prantdl numberS+S/H, spacing to height ratioS Spacing between the finsX,Y,Z Cartesian coordinatesGreek Symbolsβ Volumetric expansion coefficient,K-1µ Dynamic viscosity of air, N-s/ m2ν Kinematic viscosity of air ,m2/sρ Density of air, kg/m3Superscripts/Subscriptsa Average value, ambientb Base valueH Height of fins, mm Mean filmREFERENCES1. W. Elenbass,The Heat Dissipation of Parallel Plates by Free Convection’,Physica vol.IX, No.1 pp 1-28, 1942.2. Starner and Mcmanus ,An Experimental Investigation of Free Convection HeatTransfer from Rectangular Fin Arrays, Journal of Heat Transfer, Trans.ASME seriesC,85,273,1963.3. Herhap & Mcmanus, Natural Convection Heat Transfer From Horizontal RectangularFin Array’, Journal of Heat Transfer, Trans.ASME,Series C,89,32 ,1967.4. Chaddock J.B., Free Convection Heat Transfer from Vertical RectangularFin Arrays, Journal of Heat Transfer, Trans.ASME Series C.89,439 ,1965.5. Welling & Wooldridge , Free Convocation Heat Transfer from Vertical RectangularFin Arrays, Journal of Heat Transfer, Trans.ASME Series, C.87,439 ,1965.6. T.Ahira,‘Natural Convection Heat Transfer from VerticalRectangular Fin Arrays, Bulletin of the ISME,3,NO.64,1182- 1191,1970.7. N.K.Sane, Natural Convection Heat Transfer from HorizontalRectangular Fin Array, PhD.Thesis at IIT Bombay, 1973.8. N.Saikhedkar,Natural Convection Heat Transfer from Vertical Rectangular crosssectional Fin Arrays’ PhD.Thesis at IIT Bombay, 1980.
  11. 11. International Journal of Advanced Research in Engineering and Technology (IJARET), ISSN 0976 –6480(Print), ISSN 0976 – 6499(Online) Volume 4, Issue 2, March – April (2013), © IAEME2819. G.K.Kulkarni, Natural Convection Heat Transfer from Vertical Rectangular Fin Arrays,ME Thesis at WCE Sangali, 1983.10. H.S.Deshmukh, Natural Convection Heat Transfer from Vertical Tapered Fin Arrays,ME Thesis at WCE Sangali, 1989.11. N.B.Joshi, Natural Convection Heat Transfer from Triangular shaped Vertical FinArray, ME Thesis at WCESangali,1989.12. J.P.Holman, Heat Transfer, Tata McGraw Hill.13. A.K.Runchal and M. Wolfshtein,Numerical Integration procedure for the Steady stateNavier Stoke equations,Journal of Mechanical Engg Science , Vol. 11 no 5,1969.14. Dr.N.G.Narve and Dr.N.K.Sane, “Experimental Investigation of Laminar MixedConvection Heat Transfer in the Entrance Region of Rectangular Duct”, InternationalJournal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 1, 2013,pp. 127 - 133, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359.15. Er. Pardeep Kumar, Manoj Sain and Shweta Tripathi, “Enhancement of Heat Transferusing Wire Coil Insert in Tubes”, International Journal of Mechanical Engineering &Technology (IJMET), Volume 3, Issue 2, 2012, pp. 796 - 805, ISSN Print: 0976 – 6340,ISSN Online: 0976 – 6359.

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