AC : USE OF SPREADSHEETS IN SOLVING HEAT CONDUCTION PROBLEMS IN FINS

Transcription

1 AC : USE OF SPREADSHEETS IN SOLVING HEAT CONDUCTION PROBLEMS IN FINS Amir Karimi, University f Texas-San Antni AMIR KARIMI Amir Karimi is a Prfessr f Mechanical Engineering and an Assciate Dean f Undergraduate Studies at The University f Texas at San Antni (UTSA). He received his Ph.D. degree in Mechanical Engineering frm the University f Kentucky in His teaching and research interests are in thermal sciences. He has served as the Chair f Mechanical Engineering (1987 t 1992 and September 1998 t January f 2003), Cllege f Engineering Assciate Dean f Academic Affairs (Jan April 2006), and the Assciate Dean f Undergraduate Studies (April 2006-present). Dr. Karimi is a Fellw f ASME, senir member f AIAA, and hlds membership in ASEE, ASHRAE, and Sigma Xi. He is the ASEE Campus Representative at UTSA, ASEE-GSW Sectin Campus Representative, and served as the Chair f ASEE Zne III ( ). He chaired the ASEE-GSW sectin during the academic year. American Sciety fr Engineering Educatin, 2008 Page

2 Use f Spreadsheets in Slving Heat Cnductin Prblems in Fins Abstract Excel is an effective and inexpensive tl available n all cmputers equipped with Micrsft Office. This sftware has the necessary functins fr slving a large class f engineering prblems, including thse related t heat transfer. This paper prvides several examples t demnstrate the applicatin f Excel in slving prblems invlving ne-dimensinal heat cnductin in varius fin cnfiguratins. It prvides frmulas fr the temperature distributin and heat transfer fr several different fin prfiles. Intrductin An intrductry curse in heat transfer typically cvers the basic analysis f ne-dimensinal heat cnductin prblems invlving fins with simple gemetrical cnfiguratins. The analytical cverage is usually limited t fins f unifrm crss-sectinal area. Fr mre cmplex fin cnfiguratins, nly fin efficiency charts are prvided in mst heat transfer textbks These charts apprximate the rate f heat transfer, but d nt prvide any infrmatin n the temperature distributin in fins. Micrsft Excel, can be a useful tls in slving heat cnductin prblems fr a variety f fin cnfiguratins. Numerical slutin f heat transfer prblems has been an evlving phenmenn. In the early stages, the applicatin tls have undergne a gradual prgressin thrugh slide rules, simple calculatrs, mainframe cmputers, desktp and laptp cmputers. Prir t 1972, slide rules were the essential cmputatinal tls fr slving engineering prblems and cmputers were seldm used fr analysis f heat transfer prblems at the undergraduate level. Many f the mre cmplex analytic slutins t heat transfer prblems were given in graphs r charts. A few examples include graphs fr fin efficiencies, transient temperature distributin charts fr heat transfer in slabs, cylinders, r spheres (Heisler Charts), and radiatin shape (view) factr charts. In the early 1970 s calculatrs replaced slide rules as the basic cmputatinal tl fr slving engineering prblems. A few years later prgrammable calculatrs were available. Mdules cntaining basic slutins t heat transfer prblems were develped fr these calculatrs. Authrs included sectins in their textbks t intrduce students t numerical techniques fr slving heat transfer prblems. The cmputer applicatin sftware fr slving engineering prblems has als changed. Prir t the intrductin f persnal cmputers (PCs) in the early 1980 s, cmplex cmputer cdes were needed fr numerical slutin f heat transfer prblems and the knwledge f a cmputer prgramming language was essential fr integrating numerical slutins int heat transfer curses. Access t mainframe cmputers and prficiency in such prgramming languages as FORTRAN and PASCAL were necessary fr slving cmplex heat transfer prblems. Therefre, mechanical engineering prgrams required a curse in ne f the structured cmputer prgramming languages. Page

3 As persnal cmputers became mre available and affrdable, and as the perating systems became mre user friendly, their applicatins were gradually integrated int intrductry heat transfer curses. BASIC prgramming language was used fr slving simple heat transfer prblems. In mre recent years, the trend has shifted tward using sftware packages fr slving numerical prblems and many mechanical engineering degree prgrams n lnger require a curse in cmputer prgramming. Integratin f cmputer sftware in heat transfer curses enhances the student learning experience and aids the understanding f basic cncepts and fundamental physical laws. Many publishing cmpanies prvide cmputer sftware with heat transfer textbks Mst cmmnly used sftware tls in heat transfer curses are Interactive Heat Transfer (IHT) 16 and Engineering Equatin Slver (EES). 17 These prgrams are general purpse, nn-linear equatin slvers with built-in prperty functins. They are capable f explring and graphing the effects f change in variables n the slutin t a given prblem. There are ther sftware packages available in the market that can be integrated int a heat transfer curse. These include Micrsft Excel spreadsheet, Mathcad, MATLAB, and Maple. All available sftware packages are extremely useful tls fr analysis and design in undergraduate r graduate intrductry heat transfer curses. The mst significant advantage f these sftware prgrams is that n prir knwledge f prgramming language is necessary in their applicatins. Excel is an example f these applicatin sftware prgrams. The fcus f this paper is t describe hw Excel culd be used in the heat transfer analysis f extended surfaces. Excel Spreadsheet It has been shwn 18, 19 that Excel is an effective cmputatinal tl in slving bundary layer prblems. Excel perates with data entered by the user int a spreadsheet. This sftware recgnizes 39 engineering functins, as well as varius math and trignmetry functins. The engineering functins include Bessel functins, errr functins, and ther functins used in heat transfer. T insert engineering functins in the frmulas, the insert buttn n the Excel spreadsheet culd be used. Then clicking n functin, a windw appears n the screen, as shwn n Fig. 1. One can search fr the desired functin by typing a descriptin f the functin (financial, engineering, etc.) in the search bx r using the select categry bx by scrlling thrugh ptins fr the desired functin. Fr prblems requiring iterative calculatins, the Gal Seek r Slver tls can be emplyed. By using the tl menu, and selecting the slver ptin a dialg bx appears, as shwn in Fig 1. By selecting the target cell and fixing the desired value fr that cell, values in the selected cells autmatically change t crrespnd t the slutin given fr the target cell. This will be demnstrated later in an example. Page

4 Fig. 1. Excel wrksheet, functin selectin menu, and slver windw Page

5 One Dimensinal Heat Cnductin in Fins Heat transfer analysis f heat cnductin in straight fins f unifrm crss-sectinal area is included in heat transfer text bks. The analysis results in frmulas fr temperature distributin, the rate f heat exchange with the surrunding envirnment, and the fin efficiency. Bundary cnditins used in the analysis will influence the resulting equatins. Fr example, fr an infinitely lng fin f a unifrm crss-sectinal area the temperature distributin and heat transfer are given by the fllwing equatins. θ = = e mx (1) q ( ) = hpka T T (2) where, m = hp / ka, P dentes the perimeter, h is the heat transfer cefficient, k is the thermal cnductivity, A is the crss-sectinal area, T is the temperature at the base f fin, and T is the ambient temperature. Fr a fin f unifrm crss-sectinal area with insulated tip, the temperature distributin and heat transfer can be expressed, respectively, as [ m( L x) ] ( ml) csh θ = = (3) csh ( ) ( ml) q = hpka tanh (4) where, L is the length f the fin. If the tip f the fin is expsed t a cnvective envirnment, equatins fr the temperature distributin and heat transfer are given by [ m( L x) ] + ( h / mk) sinh[ m( L x) ] csh tip θ = = (5) csh ( ml) + ( h / mk) sinh( ml) tip q = ml + ( htip / mk) sinh( ml) ( ml) + ( h / mk) sinh( ml) csh hp / ka( ) (6) csh tip Where, h tip represents the heat transfer cefficient at the tip f fin. If the temperature at the tip f the fin is given, the temperature distributin and heat transfer equatins are given by Page

6 [( TL T ) ( T T )] sinh( mx) + sinh[ m( L x) ] sinh( ml) θ = = (7) q = [( TL T ) ( T T )] sinh( ml) csh ml + hp / ka( ) (8) where, T L represents the temperature at the tip f the fin. It shuld be nted that the equatins presented in this sectin fr the temperature prfile and heat transfer are based n the assumptin f ne-dimensinal heat cnductin in the axial directin f the fin. Fr this assumptin t be valid, the Bit number, Bi, must satisfy the fllwing cnditin ( A P) hl h Bi = ch = π 0.1 (9) k k where, L ch is a characteristic length, A is the crss sectinal area, and P is the perimeter f the fin. Fr fins f circular crss sectinal area, L ch can be represented by the radius, R. Analysis fr fins f variable crss-sectinal areas r annular fins results in mre cmplex differential equatins. The slutins include mre cmplex functins, including Bessel functins. The analyses fr these types f fins are nt typically fully cvered in an intrductry heat transfer curse. Instead the results are shwn in the frm f fin efficiency charts. In very few cases, equatins r graphs fr temperature distributins are prvided. Fr several cmmn fins, the equatins fr temperature prfile are included in the appendix. Several mdern textbks 9-12 prvide expressins fr the efficiency f mst cmmn fin shapes. A list f equatins fr fin efficiencies is included in the appendix. The fin efficiency is defined as qact qact η f = = (10) qmax ha( T T ) where, q act dentes the actual heat transfer, q max represents the maximum theretical heat transfer by assuming that the entire fine is at the base temperature. Fr types f fins nt included in the appendix, additinal expressins fr temperature distributin and efficiency are included in the bk n Extended Surface Heat Transfer. 20 A few examples are prvided in the fllwing sectin t demnstrate the use f Excel spreadsheets in slving heat transfer prblems invlving fins. Example 1: A straight fin f triangular prfile (axial sectin) 0.1 m in length, 0.02 m thick at the base, and 0.2 m in depth is used t extend the surface f a wall at 200 C. The wall and the fin are made f mild steel (k = 54 W/m. C). Air at 10 C (h = 200 W/m 2 C) flws ver the surface f the fin. Evaluate the temperature at 0.05 m frm the base and at the tip f the fin. Determine the rate f Page

7 heat remval frm the fin and the fin efficiency 21. A (x) =2 δ w (x/l) w δ x Fig. 2. Sketch f triangular fin in Example 1 x=0 x=l P 2 w Slutin An analytical slutin t this prblem 7 gives the fllwing expressin fr the dimensinless temperature distributin θ = I I ( 2 hlx kδ ) 2 ( 2 hl kδ ) = I I ( m Lx ) ( m L ) where, L is the length f the fin, x is the distance frm the tip f the fin, δ is ne half f the thickness at the base, m = 2 h kδ, and I is the mdified Bessel functin f the first kind f rder zer. The rate f heat remval can be calculated by evaluating heat transfer at the base f the fin, where x=l. (11) dt q = ka (12) dx x= L Thus the rate f heat transfer at the base can be expressed by 2 I1 ( ) ( 2 hl / kδ ) T T 2 I ( 2 hl / kδ ) q = 2w hkδ (13) Where, w represents the width f the Then the rate f heat remval frm the base is equal t q. Therefre, the fin efficiency can be determined by the fllwing relatin Page

8 q η f = (14) 2 2 2h L + δ [ w( )] The frmulatin f slutin in Excel fr this prblem is shwn in Fig. 3. The data given in the prblem statement are first entered int the cells f the wrksheet. Using these data, the frmulas fr evaluating m, ml, m xl, I (ml), I 1 (ml), I (ml), I ( m xl ), θ, T, q, and η are entered int apprpriate cells f the wrksheet. T enter frmulas an = sign is first entered int the cell fllwed with the terms needed fr the evaluatin f the frmula. The basic mathematic peratrs used are +, -, * (multiplicatin), /, and ^ (pwer). The calculated results are presented in Fig. 4. By pressing CTRL + ` (grave accent) ne can switch between the wrksheet displaying frmulas and their resulting values. Fig. 3. Excel frmulatin f the slutin fr prblem in Example 1. The wrksheet shwn in Fig. 4 can be expanded t evaluate the temperature prfile in the fin a, and plt the results. T achieve this, the values fr x ranging between 0 and 0.1 are entered in clumn A (cells A16 thrugh A-26), as shwn in Fig. 5. Then the cntent f cells B16 thrugh E16 is highlighted and cpied int lwer rws, by clicking n the bttm bundary crner f cell Page

9 E16 and dragging it all the way t cell E26. By this cpying actins the values f m xl, I ( m xl ), θ, and T are autmatically calculated fr each value f x listed in clumn A. T plt T as a functin f x, the cells A15 thrugh A26 and E15 thrugh E26 were first highlighted by pressing the Ctrl key. Then by clicking the chart wizard icn n the menu bar f the wrksheet, a menu appears ffering several standard ptins fr pltting data. The x-y (scatter) ptin was selected and the fur steps f chart wizard were prefrmed by prviding the necessary infrmatin in each step and pressing the next buttn. Finally the Finish buttn was pressed t shw the results in the wrksheet. Fig. 4. Slutin t Example prblem 1 Page

10 Fig. 5. Prcedure fr the evaluatin and pltting f the temperature prfile in Example 1 Example 2: A fin f triangular prfile (axial sectin) 0.1 m in length, 0.02 m thick at the base, 0.2 m in depth is used t extend the surface f a wall at 200 C. The wall and the fin are made f mild steel (k = 54 W/m C). Air at 10 C (h = 200 W/m 2 C) flws ver the surface f the fin. Evaluate the distance frm the base where the temperature is 175 C. Page

11 Slutin: The slutin t this prblem is based n the same equatins used in the previus example, except in this case the distance, x, cannt explicitly be determined, since it is a part f the Bessel functin argument in Eq. 11. A trial and errr prcedure is required fr the slutin f this prblem. An Excel spreadsheet can be emplyed t slve this prblem. One methd is t use the same slutin used in example 1, but fr this prblem the desired temperature culd be achieved by changing the values f x in the spreadsheet. The result f this prcedure is shwn in Fig. 6. Fig. 6. Slutin f Example 2 by a trial and errr prcedure Page

12 A simpler way t slve the prblem in Example 2, is t take the advantage f Gal Seek tl in Excel. Figure 7 shws the value f the temperature at an arbitrary psitin in the fin. By using the tl menu, and selecting the Gal Seek ptin a dialg bx appears, as shwn in Fig 7. The target cell (temperature in this case, cell E16) then is selected and its value is set t a desired value fr that cell (175). The cell that its value must be changed is identified (cell A16). After clicking the Slve buttn, the value in the selected cell A16 (x) is autmatically changes t a value that gives the desired slutin in the target cell (E16) Nw the value f x in Fig. 6 changes s that the temperature is equal t 175 C. The prcedure and the final slutin are shwn in Fig.7. a) Initial guess b) Final slutin Fig 7. Prcedure f using the Gal Seek tl t find x where T =175 C. Page

13 Example 3: An annular fin f the rectangular prfile having a thickness f 2.0 mm is attached t tube maintained at 120 C. The inner radius f fin, r 1, is 2.0 cm. The envirnmental temperature is 20 C, and h = 70 W/(m 2. C). The fin is made f 40% nickel steel rd [k=10.0 W/m. C]. Evaluate the uter radius, r 2, if the temperature at the tip f the fin is required t be maintained at 95 C. Als determine the rate f heat transfer and the fin efficiency. Slutin: The temperature distributin, assuming insulated tip is given by ( mr ) I ( mr) + I ( mr ) K ( mr) K1 θ = = (14) K ( mr ) I ( mr ) + I ( mr ) K ( mr ) where, r 1 and r 2 are the inner and uter radius f the fin, respectively, m = 2h kt, I and K are mdified, zer-rder Bessel functin f the first and secnd kind, respectively, I 1 and K 1 are mdified, first-rder Bessel functin f the first and secnd kind, respectively. Using dt q = ka results in the fllwing expressin fr the rate heat transfer at the base dr r= r 1 q ( mr ) I ( mr ) I ( mr ) K ( mr ) K1 2π r1 t k ( T T ) m (15) K = ( mr ) I ( mr ) + I ( mr ) K ( mr ) 2 The fin efficiency can be calculated by substituting Eq. 15 int Eq. 10 qact qact η f = = (10) qmax ha( T T ) where, A in this prblem represent the surface area fr the cnvective heat transfer. 2 2 ( r ) A = 2π r (16) 1 2 Again t slve this prblem, ne must emply an iterative trial and errr prcedure, since value f r 2 used in Eqs. 15 thrugh 16 is unknwn. Gal Seek tl was used t cnduct the iteratin prcess. The infrmatin given in the prblem statement and the frmulas fr the evaluatin f m, mr 1, mr 2, I (mr 1 ), I (mr 2 ), I 1 (mr 1 ), I 1 (mr 2 ), K (mr 1 ), K (mr 2 ), K 1 (mr 1 ), K 1 (mr 2 ), θ, T (r 2 ), q, and η f were entered int an Excel wrksheet. A value f r 2 = 0.04 was used t evaluate the temperature at the tip f the fin. A temperature f 89.3 C was calculated by the frmulas in Excel, as shwn in Fig. 8. The Gal Seek tl was used by setting a target value f 95 C fr the tip f the fin (Cell H17). The gal Seek calculated a value f r 2 = m, as shwn in Fig. 9 and q, and η f were recalculated by Excel, based n the new value fr r 2. Page

14 Fig 8. Initial guess fr r 2, in slving the prblem in Example 3. Example 4: A carbn steel bar (k=54.0 W/m. C) 40 cm lng cnnects tw thermal reservirs, ne at 200 C and the ther at 100 C. The bar has a diameter f 1.5 cm. Air at 25 C flws acrss the bar. Evaluate the desired air free stream velcity, if the temperature f the fin at a lcatin 30 cm frm the 200 C wall must be equal 68 C. The fllwing infrmatin is prvided fr air at 450 K: ρ a = kg/s, c p,a =1020 J/kg.K, µ a =2.493E -05 N.s/m, ν a =3.177E -05 m 2 /s, k a = W/m.K, α a m 2 /s = 0.698, and Pr a = Page

15 Fig. 9. Slutin t Example 3 prblem, using Gal Seek Slutin: In this prblem, the temperature at x = 0.3 m culd be calculated frm Eq. 7 if the value fr heat transfer cefficient was [( TL T ) ( T T )] sinh( mx) + sinh[ m( L x) ] θ = = (7) sinh ml where, ( ) m = 2h k R, k m is the thermal cnductivity f the bar. m Fr air flw ver a cylinder the Reynlds number is defined as hd Re D = (17) ka where, k a is the thermal cnductivity f air. The Nusselt number can be btained frm the relatinship given by Churchill and Bernstine / 3 5 / 8 hd 0.62 Re Pr Re D D Nu = = (18) D 3 1/ 4 k [ 1 + ( 0.4 / Pr) ] 2 / 282,000 fr 100<Re D <10 7 ; Pe D = Re D *Pr> Page

16 The data given in the prblem statement fr the bar and air were entered int cells f a wrksheet as shwn in Fig. 10. A free stream velcity f 1.0 m/s was assumed fr air (Cell A13). The apprpriate frmulas fr the evaluatin f Re D, Nu D h, Bi, m, ml, mx, θ, and T (x=0.3 m) were entered int the spreadsheet cells. Based n the assumed value f the free stream velcity a temperature f 79.8 was calculated fr T. Then the Slver tl was used t set the target value in cell F16 t 68 C. As shwn in Fig 11, the Slver calculated an air velcity f 4.69 m/s necessary fr the temperature t be 68 C at x= 0.3 m. Fig. 10. The temperature at x = 0.3 m, based n an initial guess f 1.0 m/s fr the air velcity Discussin The examples used abve, demnstrated that excel is an effective tl in slving fin heat cnductin prblems. Its applicatin simplifies the evaluatin f temperature distributin and Page

17 the rate f heat transfer when slving prblems invlving extended surfaces. The heat transfer slutins btained frm the applicatin f Excel is mre accurate than thse calculated frm fin efficiency charts. Fig. 11. Slutin t Example prblem 3, using the Excel s Slver tl. Excel can easily be integrated int a heat transfer curse. Mst students are already familiar with the basic peratin f Excel. This includes entering data int the spreadsheet and pltting simple graphs. Hwever, it is pssible that few students might nt knw hw t enter frmulas int the wrksheet cells. Perhaps, mst have never used the Gal Seek, and Slver tls in slving engineering prblems. An intrductin t applicatin f Excel in slving heat transfer prblem takes 30 t 45 minutes f class time t demnstrate hw t enter frmulas int cells f Excel wrksheet. The Page

18 applicatin f Gal Seek and Slver tls culd be demnstrated thrugh simple examples. We have nt yet integrated Excel int ur undergraduate heat transfer curse. Hwever in teaching a heat cnductin curse t a small class f first year graduate curse, students were asked t plt the fin efficiency curves fr several types f fins. Nne f the students had used the Gal Seek r Slver tls f Excel. A shrt lecture was given n the use f these tls. Students were given the ptin f using Excel, IHT, EES, r similar sftware fr pltting the curves. All students selected t use Excel t cmplete their assigned prject. The main reasn was the cnvenience and the availability f Excel f student persnal cmputers. Summary The applicatin f Excel spreadsheet in slving ne dimensinal heat cnductin prblems was demnstrated thrugh several examples. It was shwn that Excel is a useful cmputatinal tl when the slutin t prblems requires (a) varying ne f the parameters, (b) pltting the results f calculatins, and (c) an iteratin prcess. References 1. Kreith, F., 1965, Principles f Heat Transfer, Secnd Editin, Internatinal Bk Cmpany, New Yrk. 2. Bayley, F. J, Owen, M.J, and Turner, A. B, 1972, Heat Transfer, Barnes and Nble, New Yrk. 3. Chapman, A. J, 1974, Fundamentals f Heat Transfer, Macmillan, New Yrk. 4. Wlf, H., 1983, Heat Transfer, Harper and Rw Publishers, New Yrk 5. White, F., 1984, Heat Transfer, 1984, Addisn-Wesley Publishing, Reading, Massachusetts. 6. Ozisik, M. N., 1985, Heat Transfer, A Basic Apprach, McGraw Hill, New Yrk. 7. Lienhard, Jhn, 1981, A Heat Transfer Textbk, Prentice-Hall. 8. Thmas, A.C., 1992, Heat Transfer, Prentice Hall, New Jersey. 9. Hlman, J.P., 2002, Heat Transfer, Ninth Editin, New Yrk. 10. Mills, A.F., 1999, Basic Heat and Mass Transfer, 2 nd editin, Prentice Hall, New Jersey. 11. Incrpera, F. P., De Witt, D.P., Bergman, T. L., Lavin, A.S., 2007, Intrductin t Heat Transfer, Fifth Editin, Jhn Wiley, New Yrk. 12. Cengel, Y. A., 2003, Heat Transfer, A Practical Apprach, Secnd Editin, McGraw Hill, New Yrk. 13. Cengel Y. A., Turner, R. H., 2005, Fundamentals f Thermal-Fluid Sciences, 2 nd Editin, McGraw Hill, New Yrk. 14. Mran, M. J., Shapir, H. N., Munsn, B. R., and DeWitt, D. P., 2003, Intrductin t Thermal Systems Engineering: Thermdynamics, Fluid Mechanics, and Heat Transfer, Jhn Wiley, New Yrk. 15. Kaviany, M., 2002, Principles f Heat Transfer, Jhn Wiley, New Yrk. 16. Smith, T. F. and Wen, J. 2002, Interactive Heat Transfer, V.2.0, Jhn Wiley, New Yrk. 17. Beckman, W. A., and Klein, S. A., 2004, Fakhri, A., 2004, Spreadsheet Slutin f the Bundary Layer Equatins, IMECE , Prceedings f 2004 Internatinal Mechanical Engineering Cngress and Expsitin, Anaheim. Califrnia. 19. Naraghi, M H. 2004, Slutin f Similarity Transfrm Equatin fr Bundary Layers Using Spreadsheets, IMECE , Prceedings f 2004 Internatinal Mechanical Engineering Cngress and Expsitin, Anaheim. Califrnia. 20. Karimi, A, Delen, J., and Hannan, M. 2007, A Review f Available Cmputer Sftware Packages fr Use in an Undergraduate Heat Transfer Curse, IMECE , Prceedings f 2007 Internatinal Mechanical Engineering Cngress and Expsitin, Seattle, Washingtn. 21. Kraus, A., Aziz, A., and Welty, J., 2001, Extended Surface Heat Transfer, Wiley Inter-Science, New Yrk. 22. Churchill, S. W., and Bernstein, 1977, A Crrelatin Equatin fr Frced Cnvectin frm Gases and Liquids t a Circular Cylinder in Crssflw J. Heat Transfer, vl. 99, pp Page

19 Nmenclature A = surface area r crss-sectinal area, m 2 h = heat transfer cefficient, W/m 2 K I (x), I 1 (x) = mdified Bessel functin f the first kind f rder zer, rder ne K (x), K 1 (x) = mdified Bessel functin f the first kind f rder zer, rder ne J (x) r J 1 (x) =Bessel functin f the first kind f rder zer r rder ne k = thermal cnductivity, W/m K L = length, m Nu = Nusselt number Re = Reynld number q = heat transfer rate, W T = temperature, C r K Greek letters α = thermal diffusivity, m 2 /s θ = dimensinless temperature parameter, a rati f temperature differences η f = fin efficiency Subscripts f = fin = lcatin at x=0 = ambient cnditin Greek letters difference ε heat exchanger effectiveness µ = viscsity, N.s/m ν kinematic viscsity, m 2 /s Page

20 Appendix Table A. Temperature distributin equatin fr cmmn fins Straight fin, rectangular prfile Straight fin, triangular prfile Straight fin, cncave parablic prfile Annular fin, rectangular prfile Pin fin, rectangular prfile Pin fin, trianglular prfile Pin fin, cncave parablic prfile θ = csh = csh [ m( L x) ] ( ml) m = 2h kt x is measured frm the base ( 2 hl( L x) kt ) 2 ( 2 hl kt ) ( ml ( x L) ) I ( ml) I I 1 θ = = = T T I m = 2h kt x is measured frm the base θ = = 1 x L ( ml) m = 2h kt x is measured frm the base K1 mr2 I mr θ = = K mr I mr m = 2h θ = m = ( ) ( ) + I ( mr ) K ( mr) 1 2 ( ) ( ) + I ( mr ) K ( mr ) h kt θ = kd csh = csh = [ m( L x) ] ( ml) L I L x 1 2 m = 2 hl kd x is measured frm the base θ = = 1 x L 9+ 4 m = 2 h kd x is measured frm the base ( 2m L x ) I ( m L ) ( ml) Page

21 Table B. Fin Efficiency Equatins 10 Page