Numerical Simulation on Unsteady Heat Transfer of a Sphere

Transcription

1 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. Numerical Simulation on Unteady Heat Traner o a Sphere r. Aih Mitra Aociate oeor & HO, ept o BSH, College o Engineering & Management, Kolaghat. Eat Midnapur, Wet Bengal, India ABSTRACT: In the preent numerical tudy, a well-nown analytical model or tranient, one-dimenional heat traner i conidered. A phere o uniorm temperature T i at time t = i immered in a tagnant luid at dierent temperature T. Heat i tranerred within the phere by conduction and rom the outer urace o the phere to the luid by convection. Analytical model i dicued irt and then a detailed numerical analyi ha been done or accurate olution. The computer code are developed or thi numerical analyi in Matlab environment. In the econd part o the preent tudy, the computer code developed are applied to a model problem. The reult are then compared with thoe publihed in literature and ind a good agreement. KEYWORS: Biot number, Conduction, Convection, Matlab, Numerical Solution, Unteady. Symbol a, b coeicient o Eq (3) Bi Biot number, hr /, dimenionle C, C contant in Eq () C v, peciic heat capacity o olid phere, J/Kg. C diameter o the phere, m g gravitational acceleration, 9.8 m/ G unction deined in Eq () g ( T T ) Gr Graho number or phere,, dimenionle h mean heat traner coeicient, W/m. C H unction deined in Eq (9) thermal conductivity o water, W/m. C h Nu mean Nuelt number or phere,, dimenionle andtl number, dimenionle r radial ditance, m r radiu o phere, m Ra Rayleigh number or phere, Gr, dimenionle S urace area o phere, m t time, T temperature, C T i initial temperature, C T urace temperature, C T urace temperature, C V volume o phere, m 3 3 Copyright to IJIRSET

2 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. Gree thermal diuivity o phere, m / coeicient o volumetric thermal expanion, / C dimenionle temperature deined in Eq (5) dimenionle coordinate deined in Eq (5) dimenionle time deined in Eq (5) inematic vicoity, m / denity o phere, Kg/m 3 unction deined in Eq () root o Eq (4) Subcript water olid I.INTROUCTION Many heat conduction problem encountered in engineering application involve time a an independent variable. The goal o analyi i to determine the variation o the temperature a a unction o time and poition T (r, t) within the heat conducting body. In general, we deal with conducting bodie in a three dimenional pace and the goal i to predict the evolution o the temperature ield or uture time (t > ) [-7]. Heat i tranerred when a hot olid phere i immered in a cold tagnant luid by either orced convection or natural convection [8-5]. Heat i tranerred within the phere by conduction and rom the outer urace o the phere to the luid by convection. Thi tudy deal with the tranient heat tranerred o a phere. The objective o the preent tudy i two-old. The irt i to develop the analytical model o the problem and Matlab code or the numerical technique or the accurate olution. The econd i the validation o the code by comparing it reult applied to a model problem with the empirical correlation available in literature [6]. We conider the tranient heat conduction in a phere o radiu r. Initially, the temperature o the phere i T i. At time t =, the phere i immered into a well mixed cooling bath at temperature T. The heat traner tae place by conduction within the phere and by convection between the urace o the phere and the liquid. The phyical experiment depicted in the etch below. Fig. A heated olid phere immered in a cold tagnant luid The temperature ditribution mut obey the governing equation Copyright to IJIRSET

3 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) T t Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. T <r<r ubject to the ollowing initial and boundary condition IC: T(r, t) = Ti at t= T h( T T ), r r BC: r Eq () can be written in pherical coordinate 3 T T t r ( r r ) r <r<r 4 Analytical Solution The analytical olution or equation (4), ubject to equation () and (3) i available in everal textboo [-7]. We introduce the ollowing dimenionle variable T T r r i t r T T 5 So that the dimenionle temperature i a unction o dimenionle coordinate and time : (, ) 6 The initial and boundary condition () and (3) become IC: (, ) at = 7 BC: Bi at = 8 where Bi the Biot number Bi and hr T T Ti T The PE (4) become 9 Copyright to IJIRSET

4 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. ( ) < < We loo or a olution uing the method o eparation o variable: (, ) ( ) G( ) Subtituting the above expreion in the PE (), we get dg d d G d d ( ) d dg d d G d d ( ) d 3 The let ide depend on wherea the right ide depend on and both are independent o each other. Thereore both ide hould be equal to a negative contant with time a it cool). - (negative ign a the temperature o the phere will decreae G dg d 4 and d d ( ) d d 5 Solution o (4) i G ( ) ~ e Eq (5) can be written a 6 d d d ( ) d d d d d with boundary condition a 7 Copyright to IJIRSET 3

5 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. d Bi d at = 8 Now we introduce a new variable: H( ) ( ) or ( ) ( ) H d dh H( ) d d 9 d d dh dh d H ( ) H( ) 3 d d d d d Subtituting thee expreion in Eq(7), we get d H H d It general olution i H( ) C co( ) C in( ) So H( ) ( ) C co( ) C in( ) To ind the integration contant C and C, we conider the boundary condition or. At, ( ) mut be inite. But co( ) and in( ) a So co( ) in( ) and a Applying L Hopital rule to the econd term, we ind Lim ( in( )) Lim ( in( )) Copyright to IJIRSET 4

6 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. Hence or to be inite at, we require that C =. Thereore, ( ) C in( ) And the BC Eq (8) give 3 C in( ) C BiC co( ) in( ) At, it give the ollowing characteritic equation or the eigenvalue o : cot( ) Bi 4 It i a non linear trancendental equation, or a given Bi, the root n mut be ound out numerically. Finally, the olution to our PE i (rom (6) and (3)) (, ) ( ) G( ) c n e in( n) n 5 From the orthogonally condition, the value o n can be written a 4[in( ) co( )] in( ) n n n C n n n 6 Eq (4) through (6) provide a complete decription o the olution. II.NUMERICAL TECHNIQUES USING MATLAB i) Finding the root o Eq (4) The irt ta i to ind the root, n, o Eq (4), or a given Bi. Thi i a root-inding problem. There are an ininite number o n on the poitive real line. Matlab code myun.m and root.m were developed to do thi job. Fig Plot o 5. Bi or Bi = 5 howing root (*) and ingularitie () o Eq (4) in the range ( ) cot( ) Copyright to IJIRSET 5

7 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. (, ) ii) Evaluating Once the n are nown or a given Bi, C n value are computed rom Eq(6) and then the non-dimenional temperature ditribution (, ) uing Eq (5). Finally, the temperature ditribution T(r, t) rom the deinition o, (Eq 5) viz., T( r, t) T ( Ti T ) i computed. Related matlab code are nondtem.m and example.m. III.RELEVANT EMPIRICAL CORRELATION The empirical correlation [6] or natural convection o a heated olid phere in a cold tagnant luid i Ra Nu [ (. ) 6 ] 9 or Ra < 7 h Nu From the relation, we can write Ra h [ (. ) ] g 589. ( ) 4 g ( ) ( T T ) [ (. ) 6 ] [ (. ) ] T ( T ) [ Ra Gr g( T T ) 3 ] 8 The applicable lumped heat equation i v, hs( T T ) c V dt dt T() =T i 9 dt dt h S V ( )( T T ) c v, 6 h ( )( T T ) c v, 3 Copyright to IJIRSET 6

8 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. Putting the value o h rom Eq (8), we get dt ( )( T T ) dt c v, g ( )[ 5 ( ) 4 c 469 v [ (., ) ] ]( T T ) The above eqn can be written a dt dt where a( T T ) b( T T ) a ( ) c v, g b ( )[ 5 ( ) 4 c 469 v [ (., ) ] ] The exact analytical olution o Eq (3) [7] i b T t T T T a e at ( ) ( i )[ ( )] 4 4 Thermo-phyical property value are evaluated at the ilm temperature 33 ( T T ) Validation We conider the ollowing model problem: A 5 mm diameter metal phere initially at 9 C i plunged into a water bath at 3 C. Aume that the heat traner coeicient i W/(m. C), the thermal conductivity o the metal i = W/m. C and the thermal diuivity o the metal i = m /. IV.RESULTS AN ISCUSSION The temperature ditribution o the outer urace o the phere i obtained uing the matlab code developed. Thi i then compared with that obtained rom Eq (33). Figure 3 demontrate the comparion. Copyright to IJIRSET 7

9 ISSN (Online) : ISSN (int) : International Journal o Innovative Reearch in Science, Engineering and Technology An ISO 397: 7 Certiied Organization, Volume3, Special Iue 6, February 4 National Conerence on Emerging Technology and Applied Science-4 (NCETAS 4) Modern Intitute o Engineering and Technology, Bandel, Hooghly 73, Wet Bengal, India. 9 preent tudy empirical correlation 8 urace temp. ( C) time (ec) Fig 3 Variation o Temperature o the outer urace o the phere at dierent time V.CONCLUSION In thi numerical reearch wor, the tranient heat traner i conidered when a olid phere at a uniorm temperature i immered in cold tagnant luid. In the olution procedure, irt analytical model and then numerical technique have been dicued or the olution. Computer code in Matlab are developed accordingly. Applying thee code to a model problem, temperature ditribution are obtained. Thee are compared with thoe obtained rom empirical correlation available in literature and ind a good agreement. REFERENCES [] Welty, J. R., Engineering Heat Traner, John Wiley, New Yor, 974. [] Özişi, M.N., Baic Heat Traner, McGraw Hill, New Yor, 977. [3] White, F.M., Heat Traner, Addion-Weley, Reading, MA, 984. [4] Sucec, J., Heat Traner, Brown Publiher, ubuque, IA, 985. [5] Chapman, A.J., Fundamental o Heat Traner, Fith edition, Macmillan, New Yor, 987. [6] Bejan, A., Heat Traner, John Wiley, New Yor, 993. [7] Suryanarayana, N.V., Engineering Heat Traner, Wet Publihing Co. New Yor, 995. [8] Mill, A.F., Baic Heat Traner, Second edition, entice-hall, Upper Saddle River, NJ, 999. [9] Thoma, L., Heat Traner, Second edition, Captone Co., Tula, OK,. [] Kreith, F. and Bohn, M.S., inciple o Heat Traner, Sixth edition, Broo/Cole, Paciic Grove CA,. [] Holman J.P., Heat Traner, Ninth edition, McGraw Hill, New Yor,. [] Incropera, F.P. and ewitt.p., Introduction to Heat Traner, Fourth edition, John Wiley, New Yor,. [3] Çengel, Y.A., Heat Traner, Second edition, McGraw Hill, New Yor, 3. [4] Lienhard IV, J.H. and Lienhard V, J.H., A Heat Traner Textboo, Phlogiton e, Cambridge, MA, 3. [5] Nelli, G. and Klein, S., Heat Traner, Cambridge Univerity e, London, UK, 8. [6] Churchill, S.W., Free Convection Around Immered Bodie, in G.F. Hewitt, Executive Editor, Heat Exchanger eign Handboo, Section.5.7, Begell Houe, New Yor,. [7] Polyanin, A.. and Zaitev, V.F., Handboo o Exact Solution or ierential Equation, CRC e, Boca Raton, FL, 995. BIOGRAPHY r. Aih Mitra did M.Sc. in Phyic in eidency College, and Ph and pot Ph reearch wor on Heat Traner and CF in Jadavpur Univerity. He i preently Aociate oeor & HO o the epartment o Baic Science & Humanitie o College o Engineering & Management, Kolaghat, Eat Midnapur, Wet Bengal. He i having more than year pot-ph teaching experience at UG level. He i a regular reviewer o the International Journal Heat and Ma Traner and WSEAS (World Scientiic & Engineering Academy & Society). He i the author o 6 paper in International Journal, 7 paper in International Conerence and 5 paper in Seminar. He ha completed a CSIR-ponored project a Co-invetigator and organized 4 National Worhop and International Conerence a convener. Hi ield o reearch i Computational Heat Traner and Fluid Flow. Hi wor on energy eiciency in automobile i in the proce o patenting. Copyright to IJIRSET 8