BRIEF NOTE ON HEAT FLOW WITH ARBITRARY HEATING RATES IN A HOLLOW CYLINDER

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1 BRIEF NOTE ON HEAT FLOW WITH ARBITRARY HEATING RATES IN A HOLLOW CYLINDER ishor C. Deshukh a*, Shrikan D. Warhe a, and Vinayak S. ulkarni a Deparen of Maheaics, R. T. M. Nagpur Universiy, Nagpur, Maharashra, India Deparen of Maheaics, Governen College of Engineering, Aurangaad, Maharashra, India Shor paper UDC: 536.:57.93 DOI:.98/TSCI8763D In his paper he eperaure disriuion is deerined hrough a hollow cylinder under an arirary ie dependen hea flux a he ouer surface and zero hea flux a he inernal oundary due o inernal hea generaion wihin i. To develop he analysis of he eperaure field, we inroduce he ehod of inegral ransfor. The resuls are oained in a series for in-ers of Bessel s funcions. ey words: hea conducion prole, hoogeneous hea conducion prole, non-hoogeneous prole, hea generaion Inroducion Nowacki [] has deerined he eperaure disriuion on he upper face, wih zero eperaure on he lower face and he circular edge herally insulaed. Roy Choudhury [] has deerined he ransien eperaure along he circuference of a circle over he upper face wih lower face a zero eperaure and he fix circular edge of he plae herally insulaed. Wankhede [3] deerined he arirary eperaure on he upper face wih he lower face a zero eperaure and he fix circular edge herally insulaed. Deshukh e al. [4] deerined he eperaure disriuion sujeced o arirary iniial eperaure on he lower face wih he upper face a zero eperaure and fix circular edge herally insulaed. Phyhian [5] sudied he cylindrical hea flow wih arirary heaing raes a he ouer surface and zero hea flux a he inernal oundary. In his paper we consider a non-hoogeneous hollow cylinder and deerine he eperaure disriuion under an arirary ie dependen hea flux a he ouer surface and zero hea flux in he inernal surface wih he help of inegral ransfor echnique. The resuls are oained in a series for in ers of Bessel s funcions. We also discuss he liiing case of hoogeneous oundary value prole in a solid cylinder. The soluion should e useful in curren aerospace proles, for saions of a issile ody no influenced y nose apering. The issiles skill aerial is assued o have physical properies independen of eperaure, so ha he eperaure T(r, ) is a funcion of radius r and ie only. *ncorresponding auhor: e-ail: kcdeshukh@rediffail.co

2 Deshukh,. C., e al.: Brief Noe on Hea Flow wih Arirary 76 THERMAL SCIENCE, Year, Vol. 5, No., pp Forulaion of he prole Figure. The geoery of he prole Consider a hollow cylinder of radius a r, which is iniially a eperaure F(r). For ie > hea is generaed wihin he solid a a rae of g(r, ) [Buh f 3 ] *. The inner circular oundary surface a r = a is insulaed, while he arirary ie dependen hea flux Q() is applied a ouer circular oundary r = (fig. ). The arirary ie dependen hea flux eans radial hea flux prescried on oundary surface, which is funcion of ie only (for all ) and converges rapidly for large values of. Ofen a rapidly convergen series is required for sall values of ie. The oundary value prole of he hea conducion is given as: sujec o he oundary condiions and iniial condiion T T g( r, ) T r r r a r, T r T Q() r () a r = a, > () a r =, > (3) T(r, ) = F(r) when =, a r (4) where, and are he heral conduciviy and heral diffusiviy of he aerial of he hollow cylinder, respecively. Equaions () o (4) consiue he aheaical forulaion of he hea conducion prole in a hollow cylinder. The soluion Following he general procedure of Ozisik [6], we develop he finie Hankel ransfor and is inversion o he aove saed prole. On applying he finie Hankel ransfor and is inverse ransfor o he eqs. () o (4), one oains, he expression for he eperaure funcion of a non-hoogeneous oundary value prole of hea conducion in a hollow cylinder as: *nbu = J; f =.348 ; F = (9/5) C + 3

3 Deshukh,. C., e al.: Brief Noe on Hea Flow wih Arirary THERMAL SCIENCE, Year, Vol. 5, No., pp where T ( r, ) (, r)e r (, r ) F( r )dr a (5) e r (, r) g( r, )d r(, ) Q( ) d a J( ) Y( ) J ( r) Y ( r) (, r) ( ) (6) J J( ) Y ( ) J ( ) and λ s are he posiive roos of ranscendenal of equaion: J( a) Y( a) J( ) Y( ) (7) Special cases and nuerical calculaions Diension Inner radius of a hin hollow cylinder a = f, Ouer radius of a hin hollow cylinder = f, and Cenral circular pah of hollow cylinder r =.5 f. Maerial properies The nuerical calculaion has een carried ou for an aluinu (pure) hollow cylinder wih he aerial properies as: heral conduciviy = 7 Bu/hf F, heral diffusiviy a = 3.33 f /h, Roos of ranscendenal equaion Le l = 3.965, l = 6.33, l 3 = , l 4 =.58, and l 5 = are he posiive roos of ranscendenal equaion: J( ) Y( ) J( ) Y( ) Case : If F(r) = g(r,) =, Q() = e, where d is he Dirac-dela funcion, w >, and a =, i. e. zero iniial eperaure wih no inernal hea generaion, only arirary ie dependen hea flux Q() is applied a ouer circular oundary r =. Using in eq. (5), hen eperaure disriuion is oained as: T( r, ) (, r)e (, ) e Q( )d (8)

4 Deshukh,. C., e al.: Brief Noe on Hea Flow wih Arirary 78 THERMAL SCIENCE, Year, Vol. 5, No., pp where (, r) J ( r) (9) J( ) and ' s are posiive roos of he ranscendenal equaion J ( ). () Le l = 3.837, l = 7.56, l 3 =.735, l 4 = 3.337, l 5 = 6.47, l 6 = 9.659, l 7 =.76, l 8 = 5.937, l 9 = 9.468, and l = 3.8 are he roos of ranscendenal equaion J (l) =. Case : If F(r) = g(r,) = [(g cyl,i )/(pr )] d(r r ) d( ), Q() = e where d is he Dirac-dela funcion, w >, i. e. zero iniial eperaure wih inernal hea generaion g(r,) and arirary ie dependen hea flux Q() is applied a ouer circular oundary r =. Using eq. (5), hen he eperaure disriuion is oained as: (, ) cyl,i e T r g (, r) (, r ) () (, r)e (, ) e Qd Case 3: If F(r) = r g(r,) = g i d(r r ) d( ), Q() = e, where r is he radius easured in fee, d is he Diracdela funcion, w >. The arirary iniial eperaure F(r) = r is applied wih inernal hea source g(r,) is an insananeous line hea source of srengh g i = 5 Bu/hf, siuaed a he cenre of he circular cylinder along radial direcion and releases is hea insananeously a he ie = = hours. Also ie dependen hea flux Q() is applied a ouer circular oundary r =. Conclusions ( ) ( )e αλ ( ) d a T r, λ,r r λ,r r r α g αλ e τ i ( λ,r ) ( λ,r ) αλ αλ α ( λ,r)e ( λ,) e Q( )d In his paper he work of Phyhian [5] is exended o hea generaion for non-hoogeneous oundary value prole in a hollow cylinder. The eperaure disriuion under an arirary ie dependen hea flux a r = is oained wih he help of inegral ransfor echnique. Three cases have een discussed in he given prole: (a) F(r) =, g(r, ) =, and a = In his case zero iniial eperaure wih no Figure. Teperaure a = hours in radial direcion () inernal hea generaion, only arirary ie dependen hea flux Q() is applied a ouer

5 Deshukh,. C., e al.: Brief Noe on Hea Flow wih Arirary THERMAL SCIENCE, Year, Vol. 5, No., pp circular oundary r =. So his is he liiing case of hoogeneous oundary value prole in a solid cylinder. Fro fig., one can oserve hea flows fro ouer circular surface o inner circular surface of a hollow cylinder, i. e. in he direcion of cener. gcyl,i () F( r), g( r,) δ( r r ) δ( τ) r In his case zero iniial eperaure wih inernal hea generaion g(r,) and arirary ie dependen hea flux Q() is applied a ouer circular oundary r =. The eperaure disriuion is oained as in eq. (). This soluion represens he case where he cylinder is iniially a zero eperaure. An insananeous cylindrical surface hea source of radius r = r (i. e. a < r < ) and of srengh g cyl, i [Buf ] and sae of linear lengh as he cylinder, releases is hea sponaneously a ie. The insananeous cylindrical surface hea source is relaed o he volue hea source y eans of Dirac-dela funcion. The srengh of he insananeous cylindrical surface source g cyl, i is expressed as: S α cyl.i gcyl,i [ Ff ] Susiuing his ino eq. (), one oains: The er Scyl,i T ( r,) e π α ( λ,r)e αλ ( τ) αλ αλ e ( λ,r) ( λ,r ) ( λ,) e τ ( λ,r) ( λ αλ,r ) Q( )d represens he eperaure a ie due o insananeous cylindrical surface hea source of srengh S cyl, i = [ Ff ] siuaed a r = r and releasing is hea sponaneously a ie = in he region a r. (c) F(r) = r, g(r,) = g i d(r r )d( ) [Buh f 3 ] and Q() = e w, The hea source g(r,) is an insananeous volue hea source of srengh g i = 5 Buh f, siuaed a he cener of he hollow cylinder along radial direcion and releases is hea insananeously a he ie = = hours. Fro figs. 3 and 4, i is oserved ha due o inernal hea generaion, he eperaure funcion T increases non-uniforly fro inner circular oundary o he ouer circular oundary. The soluions of he hoogeneous oundary value prole are valid for all values of ( ), u converge rapidly for large values of only. Ofen a rapidly convergen series is required for sall values of ie a he surface r =. (3) Figure 3. Teperaure a = hours in radial direcion

6 Deshukh,. C., e al.: Brief Noe on Hea Flow wih Arirary 8 THERMAL SCIENCE, Year, Vol. 5, No., pp Physically, his shows ha, for he iniial eperaure changes only, he reflecion of he eperaure wave a he inner oundary ay e negleced and he eperaure ay e oained as if he cylinder were solid. The eperaure oained for he hollow cylinder and solid cylinder is paricularly useful for evaluaing he axiu eperaure gradien and he heral sresses. Figure 4. Teperaure a = hours in radial direcion Any paricular case of special ineres can e derived y assigning suiale values o he paraeer and funcion in eqs. (8), (), and (). Acknowledgen The auhors are hankful o Universiy Gran Coission, New Delhi, India, o provide he financial assisance under Major Research Projec Schee. We offer our graeful hanks o he referee for heir kind help and acive guidance in he preparaion of his revise paper. References [] Nowacki, W., The Sae of Sress in a Thick Circular Plae Due o a Teperaure Field, B. Acad. Pol. Sci. Tech., 5 (957), pp. 7 [] Roy Choudhuri, S.., A Noe on he Quasi-Saic Sress in a Thin Circular Plae Due o Transien Teperaure Applied Along he Circuference of a Circle over he Upper Face, B. Acad. Pol. Sci. Tech., (97),, pp. -4 [3] Wankhede, P. C., On he Quasi-Saic Theral Sresses in a Circular Plae, Ind. J. Pure and Appl. Mah. 3 (98),, pp [4] Deshukh,. C., horagade, N. L., Quasi-Saic Theral Deflecion in a Thin Circular Plae, for Eas J. Appl. Mah, 9 (), 3, pp [5] Phyhian, J. E., Cylindrical Hea Flow wih Arirary Heaing Raes, AIAA Journal, (963), 4, pp [6] Ozisik, M. N., Boundary Value Prole of Hea Conducion, Inernaional Tex Book Cop. Scranon, Penn., USA, 968 Paper suied: Augus 7, Paper acceped: Augus,