A Steady State Heat Conduction Problem in. a Thick Annular Disc Due to Arbitrary. Axisymmetric Heat Flux

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1 Noliear Aalysis ad Differetial Equatios, Vol. 4, 016, o. 3, HIKARI Ltd, A Steady State Heat Coductio Proble i a Thick Aular Disc Due to Arbitrary Axisyetric Heat Flux Ishaque A. Kha 1, Lalsigh Khalsa ad Viod Varghese 3 1, M. G. College, Arori, Gadchiroli (MS), Idia 3 RTM Nagpur Uiversity, Nagpur (MS), Idia Copyright 015 Ishaque A. Kha, Lalsigh Khalsa ad Viod Varghese. This article is distributed uder the Creative Coos Attributio Licese, which perits urestricted use, distributio, ad reproductio i ay ediu, provided the origial work is properly cited. Abstract I this paper attept has bee ade to discuss steady state quasi static theral stresses i a thick aular disc a r b h z h subjected to arbitrary heat flux at upper ad lower surface of aular disc while ier ad outer circular surface of thick aular disc are aitaied at zero degree teperature. The goverig heat coductio have bee solved by usig itegral trasfor techique.the results are obtaied i series for i ter of Bessel s fuctios, The result of displaceet ad stresses have bee coputed uerically ad illustrated graphically. Keywords: Quasi-static, Theroelastic proble, Theral Stresses, Axisyetric Theral Stresses 1. Itroductio Nowacki (1957) [6] has deteried steady-state theral stresses i circular plate subjected to a axisyetric teperature distributio o the upper face with zero teperature o the lower face ad the circular edge. Roy Choudhary (197) (1973) ad Wakhede (198) [13] deteried Quasi static theral stresses i thi circular plate. Gogulwar ad Deshukh (005) [1] deteried theral stresses i thi circular plate with heat sources. Where as Qia & Batra (004) [7] studied trasiet theroelastic deforatio of thick fuctioally graded plate uder lateral

2 1 Ishaque A. Kha et al. loads ad obtaied the results for radial ad axial displaceets ad teperature chage oreover Shara et al (004) [10] studied the behavior of theroelastic thick plate uder lateral loads ad obtaied the results for radial ad axial displaceets ad teperature chage have bee coputed uerically ad illustrated graphically for differet theories of geeralized theroelasticity. Recetly Ruhi et al (005) [9] did theroelastic aalysis of thick walled fiite legth cyliders of fuctioally graded aterials ad obtaied the results for stress, strai ad displaceet copoets through the thickess ad alog the legth are preseted due to uifor iteral pressure ad theral loadig. V. S. Kulkari, K. C. Deshukh [1] cosidered a thick aular disc which is subjected to a trasiet axisyetric teperature field o the radial ad axial directios of the cylidrical coordiate syste ad deteried the expressio for teperature, displaceet ad stress fuctios due to arbitrary heat flux o the upper ad lower surface. Deshukh et al [3] studied two diesioal ohoogeeous boudary value proble of heat coductio ad it s theral deflectio of a sei ifiite circular plate o the outer curved surface for a ifiite legth. Deshukh et al [4] deteried the theral stresses iduced by a poit heat source i a circular plate by quasi static approach.. Forulatio of the Proble Cosider a thick aular disc defied by a r b h z h. Let the disc be subjected to a axisyetric teperature field o the radial directio i the cylidrical co-ordiate syste. Iitially the plate kept at zero teperature the arbitrary the heat flux Qf r is prescribed over the upper surface z h ad the lower surface z h the fixed circular edges r a, r b are at zero teperature. Assue the lower ad upper surface of thick aular disc are tractio free. Uder this ore realistic prescribed coditio, the quasi state theral stresses are required to be deteried. The differetial equatio goverig displaceet Potetial fuctio as 1 K r r r z ( rz, ) is give Where K is restrait coefficiet ad teperature chages T Ti, Ti is the iitial teperature. Displaceet fuctio is kow as Goodier s theroelastic displaceet Potetial. Heat Coductio Equatios The teperature of the disc at a tie t satisfies the heat coductio equatio, (1) T 1 T T 0 r r r z ()

3 A steady state heat coductio proble 13 With the boudary coditio T Qf r For zh a r b (3) z T=0 at r=a h z h (4) T=0 at r=b h z h (5) Displaceet Potetial ad Theral Stresses The displaceet fuctio i a cylidrical co-ordiate syste is represeted by the Michell s fuctio defied as i [5]. U r M r rz (6) (1 ) M Uz M z z (7) The Michell s fuctio M ust satisfy M 0 (8) Where (9) 1 r r r z The copoet of the stresses are represeted by the Thero-elastic displaceet Potetial ad Michell s fuctio M as ad M rr G K M r z r 1 1M G K M r r z r r zz ( ) M G K M z z z (1 ) M G M rz rz r z (10) (11) (1) (13) G ad are shear odulus ad Poisso ratio respectively. For tractio free surface of the stress fuctio

4 14 Ishaque A. Kha et al. rr The equatio (1) to (14) costitute atheatical forulatio of the proble. 3. Solutio 0 at zh (14) rz The teperature chage Itroducig the fiite the Hakel trasfor over the variable r ad its iverse its trasfor defied as i. [11] b T, (15) z rk 0 r T r, z dr a, T(, z) K ( r) 1 T r z 0 (16) Where R0 ( r K r ) (17) 0 ( ) N R J0( ) 0( ) 0( ) r Y r r J0( b) Y0( b) the Norality costat a R0( ) R0( a) b N b (18) (19) Where Represets differetiatio w.r.t space variable r, ad 1,. are roots of the trascedetal equatio J0( a) Y0( a) 0 J0( b) Y( b) J(x) is the Bessel fuctio of the first kid of order ad Y(x) is the Bessel fuctio of the secod kid of order. The trasfor satisfies the relatio T 1 T H (, ) T z r r r (0) (1) ad T d T H () z dz O applyig the fiite Hakel trasfor defied i the equatio (15) to equatio () oe obtais T z T 0 (3)

5 A steady state heat coductio proble 15 Here T is the Hakel trasfor of T o solvig the equatio (3) oe obtais T Ae. z B. e z (4) O solvig the equatio (4) uder the coditio give i the equatio (3) oe obtais Qf( ) cosh h AB sih h Usig the equatio (4) oe obtais h cosh z sih h Qf( ) cosh T (5) f is the Hakel trasfor of f r O Applig the iverse Hakel trasfor defied is the equatio (16) oe obtais T N Q f J0 r Y0 r cosh z 1 J0 b Y0 b sih (6) Where f is Hakel trasfor of f r Sice the iitial tep Ti=0 T Ti T (7) Michell s Fuctio M Now assue Michell s fuctio M which satisfy the coditio (8) N QK f J0 r Y0 r M cosh z R z sih( z) 1 J 0 b Y0 b (8) Where H ad R are arbitrary fuctio which ca be deteried fially usig the coditio (14) Goodier s Theroelastic Displaceet Potetial To obtais the displaceet potetial equatio (1) oe have usig the equatio (6) ad (7) i QK f J0 r Y0 r zsih z 1 N J 0 b Y0 b zsih h (9)

6 16 Ishaque A. Kha et al. Displaceet ad Theral Stresses Now usig the equatio (6), (7), (8) ad (9) i the equatio (6), (7), (10), (11), (1) ad (13) oe obtais the expressio for displaceet ad stresses respectively as Ur N QK f J1 r Y1 r J 1 0 b Y0 b h zsih z sih H z R sih z z cosh z sih QK f J0r Y0 r sih z z cosh z U z J 1 N 0 b Y0 b sih h H cosh z R 1 vcosh( z) z sih( z) (30) (31).. GQK f J1 r Y1 r z sih z rr N J b Y b sih z J0r Y0 r cosh z J0b Y0 b sih h.. J1r Y1 r H 0 0 sih z R J b Y b J0 r Y0 r sih( z) J0 b Y0 b.. J1r Y1 r 0 0 sih z z cosh z J b Y b (3) N GQK f J1 r Y1 r z sih z J 1 0 b Y0 b sih h J0r Y0 r cosh z J0b Y0 b sih h 1 J1 r Y1 r H sih z R r J 0 b Y0 b

7 A steady state heat coductio proble 17 J0 r Y0 r J0 b Y0 b 1 J 1 r Y 1 r Sih z sih( z) ( z)cosh( z) r J 0 b Y0 b N GQK f J0 r Y0 r z sih z zz J 1 0 b Y0 b sih h N 3 3 H sih z R 1 v sih( z) ( z)cosh( z) GQK f J1 r Y1 r rz J 1 0 b Y0 b sih( z) ( z)cosh( z) sihh 3 3 H cosh( z) R cosh( z) ( z)sih( z) (33) (34) (35) Deteriatio of ukow arbitrary fuctio H ad R. I order to satisfy the coditio (14) solvig the equatio (1) ad () for H ad R oe obtais. H (1 )sih( h)cosh( h) sih( h) ( h)cosh( h) 4 sih( h) sih( h)cosh( h) ( h) sih( h) R 4 sih( h)cosh( h) ( h) (36) (37) Usig the value of H ad R i the equatio (15) to () oe obtais the expressio for displaceet ad stresses 4. Special Case Settig f r ( r ) ( r b ) (38) Applyig fiite Hakel trasfor to (38), oe obtais b 1 J r ( ) Y r f N J b Y b f ( ) a 0 0 r( r )( r b ) dr, 0 0 a b J0a b a J0b NaJ 0 a J 0 by 0 b (39)

8 18 Ishaque A. Kha et al. 5. Nuerical Calculatios The uerical calculatios have bee carried out for steel (SN 50C) plate with the 6 1 paraeters a = 1, b =, h = 0.3, theral diffusivity K ( s ) ad Poisso ratio 0.81 with ,, 6.734, , beig the positive roots of trascedetal equatio J0( a) Y0( a) 0. J0( b) Y0( b) I order to exaie the ifluece of heat flux o the upper ad lower surface of thick plate, oe perfored the uerical calculatios r1,1.,1.4,1.6,1.8, ad z 0.3, 0.15, 0, 0.15, 0.3. Nuerical variatios i radial ad axial directios are show i the figures with the help of coputer progra. 6. Cocludig Rearks I this proble, a thick aular disc is cosidered which is free fro tractio ad deteried the expressios for teperature, displaceet ad stress fuctio due to arbitrary heat flux uder steady state. As a special case atheatical odel is costructed for f r ( r a ) ( r b ) ad perfored uerical calculatios. The theroelastic behavior is exaied such as teperature, displaceet ad stresses with the help of arbitrary heat applied. Figurer 1: The teperature is axiu at the iddle of the thick aular disc ad syetrical to words outer ad ier circular surfaces. 1 3 Figure : The stress fuctio rr is axiu i the the iddle regio of 4 4 the thick aular disc ad reduces to zero towards ier ad outer surfaces of the thick aular disc. Figure 3: The stress fuctio zz is axiu at r = 1.4 ad varies i the thickess. Figurer 4: The stress fuctio is axiu ear to 1.5 ad varies with the thickess of the plate.

9 A steady state heat coductio proble 19 1,00E-01 T 0,00E+00-1,00E-01 -,00E-01-3,00E-01-4,00E-01-5,00E-01-6,00E , 1,4 1,6 1,8 z= 0.3 z= 0.15 z= 0.3 r Fig. 1: Teperature T versus r for differet values of z at z = - 0.3, -0.15, , 0.3 3,00E-05,00E-05 z= ,00E-05 0,00E+00-1,00E-05 -,00E-05-3,00E-05 z= z=0 1 1, 1,4 1,6 1,8 r z= 0.15 z= 0.3 Fig. : Stress fuctios rr versus r for differet values of z at z = - 0.3, -0.15, , 0.3 Fig. 3: Stress fuctio zz versus r for differet values of z at z = - 0.3, -0.15, , 0.3

10 130 Ishaque A. Kha et al. Fig. 4: Stress fuctio versus r for differet values of z at z = - 0.3, -0.15, , 0.3 Refereces [1] V. S. Gogulwar ad K. C. Deshuh, Theral stresses i a thi circular plate with heat sources, Joural of Idia Acadey of Matheatics, 7 (005), o.1. [] N. L. Khobragade ad K.C. Deshuh, Theroelastic proble of a thi circular plate subject to a distributed heat supply, Joural of Theral Stresses, 8 (005), [3] K. C. Deshukh, S. D. Warbhe, G. D. Kedar ad V. S. Kulkari, Iverse Heat Coductio Proble i a Sei-Ifiite Circular Plate ad its Theral Deflectio by Quasi-Static Approach, Applicatios ad Applied Matheatics, 5 (010), [4] K. C. Deshukh, Y. I. Quazi, S. D. Warbhe ad V. S. Kulkari, Theral stresses iduced by a poit heat source i a circular plate by quasi-static approach, Theoretical ad Applied Mechaics Letters, 1 (011), [5] Naotake Noda, Richard B Hetarski ad Yoshiobu Taigawa, Theral Stresses, d ed., Taylor ad Fracis, New York, 003 [6] W. Nowacki, The state of stresses i a thick circular plate due to teperature field, Bull. Acad. Polo, Sci., Scr. Scl. Tech., 5 (1957), 7. [7] L. F. Qia ad R. C. Batra, Thsiet theroelastic deforatio of a thick fuctioally graded plate, J. Ther. Stresses, 7 (004), [8] S. K. Roy Choudhary, A ote of quasi static stress i a thi circular plate due to trasiet teperature applied alog the circuferece of a circle over the

11 A steady state heat coductio proble 131 upper face, Bull Aca. Polo Sci, Ser, Scl, Tech., (197), 0-1. [9] M. Ruhi, A. Agoshatari ad R. Naghdabadi, Theroelastic aalysis of thick walled fiite legth cyliders of fuctioally graded aterial, Joural of Theral Stresses, 8 (005), [10] J. N. Shara, P. K. Shara ad R. L. Shara, Behavior of theoelastic thick plate uder lateral loads, Joural of Theral Stresses, 7 (004), [11] I.N. Seddor, The Use of Itegral Trasfor, McGraw Hill, New York, 197. [1] V. S. Kulkari, K. C. Deshukh, Quasi-static trasiet theral stresses i a thick aular disc, Sadhaa, 3 (007), o. 5, [13] P.C. Wakhede, O the Quasi static theral stresses i a circular plate, Idia Joural of Pure ad Applied Matheatics, 13 (198), o. 11, Received: Noveber 5, 015; Published: February 8, 016