Inverse Heat Conduction Problem in a Semi-Infinite Circular Plate and its Thermal Deflection by Quasi-Static Approach

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Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: 93-9466 Vol. 5 Issue ue pp. 7 Previously Vol. 5 No.

1 Available a hp://pvamu.edu/aam Appl. Appl. Mah. ISSN: Vol. 5 Issue ue pp. 7 Previously Vol. 5 No. Applicaios ad Applied Mahemaics: A Ieraioal oural AAM Iverse Hea Coducio Problem i a Semi-Ifiie Circular Plae ad is Thermal Deflecio by Quasi-Saic Approach K.C. Deshmuh S.D. Warbhe Deparme of mahemaics Nagpur Uiversiy Nagpur-4433 Maharashra Idia cdeshmuh@rediffmail.com G.D. Kedar Deparme of Mahemaics Kamla Nehru Mahavidyalaya Saardara Nagpur-449 Maharashra Idia V.S. Kulari Deparme of Mahemaics Goverme College of Egieerig Auragabad-435 Maharashra Idia viayasulari@rediffmail.com Received: November 3 9; Acceped: March 7 Absrac This paper cocers he iverse hea coducio problem i a semi-ifiie hi circular plae subjeced o a arbirary ow emperaure uder useady codiio ad he behavior of hermal deflecio has bee discussed o he ouer curved surface wih he help of mahemaical modelig. The soluios are obaied i a aalyical form by usig he iegral rasform echique. Keywords: Iverse hermoelasic problem Thermal deflecio Circular plae AMS Classificaio: 34B7 34B4 35G3 35K5

2 AAM: Ier.. Vol. 5 Issue ue [Previously Vol. 5 No. ]. Iroducio The iverse hermoelasic problem cosiss of deermiaio of he emperaure of he heaig medium he hea flux o he boudary surfaces of he solid whe he codiios of he displaceme ad sresses are ow a he some pois of he solid uder cosideraio. Sabherwal [ ] sudied iverse problem i hea coducio. Cialowsi ad Graya 98; Graya ad Cialowsi 98; Graya ad Kolowsi 98 ivesigaed oe dimesioal rasie hermoelasic problem ad derived he heaig emperaure ad hea flux o he surface of isoropic ifiie slab. Ashida e al. 6 aemped a iverse hermoelasic problem i a isoropic plae associaed wih a pieoelecric ceramic plae. Deshmuh ad Wahede 996 solved a iverse problem of hermo elasiciy i a hi circular plae by deermiig he emperaure o he curved surface of he plae displaceme ad hermal sresses usig quasi-saic approach by employig iegral rasform echiques. Khobragade ad Deshmuh 998 sudied a iverse axially symmeric quasi-saic problem of hermo elasiciy for a hi clamped circular plae i which a hea flux is prescribed o a ieral cylidrical surface of he plae ad suiable hea exchage codiios are me o he upper ad lower surfaces of he plae is solved wih he help of a geeralied iegral rasform echique. Tihe ad Deshmuh 968 sudied he iverse hea coducio problem i a hi circular plae ad is hermal deflecio o he ouer curved surface. Recely Deshmuh e al. 998 sudied a quasisaic hermal deflecio of a hi clamped circular plae due o hea geeraio. I his paper we cosider wo dimesioal o-homogeeous boudary value problem of hea coducio ad sudied he hermal deflecio of a semi-ifiie circular plae o he ouer curved surface for a ifiie legh. A circular plae is subjeced o arbirary ow emperaure uder useady sae codiio. Iiially he plae is a ero emperaure ad he lower surface is a ero emperaure. The goverig hea coducio equaio has bee solved by usig iegral rasform mehod. The resuls are obaied i series form i erms of Bessel s fucios. Mahemaical model has bee cosruced of a semi-ifiie circular plae wih he help of umerical illusraio. No oe previously sudied such ype of problem. This is a ew coribuio o he field. The iverse problem is very impora i view of is relevace o various idusrial mechaics subjeced o heaig such as he mai shaf of lahe urbies ad he role of he rollig mill. Also arise he quechig sudies he aalysis of experimeal daa ad measureme of aerodyamic heaig.. Formulaio of he Problem Cosider a semi-ifiie circular plae defied by r a. Le he plae be subjeced o arbirary ow ierior emperaure f wihi he regio r a wih lower surface is a ero emperaure. Uder hese more realisic prescribed codiios he uow

3 Deshmuh e al. emperaure o he ouer curved surface of plae a r a ad quasi-saic hermal deflecio due o uow emperaure g are required o deermie. The differeial equaio saisfyig he deflecio fucio r as i 3 is give as 4 M T D where M T is he hermal mome of he plae defied as M T h ae T r d D is he flexural rigidiy of he plae deoed as 3 Eh D 3 a E ad are he coefficies of he liear hermal expasio he Youg s modulus ad Poisso s raio of he plae maerial respecively ad. r r r 4 Sice he edge of he circular plae is fixed ad clamped; a r a. 5 r Iiially T whe. 6 The emperaure of he semi-ifiie circular plae saisfies he hea coducio equaio as i Deshmuh e al. 9 T T T r r r wih he codiios T i r a 7 T r g uow a r a 8 T r a r a 9

4 AAM: Ier.. Vol. 5 Issue ue [Previously Vol. 5 No. ] 3 r T a r a f r T ow a r a where is hermal diffusiviy of he semi-ifiie circular plae. Equaios - cosiue he mahemaical formulaio of he iverse hermoelasic problem i a semi-ifiie circular plae. 3. The Soluio Uow Temperaure By applyig Fourier sie iegral rasform ad is iversio over he hea coducio equaio oe obaied he expressio for emperaure disribuio fucio ad uow heaig emperaure as d d e f r r T si d d e f a g si 3 where Tr K Trd Tr K Tr d si K ad are he posiive roos of he rascedeal equaio. Thermal Deflecio Usig equaio io equaio oe obais

5 4 Deshmuh e al. r M ae f e d d d T si. 4 Assume he soluio of he equaio saisfy codiio 5 as r C a r r a a a a a. 5 I ca be easily see ha he plae has bee buil-i-edge r a ra. r Usig equaio 5 ad 6 i equaio oe obais ae C si f e d d d. ad 6 Subsiuig equaio 6 io equaio 5 oe obais he expressio for hermal deflecio as ae r a r r a a a a a ad si f e d d d Special Case ad Numerical Calculaios To cosruc he mahemaical hermoelasic behavior of a semi-ifiie circular plae we cosidered he followig fucios ad parameers: Se f e. e wih 5 sec. Dimesios Radius of he circular plae a m. Thicess of a circular plae. m.

6 AAM: Ier.. Vol. 5 Issue ue [Previously Vol. 5 No. ] 5 Maerial properies The umerical calculaio has bee carried ou for a alumium pure circular plae wih he maerial properies as 6 Thermal diffusiviy 84.8 m s. 3 Desiy 77 g / m. Specific hea c p 896 / gk Poisso raio Coefficie of liear hermal expasio a. ad K Lamé cosa Roos of he rascedeal equaio: The are he posiive roos of he rascedeal equaio We se for coveiece 7 X ad Y 5 a Eh D which assume o be he cosas. The umerical calculaio has bee carried ou wih he help of compuaioal mahemaical sofware Mahcad- ad graphs are ploed wih he help of Excel MS Office-3..5E- Temperaure K.E- 5.E-.E+ -5.E r m Figure. Temperaure disribuio

7 6 Deshmuh e al. From he Figure i ca be observed ha he emperaure decreases from he cere r of a plae o he circular boudary r. I is ero a he ouer circular boudary r. Uow emperaure r m Figure. Uow Temperaure disribuios From he Figure i ca be observed ha he uow emperaure icreases i he axial direcio a. 8 ad he remais cosa hereafer. Thermal deflecio wr r m Figure 3. Thermal deflecio From he Figure 3 i ca be observed ha for alumium circular plae he deflecio is maximum a he cere ad i is ero a he ouer circular edge. Also he variaio i he deflecio has show whe he plae is fixed ad clamped. 5. Discussio ad Resuls I his paper we have exeded he wor of Deshmuh e al. 996 for a wo dimesioal ohomogeeous boudary value problem of hea coducio ad have deermied he expressios of emperaure uow emperaure ad hermal deflecio. As a special case he mahemaical model is cosruced for a semi-ifiie hi circular plae made up of pure alumium meal.

8 AAM: Ier.. Vol. 5 Issue ue [Previously Vol. 5 No. ] 7 From he Figures ad 3 i is observed ha he emperaure uow emperaure ad he hermal deflecio respecively are egligible for a ifiie legh ad also observed ha he variaios i emperaure disribuio uow heaig emperaure ad hermal deflecio ae place for fiie values of parameers ad fucios. This ype of a problem has he applicaios i various idusrial machies such as lahe machie ad urbie. Also ay paricular case of special ieres ca be derived by assigig suiable values o he parameer ad fucio i he expressios 3 ad 7. Acowledgeme The auhors are very haful o Uiversiy Gras Commissio New Delhi o provide he parial fiacial assisace uder major research projec scheme. We offer our graeful has o he referees for heir id help ad acive guidace i he preparaio of his revised paper. REFERENCES Ashida F. Choi. Noda N A iverse hermoelasic problem i a isoropic plae associaed wih a pieoelecric ceramic plae. Thermal Sresses Deshmuh K.C. ad Wahede P.C A iverse quasi-saic rasie hermoelasic problem i a hi circular plae Bull. Pure ad Appl. Sci Deshmuh K. C. Warbhe S.D. Kulari V.S. 9. Quasi-saic hermal deflecio of a hi clamped circular plae due o hea geeraio. Thermal Sresses Grysa K. ad Cialowsi M O a cerai iverse problem of emperaure ad hermal sress field Aca Mechaica Grysa K. ad Cialowsi M Nuclear Egg. Dec. 64 pp Grysa K. ad Kolowsi Z. 98. Oe dimesioal problems of emperaure ad hea flux deermiaio a he surfaces of hermoelasic slab Nuclear Egg. Dec Khobragade N.L. ad Deshmuh K.C. 5. A iverse quasi-saic hermal deflecio problem for a hi clamped circular plae. Thermal Sresses Oisi M.N Boudary value problems of hea coducio Ieraioal Tex Boo Compay Scrao Pesylvaia. Sabherwal K.C Idia oural of Pure ad Applied Physics vol.3 No.; pp Sabherwal K.C Idia oural of Pure ad Applied Physics Vol. XL No.8 pp Tihe A.K. ad Deshmuh K.C. 6. Iverse hea coducio problem i a hi circular plae ad is hermal deflecio. Applied Mahemaical Modellig

Available online at J. Math. Comput. Sci. 4 (2014), No. 4, ISSN:

Available online at J. Math. Comput. Sci. 4 (2014), No. 4, ISSN: Available olie a hp://sci.org J. Mah. Compu. Sci. 4 (2014), No. 4, 716-727 ISSN: 1927-5307 ON ITERATIVE TECHNIQUES FOR NUMERICAL SOLUTIONS OF LINEAR AND NONLINEAR DIFFERENTIAL EQUATIONS S.O. EDEKI *, A.A.