Keywords: thermal stress; thermal fatigue; inverse analysis; heat conduction; regularization

Transcription

1 Proceedings Inverse Analysis for Esimaing Temperaure and Residual Sress Disribuions in a Pipe from Ouer Surface Temperaure Measuremen and Is Regularizaion Shiro Kubo * and Shoki Taguwa Deparmen of Mechanical Engineering, Faculy of Science and Engineering, Sesunan Universiy, 17-8, Ikedanakamachi, Neyagawa, Osaka , Japan; s_08_aob-15@docomo.ne.jp * Correspondence: kubo@mech.eng.osaka-u.ac.jp; Tel.: Presened a he 18h Inernaional Conference of Experimenal Mechanics, Brussels, Belgium, 1 5 July 018. Published: 30 May 018 Absrac: This paper presens an inverse analysis mehod for esimaing he emperaure and hermal residual sress disribuions in he pipe from he emperaure hisory measured on he ouer surface. A regularizaion mehod was inroduced. I is found from numerical simulaions ha he proposed inverse analysis mehod wih regularizaion is useful for obaining a reasonable esimae of he inner surface emperaure and hermal sress. Keywords: hermal sress; hermal faigue; inverse analysis; hea conducion; regularizaion 1. Inroducion I was revealed ha flucuaion of fluid emperaure in a pipe may induce high cycle hermal faigue in he pipe, which someimes resuls in fracure of he pipe [1 4]. The informaion abou he emperaure and hermal residual sress disribuions is imporan o preven he high cycle hermal faigue. I is, however, difficul o measure he emperaure in he pipe direcly. The monioring of he emperaure on accessible ouer surface is promising o solve he problem. This paper presens an inverse analysis mehod for esimaing he emperaure and hermal sress disribuions in he pipe from he emperaure hisory measured on he ouer surface. A regularizaion mehod is proposed for obaining a reasonable soluion.. Mahemaical Srucure of Temperaure Disribuion in a Thin Pipe Consider a pipe subjeced o hermal flucuaion on is inner surface. As an exreme he hickness of he pipe h is assumed o be small compared wih he radius of he pipe. The governing equaion of emperaure disribuion ( x, is expressed by he following equaion of hermal conducion. ( x, ( x, k x Here x is he disance from he inner surface of he pipe, is ime and he k is he hermal diffusiviy. Suppose ha he inner surface emperaure ( 0, is given by a sinusoidal funcion of angular velociy as: ( 0, a cos b sin () (1) Proceedings 018,, 407; doi: /icem

2 Proceedings 018,, 407 of 6 Taking accoun of he fac ha he emperaure is supposed o decay wih x, we are ineresed in he following soluion of Equaion (1): x x k ( x, ae cos x be sin x k Then he emperaure on he ouer surface of he pipe ( h, is given as, h h k ( h, ae cos h be sin h k (3) (4) From Equaions () and (4) i is seen ha he ampliude of he ouer surface emperaure compared wih ha of he inner surface emperaure is reduced by raio R, and ha he phase of he ouer surface emperaure is delayed by Δp compared o he inner surface emperaure, and R and Δp are given as [5,6]: h R e (5) p h (6) 3. Inverse Mehod for Esimaing Temperaure and Thermal Sress Disribuions When he emperaure on he ouer surface ( h, is expressed by periodical funcion of ime wih cycle ime T, ( h, is expressed by he Fourier expansion as: A0 m m ( h, Am cos Bm sin (7) m1 T m1 T Based on he mahemaical relaionship beween he emperaure on he ouer surface ( h, and ha on he inner surface ( 0, discussed in he foregoing secion, he inner surface emperaure hisory is esimaed by amplifying ( h, by a raio 1/R and also by advancing he phase by Δp: A m (0, pm (8) T 0 Am m Bm cos pm sin m1 Rm T m1 Rm Here he reducion raio Rm and he phase lag Δpm for he m-h erm in he series are expressed as, R h k m e e h m kt (9) p m h h m kt (10) 4. Regularizaion Mehod The reducion facor Rm given by Equaion (9) is very small for very high frequency componen, or large value of m. Then high frequency flucuaion measured on he ouer surface is exaggeraed in he esimaion of he inner surface emperaure hisory. In he presence of measuremen noise, incorrec high frequency componens deeriorae he esimaion. We inroduce a regularizaion mehod o exclude he incorrec componens. This limi is deermined by considering he effec of noise o he coefficien of he Fourier coefficiens. When he error in ouer surface emperaure measuremen is of he order of, he expeced ampliude of he Fourier coefficien is in he order of

3 Proceedings 018,, of 6 / 6. Then regularizaion was conduced by ignoring he Fourier coefficien of he ouer surface emperaure, whose absolue value is smaller han his value. 5. Esimaion of Thermal Sress Disribuion The oal srain is composed of hermal srain 0 0 m / and mechanical srain : m E (11) Here E is Young s modulus and denoes hermal sress. The oal srain is independen of x due o uniform deformaion of he long pipe. Then he equilibrium of hermal sress gives rise o he following expression of he hermal sress. x h dx x0 E h (1) 6. Numerical Simulaions Numerical simulaions were made o examine he applicabiliy of he proposed mehod. A pipe 6 made of ype 316 sainless seel is considered. The hermal diffusiviy is m /s. Young s modulus is 193 GPa. The hickness of he pipe h is m, he cycle ime T is 100 s. I is assumed ha he inner surface emperaure is given by superposing hree waves, whose ampliude and cycle period are given in Table 1. As he error in he ouer surface emperaure measuremen, C, 5 C and 8 C were employed. Arificial error was inroduced using uniform random number. Table 1. Ampliude and cycle period of hree waves inroduced. Wave 1 3 Ampliude [ C] Cycle Period [s] Figure 1a c show he inner surface emperaure hisory by applying he esimaion mehod wihou regularizaion for he measuremen error of C, 5 C and 8 C, respecively. I is seen ha he esimaed inner surface emperaure (blue line) is flucuaed around he real one (red line). As he noise level is increased high frequency componens give rise o big flucuaion in he esimaed inner surface emperaure hisory. The flucuaion is pronounced for large measuremen error. Figure a c show he inner surface emperaure hisory by applying he esimaion mehod wih regularizaion for he measuremen error of C, 5 C and 8 C, respecively. I is found ha a reasonable esimaion is made by applying he regularizaion mehod even in he exisence of considerable observaion error, alhough he informaion abou he very high frequency componen is los. Using he emperaure hisory he esimaion of he hermal sress in he pipe was made. As an example he hermal sress a = 5 s for he hree waves examined in he foregoing are shown in Figure 3. Thermal faigue life can be evaluaed using he esimaed hermal sress and couning he number of cycles.

4 Proceedings 018,, of 6 (a) (b) (c) Figure 1. Esimaed emperaure variaion on inner surface wihou regulaion. (a) Observaion noise of C; (b) Observaion noise of 5 C; (c) Observaion noise of 8 C.

5 Proceedings 018,, of 6 (a) (b) (c) Figure. Esimaed emperaure variaion on inner surface wih regulaion. (a) Observaion noise of C; (b) Observaion noise of C; (c) Observaion noise of 8 C.

6 Proceedings 018,, of 6 7. Conclusions Figure 3. Thermal sress disribuion a = 5 s. An inverse analysis mehod was presened for esimaing he emperaure and hermal sress disribuions in he pipe from he emperaure hisory measured on he ouer surface. Regularizaion mehod was inroduced. I was found ha he proposed inverse analysis mehod wih regularizaion was useful for obaining a reasonable esimae of he inner surface emperaure and hermal sress. References 1. Meeker, J.; Segall, A.E.; Gondar, E. An Inverse Mehod for he Deerminaion of Thermal Sress-Inensiy Facors under Arbirary Thermal-Shocks. J. Press. Vessel Technol. Trans. ASME 018, 130, Segall, A.E.; Engels, D.; Hirsh, A. Transien Surface Srains and he Deconvoluion of Thermoelasic Saes and Boundary Condiions. J. Press. Vessel Technol. Trans. ASME 019, 131, Kasahara, N.; Takasho, H.; Yacumpai, A. Srucural Response Funcion Approach for Evaluaion of Thermal Sriping Phenomena. Nucl. Eng. Des. 00, 1, Kamide, H.; Igarashi, M.; Kawashima, S.; Kimura, N.; Hayashi, K. Sudy on Mixing Behavior in a Tee Piping and Numerical Analysis for Evaluaion of Thermal Sriping. Nucl. Eng. Des. 009, 39, Ioka, S.; Kubo, S.; Ochi, M.; Hojo, K. Developmen of Inverse Analysis Mehod of Hea Conducion and Thermal Sress for Elbow (Par I). J. Press. Vessel Technol. Trans. ASME 016, 138, Hojo, K.; Ochi, M.; Ioka, S.; Kubo, S. Developmen of Inverse Analysis Mehod of Hea Conducion and Thermal Sress for Elbow Par II. J. Press. Vessel Technol. Trans. ASME 016, 138, by he auhors. Licensee MDPI, Basel, Swizerland. This aricle is an open access aricle disribued under he erms and condiions of he Creaive Commons Aribuion (CC BY) license (hp://creaivecommons.org/licenses/by/4.0/).