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An iterative finite-element algorithm for solving two-dimensional nonlinear inverse heat conduction problems
KTH, School of Engineering Sciences (SCI), Physics, Nuclear Engineering.ORCID iD: 0000-0001-8743-7157
KTH, School of Engineering Sciences (SCI), Physics, Nuclear Engineering.
KTH, School of Engineering Sciences (SCI), Physics, Nuclear Engineering. Warsaw University of Technology, Poland.ORCID iD: 0000-0001-5595-1952
2018 (English)In: International Journal of Heat and Mass Transfer, ISSN 0017-9310, E-ISSN 1879-2189, Vol. 126, p. 281-292Article in journal (Refereed) Published
Abstract [en]

It is often useful to determine temperature and heat flux in multidimensional solid domains of arbitrary shape with inaccessible boundaries. In this study, an effective algorithm for solving boundary inverse heat conduction problems (IHCPs) is implemented: transient temperatures on inaccessible boundaries are estimated from redundant simulated measurements on accessible boundaries. A nonlinear heat equation is considered, where some of the material properties are dependent on temperature. The IHCP is reformulated as an optimization problem. The resulting functional is iteratively minimized using a conjugate gradient method together with an adjoint (dual) problem approach. The associated partial differential equations are solved using the finite-element package FEniCS. Tikhonov regularization is introduced to mitigate the ill-posedness of the IHCP. The accuracy of the implemented algorithm is assessed by comparing the solutions to the IHCP with the correct temperature values, on the inaccessible boundaries. The robustness of our method is tested by adding Gaussian noise to the initial conditions and redundant boundary data in the inverse problem formulation. A mesh independence study is performed.

Place, publisher, year, edition, pages
Elsevier, 2018. Vol. 126, p. 281-292
Keywords [en]
nonlinear inverse problem, Tikhonov regularization, finite element, FEniCS, adjoint
National Category
Energy Engineering
Identifiers
URN: urn:nbn:se:kth:diva-227719DOI: 10.1016/j.ijheatmasstransfer.2018.04.104ISI: 000442979300022Scopus ID: 2-s2.0-85047095749OAI: oai:DiVA.org:kth-227719DiVA, id: diva2:1205258
Funder
Swedish Radiation Safety Authority, 47385
Note

QC 20180530

Available from: 2018-05-12 Created: 2018-05-12 Last updated: 2018-10-20Bibliographically approved
In thesis
1. Experimental and analytical study of thermal mixing at reactor conditions
Open this publication in new window or tab >>Experimental and analytical study of thermal mixing at reactor conditions
2018 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

High-cycle thermal fatigue due to turbulent mixing of streams at distinct temperatures is an interdisciplinary issue affecting safety and life extension of existing reactors together with the design of new reactors. It is challenging to model damage and thermal loads arising from the above mixing.

In order to collect vast data sets for the validation of codes modeling turbulent thermal mixing under reactor conditions, temperatures were sampled at the inner surface of the vertical annular volume between two concentric 316LN stainless steel tubes. This annulus simplifies that between control-rod guide tube and stem in Swedish boiling water reactors (BWRs) Oskarshamn 3 and Forsmark 3. In 2008, several stems there were reported as broken or cracked from thermal fatigue. Cold water entered the annulus at 333 K, at axial level z = 0.15 m. It moved upward and mixed with hot water, which entered the annulus at 549 K, at z = 0.80 m. Pressure read 7.2 MPa. Hot and cold inlet temperatures and pressure match BWR conditions. The thermocouples attached to the inner tube could only acquire inner-surface temperatures at six locations, so the inner tube was translated and rotated about the z-axis to expand the measurement zone.

Mixing inhomogeneity was estimated from such measurements. In the cases examined, the inner-surface temperatures from areas with the highest mixing inhomogeneity show dominant frequencies lower than ten times the inverse of the experiment time.

The uncertainty of this temperature measurement appears to be equal to 1.58 K.

A large eddy simulation (LES) of mixing in the above annulus was conducted. Experimental boundary conditions were applied. The conjugate heat transfer between water and tubes was modeled. The wall-adapting local eddy viscosity (WALE) subgrid model was adopted. A finite element analysis (FEA) of the inner tube was performed using LES pressure and temperature as loads. Cumulative fatigue usage factors (CUFs) were estimated from FEA stress histories. To this end, the rainflow cycle-counting technique was applied. CUFs are highest between z = 0.65 m and z = 0.67 m. Cracking is predicted to initiate after 97 h. LES and experimental inner-surface temperatures agree reasonably well in relation to mean values, ranges, mixing inhomogeneity, and critical oscillation modes in areas sensitive to fatigue. LES inner-surface temperatures from areas with the highest CUFs show dominant frequencies lower than ten times the inverse of the simulation time.

A robust, effective iterative algorithm for reconstructing the transient temperature field in the inner tube from redundant boundary data was implemented and verified. Temperature-dependent properties were included. Initial conditions and over-specified boundary data in the inverse problem were perturbed with Gaussian noise to check the robustness of the solving method to noise.

Abstract [sv]

Termisk högcykelutmattning på grund av turbulent blandning av flöden av olika temperaturer är en tvärvetenskaplig fråga som har bäring på säkerhet, underhåll och livstidsförlängning av kärnkraftverk i drift tillsammans med utvecklingen av nya reaktorer. Det är utmanande att modellera delskador och termiska belastningar som härrör från denna blandning.

För att samla stora datamängder för validering av koder modellerande turbulent termisk blandning vid reaktordriftförhållanden, mättes temperaturer vid den inre ytan av den vertikala annulära (ringformade) kanalen mellan två koncentriska rör i 316LN rostfritt stål. Denna annulära kanal representerar den verkliga geometrin mellan styrstavsledrör och styrstavsförlängare i svenska kokvattenreaktorer (BWR) Oskarshamn 3 och Forsmark 3. Vid dessa reaktorer uppvisade ett antal styrstavsförlängare sprickbildning till följd av termisk utmattning under år 2008. I den annulära kanalen strömmade kallare vattenflöden vid 333 K, vid axiell nivå z = 0.15 m. Flödena rörde sig uppåt och blandades med varmare vattenflöden, vilka strömmade i den annulära kanalen vid 549 K, vid z = 0.80 m. Trycket sattes till 7.2 MPa. 333 K, 549 K och 7.2 MPa överensstämmer med BWR-förhållanden. Termoelement fastlödda vid det inre röret kunde endast mäta innerytans temperatur på sex platser. För att kunna mäta temperaturen i hela blandningsområdet kunde röret roteras från 0° till 360° och förflyttas vertikalt över en sträcka av 387 mm.

Blandningsinhomogenitet uppskattades från sådana mätningar. I de undersökta fallen visar innerytans temperatur från områden där blandningen är som minst homogen dominerande frekvenser som är lägre än tio gånger inversen till experimenttiden.

Osäkerheten i dessa temperaturmätningar visar sig vara 1.58 K.

En storvirvelsimulering (LES) av blandning i den annulära kanalen utfördes. Experimentella randvillkor applicerades. Den konjugerade värmeöverföringen mellan vatten och rör modellerades. De små virvlarna (oupplösta skalor) approximerades med hjälp av en turbulensmodell kallad WALE, som ger rätt asymptotiskt beteende för turbulent viskositet nära väggen. En finitelementanalys (FEA) av det inre röret utfördes med LES-tryck och temperatur som belastningar. Totala delskador uppskattades från FEA-spänning/tid kurvor. För detta ändamål applicerades regndroppsmetoden. De totala delskadorna når sina högsta värden mellan z = 0.65 m och z = 0.67 m. Sprickinitiering förväntas inträffa efter 97 timmar. Innerytans CFD-temperatur är i en rimlig överensstämmelse med experimentella data med avseende på medelvärde, omfång, blandningsinhomogenitet och kritiska svängande modalfunktioner som utgör temperaturtidsserier i områden som är utsatta för utmattning. Innerytans CFD-temperatur från områden med högsta totala delskador visar dominerande frekvenser som är lägre än tio gånger inversen till simuleringstiden.

En robust, effektiv, iterativ algoritm för att beräkna det transienta temperaturfältet i det inre röret utifrån överflödiga randdata implementerades och verifierades. Temperaturberoende egenskaper inkluderades. Initiala förhållanden och överflödiga randdata i det inversa problemet stördes av gaussiskt brus för att undersöka lösningsmetodens robusthet mot brus.

Place, publisher, year, edition, pages
Stockholm: KTH Royal Institute of Technology, 2018. p. 113
Series
TRITA-SCI-FOU ; 2018:45
Keywords
High-cycle thermal fatigue, Hilbert-Huang transform, large eddy, WALE, rainflow, inverse heat conduction, adjoint conjugate gradient
National Category
Energy Engineering
Research subject
Energy Technology; Physics
Identifiers
urn:nbn:se:kth:diva-236530 (URN)978-91-7873-001-8 (ISBN)
Public defence
2018-12-06, FB53, AlbaNova University Center, Roslagstullsbacken 21, Stockholm, 13:00 (English)
Opponent
Supervisors
Funder
Swedish Radiation Safety Authority, 47385
Note

QC 20181022

Available from: 2018-10-22 Created: 2018-10-20 Last updated: 2018-11-13Bibliographically approved

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