2. PANM 16 June 3-8, 2012, Dolní Maxov Jizera (1122) Ještěd (1012) Jiří Vala (vala.j@fce.vutbr.cz) Brno University of Technology, Faculty of Civil Engineering Numerical aspects of the identification of thermal characteristics using the hot-wire method

3. Numerical aspects of the identification of thermal characteristics using the hot-wire method 1.Motivation from production of refractory materials 2.Calculations due to Czech and European technical standards 3.Application of Bessel functions in polar coordinates 4.Experimental and numerical results for a model problem 5.General approach to identification problems in heat transfer

4. 1.Motivation from production of refractory materials (P-D Refractories CZ a.s., Moravské šamotové a lupkové závody Moravské šamotové a lupkové závody Velké Opatovice) Velké Opatovice)

6. 2. Calculations due to Czech and European technical standards Assumptions hidden in ČSN ISO 8894-1 Refractory materials – Determination of thermal conductivity – Part1: Hot-wire methods (cross-array and resistance thermometer): heat source Q [W/m] and thermal properties, both material characteristics and environmental conditions, are constant thermal mass of the heater is negligible heat conduction is only in radial direction, thus temperature can be expressed as T(r,t) [K], related to some initial status T 0 (r,t) [K] Heat conduction κ ∂T / ∂t = λ ∆T λ thermal conductivity[W/(mK)] κ volumetric heat capacity[J/m 3 ] α = λ / κ thermal diffusivity[m 2 /(sK)]

8. 3. Applications of Bessel functions of Bessel functions in polar coordinates in polar coordinates Bessel functions of the 1 st kind Bessel functions of the 2 nd kind

16. 4. Experimental and numerical results for a model problem

19. experimental results first (very rough) numerical estimate improved computational predictions (with increasing number of Bessel functions)

20. Direct, sensitivity and adjoint problems little number of parameters - classical Newton method acceptable … various improvements … parameters from spaces of infinite dimensions of parameters – conjugate gradient (or similar) techniques needed 5. General approach to identification problems in heat transfer

22. differential formulations

23. Least squares optimization and conjugate gradient algorithm Newton iterations conjugate gradient technique + Rothe sequences,Crank-Nicholson scheme + finite element methodwith Hermitean elements

24. Uncertainty analysis first observations – values of J

25. basalt volcano Bukovec (1005 ) ANK YOU FOR YOU ATTENTION. T HANK YOU FOR YOU ATTENTION. Q UESTIONS AND REMARKS ARE WELCOME.