PANM 16PANM 16 June 3-June 3-8, 2012, Doln8, 2012, Dolní Maxoví Maxov

JizeraJizera (1122) (1122) Je Ještědštěd (1012) (1012)

JiJiří Vala (valaří Vala (vala..jj@fce.vutbr.cz)@fce.vutbr.cz)Brno University of Technology, Faculty of Civil EngineeringBrno University of Technology, Faculty of Civil Engineering

Numerical aspects of the identification Numerical aspects of the identification of thermal characteristics using the hot-wire methodof thermal characteristics using the hot-wire method

Numerical aspects of the identification of thermal characteristics using the hot-wire method

1.Motivation from production of refractory materials2.Calculations due to Czech and European technical standards3.Application of Bessel functions in polar coordinates4.Experimental and numerical results for a model problem5.General approach to identification problems in heat transfer

1.1. Motivation from pMotivation from production roduction of refractory materialsof refractory materials(P(P--D RefractoriesD Refractories CZ a.s., CZ a.s., Moravské šamotové a lupkové závodyMoravské šamotové a lupkové závody Velké Opatovice)Velké Opatovice)

2. Calculations due to Czech and European technical standards2. 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 T0(r,t) [K]

Heat conductionκ ∂T / ∂t = λ ∆Tλ thermal conductivity [W/(mK)]κ volumetric heat capacity [J/m3]α = λ / κ thermal diffusivity [m2 /(sK)]

3. Applications 3. Applications of Bessel functions of Bessel functions in polar coordinatesin polar coordinates

Bessel functions of the 1Bessel functions of the 1stst kind kind

Bessel functions of the 2Bessel functions of the 2nd nd kindkind

4. Experimental and numerical results for a model problem

experimental results

first (very rough) numerical estimate

improved computational predictions(with increasing number of Bessel functions)

Direct, sensitivity and adjoint problemsDirect, sensitivity and adjoint problems

little numberof parameters -classical Newton method acceptable

…

various improvements

…

parameters from spaces of infinite dimensionsof parameters –conjugate gradient (or similar) techniques needed

5. General approach to identification problems in heat transfer

differential formulations

Least squares optimization Least squares optimization and conjugate gradient algorithmand conjugate gradient algorithm

Newton iterations

conjugate gradient technique

+ Rothe sequences,Crank-Nicholson scheme

+ finite element methodwith Hermitean elements

Uncertainty analysisUncertainty analysis

first observations – values of J

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THANK YOU FOR YOU ATTENTION.ANK YOU FOR YOU ATTENTION.QUESTIONS AND REMARKS ARE WELCOME.QUESTIONS AND REMARKS ARE WELCOME.