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A boundary identification method for an inverse heat conduction problem with an application in ironmaking

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Abstract

A boundary identification problem in inverse heat conduction is studied, based on data from internal measurement of temperature and heat flux. Formulated as a sideways heat conduction equation, a spatial continuation technique is applied to extend the solution to a known boundary condition at the desired boundary position. Recording the positions traversed in the continuation for each time instant yields the boundary position trajectory and hence the solution of the identification problem. A prospective application of the method can be found in the ironmaking blast furnace, where it is desired to monitor the thickness of the accreted refractory wall based on measurement of its internal state. Simulations featuring noisy measurement data demonstrate the feasibility of the identification method for blast furnace wall thickness estimation.

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Abbreviations

C :

space-marching matrix, Eq. 15

\(\mathcal{D}\) :

computational domain, Fig. 1

I :

identity operator/matrix, Eq. 13

L 2 :

Hilbert space, Eq. 16

M :

bound for boundary temperature, Eq. 17

N :

discrete time interval number, Eq. 8

q :

measured heat flux, Eq. 2

s :

boundary trajectory, Eq. 1

S :

fixed boundary position, Eq. 17, Fig. 1

t :

time, Eq. 1

T :

temperature, Eq. 1

u :

measured temperature, Eq. 2

U :

temperature, Eq. 8

x :

co-ordinate, Eq. 1

Δ:

difference, Eq. 10

ε:

measurement error, Eq. 16

m:

measurement, Eq. 2

N :

endpoint, Eq. 1

q :

heat flux, Eq. 16

s :

boundary, Eq. 3

T:

transpose, Eq. 9

u :

temperature, Eq. 16

References

  1. Alifanov OM (1994) Inverse heat transfer problems, international series in heat and mass transfer. Springer, Berlin Heidelberg New York

    Google Scholar 

  2. Engl HW (1993) Regularization methods for the stable solution of inverse problems. Surv Math Ind 3:71–143

    Google Scholar 

  3. Berntsson F (2001) Numerical methods for solving a non-characteristic Cauchy problem for a parabolic equation. Technical Report LiTH-MAT-R-2001-17, Department of Mathematics, Linköping University, Linköping

  4. Eldén L (1997) Solving an inverse heat conduction problem by a “method of lines”. Trans ASME 119:406–412

    Google Scholar 

  5. Richtmyer RD, Morton KW (1967) Difference methods for initial-value problems, tracts in pure and applied mathematics, 2nd edn. Wiley Interscience, New York

    Google Scholar 

  6. Carasso A (1982) Determining surface temperatures from interior observations. SIAM J Appl Math 42:558–574

    Google Scholar 

  7. Levine HA (1983) Continuous data dependence, regularization, and a three lines theorem for the heat equation with data in a space-like direction. Ann Mat Pura Appl 134:267–286

    Google Scholar 

  8. Eldén L (1995) Numerical solution of the sideways heat equation. In: Engl HW, Rundell W (eds) Proceedings of the GAMM-SIAM Symposium, Gesellschaft für Angewandte Mathematik und Mechanik. GAMM-SIAM, Regensburg

  9. Carasso AS (1994) Overcoming Hölder continuity in ill-posed continuation problems. SIAM J Num Anal 31:1535–1557

    Google Scholar 

  10. Carasso AS (1999) Logarithmic convexity and the “slow evolution” constraint in ill-posed initial value problems. SIAM J Math Anal 30:479–496

    Article  Google Scholar 

  11. Carasso AS (1992) Space marching difference schemes in the nonlinear inverse heat conduction problem. Inverse Probl 8:25–43

    Article  Google Scholar 

  12. Carasso AS (1993) Slowly divergent space marching schemes in the inverse heat conduction problem. Numer Heat Transfer B 23:111–126

    CAS  Google Scholar 

  13. Fredman TP (2002) Accretions in the blast furnace stack—background factors. Can Metallurgical Q 41:475–486

    CAS  Google Scholar 

  14. Fredman TP (2002) Accretions in the blast furnace stack—diagnosis, modeling and control. Can Metallurgical Q 42:61–70

    CAS  Google Scholar 

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Authors and Affiliations

  1. Heat Engineering Laboratory, Åbo Akademi University, Biskopsgatan 8, 20500, Åbo, Finland

    T. P. Fredman

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  1. T. P. Fredman
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Correspondence to T. P. Fredman.

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Fredman, T.P. A boundary identification method for an inverse heat conduction problem with an application in ironmaking. Heat Mass Transfer 41, 95–103 (2004). https://doi.org/10.1007/s00231-004-0543-3

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  • DOI: https://doi.org/10.1007/s00231-004-0543-3

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