Estimation of front surface temperature and heat flux of a locally heated plate from distributed sensor data on the back surface

Abstract We present a new method of solving the three-dimensional inverse heat conduction (3D IHC) problem with the special geometry of a thin sheet. The 3D heat equation is first simplified to a 1D equation through modal expansions. Through a Laplace transform, algebraic relationships are obtained that express the front surface temperature and heat flux in terms of those same thermal quantities on the back surface. We expand the transfer functions as infinite products of simple polynomials using the Hadamard Factorization Theorem. The straightforward inverse Laplace transforms of these simple polynomials lead to relationships for each mode in the time domain. The time domain operations are implemented through iterative procedures to calculate the front surface quantities from the data on the back surface. The iterative procedures require numerical differentiation of noisy sensor data, which is accomplished by the Savitzky–Golay method. To handle the case when part of the back surface is not accessible to sensors, we used the least squares fit to obtain the modal temperature from the sensor data. The results from the proposed method are compared with an analytical solution and with the numerical solution of a 3D heat conduction problem with a constant net heat flux distribution on the front surface.

[1]  J. K. Chen,et al.  Real-time solution of heat conduction in a finite slab for inverse analysis , 2010 .

[2]  Kuo-Chi Liu,et al.  Solution of an Inverse Heat Conduction Problem in a Bi-Layered Spherical Tissue , 2010 .

[3]  M. S. Gadala,et al.  Effectiveness and Efficiency of Particle Swarm Optimization Technique in Inverse Heat Conduction Analysis , 2009 .

[4]  A. Savitzky,et al.  Smoothing and Differentiation of Data by Simplified Least Squares Procedures. , 1964 .

[5]  Małgorzata Jakubowska,et al.  Adaptive-degree polynomial filter for voltammetric signals , 2004 .

[6]  Masanori Monde,et al.  Estimation of Surface Temperature and Heat Flux Using Inverse Solution for One-Dimensional Heat Conduction , 2003 .

[7]  Cha'o-Kuang Chen,et al.  The boundary estimation in two-dimensional inverse heat conduction problems , 1996 .

[8]  K. Brown,et al.  Graduate Texts in Mathematics , 1982 .

[9]  J. K. Chen,et al.  Inverse Heat Conduction in a Composite Slab With Pyrolysis Effect and Temperature-Dependent Thermophysical Properties , 2010 .

[10]  Aleksey V. Nenarokomov,et al.  Uncertainties in parameter estimation: the optimal experiment design , 2000 .

[11]  Alok Sutradhar,et al.  Transient heat conduction in homogeneous and non-homogeneous materials by the Laplace transform Galerkin boundary element method , 2002 .

[12]  X. Xue,et al.  Comparisons and improvements concerning the accuracy and robustness of inverse heat conduction algorithms , 2005 .

[13]  B. Blackwell,et al.  Inverse Heat Conduction: Ill-Posed Problems , 1985 .

[14]  Boundary element approach for inverse heat conduction problems; Application to a bidimensional transient numerical experiment , 1991 .

[15]  Jianwen Luo,et al.  Properties of Savitzky-Golay digital differentiators , 2005, Digit. Signal Process..

[16]  J. K. Chen,et al.  Temperature and heat flux estimation from sampled transient sensor measurements , 2010 .

[17]  O. Burggraf.  An Exact Solution of the Inverse Problem in Heat Conduction Theory and Applications , 1964 .

[18]  J. Howell,et al.  Advanced Heat and Mass Transfer , 2010 .

[19]  S. K. Kim.  Resolving the Final Time Singularity in Gradient Methods for Inverse Heat Conduction Problems , 2010 .

[20]  D. Maillet,et al.  Thermal Quadrupoles: Solving the Heat Equation through Integral Transforms , 2000 .

[21]  G. E. Myers,et al.  Analytical Methods in Conduction Heat Transfer , 1998 .

[22]  D. P. Sekulic,et al.  Extended surface heat transfer , 1972 .

[23]  Cheng-Hung Huang,et al.  A three-dimensional inverse heat conduction problem in estimating surface heat flux by conjugate gradient method , 1999 .

[24]  E. Sparrow,et al.  The Inverse Problem in Transient Heat Conduction , 1964 .

[25]  Tahar Loulou,et al.  A Comparative Study of Heat Flux and Temperature Based Objective Functional to Solve Inverse Heat Conduction Problems , 2009 .

[26]  S. Hannani,et al.  Structured Multiblock Body-Fitted Grids Solution of Transient Inverse Heat Conduction Problems in an Arbitrary Geometry , 2008 .

[27]  J. K. Chen,et al.  Inverse Heat Conduction Using Measured Back Surface Temperature and Heat Flux , 2010 .

[28]  O. Alifanov.  Inverse heat transfer problems , 1994 .

[29]  A. Segall.  An Inverse Solution for Determining Arbitrary Boundary-Conditions using a Least-Squares Approach , 2005 .

[30]  M. N. Özişik,et al.  Inverse Heat Transfer: Fundamentals and Applications , 2000 .

[31]  P. A. Gorry.  General least-squares smoothing and differentiation by the convolution (Savitzky-Golay) method , 1990 .