Two-dimensional linear transient inverse heat conduction problem - Boundary condition identification

Abstract

This article deals with the identification of unknown time-and space-dependent boundary conditions for systems driven by the heat equation. We first consider a one-dimensional and single-input problem, dealing with the identification of the time-dependent heat flux on one side of a one-dimensional linear thermal wall, from temperature and heat flux measurements on the other side. We then focus on a quenching process; our interest is to identify the time-and space-dependent heat flux on the boundary of a metal piece from temperature measurements performed inside the material (two-dimensional geometry). Those inverse problems are solved by use of a regularization method, and the solution is obtained by minimization of a quadratic criterion. Because of the linearity of the input-output relationship, the solution of this minimization is derived from the linear quadratic optimal control theory (resolution of a nonstationary Riccati equation). The robustness of the method for very small signal-to-noise ratio is shown. In the two-dimensional multi-input problem, the identification sensitivity to the localization of the measurement points is analyzed. In both cases, we consider input that are discontinuous in time, in order to show the method accuracy in the high-frequency domain. NomenclatureA = state equation matrix a = thermal diffusivity B = input matrix C = output matrix ERu = input error ERy = output error Id = identity matrix / = optimization criterion / = thickness of the thermal wall (one-dimensional geometry) MU = regularization matrix (two-dimensional geometry) n = number of spatial nodes P = Riccati matrix r = dimensionless radius (two-dimensional geometry) S = step function T = temperature Tf = temperature at node i t = dimensionless time variable Ucl = boundary conditions vector ( f wo-dimensional geometry) u = input: heat flux on face x = 1 (one-dimensional geometry) U f = /th input (two-dimensional geometry) M O > U 0i = constant reference values of u in criterion / x = dimensionless space variable (one-dimensional geometry) YJ = /th output (two-dimensional geometry) y = output: temperature on face x = 0 (onedimensional geometry) z = dimensionless height (two-dimensional geometry) P = regularization parameter (one-dimensional geometry) F = time horizon 8t = time step dx = space step or = standard deviation of measurement noise Subscripts m = measured variables r -r direction V = "true" variables z = z direction Superscripts T = transposed vector or matrix * = identified variables

Estimation of heat flux in array of jets quenching using experimental and inverse finite element method

“…Similar kind of works relating to the present investigation can be found in Woodbury (1991); Ramakrishnan and Howson (1992); Hernandez-Morales et al (1992); Guerrier and Benard (1993); Huang et al (1994); Filipovic et al (1995); Archambault and Azim (1995); Chen et al (1998); Prabhu and Ashish (2002), and Li and Wells (2005). In contrast to the present work, Wells et al (2001) observed the increase in heat flux when the water flow rate increases and also studied the influences of surface morphology and starting temperature in DC aluminum casting.…”

Simultaneous Reconstruction of Thermal Field and Retrieval of Parameters in a Cylindrical Enclosure

“…There are many problems in science and engineering where inverse problems find applications in which the objectives remain to predict one or more of the unknowns, such as boundary condition(s), temperature field, thermophysical properties, and geometrical parameters [1][2][3][4][5][6][7][8][9][10][11][12][13]. Guerrier and Benard [1] predicted boundary conditions in a transient conduction heat transfer problem in a 2-D rectangular enclosure.…”

“…Guerrier and Benard [1] predicted boundary conditions in a transient conduction heat transfer problem in a 2-D rectangular enclosure. Heat flux and temperature profiles at the tip of the tool during turning operation were estimated by Battaglia and Batsale [2].…”