Retrieval of multidimensional heat transfer coefficient distributions using an inverse BEM-based regularized algorithm: numerical and experimental results

doi:10.1016/j.enganabound.2004.08.006

Engineering Analysis with Boundary Elements

Volume 29, Issue 2, February 2005, Pages 150-160

https://doi.org/10.1016/j.enganabound.2004.08.006 Get rights and content

Abstract

The surface distribution of heat transfer coefficients (h) is often determined point by point using surface temperature measurements of the tested object, initially at a uniform temperature and impulsively imposed with a convective boundary condition, and the solution to the transient heat conduction equation for a semi-infinite medium. There are many practical cases where this approach fails to adequately model the temperature field and, consequently, leads to erroneous h values. In this paper, we present an inverse BEM-based approach for the retrieval of spatially varying h distributions from surface temperature measurements. In this method, a convolution BEM marching scheme is used to solve the conduction problem. At each time level, a regularized functional is minimized to estimate the current heat flux and simultaneously smooth out uncertainties in calculated h values due to experimental uncertainties in measured temperatures. Newton's cooling law is then invoked to compute h. Results are presented from a numerical simulation and from an experiment. It is also shown that the method can be readily applied to steady-state.

Section snippets

Introduction and problem definition

Non-intrusive experimental techniques to determine convective heat transfer coefficients (h) currently in use both in industry and academia essentially rely on certain theoretical models and temperature driven physical phenomenon to accurately determine surface temperature histories. For instance, techniques which rely on characteristic color changes of liquid crystal films at a given temperature [1], [2] and the more recent laser induced fluorescence-based method [3], both rely on the

Time-dependent boundary element method for diffusion

The boundary element method (BEM) is a numerical implementation of boundary integral methods for solution of field problems and is described in detail in classic texts [19]. The BEM is now a well established numerical method which can be efficiently used to solve heat conduction problems in linear and nonlinear media [20], non-homogeneous isotropic media [21], [22], and non-homogenous anisotropic media [23], [24], using boundary-only discretization. As discussed previously, a distinct feature

Inverse problem definition and functional regularization

Film coefficient modeling is actually a direct problem due to the fact that time history temperature measurements are used as imposed first kind boundary conditions at the convective boundaries. However, small measurement noise at such boundaries reflect into large deviations of computed heat fluxes and consequently erroneous values of h. A remedy for this sensitivity problem is to reformulate h retrieval as an inverse problem and to regularize the functional. To this end, the following

Numerical implementation

Following standard BEM [19] the domain boundary, Γ, is discretized using N constant elements modeling the geometry as piecewise linear and the temperature and its flux as constant. Influence coefficients $H_{i j}^{F p}$ and $G_{i j}^{F p}$ are integrated analytically in time and via quadratures in space, and, in 2-D, $H_{i j}^{F p} = \int_{Γ_{j}} \frac{ρ c}{2 π r^{2}} {\exp (\frac{- ρ c r^{2}}{4 k (t_{F} - t_{p - 1})}) - \exp (\frac{- ρ c r^{2}}{4 k (t_{F} - t_{p})})} [(x - x_{i}) n_{x} + (y - y_{i}) n_{y}] d Γ$ where n_x and n_y are the x- and y-components of the outward-drawn normal, while analytical integration of coefficients $G_{i j}^{F p}$

Numerical example in a backward-facing step under convection

A numerical example is now presented to validate the inverse BEM formulation. The backward-facing step geometry is shown in Fig. 3 along with imposed boundary conditions. Fig. 4(a) presents the backward-facing step discretization consisting of 60 equally spaced constant boundary elements. Thermophysical properties of pure aluminum are used: k=2.37 W/cm K, ρ=0.002702 kg/cm³, and c=903 J/kg. The initial temperature of the field is T_i=30 °C and the convective ambient temperature is T_a=800 °C. The

Experiment

A schematic of the test set up is displayed in Fig. 9. A blower supplies ambient air to the flow conditioning section. Upon exiting the blower, inlet air is heated from room temperature to 60 °C via a heat exchanger. Water is supplied to the heat exchanger through an externally heated water supply and pump system that is controlled to maintain constant flow temperature based on a flow temperature measurement prior to the test section. The flow then passes through a flow conditioning section and

Conclusion

In this paper, we develop a BEM-based inverse algorithm to retrieve multi-dimensional h distributions from transient surface temperature measurements. A regularization procedure serves to provide a ‘best fit’ through noisy input data. Results presented for studies over a square, a backward facing step, and a rib-like structure demonstrate accuracy and robustness of the method. Results from an experiment using TLC are also presented.

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These parameters may be estimated in an inexpensive manner using inverse methods. Over the past decades, inverse analysis has been extensively used to determine the thermal conductivity (constant, temperature-dependent, and spatially varying parameter) and the convection heat transfer coefficient [1–31], the heat flux [32–34], and the boundary shape of bodies [35–39] using temperature measurement taken at some points inside the body or on some part of the boundary. However, the inverse heat transfer problems are ill-posed and mathematically challenging because the ill-posed problems are inherently unstable and very sensitive to small errors in input data.
A numerical inverse analysis based on explicit sensitivity coefficients is developed for the simultaneous estimation of heat flux and heat transfer coefficient imposed on different parts of boundary of a general irregular heat conducting body made of functionally graded materials with spatially varying thermal conductivity. It is assumed that the thermal conductivity varies exponentially with position in the body. The body considered in this study is an eccentric hollow cylinder. The heat flux is applied on the cylinder inner surface and the heat is dissipated to the surroundings through the outer surface. The numerical method used in this study consists of three steps: 1) to apply a boundary-fitted grid generation (elliptic) method to generate grid over eccentric hollow cylinder (an irregular shape) and then solve for the steady-state heat conduction equation with variable thermal conductivity to compute the temperature values in the cylinder, 2) to propose a new explicit sensitivity analysis scheme used in inverse analysis, and 3) to apply a gradient-based optimization method (in this study, conjugate gradient method) to minimize the mismatch between the computed temperature on the outer surface of the cylinder and simulated measured temperature distribution. The inverse analysis presented here is not involved with an adjoint equation and all the sensitivity coefficients can be computed in only one direct solution, without the need for the solution of the adjoint equation. The accuracy, efficiency, and robustness of the developed inverse analysis are demonstrated through presenting a test case with different initial guesses.
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Eduardo Divo used an inverse BEM-based approach to obtain the heat flux and the heat transfer coefficient over the surface of a backward-facing step geometry. It was also found that the 1D transient conduction model failed to accurately reproduce the heat transfer coefficient [9]. Sathyanarayanan and Shih measured the heat transfer coefficient over a channel with pin fins and a duct with ribs using time-accurate and steady conjugate computational fluid dynamics analyses and transient methods [10].
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Contrary to IHTP, the well-posed direct heat transfer problems are concerned with the determination of the temperature distribution in the body by having the boundary conditions, the thermo-physical properties, the body geometrical configuration, and the heat flux applied at some part of the body boundary [1,2]. The inverse analysis has been extensively employed in the estimation of the thermal conductivity and the heat transfer coefficient [3–32], the heat flux [16,33–39], and the determination of the boundary shape of bodies [40–46], to name a few. The numerical algorithm presented in this study is sufficiently general by which we can determine the variable thermal conductivity of a general two-dimensional region (heat conducting body) with Neumann and Robin conditions at the boundaries as long as the general two dimensional region can be mapped onto a regular computational domain.
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Local differences up to 15% between the 1D analysis and the consideration of lateral heat conduction effects were detected. A combination of a BEM as evaluation method and wideband TLC for measurements of TS was presented by Divo et al. [12]. In contrast Lin and Wang [13] took the measurements of TS as a boundary condition for an implicit second order FDM.
The standard transient evaluation technique for the experimental determination of heat transfer coefficients with Thermochromic Liquid Crystals is based on a one-dimensional heat conduction model inside a semi-infinite wall. However, complex flow situations induced by obstacles like cylinders, skewed ribs or vortex generators lead to spatially varying heat transfer coefficients and therefore to lateral conduction effects. Kingsley-Rowe et al. [1] developed an analytical empirical correction method considering these effects for flat two-dimensional heat conduction problems and an ideal step change of the fluid reference temperature. This method is expanded to capture three-dimensional heat conduction situations and slow transient fluid reference temperature changes. Transient simulations for twelve different heat transfer coefficient distributions and three different reference temperature histories constitute the pool of data for the determination of the empirical constants and the validation of the method. The results showed an appropriate correction of the heat transfer coefficients received by the standard evaluation method. Therefore the presented method offers the opportunity to consider three-dimensional heat conduction effects in a fast and efficient way in combination with the standard evaluation process.

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Retrieval of multidimensional heat transfer coefficient distributions using an inverse BEM-based regularized algorithm: numerical and experimental results

Abstract

Section snippets

Introduction and problem definition

Time-dependent boundary element method for diffusion

Inverse problem definition and functional regularization

Numerical implementation

Numerical example in a backward-facing step under convection

Experiment

Conclusion

Int J Heat Mass Transfer

Int J Solids Struct

Int J Solids Struct

Eng Anal

Int J Heat Mass Transfer

Eng Anal

Eng Anal

Evaluation of a method for heat transfer measurement and visualization using a composite of a heater and liquid crystals

ASME J Heat Transfer

A new hue capturing technique for the quantitative interpretation of liquid crystal images used in convective heat transfer studies

ASME J Turbomach

Automated cubic spline anchored grid pattern algorithm for the high resolution detection of subsurface cavities by the IR-CAT method

Numer Heat Transfer, Part B: Fundam

Solution of the inverse geometric problem for the detection of subsurface cavities by the IR CAT method

Nondestructive detection of cavities by an inverse elastostatics method

Eng Anal

Shape design sensitivity analysis for non-linear anisotropic heat conducting solids and shape optimization by the BEM

Int J Numer Meth Eng

Differential and BIE formulations of optimal heating of solids