Inverse analysis of forced convection in micro-channels with slip flow via integral transforms and Bayesian inference

The present work addresses the direct and inverse problems for convective heat transfer with incompressible laminar gas flow in micro-channels, within the range of validity of the slip-flow regime. The direct problem analysis combines the classical integral transform method and the generalized integral transform technique (GITT), by analytically solving the two-dimensional steady-state convection problem and finding a hybrid numerical-analytical solution for the required eigenvalue problem. The inverse problem analysis makes use of the accuracy and robustness of the direct problem solution and focus on the simultaneous identification of the momentum and thermal accommodation coefficients, related to gas flow and heat transfer within micro-channels, besides the usually unknown boundary condition parameters, here represented by the external Biot number. The inverse analysis is based on the availability solely of temperature measurements at the channel external wall, along its length, as obtained for instance via infrared camera thermography. A Bayesian inference approach is adopted in the solution of the identification problem based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings sampling algorithm. A typical example of slip flow in parallel-plates micro-channel is selected to illustrate both the direct and inverse problems solution approaches.     

Application of Wavelet in Highly Ill-Posed Inverse Heat Conduction Problem

In this work, the prefiltering of the sensor data is taken into consideration when solving an inverse heat conduction problem. The temperature data obtained from each sensor is considered as a discrete signal, and discrete wavelet transform in a multi-resolution filter bank structure is utilized for the signal analysis, after which wavelet denoising algorithm is applied to remove noise from data signal. Subsequently, noisy and denoised temperatures are separately used as input data to an inverse heat conduction problem for comparison. The inverse heat conduction problem considered in this article is an inverse volumetric heat source problem, and it is solved using the conjugate gradient method along with the associated adjoint problem used to obtain the gradient of the objective function. Three sets of results in two case studies are compared (i.e., the result obtained from non-noisy data, noisy data, and denoised data). In the case of noisy data, iterative regularization is used to regularize the solution. The root mean square error of the estimated heat source from denoised data is reduced approximately by a factor of seven to nine as compared to those obtained from noisy data.     

The BEM for point heat source estimation: application to multiple static sources and moving sources

This paper deals with an inverse problem, which consists of the identification of point heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The two-dimensional and three-dimensional linear heat conduction problems are considered here. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a quadratic norm. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. A numerical example is presented for the 3D application.     

Multiple transient point heat sources identification in heat diffusion: application to experimental 2D problems

This paper deals with an inverse problem, which consists of the experimental identification of line heat sources in a homogeneous solid in transient heat conduction. The location and strength of the line heat sources are both unknown. For a single source we examine the case of a source which moves in the system during the experiment. The identification procedure is based on a boundary integral formulation using transient fundamental solutions. The discretized problem is non-linear if the location of the line heat sources is unknown. In order to solve the problem we use an iterative procedure to minimize a cost function comparing the modelled heat source term and the measurements. The proposed numerical approach is applied to experimental 2D examples using measurements provided by an infrared scanner for surface temperatures and heat fluxes. In some particular examples, internal thermocouples can be used. A time regularization procedure associated to future time-steps is used to correctly solve the ill-posed problem.     

Inverse problem on the identification of temperature and thermal stresses in an FGM hollow cylinder by the surface displacements

A problem on the identification of time-dependent temperature on one of the limiting surfaces of a radially inhomogeneous hollow cylinder is formulated and solved under the temperature and radial displacement given on the other limiting surface. The analysis of temperature and thermal stress distribution in the cylinder is performed. The solution has been constructed by the reduction to an inverse thermoelasticity problem. By making use of the finite difference method, a stable solution algorithm is suggested for the analysis of inverse problem. The solution technique is verified numerically by making use of the solution to a relevant direct problem. It is shown that the proposed technique can be e?ciently used for the identification of a heat flux or unknown parameters (the surrounding temperature or the heat-exchange coe?cient) in the third-kind boundary conditions.     

A spectral method for the estimation of a thermomechanical heat source from infrared temperature measurements

The inverse problem of 2D time-dependent heat source reconstruction is solved. The scientific objectives are the quantification of thermal effects associated to the mechanical deformation of materials during tensile tests. The experiment provides infrared measurements of the specimen’s surface temperature and the inverse algorithm aims at providing a volumic heat source that is free of errors due to heat diffusion. This algorithm is based on an analytical solution of the direct problem in the Laplace-Fourier domain. The solution proposed here is compared to a previously used method [1] based on an adjoint formulation and a regularization of Tikhonov type. This allows to check the validity of the results.     

Reconstruction of a space and time dependent heat source from finite measurement data

This paper deals with an inverse problem of determining a heat source function in heat conduction equations when the solution is known in a discrete point set. Being different from other ordinary inverse source problems which are often dependent on only one variable, the unknown coefficient in this paper not only depends on the space variable x, but also depends on the time t. On the basis of the optimal control framework, the inverse problem is transformed into an optimization problem. The existence and necessary condition of the minimizer for the cost functional are established. The convergence of the minimizer as the mesh parameters tend to zero is also proved. The conjugate gradient method is applied to the inverse problem and some typical numerical experiments are performed in the paper. The numerical results show that the proposed method is stable and the unknown heat source is recovered very well.     

Estimation of unknown heat source function in inverse heat conduction problems using quantum-behaved particle swarm optimization

The estimation of temporal dependent heat source in transient heat conduction problem is investigated. A stochastic method known as quantum-behaved particle swarm optimization (QPSO) is used to estimate the heat source without a priori information on its functional form, which is classified as the function estimation by inverse calculation. Because of the ill-posedness of this kind of inverse problems, Tikhonov regularization method is applied in this paper to stabilize the solution. Numerical experiments indicate the validity and stability of the QPSO method. Comparison with the conjugate gradient method (CGM) is also presented in this paper.     

INVERSE FORCED CONVECTION PROBLEM OF SIMULTANEOUS ESTIMATION OF TWO BOUNDARY HEAT FLUXES IN IRREGULARLY SHAPED CHANNELS

This article deals with the use of the conjugate gradient method of function estimation for the simultaneous identification of two unknown boundary heat fluxes in channels with laminar flows. The irregularly shaped channel in the physical domain is transformed into a parallel plate channel in the computational domain by using an elliptic scheme of numerical grid generation. The direct problem, as well as the auxiliary problems and the gradient equations, required for the solution of the inverse problem with the conjugate gradient method are formulated in terms of generalized boundary-fitted coordinates. Therefore, the solution approach presented here can be readily applied to forced convection boundary inverse problems in channels of any shape. Direct and auxiliary problems are solved with finite volumes. The numerical solution for the direct problem is validated by comparing the results obtained here with benchmark solutions for smoothly expanding channels. Simulated temperature measurements containing random errors are used in the inverse analysis for strict cases involving functional forms with discontinuities and sharp corners for the unknown functions. The estimation of three different types of inverse problems are addressed in the paper: (i) time-dependent heat fluxes; (ii) spatially dependent heat fluxes; and (iii) time and spatially dependent heat fluxes.     

Implementation of an analytical two-dimensional inverse heat conduction technique to practical problems

Two improvements to practical implementation of a solution to the two-dimensional inverse heat conduction problem are presented. The first concept is useful for experimental data with strong or irregular fluctuations in time. The second procedure improves the spatial resolution for problems where the source of the surface heat flux distribution is moving along the surface. The method is tested against analytical solutions and data from quench cooling experiments. Both procedures are found to enhance the quality of the inverse solution results.     

Analysis of boundary inverse heat conduction problems using space marching with Savitzky-Gollay digital filter

This paper presents a method by which boundary inverse heat conduction problems can be analyzed. A space marching algorithm is used for formulating and solving parabolic and hyperbolic inverse heat conduction problems. The solution of numerical examples shows that a combination of the digital filter with the hyperbolic approximation of inverse heat conduction problem increases the stability of the results without loss of resolution. The validity of numerical solution for the inverse problem is examined by comparing the obtained results with the direct solution of the problem.     

MULTIPLE TRANSIENT POINT HEAT SOURCES IDENTIFICATION IN HEAT DIFFUSION: APPLICATION TO NUMERICAL TWO- AND THREE-DIMENSIONAL PROBLEMS

This article deals with an inverse problem, which consists of the location and strength identification of multiple-point heat sources in transient heat conduction. The identification procedure is based on a boundary integral formulation using space and time Green functions. The discretized problem is nonlinear if the location of the point heat sources is unknown. In order to reduce the sensitivity of the solution to errors, we use the future time step procedure associated to a Tikhonov regularization procedure. The proposed numerical approach is applied to numerical two- and three-dimensional examples.     

Bayesian Estimation of Temperature-Dependent Thermophysical Properties and Transient Boundary Heat Flux

In this article, we apply a Bayesian approach for the simultaneous identification of volumetric heat capacity, thermal conductivity, and boundary heat flux, in a one-dimensional nonlinear heat conduction problem. The Markov chain Monte Carlo sampling approach, implemented in the form of the Metropolis–Hastings algorithm, was used for the solution of the inverse problem. Simulated temperature measurements were used in the inverse analysis in order to examine the accuracy and stability of the overall approach. Independent measurement data were used to construct the prior model for the coefficients to be estimated. The approach is also applied to experiments involving the heating of a reference material with an oxyacetylene torch.     

Inverse determination of heat generation sources in two-dimensional homogeneous solids: Application to orthotropic medium

An implementation of the Network Simulation Method for determining the time-dependent heat generation of a source in a 2-D heat conduction problem under first and second type boundary conditions is presented. The thermal properties of the orthotropic solid are temperature-dependent. No prior information of the functional dependence of the time-dependent source was required. Input data for this inverse problem comprise the temperature history (measurements) at a particular location of the boundary. The common iterative least-squares approach, typical for this kind of ill-posed problem, is used in the estimation, while a piecewise continuous function is proposed as an approximate solution for the estimations. An application to estimate the triangular time dependence of the heat source generation, with error-affected measurements, is shown.     

Recovering a general space-time-dependent heat source by the coupled boundary integral equation method

As to recover a time-dependent heat source under an extra temperature measured at an interior point, we can reformulate it to be a three-point boundary value problem. We can develop a coupled boundary integral equation method, wherein by selecting two sets of adjoint test eigenfunctions in two sub domains and using polynomials as the trial functions of unknown heat source, the time-dependent heat source is recovered very well and quickly. Four numerical examples, including a discontinuous one, demonstrate the efficiency for the ill-posed inverse heat source problem in a large time duration and under a large noise up to 10–30%. Then, selecting three sets of adjoint test eigenfunctions in three domains: problem domain and two sub domains, and using the Pascal polynomials as trial functions, the unknown space-time-dependent heat source is recovered very fast and accurately from the solution of three coupled boundary integral equations.     

On estimating thermal diffusivity using analytical inverse solution for unsteady one-dimensional heat conduction

A modified procedure for calculating the thermal diffusivity of solids based on temperature measurements at two points and the semi-infinite boundary condition is presented. The method makes use of a solution to the unsteady one-dimensional inverse heat conduction problem for the semi-infinite solid. The procedure gives accurate results based on temperature changes produced by an arbitrary fluctuating heat flux source at the boundary.     

Application of optimization techniques and the enthalpy method to solve a 3D-inverse problem during a TIG welding process

Tungsten inert Gas (TIG) welding takes place in an atmosphere of inert gas and uses a tungsten electrode. In this process heat input identification is a complex task and represents an important role in the optimization of the welding process. The technique used to estimate the heat flux is based on solution of an inverse three-dimensional transient heat conduction model with moving heat sources. The thermal fields at any region of the plate or at any instant are determined from the estimation of the heat rate delivered to the workpiece. The direct problem is solved by an implicit finite difference method. The system of linear algebraic equations is solved by Successive Over Relaxation method (SOR) and the inverse problem is solved using the Golden Section technique. The golden section technique minimizes an error square function based on the difference of theoretical and experimental temperature. The temperature measurements are obtained using thermocouples at accessible regions of the workpiece surface while the theoretical temperatures are calculated from the 3D transient thermal model.     

A parameter estimation approach to solve the inverse problem of point heat sources identification

This paper deals with an inverse problem that consists of the identification of multiple line heat sources placed in a homogeneous domain. In the inverse problem under investigation the location and strength of the line heat sources are unknown. The estimation procedure is based on the boundary element method. As the discrete problem is non-linear if the location of the line heat sources is unknown, an iterative procedure has to be applied to find out the location of the sources. The proposed approach has been tested for steady and transient experiments. In the present study we propose an original approach to solve the steady problem. As in the steady heat conduction case we have a limited number of unknown for each source, a “parameter estimation” approach can be applied to estimate the sources. Using the techniques of parameter estimation, we can also estimate the confidence interval of the estimated locations, which permits to design an optimal experiment. We intend to present some numerical and experimental 2D results.     

Inverse shape design for heat conduction problems via the ball spine algorithm

ABSTRACT

In this article, a novel iterative physical-based method is introduced for solving inverse heat conduction problems. The method extends the ball spine algorithm concept, originally developed for inverse fluid flow problems, to inverse heat conduction problems by employing a subtle physical-sense rule. The inverse problem is described as a heat source embedded within a solid medium with known temperature distribution. The object is to find a body configuration satisfying a prescribed heat flux originated from a heat source along the outer surface. Performance of the proposed method is evaluated by solving many 2-D inverse heat conduction problems in which known heat flux distribution along the unknown surface is directly related to the Biot number and surface temperature distribution arbitrarily determined by the user. Results show that the proposed method has a truly low computational cost accompanied with a high convergence rate.     

Detection of hot spot through inverse thermal analysis in superconducting RF cavities

An inverse heat conduction problem in a superconducting radio frequency (SRF) cavity is examined. A localized defect is simulated as a point-heating source on the inner surface (RF surface) of the evacuated niobium cavity. Liquid helium acts as a coolant on the outer surface of the cavity. By measuring the outer surface temperature profile of the cavity using relatively few sensors, the temperature and location of a hot spot on the inner surface of the niobium are calculated using an inverse heat conduction technique. The inverse method requires a direct solution of a three-dimensional heat conduction problem through the cavity wall thickness along with temperature measurements from sensors on the outer surface of the cavity, which is immersed in liquid helium. A non-linear parameter estimation program then estimates the unknown location and temperature rise of the hot spot inside the cavity. The validation of the technique has been done through an experiment conducted on a niobium sample at room temperature.