Three-dimensional Inverse Heat Conduction Problem Research Articles

Inverse Estimation of Effective Thermal Conductivity of Multilayer Materials Considering Thermal Contact Resistance

Abstract Multilayer materials have been widely used in engineering applications, attributed to excellent thermo-mechanical performances. However, the thermal or mechanical properties of multilayer materials remain elusive, owing to contact behaviors etc. factors. In order to address this issue, an innovative method is employed to estimate the effective thermal conductivity (ETC) of multilayer materials considering thermal contact resistance (TCR) between layers, and the equivalence performance is investigated by solving three-dimensional inverse heat conduction problems. First, the equivalence method is validated by available experimental data of a multilayered insulation composite material. Then, the precision of different equivalence methods is compared, and the results indicate that the anisotropic equivalence method has higher accuracy than the isotropic and orthotropic equivalences for the five-layer material in the present work. Finally, the robustness and stability of the anisotropic equivalence method are evaluated in detail. The present work provides a new alternative method for predicting the effective thermal conductivity of multilayer materials.

Numerical and experimental studies of the hypersonic flow around a cube at incidence

In order to improve predictions of the on-ground casualty risk associated with the uncontrolled atmospheric re-entry of satellites from Low Earth Orbit, there is significant research interest in the development of engineering models of hypersonic heating rates to faceted shapes. A key part of developing such models is generating accurate datasets of the heat fluxes experienced by faceted shapes at various orientations in hypersonic flows. In this work, we use wind tunnel experiments and CFD simulations to study the hypersonic flow around a cuboid geometry at 5° incidence in a Mach 5 flow at Reynolds numbers of 79.5×103, 109×103 and 148×103. The wind tunnel data are obtained in the University of Manchester's High SuperSonic Tunnel and consist of schlieren images and temperature histories collected using infrared thermography. These temperature histories are used to calculate experimental heat fluxes by solving a three-dimensional inverse heat conduction problem. CFD simulations around the same geometry at equivalent free-stream conditions are calculated with the DLR-TAU code. The experimental and CFD results show good agreement both in terms of heat fluxes as well as flow structure. Notable flow structures include wedge-shaped regions of high heat flux which emanate from the windward corners of the cube. Analysis of numerical Q-criterion contours show that these high heat flux regions are caused by vortex structures generated by the expansion at the cube corner. Analysis of the numerical skin friction coefficient shows that even at incidence there is no breakaway separation from the expansion edges of the cube and the flow remains attached throughout. We show that although there is little change in the average heat flux experienced by a cube at 5° incidence to the free-stream compared to one at 0° incidence, there are significant changes in the heat flux contours over the cubes at these two incidences. Finally, we calculate a number of heating shape factors which can easily be implemented in satellite re-entry and demise prediction analysis tools.

Inverse Estimation of Heat Transfer Coefficient and Reference Temperature in Jet Impingement

Abstract Applications of impinging jets are wide-ranging from cooling to heating in industrial as well as domestic field. Most of the reported heat transfer distribution data to and from impinging jets have been found from steady-state measurements. This study utilizes the solution to three-dimensional (3D) inverse heat conduction problem to estimate transient temperatures on the impingement side. Then, the temperature gradient is determined near the impingement wall (∼0.01 mm inside) with which transient heat flux is estimated on the impingement side. Instead of steady-state values, transient heat flux and corresponding wall temperatures are utilized in a thin foil technique to find out heat transfer coefficient and reference temperature simultaneously. The scope of the present technique is examined through its application to impinging jets with various configurations such as laminar jet, turbulent jet, hot jet, cold jet, and multiple jets. In all cases, estimations are reasonably close. The application of this inverse technique can be extended to any configuration of jet impingement irrespective of geometry of nozzle (circular/rectangular), the orientation of nozzle (orthogonal/inclined), the temperature of a jet (hot/cold), Reynolds numbers (laminar/turbulent), the nozzle-to-plate spacing (any Z/d), and roughness of the plate surface. The effect of plate thickness on the accuracy of the present technique is also studied. Up to 5 mm thick plates can be used in impinging jet applications without worrying much on accuracy. The use of the present technique significantly reduces the experimental cost and time since it works on transient data of just a few seconds.

A general approach for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures

A novel general approach is proposed for solving three-dimensional transient nonlinear inverse heat conduction problems in irregular complex structures. The complex-variable-differentiation method (CVDM) is introduced into the commercial finite element method (FEM) software ABAQUS, for solving three-dimensional inverse heat conduction problems for the first time. First, the complex variable finite elements are set up, and the complex variable FEM is innovatively developed through the user element subroutine (UEL) in ABAQUS. Then, the key parameters in the Levenberg-Marquardt algorithm for solving inverse problems – sensitivity matrix coefficients – are accurately evaluated by utilizing the CVDM and ABAQUS standard solvers with the UEL. Finally, several numerical tests are carried out to verify the performances of the new approach for solving the three-dimensional transient nonlinear inverse heat conduction problem in an irregular complex structure.

A Virtual Boundary Element Method for Three-Dimensional Inverse Heat Conduction Problems in Orthotropic Media

This paper aims to apply a virtual boundary element method (VBEM) to solve the inverse problems of three-dimensional heat conduction in orthotropic media. This method avoids the singular integrations in the conventional bound... | Find, read and cite all the research you need on Tech Science Press

Virtual boundary element method in conjunction with conjugate gradient algorithm for three-dimensional inverse heat conduction problems

ABSTRACTIn this article, the virtual boundary element method (VBEM) in conjunction with conjugate gradient algorithm (CGA) is employed to treat three-dimensional inverse problems of steady-state heat conduction. On the one hand, the VBEM may face numerical instability if a virtual boundary is improperly selected. The numerical accuracy is very sensitive to the choice of the virtual boundary. The condition number of the system matrix is high for the larger distance between the physical boundary and the fictitious boundary. On the contrary, it is difficult to remove the source singularity. On the other hand, the VBEM will encounter ill-conditioned problem when this method is used to analyze inverse problems. This study combines the VBEM and the CGA to model three-dimensional heat conduction inverse problem. The introduction of the CGA effectively overcomes the above shortcomings, and makes the location of the virtual boundary more free. Furthermore, the CGA, as a regularization method, successfully solves the ill-conditioned equation of three-dimensional heat conduction inverse problem. Numerical examples demonstrate the validity and accuracy of the proposed method.

Shape identification for inverse geometry heat conduction problems by FEM without iteration

ABSTRACTThe boundary geometry shape is identified by the finite element method (FEM) without iteration and mesh reconstruction for two-dimensional (2-D) and three-dimensional (3-D) inverse heat conduction problems. First, the direct heat conduction problem with the exact domain is solved by the FEM and the temperatures of measurement points are obtained. Then, by introducing a virtual boundary, a virtual domain is formed. By minimizing the difference between the temperatures of measurement points in the exact domain and those in the virtual domain, the temperatures of the points on the virtual boundary are calculated based on the least square error method and the Tikhonov regularization. Finally, the objective geometry shape can be estimated by the method of searching the isothermal curve or isothermal surface for 2-D or 3-D problems, respectively. In the process, no iterative calculation is needed. The proposed method has a tremendous advantage in reducing the computational time for the inverse geometry problems. Numerical examples are presented to test the validity of the proposed approach. Meanwhile, the influences of measurement noise, virtual boundary, measurement point number, and measurement point position on the boundary geometry prediction are also investigated in the examples. The solutions show that the method is accurate and efficient to identify the unknown boundary geometry configurations for 2-D and 3-D heat conduction problems.

A meshless method based on the method of fundamental solution for three-dimensional inverse heat conduction problems

This paper documents a meshless method for the three-dimensional inverse heat conduction problems based on the method of fundamental solution (MFS). In order to overcome the ill-posedness of the corresponding problem, the Tikhonov regularization method, as well as Morozov’s discrepancy principle for selecting an appropriate regularization parameter are used. Hence there is to produce a stable and accuracy numerical results. Then some examples are given to check the effectiveness of this method, whilst the sensitive analysis is given. The numerical convergence and stability of this method are also analyzed.

Local measurements and a new flow pattern based model for subcooled and saturated flow boiling heat transfer in multi-microchannel evaporators

A comprehensive experimental campaign has been conducted to measure the local heat transfer coefficients during flow boiling of refrigerants in multi-microchannel evaporators. Two additional refrigerants (R245fa and R236fa) were tested in two silicon evaporators at three inlet subcoolings and at three outlet saturation temperatures. The test section backside temperatures were measured by a fine-resolution infrared (IR) camera providing a two-dimensional thermal map, which was used by solving the three-dimensional inverse heat conduction problem to obtain the local heat transfer coefficients on a pixel-by-pixel basis. The experimental results revealed different trends along the flow direction. The decreasing trend (when existing) at the beginning of the channel was attributable to the single-phase thermal developing flow, then heat transfer increased from the onset of subcooled flow boiling up until the onset of saturated flow boiling, and afterwards it decreased again until entering the annular flow regime where it started to pick up and rose significantly. Combining our new data together with our recent experimental work of Huang et al. (2016), a new flow pattern based model has been proposed for local heat transfer prediction, starting from single-phase flow all the way through to annular flow. This new model also included a new local heat transfer method for the subcooled flow boiling region since no truly local subcooled heat transfer prediction method can be found in the literature for microchannels. This new flow pattern based model predicted the total local heat transfer database (1,941,538 local points) well with a MAE of 14.2% and with 90.1% of the data predicted within ±30%. It successfully tracks the experimental trends without any jumps in predictions when changing flow patterns.

Experimental investigation on flow boiling pressure drop and heat transfer of R1233zd(E) in a multi-microchannel evaporator

An experimental study on flow boiling pressure drop and heat transfer of a new environmentally friendly refrigerant R1233zd(E), in a parallel multi-microchannel evaporator was carried out. The silicon microchannels evaporator was 10mm long and 10mm wide, having 67 parallel channels, each 100×100μm2, separated by a fin with a thickness of 50μm. Upstream of each channel, a micro-orifice was placed to stabilize the two-phase flow and to obtain good flow distribution. The operating conditions for flow boiling tests were: mass fluxes from 500 to 2750kgm−2s−1, heat fluxes from 6 to 50Wcm−2, inlet subcooling of 5.8K, and a nominal outlet saturation temperature of 35°C for stable flow boiling. The test section’s backside base temperatures were measured by an infrared (IR) camera. The stable flow boiling data without back flow was selected through flow visualization recorded by a high-speed camera coupled with a microscope. These data were then used to assess the applicability of existing two-phase pressure drop models, and to further develop a new empirical model suitable for the high mass flux operating conditions. This new pressure drop model was used to predict the local fluid temperature for the further heat transfer data identification. The fine-resolution local heat transfer coefficients were obtained by solving the three-dimensional inverse heat conduction problem. The experimental results showed that in the saturated flow boiling region the local heat transfer coefficient first decreased moderately in the very low vapor quality region (x<0.05), then increased significantly but monotonically along the flow direction. The fine-resolution local heat transfer data at the saturated flow boiling region were compared with two groups of heat transfer correlations. The first one considered the flow boiling mechanism occurring in muliti-microchannels as a combination of nucleate boiling and forced convection boiling, while the other one associated this mechanism to liquid thin film evaporation, thus indicating a controversy. It is found that the flow pattern based model belonging to the second group yielded the best agreement with the experimental data, predicting 92.0% of this new database within ±30%.

A BEM formulation in conjunction with parametric equation approach for three-dimensional Cauchy problems of steady heat conduction

This study documents the first attempt to apply a nonsingular indirect boundary element method (BEM) for the solution of three-dimensional (3D) inverse heat conduction problems. The present BEM formulation avoids the calculation of hyper-singular integrals. Furthermore, the exact geometrical representation of computational domain is adopted by parametric equations to eliminate the errors in traditional approaches of polynomial shape functions. Due to its boundary-only discretizations and semi-analytical nature, the proposed method can be viewed as a competitive candidate for the solution of inverse problems. Four benchmark numerical examples indicate that the proposed method, in conjunction with proper regularization techniques, is accurate, computationally efficient and numerically stable for the solution of 3D inverse problems subjected to various levels of noise in input data.

A meshless singular boundary method for three-dimensional inverse heat conduction problems in general anisotropic media

The singular boundary method (SBM) is a relatively new meshless boundary collocation method for the numerical solution of certain elliptic boundary value problems. The method, based on the notion of the boundary element method (BEM) and method of fundamental solutions (MFS), fully inherits the merits of both and in the meantime possessing its unique advantages. Due to the boundary-only discretizations and its semi-analytical nature, the method can be viewed as an ideal candidate for the solution of inverse problems. In this study, we document the first attempt to apply the SBM, together with several regularization techniques, for the solution of inverse heat conduction problems in three-dimensional (3D) anisotropic media. Four benchmark numerical examples are well-studied which indicate that the proposed scheme is accurate, computationally efficient and numerically stable for the solution of 3D inverse problems with various levels of noisy input data.

Inverse Estimation of the Front Surface Temperature of a 3-D Finite Slab Based on the Back Surface Temperature Measured at Coarse Grids

The accuracy of the solution of the inverse heat conduction problem (IHCP) can be improved by using fine grids, but measured temperature data on fine grids are usually not available due to limits imposed by the size and spacing of the temperature sensors. A new method is proposed to recover the front surface temperature of a finite slab using fine grids based on the back surface temperature measured with coarse grids. Effects of heat conduction in the sensor substrate and contact thermal rsesistance are also taken into account. The proposed method is applied to solve a three-dimensional IHCP with stainless steel and aluminum target slabs. The results show that the front surface temperature can be recovered with satisfactory accuracy. Both the distortion of the temperature distribution and the discrepancy from the exact solutions are reduced by using the new method.

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

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.

Inverse estimation of the inner wall temperature fluctuations in a pipe elbow

Thermal stratification occurs in flows through elbows used for mixing hot and cold fluids. The movement of the interface between the hot and cold fluids over time causes fluctuations of the inner wall temperature, which can lead to the stress variations and structural thermal fatigue. Therefore, accurate inner wall temperature data detailing the fluctuations is essential for analysis and predictions of thermal fatigue of the piping. In the present study, the conjugate gradient method (CGM) is applied to solve the three-dimensional inverse heat conduction problem (IHCP) to estimate the unknown temperature fluctuations on the inner wall of a pipe elbow from simulated outer temperature measurements. First, the direct heat conduction problem is solved using the finite element method (FEM) to produce the outer and inner wall temperatures. Then, the inverse heat conduction problem is solved to estimate the inner wall temperatures based on the outer wall temperature data. The accuracy of the inverse algorithm is then examined by comparing the estimated inner wall temperatures with the exact temperatures from the direct solution. The numerical results show that the inner wall temperature of the elbow can be accurately estimated by using the present algorithm for the test case considered.

Efficient reconstruction of local heat fluxes in pool boiling experiments by goal-oriented adaptive mesh refinement

In this paper, we consider the efficient estimation of local boiling heat fluxes from transient temperature measurements in the heater close to the heater surface. For accurate prediction, heat flux estimation is formulated as a transient three-dimensional (3D) inverse heat conduction problem (IHCP). This inverse problem is ill-posed and cannot be treated straightforwardly by established numerical methods. In order to obtain a regularized stable solution, a large-scale time-dependent PDE-constrained optimization problem has to be solved and an appropriate stopping criterion for the termination of the iterative solution process has to be chosen. Since the boiling heat flux is non-uniformly distributed on the heater surface due to the strong local activity of the boiling process, the use of a fixed uniform spatial discretization is not efficient. Instead, an adaptive mesh refinement strategy can be used to obtain an appropriate discretization which significantly reduces the total computational effort. In this work, we present an automatic algorithm incorporating an adaptive mesh refinement via a heat flux-based a-posteriori error estimation technique. The suggested algorithm can cope with both spatially point-wise or highly resolved temperature observations efficiently. It is applied to real measurement data obtained from two different types of pool boiling experiments. The numerical results show that the computational effort can be reduced significantly for given estimation quality. This adaptive IHCP solution technique can be also viewed as an efficient soft sensor to deduce unmeasurable local boiling heat fluxes.

An Inverse Problem in Estimating the Volumetric Heat Generation for a Three-Dimensional Encapsulated Chip

A three-dimensional inverse heat conduction problem is solved in the present study by using the conjugate gradient method (CGM) and the general-purpose commercial code CFD−ACE+ to estimate the strength of the unknown heat generation for an encapsulated chip in a three-dimensional irregular domain. The advantage of calling CFD−ACE+ code as a subroutine in the present inverse calculation lies in that many difficult but practical 3D inverse problem can be solved under this construction since the general-purpose commercial code has the ability to solve the direct problem easily. The results obtained by using the CGM to solve this 3D inverse problem are justified based on the numerical experiments using the simulated exact and inexact measurements. It is concluded that reliable heat generation can be estimated by the present inverse algorithm.

Estimation of local nucleate boiling heat flux using a three-dimensional transient heat conduction model

Boiling heat transfer is still hard to model and to predict. Among all kinds of studies on boiling processes, the estimation of the local heat fluxes at the boiling surface is fundamental. In this article, we solve a three-dimensional transient inverse heat conduction problem (IHCP) arising from a single-bubble nucleate boiling experiment to reconstruct the local boiling heat flux from a high-resolution temperature field measured at the back side of a thin heating foil. From the mathematical point of view, this problem belongs to the class of ill-posed inverse problems and cannot be solved straightforwardly by classical numerical methods. Thus, an iterative regularization strategy based on the conjugate gradient method is applied to the solution of the considered IHCP. Ring-shaped local boiling heat fluxes which undergo a significant change during a single-bubble cycle are observed.

Efficient solution of a three-dimensional inverse heat conduction problem in pool boiling

In this paper, we consider a three-dimensional transient inverse heat conduction problem arising in pool boiling experiments, i.e., the reconstruction of the surface heat flux from pointwise temperature observations inside a heater. We show that the inverse problem is ill-posed and utilize Tikhonov regularization and conjugate gradient methods together with a discrepancy stopping rule for a stable solution. We investigate the proper choice of regularization terms, which not only affects stability of the reconstructions but also greatly influences the quality of reconstructions in the case of limited observations. For the numerical solution of the governing partial differential equation, a space-time finite element method is used. This allows us to compute exact gradients for the discretized Tikhonov functional, and enables the use of conjugate gradient methods for the solution of the regularized inverse problem. We discuss further aspects of an efficient implementation, including a multilevel optimization strategy, together with an implementable stopping criterion. Finally, the proposed algorithms are applied to the reconstruction of local boiling heat fluxes from experimental data.

Determination of the convective heat transfer coefficient in three-dimensional inverse heat conduction problems

A thin steel plate probe (24 mm wide×24 mm long×3 mm thick) surrounded by insulation except for the exposed surface was constructed. Using the temperature measurements of the steel plate and the heat flux measured by a Gardon gauge and with the assistance of numerical calculations, two methods based on three-dimensional inverse conduction problems are developed to determine the convective heat transfer coefficient in fire experiments. The first one is based on optimisation of the predictions of the temperature of the steel plate using different heat transfer coefficients, while the second adopts a “predictor–corrector” method to determine the instantaneous heat transfer coefficient. Validation of both methods is accomplished by performing experiments in the cone calorimeter at a known constant heat flux. Subsequently, the two methods are applied to experiments in enclosures to examine the sensitivities of the two methods.