Transient Inverse Heat Conduction Problem Research Articles

Detection of surface moving heat source using experimental temperature measurements on the opposite surface and inverse techniques

The identification of mobile sources is a widely researched area in multiple scientific and technical domains. Detecting mobile heat sources is an essential task in different fields, including fire safety, environmental monitoring, material science, and energy systems. In this investigation, we present an innovative solution to solve the inverse problem of detecting mobile heat sources, utilizing a three-dimensional transient inverse heat conduction problem (IHCP) formulation. The goal is to identify a moving heat source from thermographic infrared measurements of the rear face. The proposed method includes a finite volume approach with model reduction techniques to improve computational efficiency. Additionally, we use the conjugate gradient method to solve the inverse problem. We verified the proposed approach by conducting experiments on a metal plate with a moving heat source from a laser and comparing recalculated temperature of the front face with thermocouple measurements. Our results indicate that our method can detect the location and trajectory of the mobile heat source with high accuracy and efficiency. This study also highlighted the limits of source speed that can be detected by this method. This technique could be considered as an excellent option for real-time monitoring of mobile heat sources in practical applications.

Sequential Regularization Method for the Identification of Mold Heat Flux during Continuous Casting Using Inverse Problem Solutions Techniques

A two-dimensional transient inverse heat-conduction problem (2DIHCP) was established to determine the mold heat flux using observed temperatures. The sequential regularization method (SRM) was used with zeroth-, first-, and second-order spatial regularization to solve the 2DIHCP. The accuracy of the 2DIHCP was investigated under two strict test conditions (Case 1: heat flux with time-spatial periodically varying, and Case 2: that with sharp variations). The effects of the number of future time steps, regularization parameters, order of regularization, discrete grids, and time step size on the accuracy of the 2DIHCP were analyzed. The results showed that the minimum relative error (epred) of the predicted Case 1 heat flux is 5.05%, 5.39%, and 5.88% for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 2 heat flux are 6.31%, 6.30%, and 6.36%. Notably, zeroth- and first-order spatial regularization had higher accuracy than second-order spatial regularization, while zeroth-order spatial regularization was comparable to first-order. Additionally, first-order spatial regularization was more accurate in reconstructing heat flux containing sharp spatial variations. The CPU time of the predicted Case 2 heat flux is 1.71, 1.71, and 1.70 s for zeroth-, first-, and second-order spatial regularization, respectively. The corresponding values for the predicted Case 1 heat flux are 6.18, 6.15, and 6.17 s. It is noteworthy that the choice of spatial regularization order does not significantly impact the required computing time. Lastly, the minimum epred of Case 2 heat flux with zeroth-order spatial regularization is 7.96%, 6.42%, and 7.87% for time step sizes of 1/fs, 1/2fs, and 1/5fs, respectively. The accuracy of the inverse analysis displays an initial improvement followed by degradation as the time step size decreases. A recommended time step size is 1/2fs, where fs denotes the temperature-sampling rate.

Solution of Inverse Problem of Non-Stationary Heat Conduction Using a Laplace Transform

This paper presents some solutions to the one-dimensional Cauchy-type transient inverse heat conduction problem. Based on known courses of temperature and the heat flux on one of the boundaries of the region, courses of temperature and heat flux on the opposite boundary of the region were determined with the use of the Laplace’s transform. Solution of the inverse problem does not require determination of temperature distribution on subsequent time moments inside the region. Stability of the solution to the inverse problem of the Cauchy type was investigated by means of determining the minimal time step ensuring the stability of the solution. Two examples of one-dimensional transient inverse problems were solved. The first one is related to the reconstruction of shock temperature change on the boundary of the region based on known courses of temperature and heat flux on the opposite boundary. The second one concerned the reconstruction of the heat flux acting on one of the boundaries through a period of time. In both considered problems the reconstructed boundary conditions were discontinuous. Stable results were obtained for both investigated examples, however, the stability of the inverse problem depended on the length of time step.

Explicit Sensitivity Coefficients for Estimation of Temperature-Dependent Thermophysical Properties in Inverse Transient Heat Conduction Problems

Explicit expressions are obtained for sensitivity coefficients to separately estimate temperature-dependent thermophysical properties, such as specific heat and thermal conductivity, in two-dimensional inverse transient heat conduction problems for bodies with irregular shape from temperature measurement readings of a single sensor inside the body. The proposed sensitivity analysis scheme allows for the computation of all sensitivity coefficients in only one direct problem solution at each iteration with no need to solve the sensitivity and adjoint problems. In this method, a boundary-fitted grid generation (elliptic) method is used to mesh the irregular shape of the heat conducting body. Explicit expressions are obtained to calculate the sensitivity coefficients efficiently and the conjugate gradient method as an iterative gradient-based optimization method is used to minimize the objective function and reach the solution. A test case with different initial guesses and sensor locations is presented to investigate the proposed inverse analysis.

Sequential particle filter estimation of a time-dependent heat transfer coefficient in a multidimensional nonlinear inverse heat conduction problem

In the applied mathematical modelling of heat transfer systems, the heat transfer coefficient (HTC) is one of the most important parameters. This paper proposes a combination of the Method of Fundamental Solutions (MFS) with particle filter Sequential Importance Resampling (PF-SIR) to estimate the time-dependent HTC in two-dimensional transient inverse heat conduction problems from non-standard boundary integral measurements. These measurements ensure the unique solvability of the boundary coefficient identification problem. Numerical results show high performance on several test cases with both linear and nonlinear Robin boundary conditions, supporting the synergy between the MFS and simulation-based particle filter sequential analysis methods.

Analytical solution to transient inverse heat conduction problem using Green’s function

A transient inverse heat conduction problem concerning jet impingement heat transfer has been solved analytically in this paper. Experimentally obtained transient temperature history at the non-impinging face, assumed to be the exposed surface in real practice, is the only input data. Aim of this study is to estimate two unknown thermo-physical parameters—overall heat transfer coefficient and adiabatic wall temperature—at the impinging face simultaneously. The approach of Green’s Function to accommodate both the transient convective boundary conditions and transient radiation heat loss is used to derive the forward model, which is purely an analytical method. Levenberg–Marquardt algorithm, a basic approach to optimisation, is used as a solution procedure to the inverse problem. An in-house computer code using MATLAB (version R2014a) is used for analysis. The method is applied for a case of a methane–air flame impinging on one face of a flat 3-mm-thick stainless steel plate, keeping Reynolds number of the gas mixture 1000 and dimensionless burner tip to impinging plate distance equals to 4, while maintaining the equivalence ratio one. Inclusion of both radiation and convection losses in the Green’s function solution for the forward problem enhances the accuracy in the forward model, thereby increasing the possibility of estimating the parameters with better accuracy. The results are found to be in good agreement with the literature. This methodology is independent of flow and heating conditions, and can be applied even to high-temperature applications.

Determination of thermophysical properties and density volume fractions of Al2O3/Y-ZrO2 layered composite materials using transient thermography and two-stage inverse nonlinear heat conduction analysis

In the present study, transient thermography, a nondestructive imaging technique, is applied to evaluate the transient temperature response in a graded medium without the use of embedded thermocouples. A layered composite sample was fabricated from Al2O3 and Y-ZrO2 powders using powder metallurgy (PM). This sample was irradiated on one side with a direct current laser while the transient temperature was measured along its depth by a midinfrared camera. Also, a MATLAB code based on the truly meshless radial point interpolation method (t-RPIM) was developed and implemented to solve the problem of quasilinear transient heat transfer in PM solids. In the t-RPIM formulation, the Cartesian transformation method and the Crank-Nicolson scheme were used for the evaluation of domain integrals and time discretization, respectively, thereby yielding a truly mesh-free technique. In the conducted experiment, the thermophysical properties were assumed to be independent of temperature because of the small amount of temperature increase. These properties and the volume fractions of the constituent powders were determined using a combination of the t-RPIM and the damped Gauss-Newton method in an inverse analysis. Good agreement was found between the measured temperature and the reconstructed temperature profile using the identified thermal parameters and volume fractions, thus validating the accuracy and ability of the applied t-RPIM as a tool in an inverse scheme to solve the inverse transient heat conduction problem in nonhomogeneous media.

Three-dimensional transient inverse heat conduction problems in the enhanced pool boiling heat transfer

In this paper, the solution methods of three-dimensional transient inverse heat conduction problems (3D IHCP) arising from pool boiling applications that need to reconstruct Neumann boundary conditions are summarized. Due to their mathematical ill-posedness, it is still non-trivial to solve these 3D IHCP at a reasonable cost. Firstly, we introduce the optimization-based regularization methods, such as Tikhonov regularization and iterative regularization. In general, it requires to solve a large number of optimization problems constrained by 3D partial differential equations (PDE). In this case, an improved Tikhonov regularization method is developed to quickly obtain optimal Tikhonov regularization parameter and thereby can accelerate the entire inverse calculation process. Furthermore, an iterative regularization method that is based on the conjugate gradient method is proposed to handle the high-resolution measurement data and many pointwise measurement data. In pool boiling experiments, they can be obtained by an infrared camera and mirco-thermocouples, respectively. Secondly, a completely different approach, a direct two-step strategy is presented to solve the same 3D transient ICHP on thin-plates that commonly occur in engineering applications. The main idea of the approach is to transform the inverse problem into a forward problem that is easier to solve by replacing the unknown Neumann boundary condition with an approximated Dirichlet boundary condition. The method has great advantage in efficiency compared with the optimization-based method that often requires to solve dozens or hundreds of direct, adjoint and sensitivity PDE problems, and therefore has the potential to be developed as efficient online heat-flux soft sensors for many practical applications. Thirdly, to further speed up the numerical solution procedure we develop a time-space dependent adaptive mesh refinement strategy due to the fact that the heat-flux and temperature dynamics related to bubble structures rapidly change in space and time on the boiling surface. A few classical engineering benchmark problems are considered to validate the aforementioned solution approaches, and the proposed approaches are also successfully applied to reconstruct the transient heat-flux distributions on the boiling surface from the noisy experimental data obtained in a series of real pool boiling experiments along the entire boiling curve. On the one hand, the corresponding research works reviewed can be used to explore the heat-transfer mechanism and establish a more accurate heat transfer model enabling the fast, robust and accurate reconstruction of local heat fluxes, which can be considered as realistic boundary conditions on the boiling surface to further study the two-phase flow and heat transfer process of boiling fluid above the heater. On the other hand, it can be also applied to explore the local details of the enhanced pool boiling heat transfer for the design of new heaters such as micro/nano structured surfaces, provide predictions related to the boiling process and conduct optimal experimental design in a systematic way. In the future work, a large number of 3D IHCP for different test regularization parameters will be calculated through a high-throughput parallel computational strategy. Besides, for thick heaters the direct two-step strategy can be used to solve the 3D ICHP accurately with an initial heat-flux guess obtained by the proposed optimization-based inversion approach. This way, the computational efficiency can be further improved. Furthermore, other new inversion methods, for example, which are based on artificial neural network method are being developed to solve 3D transient IHCP for more complex structured heater surfaces such as the micro/nano structures.

Application of numerical procedure for thermal diagnostics of the delamination of strengthening material at concrete construction

PurposeLarge structural objects, primarily concrete bridges, can be reinforced by gluing to their stretched surface tapes of fiber-reinforced polymer (FRP). The condition for this technology to work requires the quality of the bonding of FRP and the concrete to be perfect. Possible defects may arise in the phase of construction but also as a result of long-term fatigue loads. These defects having different forms of voids and discontinuities in the bonding layer are difficult to detect by optical inspection. This paper aims to describe the development of a rapid and nondestructive method for quantitative assessment of the debonding between materials.Design/methodology/approachThe applied technique belongs to the wide class of active infrared (IR) thermography, the principle of which is to heat (or cool) the investigated object, and determine the properties of interest from the recorded, by an IR camera, temperature field. The methodology implemented in this work is to uniformly heat for a few seconds, using a set of halogen lamps, the FRP surface attached to the concrete. The parameter of interest is the thermal resistance of the layer separating the polymer tape and the concrete. The presence of voids and debonding will result in large values of this resistance. Its value is retrieved by solving an inverse transient heat conduction problem. This is accomplished by minimizing, in the sense of least squares, the difference between the recorded and simulated temperatures. The latter is defined as a solution of a 1D transient heat conduction problem with the already mentioned thermal resistance treated as the only decision variable.FindingsA general method has been developed, which detects debonding of the FRP tapes from the concrete. The method is rapid and nondestructive. Owing to a special selection of the compared dimensionless measured and simulated temperatures, the method is not sensitive to the surface quality (roughness and emissivity). Measurements and calculation may be executed within seconds. The efficiency of the technique has been shown at a sample, where the defects have been artificially introduced in a controlled manner.Originality/valueA quantitative assessment procedure which can be used to determine the extent of the debonding has been developed. The procedure uses inverse technique whose result is the unknown thermal resistance between the member and the FRP strip.

A novel formulation and sequential solution strategy with time-space adaptive mesh refinement for efficient reconstruction of local boundary heat flux

Inverse heat conduction problems appear in a variety of applications in industry and academia. Due to their mathematical challenges, they have been studied extensively over the last decades. Nevertheless, the solution of three dimensional transient inverse heat conduction problems are still computationally expensive. Commonly used solution approaches include the regularization-based approaches such as Tikhonov regularization or iterative regularization that result in formulating and solving optimization problems. A radically different solution approach is presented in this paper to provide efficient solutions of three dimensional transient inverse heat conduction problems on thin-plates that are used very often in industrial applications. Instead of solving optimization problems subject to constraints given in terms of three dimensional partial differential equations, the suggested approach transforms the inverse problem into an easy-to-solve direct problem that only needs the reconstruction of a Dirichlet boundary condition. This yields an approximated solution. Furthermore, a new time-space adaptive mesh refinement strategy is proposed to further reduce computational effort at an acceptable level of estimation precision. The accuracy and efficiency of the proposed sequential approach as well as the time-space adaptive mesh refinement strategy are validated by four representative case studies arising from practice. Estimation results with experimental data are also presented. The current work is considered as an important step forward towards the efficient solution of transient three dimensional inverse heat conduction problems, which is particularly useful for developing heat-flux soft sensors in practical problems arising from different fields such as pool boiling, falling films and steelmaking.

A new modified conjugate gradient method to identify thermal conductivity of transient non-homogeneous problems based on radial integration boundary element method

In order to analyze the heat conduction problem of composite materials in aerospace engineering, a modified conjugate gradient method is proposed to identify the physical parameters of transient heat conduction problems in this paper. In the positive problem, the boundary element method based on radial integration method is used to obtain the measured point temperature of the transient heat conduction problem with varying thermal conductivity. In the inverse problem, based on the advantages of high accuracy and fast convergence rate of the conjugate gradient method, a modified conjugate gradient method is used. On the one hand, the complex variable derivative method is introduced into the traditional conjugate gradient method, and then the sensitivity coefficient matrix can be accurately calculated. On the other hand, based on the linear convergence property of the conjugate gradient method, a restart factor is introduced in this paper. For an inverse problem containing n unknown parameters, the algorithm restarts from the steepest descent method after every n + 1 iterations to identify physical properties of transient non-homogeneous thermal conduction. The complex variable derivation method overcomes the problem that the traditional conjugate gradient method needs differential processing and low derivative accuracy in solving the inverse transient heat conduction problem. When solving the complicated inverse problems, conjugate gradient method, which includes restart factor, solves the problem of slow convergence in traditional conjugate gradient methods. Numerical examples are presented to demonstrate the accuracy and efficiency of the present method in identifying the parameters of transient non-homogeneous thermal conductivity.

Estimation of the time-dependent convective boundary condition in a horizontal pipe with thermal stratification based on inverse heat conduction problem

The time-dependent convective boundary condition on the inner wall are the essential causes of flow and heat transfer that induce thermal fatigue in pipes with thermal stratification. A three-dimensional transient inverse heat conduction problem (IHCP) was developed in this work for estimating simultaneously the multi-variables of the time-dependent convective boundary condition. Conjugate gradient method (CGM) is chosen as the optimization method. The known parameter in the IHCP is the measuring temperature on the outer wall. Experiments under different seven operating conditions have been taken for the validation of the method. Based on the estimated results, the influence of volume flow rate of cold water on the conductive heat transfer coefficient near the inner wall in the test section and the temperature distribution of the pipe wall were discussed.

A new algorithm for solving an inverse transient heat conduction problem by dividing a complex domain into parts

The purpose of this work is to create a new algorithm for solving an inverse transient heat conduction problem in any extended complex domain. If the problem cannot be solved in the entire region under analysis, the domain is divided into parts and free boundary conditions are introduced between them. Then the solution is carried out in subsequent parts of the field. Depending on the shape and dimensions of the analysed part, an appropriate location of the temperature sensors, the amount and size of the control volumes, the time step and a convenient way to filter the temperature distribution over time and space can be proposed. Such actions make it possible to achieve better stabilization of the inverse method and obtain the solution in the whole area. The proposed procedure in subsequent parts of the domain belongs to the group of space marching methods. The analysis starts on the surface where temperature sensors are located and marches through space sequentially to the surface with an unknown boundary condition.The proposed algorithm can be used to optimize the power unit start-up and shutdown operation. It may also enable a reduction in the heat loss arising during the process and extend the power unit life. The presented procedure can be applied in monitoring systems working both in conventional and nuclear power plants.

Solution of an inverse transient heat conduction problem in a part of a complex domain

The purpose of this work is to formulate two simple methods which can be used to solve nonlinear inverse heat conduction problems in a part of a complex-shaped component in the on-line mode. The proposed methods can be useful if temperature measurements are carried out only in a selected part of an outsize element or if the whole element cannot be analysed due to the inverse heat conduction problem being ill-conditioned. It is assumed that the conductive heat transfer occurs through the surfaces separating the domain from the rest of the component. It is shown that the simplifying assumption of thermal insulation on these surfaces, which is often presented in literature, can cause significant errors. Compared to works published previously, if two additional unknown boundary conditions are introduced on the separating surfaces, the inverse problem conditioning deteriorates substantially. Despite this, stable solutions are achieved for “noisy measured data”.The presented methods can be used to optimize the power unit start-up and shutdown operation. They may also enable a reduction in the heat loss arising during the process and extend the power unit life. The methods presented herein can be applied in monitoring systems working both in conventional and nuclear power plants.

Effect of mass diffusion upon the thermal-diffusive behavior of a dry vibrated granular bed

We report the results of an experimental study on the thermal-diffusion mechanisms of particles with fluctuant motion in a dry granular system. Granular beds with different intensities of particle self-diffusion were produced by exerting vertical vibration-driving forces of different strengths. The effective thermal diffusivity, αe, was determined by solving the inverse transient heat conduction problem. The self-diffusion coefficient in the vertical direction, Dyy, was calculated to describe the diffusive motions of the particles in the bed. The dimensionless Lewis number, Le, was calculated to investigate the relationship between mass diffusion and thermal diffusion. First, we demonstrated that mass diffusion increased the thermal-diffusive behavior of the dry granular system. Second, we identified a positive linear correlation with the specific slope Sαd between αe and Dyy. In particular, the value of Sαd represents a characteristic of the granular system related to the relationship between thermal diffusivity and the mass diffusivity; this means that Le may approach the value of Sαd if the intensity of particle self-diffusion in the bed is sufficiently strong.

Inverse identification of boundary conditions in a scramjet combustor with a regenerative cooling system

Accurately determining boundary conditions in a scramjet combustor is of great importance for modeling the coupled process of fuel burning, fluid flow and heat transfer in the scramjet combustor, and for design and optimization of the cooling system. In this paper, a new methodology is proposed for determining boundary conditions at inaccessible surfaces of a scramjet combustor with a regenerative cooling system, which are identified by solving a three-dimensional transient inverse heat conduction problem. The finite element method in the ABAQUS is employed to solve the direct heat conduction problem in the scramjet combustor with the regenerative cooling system. A gradient-based method is used to solve the inverse problem attributed to its high accuracy and efficiency, and temperatures at accessible positions provide additional information for the inverse analysis. Examples are given to examine the performances for identifying boundary conditions in a scramjet combustor with a regenerative cooling system.

Transient inverse heat conduction problem of quenching a hollow cylinder by one row of water jets

In this study, a two-dimensional linear transition inverse heat conduction problem (IHCP) was solved using the Generalized Minimal Residual Method (GMRES) in quenching process by water jets. The inverse solution method was validated by set of artificial data and solution sensitivity analysis was done on data noise level, regularization parameter, cell size, etc. An experimental study has been carried out on quenching a rotary hollow cylinder by one row of subcooled water jets. The inverse solution approach enabled prediction of surface temperature and heat flux distribution of test specimen in the quenching experiments by using measured internal specimen temperature. Three different boiling curves were defined in the quenching process of a rotary cylinder. Result obtained by the inverse solution showed clear footprint of rotation in surface temperature and heat flux on each revolution of cylinder and temperature variation damping from quenching surface toward interior of specimen.

Multi-model method for solving nonlinear transient inverse heat conduction problems

The multi-model inverse method for nonlinear inverse problems is established based on the multi-model control theory. First the model switching variable is chosen and several typical operating balance points in the workspace of the balance variable are selected. Then for each operating balance point the linear local model is established, and the local controller is designed for each linear local model. Finally, according to the instantaneous matching degree between the actual model and the local models, the inversion results of each local controller are weighted and synthesized to obtain the final inversion result. Numerical tests are implemented to solve the one-dimensional nonlinear inverse heat conduction problem by the multi-model inverse method associated with the dynamic matrix control (DMC) and DMC filter, respectively. Numerical results by the multi-model inverse method based on DMC demonstrate that the multi-model inverse method is a highly computationally efficient and accurate algorithm for inverse problems with complicated direct problems. Numerical results by the multi-model inverse method based on DMC filter show that the presented method can extend the applied field of the complicated linear inverse algorithms such as digital filter to the nonlinear inverse problems and it can obtain satisfactory inversion results.

Transient Data Processing of Flow Boiling Local Heat Transfer in a Multi-Microchannel Evaporator Under a Heat Flux Disturbance

Multi-microchannel evaporators are often used to cool down electronic devices subjected to continuous heat load variations. However, so far, rare studies have addressed the transient flow boiling local heat transfer data occurring in such applications. The present paper introduces and compares two different data reduction methods for transient flow boiling data in a multi-microchannel evaporator. A transient test of heat disturbance from 20 to 30 W cm−2 was conducted in a multi-microchannel evaporator using R236fa as the test fluid. The test section was 1 × 1 cm2 in size and had 67 channels, each having a cross-sectional area of 100 × 100 μm2. The micro-evaporator backside temperature was obtained with a fine-resolution infrared (IR) camera. The first data reduction method (referred to three-dimensional (3D)-TDMA) consists in solving a transient 3D inverse heat conduction problem by using a tridiagonal matrix algorithm (TDMA), a Newton–Raphson iteration, and a local energy balance method. The second method (referred to two-dimensional (2D)-controlled) considers only 2D conduction in the substrate of the micro-evaporator and solves at each time step the well-posed 2D conduction problem using a semi-implicit solver. It is shown that the first method is more accurate, while the second one reduces significantly the computational time but led to an approximated solution. This is mainly due to the 2D assumption used in the second method without considering heat conduction in the widthwise direction of the micro-evaporator.

Application of conjugate gradient method for estimation of the wall heat flux of a supersonic combustor

In this paper, conjugate gradient method (CGM) considering variations in material properties with temperature is developed to solve transient inverse heat conduction problems. With CGM, the measured outer wall temperatures of supersonic combustor are used as input data to recover the heat flux and temperature on the inner wall. Numerical test has been done to study the effect of temperature measurement noises on accuracy of the inversion result. It indicates that the inversion results are very close to the exact heat flux with a maximum error of less than 5% when the measurement noise is less than ±3K. At the same time, a series of experiments are conducted on a Mach 3 supersonic combustor test facility with varied fuel injection conditions. The heat fluxes on the inner wall are recovered by measured outer wall temperatures via CGM. The inversion results of heat flux agree satisfactorily with the values measured by heat flux sensors and a maximum difference of less than 5% is found. The present comparison results prove the validity and accuracy of the CGM developed in the paper for estimation of the wall heat flux of supersonic combustor.