Inverse Heat Conduction Problem Research Articles

A new method to identify non-steady thermal load based on element differential method

Inverse heat conduction problems (IHCPs) are widely used to identify the thermal load imposed on the components in aerospace fields, and the accuracy and robustness of identification algorithm are the keys to IHCP. In this paper, the element differential method (EDM), Levenberg-Marquardt method (LM) and future time steps method are combined to identify the thermal load. Firstly, the reliability and accuracy of the element differential method are fully verified in solving complex transient heat conduction problems of a variety of temperature-dependent materials. Then, the time-variant temperatures are obtained by the EDM with guessed thermal loads. And complex variable derivative method (CVDM) is used to compute the sensitivity matrix in LM. In addition, the influence of different initial values on the identified thermal loads of the considered 2D and 3D models is also fully analyzed. To prove the robustness of the proposed method, the noises are added to the measured data. Finally, the results show that the proposed method is effective in identifying complex temporal and spatial distributions of heat fluxes, and it shows good accuracy, robustness and noise resistance.

Thermal conductivity reconstruction method with application in a face milling operation

PurposeThis paper aims to reconstruct the spatially varying orthotropic conductivity based on a two-dimensional inverse heat conduction problem described by a partial differential equation (PDE) model with mixed boundary conditions. The proposed discretization uses a highly accurate technique and allows simple implementations. Also, the authors solve the related inverse problem in such a way that smoothness is enforced on the iterations, showing promising results in synthetic examples and real problems with moving heat source.Design/methodology/approachThe discretization procedure applied to the model for the direct problem uses a pseudospectral collocation strategy in the spatial variables and Crank–Nicolson method for the time-dependent variable. Then, the related inverse problem of recovering the conductivity from temperature measurements is solved by a modified version of Levenberg–Marquardt method (LMM) which uses singular scaling matrices. Problems where data availability is limited are also considered, motivated by a face milling operation problem. Numerical examples are presented to indicate the accuracy and efficiency of the proposed method.FindingsThe paper presents a discretization for the PDEs model aiming on simple implementations and numerical performance. The modified version of LMM introduced using singular scaling matrices shows the capabilities on recovering quantities with precision at a low number of iterations. Numerical results showed good fit between exact and approximate solutions for synthetic noisy data and quite acceptable inverse solutions when experimental data are inverted.Originality/valueThe paper is significant because of the pseudospectral approach, known for its high precision and easy implementation, and usage of singular regularization matrices on LMM iterations, unlike classic implementations of the method, impacting positively on the reconstruction process.

Research on Unsteady Inverse Heat Conduction Based on Dynamic Matrix Control

For the unsteady multi-boundary inverse heat conduction problem, a real-time solution method for boundary heat flux based on dynamic matrix control is proposed in the paper. The method solves the heat flux at the boundary in real-time by measuring the temperature information at the measurement points of the heat transfer system. A two-dimensional direct heat conduction model of the heat transfer system is established in the paper, and is solved by the finite difference method to obtain the temperature information of the measurement points under any heat flux boundary. Then, the correspondence between the heat flux of boundary and the temperature information is presented by means of a step-response model. The regularization parameters are introduced into the method to improve the stability of the inversion process, and the effect of real-time inversion on the heat flux of the boundary is achieved through rolling optimization. The experimental results show that the proposed method can achieve real-time inversion of the heat fluxes of the two-dimensional boundary with good accuracy.

Thermographic Investigation on Fluid Oscillations and Transverse Interactions in a Fully Metallic Flat-Plate Pulsating Heat Pipe

The present investigation deals with the quantification of fluid oscillation frequencies in a metallic pulsating heat pipe tested at varying heat loads and orientations. The aim is to design a robust technique for the study of the inner fluid dynamics without adopting typical experimental solutions, such as direct fluid visualizations through transparent inserts. The studied device is made of copper, and it is partially filled with a water–ethanol mixture (20 wt.% of ethanol). Heat fluxes locally exchanged between the working fluid and the device walls are first assessed through the inverse heat conduction problem resolution approach by processing outer wall temperature distributions acquired by thermography. The estimated local heat transfer quantities are therefore processed to quantify the fluid oscillatory behavior in every device branch during the intermittent flow and full activation regimes, thus providing a deeper insight into the heat transfer modes. After dealing with a further validation of the inverse approach in terms of oscillation frequency restoration capability, the wall-to-fluid heat fluxes referred to each channel are processed by means of the wavelet method. Scalograms and power spectra of the considered signals are presented for a time-based analysis of the working fluid oscillations, as well as for the identification of dominant oscillation frequencies. Fluid motion is then quantified in terms of the continuity of fluid oscillations and activity of channels by applying a scalogram denoising technique named K-means clustering method. Moreover, a statistical reduction of the channel-wise dominant oscillation frequencies is performed to provide useful references for the interpretation of the overall oscillatory behavior. The link between oscillations and transverse interactions is finally investigated. The vertical bottom-heated mode exhibits stronger fluid oscillations with respect to the horizontal mode, with fluid oscillation frequencies ranging from 0.78 up to 1 Hz. Nonetheless, the fluid motion is more stable in terms of oscillation frequency between channels when the device operates in the horizontal orientation probably due to negligible buoyancy effects. Moreover, thermal interactions between adjacent channels are found to be stronger when the oscillatory behavior presents similar features from channel to channel in horizontal orientation. The proposed method for fluid oscillation analyses in fully metallic flat-plate pulsating heat pipes can be effectively adopted to other flat-plate layouts without any need for transparent windows, thus reducing the overall complexity of experimental set-ups and providing, at the same time, a good insight into the inner fluid dynamics.

Inverse estimation of unknown thermophysical properties of green facades using the Levenberg-Marquardt algorithm

Green facades are becoming increasingly popular in building design, and accurate thermophysical properties of the plant canopy are important to quantitatively evaluate the performance of green facades. The purpose of this paper is to propose an inverse heat conduction problem for estimating the unknown thermophysical properties of building green facades using Maple as the direct solver and the Levenberg Marquardt (LM) algorithm as the inverse method. The proposed method is applied to calculate the thermophysical properties of a green facade with greenery layer thickness of 0.4 m and 0.1 m air gap, and the measured leaf area index is 3.4. The results show that the thermophysical properties of the experimental green facade have no obvious functional dependence on solar radiation (Iglob,90) or ambient air temperature (Ta). The mean values of the thermal conductivity (λ), the density (ρ) and the specific heat (cp) were determined to be 4.4 W/m·K, 6.1 kg/m3 and 7171.2 J/kg·K, respectively. The thermophysical properties calculated in the computational experiment were verified by measured data. The results indicate that the overall trends are similar and show relatively good agreement, but there are still deviations between them, especially during daytime, in the presence of solar radiation. The main influencing factor for errors is the volumetric heat generation (qen). In this study, heat generation is assumed to be the transmitted solar energy totally absorbed by the homogenous layer. This assumption needs be refined in future studies, especially in the presence of higher global solar radiation. The proposed inverse method can be used not only in the analysis of the thermal response of green facades or vertical greenery systems, but also to analyze other building envelopes of unknown thermophysical properties.

A quasi-reversibility method for solving a two-dimensional time-fractional inverse heat conduction problem

In this paper, we consider a two-dimensional time-fractional inverse heat conduction problem, which is severely ill-posed. A quasi-reversibility method is proposed to solve this problem with disturbed boundary value. We give the selection of regularization parameters of quasi-reversibility method both under a priori and a posteriori rules, and give the proof of error estimates between the exact solution and its regularization approximation. Further more, numerical results are included to verify the effectiveness of the proposed method.

Identification of the effective thermal conductivity law in a nuclear fuel rod for the evaluation of neutronic feedbacks

One of the features of nuclear reactor simulation codes is the determination of the temperature distribution inside the uranium fuel rod. When modeling neutronic feedbacks in a nuclear reactor code, the methodology currently used to perform this kind of calculation is based on a simplified model which represents the fuel rod as an infinite cylinder with a one dimensional radial mesh. This choice is due to computation time constraints. Among the data needed by the model is the law giving the thermal conductivity as a function of the temperature. According to standard methodology, an effective thermal conductivity is used and the coefficients of the law are determined by best-fit of results obtained using detailed correlations. An alternative method is presented here to determine this simplified law by solving an inverse problem of heat conduction, formulated in the framework of the nodal expansion method. The results obtained using the simplified law are compared to the ones obtained with two detailed correlations.

Boundary element method for contactless estimation of spatially varying internal heat transfer coefficient in circular pipes

In the present study it is proposed a solution approach of the Inverse Heat Conduction Problem (IHCP) intended at assessing the local heat transfer coefficient on the inner wall of a tube, under a forced convection problem. The estimation method is established on the Boundary Element Method (BEM) combined with the Truncated Singular Value Decomposition (TSVD) methodology, employed to manage the ill-conditioned nature of the problem. The numerical results of the direct problem, built by the BEM, are firstly validated, and consequently adopted as synthetic data inputs to resolve the IHCP and corroborate the whole assessment procedure. This approach is also tested with experimental data about forced convection problem in coiled pipes. In these geometries, the convective heat transfer coefficient changes considerably along the wall periphery and for this reason, it constitutes a perfect example to test the ability of the presented method to infer a spatially varying internal heat transfer coefficient distribution.

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.

Influence of the Slag Rim on the Heat Transfer Behavior of a Mold

To explore the influence of the slag rim on the heat transfer behavior of a mold, a slag rim was extracted from a stainless steel production factory for high-temperature continuous casting simulation experiments. The 2DIHCP (Two-Dimensional Inverse Heat Conduction Problem) two-dimensional inverse problem calculation model was used to analyze the heat transfer characteristics in the mold, and the temperature and heat flux distribution of the hot surface of the mold copper wall under different working conditions were investigated. The results showed that under a casting speed of 0.5 Â m/min, vibration frequency of 2 Hz, and amplitude of 10 mm, the seepage flow of the mold flux was 9.314 and 6.326 g with and without the slag rim, respectively, the thickness of the slag film was 2.1 and 1.3 mm, respectively, the temperature of the mold hot surface was 118.24 and 134.83°C, respectively, and the maximum heat flux density of the mold was 0.98 and 1.95 MW/m2, respectively. Through data comparison, when there is a slag rim, the amount of slag infiltration decreased by 32%, the thickness of the slag film decreased by 38%, the temperature of the hot surface of the mold increased by 14%, the heat flux density increased by 99%, and the internal crystal morphology of the slag film was complete. This indicates that the slag rim affects the heat transfer behavior in the mold by restricting the inflow of the mold flux and affecting the formation of the slag film.

Solutions of one-dimensional inverse heat conduction problems: a review

Estimations using the inverse conduction approach to predict temperature and heat flux at the exposed surface lead to indirect measurements away from the exposed surface within the solid. The approach is extremely useful when access to direct measurements is not possible due to various working conditions, and thereby provides estimates without disturbing the flow under the real flow condition over the surface. The approach is useful not only for heat transfer applications but also for numerous engineering applications, including fluid mechanics and furnace applications. The approach requires the time history of effective parameters at the strategic locations away from the exposed surface to be known. In the present paper, a review of sequential development of various inverse heat conduction methods is presented to get solutions in different geometries.

Shannon-Taylor technique for solving one-dimensional inverse heat conduction problem

The Shannon-Taylor interpolation technique was introduced by Butzer and Engels in 1983. In this work, the sinc-function is replaced by a Taylor approximation polynomial. In this work, we implement the Shannon-Taylor approximations to solve a one-dimensional heat conduction problem. One of the major advantages of this approach is that the resulting linear system of equations of the approximation procedure has an explicit coefficient matrix. This is not the case of the classical sinc methods due to finite integrals involving e^{-x^2}. We establish rigorous error estimates involving an additional Taylor’s series tail. Numerical illustrations are depicted.

Liquid film and heat transfer characteristics during superheated wall cooling via pulsed injection of a liquid jet

Pulse firing is one of the major operation modes of bipropellant thrusters for the attitude control of small-scale spacecrafts. In the pulse-firing mode, the operational range of the thruster depends on the deterioration of liquid film cooling performance. Because cooling performance is determined by the balance of heat removed by the liquid film and transferred to the manifold, the behavior and heat transfer characteristics of liquid films under the intermittent injection of liquid jets must be understood to enable a wider range of operational patterns. We conducted cooling tests on a superheated metal plate via the pulsed injection of a liquid jet for better understanding of the pulsed cooling process. Different injection patterns, including continuous injection, with the same injection quantity were examined on two types of metal plates (aluminum alloy and copper), because the thermal properties of metal plates affect both the heat transfer characteristics of the liquid film and temperature rise of the plate during the inter-pulse duration. The cooling process was evaluated based on the evolution of the liquid film on the metal plate and rear-side temperature of the metal plate. The evolution and temperature were simultaneously visualized using a high-speed camera and infrared camera, respectively. To link the liquid film state to the heat transfer process, the temperature and heat flux on the cooled surface were estimated by solving the inverse problem of three-dimensional transient heat conduction. The results indicate that the wetting front position corresponds to the position of the maximum temperature gradient. Additionally, the residual liquid film is consumed through the evaporation and the droplet dispersion related to the nucleate boiling. The effects of thermal inertia and diffusivity of the metal plate highlight the differences in the amount of heat removed by the liquid film. The heat removed by the liquid film during pulsed cooling exhibited a peak and was higher than that removed by continuous injection on the aluminum alloy plate. Less heat was removed under low-duty-cycle conditions, and the amount of heat removed by the liquid film was lower than that removed by continuous injection on the copper plate.

A novel defect identification design of gas pipeline based on inverse heat conduction problem

A novel defect identification design of gas pipeline based on inverse heat conduction problem

Estimation of the local instantaneous heat flux inside a pulsating heat pipe for space applications

Pulsating Heat Pipes (PHP) are promising and effective passive two-phase heat transfer devices in terms of high heat transfer capability, efficient thermal control, adaptability and low cost and therefore they have been extensively studied in the last years. Many authors have estimated the heat fluxes at the evaporator and at the condenser area only in terms of the average values based on first principle considerations. In the present study the application of an inverse analysis technique to experimental infrared temperature data is proposed to investigate the local convective heat flux for forced convection flow inside these devices along the adiabatic zone. A PHP specifically designed to be hosted on board the heat transfer host of the International Space Station was tested in microgravity during the 67th Parabolic Flight Campaign organized by the European Space Agency. The device consists of an aluminium tube closed in a loop with 14 turns in the evaporator section, 3 mm inner diameter, half filled with FC-72 fluid. The external temperatures of the device are measured in the adiabatic zone with a high-speed infrared camera (50 Hz, 1280x1024 pixels). The images are thereafter post-processed and adopted as input data for the solution of the inverse heat conduction problem in the pipe wall (Tikhonov regularisation method) to extrapolate time-varying local heat fluxes on the tube internal surface in contact with the fluid.

Prediction identification method of time-varying internal heat source based on inverse heat transfer problem

Purpose This paper aims to discuss the inverse problems that arise in various practical heat transfer processes. The purpose of this paper is to provide an identification method for predicting the internal boundary conditions for thermal analysis of mechanical structure. A few examples of heat transfer systems are given to illustrate the applicability of the method and the challenges that must be addressed in solving the inverse problem. Design/methodology/approach In this paper, the thermal network method and the finite difference method are used to model the two-dimensional heat conduction inverse problem of the tube structure, and the heat balance equation is arranged into an explicit form for heat load prediction. To solve the matrix ill-conditioned problem in the process of solving the inverse problem, a Tikhonov regularization parameter selection method based on the inverse computation-contrast-adjustment-approach was proposed. Findings The applicability of the proposed method is illustrated by numerical examples for different dynamically varying heat source functions. It is proved that the method can predict dynamic heat source with different complexity. Practical implications The modeling calculation method described in this paper can be used to predict the boundary conditions for the inner wall of the heat transfer tube, where the temperature sensor cannot be placed. Originality/value This paper presents a general method for the direct prediction of heat sources or boundary conditions in mechanical structure. It can directly obtain the time-varying heat flux load and thtemperature field of the machine structure.

Experimental estimation of the local heat transfer coefficient for thin liquid film evaporation in a capillary tube

• Local heat transfer coefficient was estimated for thin liquid film evaporation in a capillary tube. • Temperature infrared measurements were employed as input data of the inverse heat conduction problem. • A robust estimation procedure was adopted. • Procedure validation through the application to both synthetic and experimental data. • Original insight of thin liquid film evaporation was given. Two-phase heat transfer devices consisting of mini and microtubes represent a powerful solution to the continuous reduction of electronic components' dimensions and to the increase of the power to dissipate. Despite many studies on the two-phase heat transfer phenomenon in minitubes, most provided outcomes regarding the average heat transfer coefficient or Nusselt number. A technique to assess the local convective wall heat flux and heat transfer coefficient is proposed, validated, and then applied to the experimental case of thin-film evaporation in a heated capillary tube. The evaporation of the thin film of liquid plays a vital part in the two-phase heat transfer phenomenon inside mini/microtubes. Our study investigated this phenomenon by examining a semi-infinite slug flow. The estimation method used is represented by the inverse heat conduction problem solution in the tube wall, applying the Tikhonov regularisation method. The temperature distributions on the outer surface of the channel measured by an infrared camera were employed as input data for the inverse problem. The estimated heat transfer coefficient distributions confirm the available correlations in average values and provide insight into these complex phenomena in local and instantaneous quantities.

PHYSICS-INFORMED NEURAL NETWORK FOR INVERSE HEAT CONDUCTION PROBLEM

PHYSICS-INFORMED NEURAL NETWORK FOR INVERSE HEAT CONDUCTION PROBLEM

Data-measuring device to determine the thermophysical properties of polymer composites: information support part

This paper deals with the problem of developing an information support subsystem of the data-measuring device for determining the thermophysical properties of polymer composites during cure, designed to identify the parameters of a mathematical model necessary to optimize the curing process of composites. Methods and algorithms for processing experimental data and calculating the thermophysical properties of polymer composites, based on an integral representation of the solution of the inverse heat conduction problem and a Kalman filter, has been developed. The structure of the construction, content, goals, parameters and functionality of the basic modules of the data-measuring device application software are presented. A subsystem of informational decision-making support has been developed for choosing the optimal method and algorithm for calculating the thermophysical properties of polymer composites, which allows minimization of the errors in determining properties under given experimental conditions. The subsystem is built on the alternative use of four algorithms for calculating thermophysical properties and is included as a module into the application software of an intelligent data-measuring device for studying the process of curing polymer composites.

Using a linear inverse heat conduction model to estimate the boundary heat flux with a material undergoing phase transformation

Modeling heat conduction during steel-cooling processes is challenging because solid–solid phase transformations can take place, introducing highly non-linear phenomena to the heat equation. Using a linear model for the inverse problem can be helpful for estimating the dissipated heat flux, especially because complex phase transformations models that use empirical parameters are not always available. However, the performance and applicability of such a simplified inverse model for a strongly non-linear process is still unclear. This study presents several numerical simulations to evaluate the accuracy and limitations of solving a linear inverse heat conduction problem (i.e. constant thermophysical properties and no internal heat source) to estimate the boundary heat flux during cooling of a material undergoing phase transformations. Preliminary simulations with stable materials but with temperature-dependent thermophysical properties showed that the linear model performs well to estimate the boundary heat flux, except when the material has a highly temperature-dependent specific heat, like pure iron near its Curie temperature. Then, we performed several simulations for 42CrMo4 steel, which undergoes phase transformations and, hence, has phase- and temperature-dependent thermophysical properties and latent heat of phase transformations as an internal heat source. The latent heat has a very small effect on the heat flux estimation for fast cooling conditions; however, it can lead to an interpretation bias in medium and slow cooling conditions due to temporary underestimates of the heat flux, reaching errors up to 100%. Even when the estimated heat fluxes seem accurate (average errors smaller than 10%), further estimates of the temperature evolution or phase transformations kinetics are inaccurate (range of uncertainties as large as 200 °C and 20%, respectively) because of the phase-dependent thermophysical properties. Hence, using a linear inverse heat conduction problem for a material undergoing phase transformations is acceptable only for fast cooling conditions ( h > 1500 Wm −2 K −1 ) and exclusively to estimate the boundary heat flux without further simulations. • Numerical simulations of a coupled heat conduction-phase transformation model. • Simulated temperature data were used in a linear inverse model to estimate the heat flux. • The inverse model performs well for AISI 304 and Ni201, which do not undergo phase transformations. • For the 42CrMo4 steel, which undergoes phase transformation, it performs well only for h > 1500 Wm − 2 K − 1 . • For lower h, the latent heat plays an important role, so the simplified model is no longer acceptable.