Inverse Heat Transfer Problem Research Articles

A Physics Informed Neural Network for Solving the Inverse Heat Transfer Problem in Gas Turbine Rotating Cavities

Abstract Recently, Physics-Informed Neural Networks have displayed great potential in delivering swift and accurate solutions for inverse problems. Here, a Physics-Informed Neural Network (PINN) was used to solve the inverse heat conduction problem in the rotating cavities of a aeroengine high-pressure compressor internal air system. The neural network was designed to receive experimentally captured radially distributed temperature profiles as inputs and predict the associated surface heat fluxes. The correctness of these predicted heat fluxes are assessed by numerically solving the direct heat conduction equation using a 2D Finite-Element model, thereby recovering the original temperature profiles. A comparative analysis is conducted between the predicted temperature profiles and the initial inputs. The physics informed neural network is trained using noise free synthetic data, created from a range of radial temperature curve fit coefficients, and subsequently tested on noisy experimental data at engine representative conditions. The predicted temperature values exhibit some good agreement with their respective actual counterparts. Furthermore, the sensitivity of model hyperparameters are explored to showcase the capability of the proposed approach. The results show that Physics- Informed Neural Networks exhibit reduced susceptibility to experimental uncertainties when addressing inverse problems, in contrast to inverse solution methods, and offer a possible new approach for analysis of experimental data if trained on a sufficiently large dataset.

Inverse algorithm for boundary heat flux density based on the NARX neural network

Abstract The inverse heat transfer problem is vital for scientific research and engineering applications. This paper introduces a method using the Nonlinear Autoregressive with Exogenous Inputs (NARX) neural network to identify heat boundary conditions in nonlinear transient heat transfer processes in real time. This method has two notable advantages: (1) It relies solely on surface temperature time series to obtain inversion results; (2) Even in the absence of knowledge regarding the system’s state equations, it can estimate heat flux density. The NARX neural network is trained by using Bayesian regularization with surface temperature and heat flux data. (3) As per the inversion results, the NARX neural network’s accuracy in predicting the boundary heat flux density (BHFD) increases as the temperature measurement points approach the heat flux boundary. This neural network calculates the current heat flux density by incorporating both present and past surface temperature measurements as inputs. Through numerical simulation experiments, the efficacy of the NARX method is confirmed, showcasing its exceptional accuracy, robustness against noise, and broad suitability.

Addressing multi-step inverse heat transfer problems via reduced order models in a cooling process for polymer pipes with sparse measurements

In modern polymer pipe extrusion processes with connected post-processing steps, the intermediate cooling and conditioning process is essential for the resulting product quality. In such cases, computationally efficient models are required for real-time process control and optimization. Unfortunately, heat transfer coefficients are rarely known for actual production processes, leading to an inverse heat transfer problem. Existing methods mainly rely on computationally intensive numerical models or on data intensive neural networks. Both are not suitable, if only sparse measurements are available and a fast execution time is required. In contrast to that, we propose to use a proper orthogonal decomposition and project the governing heat equation onto extracted basis functions via the Galerkin method. This leads to an efficient and accurate reduced order model, which can be used for parameter identification and subsequent process control. Regarding the identification, we address the ill-posed problem with well-known relative constraints, l2 regularization and decomposition of coefficients based on the Nusselt number. Simulation and experimental results show that the heat transfer coefficients are consistently identified and robust against initial parameter guesses despite sparse and noisy temperature measurements. Consequently, the proposed method can be efficiently applied in cooling and conditioning processes in polymer extrusion and related applications such as identifying the heat transfer coefficients and thermal resistance in pressurized surge lines of nuclear power plants and brick walls in melting furnaces respectively.

Local heat transfer measurements of the fin on the blade crown by infrared thermography

Current research on heat transfer of the labyrinth surface mainly focuses on the whole structure made of the same material, but the heat conduction from the tip to the root will affect the measurement of the heat transfer coefficient. To minimize the influence of heat conduction, copper strips are embedded on the surface of the fin made of bakelite. The inverse heat transfer problem is solved using the temperatures collected by an infrared thermal imager to determine the local convective heat transfer coefficient. The error of the lumped parameter method is smaller compared to the differential solution. Constant heat flux experiments are used to validate the experimental results, and numerical analysis is performed to analyze the heat transfer mechanism. The flow field, the Nu number distribution, and fitting correlations of the fins are obtained. By comparing and analyzing the heat transfer coefficients obtained by the three methods, it is found that the lumped parameter method can eliminate the influence of heat conduction on the surface. Taking the lumped parameter method as a reference, the errors of stable heat flow and numerical calculation were 21 % and 4.7 %, respectively. Due to the influence of the vortex direction, the Nu number distribution along the height of the windward surface of the second fin is different from the others. The Nu number distribution along the x-direction on the top surface of the fin is similar to the tip of the windward surface. Along the flow direction, the average Nu number on the top surface of the first fin is 1.2 times that of the second one.

An improved sequential quadratic programming method for identifying the total heat exchange factor of reheating furnace

Accurate identification of the total heat exchange factor (THEF), which serves as the foundation for online control data in the reheating furnace, holds significant importance for enhancing the digitization and intelligence level of the furnace, mitigating environmental pollution, and improving the economic performance of enterprises. For this purpose, based on the theory of inverse heat transfer problem, a solver combining Sequential Quadratic Programming and Broyden Combined Method (SQPBC) was developed to accurately identify high-dimensional and strongly nonlinear THEF in the reheating furnace. Integrating the efficient Broyden Combined Method (BCM) into Sequential Quadratic Programming (SQP) enables the rapid and accurate estimation of the Jacobian matrix to effectively enhance computational speed without compromising accuracy. Experimental data has been employed to validate the accuracy of the inversion results. The performance of SQPBC has been comprehensively analyzed through a series of numerical experiments, with a particular focus on investigating the impact of sampling frequency and sensor location on the inversion results. The findings reveal that after reducing the sampling period to 2 min, further decreasing the sampling period has a minor effect on the average relative error of the identification results; the closer the measurement points are to the surface of the slab, the more accurate the identification of THEF on the slab's surface.

A random walk based methodology to calculate the sensitivity coefficients in inverse radiant boundary design problems

Radiative heating furnaces are widely used for heat treatment applications in manufacturing industries. Strict requirements on heat treatment characteristics of the workpieces require precise control of furnace heater temperatures. Numerical techniques for the solution of inverse heat transfer problems can be used for the optimization of the radiant heater temperatures. In gradient-based optimization techniques, the optimization is carried out using the sensitivity coefficients between the heater and workpiece temperatures. In this study, we present a methodology for calculating the sensitivity coefficients for the solution of inverse boundary design problems involving transient heating of solid workpieces in a three-dimensional radiant furnace. In our methodology, the sensitivity coefficients on the selected design points within the workpieces are calculated by analytical differentiation of design point temperature formulations derived utilizing the floating random walk method and radiative heat flux formulation with exchange factors. The methodology for calculating sensitivity coefficients has been verified through applications to both a one-dimensional scenario and the three-dimensional radiant furnace model. The developed methodology was integrated into a gradient-based optimization algorithm in two test cases, to determine the furnace heater temperatures required to achieve the desired temperature histories at design points within the workpieces. First test case focuses reconstruction of an assumed heater temperature and examines the impact of design point configuration and input data noise on convergence behavior and calculated estimation. The findings reveal that increasing the number of design points in the workpieces increases solution accuracy, with transverse positioning of the design points in the workpieces resulting in heater temperatures closest to the assumed values. Second test case focused on estimating required heater temperatures for spatially uniform heating pattern on the workpieces. With 72 transversely positioned design points, we achieved uniform heating at these points with a maximum relative temperature deviation of 0.26%.

Inverse heat transfer problem for the characterization of nanofluids produced with different types of palladium nanoparticles

The objective of this work is the measurement of physical properties of distilled water nanofluids containing palladium nanocubes, palladium cerium oxide nanoparticles, and their respective hydrides. Due to their biocompatibility and favorable photothermal effects, palladium nanoparticles can be used to promote localized absorption of external energy sources in the thermal treatment of cancer, aiming at a thermal damage constrained to the tumor region without significant effects to the healthy tissues. An inverse problem is solved here within the Bayesian framework of statistics with the Markov Chain Monte Carlo method, by using nonintrusive transient measurements taken with an infrared camera during the heating of different water-based nanofluids. The mathematical model used in this work takes into account natural convection effects, due to the nonuniform heat source caused by the diode-laser that heats the nanofluids. Prior distributions for the model parameters were selected based on additional independent measurements, theoretical models and by the careful implementation of the experiments. The proposed model and the estimated parameters were validated, with an excellent agreement between the measured temperatures and those obtained from stochastic simulations during the solution of the inverse problem.

Estimation of Thermal Properties of Solid–Liquid Phase Change Material Using Fuzzy Inference Methods

The accurate measurement of thermal properties in phase change materials holds significant importance for engineering applications. This research introduces fuzzy inference methods to estimate the thermal properties of phase change materials. The solution to the coupled heat transfer involving radiation and conduction in material is achieved through a hybrid approach, which combines the finite volume method with the discrete ordinate method. The estimation process is structured as an inverse problem, where the temperature on the material surface acts as the measurement signal for conducting the inverse analysis. Both the fuzzy inference method and the decentralized fuzzy inference method are utilized to address the inverse heat transfer problem. This enables the determination of latent heat and thermal conductivities for both solid and liquid regions within the phase change material. Retrieval results demonstrate that the thermal properties of phase change materials can be accurately estimated using fuzzy inference methods. While both two fuzzy inference methods perform similarly in estimating a single parameter, the fuzzy inference method has limitations in multiparameter estimation tasks. Conversely, the decentralized fuzzy inference method yields accurate results in simultaneous estimation problems. Furthermore, this method proves robust in estimating the thermal properties of phase change materials, even in the presence of noisy data.

An Inverse Problem for Estimating Spatially and Temporally Dependent Surface Heat Flux with Thermography Techniques

In this study, an inverse conjugate heat transfer problem is examined to estimate temporally and spatially the dependent unknown surface heat flux using thermography techniques with a thermal camera in a three-dimensional domain. Thermography techniques encompass a broad set of methods and procedures used for capturing and analyzing thermal data, while thermal cameras are specific tools used within those techniques to capture thermal images. In the present study, the interface conditions of the plate and air domains are obtained using perfect thermal contact conditions, and therefore we define the problem studied as an inverse conjugate heat transfer problem. Achieving the simultaneous solution of the continuity, Navier–Stokes, and energy equations within the air domain, alongside the heat conduction equation in the plate domain, presents a more intricate challenge compared to conventional inverse heat conduction problems. The validity of our inverse solutions was verified through numerical simulations, considering various inlet air velocities and plate thicknesses. Notably, it was found that due to the singularity of the gradient of the cost function at the final time point, the estimated results near the final time must be discarded, and exact measurements consistently produce accurate boundary heat fluxes under thin-plate conditions, with air velocity exhibiting no significant impact on the estimates. Additionally, an analysis of measurement errors and their influence on the inverse solutions was conducted. The numerical results conclusively demonstrated that the maximum error when estimating heat flux consistently remained below 3% and higher measurement noise resulted in the accuracy of the heat flux estimation decreasing. This underscores the inherent challenges associated with inverse problems and highlights the importance of obtaining accurate measurement data in the problem domain.

Accurate Temperature Reconstruction in Radiofrequency Ablation for Atherosclerotic Plaques Based on Inverse Heat Transfer Analysis.

Radio frequency ablation has emerged as a widely accepted treatment for atherosclerotic plaques. However, monitoring the temperature field distribution in the blood vessel wall during this procedure presents challenges. This limitation increases the risk of endothelial cell damage and inflammatory responses, potentially leading to lumen restenosis. The aim of this study is to accurately reconstruct the transient temperature distribution by solving a stochastic heat transfer model with uncertain parameters using an inverse heat transfer algorithm and temperature measurement data. The nonlinear least squares optimization method, Levenberg-Marquardt (LM), was employed to solve the inverse heat transfer problem for parameter estimation. Then, to improve the convergence of the algorithm and reduce the computational resources, a method of parameter sensitivity analysis was proposed to select parameters mainly affecting the temperature field. Furthermore, the robustness and accuracy of the algorithm were verified by introducing random noise to the temperature measurements. Despite the high level of temperature measurement noise (ξ = 5%) and larger initial guess deviation, the parameter estimation results remained closely aligned with the actual values, with an overall ERMS consistently below 0.05. The absolute errors between the reconstruction temperature at the measurement points TC1, TC2, and TC3, and the actual temperature, remained within 0.33 °C, 2.4 °C, and 1.17 °C, respectively. The Levenberg-Marquardt algorithm employed in this study proficiently tackled the ill-posed issue of inversion process and obtained a strong consistency between the reconstructed temperature the actual temperature.

A fast Bayesian parallel solution framework for large-scale parameter estimation of 3D inverse heat transfer problems

The inverse heat transfer problems (IHTP) have a wide range of applications in the engineering field. Bayesian methods using Markov Chain Monte Carlo (MCMC) have long been considered as a robust and effective method for solving inverse problems. However, the discretization of the problem domain by the spatio-temporal Galerkin skill, i.e., the finite element interpolation also includes the time dimension, making the scale of the unknown parameters extremely difficult for Bayesian calculations. In this paper, a fast Bayesian parallel sampling (FBPS) framework is proposed for large-scale parameter estimation of benchmark three-dimensional inverse heat transfer problems (3D-IHTP). The FBPS we developed achieves a parameter computation scale of 105 magnitude within minutes, through dimensionality reduction of the space-time dependent problem domain. The Hamiltonian Monte Carlo (HMC) sampler, which is proven to be more efficient for high-dimensional parameter estimation, is employed. Through several simulation tests of IHTP, it was confirmed that the solving efficiency of FBPS surpasses that of the traditional Bayesian strategy significantly. Finally, FBPS is successfully developed to estimate the unknown heat flux on the chip heat sink and pack interface effectively, given some simulated high resolution measurement data. The reliability and efficiency show that FBPS has the potential to support efficient prediction techniques for a class of IHTPs in engineering applications.

Application of the adaptive method to determine the process noise in the extended Kalman filter to estimate the parameters of the two dimensional inverse heat transfer problem

To deal with inverse nonlinear heat transfer problems in a two-dimensional heat conduction problem, this paper presents a hybrid approach combining fuzzy logic and the extended Kalman filter (EKF). The proposed algorithm has been applied to the real-time reconstruction of the temperature field, time-varying convective heat transfer coefficient, and time-varying boundary heat flux using temperature data from a set of sensors. The effects of the covariance of the initial estimation error, the time step of data acquisition by the sensors, and the number of installed sensors on the precision and stability of the estimation results has been investigated by numerical experiments. To compare and validate the results, all these parameters were investigated using the EKF method. The results show that the adaptive fuzzy extended Kalman filter (FEKF) method for estimating temperature, convection coefficient, and heat flux at the boundaries is precise and stable. The comparison shows that the performance of the FEKF method in parameter estimation is better than the EKF method, and the highest increase in precision and stability of the results is related to the convective heat transfer coefficient.

A new stacking model method to solve an inverse flow and heat coupling problem for aero-engine turbine blades

The whole temperature distribution of a turbine blade is an important information for heat transfer design for turbine, but it is very difficult and expensive to measure the whole temperature filed, especially for the internal points of a blade. It is necessary to reconstruct the whole temperature distribution by calculating reversely unknown parameters based on sparse temperature values on the blade surface. This paper proposes a new stacking model method for solving inverse heat transfer problems of boundary conditions and it can be applied to cope with multi-input and multi-output regression problems. In order to improve the prediction performance of the stacking model, the Random Forest Regression, the XGBoost Regression and the K-Nearest Neighbors Regression are selected as the base estimators, and the Linear Regression is selected as the meta estimator. An application example for an aero-engine turbine blade is modeled and solved, in which the boundary condition parameters of the exhaust flows are reversely calculated and temperature distributions of a turbine blade surface can be obtained. Numerical experiments have demonstrated that the stacking model outperforms individual base models. This research is of considerable significance to heat transfer design, experimental research and decision support for real-time remote monitoring of aero-engines.

Soft computing methods in the solution of an inverse heat transfer problem with phase change: A comparative study

Inverse heat transfer problems are ill-posed problems and their solution is challenging. Conventional (hard computing) solution methods were developed for this purpose in the past, but they are not well applicable in cases including phase change, which contain strong non-linearity and bring additional computational difficulties. Soft computing methods, which currently experience very rapid development, are a promising tool for the solution of such problems. This paper addresses an inverse heat transfer problem with phase change, in which the boundary heat flux is estimated. Four methods based on distinct mathematical principles are applied to this problem and thoroughly compared. These methods include a conventional Levenberg–Marquardt method (LMM), a predictive fuzzy logic (PFL)-based method, a population-based meta-heuristic method called LSHADE (a state-of-the-art differential evolution variant), and a recently developed surrogate-assisted method coupled with differential evolution, referred to as LSADE method. Furthermore, a reformulation of the problem was developed, utilising a dimension reduction scheme and a decomposition scheme that led to sub-problems with different time frames. This reformulation brought extensive computational improvements. Results of the comparison of the methods then showed that the LMM and the PFL behave well in case without phase change but their performance deteriorates substantially in case with phase change. The LSHADE and the LSADE showed superior performance in the solution of the inverse problem with the phase change. Moreover, their performance was rather stable and insensitive to the location of the temperature sensor, which was the source of data for the estimation.

Thermal relaxation parameters estimation of non-Fourier heat transfer in laser-irradiated living tissue using the Levenberg-Marquardt algorithm

The present work is concerned with solving the inverse heat transfer problem (IHTP) to estimate the thermal relaxation parameters corresponding to the dual phase lag (DPL) model-based bio-heat transfer in laser-irradiated living tissue. This problem is divided into two parts: direct and inverse problems. The solution of the inverse problem has been obtained using the Levenberg–Marquardt algorithm. On the other hand, the solution to the direct problem is obtained by solving the DPL model-based bio-heat transfer equation (BHTE) with the transient radiative transfer equation (TRTE). So, the TRTE has been numerically solved by the modified discrete ordinate method (DOM) to find the intensity field in the living tissue subjected to short-pulsed laser radiation, and it acts as a heat source in the DPL model-based BHTE. So, the DPL model-based BHTE has been analytically solved by employing the finite integral transform technique to get the temperature variation inside the laser-irradiated living tissue. First, the computer code developed for estimating unknown thermal relaxation parameters using the Levenberg–Marquardt algorithm has been verified against the data given in the literature, and found good agreement between them. Then, the thermal response of laser-irradiated living tissue has been investigated. After that, the influence of sensor position on the sensitivity of the IHTP solution has been discussed. The impact of the total transient readings and the measurement error on estimating the unknown parameters have been investigated. The present study’s findings can significantly contribute to various parameter estimation problems where the conventional methods for direct measurements of parameters are impossible or very difficult. Furthermore, the current results may considerably impact the study of the thermic response of living tissue during laser-based photothermal therapy.

Implementation of the inverse problem for on-chip PCR diagnostics with rapid and precise thermal cycling

Microfluidic devices have emerged as a promising platform for point-of-care biological analysis, including PCR-based molecular diagnostics. However, the multi-layer characteristics of existing microfluidic chips introduce significant thermal inertia, resulting in temperature disparities between the PCR mixture and the heating element. This issue poses a substantial obstacle to the efficient and accurate deployment of point-of-care technology. Herein, a practical approach for accurate and fast PCR on microchips based on the inverse heat transfer problem (IHTP) is introduced and solved by incorporating computational fluid dynamics (CFD) simulations and the particle swarm optimization (PSO) algorithm. The methodology successfully resulted in a 46% reduction in PCR duration with less than 0.1 ℃ temperature discrepancy for our custom-built microfluidic device. The optimization performance was further evaluated by conducting biological tests at varying concentrations of the genetic sequence, and the standard curve of our optimized PCR system is presented. According to the standard curve of the system, using an optimized boundary temperature profile, the fabricated benchtop testing device performed COVID-19 molecular detection with 104% efficiency. The numerical approach can be extended to a variety of microfluidic devices and PCR temperature protocols, thereby enhancing the PCR diagnosis in microfluidic devices.

Автоматизация расчета реактора для синтеза серосодержащего сорбента, предназначенного для извлечения из сточных вод ионов тяжелых металлов

An automated algorithm for calculating the time of the full cycle of operation, the stages of thermal stabilization and cooling of the reaction mixture in a reactor designed for the synthesis of a sulfur-containing sorbent produced on the basis of waste products from metallurgy, petrochemistry (sulfur), epichlorohydrin (1, 2, 4-trichloropropane) and the pulp and paper industry (lignin) for extraction from wastewater of heavy metal ions. The use of algorithms and a program for automated calculation of the reactor helps to reduce the complexity of production costs and the cost of the finished sorbent, increases the reliability of calculations and the quality of design solutions. The developed algorithms and program include the following calculation routines: physico-chemical properties of the components of the reaction mixture and the choice of a mixing device (propeller three-bladed agitator), taking into account the viscosity of the mixture; hydrodynamic calculation of the mixing device and heat transfer when heating the mixture from 20 to 45 °C; heat transfer during thermal stabilization and cooling of the working mixture, as well as the time of the full cycle of the reactor. The proposed algorithm of heat transfer during thermal stabilization of the reaction mass is based on the determination of the temperature range of water heating, compensating for heat losses. For this purpose, the inverse problem of heat transfer with unknown temperatures over a hot heat carrier is formulated and solved, such that the average value between them is a thermostabilizable value. An algorithm for calculating the cooling process of the reaction mixture in a sulfur-containing sorbent synthesis reactor, taking into account heat losses to the environment in the amount of 5%, is presented. A program has been developed that implements the presented algorithms in C#, designed to automate the calculation of a sulfur-containing sorbent synthesis reactor designed to extract heavy metal ions from wastewater.

The reduction of ill-posedness of the inverse heat transfer problem in rotating disc cavities using the wall temperature characteristics as the priori knowledge

Solving the heat conduction equation using surface measurement temperature as boundary conditions to obtain surface heat flux is an ill-posed inverse heat transfer problem, in which small temperature errors can create large uncertainties in heat flux. This paper proposes a RBF (Radial Basis Function) + BP (Back Propagation) neural network approximation method that uses the wall temperature characteristics as priori knowledge to reduce the ill-posedness of the inverse heat transfer problem in rotating disc cavities of aero-engines. The wall temperature characteristics are described using first-order derivative, which are discussed and obtained through analytical solution and fluid-thermal coupling numerical simulation. Bayesian inference theory is used to construct neural network training objective function with regularization term and the RBF + BP neural network approximation method with uncertain regularization coefficient is proposed. Numerical experiments were conducted based on numerical simulation data to verify the effectiveness of the proposed method. The results show that the method can effectively suppress the ill-posed nature of the inverse heat transfer problem and can reduce the heat flux relative error level by 1 ∼ 2 orders of magnitude compared with several traditional data fitting methods commonly used in rotating disc cavities. The method of densifying measurement points combined with priori knowledge revision near the air impact point can effectively capture the local heat flux characteristics caused by airflow impact. The RBF + BP approximation method can reduce the demand for measurement points by approximately 44 ∼ 60 % under the error of 3σ = 0 K ∼ 0.9 K compared with Bayesian inference method.

Inverse estimation of heat flux and temperature field for a nonlinear heat transfer system using step-renewed two-stage Kalman filter

Inverse estimation of the thermal input and states is a useful tool for the monitoring of a thermal system. Nonlinear heat transfer is a quite common issue in this kind of problem. As the thermal properties are temperature-dependent, it is difficult to inversely estimate the input and temperature field. To deal with this problem, a step-renewed scheme is adopted into the weighted optimal two-stage Kalman filter (WOTSKF), forming step-renewed WOTSKF (SR-WOTSKF). This strategy is a marching method, without solving the Jacobian matrix in extended Kalman filter. The feasibility of dealing with the nonlinear inverse heat transfer problem is implemented by comparing WOTSKF and SR-WOTSKF. Then, the weighting factor, initial guess of the input heat flux and measurement noise are changed to investigate their effects on the simultaneous estimation results of heat flux and temperature field. It is found that when the weighting factor is 0.95, the relative error of the estimated heat flux using SR-WOTSKF is approximately 2.73% which is much lower than that using WOTSKF. Results using SR-WOTSKF demonstrate that the estimation results are independent of the initial guess and the estimated heat flux can track the exact one in a short period. An incorrect initial guess causes the initial estimation error of the temperature and heat flux and it makes no sense to following period. When the standard deviations of the measurement noise are adopted as 0.01, 0.10, 0.50 and 1.00, the mean relative error of the heat flux is 2.73%, 4.79%, 7.71% and 9.43%, demonstrating the strong robustness of SR-WOTSKF to resist the measurement noise.

Experimental set-up for the validation of phase change models in case of direct and inverse heat transfer problems

Experimental set-up for the validation of phase change models in case of direct and inverse heat transfer problems