Nonlinear Heat Transfer Problems Research Articles

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.

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.

A Neural Network-Based Method for Real-Time Inversion of Nonlinear Heat Transfer Problems

Inverse heat transfer problems are important in numerous scientific research and engineering applications. This paper proposes a network-based method utilizing the nonlinear autoregressive with exogenous inputs (NARX) neural network, which can achieve real-time identification of thermal boundary conditions for nonlinear transient heat transfer processes. With the introduction of the NARX neural network, the proposed method offers two key advantages: (1) The proposed method can obtain inversion results using only surface temperature time series. (2) The heat flux can be estimated even when the state equation of the system is unknown. The NARX neural network is trained using the Bayesian regularization algorithm with a dataset comprising exact surface temperature and heat flux data. The neural network takes current and historical surface temperature measurements as inputs to calculate the heat flux at the current time. The capability of the NARX method has been verified through numerical simulation experiments. Experimental results demonstrate that the NARX method has high precision, strong noise resistance, and broad applicability. The composition of the input data, the surface temperature measurement noise, and the boundary heat flux shape have been studied in detail for their impact on the inversion results. The NARX method is a highly competitive solution to inverse heat transfer problems.

Estimation of soils thermophysical characteristics in a nonlinear inverse heat transfer problem

It is well known that knowledge of thermophysical parameters is a leading strategy to research effects of energy transfer in soils. The present article proposes an inverse analysis for numerical solving of nonlinear heat transfer problem to determine the thermophysical properties of two different soil types: sand and chernozem. First, estimation of thermophysical parameters is performed using temperature data from experimental set-up, which is two-chambered container for two soil types. Second, numerical algorithm is based on implicit Euler scheme for discretization, Newton method to solve nonlinear system of equations and Levenberg-Marquardt method to minimize nonlinear estimator with Tikhonov's regularization technique. Simulations have been efficiently carried out for two different soil types, showing that the reliability of the model is satisfying with a discrepancy between numerical predictions and experimental observations remaining within the measurement error.

COUPLED HEAT AND MASS TRANSFER PROBLEM WITH DEPENDENT HEAT CONDUCTIVITY PROPERTIES AND ITS SEMI-ANALYTICAL SOLUTION

Present paper proposes a computation method for a coupled heat and mass transfer problem within the porous media where the heat conductivity properties of the media undergo changes caused by the mass transfer. Heat and mass transfer processes are coupled through the inclusion of evaporation and condensation phenomena which in turn require the solution to an another the vapour transfer problem. This complex coupled problem re-sults in a system of both linear and non-linear second order partial differential equations that are spatially discre-tized by Finite Element Method. Temporal integration is carried out analytically. Thus, the proposed system of equations covering linear vapor transfer problem, linear filtration problem and non-linear heat transfer problem is transformed into a system of both linear and non-linear first order ordinary differential equations being solved by semi-analytical method. Picard approach of successive iterations is used for linearization of the equations. Convergent solution is achieved which is demonstrated on a sample problem herein below. Proposed method gives good insight on the processes taking place within the structures being subjected to temperature and vapor pressure gradients, including the residual effects of moisture accumulation, and assesses its impact on heat con-ductivity of materials that the structures consist of. Present study is a part of more extensive research on the ap-plication of semi-analytical methods in heat transfer problems, therefore it is not exhaustive and complete. Shortcomings of the method and its possible work-arounds as well as the topics for further studies are discussed in Conclusions and Further Studies section.

Explicit spectral element collocation method for nonlinear transient heat transfer

In this paper, we proposed a novel collocation method to solve nonlinear transient heat transfer problems in typical intelligent materials. The new method is derived from the weighted residual method in which the weight of Jacobian is proven to be the best choice. In the paper, the physical variables and their fluxes are respectively approximated by two series of Lagrange interpolation polynomials at Chebyshev-Gauss-Lobatto nodes. In the proposed method, the mass matrices of the final linear system of equations are diagonal and the explicit time discretization method can be employed. Using the proposed method, we solved the heat transfer in intelligent materials and it is proven to be quite efficient and accurate, which benefits from that no large sparse matrix is generated and solved. By the proposed method, one can easily solve the problems with complex geometries.

A cheap preconditioner based on fast diagonalization method for matrix-free weighted-quadrature isogeometric analysis applied to nonlinear transient heat transfer problems

In this paper, IsoGeometric Analysis is applied to a transient nonlinear heat conduction problem using a predictor-corrector scheme. To obtain a better approximation of the solution, we propose the k-refinement method coupled to a Matrix-Free Weighted Quadrature approach which speeds up matrix-vector operations and also reduces memory consumption. The main goal of this work is to introduce a preconditioning strategy to improve the convergence rate of the iterative solver. This method is based on Fast Diagonalization technique which exploits the tensor structure of basis functions. We show that our preconditioner is inexpensive, easy to code and as effective as other more sophisticated methods. It is also robust with respect to polynomial degree, mesh size and time-integration scheme parameters. Numerical experiments for different geometries and materials are presented, which support our theoretical results.

An Adaptive Matrix Method for the Solution of a Nonlinear Inverse Heat Transfer Problem and Its Experimental Verification

An adaptive matrix inverse (AMI) method is presented to identify the temperature and unknown boundary heat flux in a domain of a regular or irregular shape with temperature-dependent properties. The nonlinear problem is broken down into a number of linear submodels, and for each submodel, the temperature is obtained in measuring points. Next, based on the matching degree between the temperatures measured and calculated by each prediction submodel, the submodels are weighted and combined to create the full model for the solution of an inverse nonlinear heat transfer problem. Comparisons are also made with the existing multiple model adaptive inverse (MMAI) algorithm and method based on the Levenberg–Marquardt algorithm (LMA). The results of the presented numerical tests for undisturbed and disturbed “measuring” data indicate that the heat fluxes identified by the AMI method are close to the exact values. The application of the presented method for bodies with an irregular shape is also demonstrated. The AMI method has been experimentally verified during the thick-walled cylinder cooling process. The proposed method can be applied in online diagnostic systems for thermal state monitoring.

A fast inversion approach for the identification of highly transient surface heat flux based on the generative adversarial network

The efficient identification of surface heat flux during highly transient pool boiling process is an essential task for engineering production and manufacture, which can be mathematically defined as nonlinear inverse heat transfer problems (IHTP) of Neumann boundary conditions. Conventional regularization-based inversion approaches for these types of problems are computationally expensive since packs of nonlinear partial differential equations (PDE) constrained optimization problems in complex computational domains need to be solved. In this paper, an operator-learning method is proposed for the efficient solution of complex three-dimensional (3D) transient nonlinear IHTP. FluxNet, a generated adversarial network composed of a discrete wavelet transform (DWT) layer, a “generator” and a “discriminator”, is constructed to significantly improve the computational efficiency, reaching sub-second for the first time. The DWT layer is capable of extracting detailed information of the temperature distribution while the “generator” approximates the inverse operator of the governing PDE system with its convolution kernels. A minimax optimization strategy is adopted through adversarial training between the “generator” and the “discriminator” to encourage deeper understanding of data distribution. Compared with other types of deep learning methods, the proposed method shows its superiority when dealing with highly transient experimental data and micro-nano porous structured geometry. Based on the developed sub-second light-weight numerical solver with operator learning, the development of advanced real-time soft sensor techniques for a class of 3D transient IHTP in practical engineering applications is possible.

Determination of thermophysical characteristics in a nonlinear inverse heat transfer problem

This paper presents a detailed and developed method for finding soil nonlinear thermophysical characteristics. Two-layer container complexes are constructed. The side faces of which are thermally insulated, so that one can use the one dimensional heat equation. In order not to solve the boundary value problem with a contact discontinuity and not to lose the accuracy of the solution method, a temperature sensor was placed at the junction of the two media, and a mixed boundary value problem is solved in each region of the container. In order to provide the initial data for the inverse coefficient problem, two temperature sensors are used: one sensor was placed on the open border of the container and recorded the temperature of the soil at this border, and the second sensor was placed at a short distance from the border, which recorded the air temperature. The measurements were carried out in the time interval (0,4tmax). At first, the initial-boundary value problem of nonlinear thermal conductivity equation with temperature dependent coefficients of thermal conductivity, heat capacity, heat transfer and material density is investigated numerically. The nonlinear initial-boundary value problem is solved by the finite difference method. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar Thomas’ method, and the nonlinear difference problem is solved by the Newton’s method. Assuming the constancy of all thermophysical parameters, except for the thermal conductivity of the material, on the segment (0,tmax), the quadratic convergence of Newton’s method is proved and the thermal conductivity is calculated, while the solution of the linearized difference problem was taken as the initial approximation for the Newton’s method. By freezing all other thermophysical parameters, and using the measured temperatures in the segment (tmax,2tmax), the coefficient of specific heat of the material is found. Proceeding in a similar way, using the initial data on the segment (2tmax,3tmax), the density of the soil is determined. Also, based on the measured initial data on the segment (3tmax,4tmax), the heat transfer coefficient is determined.Based on the experimentally measured data, at each time interval, the corresponding functional is minimized using the gradient descent method. The differentiation of a nonlinear difference problem with respect to the desired parameter method is used to find the gradient of the functional. This method allows you to find the functional gradient and the damping coefficient of the gradient descent in an explicit form. In this case, the compilation and solution of the conjugate problem, which is a frequently used method for solving the inverse problems, is not needed.The performed numerical calculations show that for small time intervals the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%. In addition, all thermophysical characteristics (8 coefficients) were found for a two-layer container with sand and black soil. Also the dependence of the temperature difference at the boundary between the environment and the soil for the heat transfer coefficient was shown.

On the fast assemblage of finite element matrices with application to nonlinear heat transfer problems

The finite element method is a well-established method for the numerical solution of partial differential equations (PDEs), both linear and nonlinear. However, the repeated reassemblage of finite element matrices for nonlinear PDEs is frequently pointed out as one of the bottlenecks in the computations. The second bottleneck being the large and numerous linear systems to be solved arising from the use of Newton’s method to solve nonlinear systems of equations. In this paper, we will address the first issue. We will see how under mild assumptions the assemblage procedure may be rewritten using a completely loop-free algorithm. Our approach leads to a small matrix-matrix multiplication for which we may count on highly optimized algorithms.

Solution of a nonlinear heat transfer problem based on experimental data

The paper develops a method for solving nonlinear equations of heat conduction. Two-layer container complexes have been created, the side faces of which are thermally insulated so that the 1D heat equation can be used. In order not to solve the boundary value problem with a contact discontinuity and lose the accuracy of the solution method, a temperature sensor was placed at the junction of two media, and a mixed boundary value problem is solved in each area (container). To provide initial data for the initial boundary value problem, three temperature sensors were used: two sensors measure air temperatures at the left and right boundaries of the container complex; the third sensor measures the temperature of the soil at the junction of two media. The paper numerically investigates the initial-boundary problem of heat conduction with nonlinear coefficients of heat conduction, heat capacity, heat transfer and density of the material. To solve a nonlinear initial-boundary value problem, the grid method is used. Two types of difference schemes are constructed: linearized and nonlinear. The linearized difference scheme is implemented numerically by the scalar sweep method, and the nonlinear difference problem is solved by the Newton method. On the basis of an a priori estimate of the solution of a nonlinear difference problem, we prove the convergence of the second degree of Newton’s method. The numerical calculations carried out show that, for small time intervals, the solutions of the linearized difference problem differ little from the solution of the nonlinear difference problem (1 - 3%). And for long periods of time, tens of days or months, the solutions of the two methods differ significantly, sometimes exceeding 20%

ADAPTATION OF TEMPERATURE WALL FUNCTION FOR CALCULATION THE HEAT TRANSFER AT SUPERCRITICAL COOLANT PARAMETERS

To date, the engineering community pays special attention to the study, research and development of methods for predicting the behavior of the coolants at supercritical parameters. This paper discusses the problem of adaptation of the existing universal or specialized tools of thermohydraulic analysis for nonlinear heat transfer problems at supercritical coolant parameters with deterioration of heat transfer. Complexities of prediction the nonlinear heat transfer at supercritical parameters of the coolant by engineering methods of computational fluid dynamics are considered. A simple way to adapt the dual-zone Kader's temperature wall function on the basis of existing probe measurements for carbon dioxide is proposed. The problem of implementation into universal packages of computational fluid dynamics (CFD), which is based on the method of choosing the reference coordinate of the near-wall zone to determine the dynamic speed and dimensionless temperature, is discussed. The example on the ANSYS CFX using shows one of the ways to create a special user procedure, which has an improved tendency to predict the axial temperature profile with deteriorated heat transfer. Calibration and validation of the obtained results on the basis of experimental investigations for vertical pipes and rod assemblies of fuel simulators is carried out in the work. The paper also discusses the features of the proposed implementation, and formed recommendations for the application and further improvement of engineering approaches to predict the heat transfer deterioration at supercritical parameters of the coolant.

Semianalytical Solution of Transient Heat Transfer for Laminated Structures under Time-Varying Boundary Conditions

In order to solve the transient heat transfer problem of the laminated structure, a semianalytical method based on calculus is adopted. First, the time domain is divided into tiny time segments; the analytical solution of transient heat transfer of laminated structures in the segments is derived by using the method of separation of variables. Then, the semianalytical solution of transient heat transfer in the whole time domain is obtained by circulation. The transient heat transfer of the three-layer structure is analyzed by the semianalytical solution. Three time-varying boundary conditions (a: square wave, b: triangular wave, and c: sinusoidal wave) are applied to the surface of the laminated structure. The influence of some key parameters on the temperature field of the laminated structure is analyzed. It is found that the surface temperature of the laminated structure increases fastest when heated by square wave, and the maximum temperature can reach at 377°C, the temperature rises the most slowly when heated by the triangular wave, and the maximum temperature is 347°C. The novelty of this work is that the analytical method is used to analyze the nonlinear heat transfer problem, which is different from the general numerical method, and this method can be applied to solve the heat transfer problem of general laminated structures.

Application of Euler Neural Networks with Soft Computing Paradigm to Solve Nonlinear Problems Arising in Heat Transfer.

In this study, a novel application of neurocomputing technique is presented for solving nonlinear heat transfer and natural convection porous fin problems arising in almost all areas of engineering and technology, especially in mechanical engineering. The mathematical models of the problems are exploited by the intelligent strength of Euler polynomials based Euler neural networks (ENN’s), optimized with a generalized normal distribution optimization (GNDO) algorithm and Interior point algorithm (IPA). In this scheme, ENN’s based differential equation models are constructed in an unsupervised manner, in which the neurons are trained by GNDO as an effective global search technique and IPA, which enhances the local search convergence. Moreover, a temperature distribution of heat transfer and natural convection porous fin are investigated by using an ENN-GNDO-IPA algorithm under the influence of variations in specific heat, thermal conductivity, internal heat generation, and heat transfer rate, respectively. A large number of executions are performed on the proposed technique for different cases to determine the reliability and effectiveness through various performance indicators including Nash–Sutcliffe efficiency (NSE), error in Nash–Sutcliffe efficiency (ENSE), mean absolute error (MAE), and Thiel’s inequality coefficient (TIC). Extensive graphical and statistical analysis shows the dominance of the proposed algorithm with state-of-the-art algorithms and numerical solver RK-4.

Heat transfer for nonlinear boundary conditions involving radiation for constant and temperature related thermal conductivity material using the MWLS method

The paper presented the heat transfer analysis for nonlinear boundary conditions involving radiation with constant and temperature related thermal conductivity material. For the constant thermal conductivity, linearizing the fourth power of radiation item, solve the temperature distribution through the global iteration coupled with the meshless weighted least squares (MWLS) method. For the temperature related variable thermal conductivity, the Kirchhoff transformation is usually employed to transform the nonlinear heat transfer problem into the linear Laplace's equation. Through the two-step iteration method, the convection and radiation boundary are converted into Dirichlet and Neumann boundary which is the easy-to-program linear condition after the transformation. In the end, the effectiveness and accuracy of the method were illustrated by some numerical examples .

Full-Scale Bounds Estimation for the Nonlinear Transient Heat Transfer Problems with Interval Uncertainties

A package solution is presented for the full-scale bounds estimation of temperature in the nonlinear transient heat transfer problems with small or large uncertainties. When the interval scale is relatively small, an efficient Taylor series expansion-based bounds estimation of temperature is stressed on the acquirement of first and second-order derivatives of temperature with high fidelity. When the interval scale is relatively large, an optimization-based approach in conjunction with a dimension-adaptive sparse grid (DSG) surrogate is developed for the bounds estimation of temperature, and the heavy computational burden of repeated deterministic solutions of nonlinear transient heat transfer problems can be efficiently alleviated by the DSG surrogate. A temporally piecewise adaptive algorithm with high fidelity is employed to gain the deterministic solution of temperature, and is further developed for recursive adaptive computing of the first and second-order derivatives of temperature. Therefore, the implementation of Taylor series expansion and the construction of DSG surrogate are underpinned by a reliable numerical platform. The parallelization is utilized for the construction of DSG surrogate for further acceleration. The accuracy and efficiency of the proposed approaches are demonstrated by two numerical examples.

A new optimum numerical method for analysis of nonlinear conductive heat transfer problems

In this research, a new optimum numerical method is introduced to analyze nonlinear conductive heat transfer problems. The governing equation and the boundary conditions are combined using the penalty method to define a proper cost function. Furthermore, a Moving Least Squares (MLS) approach is applied to approximate the derivatives in the introduced nonlinear objective function. The objective function is minimized using particle swarm optimization (PSO) to find un-known nodal values. Obtained numerical results are compared with those reported in literature. The comparisons clearly illustrate the successfulness of the introduced strategy to solve the nonlinear heat transfer and its superiority in comparison with other well-known schemes.

A model of micro lubrication between two walls with unequal temperature distribution based on kinetic theory

Micro lubrication of a gas between two walls with arbitrary and independent temperature distributions is studied on the basis of the Bhatnagar–Gross–Krook–Welander (BGKW) model of the Boltzmann equation. The BGKW equation is studied analytically using the slowly varying approximation. Following the author's previous study [T. Doi, “A model of micro lubrication between two walls with an arbitrary temperature difference based on kinetic theory,” Phys. Fluids 32, 052005 (2020)], the leading-order approximation, which ought to be the solution of the nonlinear heat transfer problem, is replaced by its free molecular solution. A lubrication model of the Reynolds-type equation is derived in closed form. A direct numerical analysis of the lubrication flow subject to localized heating or cooling of the walls is conducted for an assessment of the lubrication model. The lubrication lift calculated using the model agrees with that of the direct numerical solution within an error of 5% when the Knudsen number based on the gap size lies between 0.1 and 10. The result of the lubrication model agrees also with that of the Boltzmann equation for a variable hard sphere gas. A sharp peak arises in the pressure distribution for large Knudsen numbers owing to the effect of thermal creep flows induced by localized heating.

A novel numerical method for the solution of nonlinear equations with applications to heat transfer

PurposeThe purpose of this study is to extend a novel numerical method proposed by the first author, known as the dual mesh control domain method (DMCDM), for the solution of linear differential equations to the solution of nonlinear heat transfer and like problems in one and two dimensions.Design/methodology/approachIn the DMCDM, a mesh of finite elements is used for the approximation of the variables and another mesh of control domains for the satisfaction of the governing equation. Both meshes fully cover the domain but the nodes of the finite element mesh are inside the mesh of control domains. The salient feature of the DMCDM is that the concept of duality (i.e. cause and effect) is used to impose boundary conditions. The method possesses some desirable attributes of the finite element method (FEM) and the finite volume method (FVM).FindingsNumerical results show that he DMCDM is more accurate than the FVM for the same meshes used. Also, the DMCDM does not require the use of any ad hoc approaches that are routinely used in the FVM.Originality/valueTo the best of the authors’ knowledge, the idea presented in this work is original and novel that exploits the best features of the best competing methods (FEM and FVM). The concept of duality is used to apply gradient and mixed boundary conditions that FVM and its variant do not.