Direct Heat Conduction Problem Research Articles

Determination of the coefficient of thermal effusivity ε for a semi-confined rod

The article provides methods and formulas for calculating the coefficient of thermal effusivity ε (coefficient of heat accumulation) of solid bodies using known solutions of direct heat conduction problem. The work describes a method by which the final formulas are simplified to algebraic equations using the relative temperature θM of heating semi-confined rod from the end. The temperature Tc at the beginning of the rod is measured — x0 and time — τ0 from the start of heating. For calculation, the amount of heat Q accumulated during heating by a heater with a constant temperature at the end of the rod is used for the entire heating time. The technique is physically implemented during the experiment by bringing the ends of two semi-confined rods into contact. A flat, low-inertia electric heater is placed between the rods. The amount of heat ∆Q used in the calculations is half of the total heat of the heater for the entire heating time.

Thermal modeling of friction stir welding of thick high-density polyethylene plates

The process temperatures in the friction stir welding of thick polymer plates play a significant role in the joint's quality since the process is characterized by mixed solid and viscous flow states. The heat generation mechanism in each state is fundamentally different, with heat being generated by friction in the solid-state and by viscous shear flow in the viscous state. In this study, the heat generation and dissipation in the friction stir welding of 14 mm thick high-density polyethylene plates were studied numerically through solving the direct heat conduction problem. Two models of heat generation were used in the numerical solution and the effect of the pin rotational speed on the process temperatures was investigated. It was shown that the utilization of a mixed heat generation model consisting of both the solid state and the viscous shear flow considerably improves the numerical model predictions. The temperature predictions were validated through welding experiments and showed a temperature difference of 3 %. Furthermore, it was found that the welding process stabilizes at rotational speeds higher than 800 rpm, where no considerable change occurs in the volume of the viscous flow region and the welding power requirement. The numerical results based on the combined solid-viscous heat model were in good agreement with the experimental thermal histories.

DETERMINATION OF THE TEMPERATURE CONDUCTIVITY COEFFICIENT a GRAPHICAL METHOD BASED ON NOMOGRAM FOR A SEMI-CONFINED ROD

The article presents a technique and formula for calculating the thermal diffusivity of solids using nomograms of temperature changes. The paper describes a technique by which the final formulas are simplified to algebraic equations using the relative temperature θM of heating a semi-restricted rod from the end.Known equations for solving direct problems of heat conduction using regular modes of the 1-st, 2-nd and 3-rd kind, in which the final formulas are simplified.The article provides methods and formulas for calculating the coefficient of thermal conductivity using nomograms.The method uses the maximum of the first derivative of θM and tabular values of the function erf (1/(2 x Fo )), which is a solution to the direct problem of thermal conductivity. The temperature at a specific point of the rod is measured – x1 and time –τ1 with start of heatingent of thermal conductivity of solid bodies using nomograms of temperature changes.The work describes a method by which the final formulas are simplified to algebraic equations using the relative temperature θM of heating a semirestricted rod from the end. Known equations for solving direct heat conduction problems using regular regimes of the 1-st, 2-nd and 3-rd kind, in which the final formulas are simplified. In work Kondratiev H.M. notes: "The theory of the regular thermal regime is one of the sections of the study of heat transfer insolids. The theory of the regular mode considers the process of cooling or heating not throughout, but only at the stage that has ceased to be affected by the initial state of the body.In work of Lykov O.V. «Theory of thermal conductivity» the solution of the problem is done by three methods: classical, operational and the method of Fourier transformation. Not all methods of pulsed heating are related to these types of regular mode, but it can be used due to the simple form of the exponent. In this work, the proposed method does not belong to regular mode methods. It is done by solving the inverse problem with the help of a well-developed graphic nomogram.According to the accepted classification, the problem is solved under boundary conditions of the 1-st kind, in which the surface temperature is specified as a function of time. Considered a simpler case when surface of the bodyremains constant throughout the heat exchange process.This is achieved with the help of special devices that maintain a constant temperature of a semi-restricted rod with thermal insulation of the side surface.

Real-Time Solution of Unsteady Inverse Heat Conduction Problem Based on Parameter-Adaptive PID with Improved Whale Optimization Algorithm

To solve the problem of the common unsteady inverse heat conduction problem in the industrial field, a real-time solution method of improving the whale optimization algorithm (IWOA) and parameter-adaptive proportional-integral-differential (PID) is proposed in the paper. A feedback control system with IWOA-PID, which can inversely solve the boundary heat flux, is established. The deviation between the calculated temperature and the measured temperature of the measured point obtained by solving the direct heat conduction problem (DHCP) is used as the system input. The heat flux which is iteration-solved by IWOA-PID is used as system output. The method improves the initial solution distribution, global search capability and population diversity generalization of the traditional whale optimization algorithm (WOA), which effectively improves the parameter-adaptive capability of PID. The experimental results show that the solution method of inverse heat transfer proposed in the paper can accurately retrieve the variation of the boundary heat flux in real time and has good resistance and self-adaptability.

A review of methods for calculating of the interfacial heat transfer coefficient

As an important parameter for quenching and simulation, the interfacial heat transfer coefficient (IHTC) is hard to be directly measured through experiments. Researchers have come up with different methods to solve the problem. Currently, the numerical method, the heat balance method (HBM), the Beck’s non-linear estimation method (BEM), and the lumped heat capacity method (LCM) are the main methods of calculating the IHTC. In this paper, the direct heat conduction problem and the solution are firstly analysed, then the solutions and applications of different methods for calculating IHTC are summarized and reviewed. The different methods are compared at the end of the paper.

Enhanced Conjugate Gradient Method for Heat Flux Estimation in Laminar and T-Junction Turbulent Mixing Flow

In this paper, the Jaya algorithm is coupled with the traditional conjugate gradient method (CGM) for solving the inverse heat transfer problem. The coupling of two algorithms is termed as enhanced conjugate gradient method (ECGM). Two different applications have been considered to determine the accuracy and stability of ECGM. First, the algorithm is applied in a steady-state two-dimensional laminar flow through a duct for the boundary heat flux estimation. The accuracy of estimation has been compared with traditional CGM by calculating the root mean square (RMS) error. The ECGM algorithm (RMS =0.142 kW/m2) is found to be more accurate than conventional CGM (RMS = 0.198 kW/m2). ECGM is then applied to determine the peripheral variation of heat flux at the fluid-solid interface of a tube at a section downstream to the T-junction. The experimental temperature readings at the outer periphery of the tube required in the inverse algorithm are simulated by solving the direct heat conduction problem. Here, the heat flux at the fluid-solid interface is obtained by solving the governing equations of the turbulent flow by using commercial computational fluid dynamics software. ECGM is found to be a stable and accurate inverse heat transfer algorithm.

Application of inverse heat conduction calculation method for fast-transient flow boiling heat transfer analysis

ABSTRACT A noise robust inverse heat conduction calculation method is applied for the study of fast-transient heat transfer phenomena. This method is useful when the surface of interest is physically inaccessible. A typical finite difference method was modified with a spatial node marching scheme to perform inverse calculations. The Gaussian low-pass frequency filter was adopted to avoid noise induced oscillations during the calculations. Numerical validation of the method, including sensitivity analysis, was performed using a direct one-dimensional heat conduction problem. The developed method predicts the wall temperature and the heat flux with good accuracy, although the inputs to the calculation contain noises. The developed method is demonstrated experimentally using a fast-transient flow boiling test facility. Applying a pulse power on a metallic tube, the resulting temperature response of the tube outer wall was measured with the power transient information simultaneously. The temperature and the heat flux variations on the tube’s inner wall were calculated inversely from the measured data. Finally, its potential to fast-transient boiling heat transfer research was emphasized by showing the clearly different transient boiling heat transfer characteristics according to the applied power.

A physics-informed deep learning method for solving direct and inverse heat conduction problems of materials

Due to the complex physical and chemical changes of thermal conductive materials during the heat transfer process, the research on heat transfer performance has inevitably been a hot point in thermal analysis. In this paper, a novel data-driven framework based on Physical Information Neural Networks (PINNs) is proposed to accomplish the direct analysis and parameter inversion of heat conduction problems. For the first time, two kinds of heat conduction problems are discussed under the PINNs framework simultaneously. In the phase of direct analysis, a modified PINNs framework based on adaptive activation functions is proposed. The case studies of wood and steel indicate that the proposed framework can achieve satisfactory accuracy and has the potential to replace finite element modeling to a certain extent. In the inverse analysis, inversion problems of constant and variable parameters are studied in detail. Coupled neural network frameworks with skip connections are proposed to predict unknown parameters. Experimental results represent that unknown parameters in the heat conduction equation can be accurately inversed with high computational efficiency. Compared with conventional methods, the proposed framework can solve both the direct and inverse heat conduction problems in a unified and concise form and has a broad application prospect in materials science.

Estimation of Functional Form of Time-Dependent Heat Transfer Coefficient Using an Accurate and Robust Parameter Estimation Approach: An Inverse Analysis

This paper presents a numerical method to address function estimation problems in inverse heat transfer problems using parameter estimation approach without prior information on the functional form of the variable to be estimated. Using an inverse analysis, the functional form of a time-dependent heat transfer coefficient is estimated efficiently and accurately. The functional form of the heat transfer coefficient is assumed unknown and the inverse heat transfer problem should be treated using a function estimation approach by solving sensitivity and adjoint problems during the minimization process. Based on proposing a new sensitivity matrix, however, the functional form can be estimated in an accurate and very efficient manner using a parameter estimation approach without the need for solving the sensitivity and adjoint problems and imposing extra computational cost, mathematical complexity, and implementation efforts. In the proposed sensitivity analysis scheme, all sensitivity coefficients can be computed in only one direct problem solution at each iteration. In this inverse heat transfer problem, the body shape is irregular and meshed using a body-fitted grid generation method. The direct heat conduction problem is solved using the finite-difference method. The steepest-descent method is used as a minimization algorithm to minimize the defined objective function and the termination of the minimization process is carried out based on the discrepancy principle. A test case with three different functional forms and two different measurement errors is considered to show the accuracy and efficiency of the used inverse analysis.

Inverse determination of sliding surface temperature based on measurements by thermocouples with account of their thermal inertia

This study developed an inverse heat conduction algorithm to determine temperature at a sliding surface taking account of thermocouple thermal inertia. The direct heat conduction problem was solved analytically based on the Laplace integral transform approach. The inverse algorithm was applied to the problem of friction of a brake material against a steel. The experiments were conducted on a pin-on-disc tribometer for three short-time sliding regimes: velocity step, acceleration and deceleration. Temperature in the pin sample was measured by two identical miniature thermocouples installed at different distances from the friction surface. It was found that the two inverse surface temperatures agree well between each other. The inverse algorithm allows predicting the contact temperature measured by infrared thermography with accuracy 5–7%.

Inverse Approach for the Average Convection Coefficient Induced by a Forced Fluid Flow Over an Annular Fin of Rectangular Profile Using Tip Temperature Measurements

The objective of the present paper is to develop a simple algebraic computational procedure for the estimation of the average convection coefficient of a forced fluid flow over an annular fin of rectangular profile within the platform of inverse heat conduction problems. The data required is the tip temperatures of an annular fin of rectangular profile, which are measured in an experimental setup. Based on nonlinear regression analysis, an empirical correlation equation is constructed for the dimensionless average tip temperature depending upon the dimensionless thermo–geometrical parameter and the radius ratio. When compared against the outcome of a direct heat conduction problem, the good quality of the estimated average convection coefficient for the annular fin of rectangular profile demonstrates the feasibility of the simple algebraic computational procedure.

Uniqueness, existence and solution of the direct boundary heat exchange problem for a weakly non-linear temperature conductivity coefficient

This work is a preliminary result necessary for the formulation and solution of the inverse heat conduction problem for a weakly quasilinear equation. Nevertheless, the article has independent significance. For the direct heat conduction problem a solution is constructed in the form of a series with a small parameter, the existence of the solution, its uniqueness are proved, and a number of estimates are made - in particular, the rate of decrease of the solution at large times. The solution is built for a medium, part of which has a piecewise constant coefficient of thermal conductivity, and part of the medium has slightly non-linear thermal diffusivity. The construction of a classical solution is impossible due to the discontinuity of the thermal diffusivity, therefore, on the sections of the media the matching conditions are set - the conditions for the continuity of the solution and the condition for the continuity of the heat flux. The solution to the weakly quasilinear equation is constructed in the explicit form of a series with a small parameter, for which uniform convergence is proved. The decreasing of the solution at large times is proved as of time raised to the negative third power.

Estimation of heat transfer parameters by using trained POD-RBF and Grey Wolf Optimizer

The article presents a numerical model for estimation of heat transfer parameters, e.g. thermal conductivity and convective coefficient, in two-dimensional solid bodies under steady-state conduction. This inverse problem is stated as an optimization problem, in which input is reference temperature data and the output is the design variables, i.e. the thermal properties to be identified. The search for optimum design variables is conducted by using a recent heuristic method, namely Grey Wolf Optimizer. During the heuristic search, direct heat conduction problem has to be solved several times. The set of heat transfer parameters that lead to smallest error rate between computed temperature field and reference one is the optimum output of the inverse problem. In order to accelerate the process, the model order reduction technique Proper-Orthogonal-Decomposition (POD) is used. The idea is to express the direct solution (temperature field) as a linear combination of orthogonal basis vectors. Practically, a majority of the basis vectors can be truncated, without losing much accuracy. The amplitude of this reduced-order approximation is then further interpolated by Radial Basis Functions (RBF). The whole scheme, named as trained POD-RBF, is then used as a surrogate model to retrieve the heat transfer parameters.

Solving direct and inverse heat conduction problems in functionally graded materials using an accurate and robust numerical method

In this study we present a numerical approach to solve steady-state heat conduction problems in functionally graded materials (FGMs). Two different types of material gradations are considered for spatially varying thermal conductivity of FGM: (1) Quadratic material gradation; (2) Exponential material gradation. The proposed numerical procedure is based on finite-difference method and is developed to solve the steady-state heat conduction equation over a general two-dimensional (irregular) heat conducting body (FGM) with Dirichlet, Neumann, and Robin boundary conditions. In addition to presenting an accurate heat conduction equation solution considering an irregular shape and a variety of boundary conditions, the other novel aspect of this study is to identify the constant parameters in the material gradations accurately by an inverse analysis thereby determining the accurate form of gradation. The novelty of the inverse analysis lies in proposing an accurate and efficient explicit sensitivity analysis scheme. The main advantage of the sensitivity analysis scheme is that it is not involved with an adjoint equation and all the sensitivity coefficients can be explicitly computed in only one direct solution. The conjugate gradient method (CGM) is used to reduce the mismatch between the computed temperature on part of the boundary and the simulated measured temperature distribution. The accuracy, efficiency, and robustness of the proposed numerical approach are demonstrated through presenting two test cases.

Parametric identification of differencial-difference models of heat transfer in one-dimensional bodies based on Kalman filter algorithms

The paper considers the solution of the inverse heat conduction problem by parametric identification of differential- difference models of heat transfer in one-dimensional bodies. Differential-difference models is a system of differential ordinary equations of the first order with respect to the state vector. In this case, the direct and inverse heat conduction problems are solved, and the Kalman recurrent digital filter algorithm is used in terms of parameters for discrepancy minimization between the measured and model parameters values. The paper considers the Kalman algorithm application for two specific problems, namely: an experiment estimation and planning for restoration of heat transfer boundary conditions for a system of bodies. When planning an experiment or during field studies, the task parametrization is carried out initially and then parametric identification, as well. To determine the confidence area for measuring the desired parameters, the Gram matrix (Fisher information matrix) is used, involving the components which are sensitivity functions that represent all significant factors of heat metering: the type of heat transfer in the system, the number and location of temperature measurement points, the quality of channels for recording measured values, and the features of input actions, time measurement section and the number of measurement time points in this section. The paper gives an example of Kalman recurrent digital filter, considers the battery transformer of unsteady heat flux, for which the creation of the differential-difference model is carried out, shows the results of unsteady heat flux restoration, changing according to an arbitrary law. Confidence areas of the desired parameters are established.

Solving Backward Heat Conduction Problems Using a Novel Space–Time Radial Polynomial Basis Function Collocation Method

In this article, a novel meshless method using space–time radial polynomial basis function (SRPBF) for solving backward heat conduction problems is proposed. The SRPBF is constructed by incorporating time-dependent exponential function into the radial polynomial basis function. Different from the conventional radial basis function (RBF) collocation method that applies the RBF at each center point coinciding with the inner point, an innovative source collocation scheme using the sources instead of the centers is first developed for the proposed method. The randomly unstructured source, boundary, and inner points are collocated in the space–time domain, where both boundary as well as initial data may be regarded as space–time boundary conditions. The backward heat conduction problem is converted into an inverse boundary value problem such that the conventional time–marching scheme is not required. Because the SRPBF is infinitely differentiable and the corresponding derivative is a nonsingular and smooth function, solutions can be approximated by applying the SRPBF without the shape parameter. Numerical examples including the direct and backward heat conduction problems are conducted. Results show that more accurate numerical solutions than those of the conventional methods are obtained. Additionally, it is found that the error does not propagate with time such that absent temperature on the inaccessible boundaries can be recovered with high accuracy.

Methodology for assessment of the fire-resistant quality of reinforced-concrete floors protected by fire-retardant coatings

In the work, the tests have been analysed for fire-resistant quality of hollow-core reinforced-concrete floors with fire-retardant plaster covering. A two-dimensional physical model and computer model have been developed in the ANSYS FLUENT software environment, which includes a system of equations and boundary conditions, taking into account the thermal conductivity in the coating and concrete, as well as complex convection-radiation heat transfer in air voids. A one-dimensional multilayer mathematical model has been developed, equivalent to a two-dimensional model of the thermal state of hollow-core reinforced-concrete floor with fire-retardant plaster covering, with the specific these layers’ thicknesses. A methodology has been developed in the article, based on solving the direct and inverse heat conduction problems, with the help of which it is possible to assess the fire-resistant quality of hollow-core reinforced-concrete floors with and without fire-retardant coatings. The developed methodology was applied to determine the thermophysical (heating constant and specific heating capacity per unit volume) and fire-retardant characteristics of the studied plaster covering according to the results of tests for fire-resistant quality of reinforced-concrete floors with this fire-retardant coating. The conclusion was made about the effectiveness of this coating and the boundaries of using the fire-retardant coatings to ensure normalizable values for the limit of the fire-resistant quality of hollow-core reinforced-concrete floors.

An inverse analysis for determination of space-dependent heat flux in heat conduction problems in the presence of variable thermal conductivity

This article presents an inverse problem of determination of a space-dependent heat flux in steady-state heat conduction problems. The thermal conductivity of a heat conducting body depends on the temperature distribution over the body. In this study, the simulated measured temperature distribution on part of the boundary is related to the variable heat flux imposed on a different part of the boundary through incorporating the variable thermal conductivity components into the sensitivity coefficients. To do so, a body-fitted grid generation technique is used to mesh the two-dimensional irregular body and solve the direct heat conduction problem. An efficient, accurate, robust, and easy to implement method is presented to compute the sensitivity coefficients through derived expressions. Novelty of the study is twofold: (1) Boundary-fitted grid-based sensitivity analysis in which all sensitivities can be obtained in only one direct solution (at each iteration), irrespective of the number of unknown parameters, and (2) the way the measured temperatures on part of boundary are related to a variable heat flux applied on another part of boundary through components of a variable thermal conductivity. The conjugate gradient method along with the discrepancy principle is used in the inverse analysis to minimize the objective function and achieve the desired solution.

Mathematical Modeling of the Thermal Response of Laboratory-Scale Probes

Abstract Experiments for cooling curve analysis of laboratory-scale quenched probes produce a wetting front that advances at a finite velocity along the longitudinal axis of the probe as cooling progresses. The wetting front velocity depends on probe material and geometry as well as the type of quenchant, its temperature, and agitation. Because of the wetting front, the mathematical model of the direct heat conduction problem to compute the evolution of the thermal field within the probe may be characterized as a moving front boundary problem. In this work, cooling curves at three subsurface locations along the length of conical-end cylindrical probes, fabricated with AISI 304 stainless steel, during quenching from 850°C with water at 60°C flowing parallel to the probe’s longitudinal axis were measured. A fourth thermocouple located at the geometrical center of the probe was used to validate the model. Two values of free-stream water velocity (0.2 and 0.6 m/s) were studied. As a first approximation, the measured cooling curves were used to estimate the corresponding surface heat flux histories by solving three separate 1-D inverse heat conduction problems. Then, the thermal field evolution within the probe was computed by solving a 2-D axis-symmetrical heat transfer model. The boundary condition at the probe surface was modeled with a composite mapping of the surface heat flux histories. The surface heat flux history estimated for the lower section of the cylindrical part of the probe was applied as a boundary condition for the probe tip. It was found that the cooling curve at the lowest thermocouple position was overestimated; thus, the surface heat flux history for the probe tip was divided by 1.2 to correctly model the thermal response. The model was successfully validated by comparing measured and computed cooling curves for the thermocouple located at the probe’s geometrical center.

FEATURES OF THE APPLICATION OF THE IQLAB PROGRAM FOR SOLVING THE INVERSE HEAT CONDUCTION PROBLEM FOR CHROMIUM-NICKEL CYLINDRICAL THERMOSONDES

The paper presents an experimental-computational study of the results of using the IQLab program to solve inverse heat conduction problem and restore the surface temperature of cylindrical thermosondes from heat-resistant chromium-nickel alloys while cooling them in liquid media. The purpose of this paper is to verify the correct operation of the IQLab program when restoring the surface temperature of thermosondes with 1-3 thermocouples. The IQLab program is also designed to solve one-dimensional nonlinear direct lines and inverse heat conduction problems with constant initial and boundary conditions specified as a function of time in a tabular form with a constant and variable time step. A finite-difference method is used to solve the heat equation. Experiments were carried out on samples D = 10-50 mm in liquids with different cooling capacities such as aqueous solutions of NaCl and Yukon-E polymer, rapeseed oil and I-20A mineral oil. For the calculation we used the readings of thermocouples installed at internal points of cylindrical thermosondes. The advantages of solving inverse heat conduction problems with the IQLab program include the possibility of restoring the surface temperature for cylindrical samples with a diameter of 10 mm to 50 mm with practical accuracy according to the indications of a single thermocouple located in the geometrical center of the thermosonde, which simplifies the manufacture of the probe. For larger dimensions with a diameter D ≥ 50 mm, it is necessary to install control intermediate thermocouples and perform additional tests. The solution of inverse heat conduction problems and restoration of the surface temperature of the sample makes it possible to calculate other important characteristics of the cooling process: the heat flux density and the heat transfer coefficient.