Unsteady Heat Conduction Equation Research Articles

Heat Transfer Coefficient Distribution—A Review of Calculation Methods

Determination of the heat transfer coefficient (HTC) distribution is important during the design and operation of many devices in microelectronics, construction, the car industry, drilling, the power industry and research on nuclear fusion. The first part of the manuscript shows works describing how a change in the coefficient affects the operation of devices. Next, various methods of determining the coefficient are presented. The most common method to determine the HTC is the use of Newton’s law of cooling. If this method cannot be applied directly, there are other methods that can be found in the open literature. They use analytical formulations, the lumped thermal capacity assumption, the 1D unsteady heat conduction equation for a semi-infinite wall, the fin model, energy conservation and the analogy between heat and mass transfer. The HTC distribution can also be calculated by means of computational fluid dynamics (CFD) modelling if all boundary conditions with fluid and solid properties are known. Often, the surface on which the HTC is to be determined is not accessible for any measuring sensors, or their installation might disturb the analysed phenomenon. It also happens that calculations using direct or CFD methods cannot be performed due to the lack of required boundary conditions or sufficiently proven models to analyse the considered physical phenomena. Too long a calculation time needed by CFD tools may also be problematic if the method should be used in the online mode. One way to solve the above problem is to assume an unknown boundary condition and include additional information from the sensors located at a certain distance from the investigated surface. The problem defined in this way can be solved by inverse methods. The aim of the paper is to show the current state of knowledge regarding the importance of the heat transfer coefficient and the variety of methods that can be used for its determination.

О прямом методе решения задачи сопряженного теплообмена газовой смеси и твердого тела

A novel methodology of numerical solving is developed for applications of unsteady conjugate heat transfer. It is based on a parallel integration strategy of governing equations in fluid and solid domains. The 3D numerical model takes into account an unsteady thermal interaction of viscous multicomponent flow and a solid body. The fluid dynamic model is based on an extended system of the compressible Navier–Stokes equations with the multicomponent diffusion. In a solid, the unsteady heat conduction equation is stated. In a fluid–solid system the heat transfer is fully coupled. Normal heat flux and temperature are continuous across an interface. The method is based on direct coupling heat transfer due to the time integration of the heat equation in fluid and in solid with an automatic approximation of the interfacial condition. This approach is especially effective for unsteady computations, since it does not require the use of an iterative method at each time step. The proposed method is generalized for multiblock conformal unstructured grids. This approach is especially effective for non-stationary calculations, since it does not require the use of an iterative method at each time step. The results of comparison with a model problem analytical solution confirm an efficiency of the proposed method.

Study on Heat Transfer Characteristics and Intelligent Optimization of Protective Clothing Materials in front of the Blast Furnace.

Special protective clothing should have a certain thermal insulation and protective effect in the production in front of the blast furnace. In order to study the design of special protective clothing for blast furnace foreman, its temperature distribution and internal thickness structure are deeply analyzed. First, based on the existing experimental data, combined with heat transfer and the Fourier law theory, a one-dimensional unsteady heat conduction equation is established. Using the explicit finite difference solution algorithm, the distribution law of the temperature of each fabric layer with time is obtained and the error analysis is performed. The error result is 0.42877%, which shows that the numerical solution is stable and convergent. Then, through the intelligent optimization model of a simulated annealing ant colony (SA-ACO), compared with the single ant colony algorithm (ACO) and a simulated annealing algorithm (SA), it avoids falling into the local optimal solution, reversely solves the optimal thickness of two layers of protective clothing in front of the blast furnace, improves the convergence speed of optimization, and verifies the reliability of the model with a sensitivity test. Finally, this paper actually solves the problem of the protective clothing design and mathematical intelligent model in front of the blast furnace and combines it with numerical simulation software to realize the numerical simulation of temperature distribution and thickness optimization of special protective clothing in front of the blast furnace, so as to provide theoretical guidance for enterprise blast furnace production and development of special protective clothing.

Numerical simulation of flow and heat transfer performance during supercritical water injection in vertical wellbore: A parameter sensitivity analysis

Supercritical water injection is a promising technology for heavy oil thermal recovery. Predicting and regulating the thermophysical parameters of supercritical water at bottomhole are the prerequisite for achieving high recovery efficiency. In this paper, a novel numerical model was proposed to simulate wellbore flow and heat transfer of supercritical water injection. A modified correlation of frictional coefficient was developed to calculate water flow resistance near its critical point, where its properties change abruptly. The unsteady heat loss to the formation was calculated directly by solving two-dimensional unsteady heat conduction equations. They were respectively coupled in momentum and energy balance equations using an iterative scheme. This model was proved to be accurate by two oilfield cases in which the relative errors of wellbore fluid pressure and temperature are less than 1%. Then parameters sensitivity analysis of the injection pressure, temperature, mass flux and the apparent heat conductivity of insulating tube was conducted. The results indicated that the temperature variation of wellbore fluid depended on both enthalpy drop (or heat loss) and Joule-Thomson effect. An abnormal phenomenon that the fluid temperature increased with wellbore depth near the critical and pseudo-critical points was found because of the sudden increase in high heat capacity and Joule-Thomson coefficient of water. Raising the bottomhole fluid temperature was the key to enhanced oil recovery by supercritical water injection. Low apparent heat conductivity of insulating tube contributed richly to raise bottomhole fluid temperature by enlarging thermal resistance and reducing wellbore heat loss. There existed an optimal mass flux for maximizing bottomhole temperature, because when the mass flux increased, the shortened resident time within wellbore and the decreased fluid pressure favored temperature increase and decrease respectively. Selecting an injection pressure near the critical or pseudo-critical point and raising the injection temperature would increase the bottomhole temperature and reduce relative fluid heat loss.

Efficient Simulations of Large Scale Convective Heat Transfer Problems

We describe an approach for efficient solution of large scale convective heat transfer problems, formulated as coupled unsteady heat conduction and incompressible fluid flow equations. The original problem is discretized in time using classical implicit methods, while stabilized finite elements are used for space discretization. The algorithm employed for the discretization of the fluid flow problem uses Picard's iterations to solve the arising nonlinear equations. Both problems, heat transfer and Navier-Stokes quations, give rise to large sparse systems of linear equations. The systems are solved using iterative GMRES solver with suitable preconditioning. For the incompressible flow equations we employ a special preconditioner based on algebraic multigrid (AMG) technique. The paper presents algorithmic and implementation details of the solution procedure, which is suitably tuned, especially for ill conditioned systems arising from discretizations of incompressible Navier-Stokes equations. We describe parallel implementation of the solver using MPI and elements of PETSC library. The scalability of the solver is favourably compared with other methods such as direct solvers and standard GMRES method with ILU preconditioning.

Hybrid Analytical/Numerical Solution of the Unsteady Heat Conduction Equation Subject to Unequal Robin Boundary Conditions

Hybrid Analytical/Numerical Solution of the Unsteady Heat Conduction Equation Subject to Unequal Robin Boundary Conditions

A COMPARATIVE STUDY OF NEW HOMOTOPY PERTURBATION METHOD AND FINITE DIFFERENCE METHOD FOR SOLVING UNSTEADY HEAT CONDUCTION EQUATION

Calendering is a mechanical finishing technology used in the leather, textile, and paper industry, with the mechanism of reshaping and deforming fibers of the web to be calendered with the help of pressure and heat transfer. In this paper, a comparative study has been done to examine the flow of heat inside the calender nip using an unsteady heat conduction equation with specific initial and boundary conditions. The outcomes achieved by utilizing the new homotopy perturbation method and finite difference method have been compared with the exact solution. The achieved outcomes reveal the accuracy, effectiveness, and reliability of techniques applied. It is observed that the outcomes achieved by the new homotopy perturbation method are more accurate than those obtained by the finite difference method.

Effect of Thermal Stresses and Reduced Differential Transform on Bending of The Rectangular Plate

The bending of the rectangular plate exists for a large length comparing with its width as we have considered the simply supported plate. Also, it gets the accuracy, if the plate is attached between enough distance from the ends. This paper concern with the study of bending moment in the rectangular plate by using the thermal stresses. In this paper, we have considered the mathematical model of the unsteady state three-dimensional heat conduction equation for a thick rectangular plate with non-zero temperature initially. We have determined the thermal stresses and the bending moment in the rectangular plate by using the reduced differential transform method. The results are discussed by considering the special cases for copper material. Also, results are shown graphically. The graphs are drawn by using mathematical software Scilab.

Analysis of Thermodynamic Parameter Variability in a Chamber of a Furnace for Thermo-Chemical Treatment

This paper presents results of research on unevenness of cylinder heating in a furnace for thermo-chemical treatment. Experimental research was conducted with respect to nitriding. Various heating speeds and settings of the fan operation in the furnace were considered. Boundary conditions were calculated in the form of temperature and the heat transfer coefficient (HTC) on the cylinder boundary in four planes along the cylinder length. Calculations were performed with the use of the inverse problem for non-linear and unsteady heat conduction equations. Boundary conditions from individual planes were compared with the mean value of them all. The variability of the calculated boundary conditions (temperature and HTC) along the cylinder length was investigated based on values of the absolute and relative differences for temperature and HTC. Estimates: mean value, mean value from the absolute value and the maximum values for the absolute and the relative differences of temperature and HTC were also calculated. Estimates were the measurements of the unevenness of cylinder heating in a furnace for thermo-chemical treatment. Based on the results of our research, it was found that an increase of the fan rotational speed from 50% to 100%, with the same heating speed, resulted in a significant leveling of temperature in the analyzed planes. The difference in temperature along the cylinder length was reduced from 6.8 °C to 3.3 °C. The increase of the heating speed from 5 °C/min to 10 °C/min resulted in an increase of the unevenness of the cylinder heating. Values of the absolute differences of temperature in the analyzed planes with reference to the mean temperature changed from an interval from −2.7 °C to 2.3 °C to a range from −4 °C to 5 °C. In processes with a heating speed greater than 5 °C/min, more intensive heating in the end part of the cylinder (close to the cylinder) was achieved than it was in other planes. It was proven by temperature values, which were higher, even, by 5.4 °C, and by HTC values, higher by 11.4 W/m2K, when compared with mean values. Obtained results can form the basis for nitriding process optimization.

Impact of ELM mitigation on the ITER monoblock thermal behavior and the tungsten recrystallization depth

Type I Edge Localized Modes (ELMs) occur naturally in H-mode plasmas, the operational regime envisaged for ITER, and lead to high temporal heat load peaks on plasma-facing components that might induce surface melting and deteriorated material properties due to recrystallization. ELM mitigation techniques are being developed that either attempt to fully suppress the ELM, e.g. using resonant magnetic perturbations (RMP), or to trigger the ELM at higher frequencies, intending to reduce its energy content and hence the heat load peak. In this paper, the effect of the increased ELM frequency achieved by ELM mitigation techniques on the ITER tungsten divertor monoblock thermal and recrystallization behavior is numerically analyzed by solving the unsteady heat conduction equation when repeatedly exposing the monoblock to ELM-like heat loads under ITER baseline burning plasma operating conditions. The impinging heat load is based on a set of empirical relations readily found in literature and the recrystallization fraction is computed using an empirical data fit which is based on the non-isothermal JMAK equation as function of temperature and time. Uncertainties such as the statistical spread on empirical relations and the unknown inter-ELM heat load, as well as the effect of the ELM frequency, are quantified by a parameter scan. Adopting the recently developed concept of a ‘recrystallization budget’ to determine the allowed monoblock heat loads [G. De Temmerman et al., PPCF 60 (2018) 0044018], it was found that ELM frequencies between 25 – 50 Hz are allowed for inter-ELM heat loads between 9 – 6 MW m − 2 , respectively, when assuming that the scaling for the peak ELM target energy density in [T. Eich et al., NME 12 (2017) 84-90] holds. This provides a new limit for ELM mitigation strategies in ITER. • The ITER monoblock is analyzed numerically under repeated mitigated ELM heat loads. • An empirical heat load model is developed for mitigated ELMs using Eich’s scaling. • ELM mitigation techniques increase the time-averaged peak heat load. • A novel recrystallization budget limits the permissible ELM mitigation frequency.

Research on Measuring Thermal Conductivity of Quartz and Sapphire Glass Using Rear-Side Photothermal Deflection Method

As the display industry continues to advance, various new materials are being developed for utilizing microtechnology and nanotechnology in display panels. Among these, transparent materials have been widely applied to the internal wiring of displays and flexible substrates, owing to their high optical transmittance, isotropy, and anisotropy. Thus, measurement of the thermophysical properties of various transparent materials is important. This study measured thermal conductivity by selecting quartz, a transparent isotropic material, and sapphire glass, a transparent anisotropic material, as measurement target materials using a rear-side photothermal deflection method. Measurements were made via a three-dimensional unsteady heat conduction equation, to which complex transformation was applied and numerically analyzed using COMSOL Multiphysics. Phase delays for a pump beam and a probe beam for a relative position were derived through a deflection analysis. From the derived phase delays between the numerical analysis and experimental result with optical alignment, the absolute and relative errors of quartz were appropriately confirmed to be 0.069 W/m-K and 5%, respectively, while those of the sapphire glass were likewise confirmed to be 0.55 W/m-K and 1.5%, respectively.

Enhancement of methodology for protection of structures in contradiction of thermal effects

The research paper is devoted to protection of structures against heat and temperature effects. The necessity of improving the calculation of multilayered fence structures is shown. The solution of a one-dimensional unsteady heat conduction equation with constant and variable coefficients allowing to use of inhomogeneous and anisotropic materials as the fence material is given. An example of the solution of a fence made of inhomogeneous and anisotropic material is given. Solution of heat conduction equation is obtained by the recurrence-operator method. The solution of one-dimensional unsteady heat conduction equation with variable coefficients is obtained using the recurrence-operator method. The possibility of using the solution of the equation for multilayered inhomogeneous anisotropic fence materials is indicated.

Lucid characterization of unsteady heat conduction in large, square crossbars and cubes subject to uniform surface heat flux by way of “quasi–steady” Helmholtz equations

The technical paper addresses unsteady heat conduction in large, square crossbars and cubes heated continually with uniformly surface heat flux. The uniform surface heat flux is customarily provided by electrical, radiative and pool fire heating in engineering applications. This type of heating fall under the category of non-homogeneous Neumann boundary conditions. The existing traditional method for solving the unsteady, heat conduction equation in two and three dimensions with constant thermo-physical properties and subject to non-homogeneous uniform heat flux boundary conditions is the principle of superposition of partial solutions of simpler unsteady, heat conduction equations. For the latter, the exact, analytical temperature distributions are customarily expressed by infinite series regardless of the solution method. For the numerical evaluation of spatio-temporal temperatures, the main inconvenience of the infinite series lies in the poor convergence when the time is small. To prevent the impeding obstacle, the Transversal Method Of Lines (TMOL) is implemented in this work to transform the unsteady, two- and three-dimensional heat conduction equations into degraded “quasi steady”, two- and three-dimensional heat conduction equations, namely the homogeneous Helmhotz equations. Thereby, the exact, analytical solutions of the converted partial differential equations of elliptic type and homogeneous were determined by the standard method of separation of variables. The resulting approximate, semi-analytical two- and three-dimensional temperature distributions for large, square crossbars and cubes do not contain elaborate infinite series. Instead, the temperature disdtributions contain hyperbolic functions elucidating worthy compactness and superb quality, regardless of the time, either “small time”, “moderate time” or “large time”.

On the prediction of hot spot induced ignition by the Livengood-Wu integral

The Livengood-Wu (L-W) integral is popularly used to predict the occurrence of homogeneous autoignition without heat and mass transport; yet it has not been used for non-homogeneous forced ignition which usually occurs in fire or explosion accidents. In this study, the forced ignition process caused by a hot spot in a flammable mixture is considered, and a method using the L-W integral is developed to predict the critical ignition temperature. In this method, the evolution of the temperature at the hot spot center is first obtained by solving the one-dimensional unsteady heat conduction equation. Then the L-W integral is evaluated based on the central temperature evolution and the corresponding homogenous ignition delay time. The critical ignition temperature is determined at the condition when the L-W integral reaches unity. The present method based on the L-W integral can reduce the computational cost by two to three orders compared to the transient simulation considering detailed chemistry and transport. Different fuels including methane, hydrogen, n-heptane, and dimethyl ether are considered. The predicted critical ignition temperature based on the L-W integral is compared with that predicted by one-dimensional transient simulations considering detailed chemistry and transport. For hydrogen, the present method based on the L-W integral can accurately predict the critical ignition temperature. However, for the other three fuels, the predicted critical temperature is lower than that from detailed simulation. Such under-prediction is interpreted by the ratio between the time scales for ignition and mass diffusion. Nevertheless, the method based on the L-W integral can give a conservative prediction of the critical ignition temperature, which is important for fire safety considerations. Besides, the effects of low temperature chemistry (LTC) on hot spot induced ignition are examined. It is found that the hot spot induced ignition is mainly controlled by high temperature chemistry.

Semi-analytical solution of unsteady heat conduction in plain walls with equal surface temperatures: The Transversal Method Of Lines (TMOL) delimited to the “small time” sub-domain

The present paper addresses uni-directional, unsteady, heat conduction in a plane wall with uniform initial temperature and equal temperature prescribed at the surfaces. The thermal diffusivity of the solid is assumed to be nearly invariant with temperature. A hybrid analytical /numerical procedure named the Transversal Method Of Lines (TMOL) transforms the one-dimensional, unsteady heat conduction equation along with the initial temperature in rectangular coordinates into an equivalent one-dimensional, “quasi-steady” heat conduction equation applied at a pre set time. In other words, a partial differential equation of parabolic type is converted into an ordinary differential equation of second order with the space coordinate as the independent variable and the time absorbed as an embedded parameter. In this work, the analytical/numerical mid-plane and mean temperature profiles, as well as the analytical-numerical total heat transfer are produced by various time levels within TMOL framework, namely 1−TMOL, 2−TMOL and 3−TMOL. Overall, from a qualitative standpoint, the analytical/numerical mid-plane and mean temperature profiles and total heat transfer with the the 3−TMOL approach exhibit excellent quality over the crucial “small time” sub-domain evidencing relative errors that stay within a 5% margin.

Analytical model for prediction of temperature distribution in early age mass concrete

Thermally induced cracks have far-reaching implications on the durability of concrete structures. When cement mixes with water, the reaction is exothermic implying the release of heat. In the case of mass concrete structures, quite a substantial increase in internal temperature may be experienced depending on the ambient temperature and cement content in the mix. The objective of the paper is to develop a mathematical model to predict the time dependent temperature profile in early age mass concrete. Mass concrete block was used to verify the model. Type-K thermocouples placed at various positions and digital thermometer was used to monitor the temperature distribution within the mass concrete block at intervals. The highest temperature values occurred within the core of the mass concrete after one day of concrete placement. Analytical model was developed by applying method of separation of variables and orthogonality relation to two dimensional unsteady state heat conduction equations. The model equation was evaluated and using MATLAB based computer programe. The model successfully predicted the temperature variation within the mass concrete with time. It is therefore suitable for use in the assessment of thermal cracks potential in mass concrete structures.Â

The Numerical Method of Lines facilitates the instruction of unsteady heat conduction in simple solid bodies with convective surfaces

The present study on engineering education addresses the Method of Lines and its variant the Numerical Method of Lines as a reliable avenue for the numerical analysis of one-dimensional unsteady heat conduction in walls, cylinders, and spheres involving surface convection interaction with a nearby fluid. The Method of Lines transforms the one-dimensional unsteady heat conduction equation in the spatial and time variables x, t into an adjoint system of first-order ordinary differential equations in the time variable t. Subsequently, the adjoint system of first-order ordinary differential equations is channeled through the Numerical Method of Lines and the powerful fourth-order Runge–Kutta algorithm. The numerical solution of the adjoint system of first-order ordinary differential equations can be carried out by heat transfer students employing appropriate routines embedded in the computer codes Maple, Mathematica, Matlab, and Polymath. For comparison, the baseline solutions used are the exact, analytical temperature distributions that are available in the heat conduction literature.

Mathematical modeling of frozen rock massifs temperature fields considering phase transitions during the shaft deepening

The object of the research is the formation of frozen rock massif during the shaft deepening. The purpose of the work is to create a mathematical model of the rock mass dimensional temperature fields in the process of freezing wells during the mine shaft penetration. This work was carried out at the Perm National Research Polytechnic University. Design calculations provided by the customer, analytical solution of the unsteady heat conduction equation to determine the preliminary calculation of time of the frozen rock massif formation were done. ANSYS application package was used to simulate the dynamics of the temperature field in this work, with which it is also possible to determine the ice formation time in the rock massif. The initial geometrical and thermophysical information was prepared in the geographic information system «ArcGIS». Dependences frozen rock massif thickness on time and the initial temperature of the environment are obtained. The approximate dates for the formation the required thickness of the frozen rock massif, which are required for the shaft deepening, are determined.

Numerical prediction of solidified shell thicknesses obtained in continuous casting with different billet shapes

Thermal physics of continuous casting is strongly dependent on the velocity of the molten metal, surface heat extraction rate and also on the mold shapes. In the present numerical study, both the moving front and the geometric boundaries of non-rectilinear molds are modeled using Immersed Boundary Method (IBM) which reduces computational time and provides simplicity as compared to the body confirming grid approaches. A second order accurate finite difference method is used to solve unsteady heat conduction equation along with lateral convection to predict the solidified shell thickness during continuous casting. This methodology shows good agreement with the reported data for a rectangular billet. A quantitative study has been performed to report the effects of casting velocity and heat extraction rate on the solidified shell thickness for different mold shapes such as circular, elliptic and I-shaped. Chipman-Fondersmith correlations for solidified shell thickness during continuous casting are assessed for these three different mold shapes and improved correlations are suggested to take into account the variations in the heat extraction rate.

Application of methods of mathematical modeling in the optimization design of technological processes of food production

Optimization of thermochemical and biological transformations occurring during the heat treatment of food products can be carried out only taking into account the dynamics of the formation of temperature fields inside the processed products. A mathematical model of the Sous-Vide processing process has been developed, which provides the determination of the dynamics of changes in the temperature field of Sous-Vide processed functional food products based on animal and vegetable raw materials. To calculate the process of Sous-Vide processing of functional products, we used the differential equation of unsteady heat conduction taking into account the Bio criterion. When solving a system of partial differential equations, the mathematical package flexPDE is used. In calculations, a spatial grid with the number of nodes 20000 was used, the accuracy of calculations was 10E-4. Experimentally determined values of thermal conductivity, specific heat capacity, density and thermal diffusivity of each of the components of the food system are used in calculations. The results of the calculated experiment in mathematical modeling of the Sous-Vide processing process show the evolution of temperature fields in different sections of the food system and on its upper surface, having the shape of an ellipsoid of rotation. The dependences of the change in the thermophysical characteristics of the longitudinal and transverse flow of the food system coolant. The results of the computational experiment show a good convergence of the mathematical model and the results obtained, which is 94%, which indicates the adequacy of the developed mathematical model of the Sous-Vide processing process. The obtained mathematical model allows, by means of a computational experiment, to analyze and optimize the Sous-Vide processing modes of a wide range of food raw materials, of various geometric shapes and sizes, with different physicochemical and thermal properties, taking into account their non-stationarity under thermal influence.