Form Of Heat Conduction Equation Research Articles

Influence of fault current on electromagnetic-thermal characteristics of HTS coated conductors

In this paper, the electromagnetic-thermal coupling numerical model is used to study the loss and temperature distribution of two typical high temperature superconducting (HTS) coated conductors (CCs) under short-circuit fault, as well as the dynamic evolution law with time. The electromagnetic module adopts the widely used H-formulation, and the thermal module adopts the general form of heat conduction equation. The coupling between the two module is realized by using the loss calculated by the electromagnetic module as the heat source of the thermal module and combining the Jc0(B, T) relationship. A modified E-J relation suitable for any current density range is used to describe the resistivity of superconductors. Due to the difference of magnetic and thermal properties of the two kinds of structural tape substrate materials and geometric dimensions of copper stabilized layer, the losses of the two kinds of structural tape under different fault currents show obvious differences. At low fault transport current, the magnetic substrate of type B tape (NiW substrate) will obviously make the magnetic flux density perpendicular to the superconducting layer greater than that of type A tape (Hastelloy substrate), thus making the loss of type B tape greater. Under the condition of high fault transport current, the fault current will be shunted between different layers of the tape. Type B tape has a thicker copper stabilizer, which makes the current density of type B tape copper layer lower than that of type A tape, and the temperature rise and loss of type B tape are lower. Specifically, when Ip/Ic0 is less than about 0.65, the total loss of type A tape is less than that of type B tape. When Ip/Ic0 is greater than about 0.65, the total loss of type A tape is greater than that of type B tape.

Analytical approach for the temperature distribution in the casting-mould heterogeneous system

PurposeThe main purpose of this paper is to calculate the analytical solution or a closed-form solution for the temperature distribution in the heterogeneous casting-mould system.Design/methodology/approachFirst, the authors formulate and analyze the mathematical formulation of heat conduction equation in the heterogeneous casting-mould system, with an arbitrary assumption of the ideal contact at the cast-mould contact point. Then, He-Laplace method, based on variational iteration method (VIM), Laplace transform and homotopy perturbation method (HPM), is used to elaborate the analytical solution of this system. The main focus of He-Laplace method is to find the Lagrange multiplier with an easy approach which enables the implementation of HPM very smoothly and provides the series solution very close to the exact solution.FindingsAn example is considered to show that He-Laplace method provides the efficient results for calculating the temperature distribution in the casting-mould heterogeneous system. Graphical representation and error distribution represents that He-Laplace method is very simple to implement and effective for casting-mould heterogeneous system.Originality/valueThe work in this paper is original and advanced. Specially, calculation of Lagrange multiplier for casting-mould system has not been reported in the literature for this work.

Fractional order heat conduction and thermoelastic response of a thermally sensitive rectangular parallelopiped

In the present paper, the problem of finite dimensional rectangular parallelepiped in isotropic thermoelastic medium with convective type heating is considered. The heat conduction equation (HCE) of the region is described by time HC of fractional order with Caputo derivative form. The non-linear form of heat conduction equation is converted to linear form with Kirchhoff’s transformation. Integral transform technique is used to deal with the spatial variables and Laplace transform technique is used to deal with Caputo type time fractional derivative. Inverse Laplace transform and inverse finite Fourier transform are employed to expose the solution in the transformed domain. Numerical results are obtained for temperature distribution, deflection, stress resultants and thermal stress distribution for different values of time fractional order parameter. These results are presented graphically and discussed for various values of time fractional parameters. The obtained results show significant influence of the time fractional order derivative on the temperature as well as stress distribution. Thermosensitivity plays a vital role in the analysis of any real thermoelastic problems and one should consider their effect while dealing with materials in high temperature environment.

Multiphysics Topology Optimization of Thermal Actuators by Using the Level Set-Based Multiple-Type Boundary Method

Thermal actuators use thermal expansion and contraction of an elastic body to produce motion at its output port. In the present study, a thermal actuator comprises an elastic body and heating/cooling devices. Such devices have a thin-layer shape and are installed on the surface of the elastic body. The design optimization of thermal actuator is a multiphysics problem, including both heat conduction and elastic deformation. The design variables include multiple types of boundaries, i.e., the temperature boundary (high temperature and low temperature) and the free boundary. In order to solve such a multiphysics optimization problem involving multiple types of boundaries, the level set-based multiple-type boundary method is employed. In the analysis for the shape derivative of the temperature boundary, the constrained variational principle is employed to explicitly include the temperature boundary condition into the weak form of heat conduction equation. Numerical examples in two dimensions are investigated.

A general form of the heat conduction equation of thermoelasticity with voids and gravity field

PurposeThe purpose of this paper is to discuss the problem of thermoelasticity with voids under the gravity field in the context of a general form of heat conduction equation with some particular cases. The normal mode method used is to obtain the physical quantities.Design/methodology/approachThe analytical method used was the normal mode.FindingsNumerical results for the physical quantities are presented graphically and the results are analyzed thereafter. The comparisons are established between the theories for the physical quantities during the constant time and gravity and were shown graphically.Originality/valueIn the present work, the authors shall establish the general form for the heat conduction equation which contains the seven theories as well as the special cases in thermoelasticity with voids. The problem is very important in many dynamical systems.

An inverse problem in estimating the laser irradiance and thermal damage in laser-irradiated biological tissue with a dual-phase-lag model

The aim of this study is to solve an inverse heat conduction problem to estimate the unknown time-dependent laser irradiance and thermal damage in laser-irradiated biological tissue from the temperature measurements taken within the tissue. The dual-phase-lag model is considered in the formulation of heat conduction equation. The inverse algorithm used in the study is based on the conjugate gradient method and the discrepancy principle. The effect of measurement errors and measurement locations on the estimation accuracy is also investigated. Two different examples of laser irradiance are discussed. Results show that the unknown laser irradiance and thermal damage can be predicted precisely by using the present approach for the test cases considered in this study.

Accuracy Analysis of Interpolating Element-Free Galerkin (IEFG) Method in Solving Transient Heat Conduction Problems

For the interpolating element-free Galerkin (IEFG) method set up by combining the shape function constructed with interpolating moving least-squares (IMLS) approximation and Galerkin integral weak form of partial differential equation, we show that the task of accuracy analysis of the IEFG method in solving transient heat conduction problems can be reduced to the estimation of errors in the procedure of IMLS approximation and the integral weak form of heat conduction equation. Following this practice, accuracy analysis of IEFG method is derived. To illustrate our results, numerical tests are discussed by two examples and give an optimal convergence rate with respect to two types of norm.

Entropy generation minimization in steady-state and transient diffusional heat conduction processes Part I – Steady-state boundary value problem

Abstract An application of the Entropy Generation Minimization principle allows new formulation of the boundary and initial boundaryvalue problems. Applying Euler-Lagrange variational formalism new mathematical form of heat conduction equation describing steady-state processes have been derived. Mathematical method presented in the paper can also be used for any diffusion heat and mass transfer process. Linear and non-linear problems with internal heat sources have been analyzed.

Research on the Relationship between Temperature and Dimension of Large-Scale Forgings in Manufacturing Process

In order to achieve online monitoring about hot-state dimension of large-scale forgings, a new monitoring method has been developed in this paper. The large-scale barrel-type forgings heat conduction differential equation is deduced on the basis of the heat transfer. The functional form of heat conduction equation for large-scale forgings is established. Then the relationship between temperature and dimension is solved according to the variational principle. So the dimension of the large-scale forgings is monitored by combining the relationship between temperature and dimension. The temperature is measured by sensor. As a result, the online monitoring about hot-state dimension of large-scale forgings is achieved.

Application of the time discontinuous Galerkin finite element method to heat wave simulation

This paper deals with the numerical simulation of heat wave propagation in the medium subjected to different kinds of heat source, particularly heat impulse. The discontinuous Galerkin finite element method (DGFEM) proposed for the stress wave propagation in solids [X.K. Li, D.M. Yao, R.W. Lewis, A discontinuous Galerkin finite element method for dynamic and wave propagation problems in non-linear solids and saturated porous media. Int. J. Numer. Meth. Eng. 57 (2003) 1775–1800] is extended to numerically solve for the non-Fourier heat transport equation constructed according to the CV model [C. Cattaneo, A form of heat-conduction equation which eliminates the paradox of instantaneous propagation, Compute Rendus 247 (1958) 431–433; P. Vernotte, Les paradoxes de la theorie continue de l’equation de la chaleur, Compute Rendus 246 (1958) 3154–3155]. Temperature and its time-derivative are chosen as primitive variables defined at each FE node. The main distinct characteristic of the proposed DGFEM is that the specific P3–P1 interpolation approximation, which uses piecewise cubic (Hermite’s polynomial) and linear interpolations for both temperature and its time-derivative, respectively, in the time domain is particularly proposed. As a consequence the continuity of temperature at each discrete time instant is exactly ensured, whereas discontinuity of the time-derivative of temperature at discrete time levels remains. Numerical results illustrate good performance of the present method in the numerical simulation of heat wave propagation in eliminating spurious numerical oscillations and in providing more accurate solutions in the time domain.

On the minimum entropy production in steady state heat conduction processes

On the basis of minimum entropy generation principle, a new formulation of the boundary value problems is proposed. Applying Euler–Lagrange variational formalism, a new mathematical form of heat conduction equation with additional heat source terms has been derived. To obtain a unique solution a special mathematical form of boundary conditions for 2D and 3D problems is required. As a result, entropy generation rate of the process can significantly be reduced, which leads to the decrease of the irreversibility ratio according to the Gouy–Stodola theorem. Minimization of entropy generation in heat conduction process is always possible by introducing additional heat sources.

Study on the intrinsic thermal stability of anisotropic thin film superconductors with a line heat source

This paper presents an analytical method to investigate the intrinsic thermal stability of anisotropic thin film superconductors. A three-dimensional analysis is introduced to consider an instantaneous release of energy in the form of a finite length line source. Through the transformation of coordinates, an isotropic form of heat conduction equation is obtained. To solve this initial and boundary value problem, the method of separation of variables is employed, which gives an exact solution. The results of limiting cases of this study are compared with those of two previous investigations. Results from this study also show that the anisotropy of highly oriented films of superconductors has a great impact on the thermal stability. The stability parameter also depends on the location of the line source, the length of the line source, the magnitude of disturbance energy, the Biot number and the aspect ratio in the plane of thin film and normal to it.

The numerical solution of the heat conduction equation in one dimension

AbstractThe replacement of the second space derivative by finite differences reduces the simplest form of heat conduction equation to a set of first-order ordinary differential equations. These equations can be solved analytically by utilizing the spectral resolution of the matrix formed by their coefficients. For explicit boundary conditions the solution provides a direct numerical method of solving the original partial differential equation and also gives, as limiting forms, analytical solutions which are equivalent to those obtainable by using the Laplace transform. For linear implicit boundary conditions the solution again provides a direct numerical method of solving the original partial differential equation. The procedure can also be used to give an iterative method of solving non-linear equations. Numerical examples of both the direct and iterative methods are given.