Linear Heat Conduction Equation Research Articles

A new solution to a weakly non-linear heat conduction equation in a spherical droplet: Basic idea and applications

A new analytical solution to a non-linear heat transfer equation in a spherically-symmetric droplet is suggested. All thermophysical properties inside the droplet are considered to be close to their average values. This allows us to consider the non-linearity of this equation as weak. The solution is presented as T=T0+T1, where T0 is the solution to a linear heat conduction equation, and T1≪T0. The equation for T1 is presented as a linear heat conduction equation with a source term depending on the distribution of T0 and its spatial derivatives inside the droplet. The latter equation is solved analytically alongside the linear equation for T0, and the final solution is presented as T=T0+T1. The predictions of the numerical code in which this solution was implemented are verified based on a comparison of those predictions with the predictions of COMSOL Multiphysics code using input parameter values that are typical for nanofluid (water and SiO2 nanoparticles) droplet evaporation in atmospheric conditions. It is demonstrated that for these experiments T1≪T0 which justifies the applicability of the linear heat conduction equation used for the analysis of this process. Small differences in the temperatures predicted by both non-linear and linear models lead to a much more noticeable difference in integral characteristics such as time before the start of the formation of the cenosphere when the mass fraction of nanoparticles at the droplet surface reaches about 40%.

Synthesis of a feedforward control system for object with distributed parameters

This study is researching a synthesis of an automatic control system for maintaining the temperature of petroleum products during storage in tanks. A new approach for the system construction has been developed during this research. It helps to get rid of disadvantages of existing methods of heating tanks. Developed control system is built on the basis of the feedforward control principle. The object of control is the tank insulation layer, considered as an object with distributed parameters, which provides opportunity to take into account the high inertia of the object. The uneven distribution of temperature is taken into account only over the thickness of the thermal insulation layer, thus, when modeling the control object, a linear one-dimensional heat conduction equation was used, supplemented by initial and boundary conditions of the third kind.
 To ensure invariance to a disturbing effect in the form of a negative ambient temperature, a compensation element is built using the method of periodic structures. The required heating power dynamics of the heating cable is calculated to compensate for heat losses by the compensation element, depending on the current ambient temperature. Result of this study is a high-performance system that allows maintaining the high precision temperature of the oil in the tank. The article also provides research of the periodic structure parameters impact on the quality of the control algorithm.

A Comparative Study of Explicit and Stable Time Integration Schemes for Heat Conduction in an Insulated Wall

Calculating heat transfer in building components is an important and nontrivial task. Thus, in this work, we extensively examined 13 numerical methods to solve the linear heat conduction equation in building walls. Eight of the used methods are recently invented explicit algorithms which are unconditionally stable. First, we performed verification tests in a 2D case by comparing them to analytical solutions, using equidistant and non-equidistant grids. Then we tested them on real-life applications in the case of one-layer (brick) and two-layer (brick and insulator) walls to determine how the errors depend on the real properties of the materials, the mesh type, and the time step size. We applied space-dependent boundary conditions on the brick side and time-dependent boundary conditions on the insulation side. The results show that the best algorithm is usually the original odd-even hopscotch method for uniform cases and the leapfrog-hopscotch algorithm for non-uniform cases.

Painlevé Test and Exact Solutions for (1 + 1)-Dimensional Generalized Broer–Kaup Equations

In this paper, the Painlevé integrable property of the (1 + 1)-dimensional generalized Broer–Kaup (gBK) equations is first proven. Then, the Bäcklund transformations for the gBK equations are derived by using the Painlevé truncation. Based on a special case of the derived Bäcklund transformations, the gBK equations are linearized into the heat conduction equation. Inspired by the derived Bäcklund transformations, the gBK equations are reduced into the Burgers equation. Starting from the linear heat conduction equation, two forms of N-soliton solutions and rational solutions with a singularity condition of the gBK equations are constructed. In addition, the rational solutions with two singularity conditions of the gBK equation are obtained by considering the non-uniqueness and generality of a resonance function embedded into the Painlevé test. In order to understand the nonlinear dynamic evolution dominated by the gBK equations, some of the obtained exact solutions, including one-soliton solutions, two-soliton solutions, three-soliton solutions, and two pairs of rational solutions, are shown by three-dimensional images. This paper shows that when the Painlevé test deals with the coupled nonlinear equations, the highest negative power of the coupled variables should be comprehensively considered in the leading term analysis rather than the formal balance between the highest-order derivative term and the highest-order nonlinear term.

The generalized Cole–Hopf transformation for a generalized Burgers–Fisher equation with spatiotemporal variable coefficients

In this paper, we investigate a generalized Burgers-Fisher equation with spatiotemporal variable coefficients(gvcBF), which can describe the nonlinear convection–diffusion phenomenon in chemical engineering and biology. It is shown that the gvc Burgers-Fisher equation can be exactly linearized as long as the variable coefficient functions satisfy some constraints conditions. We obtain a Bäcklund transformation between the gvc Burgers-Fisher equation and a linear parabolic equation with variable coefficients by the simplified homogeneous balance method. Furthermore, we derive out a generalized Cole-Hopf transformation between a class of the gvc Burgers-Fisher equation and the linear heat conduction equation with the help of a new unknown function transformation. Especially, we obtain the generalized Cole-Hopf transformation between the cylindrical and spherical Burgers-Fisher equations with algebraical decaying damping term and the heat conduction equation. Finally, we obtain a series of explicit exact solutions of the gvc Burgers-Fisher equation.

MAGNETO-THERMOELASTIC PROBLEM WITH EDDY CURRENT LOSS OF A THERMOSENSITIVE CONDUCTIVE PLATE

In this paper a two dimensional magneto-thermoelastic problem of a thermosensitive finite conducting plate with eddy current loss is considered. It is assumed that the plate is influenced by a time-varying external magnetic field and that the heating is caused by Joule heat. The fundamental equations for magnetic field, heat conduction and elastic fields are formulated. Temperature dependent material properties and heat source as eddy current loss is considered in the heat conduction equation. Kirchhoff's variable transformation is employed to convert nonlinear to linear heat conduction equation. Integral transform technique is used to solve the magnetic field and temperature distribution. The stresses in a plane state are determined by using Airy's stress function. The numerical analysis is carried out and the results are graphically displayed.

Coupled fluid-solid thermal interaction modeling for efficient transient simulation of biphasic water-steam energy systems

This paper proposes a fluid-solid coupled finite element formulation for the transient simulation of water-steam energy systems with phase change due to boiling and condensation. As it is commonly assumed in the study of thermal systems, the transient effects considered are exclusively originated by heat transfer processes. A homogeneous mixture model is adopted for the analysis of biphasic flow, resulting in a nonlinear transient advection-diffusion-reaction energy equation and an integral form for mass conservation in the fluid, coupled to the linear transient heat conduction equation for the solid. The conservation equations are approximated applying a stabilized Petrov-Galerkin FEM formulation, providing a set of coupled nonlinear equations for mass and energy conservation. This numerical model, combined with experimental heat transfer coefficients, provides a comprehensive simulation tool for the coupled analysis of boiling and condensation processes. For the treatment of enthalpy discontinuities traveling with the flow, a novel explicit-implicit time integration method based on Crank-Nicolson scheme is proposed, analyzing its accuracy and stability properties. To reduce problem size and enhance numerical efficiency, a modal superposition method with balanced truncation is applied to the solid equations. Finally, different example problems are solved to demonstrate the capabilities, flexibility and accuracy of the proposed formulation.

Development of a Numerical Method for Solving the Inverse Cauchy Problem for the Heat Equation

In this work, the initial temperature has been investigated in the Cauchy inverse problem for linear heat conduction equation that it depends on the given temperature at specification time. In this problem, the initial temperature distribution is unknown, but instead, there is a known temperature at the time, t = T > 0. The heat conduction problem can be formulated as Fredholm integral first kind equation. It is well known that this problem is an ill-posed problem and direct solution to this problem is unacceptable. An algorithm has been used to define a finite-dimensional operator for this problem also used the generalized discrepancy method to reduce the conditional extremum variation problem to unconditional extremum variation problem for the integral equation. The discretization of the integral equation has made it possible to reduce this problem to a system of linear algebraic equations. Then, Tikhonov’s regularization inversion method has been used to find an approximation solution. Finally, the numerical computation example has been presented to verify the accuracy of the estimated solution.

Decay mode solution of nonlinear boundary–initial value problems for the cylindrical (spherical) Boussinesq–Burgers equations

The major target of this paper is to construct new nonlinear boundary–initial value problems for Boussinesq–Burgers Equations, and derive the solutions of these nonlinear boundary–initial value problems by the simplified homogeneous balance method. The nonlinear transformation and its inversion between the Boussinesq–Burgers Equations and the linear heat conduction equation are firstly derived; then a new nonlinear boundary–initial value problem for the Boussinesq–Burgers equations with variable damping on the half infinite straight line is put forward for the first time, and the solution of this nonlinear boundary–initial value problem is obtained, especially, the decay mode solution of nonlinear boundary–initial value problem for the cylindrical (spherical) Boussinesq–Burgers equations is obtained.

Structural topology optimization with strength and heat conduction constraints

In this research, a topology optimization with constraints of structural strength and thermal conductivity is proposed. The coupled static linear elastic and heat conduction equations of state are considered. The optimization problem was formulated; viz., minimizing the volume under the constraints of p-norm stress and thermal compliance introducing the qp-relaxation method to avoid the singularity of stress-constraint topology optimization. The proposed optimization methodology is implemented employing the commonly used solid isotropic material with penalization (SIMP) method of topology optimization. The density function is updated using sequential linear programming (SLP) in the early stage of optimization. In the latter stage of optimization, the phase field method is employed to update the density function and obtain clear optimal shapes without intermediate densities. Numerical examples are provided to illustrate the validity and utility of the proposed methodology. Through these numerical studies, the dependency of the optima to the target temperature range due to the thermal expansion is confirmed. The issue of stress concentration due to the thermal expansion problem in the use of the structure in a wide temperature range is also clarified, and resolved by introducing a multi-stress constraint corresponding to several thermal conditions.

Non-Fourier Thermoelastic Analysis of an Annular Fin with Variable Convection Heat Transfer Coefficient

This paper numerically investigates the hyperbolic thermoelastic problem of an annular fin. The ambient convection heat transfer coefficient of the fin is assumed to be spatially varying. The major difficulty in dealing with such problems is the suppression of numerical oscillations in the vicinity of a jump discontinuity. An efficient numerical scheme involving hybrid application of Laplace transform and control volume method in conjunction with hyperbolic shape functions is used to solve the linear hyperbolic heat conduction equation. The transformed nodal temperatures are inverted to the physical quantities by using numerical inversion of the Laplace transform. Then the stress distributions in the annular fin are calculated subsequently. The results in the illustrated examples show that the application of hyperbolic shape functions can successfully suppress the numerical oscillations in the vicinity of jump discontinuities.

One Explicit Scheme for the Linear Heat Conduction Equation and the Numerical Approximation

In this paper, one explicit scheme for the linear he at conduction equation in one space d imension is conside re d. We will analysis the truncation error of the method and w ill use the super norm, Fourier series to analysis the stability of the scheme; at last the numerical experiments of initial va lue problem will be presented.

Numerical solution for the linear transient heat conduction equation using an Explicit Green’s Approach

This paper presents a novel numerical solution algorithm for the linear transient heat conduction equation using the ‘Explicit Green’s Approach’ (ExGA). The method uses the Green’s matrix that represents the domain of the problem to be solved in terms of the physical properties and geometrical characteristic. The Green’s matrix is the problem discrete Green’s function determined numerically by the Finite Element Method (FEM). The ExGA allows explicit time marching with time step larger than the one required by FEM, without losing precision. The ExGA numerical results are quite accurate when compared to analytical solutions and to numerical solutions obtained by the FEM.

Mathematical model of the heating of a dual-layer plate in a metal-elastomer system during thermal vulcanization of the elastomer

A mathematical model of the heating of an infinite dual-layer plate for analytical calculations of the nonstationary temperature fields of this plate during the course of its entire heating with allowance for various schemes employed to organize the vulcanization of a polymeric coating on the metal was built to establish basic laws governing the heating of articles and to determine the effect of the vulcanization obtained by the elastomeric lining during nonstationary heating. As a basis of the model, it is proposed to use a system of familiar linear heat-conduction equations, each of which describes a temperature function, which is dependent on the coordinates and time. The mathematical software package MathCAD was used to solve the indicated system, beginning with version MathCAD-2000. Here, the relative error of the calculations does not exceed 0.5%.

Convective corrections to the linear diffusion equation

The classical problem of the diffusion of heat in a homogeneous medium is reexamined, the medium being confined by fixed boundaries maintained at a fixed temperature. When the thermal diffusivity is small, the relaxation of the temperature of the medium to that of the boundary proceeds on two time scales, one associated with a lightly damped high-frequency acoustic mode and the other with an aperiodically damped diffusive mode. Considering for simplicity a spherical configuration, it is shown that the latter does not obey the classical linear heat conduction equation.

Self-similar solutions in the problem of generalized vortex diffusion

The analysis of the group properties and the search for self-similar solutions in problems of mathematical physics and continuum mechanics have always been of interest, both theoretical and applied [1–3]. Self-similar solutions of parabolic problems that depend only on a variable of the type η = x/√t are classical fundamental solutions of the one-dimensional linear and nonlinear heat conduction equations describing numerous physical phenomena with initial discontinuities on the boundary [4]. In this study, the term “generalized vortex diffusion” is introduced in order to unify the different processes in mechanics modeled by these problems. Here, vortex layer diffusion and vortex filament diffusion in a Newtonian fluid [5] can serve as classical hydrodynamic examples. The cases of self-similarity with respect to the variable η are classified for fairly general kinematics of the processes, physical nonlinearities of the medium, and types of boundary conditions at the discontinuity points. The general initial and boundary value problem thus formulated is analyzed in detail for Newtonian and non-Newtonian power-law fluids and a medium similar in behavior to a rigid-ideally plastic body. New self-similar solutions for the shear stress are derived.

An Adjoint-Based A Posteriori Estimation of Iterative Convergence Error

The iterative convergence error of goal functional is considered for solving the steady problem by time iterations. The functional error is calculated using adjoint parameter and time derivative. The numerical tests demonstrate the applicability of this approach for linear and nonlinear heat conduction equations.

Image based heating rate calculation from thermographic data considering lateral heat conduction

An image based method for transient surface normal heat flux calculation from thermographic data is suggested. It is based on an analytical solution of the three-dimensional linear heat conduction equation. The method yields correct results for the surface normal heat flux even in regions with strong lateral gradients by taking into account the transient surface temperature of an area surrounding each evaluation point within the thermographic image. The solution for the heat flux can be stabilized with respect to measurement errors by an iterative regularization method. The validation of the method in synthetic test cases indicates its good accuracy over a broad range of Fourier numbers.

Influence of laser pulse shape both on temperature profile and hardened layer depth

This paper presents a mathematical modelling and numerical calculations of heat conduction problems where laser generated heat is assumed as a surface heat source. Also the effect of a laser time structure on a hardened layer depth is examined. Temperature profiles for different laser pulse shapes are determined from the solution of a linear one-dimensional heat conduction equation for semi-infinite medium and discussed in terms of the parameters evolution such as dimensionless: temperature, heat flux, hardening depth, laser impulse duration and increasing time of triangular pulse shape.

Parallel Difference Schemes For Heat Conduction Equations * The work is supported by the Chinese National Key Program for Developing Basic Science, G1999032801, the National Natural Science Foundation of China (19932010) and the Foundation of CAEP (2X0107)

In this paper a parallel difference scheme based on Dufort-Frankel scheme and the classic implicit scheme for linear heat conduction equations is studied. In this procedure, the values at subdomain interfaces are calculated by using the Dufort-Frankel scheme, and then these values serve as Dirichlet boundary data for the implicit scheme in the subdomains. The weak necessary condition of the unconditional stability of the parallel difference scheme is proved. Numerical experiments indicates that the parallel difference scheme has good parallelism, and has better accuracy than the fully implicit scheme.