Two-dimensional Heat Conduction Equation Research Articles

Numerical investigations of the lateral heat transfer in coaxial thermocouples

Coaxial thermocouples have been widely used for transient heat transfer measurements in high-enthalpy shock tunnels. The one-dimensional semi-infinite heat conduction theory is typically used for temperature data processing. However, due to the material difference between the two electrodes and the junction, lateral heat transfer occurs, causing a deviation in the heat flux measurement from the prediction results of the one-dimensional semi-infinite heat conduction theory. Thus, the lateral heat transfer effect has to be investigated to improve the accuracy and reliability of heat flux measurements. In this article, the heat transfer in E-type (chromel-constantan) coaxial thermocouples was analyzed by numerically solving the two-dimensional axisymmetric heat conduction equation with the Du Fort-Frankel scheme. During the heating process, the maximum temperature point on the surface of the coaxial thermocouples moved to the positive electrode over time. The numerical simulation indicated that the surface temperature of the coaxial thermocouples and the derived heat flux were larger than the theoretical value. The heat flux measurement error of the coaxial thermocouples can be reduced by increasing the width of the positive electrode. The results of this study can provide a reference for the design and manufacture of coaxial thermocouples.

Thermo-vibrational analyses of skin tissue subjected to laser heating source in thermal therapy

Laser-induced thermal therapy, due to its applications in various clinical treatments, has become an efficient alternative, especially for skin ablation. In this work, the two-dimensional thermomechanical response of skin tissue subjected to different types of thermal loading is investigated. Considering the thermoelastic coupling term, the two-dimensional differential equation of heat conduction in the skin tissue based on the Cattaneo–Vernotte heat conduction law is presented. The two-dimensional differential equation of the tissue displacement coupled with the two-dimensional hyperbolic heat conduction equation in the tissue is solved simultaneously to analyze the thermal and mechanical response of the skin tissue. The existence of mixed complicated boundary conditions makes the problem so complex and intricate. The Galerkin-based reduced-order model has been utilized to solve the two-sided coupled differential equations of vibration and heat transfer in the tissue with accompanying complicated boundary conditions. The effect of various types of heating sources such as thermal shock, single and repetitive pulses, repeating sequence stairs, ramp-type, and harmonic-type heating, on the thermomechanical response of the tissue is investigated. The temperature distribution in the tissue along depth and radial direction is also presented. The transient temperature and displacement response of tissue considering different relaxation times are studied, and the results are discussed in detail.

Investigating two-dimensional mechanical and thermal behavior of skin tissue in confronting with various laser irradiation

In this paper, the transient temperature and vibration response of skin tissue to different types of laser heating sources are studied. Two-dimensional heat conduction equation of skin tissue based on the Cattaneo–Vernotte model is extracted considering the thermoelastic effect. Two-sided coupled differential equations governing heat transfer and displacement of skin tissue have been non-dimensionalized and solved numerically. Due to the appearance of extremely complicated boundary conditions, the Galerkin-based reduced-order model has been utilized to solve the equations properly. The effect of relaxation time on the temperature and vibration response of tissue has been studied. The thermomechanical response of tissue has been investigated considering various types of laser heat sources. Results indicated that the relaxation time creates a delay in the transient dynamic response of the tissue. It was also observed that when energy is delivered to skin tissue via the repetitive laser heating pulses, the relaxation time reduces the vibration amplitude of skin tissue in the radial direction as well as depth direction. The effect of the thermoelastic coupling term on the temperature response of tissue was not considerable.

Urban Planning Image Feature Enhancement and Simulation Based on Partial Differential Equation Method

Based on the introduction of the basic ideas and related technologies of partial differential equations, as well as the method of path planning, the application of partial differential equations in solving urban path planning is studied. The path planning model of partial differential equations and the setting of obstacle boundary conditions are introduced, and adaptive. Theoretical research and experimental results show that it is feasible and effective to solve urban path planning by partial differential equations, which provides a new way for urban path planning research ideas and methods. This paper proposes an image detection algorithm based on diffusion equation. According to the logarithmic transformation, the multiplicative speckle noise in the image can be converted into additive noise. We first perform logarithmic transformation on the image and then use the denoising model of the diffusion equation to filter out the noise in the image and then use the logarithm to recognize the image. The difference image is obtained by the domain difference method, and finally, the difference image is classified by the clustering algorithm, and the change area is extracted. Experiments show that the algorithm can effectively reduce the effect of multiplicative speckle noise on the change detection results, improve the accuracy of change detection, and shorten the change detection time. This article takes the path planning problem of a two-dimensional space city as an example to discuss the application of partial differential equations. According to the principle of energy conservation, this paper uses the two-dimensional space radiant heat conduction equation as an example to model and illustrate the solution of the path planning problem.

Numerical heat transfer in annular fins of curved profile formed with the intersection of two equal circles

Purpose The purpose of this paper is to investigate the heat transfer attributes of annular fins with quarter circle profile in terms of the Biot number Bi and the radius ratio rr. The latter corresponds to the internal radius of the tube divided by the length of the fin in question. Design/methodology/approach To this end, the governing two-dimensional (2-D) heat conduction equation in cylindrical coordinates is numerically solved via finite element analysis for different Bi (i.e., 0.1, 1 and 5) and rr (i.e., 0.5, 1 and 2). Findings The obtained results for the mid-plane and surface temperatures show that these profiles, which exhibit nearly rr-independent responses, only present one-dimensional (1-D) radially linear distributions for the case Bi = 1. For Bi = 0.1, the temperature profiles also possess a 1-D character but with a clearly defined concave pattern. Finally, for Bi = 5, a 2-D temperature field in a wide zone from the fin base is achieved with a convex pattern for the mid-plane and surface temperatures. Originality/value Exhaustive assessment of the heat transfer in annular fins with quarter circle profile in terms of different Biot numbers and radius ratios

Theoretical Investigation on Failure Behavior of Ogive-Nose Projectile Subjected to Impact Loading.

Experimental and theoretical investigations on the failure behaviors of projectile during high-speed impact into concrete slabs were performed in this study. The ogive-nose projectiles after impact experiments were recovered and their microstructures were observed by scanning electron microscope and metallographic microscope. Mass abrasion and nose blunting are the typical failure models of steel projectile. Furthermore, thermal melting and cutting are the two main failure mechanisms. Based on the microscopic experimental results, a theoretical model of ogive-nose projectile subjected to impact loading considering the melting and cutting mechanisms was proposed. A modified cap model is introduced for describing the failure behavior of concrete targets, and then the dynamic cavity expansion theory is used to determine the resistance of projectiles during penetration. Besides, combining with the two-dimensional heat conduction equation and abrasive wear theory, the two main abrasion mechanisms of melting and cutting are included in the proposed model, which breaks through the framework of previous abrasion models with single abrasion mechanism. The predicted results of the present abrasion model are in good agreement with the experimental data, which indicates that the proposed model can effectively predict the failure behavior and penetration performance parameters of high-speed projectiles during penetration into concrete targets, such as mass loss, nose blunting, and depth of penetration.

An Efficient Three-Level Explicit Time-Split Approach for Solving Two-Dimensional Heat Conduction Equation

An Efficient Three-Level Explicit Time-Split Approach for Solving Two-Dimensional Heat Conduction Equation

Conjugate heat transfer analysis of liquid metal turbulent flow through a horizontal channel by LES

Conjugate heat transfer analysis of liquid metal turbulent flow through a horizontal channel has been numerically studied using a large eddy simulation (LES) technique. The effect of wall thickness and heat conduction in the wall on flow and thermal characteristics of low Prandtl number liquid metals has been investigated and reported here. Results are presented for three different liquid metal flows with different Reynolds numbers and different wall materials with wall thickness. The flow field is modeled as transient two-dimensional incompressible turbulent forced convection flow. Wall Adapting Local Eddy Viscosity (WALE) sub-grid scale (SGS) model is used to solve the turbulence closure. Two-dimensional transient heat conduction equation is solved to know the temperature distribution in the wall. A finite difference method solver is developed for solving the governing equations using sixth-order accuracy of compact schemes. Prandtl number and Reynolds number alter the heat transfer characteristics. Wall thickness shows a significant effect on the convective heat transfer rate. The new correlation of average Nusselt number has been proposed, including the effect of wall thickness for constant heat flux boundary condition. The present results are validated with DNS data and experimental results available in the literature.

Axisymmetric vibrations of temperature-dependent functionally graded moderately thick circular plates with two-dimensional material and temperature distribution

In the present paper, free vibration analysis of bi-directional functionally graded circular plates subjected to two-dimensional temperature variation has been presented on the basis of first-order shear deformation theory. The mechanical properties of the plate material are assumed to be temperature-dependent and graded in thickness as well as radial direction. Using the thermal boundary conditions of the plate, the exact solution of two-dimensional heat conduction equation has been obtained by separation of variables technique. The thermoelastic equilibrium equations and equations of motion for such plate have been derived from an energy-based Hamilton’s principle. The generalized differential quadrature method has been applied to compute the numerical values for thermally induced displacements and natural frequencies. MATLAB has been used to obtain these values for clamped and simply supported plates. The influence of thermal environment together with various plate parameters such as power-law index, heterogeneity parameter and density parameter on the natural frequencies has been investigated for the first three modes. The results obtained herein for some special cases are in good agreement with those obtained by other numerical methods. Three-dimensional mode shapes for specified plates have been illustrated.

Effect of Pulsation and Acceleration of Liquid Metal Turbulent Flow Through a Horizontal Channel by Large Eddy Simulation

Abstract Pulsation and acceleration of liquid metal turbulent flow through a horizontal channel has been numerically studied using a large eddy simulation (LES) technique. The effect of inlet pulsation and acceleration on flow and heat transfer characteristics of low Prandtl number liquid metal flow have been investigated and reported here. Results have been presented for different Reynolds numbers, different amplitudes, and frequency with constant bottom wall thickness. The flow field is modeled as unsteady-state two-dimensional incompressible turbulent-forced convection flow. Turbulence is modeled using a LES technique. Two-dimensional unsteady-state heat conduction equation is solved to know the temperature distribution in the solid region. Finite difference method solver is developed for solving the governing equations using sixth-order accuracy of compact schemes. The average Nusselt number shows cyclic variation with respect to time in pulsation flows. The enhancement of heat transfer with pulsation at amplitude 0.4 and frequency 100 Hz is 6.51%. The rate of heat transfer increases in pulsation flow compared to quasi-steady flow. The inlet acceleration shows a significant effect on flow characteristics. The present results are compared with direct numerical simulation (DNS) results available in the literature and matching well with DNS data.

Failure behavior of projectile abrasion during high-speed penetration into concrete

The mass abrasion of projectile is an obvious failure phenomenon because of mass loss and nose blunting during the high-speed projectile deep penetration into concrete target. The thermal melting and cutting by aggregate on projectile outer surface are the main mechanisms of mass abrasion according to the experimental observation. In this paper, a coupled melting-cutting model is constructed to estimate mass abrasion and dynamic behavior of ogive-nose projectile during penetration. In this model, the abrasion volume of melting and cutting are obtained by two-dimensional heat conduction equation and a classic abrasive theory including a parameter that related to aggregate strength, respectively. Furthermore, the two mechanisms of mass abrasion interact with each other and are connected by the Johnson–Cook model. Finally, the penetration performance parameters, i.e., depth of penetration, variation of projectile shape and mass loss rate, can be obtained by the iteration calculation based on coupled abrasion model. The theoretical predictions agree well with the typical experimental data. Besides, the effects of the model parameters on the penetration performance are discussed.

Establishment and evaluation of a co-effect structure with thermal concentration–rotation function in transient regime**Project supported by the National Natural Science Foundation of China (Grant Nos. 51776050 and 51536001).

The advanced heat flux manipulating structures inspired by TO-based spatial mapping have aroused wide interests owing to huge potential in high-efficient thermal energy utilization. However, most researches are limited to the realization of single function in one specific structure and appropriate evaluation of the energy transfer process is relatively lacking. In this work, based on time-dependent two-dimensional heat conduction equation, a co-effect structure capable of accomplishing concentration and rotation functions simultaneously is established and validated by finite element simulations compared with the conventional single concentrator and singe rotator. In addition, from the perspective of thermodynamics, the transformed local entropy production rate and total entropy production are theoretically derived and applied to evaluate the quality of energy transfer processes. The proposed co-effect structure can help to explore other potential mass/flux manipulating devices and the evaluation method is valuable for the further manufacturing as well as optimization of these devices in engineering applications.

Invariant Solutions of Two Dimensional Heat Equation

The symmetry group of a given differential equation is the group of transformations that translate the solutions of the equation into solutions. If the infinitesimal generators of symmetry groups of systems of partial differential equations are known, the symmetry group can be used to explicitly find particular types of solutions that are invariant with respect to the symmetry group of the system. The class of invariant solutions includes exact solutions that have direct mathematical or physical meaning. In this paper, using the well-known infinitesimal generators of some symmetry groups of the two-dimensional heat conduction equation, solutions are found that are invariant with respect to these groups. It is considered cases when conductivity coefficients of the two-dimensional heat conduction equation are power functions of temperature and conductivity coefficients are exponential functions of temperature. In first case invariant solutions contain well known self-similar solutions which are widely used in applications.

Effect of cladding on thermal behavior of nuclear fuel element with non-uniform heat generation

In nuclear power reactors if the maximum temperature (2000ᵒC) of the fuel element like Uranium-dioxide increments and reaches 2865ᵒC then the fuel element melts down leading to heavy loss and damage. To maintain this maximum temperature well below the allowable limit the heat in the fuel element should be conducted and dissipated to the surrounding medium. In this article, the numerical prediction and comparison of thermal performance characteristics of a nuclear fuel element with and without cladding (bare fuel element) having non-uniform energy generation in axial direction is studied in detail. Accordingly, the two-dimensional heat conduction equation having a cosine variation of heat generation term in the axial direction, along with appropriate boundary conditions is solved using second order accurate finite difference schemes. The results are presented for a wide range of parameters like Aspect Ratio Ar, Biot number, Bi, Conductivity ratio, Cr, Thickness ratio, Th, and total heat generation parameter, Qt in the form of transverse temperature profiles and axial temperature profiles. Effect of Cr and Th due to presence of cladding is thoroughly studied. Different range of parameters for which the temperature of the fuel element is within the maximum temperature range is obtained. Critical values of these parameters for which the temperature increments above maximum temperature is also investigated. From the detailed discussion of the results it is concluded that the maximum temperature attained in the fuel element without cladding (bare fuel element) is very much higher than that attained in fuel element with cladding. Also it is observed that for fixed values of Ar, Cr and Th, there exists a critical value of Qt and Bi beyond and below which maximum temperature in the fuel element exceeds its allowable limit.

Corrected quasi one-dimensional heat conduction equation for the analysis of straight fins of uniform profile

Justification of the quasi one-dimensionality assumption in the analysis of fins of various shapes has been traditionally based on the criterion that fins have large slenderness ratios, i.e., large length and small thickness. This simplistic approach is based on a mere geometric consideration that obviates the implications of the two-dimensional heat paths in the axial and transversal directions that occurs in fins. Beginning with the formal differential formulation of the two-dimensional heat conduction equation for straight fins of uniform profile and the appropriate boundary conditions, the central objective of the present paper is to develop a systematic mathematical procedure that revolves around a new corrected quasi one-dimensional heat conduction equation that is more physically sound. The step-by-step mathematical development depends on two controlling parameters: (1) a thermo-geometric parameter, i.e., the transverse Biot number based on the half-thickness, and (2) a geometric parameter, i.e., the slenderness ratio. The computed two-dimensional heat transfer rates clearly demonstrate that the corrected quasi one-dimensional heat conduction equation captures the two-dimensional heat paths flawlessly and as a direct result is better than the standard quasi one-dimensional heat conduction equation. The discrepancy between the corrected quasi one-dimensional and the standard quasi one-dimensional heat transfer rates is of the order of 10%.

Mechanical and thermal stresses in radially functionally graded hollow cylinders with variable thickness due to symmetric loads

ABSTRACT In this paper, based on the higher-order shear deformation theory, a thermo-elastic analysis for axisymmetric clamped–clamped rotating pressurised thick cylindrical shells with variable thickness made of functionally graded materials subjected to thermal loading is presented. The material properties, except the Poisson’s ratio, are assumed to vary with the power law function in the radial direction of the cylinder. The governing equations are a system of non-homogeneous ordinary differential equations with variable coefficients. Based on two-dimensional steady-state heat conduction equation, temperature distribution in cylinder with variable thickness is given. Using multilayer method, these equations could be converted into systems of differential equations with constant coefficients. The effects of higher-order approximations on the radial and axial displacements, von Mises stress and shear stress have been studied. Finally, the displacements and stresses along the radius and length have been plotted, and distribution of these is compared with the solution using finite-element method.

A Revised Tikhonov Regularization Method for a Cauchy Problem of Two-Dimensional Heat Conduction Equation

In this paper we investigate a Cauchy problem of two-dimensional (2D) heat conduction equation, which determines the internal surface temperature distribution from measured data at the fixed location. In general, this problem is ill-posed in the sense of Hadamard. We propose a revised Tikhonov regularization method to deal with this ill-posed problem and obtain the convergence estimate between the approximate solution and the exact one by choosing a suitable regularization parameter. A numerical example shows that the proposed method works well.

Boundary function method for inverse geometry problem in two-dimensional anisotropic heat conduction equation

This paper develops a novel approach for detecting unknown boundaries in the two-dimensional anisotropic heat conduction equations based on the boundary function method, in which a partial homogenization function satisfied the over-specified Cauchy data on an arc is derived to effectively solve the inverse geometry problem. After the homogenized technique, the governing equation is transformed into the one in a reduced domain, whose numerical solution is expanded by a sequence of boundary functions, automatically satisfying the homogeneous boundary conditions on the arc. The nonlinear equation will be formed and then solved by the Newton iterative method. Two numerical examples are provided to demonstrate the ability and accuracy of the proposed scheme.

A Nonlocal Inverse Problem for the Two-Dimensional Heat-Conduction Equation

We consider an inverse problem of determination of the time-dependent leading coefficient of a two-dimensional heat-conduction equation with nonlocal overdetermination condition. The existence and uniqueness conditions are established for the classical solution of the analyzed problem.

Heat transfer enhancement in a modified straight fin of rectangular profile with enlarged round base and correlated reduced round tip

The present study addresses the potential heat transfer enhancement in a modified straight fin of rectangular profile with an enlarged round base compensated by a reduced round tip when compared against the standard straight fin of rectangular profile. The straight fins of rectangular profile share the profile area (volume) and are subject to two controlling parameters: the transverse Biot number and the aspect ratio. In the common straight fin of rectangular profile, a small fraction of material approximately equal to two thirds of the semi-thickness squared is removed from the tip region and is attached to the base region. Thereby, the tenuous material migration creates the modified straight fin of rectangular profile with an enlarged round base and a reduced round tip.Owing that the modified straight fin with round base and round tip possesses a pronounced irregular profile area, the one-dimensional modeling with the simplistic quasi one-dimensional heat conduction equation is inappropriate. Switching to the two-dimensional modeling, the governing two-dimensional heat conduction equation in Cartesian coordinates cannot be solved analytically and forcibly has to be done numerically with the potent finite element method. The two-dimensional temperature distributions and the companion heat transfer rates are computed for the two distinct straight fins with rectangular profile using realistic combinations of the transverse Biot number and the aspect ratio normally used in engineering practice. At the end, the maximum heat transfer enhancement granted by the modified straight fin with round base and round tip having small-to-moderate aspect ratios can reach a favorable 12% margin.