Finite element analysis for phase change problem in polymer processing

Transient analysis of heat transfer with phase change in thermoplastic injection molding plays an important role to predict solidification in many material-manufacturing processes. Solidification results in a moving solid-liquid interface during filling. Two approaches of the finite element method, namely the apparent heat capacity model and the enthalpy model, for the solution of the Stefan problem in a fixed domain are examined. A two-dimensional, transient, non-newtonian and non-isothermal thermal field of a polymer in a rectangular domain is selected as study case.     

Analytical solution for a transient,three-dimensional temperature distribution due to a moving laser beam

This paper presents an analytical solution of the transient temperature distribution in a finite solid when heated by a moving heat source. The analytical solution is obtained by solving the transient three-dimensional heat conduction equation in a finite domain by the method of separation of variables (SOV). Meanwhile previous studies focus on analytical solutions for semi-infinite domains, here an analytical solution is provided for a finite domain. The non-homogeneous equation is solved by using the Laplace transform for a unit impulse and then convoluted with the actual heat source. Two different distributions are used: a Gaussian distribution and a spatially uniform plane heat source.     

Heat flux field for one spherical inhomogeneity embedded in a functionally graded material matrix

The heat flux field for a single particle embedded in a graded material is derived by using the equivalent inclusion method. A linearly distributed prescribed heat flux field is introduced to represent the material mismatch between the particle and the surrounding graded materials. By using Green’s function technique, an explicit solution is obtained for the heat flux field in both the particle and the graded material. Comparison of the present solution with finite element results illustrates the accuracy and limitation of this solution.     

Heat flow rate at a bore-face and temperature in the multi-layer media surrounding a borehole

Assessment of the heat either delivered from high temperature rocks to the borehole or transmitted to the rock formation from circulating fluid is of crucial importance for a number of technological processes related to borehole drilling and exploitation. Normally the temperature fields in the well and surrounding rocks are calculated numerically by finite difference method or analytically by applying the Laplace-transform method. The former approach requires tedious and, in certain cases, non-trivial numerical computations. The latter method leads to rather bulky formulae that are inconvenient for further numerical evaluation. Moreover, in previous studies where the solution is obtained analytically, the heat interaction of the circulating fluid with the formation was treated on the condition of constant bore-face temperature. In the present study the temperature field in the rock formation disturbed by the heat flow from the borehole is modeled by a heat conduction equation, assuming the Newton model for the convective heat transfer on the bore-face, with boundary conditions that account for the thermal history of the borehole exploitation. The problem is solved analytically by the generalized heat balance integral method. Within this method the approximate solution of the heat conduction problem is sought in the form of a finite sum of functions that belong to a complete set of linearly independent functions defined at the finite interval bounded by the radius of thermal influence and that satisfy the homogeneous boundary conditions on the bore-face. In the present study first and second order approximations are obtained for the composite multi-layer domain. The numerical results illustrate that the second approximation is in a good agreement with the exact solution. The only disadvantage of this solution is that it depends on the radius of thermal influence, which is an implicit function of time and can only be found numerically by iterative algorithms. In order to eliminate this complication, in this study an approximate explicit formula for the radius of thermal influence and new close-form approximate solution are proposed on the basis of the approximate solution obtained by the integral-balance method. Employing the non-liner regression method the coefficients for this simplified solution are obtained. The accuracy of the approximate solution is validated by comparison with the exact analytical solution found by Carslaw and Jaeger for the homogeneous domain.     

Fourier and non-Fourier heat conduction analysis in the absorber plates of a flat-plate solar collector

This paper presents an analytical analysis of both Fourier and non-Fourier heat conduction in the absorber plates of a flat-plate solar collector. Separation of variables was employed to develop the model. For the analysis, a repetitive heat transfer module was used for the solution of parabolic and hyperbolic equations. From the practical point of view, two types of boundary conditions were separately chosen. A numerical technique based on the finite difference method was employed to determine the temperature for validation purposes. A comparative investigation was carried out to understand the requirements for use of the non-Fourier heat conduction model easily. A significant difference in the temperatures obtained from the Fourier and non-Fourier models was observed for lower values of the Fourier number and higher values of the Vernotte number. Finally, the effect of the boundary conditions on the Fourier and non-Fourier heat transfer was demonstrated.     

EXPLICIT FINITE ANALYTIC METHOD FOR UNSTEADY THREE-DIMENSIONAL NAVIER-STOKES EQUATIONS

An explicit finite analytic method (EFA) for solving unsteady Navier-Stokes equations is developed based on the finite analytic method (FA). In this method, the connective transport equation is solved with a heat analytic solution based on the characteristic method in space and time variables for the inviscid portion, while the viscous diffusion and source term are approximated by finite differences. A second-order polynomial determined by the direction of velocity is adopted to approximate the initial condition in the local element. Numerical solutions obtained for the lid-driven cavity flow are compared with those obtained by the 19-point FA method.     

Real-time solution of heat conduction in a finite slab for inverse analysis

Laplace transform is used to solve the problem of heat conduction over a finite slab. The transfer functions relating the temperature and heat flux on the front and back surfaces of the finite slab are developed. Although there are many competing methods for constructing the inverse Laplace transform, we use polynomial approximation of the transfer function. Therefore, transient solutions for given boundary conditions are easily obtained using SIMULINK. This process is much simpler than other numerical solution methods for the heat equation. Most importantly, our method of solution allows us to obtain, in real-time, the front surface temperature and heat flux based on the thermodynamic measurements on the back surface. We also demonstrate the feasibility of reconstructing the front surface temperature when sensor noise is incorporated to the back surface measurements.     

Meshless element free Galerkin method for unsteady nonlinear heat transfer problems

In this paper, meshless element free Galerkin (EFG) method has been extended to obtain the numerical solution of nonlinear, unsteady heat transfer problems with temperature dependent material properties. The thermal conductivity, specific heat and density of the material are assumed to vary linearly with the temperature. Quasi-linearization scheme has been used to obtain the nonlinear solution whereas backward difference method is used for the time integration. The essential boundary conditions have been enforced by Lagrange multiplier technique. The meshless formulation has been presented for a nonlinear 3-D heat transfer problem. In 1-D, the results obtained by EFG method are compared with those obtained by finite element and analytical methods whereas in 2-D and 3-D, the results are compared with those obtained by finite element method.     

用渐近分析法研究蜂窝蓄热体温度分布

1前言在冶金、机械和石化工业锻造炉、均热炉、连续加热炉、热处理炉、钢包烘烤炉、辐射管和熔铝炉上应用的高温空气燃烧(High Temperature Air Combustion,Hi-TAC)[1],具有热效率高、低NOx排放和燃烧放热均匀等特点。大多数的HiTAC应用了蜂窝蓄热系统[2]。温度变化和温度效率(     

模拟井筒耦合传热大涡模拟的建模与数值求解

模拟井筒加温系统是研究油田井下高温高压环境的特种实验装置,其主体部分模拟井筒为耐高温高压的厚壁圆柱形封闭腔体,模拟井筒温度场是其加温系统设计与工作参数确定的基本依据。分析加热过程中腔体内流体与厚壁腔体之间的动态耦合传热,通过合理简化建立了井筒加热物理模型,给出了其基于大涡模拟方法的数学模型,采用有限差分法和应用投影法求解模型。研究结果表明：模型和求解方法可以用于高温高压模拟井筒流固耦合传热研究,其实验误差低于16%;通过模拟计算得到了井筒及其腔体内流体的动态耦合传热过程温度场分布规律,为模拟井筒加温系统设计与工作参数确定提供理论依据。     

Weight functions from finite element displacements

The weight function method has been extensively used in fracture mechanics for stress intensity factor determination. It was shown that, if a complete solution (the crack face displacement and the stress intensity factor) to a crack problem for one loading system is known, then the solution for the stress intensity factor for the same cracked configuration, but with any other loading, may be obtained directly from the known solution. Several procedures have been proposed, which use a solution for the stress intensity factor available from handbooks. In this paper, an approach based on the crack face displacements obtained from a finite element analysis is used. This procedure can be applied without knowing a previous solution for a certain loading. The method proves to be extremely efficient, and some results and generalisations for a pressurised cylindrical shell with an external straight axial crack are presented.     

Solutions for modelling moving heat sources in a semi-infinite medium and applications to laser material processing

This study describes the analytical and numerical solution of the heat conduction equation for a localised moving heat source of any type for use in laser material processing, as welding, layered manufacturing and laser alloying. In this paper, the analytical solution for a uniform heat source is derived from the solution of an instantaneous point heat source. The result is evaluated numerically and is compared to existing solutions for the moving point source and a semi-ellipsoidal source. Next, the result is used to demonstrate how such model can be used to study the effect of the heat source geometry. Besides, this solution reveals that a melting efficiency higher than 0.37 (= 1/e, a maximum value stated by Rykalin [N. Rykalin, A. Uglov, A. Kokora, O. Glebov, Laser Machining and Welding, Mir Publishers, Moscow, 1978]) can be obtained. To investigate the effect of the temperature dependence of the material parameters, in particular the latent heat of fusion, a finite difference model is implemented. It is shown that the enthalpy method is most suited to implement the latent heat of fusion. A numerical evaluation for Ti–6Al–4V, reveals that the effect of the latent heat is rather small, except when the conductivity is very low, e.g. when scanning in a loose powder bed. The results demonstrate that analytical and numerical solutions can be effectively used to calculate the temperature distribution in a semi-infinite medium for finite 3D heat sources. In this way, a tool to investigate the importance of different processing parameters in laser manufacturing is obtained.     

Computing sensitivity coefficients by using complex differentiation: Application to heat conduction problem

Abstract

A simple and efficient computational method for Sensitivity Coefficient calculation is presented. Sensitivity analysis are carried out for a simple one-dimensional heat conduction problem with respect to two parameters: thermal conductivity (k) and volumetric heat capacity (c). Results obtained by finite difference and complex step differentiation are compared with analytical solution in non-dimensional form. With an arbitrary small step-size, the complex differentiation method can achieve near analytical accuracy in languages that support complex arithmetic. Further implementation of complex step differentiation in Matlab environment is explained with some precautions.     

地源热泵竖直埋管的有限长线热源模型

对地热换热器竖直埋管的非稳态传热模型进行了分析讨论。采用虚拟热源和格林函数法给出了半无限大介质中有限长线热源产生的非稳态温度场的解析解表达式。与稳态温度场的解进行比较,讨论了温度场达到名义上的“稳态”所需的时间,同时对于达到稳态时的温度场也进行了分析,指出了现行教科书中关于该问题的错误,提出了稳态时两个地热换热器孔壁代表性温度的定义,并对两者进行了比较,进而给出了可供工程应用的简化计算公式,并对两者进行了比较,进而给出了可供工程应用的简化计算公式。基于以上分析,进一步讨论了全年冷热负荷不平衡对地热换热器长期性能的影响。     

Thermal decay of an initially hot isothermal water body with application to thermal energy storage

The thermal decay of an initially hot isothermal water body contained in a tank was studied. Analytical solutions (including a group-theoretic solution) were obtained for the one-dimensional heat conduction equation with heat loss from the sides of the tank. The convective heat transfer coefficient was assumed not to be constant over the surface of the tank but to vary with space and time. A very good agreement was obtained between the group-theoretic solution and a numerical solution using the Crank-Nicholson finite difference scheme. The analytical results are discussed in terms of the underlying physical mechanisms.     

Free convection near a corner formed by two vertical plates embedded in porous medium

The analysis of free convection heat transfer near a corner formed by two mutually perpendicular flat plates embedded in porous medium is presented. It is found that local similarity solution exists for the case of constant wall temperature. Solutions are obtained by selecting a stream function to satisfy the continuity equation, the energy equation and the asymptotic boundary conditions at infinity. The problem is solved by finite difference method. Velocity profiles and heat transfer characteristics are also presented.     

FINITE ELEMENT ANALYSIS OF COUPLED HEAT. MASS,AND PRESSURE TRANSFER IN POROUS BIOMATERIALS

Abstract

The finite element method was used to solve Luikov's system of partial differential equations for neat, mass, and pressure transfer in capillary porous bodies. The finite element predictions were validated by comparing with exact solutions and the analytical results given by Mikhailov and Shishedjiev [1]. An application of the finite element method to the drying of wood (spruce) and a comparison based on an eigenvalue solution for simultaneous heat and mass transfer [2] are also provided. This technique was applied to study the coupled transport process in a silicon gel. The simulation indicated that the results obtained from the heat, mass, and pressure transfer model showed a marked difference from the results obtained by the heat and mass transfer model.     

Meshless analysis of unsteady-state heat transfer in semi-infinite solid with temperature-dependent thermal conductivity

In this paper, meshless element-free Galerkin (EFG) method has been extended to obtain the numerical solution of linear and non-linear heat transfer in semi-infinite solids. A model problem has been solved using constant and temperature-dependent thermal conductivities of the material. For the linearization of nonlinear equations, quasilinearization scheme is adopted to avoid iterations and for time integration, backward difference method has been used. The meshless formulation has been presented for a non-linear heat transfer in semi-infinite solids. The EFG results have been obtained for a model problem using cubic spline weight function. The results obtained by the EFG method are compared with those obtained by finite element and analytical methods.     

Thermoelastic study of a functionally graded annular fin with variable thermal parameters using semiexact solution

Abstract

In this paper, the thermoelastic behavior of a functionally graded material (FGM) annular fin is investigated. The material properties of the annular fin are assumed to vary radially. The heat transfer coefficient and internal heat generation are considered to be functions of temperature. A closed form solution of nonlinear heat transfer equation for the FGM fin is obtained using the homotopy perturbation method (HPM) which leads to nonuniform temperature distributions within the fin. The temperature field is then coupled with the classical theory of elasticity and the associated thermal stresses are derived analytically. For the correctness of the present closed form solution for the stress field, the results are compared with the ANSYS-based finite element method (FEM) solution. The present HPM-based closed form solution of the stress field exhibits a good agreement with the FEM results. The effect of various thermal parameters such as the thermogeometric parameter, conduction-radiation parameter, internal heat generation parameter, coefficient of variation of thermal conductivity, and the coefficient of thermal expansion on the thermal stresses are discussed. The results are presented in both nondimensional and dimensional form. The dimensional stress analysis discloses the suitability of FGM as the fin material in practical applications.