Finite element method formulation in polar coordinates for transient heat conduction problems

The aim of this paper is the formulation of the finite element method in polar coordinates to solve transient heat conduction problems. It is hard to find in the literature a formulation of the finite element method (FEM) in polar or cylindrical coordinates for the solution of heat transfer problems. This document shows how to apply the most often used boundary conditions. The global equation system is solved by the Crank-Nicolson method. The proposed algorithm is verified in three numerical tests. In the first example, the obtained transient temperature distribution is compared with the temperature obtained from the presented analytical solution. In the second numerical example, the variable boundary condition is assumed. In the last numerical example the component with the shape different than cylindrical is used. All examples show that the introduction of the polar coordinate system gives better results than in the Cartesian coordinate system. The finite element method formulation in polar coordinates is valuable since it provides a higher accuracy of the calculations without compacting the mesh in cylindrical or similar to tubular components. The proposed method can be applied for circular elements such as boiler drums, outlet headers, flux tubes. This algorithm can be useful during the solution of inverse problems, which do not allow for high density grid. This method can calculate the temperature distribution in the bodies of different properties in the circumferential and the radial direction. The presented algorithm can be developed for other coordinate systems. The examples demonstrate a good accuracy and stability of the proposed method.     

Solution of transient heat transfer in graded-material fins of varying thickness under step changes in boundary conditions using the Lattice Boltzmann Method

Graded materials (GM) possess superior thermo-mechanical properties, which are not feasible to obtain with homogeneous materials (HM), and hence in this paper, the transient response of a longitudinal fin of varying geometry made up of GM is reported. The temperature-dependent convection coefficient and heat generation parameters are considered to account for real-world high-temperature applications of fins. Fin material properties such as density and specific heat remain constant while thermal conductivity is assumed to vary axially based on four different physically possible variations namely, linear, quadratic, power, and exponential variations. The typical nonlinear differential equation obtained for fins was solved by using a mesoscopic scale-based particle tracking method called the Lattice Boltzmann method. The Lattice Boltzmann solver has been implemented in form of an in-house MATLAB code and validated with existing results, thereafter it is developed for solving the foregoing problems. The results obtained are reported for rectangular, triangular, convex, and concave profiles under step change in base temperature and base heat flux. The performance of graded fins is investigated in terms of time required to attain steady-state and fin tip temperature which are inherent design parameters in the case of the transient fin. Inhomogeneity index and profile function have a significant effect on the performance of fin in terms of resistance to heat flow. Hereby, in comparison with HM fins, GM fins have lower resistance to heat flow irrespective of fin profiles. Concurrently, comparative analysis for fins of different profiles made of HM and GM is also done to facilitate the designer in selecting the most appropriate fins.     

Steady and transient numerical analysis of the performance of annular fins

This article presents a simple, straightforward and accurate numerical technique that can be used to predict the steady and transient temperature distributions in an annular fin with uniform cross‐section. Both convection and/or thermal radiation ambient conditions are considered in this study. The heat transfer rate and fin efficiency are also calculated. Results are compared with the exact solution for the steady case where excellent agreement was observed for an insulated tip fin and with approximate solutions for the transient case for a non‐insulated tip condition. Copyright © 2001 John Wiley & Sons, Ltd.     

辐射对流条件下肋片散热的数值计算和结构优化

本文利用matlab对一种矩形肋片进行了散热数值计算 ,以金属热强度为结构优化的目标函数 ,通过对计算结果的分析得出了此种肋片的最佳肋高尺寸 ,同时 ,认为利用matlab来解决这类肋片优化问题是非常方便的。     

Improvement of inverse heat conduction solution using Laplace transformation: Method of partial division of time

Any solution for an inverse heat conduction problem makes the estimation of surface temperature and surface heat flux worsen in the case where these values behave like a triangular shape change with time. In order to compensate for this defect, Monde and colleagues, who succeeded in obtaining analytical inverse solutions using the Laplace transform technique, introduce a new idea where these changes over the entire measurement time can be split into several parts depending on the behavior. Therefore, an approximate equation to trace the measured temperature change can be derived, resulting in good estimation of surface temperature and surface heat flux even in the case of the triangular shape change and sharp change. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 32(7): 630–638, 2003; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10117     

The adjoint-solution technique for the calculation of the thermal properties of layers in multilayer walls under transient heat conduction

The thermal properties of the layers of a wall, whether or not exposed to solar radiation, are calculated provided that the boundary conditions and some values of the transient temperature field within the wall are known. The developed procedure is based on the adjoint-solution technique and is applicable both to walls in operation and to the design of walls that are required to meet certain temperature specifications. In the former case, temperature measurements are needed. Theoretical and experimental tests have proved the accuracy of the method. Applications may be found in energy management and thermal storage in buildings, in the improvement of passive systems and in the design of multilayer slabs forming parts of heat-transfer equipment. © 1997 by John Wiley & Sons, Ltd.     

空气横掠矩形翅片椭圆管束换热规律的数值研究

采用Fluent软件对矩形翅片椭圆管束空气侧的对流换热情况进行了三维数值模拟,获得了不同流速下翅片表面温度分布,分析了迎面风速与换热系数之间的关系,随着速度的增大,空气侧的换热系数增加,并拟合了换热计算公式。同时分析了不同翅片间距对换热的影响因素,随着翅片间距的增大,空气侧换热系数增加,而且随着Rg数的增加,换热的强化更加明显。     

肋参数对复合肋套管流动和换热的影响

同心复合套管式回热器是微型燃气轮机的重要部件—紧凑式回热器的一种类型。采用Fluent对有交错肋的同心套管中的对流与换热进行了数值模拟。结果表明：在给定的边界条件下,肋的一些参数,比如冷、热流体通道内的肋之间的夹角,肋片的导热系数和管长会对Nu数产生影响,但夹角的变化对摩擦系数几乎没影响。     

板翅式换热器翅片及隔板动态特性分析

基于板翅式换热器翅片的非稳态导热方程，计算分析了翅片的动态特性，认为翅片的不稳定传热过程相对于换热器其它过程特征时间无限小，因此可以不考虑翅片的动态特性，从而简化了板翅式换热器动态模型，通过分析换热器动态过渡过程表明：隔板的热容对板翅式换热器的动态特性的影响是不容忽略的。     

A transient liquid crystal method using a 3-D inverse transient conduction scheme

The present method utilized the hue-angle method to process the color images captured from the liquid crystal color play. Instantaneous temperature readings from embedded thermocouples were utilized for in situ calibration of hue angle for each data set. The convective heat transfer coefficient results were obtained by performing a 3-D inverse transient conduction calculation over the entire jet impingement target surface and the substrate. The results of average heat transfer coefficients agreed well with previous experimental results of point measurements by thermocouples.Comparison between 1-D and 3-D results indicates that 1-D results are higher than the 3-D results with the local maximum and minimum heat transfer values being overvalued by about 15-20% and the overall heat transfer by approximately 12%. This is due to the fact that 1-D method does not include the lateral heat flows induced by local temperature gradients.     

A new stochastic approach to transient heat conduction modeling with uncertainty

We present a generalized polynomial chaos algorithm for the solution of transient heat conduction subject to uncertain inputs, i.e. random heat conductivity and capacity. The stochastic input and solution are represented spectrally by the orthogonal polynomial functionals from the Askey scheme, as a generalization of the original polynomial chaos idea of Wiener [Am. J. Math. 60 (1938) 897]. A Galerkin projection in random space is applied to derive the equations in the weak form. The resulting set of deterministic equations is subsequently discretized by the spectral/hp element method in physical space and integrated in time. Numerical examples are given and the convergence of the chaos expansion is demonstrated for a model problem.     

Analytical solution for transient temperature and thermal stresses within convective multilayer disks with time-dependent internal heat generation,Part II: Applications

This work investigates the analytical solution for transient temperature and thermal stresses within three circular geometries. First, the transient temperature and thermal stresses within a composite disk are addressed. Then, two examples regarding transient temperature and thermal stresses throughout circular heaters are analyzed. Pulsed and sinusoidal internal heat generations are incorporated into the second and third examples, respectively. For the composite hollow-disk example, merely the separation of variables method (SVM) is used to overcome the energy partial differential equation. For the other two examples, the combination of the SVM and Duhamel's theorem are adopted to solve the partial differential equations. Accordingly, assuming plane stress formulation, the transient thermal stresses within structures are obtained.     

A two-dimensional fin efficiency analysis of combined heat and mass transfer in elliptic fins

This paper presents a two-dimensional analysis for the efficiency of an elliptic fin under the dry, partially wet and fully wet conditions of a range of value for axis ratios, Biot numbers, and air humidifies. It is shown that the fin efficiencies increase as the axis ratio Ar is increased. For a given axis ratio Ar, the fin efficiency decreases as the fin height l∗ or Biot number is increased. The conventional 1-D sector method overestimates the fin efficiency resulting in increasing error as the axis ratio Ar is increased. In addition, using experimentally determined heat transfer coefficients, it is found that both the fully dry and wet elliptic fin efficiencies are up to 4-8% greater than the corresponding circular fin efficiencies having the same perimeter.     

Thermal analysis of natural convection and radiation in porous fins

In this study, the effects of radiation and convection heat transfer in porous media are considered. The geometry considered is that of a rectangular profile fin. The porous fin allows the flow to infiltrate through it and solid-fluid interaction takes place. This study is performed using Darcy's model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered viz. long fin, finite length fin with insulated tip and finite length fin with tip exposed. The theory section addresses the derived governing equation. The effects of the porosity parameter S_h, radiation parameter G and temperature ratio C_T on the dimensionless temperature distribution and heat transfer rate are discussed. The results suggest that the radiation transfers more heat than a similar model without radiation.     

Constructal design of vertical multiscale triangular fins in natural convection

Constructal design of vertical multiscale triangular fins in natural convection is investigated in this paper. The design consists of two parts. The first part is for single-scale triangular fins. The objective in the first design is to reach to the highest heat transfer density from the fins for three fin angles (15°, 30°, and 45°). The single-scale fins are placed in a horizontal array and considered as isothermal fins. The degrees of freedom are the fin angle, and the fin-to-fin spacing. The constraint is the fin height. The second part is for multiscale fins where small fins are placed between the large fins which are optimized in the first part. In the second part, the angles of the large and small scales fins are kept constant at (15°). The optimal fin-to-fin spacing which is obtained in the first part is considered a constraint in the second part. The Rayleigh numbers in this design are (Ra = 10³, 10⁴, and 10⁵). The two-dimensional mass, momentum, and energy equations for natural convection are solved with the finite volume method. The results show that there is a benefit of placing the small-scale fins where the percentage increase in the heat transfer density is (10.22%) at (Ra = 10³), and (50.6%) at (Ra = 10⁵) due to existence of the small fins between the large fins.     

A simplified solution technique for some fluid flow and heat transfer problems

A new simple solution technique is presented for fluid flow and heat convection problems that incorporate the similarity solution. Finite-difference methods and numerical integration are used to solve the describing equations using spreadsheets. Numerical solution by spreadsheets results in a solution of good accuracy and it requires much less effort than conventional programming languages. This paper also presents the current status of using spreadsheets in solving energy-related problems © 1998 John Wiley & Sons, Ltd.     

Heat transfer and flow characteristics of offset fins in the low-Reynolds-number region

The characteristics of heat exchangers with offset-type plate fins for space stations are studied for Reynolds numbers less than 300 based on the hydraulic diameter. A three-dimensional analysis is carried out to study the effects of the following parameters on the heat transfer and the flow characteristics: (a) the thermal boundary layer developing on the bottom plate and on the fins on the plate, (b) the aspect ratio (height/pitch) of the cross section of the flow passage, the fin thickness, the fin length in the direction of the flow, the thermal conductivity of the fluid and the fins, and the Prandtl number of the fluid. The results obtained are as follows. (1) The heat-transfer coefficient on the fin surface is characterized by the thermal-conductivity ratio of fluid to fin material. When the thermal conductivity of the fin material approaches that of the fluid, the heat-transfer coefficient on the fin surface becomes low. (2) The optimum condition of the aspect ratio depends on the value of the thermal-conductivity ratio between the fluid and the fins. (3) When the aspect ratio becomes large or small, the friction factor of offset fins approaches that of fully developed duct flow with the same aspect ratio as the Reynolds number decreases. © 1998 Scripta Technica. Heat Trans Jpn Res, 26(4): 249–261, 1997     

A solution to the two‐dimensional inverse heat conduction problem

A simple method is developed in this paper to solve two‐dimensional nonlinear steady inverse heat conduction problems. The unknown boundary conditions can be numerically obtained by using the iteration and modification method. The effect of measurement errors of the wall temperature on the algorithm is numerically tested. The results prove that this method has the advantages of fast convergence, high precision, and good stability. The method is successfully applied to estimate the convective heat transfer coefficient in the case of a fluid flowing in an electrically heated helically coiled tube. © 2000 Scripta Technica, Heat Trans Asian Res, 29(2): 113–119, 2000     

A numerical study on the performance of PEMFC with wedge-shaped fins in the cathode channel

The gas flow field design has a significant influence on the overall performance of a proton exchange membrane fuel cell (PEMFC). A single-channel PEMFC with wedge-shaped fins in the cathode channel was proposed, and the effects of fin parameters such as volume (0.5 mm³, 1.0 mm³, and 1.5 mm³), number (3, 5, and 9), and porosity of the gas diffusion layer (GDL) (0.2, 0.4, 0.6, and 0.8) on the performance of PEMFC were numerically examined based on the growth rate of power density (GRPD) and polarization curve. It was shown that wedge-shaped fins could effectively improve the PEMFC performance. With an increase in fin volume, the distributions of oxygen mass fraction in the outlet area of the cathode channel were lower, the drainage effect of the PEMFC improved, and GRPD also increased accordingly. Similar results were obtained as the number of fins increased. The GDL porosity had a greater effect than the wedge-shaped fins on the improvement in PEMFC performance, but the influence of GDL porosity weakened and the GRPD of porosity decreased as the porosity increased. This study provides an effective guideline for the optimization of the cathode channel in a PEMFC.     

Peridynamic simulation of transient heat conduction problems in functionally gradient materials with cracks

In this article, a modified state-based peridynamic (PD) method is proposed to solve transient heat conduction problems in functionally gradient materials (FGMs) with extending insulated cracks. The PD formulation of transient heat conduction has been derived by using the time integration through the forward difference technique. Numerical simulations have been performed to verify the accuracy and effectiveness of the proposed method. The analytical solution and the finite element method results are used for comparison. In this work, the material properties of functionally graded materials are assumed to vary exponentially in z-direction. Our PD results show good agreement with analytical solutions and results from the finite element method. Hence the proposed PD method is suitable to deal with the transient heat conduction problem in FGMs with extending insulated cracks.