Thermo-geometric Fin Parameter Research Articles

Analysis of temperature-dependent thermal conductivity and fin efficiency: Direct Akbari-Ganji method

The vast majority of engineering issues, particularly heat transport equations, have a non-linear form. The analytical solution of temperature distribution, fin effectiveness, and fin efficiency of straight convective fins that transport heat with temperature-dependent thermal conductivity convection is derived in this work using the Direct Akbari-Ganji method (DAGM). DAGM is an effective methodology that is very simple and takes less time to solve non-linear temperature distribution fin problems. In the present investigation, Direct Akbari-Ganji technique offers desirable solutions to heat transfer problems. The round-off error is smaller compared to other methods. For the purpose of calibrating the error accuracy of the suggested direct technique, this result is compared with the numerical and earlier solution. The outcomes show that the DAGM for heat transfer equations in engineering situations has excellent validity and considerable promise. Finally, it is clear that when the thermo-geometric fin parameter increases, so does the fin efficiency. The homogeneity of the fin is due to the variable thermal conductivity. Four convection conditions with constant coefficients, growing and decaying convection coefficient with fin temperature are discussed.

Study of Thermoelastic behaviour, Efficiency and Effectiveness of Rectangular Fin with Fractional Order Energy Balance Equation

In this paper, the analytical solution is obtained for temperature distribution, efficiency, effectiveness and thermal stresses in convective fin of a rectangular profile with thermosensitive conductivity by implementing Homotopy Perturbation Method (HPM). This method is one of the effective tool to solve the fractional order non-linear diffussion equation with thermosensitive conductivity. The results obtained are illustrated graphically for better understanding of the effect of fractional parameter on fin effeciency and effectiveness with respect to thermogeometric fin parameter.

Exact solution of the nonlinear fin problem with exponentially temperature-dependent thermal conductivity and heat transfer coefficient

This article studies a class of fin problems with two nonlinear terms arising from thermal conductivity and convection heat transfer coefficient. A one-dimensional convective straight fin is analysed for exponentially temperature-dependent thermal conductivity and exponentially temperature-dependent heat transfer coefficient. The exact fin temperature excess, heat transfer rate and fin efficiency are obtained and presented graphically. The obtained results show the strong influences of exponent indexes of thermal conductivity and convection transfer coefficient as well as thermogeometric fin parameter on the fin efficiency and heat flow.

Effect of Variable Thermal Conductivity Calibration Functions on Fin Temperature Distribution

In this article, we introduce a new thermal conductivity calibration function in modeling heat transfer through extended surfaces. The variable thermal conductivity functions are studied on a stand alone basis and further compared to one another. The calculations are carried out using the Variational Iteration Method (VIM) which is an analytical solution technique. The series solutions are bench-marked against the numerical results obtained by applying the Runge-Kutta fourth order method coupled with shooting technique. The effects of some physical parameters such as the thermogeometric fin parameter and thermal conductivity gradient, on temperature distribution are illustrated and explained.

Predicting the Temperature Distribution in Longitudinal Fins of Various Profiles with Power Law Thermal Properties Using the Variational Iteration Method

In this article, the variational iteration method (VIM) is used to analyze heat transfer in longitudinal fins of various profiles with temperature dependent thermal conductivity and heat transfer coefficient. In order to show the efficiency of the VIM, the results obtained using the VIM are compared with the previously obtained results using the differential transform method (DTM) and Lie group analysis of a fin problem with the rectangular profile. After establishing confidence in VIM, heat transfer is analyzed to determine the temperature profiles in longitudinal fins of various profiles with power law thermal properties. The temperature distribution is compared between various profiles. The effects of some physical parameters such as the thermo-geometric fin parameter and thermal conductivity gradient on temperature distribution are illustrated.

A Study on Convective-Radial Fins with Temperature-dependent Thermal Conductivity and Internal Heat Generation

Abstract In this paper, the temperature distribution in a convective radial fins is analyzed through a fractional order energy balance equation with the consideration of internal heat generation and temperature dependent thermal conductivity. Adomian decomposition Sumudu transform method is used to study the influence of temperature distribution and the efficiency of radial fins for different values of thermal conductivity and to determine the role of thermal conductivity, thermo-geometric fin parameter as well as fractional order values in finding the temperature distribution and the fin efficiency of the convective radial fins. Finally, the efficiency of this proposed method has been studied by comparing the obtained results with the classical order results obtained by using numerical method and Variational Iteration Method (Coskun and Atay, 2007).

A new fractional model for convective straight fins with temperature-dependent thermal conductivity

The key aim of this work is to present a new non-integer model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo-Fabrizio fractional derivative. The fractional energy balance equation is solved by using homotopy perturbation method coupled with Laplace transform method. The efficiency of straight fin has been derived in terms of thermo-geometric fin parameter. The numerical results derived by the application of suggested scheme are demonstrated graphically. The subsequent correlation equations are very helpful for thermal design scientists and engineers to design straight fins having temperature-dependent thermal conductivity.

Exact results for efficiency and effectiveness assessment of longitudinal porous fins

In this work, thermal performance of porous fins with temperature-dependent thermal conductivity is revisited. It is shown the governing strongly nonlinear differential equation, which has been formulated by performing Darcy’s model, is exactly solvable. A full discussion and exact analytical solution (Dimensionless temperature distribution along the length of porous fin as a function of porosity and convection parameters) in the implicit form is given for further physical interpretation and it is proved the solution is mathematically unique. Furthermore, we give exact analytical expressions of fin efficiency and fin effectiveness as a function of convection parameter. Also, the exact ratio of porous fin to solid fin heat transfer rate as a function of thermo-geometric fin parameter is obtained.

Exact Closed-Form Solution of the Nonlinear Fin Problem With Temperature-Dependent Thermal Conductivity and Heat Transfer Coefficient

In the current paper, the nonlinear fin problem with temperature-dependent thermal conductivity and heat transfer coefficient is revisited. In this problem, it has been assumed that the heat transfer coefficient is expressed in a power-law form and the thermal conductivity is a linear function of temperature. It is shown that its governing nonlinear differential equation is exactly solvable. A full discussion and exact analytical solution in the implicit form are given for further physical interpretation and it is proved that three possible cases may occur: there is no solution to the problem, the solution is unique, and the solutions are dual depending on the values of the parameters of the model. Furthermore, we give exact analytical expressions of fin efficiency as a function of thermogeometric fin parameter.

A Study on Nonlinear Wet Fin Problem Using Homotopy Analysis Method

Homotopy analysis method (HAM) is used to study a rectangular wet fin with variable thermal conductivity. Unlike most theoretical investigations which have been carried out by considering a constant thermal conductivity and a linear relationship between the specific humidity of air and temperature, in this work, thermo-geometric and psychrometric variations are assumed to be nonlinear. The above considerations make the fin problem highly nonlinear due to nonlinearities in the governing equation as well as in the boundary conditions. HAM is used to determine the closed-form solution (i.e., temperature field) and the efficiency of the fin. It is noticed that the fin efficiency is obtained as a function of thermo-geometric fin parameter. The effects of the wet surface and variable conductivity on the temperature field as well as on the fin efficiency are analyzed. The present results of HAM are found to be in very good agreement with the available results of relevant literature. It is shown that the thermo-geometric fin parameter has a significant role on the fin efficiency. The present study reveals that the simultaneous heat and mass transfer across the fin surface is thermally less efficient compared to the fin surface involving heat transfer only.

Thermal analysis of straight rectangular fin using homotopy perturbation method

In this study a simple and accurate semi-analytical method called Homotopy Perturbation Method (HPM) is used for solving nonlinear energy equation in a straight rectangular fin. The thermal conductivity and surface emissivity are considered as temperature dependent with some constant source of internal heat generation. The problem is solved for two main cases of thermo geometric fin parameter Nc=1 and for Nc=0.25. The results are presented for the temperature distribution, efficiency and optimum dimensionless parameters are effective and convenient for practical fins. Also it is found that this method can achieve more suitable results as compared to the other methods available in the literature.

Analysis for fin efficiency with temperature-dependent thermal conductivity of fractional order energy balance equation using HPST Method

Radiating extended surfaces are usually utilized to enhance the heat transfer between primary surface and the environment. In this paper, temperature distribution, fin efficiency, efficacy of convective straight fins with constant and temperature-dependent thermal conductivity are solved by implementing homotopy perturbation sumudu transform method (HPSTM). The proposed method is very useful and practical for solving the fractional order nonlinear diffusion equation, which is associated with variable thermal conductivity condition. A dimensionless analytical expression has been developed for fin effectiveness. The fin efficiency and the fin effectiveness have been attained as a function of thermo-geometric fin parameter. It can be noticed that the thermal conductivity parameter has a strong influence over the fin efficiency. The analytical solutions acquired by the present method illustrate the approach is easy to implement and computationally very interesting. The obtained results are compared with previously found classical order results using variational iteration method (VIM), Adomian decomposition method, and the results from Galerkin method in order to show the competence of this present method. HPSTM is a simple and effective method for rapid assessment of physical systems although the fractional order energy balance equations comprise with strong nonlinear terms. The subsequent correlation equations can benefit thermal design engineers for designing of innovative straight fins with both constant and temperature-dependent thermal conductivity.

General exact solution of the fin problem with the power law temperature-dependent thermal conductivity

In this paper, the general exact implicit solution of the second-order nonlinear ordinary differential equation governing heat transfer in rectangular fin is obtained using Lie point symmetry method. General relationship among the fin efficiency, the rate of heat transfer from the entire fin, the fin effectiveness, and the thermo-geometric fin parameter is obtained for any value of the mode of heat transfer n and the constant β. Copyright © 2015 John Wiley & Sons, Ltd.

A successful application of homotopy perturbation method for efficiency and effectiveness assessment of longitudinal porous fins

In our previous works, thermal performance of straight fins with both constant and temperature-dependent thermal conductivity has been investigated in detail and dimensionless analytical expressions of fin efficiency and fin effectiveness have been developed for the first time in literature via homotopy perturbation method. In this study, previous works have been extended to porous fins. Governing equations have been formulated by performing Darcy’s model. Dimensionless temperature distribution along the length of porous fin has been determined as a function of porosity and convection parameters. The ratio of porous fin to solid fin heat transfer rate has also been evaluated as a function of thermo-geometric fin parameter. The results have been compared with those of finite difference method for a specific case and an excellent agreement has been observed. The expressions developed are beneficial for thermal engineers for preliminary assessment of thermophysical systems instead of consuming time in heat conduction problems governed by strongly nonlinear differential equations.

Homotopy perturbation sumudu transform method for solving convective radial fins with temperature-dependent thermal conductivity of fractional order energy balance equation

In order to enhance the heat transfer between primary surface and the environment, radiating extended surfaces are commonly utilised. In this present analysis, temperature distribution and effectiveness of convective radial fins with constant and temperature-dependent thermal conductivity are solved by applying homotopy perturbation sumudu transform method (HPSTM). The concept of homotopy perturbation sumudu transform method is briefly introduced, and then it is employed to derive solution of fractional order nonlinear governing equation. The fractional derivative is considered in the Caputo sense. The HPSTM is a combined form of sumudu transform, homotopy perturbation method, and He’s polynomials. The nonlinear terms can be easily handled by the use of He’s polynomials. The fin effectiveness has been obtained as a function of thermo-geometric fin parameter. The analytical solutions obtained by the proposed method show that the approach is easy to implement and computationally very attractive. In order to show the efficiency of this proposed method, the results are compared with previously obtained classical order results using Variational Iteration Method (VIM) (Coskun and Atay, 2007) [1]. It has revealed that homotopy perturbation sumudu transform method is a very simple and effective approach for a rapid assessment of physical systems even if the energy balance equations of fractional order include terms with strong nonlinearities. The resulting correlation equations can assist thermal design engineers for designing of radial fins with both constant and temperature-dependent thermal conductivity.

Comparison of Exact Solutions for Heat Transfer in Extended Surfaces of Different Geometries

We consider a steady state problem for heat transfer in fins of various geometries, namely, rectangular, radial, and spherical. The nonlinear steady state problem is linearizable provided that the thermal conductivity is the differential consequence of the term involving the heat transfer coefficient. As such, one is able to construct exact solutions. On the other hand, we employ the Lie point symmetry methods when the problem is not linearizable. Some interesting results are obtained and analyzed. The effects of the parameters such as thermogeometric fin parameter and the exponent on temperature are studied. Furthermore, fin efficiency and heat flux along the fin length of a spherical geometry are also studied.

Some Exact Solutions of Nonlinear Fin Problem for Steady Heat Transfer in Longitudinal Fin with Different Profiles

One-dimensional steady-state heat transfer in fins of different profiles is studied. The problem considered satisfies the Dirichlet boundary conditions at one end and the Neumann boundary conditions at the other. The thermal conductivity and heat coefficients are assumed to be temperature dependent, which makes the resulting differential equation highly nonlinear. Classical Lie point symmetry methods are employed, and some reductions are performed. Some invariant solutions are constructed. The effects of thermogeometric fin parameter, the exponent on temperature, and the fin efficiency are studied.

Analytical Solutions for Steady Heat Transfer in Longitudinal Fins with Temperature-Dependent Properties

Explicit analytical expressions for the temperature profile, fin efficiency, and heat flux in a longitudinal fin are derived. Here, thermal conductivity and heat transfer coefficient depend on the temperature. The differential transform method (DTM) is employed to construct the analytical (series) solutions. Thermal conductivity is considered to be given by the power law in one case and by the linear function of temperature in the other, whereas heat transfer coefficient is only given by the power law. The analytical solutions constructed by the DTM agree very well with the exact solutions even when both the thermal conductivity and the heat transfer coefficient are given by the power law. The analytical solutions are obtained for the problems which cannot be solved exactly. The effects of some physical parameters such as the thermogeometric fin parameter and thermal conductivity gradient on temperature distribution are illustrated and explained.

Homotopy perturbation method for temperature distribution, fin efficiency and fin effectiveness of convective straight fins with temperature-dependent thermal conductivity

Homotopy perturbation method is a novel approach that provides an approximate analytical solution to differential equations in the form of an infinite power series. In our previous work, homotopy perturbation method has been used to evaluate thermal performance of straight fins with constant thermal conductivity. A dimensionless analytical expression has been developed for fin effectiveness. In this study, homotopy perturbation method has been applied to convective straight fins considering thermal conductivity of the fin material is a function of the fin temperature. Former expression for fin effectiveness has been rearranged. The fin efficiency and the fin effectiveness have been obtained as a function of thermo-geometric fin parameter. The results have revealed that homotopy perturbation method is a very effective and practical approach for a rapid assessment of physical systems even if the energy balance equations include terms with strong nonlinearities. The resulting correlation equations can assist thermal design engineers for designing of straight fins with both constant and temperature-dependent thermal conductivity.

Transient response of longitudinal rectangular fins to step change in base temperature and in base heat flow conditions

In this paper, we construct solutions for model describing heat transfer in longitudinal rectangular fins. The imposed boundary conditions are the step change in the base temperature and the step change in base heat flow. Both the thermal conductivity and the heat transfer coefficients are assumed to be power law temperature dependent. We employ the local and nonlocal symmetry techniques to analyze the problem at hand. In one case the reduced equation transforms to the tractable Ermakov–Pinney equation. Nonlocal symmetries are admitted when some arbitrary constants appearing in the governing equations are specified. The exact analytical steady state solutions which satisfy the prescribed boundary conditions are constructed. Since the obtained general exact analytical solutions for the transient state satisfy only the zero initial temperature and adiabatic boundary condition at the fin tip, we sort numerical solutions. The effects of the thermo-geometric fin parameter and the power law exponent on temperature distribution are studied.