Analytical approaches for thermal analysis of radiative fin with a step change in thickness and variable thermal conductivity

In this paper, heat transfer in a straight fin with a step change in thickness and variable thermal conductivity which is losing heat by radiation to its surroundings is analyzed. The calculations are carried out by using the differential transformation method (DTM) and variational iteration method (VIM) that can be applied to various types of differential equations. The results obtained employing the DTM and VIM are compared with a finite difference technique with Richardson extrapolation which is an accurate numerical solution to verify the accuracy of the proposed methods. As an important result, it is depicted that the DTM results are more accurate in comparison with those obtained by VIM. After these verifications the effects of parameters such as thickness parameter α, dimensionless fin semi‐thickness δ, length ratio λ, thermal conductivity parameter β, and radiation–conduction parameter N_r, on the temperature distribution and fin efficiency are illustrated and explained. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj) . DOI 10.1002/htj.21000     

Analytical and numerical investigation of fin efficiency and temperature distribution of conductive,convective, and radiative straight fins

In this study, fin efficiency, temperature distribution, and effectiveness of conductive, convective, and radiative straight fins with temperature dependent thermal conductivity are solved using the differential transformation method (DTM).The concept of differential transformation is briefly introduced, and then it is employed to derive the solutions of nonlinear governing equations of fins with highly nonlinear terms because of existing radiation in this study. The obtained results of DTM are compared with those of the Galerkin method (GM) and numerical boundary value problem method (BVP) to verify the accuracy of the proposed method. Furthermore, the effects of some physical appropriate parameters such as thermo‐geometric fin parameters and thermal parameters are analyzed. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20341     

Series solution of convective radiative conduction equation of the nonlinear fin with temperature dependent thermal conductivity

In this study, differential transform method (DTM) is used to evaluate the analytical solution of the nonlinear fin problem with temperature dependent thermal conductivity. Results are presented for the dimensionless temperature distribution and fin efficiency for a range of values of the problem parameters. Using DTM, the differential equation and related boundary conditions are transformed into a recurrence set of equations and finally, the coefficients of power series are obtained based on the solution of this set of equations. Results of this method are compared with the variational iteration method (VIM) and numerical solutions. Here, it is shown that the results of DTM are more accurate than the VIM with lesser calculation cost, which is the main innovation of the current study.     

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.     

A decomposition method for fin efficiency of convective straight fins with temperature-dependent thermal conductivity

The Adomian decomposition method (ADM) has been used to evaluate the efficiency of straight fins with temperature-dependent thermal conductivity and to determine the temperature distribution within the fin. The method is useful and practical for solving the nonlinear heat diffusion equation, which is associated with variable thermal conductivity condition. The ADM provides an analytical solution in the form of an infinite power series. The fin efficiency of the straight fins with temperature-dependent thermal conductivity has been obtained as a function of thermo-geometric fin parameter and the thermal conductivity parameter describing the variation of the thermal conductivity. It has been observed that the thermal conductivity parameter has a strong influence over the fin efficiency. The data from the present solutions has been correlated for a wide range of thermo-geometric fin parameter and the thermal conductivity parameter. The resulting correlation equations can assist thermal design engineers for designing of straight fins with temperature-dependent thermal conductivity.     

Determination of temperature distribution for annular fins with temperature‐dependent thermal conductivity

In this paper, homotopy analysis method (HAM) has been used to evaluate the temperature distribution of annular fin with temperature‐dependent thermal conductivity and to determine the temperature distribution within the fin. This method is useful and practical for solving the nonlinear heat transfer equation, which is associated with variable thermal conductivity condition. HAM provides an approximate analytical solution in the form of an infinite power series. The annular fin heat transfer rate with temperature‐dependent thermal conductivity has been obtained as a function of thermo‐geometric fin parameter and the thermal conductivity parameter describing the variation of the thermal conductivity. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20353     

A least squares method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity

Approximate but highly accurate solutions for the temperature distribution, fin efficiency, and optimum fin parameter for a constant area longitudinal fin with temperature dependent internal heat generation and thermal conductivity are derived analytically. The method of least squares recently used by the authors is applied to treat the two nonlinearities, one associated with the temperature dependent internal heat generation and the other due to temperature dependent thermal conductivity. The solution is built from the classical solution for a fin with uniform internal heat generation and constant thermal conductivity. The results are presented graphically and compared with the direct numerical solutions. The analytical solutions retain their accuracy (within 1% of the numerical solution) even when there is a 60% increase in thermal conductivity and internal heat generation at the base temperature from their corresponding values at the sink temperature. The present solution is simple (involves hyperbolic functions only) compared with the fairly complex approximate solutions based on the homotopy perturbation method, variational iteration method, and the double series regular perturbation method and offers high accuracy. The simple analytical expressions for the temperature distribution, the fin efficiency and the optimum fin parameter are convenient for use by engineers dealing with the design and analysis of heat generating fins operating with a large temperature difference between the base and the environment.     

Application of homotopy perturbation method and inverse prediction of thermal parameters for an annular fin subjected to thermal load

In this work, application of the homotopy perturbation method (HPM) and an inverse solution for estimating unknown thermal parameters such as the variable thermal conductivity parameter (β), the thermogeometric parameter (K), and the nondimensional coefficient of thermal expansion (χ) in an annular fin subjected to thermal stresses is presented. Initially, to obtain the nondimensional temperature distribution from the heat equation, the forward method is employed using an approximate analytical solution based on HPM. Thereafter, a closed form solution for the temperature-dependent thermal stresses is obtained using the classical theory of thermoelasticity coupled with HPM solution containing the temperature distribution. Next, for satisfying a particular stress criterion which makes relevance in selecting appropriate configurations for selecting the finned system, unknown thermal parameters are obtained using an inverse approach based on the Nelder–Mead simplex search minimization technique. The objective function is taken as the sum of square of the residuals between the measured stress field and an initially guessed value which is updated iteratively. It is found that more than one type of temperature distribution may yield a given stress distribution, thereby giving rise to different fin efficiencies. The agreement between the actual and the predicted results was found to be satisfactory.     

Stresses in radiative annular fin under thermal loading and its inverse modeling using sine cosine algorithm (SCA)

This work presents a novel mathematical model for the analysis of thermal stresses in a radiative annular fin with temperature-dependent thermal conductivity and radiative parameter. An approximate analytical solution for thermal stresses is derived using a homotopy perturbation method (HPM)-based closed-form solution of steady-state nonlinear heat transfer equation, coupled with classical elasticity theory. The effect of thermal parameters on the temperature field and the thermal stress fields are discussed. The various thermal parameters, such as a parameter describing the temperature-dependent thermal conductivity, coefficient of thermal expansion, coefficient of radiative parameter, and the variable radiative parameter, are inversely estimated for a given stress field. For inverse modeling, a population-based sine cosine algorithm (SCA) was employed to estimate the thermal parameters. The inverse modeling is verified by using the estimated thermal parameters in the closed-form solution of stress field. The reconstructed stress fields obtained from the inversely estimated parameters are then compared with the reference stress field. Results show a very good agreement between the reference stress field and the inversely estimated stress fields.     

Inverse and efficiency of heat transfer convex fin with multiple nonlinearities

In this article, we first propose the novel semi‐analytical technique—modified Adomian decomposition method (MADM)—for a closed‐form solution of the nonlinear heat transfer equation of convex profile with singularity where all thermal parameters are functions of temperature. The longitudinal convex fin is subjected to different boiling regimes, which are defined by particular values of n (power index) of heat transfer coefficient. The energy balance equation of the convex fin with several temperature‐dependent properties are solved separately using the MADM and the spectral quasi‐linearization method. Using the values obtained from the direct heat transfer method, the unknown parameters of the profile, such as thermal conductivity, surface emissivity, heat generation number, conduction‐convection parameter, and radiation‐conduction parameter are inversely predicted by an inverse heat transfer analysis using the simplex search method. The effect of the measurement error and the number of measurement points has been presented. It is found that present measurement points and reconstruction of the exact temperature distribution of the convex fin are fairly in good agreement.     

Conjugate Forced Convection Heat Transfer From a Heated Flat Plate of Finite Thickness and Temperature- Dependent Thermal Conductivity

In this paper, a conjugate forced convection heat transfer from a good conducting plate with temperature-dependent thermal conductivity is studied. The semi-analytical solution for the nonlinear integro-differential equation occurring in the problem is handled easily and accurately by implementing the differential transform method (DTM). The horizontal plate is heated with uniform heat flux at the lower surface while being cooled at the upper surface under laminar forced convection flow. A numerical approach is also performed via a finite-volume method to examine the validity of the results obtained by DTM. The results of DTM show closer agreement with the results of the numerical method than the results obtained by the perturbation method existing in the literature. It is concluded that for a good conducting plate with a finite thickness the distribution of the conjugate heat flux at the upper surface is significantly affected by the plate thickness. Moreover, we conclude that in the conjugate heat transfer case the temperature distribution of the plate is flatter than the one in the nonconjugate case.     

Effect of electromagnetic field on the thermal performance of longitudinal trapezoidal porous fin using DTM–Pade approximant

The heat transfer and thermal distribution through porous fins have gotten a lot of attention in recent years due to their extensive applications in the manufacturing and engineering field. In porous fins, the impact of magnetic field aids in improved heat transfer enhancement. Also, the combination of an electric effect and a magnetic field considerably enhances heat transfer. In this direction, the thermal distribution through a convective–radiative longitudinal trapezoidal porous fin with the impact of an internal heat source and an electromagnetic field is discussed in the present analysis. The governing heat equation is nondimensionalized with nondimensional terms, and the transformed nonlinear ordinary differential equation is solved analytically using the DTM–Pade approximant algorithm. Furthermore, the graphical discussion is presented to explore the impact of various nondimensional parameters, such as convection-conduction parameter, fin taper ratio, thermomagnetic field, radiation–conduction parameter, internal heat generation parameter, and thermoelectrical field on the temperature gradient of the fin. The investigation's key findings disclose that as the magnitude of the convection–conduction parameter, fin taper ratio, and radiation–conduction parameter increase, the thermal distribution through the fin reduces. The thermal distribution inside the fin increases for the heat-generating parameter, thermoelectric, and thermomagnetic fields.     

A Double Decomposition Method for Solving the Annular Hyperbolic Profile Fins With Variable Thermal Conductivity

In this article, the double decomposition method is used to analyze the annular hyperbolic profile fins with variable thermal conductivity. The double decomposition method is an advantageous way to solve the nonlinear problem. The solution gained by the double decomposition method is in the form of infinite power series, and the variable thermal conductivity is considered to have a linear relation with temperature. The results from the exact model solution are compared with the constant thermal conductivity case. The parameters that affect the fin performance and temperature distribution strongly are identified.     

Entropy generation in a hollow cylinder with temperature dependent thermal conductivity and internal heat generation with convective–radiative surface cooling

This article investigates entropy generation in an asymmetrically cooled hollow cylinder with temperature dependent thermal conductivity and internal heat generation. The inside surface of the cylinder is cooled by convection on its inside surface while the outside surface experiences simultaneous convective–radiative cooling. The thermal conductivity of the cylinder as well as the internal heat generation within the cylinder are linear functions of temperature, introducing two nonlinearities in the one-dimensional steady state heat conduction equation. A third nonlinearity arises due to radiative heat loss from the outside surface of the cylinder. The nonlinear system is solved analytically using the differential transformation method (DTM) to obtain the temperature distribution which is then used to compute local and total entropy generation rates in the cylinder. The accuracy of DTM is verified by comparing its predictions with the analytical solution for the case of constant thermal conductivity and constant internal heat generation. The local and total entropy generations depend on six dimensionless parameters: heat generation parameter Q, thermal conductivity parameter β, conduction–convection parameters Nc₁ and Nc₂, conduction–radiation parameter Nr, convection sink temperature δ and radiation sink temperature η.     

Heat transfer enhancement in annular fins using functionally graded material

This study presents a new approach on the heat transfer enhancement of annular fins with constant thickness using functionally graded materials. The thermal conductivity of the annular fin is assumed to be graded along the fin radius as a power‐law function. The resulting fin equation is solved by an approximate analytical method using the mean value theorem. The variable coefficients of second and third terms in the second‐order differential equation of the fin are replaced with their mean values along the fin radius. Several different graphs regarding the computed temperature profile, fin tip temperature, and fin efficiency are plotted with respect to the radii ratio thermo‐geometric parameter, and inhomogeneity parameter. It is demonstrated that the inhomogeneity parameter plays an important role on the heat transfer enhancement of the annular fin. However, for large radii ratios the effect of the inhomogeneity parameter decreases. Finally, it is stated that application of the functionally graded material in the annular fins, enhances the heat transfer rate between the fin and surrounding fluid resulting from the higher fin efficiency in comparison to the homogeneous annular fin. It is hoped that the results obtained from this study arouse interest among thermal designers and heat exchanger industries. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 42(7): 603–617, 2013; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21053     

Orthotropic thermal conductivity effect on cylindrical pin fin heat transfer

Analytical equations for temperature distribution and heat transfer rate from a cylindrical pin fin with orthotropic thermal conductivity, encountered in the use of thermally enhanced polymer composites, are derived and validated using detailed finite-element results. The thermal performance of such fins was found to depart from the classical fin solution with increasing radial conductivity-based Biot number. The in depth analysis of developed orthotropic axi-symmetric pin fin temperature and heat transfer rate equation is carried out to better understand the heat flow rate in such fins.     

A decomposition method for solving the convective longitudinal fins with variable thermal conductivity

In this paper the Adomian decomposition method is used to evaluate the efficiency and the optimal length of a convective rectangular fin with variable thermal conductivity, and to determine the temperature distribution within the fin. It is a useful and practical method, which can be used to solve the nonlinear energy balance equations which are associated with variable thermal conductivity conditions. The Adomian decomposition method provides an analytical solution in the form of an infinite power series. From a practical perspective, it is necessary to evaluate this analytical solution, and to obtain numerical values from the infinite power series. This requires series truncation, and a practical procedure to accomplish the task. Together, these transform the analytical results into a solution with a finite degree of accuracy. The accuracy of the Adomian decomposition method with a varying number of terms in the series is investigated by comparing its results with those generated by a finite-difference method which uses a Newton linearization scheme.     

Thermal performance of straight porous fin with variable thermal conductivity under magnetic field and radiation effects

The present numerical study reports the thermal performance of the straight porous fin with temperature-dependent thermal conductivity, radiation, and magnetic field effects. The heat transfer model comprising the Darcy's law for simulating flow with solid-fluid interactions in porous medium, Rosseland approximation for heat transfer through radiation, Maxwell equations for magnetic field effect and linearly varying temperature dependent thermal conductivity, results into highly nonlinear ordinary differential equation. The governing equation is solved using a finite difference scheme with suitable boundary conditions. The obtained solutions are physically interpreted by considering the impact of different nondimensional parameters on thermal performance, efficiency, and effectiveness of the system through plotted graphs. A detailed result with regard to the Nusselt number at the fin base is calculated. The results obtained are observed to be in excellent agreement with previous studies. From the study, it is observed that there is a significant effect on the thermal performance of the fin in the presence of porous constraints; also, results reveal that the nonlinear thermal conductivity parameter strengthens the thermal performance, efficiency, and effectiveness of the fin. Furthermore, the results of the study reveal that the rate of heat transfer of the fin increases with the increase in the magnetic parameter and radiation parameter.     

Effects of particle fouling and magnetic field on porous fin for improved cooling of consumer electronics

The combined effect of particulate fouling and magnetic field on the efficiency of a convective–radiative porous fin heatsink with temperature‐dependent thermal conductivity is presented. The developed thermal models are solved using differential transformation method. The effects of thermal conductivity, porosity, convection, radiation parameter, and thermal fouling number on the fin thermal efficiency are investigated. The presence of thermal fouling on the surface of the fin is shown to increase the temperature distribution. The presence of particle deposition on the fin surface significantly decreases the rate of heat transfer as additional thermal resistance of the fouling layer decreases the thermal performance of porous fin heatsink. Moreover, the fin efficiency decreases as the value of fouled Biot, Darcy, radiation number, and thermogeometric parameter increases. It is established that M_f < M_c, which indicates that the efficiency of the fouled fin is greater than the efficiency of the clean fin. Furthermore, the result of the present study is validated with the established results of Chebyshev spectral collocation method and fourth‐order Runge–Kutta with shooting method and an error margin of 0.000000023 is established.     

Hybrid differential transformation and finite difference method to annular fin with temperature-dependent thermal conductivity

A hybrid numerical technique which combines the differential transformation and finite difference method is utilized to investigate the annular fin with temperature-dependent thermal conductivity. The exposed surfaces of the fin dissipate heat to the surroundings by convection and radiation. The influences of the convective heat transfer coefficient, absorptivity, emissivity and thermal conductivity parameter on the temperature distribution are examined. The results show that the convective heat transfer plays a dominant role for heat dissipation under the convection–radiation condition. The optimum radii ratio of fin which maximizes the heat transfer rate and fin efficiency is also discussed.