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 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.     

Performance of annular fins with different profiles subject to variable heat transfer coefficient

Performance of annular fins of different profiles subject to locally variable heat transfer coefficient is investigated in this paper. The performance of the fin expressed in terms of fin efficiency as a function of the ambient and fin geometry parameters has been presented in the literature in the form of curves known as the fin-efficiency curves for different types of fins. These curves, that are essential in any heat transfer textbook, have been obtained based on constant convection heat transfer coefficient. However, for cases in which the heat transfer from the fin is dominated by natural convection, the analysis of fin performance based on locally variable heat transfer coefficient would be of primer importance. The local heat transfer coefficient as a function of the local temperature has been obtained using the available correlations of natural convection for plates. Results have been obtained and presented in a series of fin-efficiency curves for annular fins of rectangular, constant heat flow area, triangular, concave parabolic and convex parabolic profiles for a wide range of radius ratios and the dimensionless parameter m based on the locally variable heat transfer coefficient. The deviation between the fin efficiency calculated based on constant heat transfer coefficient, reported in the literature, and that presently calculated based on variable heat transfer coefficient, has been estimated and presented for all fin profiles with different radius ratios.     

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.     

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.     

Analytical heat diffusion models for heat sinks with circular micro-channels

This study explores the prediction of temperature distribution in a heat sink containing an array of circular micro-channels, which is found mostly in electronic cooling applications. The analytical heat diffusion models for most common micro-channel shapes are based on one-dimensional fin models with varying degrees of complexity. Because of a singularity in the governing one-dimensional heat diffusion equation for a fin with circular profile, no exact solution is possible for the circular heat sink geometry. In this paper, an alternative analytical power series solution technique is presented in which the differential equation is recast in polynomial form. Predictions of the power series solution are validated for different channel diameters and spacings and both one-sided and two-sided heating conditions using one-dimensional and two-dimensional numerical simulations. Overall, maximum percent differences in temperature and heat transfer rate between the analytical and two-dimensional numerical results of 0.23% and 1.33%, respectively, prove that the present analytical models are very accurate and effective tools for the design and thermal resistance prediction of micro-channel heat sinks found in electronic cooling applications.     

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.     

Stability of Boiling on Annular Fin Surfaces

A theoretical investigation was conducted to understand the stability of boiling on annular fin surfaces with different boundary conditions. Lyapunov's function was derived for boiling on annular fin surfaces; its characteristics near the steady distribution are particularly investigated. With no pre-assumed functional form for the temperature perturbation to be imposed to the boiling system, a general stability analysis was established by optimizing Lyapunov's function. The steady temperature distribution with isothermal or isoflux condition was obtained numerically, and the boiling heat transfer characteristic was noted to be dependent of the fin configuration and boundary conditions. Annular fin surfaces can significantly benefit the heat transfer, and the fins with small inner radii have better performance of base heat flux or temperature. The most unstable mode and maximum eigenvalue were employed to describe the boiling stability under various boundary conditions, and characteristics of the obtained maximum eigenvalue very well match with those of the boiling heat transfer on annular fin surfaces.     

Performance and optimization analysis of a constructal T-shaped fin subject to variable thermal conductivity and convective heat transfer coefficient

In the present study, an exercise has been devoted to establish an analytical model for thermal performance and optimization of a constructal fin subject to variable thermal conductivity of fin material and convective heat transfer coefficient over the fin surface. For the adaptation of these considerations, the governing energy equation for the stem as well as the flange becomes nonlinear. A new analytical scheme based on the Adomian decomposition method has been established for the solution process. As the present study is an analytic, it can be extended to the analysis for determining the optimum dimensions of fins satisfying either the maximization of rate of heat transfer for a given fin volume or the minimization of fin volume for a desired heat transfer rate. From the results, it can be highlighted that the present model predicts the fin performance always an under value in comparison with that the published results whereas the optimum heat transfer rate determined by using the present analysis gives an over value. The effect of different geometric and thermophysical parameters on both the fin performance and optimization has been studied. For a comparative study, the present and published results are executed for a wide range of thermogeometric parameters.     

Thermal Enhancement from Pin Fins by Using Elliptical Perforations with Different Inclination Angles

Many of the proposed methods introduce the perforated fin with the straight direction to improve the thermal performance of the heat sink. The innovative form of the perforated fin (with inclination angles) was considered. Present rectangular pin fins consist of elliptical perforations with two models and two cases. The signum function is used for modeling the opposite and the mutable approach of the heat transfer area. To find the general solution, the degenerate hypergeometric equation was used as a new derivative method and then solved by Kummer's series. Two validation methods (previous work and Ansys 16.0‐Steady State Thermal) are considered. The strong agreement of the validation results (0.31% to 0.52%) lends to the reliability of the presented model. It was found that use of the perforated fin leads to decreased thermal resistance and improvement in the thermal performance of the pin fin by enhancing the heat transfer and increasing Nusselt number. Also, the increase of the inclination angle, size, and number of perforations can be used to optimize the present model by maximizing the heat transfer area and minimizing both the weight and length of the pin fins.     

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.     

A three-dimensional inverse problem in imaging the local heat transfer coefficients for plate finned-tube heat exchangers

A three-dimensional inverse heat conduction problem in imaging the local heat transfer coefficients for plate finned-tube heat exchangers utilizing the steepest descent method and a general purpose commercial code CFX4.4 is applied successfully in the present study based on the simulated measured temperature distributions on fin surface by infrared thermography.It is assumed that no prior information is available on the functional form of the unknown local heat transfer coefficients in the present study. Thus, it can be classified as function estimation for the inverse calculations.Two different heat transfer coefficients for in-line tube arrangements with different measurement errors are to be estimated. Results show that the present algorithm can obtain the reliable estimated heat transfer coefficients.     

Two-dimensional thermal analysis of a polygonal fin with two tubes on a square pitch

The boundary element method (BEM) has been used to investigate the two-dimensional temperature distribution and the flow of heat from a polygonal fin with two tubes on a square pitch. This numerical method is shown to be convergent, stable and consistent. The resultant heat flows from the fin and the tubes are presented in the form of fin performance ratios. The values of the two-dimensional fin performance ratios are almost identical to those obtained for a single radial rectangular fin of equivalent surface area. The one-dimensional fin performance indicators, fin performance ratio or fin efficiency can be used to predict the heat flows. However, the two-dimensional temperature distributions have revealed the existence of conductive paths between the two tubes depending upon the fin dimensions, the values of the heat transfer and material thermal conductivity, and the magnitude of the temperature differences between the two tubes and the surrounding air.     

Fin Efficiency of Serrated Fins

Fin efficiency of serrated fins was analyzed and an analytical solution of the theoretical fin efficiency was derived in the form of a function of modified Bessel functions. Furthermore, an approximate equation has been given which enables one to calculate the theoretical fin efficiency with a pocket calculator with an accuracy of - 1.5%. In the analysis of the theoretical fin efficiency, however, two assumptions were employed, i.e., uniform heat transfer coefficient over the fin surface and thermal insulation at the end surface of the segmented sections. To compensate for these assumptions, a correction factor was introduced and determined experimentally. Using this correction factor, together with the theoretical fin efficiency, the actual fin efficiency can be estimated for serrated fins of various fin geometries, including plain fins.     

Prediction of heat transfer coefficient on the fin inside one-tube plate finned-tube heat exchangers

The finite difference method in conjunction with the least-squares scheme and the experimental temperature data is proposed to predict the average heat transfer coefficient and the fin efficiency on the fin inside one-tube plate finned-tube heat exchangers for various air speeds and the temperature difference between the ambient temperature and the tube temperature. Previous works showed that the heat transfer coefficient on this rectangular fin is very non-uniform. Thus the whole plate fin is divided into several sub-fin regions in order to predict the average heat transfer coefficient and the fin efficiency on the fin from the knowledge of the fin temperature recordings at several selected measurement locations. The results show that the surface heat flux and the heat transfer coefficient on the upstream region of the fin can be markedly higher than those on the downstream region. The fin temperature distributions depart from the ideal isothermal situation and the fin temperature decreases more rapidly away from the circular center, when the frontal air speed increases. The average heat transfer coefficient on the fin increases with the air speed and the temperature difference between the ambient temperature and the tube temperature. This implies that the effect of the temperature difference between the tube temperature and the ambient temperature is not negligent.     

Analysis of flux-base fins for estimation of heat transfer coefficient

Exact solutions are given for the transient temperature in flux-base fins with the method of Green’s functions (GF) in the form of infinite series for three different tip conditions. The speed of convergence is improved by replacing the steady part by a closed-form steady solution. For the insulated-tip case, a quasi-steady solution is presented. Numerical values are presented and the conditions under which the quasi-steady solution is accurate are determined. An experimental example is given for estimation of the heat transfer coefficient (HTC) on a non-rotating roller bearing, in which the outer bearing race is treated as a transient fin.     

Frost behavior on a fin considering the heat conduction of heat exchanger fins

A mathematical model is proposed for predicting frost behavior on a heat exchanger fin under frosting conditions, taking into account fin heat conduction. The change in the three-dimensional airside airflow caused by frost growth is reflected in this model. The numerical estimates of frost thickness are consistent with experimental data, with an error of less than 10%. Due to fin heat conduction, frost thickness decreases exponentially toward the fin tip, while considerable frost growth occurs near the fin base. When a constant fin surface temperature is assumed, the predicted frost thickness was larger by more than 200% at maximum, and the heat flux by more than 10% on average, compared to results obtained with fin heat conduction taken into account. Therefore, fin heat conduction could be an essential factor in accurately predicting frost behavior. To improve prediction accuracy under the assumption of constant fin surface temperature, the equivalent temperature (for predicting frost behavior) is defined to be the temperature at which the heat transfer rate neglecting fin heat conduction is the same as the heat transfer rate with fin heat conduction taken into consideration. Finally, a correlation for predicting the equivalent temperature is suggested.     

Transient heat transfer in a heat‐generating fin with radiation and convection with temperature‐dependent heat transfer coefficient

The transient heat transfer in a heat‐generating fin with simultaneous surface convection and radiation is studied numerically for a step change in base temperature. The convection heat transfer coefficient is assumed to be a power law function of the local temperature difference between the fin and its surrounding fluid. The values of the power exponent n are chosen to include simulation of natural convection (laminar and turbulent) and nucleate boiling among other convective heat transfer modes. The fin is assumed to have uniform internal heat generation. The transient response of the fin depends on the convection‐conduction parameter, radiation‐conduction parameter, heat generation parameter, power exponent, and the dimensionless sink temperature. The instantaneous heat transfer characteristics such as the base heat transfer, surface heat loss, and energy stored are reported for a range of values of these parameters. When the internal heat generation exceeds a threshold the fin acts as a heat sink instead of a heat source. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21012     

Numerical study of the fin efficiency and a modified fin efficiency formula for flat tube bank fin heat exchanger

In this paper, a three-dimensional numerical heat transfer analysis has been performed in order to obtain the temperature distribution and the fin efficiency using the experimentally determined local heat transfer coefficients from the naphthalene sublimation technique and heat and mass transfer analogy. The influences of the fin material, fin thickness, and transversal tube pitch on the fin efficiency are studied for flat tube bank fin heat exchangers. The fin efficiency, obtained by a numerical method using the averaged heat transfer coefficient, is compared with that using the local heat transfer coefficient. The reliability of the generally used formula for fin efficiency is tested also, and then a modified fin efficiency formula with a new equivalent fin height is provided. The results show that the difference between the fin efficiency obtained by the numerical method using the local heat transfer coefficient and the fin efficiency using the averaged heat transfer coefficient is small, but the fin efficiency obtained by the generally used formula is lower than that obtained by the numerical method using the local heat transfer coefficient; the fin efficiency obtained by the modified formula matches very well with the fin efficiency obtained by the numerical method using the local heat transfer coefficient. The modified formula for the fin efficiency calculation is more reliable, and it can be applied directly to the design of a flat tube bank fin heat exchanger and also will be useful in engineering applications.     

A simple design of fins for boiling heat transfer

This study presents the thermal characteristics of a fin with excavation at base when various types of boiling occur simultaneously at adjacent locations on its surface experimentally and analytically. The heat transfer coefficient of each boiling mode is taken as a power function of wall superheat. Continuity of temperature and the heat transfer rate at the intersection of the two different modes on fin surface are employed to obtain the one-dimensional temperature distribution and total heat transfer of the excavated fin. Both heating and cooling cases are investigated in the analysis. Compared with solid pin fins, the proposed fins can extend the operating condition to a higher temperature of the heat transfer surface. In addition, the experimental data compare favorably with the analytical results.