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.     

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.     

Analytical Solution and Numerical Simulation for One‐Dimensional Steady Nonlinear Heat Conduction in a Longitudinal Radial Fin with Various Profiles

In this article we consider a model describing the temperature profile in a longitudinal fin with rectangular, concave, triangular, and convex parabolic profiles. Both thermal conductivity and the heat transfer coefficient are assumed to be temperature‐dependent, and given by a linear function and by power laws, respectively. In addition, the effects of the thermal conductivity gradient have been investigated. Optimal homotopy analysis method (OHAM) is employed to analyze the problem. The effects of the physical applicable parameters such as thermo‐geometric fin, thermal conductivity, and heat transfer mode are analyzed. The OHAM solutions are obtained and validity of obtained solutions is verified by the Runge–Kutta fourth‐order method and numerical simulation. A very good agreement is found between analytical and numerical results. Also for investigation of lateral effects on the accuracy of results, numerical simulation (by Ansis software) is compared with the homotopy analysis method (HAM) and numerical solution (by Runge–Kutta) of the energy balance equation. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21104     

Fin efficiency of serrated fins: Part 1. Analysis of theoretical fin efficiency and experimental results

The fin efficiency of serrated fins was analyzed, and an analytical solution was derived as a function of modified Bessel functions. Two assumptions, i.e., thermal insulation at the end surfaces of segmented sections, and a uniform heat transfer coefficient over the fin surface, were employed in this analysis. To determine the effect of these assumptions, a heat transfer experiment was performed. From a comparison of the experimentally evaluated fin efficiencies with the analytical solution, a correction factor was obtained for a typical serrated fin configuration. © 1999 Scripta Technica, Heat Trans Asian Res, 28(6): 528–540, 1999     

Heat Transfer Characteristics of Grooved Fin Under Forced Convection

Material removal from an extended surface in the form of perforations and slots is a proven technique to augment heat transfer rates with a considerable reduction in the surface weight. This work presents the outcomes of experimental investigation on heat transfer characteristics of a plate fin having grooves of various configurations on two broad faces. The experimental data pertaining to heat transfer have been collected by varying Reynolds number from 1500 to 5000, for transverse grooved, inclined grooved, V-grooved, and multi-V-grooved fin. The results of the grooved fin are compared with that of a smooth conventional fin to gauge the heat transfer performance of modified fin. The maximum enhancement in Nusselt number corresponds to the inclined groove fin, whereas the highest value of grooved fin effectiveness is obtained for the multi-V-grooved fin. The Nusselt number correlations are presented for different fin configurations tested in this work.     

一维最优直肋的肋效率分析

为研究等截面一维直翅优化问题,从Schmidt最优翅片形状假定出发,通过理论分析得到一种优化的直翅周线表达式,并找出了优化后的翅片散热效率与其几何外形之间的关系。在优化后的直翅精确解基础上,针对实际工程设计中,在散热效率与精确解偏差不大于0．12％的情况下,进一步推导出翅片所能采用的近似周线方程,为等截面直翅的工程优化设计和应用提供了依据。     

Heat transfer from a horizontal fin array by natural convection and radiation—A conjugate analysis

The problem of natural convection heat transfer from a horizontal fin array is theoretically formulated by treating the adjacent internal fins as two-fin enclosures. A conjugate analysis is carried out in which the mass, momentum and energy balance equations for the fluid in the two-fin enclosure are solved together with the heat conduction equations in both the fins. The numerical solutions by using alternating direction implicit (ADI) method yield steady state temperature and velocity fields in the fluid, and temperatures along the fins. Each end fin of the array is exposed to limited enclosure on one side and to infinite fluid medium on the other side. Hence a separate analysis is carried out for the problem of end fin exposed to infinite fluid medium with appropriate boundary conditions. From the numerical results, the heat fluxes from the fins and the base of the two-fin enclosure, and the heat flux from the end fin are calculated. Making use of the heat fluxes the total heat transfer rate and average heat transfer coefficient for a fin array are estimated. Heat transfer by radiation is also considered in the analysis. The results obtained for a four-fin array are compared with the experimental data available in literature, which show good agreement. Numerical results are obtained to study the effectiveness for different values of fin heights, emissivities, number of fins in a fixed base, fin base temperature and fin spacing. The numerical results are subjected to non-linear regression and equations are obtained for heat fluxes from the two-fin enclosure and single fin as functions of Rayleigh number, aspect ratio and fin emissivity. Also regression equations are obtained to readily calculate the average Nusselt number, heat transfer rate and effectiveness for a fin array.     

A model on the basis of analytics for computing maximum heat transfer in porous fins

This study presents an analytical work on the performance and optimum design analysis of porous fin of various profiles operating in convection environment. Straight fins of four different profiles, namely, rectangular, convex parabolic and two exponential types are considered for the present investigation. An analytical technique based on the Adomian decomposition method is proposed for the solution methodology as the governing energy equations of porous fins for all the profiles are non-linear. A comparative study has been carried out among the results obtained from the porous and solid fins, and an appreciable difference has been noticed for a range of design conditions. Finally, the result shows that the heat transfer rate in an exponential profile with negative power factor is much higher than the rectangular profile but slightly higher than the convex profile. On the other hand, the fin performance is observed to be better for exponential profiles with positive power factor than other three profiles. A significant increase in heat transfer through porous fins occurs for any geometric fin compared to that of solid fins for a low porosity and high flow parameter.     

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.     

Establishment of Modified-One-Dimensional and Two-Dimensional Models for Two-Directional Heat Conduction in a Wet Fin Assembly

This paper deals with an analysis of two-dimensional temperature distribution within a rectangular fin assembly under wet conditions and subject to convective condition at the primary inner wall. An analytical method based upon the separation of variables was suggested to determine the two-dimensional temperature field. A modified one-dimensional model was used to more closely approximate the results of the two-dimensional model; a one-dimensional classical model was also elaborated to provide a basis for comparison, allowing the two-dimensional effect to be established in the case of wet fins. Next, wet fin efficiency was determined. To establish the merit of the present work in considering the two-dimensional effect in wet fins, the results of the two-dimensional model were compared with those obtained from the one-dimensional classical models, demonstrating a considerable difference in their results for various design and psychrometric conditions. The wet-fin analysis presented herein is equally suitable for dry-surface fins by accounting for the absence of latent heat transfer.     

Optimum geometry of fixed volume plate fin for maximizing heat transfer

A method is presented for finding the plate fin geometry for maximizing total heat transfer when cooled by forced or natural convection. The method is based on an approximate treatment of conjugated heat transfer in which analytical results are utilized. As a result of this type of approach, non-dimensional variables have been found that contain the geometrical and thermal properties of a fin and the flow. An essential fact is that there is no need to evaluate convection heat transfer coefficients. Only one variable with a fixed value is needed to determine the geometry of a fixed volume fin that gives the maximum heat transfer.     

Optimum design of a longitudinal fin array with convection and radiation heat transfer using a genetic algorithm

In the present work, the optimization of a longitudinal fin array is investigated. Heat is transferred by conduction along the fins and dissipated from the fin surface via natural convection to the ambient and radiation to other fin surfaces and surrounding. The aim of the optimization is to find the optimum geometry and the number of fins in such a way that the rate of heat transfer from the array is maximized. A modified genetic algorithm is used to maximize the objective function which is defined as the net heat rate from the fin surface for a given length. The fin profile is represented by B-spline curves, where the shape of fin is determined by the positions of a set of control points. The effects of the base temperature, the fin length and the height of array on the optimum geometry and on the number of fins are investigated by comparing the results obtained for several test cases. In addition, the contributions of convective heat transfer and radiative heat transfer in net heat transfer are studied for these cases. The enhancement of heat transfer due to the optimum fin geometry is examined by comparing the results obtained for the optimum fin profile with those with conventional profiles.     

Numerical study of flow patterns of compact plate-fin heat exchangers and generation of design data for offset and wavy fins

Thermo-hydraulic design of compact heat exchangers (CHEs) is strongly dependent upon the predicted/measured dimensionless performance (Colburn factor j and Fanning friction factor f vs. Reynolds number Re) of heat transfer surfaces. Also, air (gas) flow maldistribution in the headers, caused by the orientation of inlet and outlet nozzles in the heat exchanger, affects the exchanger performance. Three typical compact plate-fin heat exchangers have been analyzed using Fluent software for quantification of flow maldistribution effects with ideal and real cases. The headers have modified by providing suitable baffle plates for improvement in flow distribution. Three offset strip fin and 16 wavy fin geometries used in the compact plate-fin heat exchangers have also been analyzed numerically. The j and f vs. Re design data are generated using CFD analysis only for turbulent flow region. For the validation of the numerical analysis conducted in the present study, a rectangular fin geometry having same dimensions as that of the wavy fin has been analyzed. The results of the wavy fin have been compared with the analytical results of a rectangular fin and found good agreement. Similarly, the numerical results of offset strip fin are compared with the correlations available in the open literature and found good agreement with most of the earlier findings.     

Efficiency of the Horizontal Single Pin Fin Subjected to Free Convection and Radiation Heat Transfer

In this communication, the results of numerical calculations of the heat transfer coefficient and temperature distribution along the horizontal single pin fin in still ambient air, as well as the fin efficiency, are presented and compared with classical analytical results in the case of the constant heat transfer coefficient fin theory. The measured temperature distributions along the two low carbon steel pin fins having a length-to-diameter ratio of 35—one covered with the polished nickel and the other painted mat black—agree very well with the numerical results and are higher than the classical results. The analytically calculated fin efficiency does not differ significantly from the results of the numerical calculations if they are compared for the same dimensionless fin parameter in which the heat transfer coefficient is determined for the fin base temperature. More extended numerical calculations showed that beyond the fin parameter of five, the analytical results of the fin efficiency are higher than the numerical results by no more than about 1%. The largest difference between the classical and numerical fin base efficiencies is about 3.5%, and it was observed at a fin parameter of about 1, where the length of the pin fin has the optimal value based on the classical theory.     

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     

Efficiency and Optimization of a Straight Rectangular Fin with Combined Heat and Mass Transfer

An analysis was carried out to study the efficiency of a straight rectangular fin with a uniform cross-section area when subjected to simultaneous heat and mass transfer mechanisms. The temperature and humidity ratio differences are the driving forces for the heat and mass transfer, respectively. Numerical solutions are obtained for the temperature distribution over the fin surface when the fin surface is dry, fully wet, and partially wet. The psychrometric correlation of an air-water vapor mixture was used to simulate the relation between the temperature and humidity ratio instead of the linear approximate correlations used in the literature. The effect of atmospheric pressure on the fin efficiency was also studied, in addition to fin optimum thickness for specific operating conditions. The numerical solution was compared with those of previous studies in order to find if the linear model in the published analytical results are near to the real situation. It is found that the linear model for the relation between the humidity ratio and the temperature used by Wu and Bong is a reasonable engineering approximation for small values of the fin parameter and at low relative humidities.     

Optimization of heat exchangers with vortex-generator fin by Taguchi method

A comparative study of effects of attack angle, length of vortex generator, height of vortex generator, fin material, fin thickness, fin pitch and tube pitch on fin performance of vortex-generator fin-and-tube heat exchanger is conducted by numerical method. The parameters of vortex-generator fin-and-tube heat exchangers are optimized by the Taguchi method. Eighteen kinds of models are made by compounding levels on each factor, and the heat transfer and flow characteristics of each model are analyzed. The results show that these six factors (fin pitch, longitudinal tube pitch, transverse tube pitch, length of vortex generator, height of vortex generator, and attack angle of vortex generator) have great influences on the JF-factor. The fin material and fin thickness have trifling effects on the JF-factor. The two optimal conditions (A1B3C3D2E1F2G1H3 and A2B2C2D3E1F2G1H3) are acquired, and the reproducibility of the results is verified by two analytical results.     

Analytical solution of forced convective heat transfer in tubes partially filled with metallic foam using the two-equation model

In this study, an analytical solution for fully developed forced convection in a tube partially filled with open-celled metallic foams is presented. In the foam region, the Brinkman flow model is used to describe the fluid transport, and the local thermal non-equilibrium model is adopted to represent the fluid–solid energy exchange. At the foam–fluid interface, interfacial coupling conditions for temperature are proposed and used to derive the analytical solution. Velocity and temperature profiles are derived from this solution, and explicit expressions for the friction factor and the Nusselt number are obtained. A parametric study is conducted to study the influences of various factors on flow resistance and heat transfer performance. The present analytical solution establishes a benchmark for similar work hereafter.     

Performance of a Circular Fin in a Cylindrical Porous Enclosure

The problem of natural convection heat transfer from a vertical cylindrical fin to a saturated porous medium in a cylindrical enclosure is solved numerically. A conjugate conduction-convection analysis is accomplished by solving the equation of heat conduction in the fin together with the mass, momentum and energy balance equations in the porous medium. Numerical results are obtained by using ADI method. The effects of the conduction-convection ratio parameter, aspect ratio and Darcy number on local heat transfer coefficients and fin efficiency are discussed. Comparison of the local heat transfer coefficients and fin efficiency is shown with those for non-porous medium.     

Heat transfer from a vertical fin array by laminar natural convection and radiation—A quasi‐3D approach

A quasi‐3D numerical model is developed to study the problem of laminar natural convection and radiation heat transfer from a vertical fin array. An enclosure is formed by two adjacent vertical fins and vertical base in the fin array. Results obtained from this enclosure are used to predict heat transfer rate from a vertical fin array. All the governing equations related to fluid in the enclosure, together with the heat conduction equation in both fins are solved by using the Alternating Direction Implicit (ADI) method for getting the temperatures along the height of the fin and the temperature of the fluid in the enclosure. Separate analysis is carried out to calculate the heat transfer rates from the end fins in the fin array. A numerical study has been carried out for the effect of fin height, fin spacing, fin array base temperature, and fin emissivity on total heat transfer rates and effectiveness of the fin array. The numerical results obtained for an eight‐fin array show good agreement with the available experimental data. Results show that the fin spacing is the most significant parameter and there exists an optimum value for the fin spacing for which the heat transfer rate from the fin array is maximum. Correlations are presented for predicting the total heat transfer rate, average Nusselt number, and effectiveness of the fin array. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20360