An analytical prediction for performance and optimization of an annular fin assembly of trapezoidal profile under dehumidifying conditions

The present study demonstrates an analysis analytically to determine the performances and optimum design of wet annular fin assemblies of the trapezoidal profile. The Frobenius power series method is adopted as an analytical tool to solve the governing differential equation of the above type of wet fin assemblies. The performance parameters, namely, the surface efficiency and augmentation factor are determined. The present model has also ability to predict the performances of a wet fin assembly of triangular fin geometry for the selection of the very small value of the tip thickness. As the present study establishes an analytical model, it can be easily employed in determination of an optimum design condition. Both the performance and optimization study have been made by two approaches of the analysis described based on the handling of the psychrometric properties at the tip as a function of temperatures. Finally, it can be highlighted that the performances and optimum conditions of a wet fin assembly are not only dependent upon the psychrometric properties of air but also dependent upon the approach selected for calculating the energy transferred by the mechanism of mass transfer.     

Approximate analytic solution for performances of wet fins with a polynomial relationship between humidity ratio and temperature

It is well known from the psychrometric properties of humid air that a linear relationship between specific humidity and dry bulb temperature (DBT) never exists. However, to establish analytical models of the performance and optimization analysis of wet fins, a linear relationship instead of psychrometric variation has been adopted by many researchers. For betterment of results in comparison with the existing value, the present study establishes a new approximate analytical model with the selection of the cubic polynomial relationship between specific humidity and dry bulb temperature. In view of this, thermal analysis of wet fins of straight profile has been addressed. The temperature profile and performance parameters of fully wet fins have been evaluated by Adomian decomposition method (ADM). A new scheme is furnished to carryout the performance analysis of partially wet fins using ADM. A notable difference in results is found while the present result has been compared with that values obtained from the published linear models.     

Performance and optimum design analysis of longitudinal and pin fins with simultaneous heat and mass transfer: Unified and comparative investigations

In the present paper, the thermal analysis and optimization of longitudinal and pin fins of uniform thickness subject to fully wet, partially wet and fully dry surface conditions are carried out analytically, and also a comparative study is made between the longitudinal and pin fin for a wide range of design parameters. From the results, a significant effect on the temperature distribution in the fin and the fin efficiency with the variations in moist air psychometric conditions is noticed. For partially wet fins, the length of the wet–dry interface depends on the relative humidity RH, fin parameter Z_d and geometry of the fin. From the results, it can also be highlighted that for the same thermo-geometric and psychometric parameters, a longitudinal fin gives higher efficiency than the corresponding pin fin irrespective of surface conditions. Next, a generalized scheme for optimization has been demonstrated in such a way that either heat transfer duty or fin volume can be taken as a constraint selected as per design requirement. From the optimization results, it can be pointed out that the optimum design of both the longitudinal and pin fin under the partially wet surface condition is only possible for a narrow range of relative humidity whereas for the fully wet surface, a wide range is noticed. Finally, design curves have been established for a wide range of thermo-psychometric parameters, which may be helpful to a designer for estimation of the optimum design variables of a fin with a minimum effort.     

Performance and optimization analysis of SRC profile fins subject to simultaneous heat and mass transfer

Fin material near the tip of a uniform cross sectional (UC) fin does not participate actively in transferring heat. This effect may seem to have progressed much with the increase in fin length. A uniform cross sectional fin with a step reduction in local cross section (SRC) not only increases the effective utilization of fin material near the tip but also it promotes the ease of fabrication. In this study, an effort has been devoted to determine analytically the overall fin performance of both longitudinal and pin fins of SRC profile under fully dry, partially wet and fully wet conditions. The effect of various design and psychometric parameters on the fin performance of SRC fins has been investigated and compared it is with the corresponding UC fin. A scheme for optimizing SRC fins has also been demonstrated in the present work. From the result, it can be highlighted that the optimum values of Biot number and aspect ratio of SRC fins increase with the increase in relative humidity for the same fin volume. In comparison with the UC fin for the identical fin volume, the SRC fin transfers more rate of heat and consequently, this difference in heat transfer rate increases slowly with the relative humidity.     

Optimal Design of Constructal T-Shaped Fin Under Partially Wet Condition for Enhanced Thermal Performance: An Analytical Approach

Abstract

In refrigeration and air conditioning systems, the fin surface may become dry, fully wet or partially wet depending upon the psychometric, thermo-physical and geometric parameters. The current work is intended to determine the length of the dry portion, temperature distribution, performance and optimum design parameters of a T-shaped fin under partially wet condition. Temperature dependent thermal conductivity of the fin material is taken and the humidity ratio of saturated air is considered to vary linearly with the corresponding fin surface temperature. Considering these the governing equation of T-shaped fin becomes nonlinear and it has been solved analytically by using Homotopy Perturbation Method. The point of separation between the dry and wet portions of a T-shaped fin may be located either in the stem or flange part and hence, two cases having different governing equations and boundary conditions are analyzed in the current study. For the optimization study, Lagrange multiplier technique is employed and the results are obtained by maximizing the heat transfer rate for a constant fin volume. Further, a comparative study is presented between insulated and convective fin tip conditions.     

Performance analysis and optimization of straight taper fins with variable heat transfer coefficient

In the present paper, the thermal analysis and optimization of straight taper fins has been addressed. With the help of the Frobenius expanding series the temperature profiles of longitudinal fin, spine and annular fin have been determined analytically through a unified approach. Simplifying assumptions like length of arc idealization and insulated fin tip condition have been relaxed and a linear variation of the convective heat transfer coefficient along the fin surface has been taken into account. The thermal performance of all the three types of fin has been studied over a wide range of thermo-geometric parameters. It has been observed that the variable heat transfer coefficient has a strong influence over the fin efficiency. Finally, a generalized methodology has been pointed out for the optimum design of straight taper fins. A graphical representation of optimal fin parameters as a function of heat duty has also been provided.     

Optimization of a Thermally Asymmetric Convective and Radiating Annular Fin

In this study, convective and radiating annular fins of rectangular profile under thermally asymmetric conditions are examined using an analytical method. For the fin base condition, it is assumed that heat transfer from the fluid to the inside surface of the pipe is equal to the heat transfer through the fin base. The temperature distribution along the fin height at the fin tip is presented to demonstrate the effects of the thermally asymmetric condition. The heat loss and fin tip radius for fixed fin height are optimized as a function of the fin top convection characteristic number. Also, for fixed fin volume, the heat loss and fin dimensions are optimized based on the top, bottom, and tip convection characteristic numbers, radiation characteristic numbers, fin base radius, and fin volume. The fin effectiveness as a function of the top convection characteristic number and annular fin length are also presented.     

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.     

Performance analysis and optimization of elliptic fins circumscribing a circular tube

The performance of elliptic disc fins has been analyzed using a semi-analytical technique. It has been shown that the efficiency of such fins can also be predicted very closely using the sector method. However, the equivalent annulus method is not suitable for this fin geometry. A method for the optimum design of fins, using a constraint of either fin volume or rate of heat dissipation has also been suggested. Optimum elliptical fins dissipate heat at a higher rate compared to an annular fin when space restriction exists on both sides of the fin. Even when the restriction is on one side only, the performance of elliptical fin is comparable to that of eccentric annular fin for a wide parametric range.     

Performance and optimum geometry of spines with simultaneous heat and mass transfer

An analysis was carried out to study the performance of spine fins of different configurations 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. Analytical solutions are obtained for the efficiency and temperature distribution over the spine surface when the surface condition is fully wet. A correction chart is developed to correct the value of the dry fin parameter if the fin surface condition is fully wet. The effect of atmospheric pressure on the spine efficiency was also studied as well as the spine optimum geometries were obtained such that a maximum amount of heat transfer rate occurs. It is shown that the closed-form solution for a dry spine case discussed in text books is a special case for the solutions presented in this paper.     

Thermal Analysis of Orthotropic Pin Fins With Contact Resistance: A Closed-Form Analytical Solution

Light weight composite fins are considered to deal with thermal management problems for many microelectronic components. These composite fins are inherently anisotropic, therefore cannot be handled by a traditional one-dimensional approach; however, these materials can be designed to provide high thermal conductivity values in the desired direction to handle application-specific demands. In this article, we present analytical solutions for temperature distribution and heat transfer rate for orthotropic two-dimensional pin fins subject to convective-tip boundary condition and the contact resistance at the fin base. The generalized results are presented in terms of fin aspect ratio (fin length-to-radius ratio) and three dimensionless fin parameters that relate the internal conductive resistance to three convective resistances discussed in terms of dimensionless variables such as contact, tip, and axial Biot numbers, in addition to the axial-to-radial conductivity ratio. Several special cases including the insulated tip boundary condition are presented. It is demonstrated that the temperature distribution and heat transfer rate from the two-dimensional isotropic annular fin introduced earlier in the literature, can easily be recovered from the benchmark solutions presented in this article. Furthermore, dimensionless heat transfer rates are presented for the pin fins with contact resistance that can help to solve design and optimization problems of many natural-to-forced convection composite fins that are typically encountered in many microelectronic applications.     

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     

Some Observations of the Frost Formation in Fin Arrays

An experimental study concerning the frost formation and the airside performance of a fin array is reported. Visual observation indicates that the frost thickness at the fin base is thicker than that at the tip. During the period of frost formation along the fin array, one can see an apparent boundary separating a region of very thin frost thickness (the region does not show appreciable frost formation) and a region with appreciable frost thickness. Furthermore, the frost-separating boundary is not horizontal but inclined toward the airflow direction. In the entrance region, the frost formation is apparently less pronounced than that at the rear region. If part of the fin surface is above the freezing point temperature but is still below the dew point temperature, the moist air condenses on the upper part of the fin while frost is formed nearby the fin base. The water condensate on the upper part of the fin array may fall off to wash away the porous structure to become a dot-like pattern. With a fixed frontal velocity, the heat transfer coefficient for a fin spacing of 3 mm is slightly increased with the elapse of time and is higher than that of 8 mm fin spacing. The frost thickness increases with time but decreases with the frontal velocity.     

Optimum Dimensions of Longitudinal Rectangular Fins and Cylindrical Pin Fins with a Prescribed Tip Temperature

This paper deals with constant-area, longitudinal fins with rectangular and circular cross-sections for the condition of prescribed tip temperature. For these fin geometries, the optimum aspect ratios for maximum heat transfer per unit volume have been determined analytically in the literature for the cases of adiabatic tip and convective tip using a one-dimensional analysis. For the case of prescribed tip temperature, a numerical two-dimensional analysis was reported for rectangular fins. In the present study, the temperature distribution and rates of heat transfer at the base and tip of both fin cross-sections were determined using one– and two-dimensional analyses. The ranges of independent parameters within which the one-dimensional solution was within 1% from the two-dimensional solution were determined, and simple analytical formulae for the optimum aspect ratios were derived based on the one-dimensional solution. An example is given showing the application of the results in design.     

A Simple Air-Side Data Analysis Method for Partially Wet Flat-Tube Heat Exchangers

An air-side data analysis method is developed for flat-tube heat exchangers under partially wet conditions. In order to simplify the combined sensible and latent heat transfer, it is assumed that condensate drainage paths develop such that, at steady state, water does not spread to noncondensing surfaces, which therefore remain dry. The air dew point is compared to local fin-tip and fin-base temperatures, and a partially wet flat-tube heat exchanger is partitioned into fully wet, partially wet, and dry-fin regions, which are subsequently analyzed as separate heat exchangers. Using an enthalpy-based effectiveness–NTU (number of transfer units) method, average fin efficiency is calculated for each region, and the locations of region boundaries are determined iteratively. The proposed data analysis method is demonstrated with experimental data for a flat-tube louver-fin heat exchanger under various latent loads. The general approach presented can be extended to other heat exchanger geometries.     

B-spline curve based optimum profile of annular fins using multiobjective genetic algorithm

Thermal stress in annular fins, which influences the life of a fin, still needs its detail analysis. Heat transfer in annular fins is studied here as a multiobjective optimization problem. Representing fin profiles by B-spline curves, fin geometries are obtained primarily by maximizing heat transfer rate and minimizing thermal stress. Fin performance is further assessed by minimizing fin volume and maximizing fin efficiency and effectiveness. Evaluating temperature and thermal stress by hybrid spline difference method, non-dominated sorting genetic algorithm II is applied to approximate the Pareto-optimal front. The proposed procedure would be helpful for designers to adopt suitable fin configurations from the Pareto front.     

Prediction of safe tip temperatures in uniform annular fins for the design of thermal exchange equipment via symbolic mathematics

The primary specification when designing annular fins of uniform profile is undoubtedly the total amount of heat transferred to the surrounding fluid. These calculations are normally done with the help of a thermal efficiency diagram. This paper shows another facet in the design of uniform annular fins related to the prediction of safe tip temperatures. In this regard, a correlation equation for the tip temperatures of these annular fins as a function of the thermogeometric parameter and the radii ratio has been obtained. This route provides a dependable alternative to evaluating the original analytical solution at the fin extreme involving Bessel functions. Thermal design engineers can use the correlation equation in conjunction with the thermal efficiency diagram.     

Simple correlation equations for optimum design of annular fins with uniform thickness

Simple correlation equations for optimum design of annular fins with uniform cross section are obtained in the present work. The fin volume is fixed to obtain the dimensionless geometrical parameters of the fin with maximum heat transfer rates. The optimum radii ratio of an annular fin which maximizes the heat transfer rate has been found as a function of Biot number and the fin volume. The data from the present solutions is correlated for a suitable range of Biot number and the fin volume. The simple correlation equations presented in this work can assist for thermal design engineers for optimum design of annular fins of uniform thickness.     

Correlations for wet surface ratio of fin-and-tube heat exchangers

Correlations are proposed for the wet surface ratio of a fin-and-tube heat exchanger with plain and wavy fin geometry under dehumidifying conditions. The ‘Finite Circular Fin Method’ (FCFM) is used for data reduction. It is found that the percentage of wet surface area increases with increasing fin spacing or number of tube rows and decreasing Reynolds number. Despite the addition of tube rows or reduced fin spacing the effective surface area is increased, and its influence on a partially wet surface is apparently the opposite. This is because adding tube rows will provide a more effective temperature drop in air flow than adding fins, Moreover, the heat and mass transfer characteristics of j_h and j_m increase with an increase in the area of wet surface. Correlations for prediction of the percentage of wet surface area are proposed. These correlations can describe 83.81% of the area of wet surface to within ±10%.     

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

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