Conjugate natural convection about a vertical cylindrical fin with lateral mass flux in a saturated porous medium

The problem of conjugate natural convection about a vertical cylindrical fin with uniform lateral mass flux in a fluid-saturated porous medium has been studied numerically. Solutions based on the third level of truncation are obtained by the local nonsimilarity method. The effects of the surface mass flux, the conjugate convection-conduction parameter, and the surface curvature on fin temperature distribution, local heat transfer coefficient, local heat flux, average heat transfer coefficient, and total heat transfer rate are presented. A comparison with finite-difference solutions for the case of constant wall temperature was made, and found in a good agreement.     

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.     

CONJUGATE FREE CONVECTION FROM VERTICAL FINS EMBEDDED IN A POROUS MEDIUM

The problem of conjugate free convection in a porous medium from a vertical plate and a vertical cylindrical fin is considered. The governing equations for the convective flow in the porous medium are coupled to the governing equation for the heat flow in the fin by the conditions of continuity of the temperature and the heat flux at the interface. The heat flow in the fin is taken to be either two-dimensional or one-dimensional, in order to compare the results provided by both approaches. The nondimensional parameters that appear are the convection-conduction parameter, N cc , and the aspect ratio of the fin, u p for the plate fin and u c for the cylindrical fin. The surface curvature parameter u C appears as a parameter which is directly dependent on u c in the case of the cylindrical fin. Considering two-dimensional conduction in the solid fin is a more realistic approach from a practical standpoint, but the shortcoming of specifying zero heat flux at the tip of the fin is discussed.     

Study of heat transfer enhancement in a nanofluid-cooled miniature heat sink

This paper reports numerical solution for thermally developing temperature profile and analytical solution for fully developed velocity profile in a miniature plate fin heat sink with SiO₂–water nanofluid as coolant. The flow regime is laminar and Reynolds number varies between 0 and 800. The heat sink is modeled using porous medium approach. Modified Darcy equation for fluid flow and the two-equation model for heat transfer between the solid and fluid phases are employed to predict the local heat transfer coefficient in heat sink. Results show that the nanofluid-cooled heat sink outperforms the water-cooled one, having a considerable higher heat transfer coefficient. The effects of channel aspect ratio and porosity on heat transfer coefficient of the heat sink are studied in detail. Based on the results of our analysis, it is found that an increase in the aspect ratio or the porosity of the plate fin heat sink enhances the heat transfer coefficient.     

Thermal Resistance Minimization of a Fin-and-Porous-Medium Heat Sink with Evolutionary Algorithms

This article presents a conceptual design of a heat sink combining a porous medium whose matrix is highly conductive and a fin. A simplified model is presented to estimate the performance of the system, relying on Darcy law and local thermal equilibrium. The objective is to minimize the hot-spot temperature under global mass constraint by using an optimization procedure based on genetic algorithms. The design variables are the porosity and material of each layer of the porous medium, the fin material, height, and width, the aspect ratio of the heat sink, and the shape of a weightless upper corner deflector which reduces the width of the inlet and outlet air slots while removing the less useful mass. Results show that the optimal porous layers were generally of copper, independent of the mass constraint. However, the fin is mostly beneficial for heavier designs, while the deflector becomes more important when lightness is required. These two special features show their efficiency by allowing a mass reduction of 95% with a decrease of only 24% in the cooling performance.     

Application of Simulated Annealing for Simultaneous Estimation of Parameters in a Cylindrical Fin

An inverse problem is solved for simultaneous three parameter estimation in a large-sized cylindrical fin involving conduction-convection heat transfer. The temperature field used for the estimation of unknowns is calculated from the finite difference method (FDM). The simulated annealing (SA) is applied in this work for minimizing the required objective function. The accuracy of the estimated parameters has been studied for different measurement points, initial guess, measurement errors, and number of iterations; CPU times are also reported. The FDM-SA is observed to reconstruct the temperature field with an excellent accuracy, but ill-posed behavior is observed.     

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 buoyancy and conjugate heat transfer on non-Darcy mixed convection about a vertical slender hollow cylinder embedded in a porous medium with high porosity

This study investigates mixed convection heat transfer about a vertical slender hollow cylinder in the buoyancy and conjugate heat transfer effects in the porous medium with high porosity. The non-similar solutions using the Keller box method are obtained. The wall conduction parameter p, the porous medium parameter k₁, the Forchheimer parameter F∗ and the Richardson number are the main parameters. For various values of these parameters the local skin friction and local heat transfer parameters are determined. The validity of the methodology is checked by comparing the results with those available in the open literature and a fairly good agreement is observed. Finally, it is determined that the local skin friction and the local heat transfer coefficients increase with an increase buoyancy parameter Ri, porous medium parameter k₁, Forchheimer parameter F∗ and decrease with conjugate heat transfer parameter p.     

Conjugate mixed convection-conduction heat transfer along a vertical cylindrical fin in non-newtonian fluids

By considering the interaction between conduction within the fin and convection to the fluid surrounding the fin, an analysis is presented to study the heat transfer characteristics of laminar mixed convection of a non-Newtonian fluid flow over a vertical cylindrical fin. Due to the compatibility conditions of heat flux and temperature at the surface of fin, the boundary layer equations of the fluid are coupled with the heat conduction equation of the fin and should be solved simultaneously. Of interest are the effects of transverse curvature parameter, bouyancy parameter, power-law viscosity index, generalized Prandtl number and conjugate convection-conduction parameter on the local heat transfer coefficient, local heat flux and temperature distribution of the fin. Comparison of the calculated results with available data sets in the open literature for a Newtonian fluid shows a very good performance of the present numerical procedure.     

Fin Heat Transfer Modeling and Its Impact on Predictions of Efficiency and Condensation in Gas-Fired Boilers

In conventional and high-efficiency boilers it is important to understand where water from the products of combustion may condense onto the heat exchanger surface. The usual fin modeling approach is inadequate because it predicts no circumferential preference for condensation, whereas spatial effects have been observed. Two alternative approaches for modeling fin heat transfer are explored: one method is based on a generalization of observed trends in local convective heat transfer coefficients, and the other on a semiempirically motivated variation in convective flow temperature. Temperature distribution and fin efficiency predictions are compared to the conventional fin modeling approach. The alternative fin heat transfer models described in this study both predict more extensive condensation on the portion of the fin within the wake of the tube. Furthermore, both models predict fin efficiencies below those obtained using an assumption of constant heat transfer coefficient and convective temperature.     

Cooling enhancement in an air-cooled finned heat exchanger by thin water film evaporation

A theoretical analysis on the cooling enhancement by applying evaporative cooling to an air-cooled finned heat exchanger is presented in this work. A two-dimensional model on the heat and mass transfer in a finned channel is developed adopting a porous medium approach. Based on this model, the characteristics of the heat and mass transfer are investigated in a plate-fin heat exchanger with the interstitial surface fully covered by thin water film. Assuming that the Lewis number is unity and the water vapor saturation curve is linear, exact solutions to the energy and vapor concentration equations are obtained. The cooling effect with application of evaporative cooling was found to be improved considerably compared with that in the sensible cooler. This is because the thermal conductance between the fin and the air increases due to the latent heat transfer caused by the water evaporation from the fin surface. It is also found that the cooling enhancement depends greatly on the fin thickness. If the fin is not sufficiently thick, the cooling enhancement by the evaporative cooling decreases since the fin efficiency drops considerably due to the water evaporation from the fin surface. The fin thickness in the evaporative cooler should be increased larger than that in the sensible cooler to take full advantage of the cooling enhancement by the water evaporation.     

An experimental study in determining the local heat transfer coefficients for the plate finned-tube heat exchangers

A three-dimensional inverse problem in determining the local heat transfer coefficients for the plate finned-tube heat exchangers utilizing the steepest descent method (SDM) and a general purpose commercial code CFX4.4 is applied successfully in the present study based on the measured temperature distributions on fin surface by infrared thermography.Two different tube arrangements (i.e. in-line and staggered) with different fin pitch and air velocity are considered and the corresponding local heat transfer coefficients are to be determined. Results show that some interesting phenomena of the local heat transfer coefficients for the finned surface are found in the work and the averaged heat transfer coefficient of the staggered configuration is about 8–13% higher than that of the in-line configuration.     

Thermal analysis of natural convection and radiation in porous fins

In this study, the effects of radiation and convection heat transfer in porous media are considered. The geometry considered is that of a rectangular profile fin. The porous fin allows the flow to infiltrate through it and solid-fluid interaction takes place. This study is performed using Darcy's model to formulate heat transfer equation. To study the thermal performance, three types of cases are considered viz. long fin, finite length fin with insulated tip and finite length fin with tip exposed. The theory section addresses the derived governing equation. The effects of the porosity parameter S_h, radiation parameter G and temperature ratio C_T on the dimensionless temperature distribution and heat transfer rate are discussed. The results suggest that the radiation transfers more heat than a similar model without radiation.     

Analysis of microchannel heat sinks for electronics cooling

This paper presents an analytical and numerical study on the heat transfer characteristics of forced convection across a microchannel heat sink. Two analytical approaches are used: the porous medium model and the fin approach. In the porous medium approach, the modified Darcy equation for the fluid and the two-equation model for heat transfer between the solid and fluid phases are employed. Firstly, the effects of channel aspect ratio (α_s) and effective thermal conductivity ratio (k?) on the overall Nusselt number of the heat sink are studied in detail. The predictions from the two approaches both show that the overall Nusselt number (Nu) increases as α_s is increased and decreases with increasing k?. However, the results also reveal that there exists significant difference between the two approaches for both the temperature distributions and overall Nusselt numbers, and the discrepancy becomes larger as either α_s or k? is increased. It is suggested that this discrepancy can be attributed to the indispensable assumption of uniform fluid temperature in the direction normal to the coolant flow invoked in the fin approach. The effect of porosity (ε) on the thermal performance of the microchannel is subsequently examined. It is found that whereas the porous medium model predicts the existence of an optimal porosity for the microchannel heat sink, the fin approach predicts that the heat transfer capability of the heat sink increases monotonically with the porosity. The effect of turbulent heat transfer within the microchannel is next studied, and it is found that turbulent heat transfer results in a decreased optimal porosity in comparison with that for the laminar flow. A new concept of microchannel cooling in combination with microheat pipes is proposed, and the enhancement in heat transfer due to the heat pipes is estimated. Finally, two-dimensional numerical calculations are conducted for both constant heat flux and constant wall temperature conditions to check the accuracy of analytical solutions and to examine the effect of different boundary conditions on the overall heat transfer.     

Simulation of turbulent impinging jet into a cylindrical chamber with and without a porous layer at the bottom

Turbulent impinging jets on heated surfaces are widely used in industry to modify local heat transfer coefficients. The addition of a porous substrate covering the surface contributes to a better flow distribution, which favors many engineering applications. Motivated by this, this work shows numerical results for a turbulent impinging jet into a cylindrical enclosure with and without a porous layer at the bottom. The macroscopic time-averaged equations for mass and momentum are obtained based on a concept called double decomposition, which considers spatial deviations and temporal fluctuations of flow properties. Turbulence is handled with a macroscopic k–ε model, which uses the same set of equations for both the fluid layer and the porous matrix. The numerical technique employed is the control volume method in conjunction with a boundary-fitted coordinate system. One unique computational grid is used to compute the entire heterogeneous medium. The SIMPLE algorithm is applied to relax the system of algebraic equations. Results indicate that the permeability of the porous layer and the height of the fluid layer significantly affect the flow pattern. The effect of the porous layer thickness was less pronounced in affecting the flow behavior in the fluid layer.     

Fin performance ratios greater than unity: Not just a theoretical aspiration

Previous work using the fin performance ratio has shown that it may not exceed unity. This study proves that this is not the case under certain conditions relating to the heat transfer from the tip of the fin. Equations for longitudinal rectangular fins have been used to demonstrate how this can be achieved and a performance ratio chart is provided showing performance ratios exceeding unity when the ratio of the heat transfer coefficient at the tip to that along the length of the fin exceeds the maximum effectiveness (a parameter of the problem). Under these conditions the maximum performance ratio is given by the ratio of the heat transfer coefficients (α_e/α) divided by the maximum effectiveness. This has relevance for fins used in boiling and condensing systems where different heat transfer coefficients may exist on different parts of the fins due to the existence of different phases.     

Predictions of phase temperatures in a porous cathode of polymer electrolyte fuel cells using a two-equation model

In this study, both solid-phase and fluid-phase temperatures inside a porous cathode of a polymer electrolyte fuel cell in contact with an interdigitated gas distributor are predicted numerically. The porous cathode consists of a catalyst layer and a diffusion layer. The heat transfer in the catalyst layer is coupled with species transports via a macroscopic electrochemical model. On the other hand, in the diffusion layer, the energy equations based on the local thermal non-equilibrium (LTNE) are derived to resolve the temperature difference between the solid phase and the fluid phase. As for the species transports, the Bruggemann model is employed to describe the effective diffusivities of the oxygen and water vapor in the porous cathode. Results show that the wall temperature decreases with increasing the intrinsic heat transfer coefficient. As the intrinsic heat transfer coefficients increase, the porous electrode becomes local thermal equilibrium with a strong thermal interaction (heat transfer) between the solid and fluid phases. Under the conditions of high intrinsic heat transfer coefficients, the temperature difference between the solid matrices and the reactant fluids are negligible.     

Numerical prediction of the instantaneous regenerator and in-cylinder heat transfer of a stirling engine

The paper is concerned with the study of the effects of the in-cylinder and regenerator heat transfer characteristics of a single-acting opposed-piston Stirling engine, with heater and cooler both omitted, for which a simulation model has been developed. The engine thermodynamic cycle is divided into a number of time-steps, and a system of nonlinear ordinary differential equations, which describe the energy balances over the three basic control volumes (hot and cold cylinders and regenerator), is solved numerically. Empirical correlations are used to determine the instantaneous heat transfer coefficients in the regenerator (flow across a porous medium) and inside the cylinder space (gas confined in a cylindrical volume with a moving boundary). Numerical results from the model are presented.     

A boundary layer analysis of heat transfer by free convection from permeable horizontal cylinders of elliptic cross-section in porous media using a thermal non-equilibrium model

This work uses a thermal non-equilibrium model to study the free convection boundary layer flow driven by temperature gradients near a permeable horizontal cylinder of elliptic cross-section with constant wall temperature in a fluid-saturated porous medium. A coordinate transformation is used to obtain the nonsimilar boundary layer equations. The transformed boundary layer equations are then solved by the cubic spline collocation method. Results for the local Nusselt numbers are presented as functions of the porosity scaled thermal conductivity ratio, the heat transfer coefficient between solid and fluid phases, the transpiration parameter, and the aspect ratio when the major axis of the elliptical cylinder is vertical (slender orientation) and horizontal (blunt orientation). An increase in the porosity scaled thermal conductivity ratio or the heat transfer coefficient between the solid and fluid phases increases the heat transfer rates. Moreover, the use of suction (positive transpiration parameter) tends to increase the heat transfer rates between the porous medium and the surface.     

Unified streamline, heatline and massline methods for the visualization of two-dimensional heat and mass transfer in anisotropic media

Many of the actual materials are anisotropic, ranging from natural products to the most sophisticated composite materials. Special emphasis needs to be devoted to the heat and mass transfer calculations in anisotropic media, and to the development of visualization tools for the transport phenomena occurring in such media, similarly to what happens with isotropic media. The most adequate tools for visualization purposes are the streamlines, the heatlines and the masslines, when dealing with two-dimensional steady problems without source terms. Moreover, further attention needs to be devoted to the diffusion coefficients for the streamfunction, heatfunction and massfunction, whose contour plots are used for visualization purposes. This is specially important for domains of marked anisotropy or for domains involving media of different properties, or even conjugate diffusion/convection heat and mass transfer problems. Once defined the proper diffusion coefficients, it is proposed a unified physical treatment, as well as a unified treatment to evaluate the function’s fields by using the same numerical procedures and code routines as for the primitive conserved variables. The unified approach is illustrated through pure conduction heat transfer problems, natural convection heat transfer in a porous enclosure, and conjugate conduction-convection heat transfer.