Convective‐radiative fins with simultaneous variation of thermal conductivity,heat transfer coefficient,and surface emissivity with temperature

This paper is a numerical study of thermal performance of a convective‐radiative fin with simultaneous variation of thermal conductivity, heat transfer coefficient, and surface emissivity with temperature. The convective heat transfer is assumed to be a power function of the local temperature between the fin and the ambient which allows simulation of different convection mechanisms such as natural convection (laminar and turbulent), boiling, etc. The thermal conductivity and the surface emissivity are treated as linear functions of the local temperature between the fin and the ambient which provide a satisfactory representation of the thermal property variations of most fin materials. The thermal performance is governed by seven parameters, namely, convection–conduction parameter Nc, radiation–conduction parameter Nr, thermal conductivity parameter A, emissivity parameter B, the exponent n associated with convective heat transfer coefficient, and the two temperature ratios, θ_a and θ_s, that characterize the temperatures of convection and radiation sinks. The effect of these parameters on the temperature distribution and fin heat transfer rate are illustrated and the results interpreted in physical terms. Compared with the constant properties model, the fin heat transfer rate can be underestimated or overestimated considerably depending on the values of the governing parameters. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20408     

Application of the Haar wavelet method on a continuously moving convective‐radiative fin with variable thermal conductivity

In this paper, we numerically investigate the heat transfer in a continuously moving convective‐radiative fin with variable thermal conductivity by using Haar wavelets. Heat is dissipated to the environment simultaneously through convection and radiation. The effect of various significant parameters—in particular the thermal conductivity parameter a, convection‐sink temperature θ_a, radiation‐sink temperature θ_s, convection‐radiation parameter N_c, radiation‐conduction parameter N_r, and Peclet number Pe—on the temperature profile of the fin are discussed and interpreted physically through illustrative graphs. Computational results obtained by the present method are in good agreement with the standard numerical solutions. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21038     

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     

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.     

Exact,analytic temperature distributions of pin fins with constant thermal conductivity and power‐law type heat transfer coefficient

In this Technical Note, the problem of determining the temperature distribution in a pin fin with power‐law heat transfer coefficients is addressed. It is demonstrated that the governing fin equation, a nonlinear second‐order differential equation, is exactly solvable for the entire range of the exponent n in the power‐law heat transfer coefficients. The exact, closed‐form analytical solutions in implicit form are convenient for physical interpretation and optimization for maximum heat transfer. Furthermore, it is proved that the exact solutions have three different structures: (1) dual in the range of , (2) unique or dual in the range of , and (3) unique in the range of . Additionally, exact analytical expressions for the fin efficiency and the fin effectiveness are provided, both as a function of the dimensionless fin parameter for the gamma of n under study.     

Numerical study of radiation‐convection heat transfer

In this study, the authors attempted to introduce a simulation technique for radiation‐convection heat transfer in the high‐temperature fields of industrial furnaces, boilers, and gas turbine combustors. The convection effect was analyzed by a differential equation, but the radiation effect was analyzed by an integral equation. Thus, it was not easy to arrange both effects using the same type of equations. Then, the authors introduced the zone method and Monte Carlo method for the integral equation of the radiation effect and the finite difference method for the differential equation of the convection effect. A three‐dimensional analysis of the high‐temperature furnace was performed by this simulation technique to obtain its temperature distribution. Furthermore, another radiation‐convection heat transfer analysis in the low‐temperature living room was performed by the same technique. Finally, the authors tried to develop a computer software for radiation‐convection heat transfer and described their idea of software construction for the above. © 2002 Wiley Periodicals, Inc. Heat Trans Asian Res, 31(5): 391–407, 2002; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10042     

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     

Numerical simulation of coupled radiation‐convection heat transfer of high‐temperature developing flow in tube

By combining the discrete ordinate method with the control volume method, the coupled radiation‐convection heat transfer of high‐temperature developing laminar flow in a tube is investigated numerically. The radiative transfer is solved by the discrete ordinate method and its contribution to thermal balance is dealt with as a source term in the energy equation, which is solved, as well as the momentum equation, by the control volume method. The effects of medium optical thickness and tube wall temperature on the temperature distribution in medium as well as the heat flux and local Nusselt number on wall are analyzed. The results show that the radiation heat transfer of high‐temperature medium influences the temperature distribution and convection heat transfer greatly, and plays an important role in the heat transfer of developing laminar flow in a tube. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(1): 53–63, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10135     

Coupled radiation‐convection heat transfer of high‐temperature participating medium in heated/cooled tubes

The coupled radiation‐convection heat transfer of high‐temperature participating medium in heated/cooled tubes is investigated numerically. The medium flows in a laminar and fully developed state with a Poiseuille velocity distribution, but the thermal status is developing. By the discrete ordinate method, the nonlinear integrodifferential radiative transfer equation in a cylindrical coordinate form is solved to give the radiative source term in the energy equation of coupled heat transfer. The energy equation is solved by the control volume method. The local Nusselt number and wall heat flux of convection as well as the total wall heat flux are employed to evaluate the influence of radiation heat transfer on convection. The analysis shows that the radiation heat transfer weakens the convection effect, promotes the temperature development, and significantly shortens the tube length with obvious heated/cooled effect. There is an obvious difference between the coupled heat transfer in a heated tube and that in a cooled tube, even though the medium properties are kept constant. The wall emissivity, the medium thermal conductivity and scattering albedo have significant influences on the coupled heat transfer, but the effect of medium scattering phase function is small. © 2003 Wiley Periodicals, Inc. Heat Trans Asian Res, 33(1): 64–72, 2004; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.10137     

Study on heat transfer characteristics of porous metallic heat sink with conductive pipe under bypass effect

The work investigated the forced convection heat transfer of the heat sink situated in a rectangular channel by considering the bypass effect. The fluid medium was air. The relevant parameters were the Reynolds number (Re), the relative top by‐pass gap (C/H), and the relative side by‐pass gap (S/L). The size of the heat sink was 60 mm (L)×60 mm(W)×24 mm(H). Two heat sinks were employed as test specimens: (A) the 0.9‐porosity aluminum foam heat sink and (B) the 0.9‐porosity aluminum foam heat sink with a 20 mm diameter copper cylinder. The copper cylinder was used as a conductive pipe of heat sink. The average Nusselt number was examined under various forced convection conditions. Experimental results demonstrate that increasing by‐pass space decreased the Nusselt number. Besides, the average Nusselt number of mode B heat sink was higher than that of mode A heat sink by 30% for the case without by‐pass flow. The heat transfer enhancement by the copper cylinder would decline as the by‐pass space grew. © 2009 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/htj.20247     

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     

Mixed convection heat and mass transfer over a vertical plate in a power‐law fluid‐saturated porous medium with radiation and chemical reaction effects

Mixed convection heat and mass transfer from a vertical plate embedded in a power‐law fluid‐saturated Darcy porous medium with chemical reaction and radiation effects is studied. The governing partial differential equations are transformed into ordinary differential equations using similarity transformations and then solved numerically using the shooting method. A parametric study of the physical parameters involved in the problem is conducted and a representative set of numerical results is illustrated graphically. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library (wileyonlinelibrary.com/journal/htj). DOI 10.1002/htj.21058     

Entropy generation in an asymmetrically cooled slab with temperature‐dependent internal heat generation

The paper presents an entropy generation analysis for steady conduction in a slab with temperature‐dependent volumetric internal heat generation. The slab experiences asymmetric convective cooling on its two faces. The exact analytical solution for the temperature distribution is used to compute dimensionless local and total entropy generation rates in the slab. The total entropy generation rate depends on five dimensionless parameters: reference heat generation temperature Q, the heat generation–temperature variation parameter a, the temperature asymmetry parameter λ, and Biot numbers Bi₁ and Bi₂. Graphs illustrating the effect of these five parameters on the local and total entropy generation rates are presented and discussed. It is found that the total entropy generation in the slab can be minimized with a suitable choice of the cooling parameters. The paper corrects the flawed entropy results published recently. The present results for the special case of uniform internal heat generation confirm the results presented in a 2003 paper. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20404     

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

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

Effects of fin shapes on heat transfer in microchannel heat sinks

In the present study, compact water cooling of high‐density, high‐speed, very‐large‐scale integrated (VLSI) circuits with the help of microchannel heat exchangers were investigated analytically. This study also presents the result of mathematical analysis based on the modified Bessel function of laminar fluid flow and heat transfer through combined conduction and convection in a microchannel heat sink with triangular extensions. The main purpose of this paper is to find the dimensions of a heat sink that give the least thermal resistance between the fluid and the heat sink, and the results are compared with that of rectangular fins. It is seen that the triangular heat sink requires less substrate material as compared to rectangular fins, and the heat transfer rate per unit volume has been almost doubled by using triangular heat sinks. It is also found that the effectiveness of the triangular fin is higher than that of the rectangular fin. Therefore, the triangular heat sink has the ability to dissipate large amounts of heat with relatively less temperature rise for the same fin volume. Alternatively, triangular heat sinks may thus be more cost effective to use for cooling ultra‐high speed VLSI circuits than rectangular heat sinks.     

Numerical study of heat transfer enhancement and fluid flow with inline common‐flow‐down vortex generators in a plate‐fin heat exchanger

Three‐dimensional numerical simulations are performed on a plate‐fin heat exchanger (with triangular fins as inserts between the plates) to evaluate the laminar heat transfer and fluid flow characteristics with longitudinal vortex generators (LVGs). The effect with an inline rectangular winglet pair (RWP) with a common‐flow‐down (CFD) configuration is studied. The numerical results indicate that the application of inline LVGs effectively enhances the heat transfer of the channel. The heat transfer further increases with the increase in the Reynolds number from 200 to 500 and angle of attack from β = 15° to 22.5°. The computations are also performed to find the best location for the second RWP. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20414     

Heat transfer from hollow cylinder using optimal homotopy asymptotic method

Optimal homotopy asymptotic method (OHAM) is employed to investigate steady‐state heat conduction with temperature dependent thermal conductivity and uniform heat generation in a hollow cylinder. Analytical models are developed for dimensionless temperature distribution and heat transfer for two cases using mixed boundary conditions (Dirichlet, Neumann, and Robin). The inner cylinder is assumed to be insulated in both cases. In the first case, the outer cylinder is assumed to be isothermal whereas in the second case, the outer cylinder is convectively cooled by a fluid of temperature T₂ through a uniform heat transfer coefficient h. The effects of Biot number, dimensionless heat generation, and thermal conductivity parameters on the temperature distribution and heat transfer are determined analytically and validated numerically using MAPLE 14. In both cases, the results obtained by OHAM are found to be in good agreement with the numerical results. It is found that as the Biot number increases, the results approach that of the isothermal case. © 2012 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20407     

Multiwalled Carbon Nanotube Nanofluid for Thermal Management of High Heat Generating Computer Processor

Minichannel heat sink geometries with varying fin spacing were tested with de‐ionized water and MWCNT (1 wt %) nanofluid to evaluate their performance with flow components of a liquid cooling kit. Four heat sinks with fin spacing of 0.2 mm, 0.5 mm, 1.0 mm, and 1.5 mm were used in this investigation. Heat sink base temperature was analogous to processor operating temperature which was the prime parameter of interest in this investigation. The base temperature decreased by reducing the fin spacing and using multiwalled carbon nanotube (MWCNT) nanofluid. The lowest value of heat sink base temperature recorded was 49.7 ^°C at a heater power of 255 W by using a heat sink of 0.2 mm fin spacing and MWCNT nanofluid as a coolant. Moreover, as a result of reduced fin spacing and using MWCNT nanofluid as a coolant the value of overall heat transfer coefficient increased from 1200 W/m²K to 1498 W/m²K, translating to about a 15% increase. The value of thermal resistance also dropped by reducing the fin spacing and using MWCNT nanofluid. The most important aspect of the study is that the heat sinks and MWCNT nanofluid proved to be compatible with the pump and radiator of the commercial CPU liquid cooling kit. The pump was capable to handle the pressure drop which resulted by reducing the heat sink fin spacing and by using MWCNT nanofluid. © 2013 Wiley Periodicals, Inc. Heat Trans Asian Res, 43(7): 653–666, 2014; Published online 11 November 2013 in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.21107     

Heat transfer augmentation in a plate‐fin heat exchanger using a rectangular winglet

This study presents numerical computation results on laminar convection heat transfer in a plate‐fin heat exchanger, with triangular fins between the plates of a plate‐fin heat exchanger. The rectangular winglet type vortex generator is mounted on these triangular fins. The performance of the vortex generator is evaluated for varying angles of attack of the winglet i.e., 20, 26, and 37° and Reynolds number 100, 150, and 200. The computations are also performed by varying the geometrical size and location of the winglet. The complete Navier–Stokes equation and the energy equation are solved by the (Marker and Cell) MAC algorithm using the staggered grid arrangement. The constant wall temperature thermal boundary conditions are considered. Air is taken as the working fluid. The heat transfer enhancement is seen by introducing the vortex generator. Numerical results show that the average Nusselt number increases with an increase in the angle of attack and Reynolds number. For the same area of the LVG, the increase in length of the LVG brings more heat transfer enhancement than increasing the height. The increase in heat transfer comes with a moderate pressure drop penalty. © 2010 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com ). DOI 10.1002/htj.20318     

Numerical investigation of turbulent heat transfer in channels with detached rib‐arrays

A numerical investigation is conducted to analyze the flow‐field and heat transfer characteristics in a rectangular passage of width‐to‐height ratio of 6:1 with detached ribs on one wall, where constant wall temperature condition is applied. The effect of detached‐rib geometry on heat transfer coefficient, friction factor, and thermal enhancement factor is investigated covering the range of the detached‐clearance ratios (c/a) of 0.1, 0.2, 0.3, and 0.4, the Reynolds number based on the channel hydraulic diameter ranges from 8000 to 24,000. The numerical results show that the flow‐field, temperature pattern, local Nusselt number distribution, average Nusselt number, and friction factor are strongly dependent on the detached‐clearance ratios. The thermal enhancement factor (TEF) under the same pumping power constraint is calculated in order to examine the overall effect of the detached‐clearance ratio. For the present range investigated, the maximum TEF of 1.22 is achieved by the use of the ribs with c/a of 0.1 at Reynolds number of 8000. © 2011 Wiley Periodicals, Inc. Heat Trans Asian Res; Published online in Wiley Online Library ( wileyonlinelibrary.com/journal/htj ). DOI 10.1002/htj.20357