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     

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     

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     

Effects of particle fouling and magnetic field on porous fin for improved cooling of consumer electronics

The combined effect of particulate fouling and magnetic field on the efficiency of a convective–radiative porous fin heatsink with temperature‐dependent thermal conductivity is presented. The developed thermal models are solved using differential transformation method. The effects of thermal conductivity, porosity, convection, radiation parameter, and thermal fouling number on the fin thermal efficiency are investigated. The presence of thermal fouling on the surface of the fin is shown to increase the temperature distribution. The presence of particle deposition on the fin surface significantly decreases the rate of heat transfer as additional thermal resistance of the fouling layer decreases the thermal performance of porous fin heatsink. Moreover, the fin efficiency decreases as the value of fouled Biot, Darcy, radiation number, and thermogeometric parameter increases. It is established that M_f < M_c, which indicates that the efficiency of the fouled fin is greater than the efficiency of the clean fin. Furthermore, the result of the present study is validated with the established results of Chebyshev spectral collocation method and fourth‐order Runge–Kutta with shooting method and an error margin of 0.000000023 is established.     

Entropy generation in a hollow cylinder with temperature dependent thermal conductivity and internal heat generation with convective–radiative surface cooling

This article investigates entropy generation in an asymmetrically cooled hollow cylinder with temperature dependent thermal conductivity and internal heat generation. The inside surface of the cylinder is cooled by convection on its inside surface while the outside surface experiences simultaneous convective–radiative cooling. The thermal conductivity of the cylinder as well as the internal heat generation within the cylinder are linear functions of temperature, introducing two nonlinearities in the one-dimensional steady state heat conduction equation. A third nonlinearity arises due to radiative heat loss from the outside surface of the cylinder. The nonlinear system is solved analytically using the differential transformation method (DTM) to obtain the temperature distribution which is then used to compute local and total entropy generation rates in the cylinder. The accuracy of DTM is verified by comparing its predictions with the analytical solution for the case of constant thermal conductivity and constant internal heat generation. The local and total entropy generations depend on six dimensionless parameters: heat generation parameter Q, thermal conductivity parameter β, conduction–convection parameters Nc₁ and Nc₂, conduction–radiation parameter Nr, convection sink temperature δ and radiation sink temperature η.     

Analytical solutions of the 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient - II. Recurrent direct solution

The recurrent direct solution of the 1-D heat conduction problem for a single straight fin and spine with power-law-type temperature dependent heat transfer coefficient has been derived using inversion of the closed-form solution obtained in the first part of the study. The expression with improving convergence to calculate accurately the dimensionless temperature excess T_e at the fin tip for a given values of the fin parameter N and exponent n in heat transfer equation has been obtained by a linearization method. Equation for the temperature excess distribution throughout the fin has also been derived. The obtained formula for T_e allows to calculate the fin base thermal conductance and augmentation factor. Obtained expressions are seen to be simple and convenient for the engineering design of the fins and finned surfaces.     

Combustion characteristics and thermal enhancement of premixed hydrogen/air in micro combustor with pin fin arrays

Targeted at improving the combustion stability and enhancing heat transfer in micro combustor, the combustion characteristics and thermal performance of micro combustor with pin fin arrays are numerically investigated by employing detail H₂/O₂ reaction mechanism. It is shown that the micro combustor with staggered pin fin arrays exhibits the highest average temperature and heat flux of external wall, while the micro combustor with in-line pin fin arrays displays the most uniform temperature distribution of external wall. When the equivalence ratio is 1.1, all micro combustors exhibit the highest mean temperature and heat flux of external wall. The micro combustor materials with high thermal conductivity can not only improve the average temperature and heat flux of external wall, but also enhance heat transfer to the upstream which can preheat the mixed gas. Therefore, the materials with high thermal conductivity, such as red copper and aluminum, can make up for the nonuniform temperature distribution of micro combustor with staggered pin fin arrays, so as to realize uniform high heat flux output of external wall.     

A least squares method for a longitudinal fin with temperature dependent internal heat generation and thermal conductivity

Approximate but highly accurate solutions for the temperature distribution, fin efficiency, and optimum fin parameter for a constant area longitudinal fin with temperature dependent internal heat generation and thermal conductivity are derived analytically. The method of least squares recently used by the authors is applied to treat the two nonlinearities, one associated with the temperature dependent internal heat generation and the other due to temperature dependent thermal conductivity. The solution is built from the classical solution for a fin with uniform internal heat generation and constant thermal conductivity. The results are presented graphically and compared with the direct numerical solutions. The analytical solutions retain their accuracy (within 1% of the numerical solution) even when there is a 60% increase in thermal conductivity and internal heat generation at the base temperature from their corresponding values at the sink temperature. The present solution is simple (involves hyperbolic functions only) compared with the fairly complex approximate solutions based on the homotopy perturbation method, variational iteration method, and the double series regular perturbation method and offers high accuracy. The simple analytical expressions for the temperature distribution, the fin efficiency and the optimum fin parameter are convenient for use by engineers dealing with the design and analysis of heat generating fins operating with a large temperature difference between the base and the environment.     

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.     

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.     

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

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

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     

The hydraulic‐thermal performance of miniature compact heat sinks using SiO2‐water nanofluids

Ever since the rapid increase in both the demand for the miniature electronic devices and their applications, heat dissipation in the electronic components has been a serious issue. A miniature plate‐pin heat sink model with square, circular, and elliptic pins is considered to enhance the hydrothermal performance of this kind of compact heat sink (CHS). Water and 3% of SiO₂‐water nanofluids of volume fraction were used with different Reynolds number ranges (100‐1000). The findings show that the base temperature of heat sink reduces while the Nusselt number enhances by using nanofluids and increasing Reynolds number. The lowest value of the base temperature is nearly 25% for the square pins and circular pins CHSs compared with a plate–fin heat sink at 3% of nanofluids. Furthermore, the highest value of the Nusselt number is about 98% at 3% SiO₂ for circular pin CHSs compared with the plate–fin heat sink. However, the pressure drop of CHSs is higher than that of plate–fin heat sink. Moreover, the most significant hydrothermal performance value is about 1.44 for water and around 1.51 for SiO₂ as using the CHS with circular and elliptic pins depends on the Reynolds number. Thus, applying CHSs with nanofluids instead of the traditional heat sinks might produce a substantial enhancement in the hydrothermal performance of heat sinks.     

Forced convection heat transfer in microchannel heat sinks

This paper presents an analysis of forced convection heat transfer in microchannel heat sinks for electronic system cooling. In view of the small dimensions of the microstructures, the microchannel is modeled as a fluid-saturated porous medium. Numerical solutions are obtained based on the Forchheimer–Brinkman-extended Darcy equation for the fluid flow and the two-equation model for heat transfer between the solid and fluid phases. The velocity field in the microchannel is first solved by a finite-difference scheme, and then the energy equations governing the solid and fluid phases are solved simultaneously for the temperature distributions. Also, analytical expressions for the velocity and temperature profiles are presented for a simpler flow model, i.e., the Brinkman-extended Darcy model. This work attempts to perform a systematic study on the effects of major parameters on the flow and heat transfer characteristics of forced convection in the microchannel heat sink. The governing parameters of engineering importance include the channel aspect ratio (α_s), inertial force parameter (Γ), porosity (ε), and the effective thermal conductivity ratio (k_r). The velocity profiles of the fluid in the microchannel, the temperature distributions of the solid and fluid phases, and the overall Nusselt number are illustrated for various values of the problem parameters. It is found that the fluid inertia force alters noticeably the dimensionless velocity distribution and the fluid temperature distribution, while the solid temperature distribution is almost insensitive to the fluid inertia. Moreover, the overall Nusselt number increases with increasing the values of α_s and ε, while it decreases with increasing k_r.     

A mobile thermal battery resembling a solar receiver: Innovative design and performance assessment

An innovative design of a mobile thermal battery resembling the solar receiver is presented. A ternary salt mixture consisting of 52% KNO₃, 18% NaNO₃, and 30% LiNO₃ by wt% is used as the thermal energy storing medium inside the thermal battery. Since the thermal conductivity of the ternary salt mixture is low, aluminum meshes are introduced to create a thermal conduction tree inside the thermal energy storing medium. The actual field data are used in the simulations to resemble the solar irradiation emanating from the parabolic trough and focusing onto the thermal battery outer surface. To improve the uniform heating at the outer surface, the thermal battery rotation along the centerline of the trough is considered. The temperature parameter is introduced to assess the uniform‐like temperature distribution inside the ternary salt mixture. It is found that the use of aluminum meshes improves the heat diffusion in the phase change material of the ternary salt mixture; in which case, it acts like a thermal conduction tree inside the thermal battery. The rotation of the thermal battery results in uniform‐like temperature distribution across the thermal battery cross section and suppresses the excessive temperature rise because of the local heating in the close region of the thermal battery outer surface.     

Optimum Thermal Analysis of a Heat Sink with Various Fin Cross-Sections by Adjusting Fin Length and Cross-Section

A theoretical analysis of a heat sink is presented in this study to pursue the purpose of maximum thermal dissipation and the least material cost. Due to the general derivation, the longitudinal fin arrays on a heat sink can have either square, rectangular, equilaterally triangular, or cylindrical cross-section. By input of the Biot number, Bi, heat transfer coefficient ratio, H, and the shape parameter, γ, the heat transfer equation, which is expressed in implicit form, can be solved by iterative method to calculate the optimum fin length and fin thickness. Meanwhile, the thermal resistance of a heat sink can be obtained to illustrate the cooling performance under various design conditions.     

Book Review Corner

The optimum dimensions of rectangular longitudinal radiating fins with radiant interaction between the fin and its base are determined. The basic assumptions are one-dimensional heat conduction and ideal black-surface radiation. The governing differential equation is formulated by means of dimensionless variables and is solved using a variable-step Runge-Kutta [1, 2] algorithm, with local extrapolation [3], in order to carry out the required minimization procedure. The optimum thickness, height, and percent contribution of the fin heat dissipation are presented in dimensionless form in several diagrams that give insight into the operational characteristics of the heat rejection mechanism. It has been found that the results are strongly affected by the surface Biot number B_r = hrR/k. Several examples that highlight design criteria with regard to the selection of the material, amount of heat rejected, and tube and fin base temperature are given. A comparison of our results with those existing in the literature shows some significant differences that are discussed.     

Analysis of the 1-D heat conduction problem for a single fin with temperature dependent heat transfer coefficient: Part I – Extended inverse and direct solutions

A closed-form inverse solution of the 1-D heat conduction problem for a single fin or spine of constant cross section with an insulated tip is generalized to account for the effect of the tip heat loss. The heat transfer coefficient (HTC) is assumed to exhibit the power-law type dependence on the local excess temperature with arbitrary value of the exponent n in the range of ?0.5 ? n ? 5. The form of the obtained inverse solution is the same as the one for a fin with an insulated tip. However, in addition to the dimensionless fin tip temperature T_e and n, the fin parameter N also depends on the complex parameter ω²Bi. Using the inversion of this solution and a linearization procedure, the recurrent direct solution with a high convergence rate is derived. Based on the latter, the explicit direct closed-form solution for the accurate determination of the temperature distribution along a fin height at the given values of N, n, and ω²Bi is obtained. This allows one to determine the base thermal conductance G of the straight plate fin (SPF) and cylindrical pin fin (CPF). The relations between the fin parameters are systematized and collected in two tables for the SPF and CPF. They permit one to determine the arbitrary dimensionless geometrical or thermal fin parameter at given value of any other of its parameters and prescribed or calculated values of the main fin parameter(s) N or (and) G.     

Analysis of three-dimensional heat transfer in micro-channel heat sinks

In this study, the three-dimensional fluid flow and heat transfer in a rectangular micro-channel heat sink are analyzed numerically using water as the cooling fluid. The heat sink consists of a 1-cm² silicon wafer. The micro-channels have a width of 57 μm and a depth of 180 μm, and are separated by a 43 μm wall. A numerical code based on the finite difference method and the SIMPLE algorithm is developed to solve the governing equations. The code is carefully validated by comparing the predictions with analytical solutions and available experimental data. For the micro-channel heat sink investigated, it is found that the temperature rise along the flow direction in the solid and fluid regions can be approximated as linear. The highest temperature is encountered at the heated base surface of the heat sink immediately above the channel outlet. The heat flux and Nusselt number have much higher values near the channel inlet and vary around the channel periphery, approaching zero in the corners. Flow Reynolds number affects the length of the flow developing region. For a relatively high Reynolds number of 1400, fully developed flow may not be achieved inside the heat sink. Increasing the thermal conductivity of the solid substrate reduces the temperature at the heated base surface of the heat sink, especially near the channel outlet. Although the classical fin analysis method provides a simplified means to modeling heat transfer in micro-channel heat sinks, some key assumptions introduced in the fin method deviate significantly from the real situation, which may compromise the accuracy of this method.     

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