heat in history Isaac Newton and Heat Transfer

Abstract

An attempt is made to provide historical perspectives on the influences of Newton's law of cooling (1701) on the development of heat transfer theory. Newton's cooling law provides the first heat transfer formulation and is the formal basis of convective heat transfer. The cooling law was incorporated by Fourier (1822) as the convective boundary condition (Biot number) in his mathematical theory of heat conduction. The decisive step in the application of the concept of heat transfer coefficient occurred with the publication of the “basic law of heat transfer” by Nusselt in 1915. Newton's law is valid only for forced convection with constant physical properties. The close relationships for various heat transfer theories are pointed out. Heat transfer phenomena can also be classified based on the relationship between surface heat flux and temperature difference as a driving force.     

内部通道肋片结构对Mark Ⅱ叶片冷却性能影响的降维模型研究

一维冷却通道气热耦合计算是分层涡轮叶片冷却结构设计的重要方法。发展了以管道网络算法为核心的内部冷却特性计算程序,并与三维传热计算进行了耦合。通过与MarkⅡ叶片特定实验工况下的结果进行对比,验证了方法的有效性。此外,进一步将带肋结构流道传热特性相关经验公式集总在一维气热耦合算法中,分析了带肋通道改型的MarkⅡ叶片冷却性能。结果显示,带肋结构相比光滑流道能显著提升换热性能,在中径截面处较原方案温度下降15～30 K。     

Analysis of temperature gradient bifurcation in porous media – An exact solution

The phenomenon of temperature gradient bifurcation in a porous medium is analyzed by studying the convective heat transfer process within a channel filled with a porous medium, with internal heat generation. A local thermal non-equilibrium (LTNE) model is used to represent the energy transport within the porous medium. Exact solutions are derived for both the fluid and solid temperature distributions for two primary approaches (Models A and B) for the constant wall heat flux boundary condition. The Nusselt number for the fluid at the channel wall is also obtained. The effects of the pertinent parameters such as fluid and solid internal heat generations, Biot number and fluid to solid thermal conductivity ratio are discussed. It is shown that the internal heat generation in the solid phase is significant for the heat transfer characteristics. The validity of the one equation model is investigated by comparing the Nusselt number obtained from the LTNE model with that from the local thermal equilibrium (LTE) model. The results demonstrate the importance of utilizing the LTNE model in the present study. The phenomenon of temperature gradient bifurcation for the fluid and solid phases at the wall for Model A is established and demonstrated. In addition, the temperature distributions for Models A and B are compared. A numerical study for the constant temperature boundary condition was also carried out. It was established that the phenomenon of temperature gradient bifurcation for the fluid and solid phases for the constant temperature boundary condition can occur over a given axial region.     

Comparative Study of Convective Heat Transfer Performance of Steam and Air Flow in Rib Roughened Channels

A comparative experimental study of heat transfer characteristics of steam and air flow in rectangular channels roughened with parallel ribs was conducted by using an infrared camera. Effects of Reynolds numbers and rib angles on the steam and air convective heat transfer have been obtained and compared with each other for the Reynolds number from about 4,000 to 15,000. For all the ribbed channels the rib pitch to height ratio(p/e) is 10, and the rib height to the channel hydraulic diameter ratio is 0.078, while the rib angles are varied from 90° to 45°.Based on experimental results, it can be found that, even though the heat transfer distributions of steam and air flow in the ribbed channels are similar to each other, the steam flow can obtain higher convective heat transfer enhancement capability, and the heat transfer enhancement of both the steam and air becomes greater with the rib angle deceasing from 90° to 45°. At Reynolds number of about 12,000, the area-averaged Nusselt numbers of the steam flow is about 13.9%, 14.2%, 19.9% and 23.9% higher than those of the air flow for the rib angles of 90°,75°, 60° and 45° respectively. With the experimental results the correlations for Nusselt number in terms of Reynolds number and rib angle for the steam and air flow in the ribbed channels were developed respectively.     

Thermal state of an incompressible pseudo-plastic fluid and Nusselt number at the interface fluid–die wall

An experimental and numerical study was made of the steady laminar convective heat transfer to polyethylene Dowlex 2042E described by the Cross model flowing through an extrusion die. The inverse heat transfer problem is formulated to reconstruct the thermal history of the melted polymer in the channel die. Local Nusselt number distributions are presented for different flow rate and thermal boundary conditions. The effect of viscous dissipation combined with the flow rate on heat transfer is discussed. It is shown that an increasing flow rate leads to different Nusselt number and correlation is proposed to compute the heat transfer.     

Empirical Correlations for Free Convection in an Isothermal Asymmetrically Heated Vertical Channel

Laminar free convection in an isothermal asymmetrically heated vertical channel has been extensively studied in the past, and empirical correlations for the overall channel convective heat transfer rate are available in the literature. However, this problem has been revisited in order to develop empirical correlations that allow the calculation of the average Nusselt number for each channel wall separately. A numerical solution has been obtained for a Prandtl number of 0.71 and for Rayleigh numbers ranging from the conduction regime to the isolated boundary layer regime. The data have been used to develop correlations for the average Nusselt numbers on the hot and cold walls. These correlations satisfy a heat balance for the overall channel.     

A Possible Strategy for the Performance Enhancement of Turbine Blade Internal Cooling With Inclined Ribs

Ribbing the internal passages of turbine blades with 45 deg inclined ribs is a common practice to achieve a good compromise between high heat transfer coefficients and not too large pressure drop penalties. Literature studies demonstrated that, for channels having a large aspect ratio, the effect of the secondary vortices induced by angled ribs is reduced and the heat transfer performance is degraded. In order to enhance the performance, a possible strategy consists in introducing one or more longitudinal ribs (intersecting ribs) aligned to the main direction of flow. The intersecting ribs cut the ribbed channel into separate sub-channels and markedly affect the secondary flows with consequent increases in heat transfer performance. Experiments were performed for a rectangular channel with a large aspect ratio (equal to five) and 45 deg inclined ribs, regularly spaced on one of the principal walls of the channel. The effect of one and two intersecting ribs on friction and heat transfer characteristics has been investigated. The ribbed surface of the channel has been electrically heated to provide a uniform heat flux condition over each inter-rib region. The convective fluid was air. Heat transfer experiments have been conducted by using the liquid crystal thermography. Results obtained for the ribbed channel without intersecting rib and with one/two intersecting ribs are compared in terms of dimensionless groups.     

Steady Conjugate Heat Transfer from X-Ray or Laser-Heated Sphere in External Flow at Low Reynolds Number

ABSTRACT

A laser or an X-ray beam is used to heat a sphere that is immersed in uniform external flow. Temperature distributions as well as local and average convective heat transfer coefficients are calculated in order to evaluate the efficacy of cooling the solid sphere. The present work extends previous studies by: (1) applying a unique heat source imposed by irradiating the sphere with an intense X-ray energy beam; (2) performing the conjugate heat transfer analysis in fluid and solid domain; and (3) calculating the internal and surface temperature distribution. Absorption of the irradiation results in nonuniform heat generation, having an exponential spatial distribution of heat source. The limiting cases of heat source distribution are localized surface “laser” heating and near-uniform heat generation throughout the sphere. Key results are reported for two different source beam sizes (small and large) striking the sphere, with comparison to the solution for the isothermal wall boundary condition.     

The coupling of conduction with mixed convection of micropolar fluids past a vertical flat plate

In this paper, the thermo-fluid-dynamic field resulting from the coupling of wall conduction with laminar mixed convection heat transfer of micropolar fluids along a vertical flat plate is studied. A conjugate heat transfer is proposed to serve as a controlling index that indicates the effect of wall conduction. After a suitable coordinate transformation to reduce the complex of the governing boundary layer equations, the resulting nonlinear differential equations were solved with an implicit finite difference method. The effects of the micropolar material parameters, the buoyancy parameter, the Prandtl number and the conjugate heat transfer parameter on the flow and the thermal fields are discussed in detail.     

Convection mixte dans un écoulement de poiseuille vertical: Étude comparative de chauffage volumique et pariétal à flux constant

Numerical study of heat transfer in a laminar flow between two vertical parallel plates is reported. Two configurations are considered: the first consists of an imposed constant wall-heat-flux boundary conditions; the second corresponds to a bulk energy dissipation by direct electric-conduction (Joule effect) in the fluidIn the two cases, we have kept the same average temperature-difference between the fluid at the entrance and at the exit of the channel. The flow rate is also the same in both cases. Comparison is made between the axial evolution of Nusselt number obtained from this computation and the Nusselt number given by a classical correlation, in forced convection regime. It has been found that they are in good agreement.Heat transfer enhancement in the mixed convection regime is analysed We show particularly that the temperature distribution at the exit section of the channel is more uniform in the bulk heating case than in the constant wall-heat-flux condition.The secondary-flow induced by free convection and its contribution to the temperature uniformity is also studied We show the similarity between the two configurations from the point of view of the secondary-flow structure. In both cases we cam observe two convective cells whose centres are located closer to the exit from the channel. It is explained that this phenomenon is due to the presence, in the two cases, of the wall temperature increase along the channel in the downstream direction.     

Axial conduction in laminar pipe flows with nonlinear wall heat fluxes

A numerical analysis to determine the heat-transfer parameters of a fluid flow rejecting heat to the surrounding medium by convection and radiation is developed. The influence of axial conduction is included and the velocity profile is taken as nonuniform in the transverse direction. Use of a transformation eliminates the required boundary conditions at infinity. Approximate numerical techniques are employed to solve the nonlinear conjugate problem. As Péclet number increases, the temperature fields simplify to those where axial conduction is excluded. The computed results indicate that the effects of axial conduction are strongly altered by the parameters responsible for the convection and radiation. Bulk fluid temperatures, wall heat fluxes and Nusselt numbers are plotted against Graetz numbers. Critical Péclet numbers for a variety of cooling conditions are presented using the bulk fluid temperature as a reference.     

Effect of substrate thickness and material on heat transfer in microchannel heat sinks

Heat and fluid flow in microchannels of size (200μm × 200 μm, 5 cm long) of different substrate thicknesses (t = 100 μm–1000 μm) and different MEMS (Microelectromechanical Systems) materials (Polyimide, Silica Glass, Quartz, Steel, Silicon, Copper) was studied to observe the effects of thermal conductivity and substrate thickness on convective heat transfer in laminar internal flows.The results of the model were first validated by the theoretical results recommended by standard forced convection problem with H1 (Constant heat flux boundary condition) condition before the results from the actual microchannel configurations were obtained. Thereafter, general Nusselt number results were obtained from the models of many microchannel configurations based on the commercial package COMSOL MULTIPHYSICS^® 3.4 and were discussed on both local and average basis.A general Nusselt number correlation for fully developed laminar flow was developed as a function of two dimensionless parameters, namely Bi, Biot number and relative conductivity k¹, to take the conduction effects of the solid substrate on heat transfer into account. It was also demonstrated when the commonly used assumption of constant heat flux boundary (H1) condition is applicable in heat and fluid flow analysis in microfluidic systems. For this, a new dimensionless parameter was employed. A value of 1.651 for this suggested dimensionless parameter (Bi^0.04k^1?0.24) corresponds to 95% of the Nusselt number associated with the constant heat flux boundary condition so that it could be set as a boundary for the applicability of constant heat flux boundary (H1) condition in microfluidic systems involving heat transfer.     

Experimental investigations on overall cooling effect of ribbed channel with air bleeds

An experimental investigation on overall heat transfer performance of a rectangular channel, in which one wall has periodically placed oblique ribs to enhance heat exchange and cylindrical film holes to bleed cooling air, has been carried out in a hot wind tunnel at different mainstream temperatures, hot mainstream Reynolds numbers, coolant Reynolds numbers and blowing ratios. To describe the cooling effect of combined external coolant film with the internal heat convection enhanced by the ribs, the overall cooling effectiveness at the surface exposed in the mainstream with high temperature was calculated by the surface temperatures measured with an infrared thermal imaging system. The total mass flow rate of cooling air through the coolant channel was regulated by a digital mass flow rate controller, and the blowing ratio passing through the total film holes was calculated based on the measurements of another digital-type mass flow meter. The detailed distributions of overall cooling effectiveness show distinctive peaks in heat transfer levels near the film holes, remarkable inner convective heat transfer effect over entire channel surface, and visible conductive heat transfer effect through the channel wall; but only when the coolant Reynolds number is large enough, the oblique rib effect can be detected from the overall cooling effectiveness; and the oblique bleeding hole effect shows the more obvious trend with increasing blowing ratios. Based on the experimental data, the overall cooling effectiveness is correlated as the functions of Re_m (Reynolds number of hot mainstream) and Re_c (Reynolds number of internal coolant flow at entrance) for the parametric conditions examined.     

An analytical solution for thermally fully developed combined pressure – electroosmotically driven flow in microchannels

An analytical solution is presented to study the heat transfer characteristics of the combined pressure – electroosmotically driven flow in planar microchannels. The physical model includes the Joule heating effect to predict the convective heat transfer coefficient in two dimensional microchannels. The velocity field, which is a function of external electrical field, electroosmotic mobility, fluid viscosity and the pressure gradient, is obtained by solving the hydrodynamically fully-developed laminar Navier–Stokes equations considering the electrokinetic body force for low wall zeta potentials. Then, assuming a thermally fully-developed flow, the temperature distribution and the Nusselt number is obtained for a constant wall heat flux boundary condition. The fully-developed temperature profile and the Nusselt number depend on velocity field, channel height, solid/liquid interface properties and the imposed wall heat flux. A parametric study is presented to evaluate the significance of various parameters and in each case, the maximum heat transfer rate is obtained.     

Conjugate Heat Transfer for Free Convection along a Vertical Plate Fin

<正>In this study,a new and effective improved Semi-Analytic and Semi-Empirical formula f(Pr)= (0.749999437Pr~(1/2))/((0.609+1.221Pr~(1/2)+1.238Pr)~(1/4))has been proposed to solve a conjugate problem with free convection in the incompressible laminar boundary layer flow and heat conduction in a solid wall for the flow passing a flat plate fin. A combination of flat-plate flow and flat-plate fin heat conduction has been considered in the present study.Finite -difference solutions for the interface temperature profiles and the heat transfer rates have been presented over the entire thermo-fluid-dynamic field for Prandtl numbers from 0.001 to 10000.First,the similar flow field has been solved by the Runge-Kutta method and the shooting methods,then the correlation equation of the local heat transfer coefficient have been obtained.Finally,the empirical formula has been substituted into the fin temperature heat conduction calculation processes to obtain the iterative solutions of the conjugate problems.     

Conjugate heat and mass transfer in membrane-formed channels in all entry regions

Membrane-based energy recovery ventilators (or total heat exchangers) are key equipments to fresh air ventilation, which is helpful for the control of respiratory diseases like Swine flu (H1N1) and SARS. Parallel-plates narrow channels are common structure for membrane-based energy recovery ventilators. In practice, the exchanger channel lengths are limited due to the confinement in pressure drops and noises. In these channels, the hydraulically, thermally and concentrationally entry regions account for a large fraction of the total duct length. However, previous investigations neglected the entry issues for simplicity. Either hydraulically fully developed, or thermally or/and concentrationally fully developed flow were assumed, which would underestimate equipments performances seriously. This study provides a more accurate methodology: fluid flow, heat and mass transport equations were solved directly as they enter into the channel. In other words, both the fluid flow and the heat and mass transport are in simultaneously developing regions. The membrane and the two neighboring flows are considered as a conjugate problem. The conjugate heat transfer problem is solved with a commercial CFD code. Then the conjugate mass transfer problem is solved by transferring it to another conjugate heat transfer problem by heat mass analogy. The Nusselt and Sherwood numbers in the entry regions are calculated. The effects of three typical flow arrangements: cocurrent, counter and cross flow, on the boundary conditions and the consequent Nusselt and Sherwood numbers in the channels are evaluated.     

等壁温条件下潜热型功能热流体换热强化机理的理论研究

从对流与导热的相似性出发，揭示了潜热型功能热流体强化换热的物理机制。基于等效比热模型，对影响速度充分发展的等壁温圆管内该类流体层流流动换热强化的各因素进行了敏感性分析，弄清了影响换热强化的主要因素及其强化换热的机理，并改进了内部流动传统的Nu定义，使之能完全表征功能热流体换热器的程度，并和外部流动的Nu定义统一，有助于内部流动和外部流动对流换热的统一评价。     

Combined Effect of Mixed Convection and Surface Radiation Heat Transfer for Thermally Developing Flow in Vertical Channels

Combined effect of laminar flow mixed convection and surface radiation heat transfer for thermally developing airflow in a vertical channel heated from a side has been experimentally examined with different thermal and geometric parameters. The channel boundary is made of two isothermal walls and two adiabatic walls, the isothermal parallel wall is heated uniformly and the opposite cold wall temperature is maintained equal to the inlet conditions. The heated wall temperature ranged from 55 to 100°C, Reynolds number ranged from 800 to 2900 and the heat flux was varied from 250 to 870 W/m². To cover the wide range of Reynolds numbers, two aspect ratios of square and rectangular section were used. Surface radiation from the internal walls is considered through two emissivities i.e. 0.05 and 0.85, to represent weak and strong radiation effects, respectively. From the experiments, surface temperature and Nusselt number distributions of convection and radiation heat transfer are obtained for different heat flux values. Flow structure inside the channel is visualized to observe the flow pattern. The results show the combined effect of laminar flow mixed convection and surface radiation on the total heat transfer rate within the channel. The accumulating buoyancy force and airflow moves together vertically in the upward direction to give significant heat transfer enhancement in the vertical orientation of the channel.     

Forced Convection Heat Transfer Simulation Using Dissipative Particle Dynamics

Dissipative particle dynamics (DPD) with energy conservation was applied to simulate forced convection in parallel-plate channels with boundary conditions of constant wall temperature (CWT) and constant wall heat flux (CHF). DPD is a coarse-grained version of molecular dynamics. An additional equation for energy conservation was solved along with conventional DPD equations, where inter-particle heat flux accounts for changes in mechanical and internal energies when particles interact with surrounding particles. The solution domain was considered to be two–dimensional with periodic boundary condition in the flow direction and additional layers of particles on the top and bottom of the channel to apply no-slip and wall temperature boundary conditions. The governing equation for energy conservation was modified based on periodic fully developed velocity and temperature conditions. The results were shown via velocity and temperature profiles across the channel cross-section. The Nusselt numbers for CWT and CHF were calculated from the temperature gradient at the wall using a second order accurate forward difference approximation. The results agreed well with the exact solutions to within 2.3%.     

Turbulent heat and momentum transfer of combined forced and natural convection along a vertical flat plate—aiding flow

An aiding flow of turbulent, combined convection along a vertical flat plate is investigated experimentally in the range of high values of Rax¹ and Rex numbers. Local Nusselt numbers in the combined convection region are found to decrease as much as 25% than those for the pure forced and natural convection. It is revealed from the measurements of velocity and temperature that the reductions in heat transfer are mainly caused by the turbulent suppression. Turbulent transport mechanisms within the combined convective boundary layers are also discussed from the visualized data of the flow and temperature fields over the test plate.