Effect of temperature gradient within a solid particle on the rotation and oscillation modes in solid-dispersed two-phase flows

Liquid–solid two-phase flow with heat transfer is simulated, and the effect of temperature gradient within a solid particle on the particle behaviour and heat transfer is studied. The interaction between fluid and particles is considered with our original immersed solid approach on a rectangular grid system. The local heat flux at the fluid–solid interface is described with an anisotropic heat conductivity matrix, and the governing equation of temperature is time-updated with an implicit treatment for the diffusion term. The method is applied to a 2-D natural convection flow of a relatively low Rayleigh number including multiple particles. Heat transfer and particle behaviours are studied for different solid heat conductivities (ratio to the fluid conductivity ranging between 10⁻³ and 10³) and solid volume fractions. Under a condition of relatively low heat conductivity ratio, the particles show a simple circulating flow. By increasing the heat conductivity ratio, a transition of the particulate flow is observed to oscillation mode around the domain centre due to the buoyancy force as a restitution force. The oscillation period is found to vary with the heat conductivity ratio, and it is related to the time scales for the heat transfer via fluid and solid.     

Mixed convection heat transfer from a horizontal plate with variable surface heat flux in a porous medium

In this article nonsimilarity solution for mixed convection from a horizontal surface in a saturated porous medium was obtained for the case of variable surface heat flux. The entire mixed convection regime, ranging from pure forced convection to pure free convection, is considered by introducing a single nonsimilarity parameter. Heat transfer results are predicted by employing four different flow models, namely, Darcy's law, the Ergun model, and the Brinkman-Forchheimer-extended Darcy model with constant and variable porosity. The variable porosity effect is approximated by an exponential function. Effects of transverse thermal dispersion are taken into consideration in the energy equation, along with variable stagnant thermal conductivities. The formulation of the present problem shows that the flow and heat transfer characteristics depend on five parameters, that is, the power in the variation of surface heat flux, the nonsimilarity mixed-convection parameter, the inertia effect parameter, the boundary effect parameter, and the ratio of thermal conductivity of the fluid phase to that of the solid phase. Numerical results for the local Nusselt number variations, based on the various flow models, are presented for the entire mixed convection regime. The impacts␣of different governing parameters on the heat transfer results are thoroughly investigated. Received on 7 August 1997     

Conjugate Natural Convection in a Porous Enclosure with Non-Uniform Heat Generation

Effects of a conductive wall on natural convection in a square porous enclosure having internal heating at a rate proportional to a power of temperature difference is studied numerically in this article. The horizontal heating is considered, where the vertical walls heated isothermally at different temperatures while the horizontal walls are kept adiabatic. The Darcy model is used in the mathematical formulation for the porous layer and finite difference method is applied to solve the dimensionless governing equations. The governing parameters considered are the Rayleigh number (0 ???Ra ???1000), the internal heating and the local exponent parameters (0 ????? ???5), (1 ????? ???3), the wall to porous thermal conductivity ratio (0.44 ???Kr ???9.9) and the ratio of wall thickness to its width (0.02 ???D ???0.5). The results are presented to show the effect of these parameters on the fluid flow and heat transfer characteristics. It is found a strong internal heating can generate significant maximum fluid temperature more than the conductive solid wall. Increasing value thermal conductivity ratio and/or decreasing the thickness of solid wall can increase the maximum fluid temperature. It is also found that at very low Rayleigh number, the heat transfer across the porous enclosure remain stable for any values of the thermal conductivity ratio.     

Numerical Study of Heat Transfer in a Rectangular Channel with Porous Covering Obstacles

A numerical study is performed to analyze steady laminar forced convection in a channel in which discrete heat sources covered with porous material are placed on the bottom wall. Hydrodynamic and heat transfer results are reported. The flow in the porous medium is modeled using the Darcy–Brinkman–Forchheimer model. A computer program based on control volume method with appropriate averaging for diffusion coefficient is developed to solve the coupling between solid, fluid, and porous region. The effects of parameters such as Reynolds number, Prandtl number, inertia coefficient, and thermal conductivity ratio are considered. The results reveal that the porous cover with high thermal conductivity enhances the heat transfer from the solid blocks significantly and decreases the maximum temperature on the heated solid blocks. The mean Nusselt number increases with increase of Reynolds number and Prandtl number, and decrease of inertia coefficient. The pressure drop along the channel increases rapidly with the increase of Reynolds number.     

Free Convection Heat Transfer From A Sphere In A Porous Medium Using A Thermal Non-equilibrium Model

This work studies the free convection heat transfer from a sphere with constant wall temperature embedded in a fluid-saturated porous medium using a thermal non-equilibrium model. The governing equations are transformed into boundary-layer partial differential equations by the coordinate transform, and the obtained governing equations are then solved by the cubic spline collocation method. The temperature distributions for fluid and solid phases are shown for different values of the porosity scaled thermal conductivity ratio, the interphase heat transfer parameter, and the streamwise coordinate. The effects of the porosity scaled thermal conductivity ratio and the interphase heat transfer parameter between solid and fluid phases on the local Nusselt numbers for fluid and solid phases are examined. Results show the local Nusset number for the porous medium can be increased by increasing the porosity scaled thermal conductivity ratio. Moreover, the thermal non-equilibrium effect is more significant for low values of the porosity scaled thermal conductivity ratio or the interphase heat transfer parameter.     

3D Numerical study of the effect of aspect ratio on mixed convection air flow in upward solar air heater

A 3D Numerical study of mixed convection air flow in upward solar air heater with large spanwise aspect ratio (A = 10 to 40) was performed using CFD commercial code Fluent 14.5 (ANSYS). The main objective of this study is to investigate the channel height's effect (aspect ratio) on flow pattern and heat transfer in upward solar air heater in the particular case of low Re and high aspect ratio. The bottom plate (absorber) was submitted to Constant Heat Flux (CHF) in the range of 200 to 1000 W/m² and Reynolds number was varied from 50 to 1000. Our results are in concordance with most of authors conclusions about Poiseuille–Rayleigh–Benard flows. In mixed convection, increasing heat flux enhances heat transfer unlike forced convection flows. Simulation results of flow visualizations and Nusselt number calculations have shown that depending on Ri*, the velocity and temperature distributions in SAH vary greatly with the channel's height. The obtained results were different from previous studies. Indeed, our investigation of channel's height was achieved for the same heat flux but different Grashof numbers. For low channel's heights (high aspect ratio), increasing heat flux has not a significant effect but for higher channel's heights, an augmentation of heat flux enhances buoyancy effects in the flow and causes high turbulence. Also, increasing Reynolds number in low channel's heights (high A), can enhance substantially heat transfer. For higher channel's heights (low A), increasing Reynolds number decreases Ri* and thus buoyancy forces. Heat transfer is reduced and so Nusselt number. The obtained results may be very useful for engineers in designing and testing solar collectors.     

Enhanced heat transfer performances of molten salt mixed convection in a vertical annular duct

The mixed convection heat transfer of upward molten salt flow in a vertical annular duct is experimentally and numerically studied. The heat transfer performances of mixed convection are measured under Reynolds number 2,500–12,000 and inlet temperature 300–400 °C, and Nusselt number of molten salt flow with cooled inner wall monotonically increases with buoyancy number. The mixed convection is further simulated by low-Reynolds number k-ε model and variable properties, and the heat transfer tendency from numerical results agrees with that from experiments. At low Reynolds number, the natural convection plays more important role in the mixed convection. As the buoyancy number rises, the thickness of flow boundary layer near the inner wall increases, while the effective thermal conductivity remarkably rises, so the enhanced heat transfer of mixed convection is mainly affected by the effective thermal conductivity due to turbulent diffusion.     

Mixed Convection Heat Transfer in the Annulus Between Two Concentric Vertical Cylinders Using Porous Layers

A numerical investigation of the steady-state, laminar, axi-symmetric, mixed convection heat transfer in the annulus between two concentric vertical cylinders using porous inserts is carried out. The inner cylinder is subjected to constant heat flux and the outer cylinder is insulated. A finite volume code is used to numerically solve the sets of governing equations. The Darcy–Brinkman–Forchheimer model along with Boussinesq approximation is used to solve the flow in the porous region. The Navier–Stokes equation is used to describe the flow in the clear flow region. The dependence of the average Nusselt number on several flow and geometric parameters is investigated. These include: convective parameter, λ, Darcy number, Da, thermal conductivity ratio, K _r, and porous-insert thickness to gap ratio (H/D). It is found that, in general, the heat transfer enhances by the presence of porous layers of high thermal conductivity ratios. It is also found that there is a critical thermal conductivity ratio on which if the values of Kr are higher than the critical value the average Nusselt number starts to decrease. Also, it found that at low thermal conductivity ratio (K _r ≈ 1) and for all values of λ the porous material acts as thermal insulation.     

浮力对混合对流流动及换热特性的影响

用热线和冷线相结合的技术测量垂直圆管内逆混合对流流体的平均速度、 温度以及它们的脉动. 较详细地研究了浮力对逆混合对流的流动特性和传热特性的影响. 评 估了实验中采用的冷线测量温度补偿速度探头温度敏感的影响. 逆混合对流的传热结果用无 量纲参数Ω (Ω= Grd / Red2 )来表示，其中，基于管道直 径的雷诺数Red变化范围为900~18000, 浮力参数Ω变化范围为 0.004899~0.5047. 研究结果表明，浮力对逆混合对流的换热有强化作用. 随着葛拉晓夫数Grd的增加，温度脉动，流向雷诺正应力和流向温度通量增 大，并且在靠近壁面的流体区域尤其明显. 热线与冷线相结合的技术适合于研究非绝热的流 动测量，可以用于研究浮力对流动和换热特性的影响.     

Fully Developed Forced Convection Heat Transfer in a Porous Channel with Asymmetric Heat Flux Boundary Conditions

An analytical study is performed on steady, laminar, and fully developed forced convection heat transfer in a parallel plate channel with asymmetric uniform heat flux boundary conditions. The channel is filled with a saturated porous medium, and the lower and upper walls are subjected to different uniform heat fluxes. The dimensionless form of the Darcy–Brinkman momentum equation is solved to determine the dimensionless velocity profile, while the dimensionless energy equation is solved to obtain temperature profile for a hydrodynamically and thermally fully developed flow in the channel. Nusselt numbers for the lower and upper walls and an overall Nusselt number are defined. Analytical expressions for determination of the Nusselt numbers and critical heat flux ratio, at which singularities are observed for individual Nusselt numbers, are obtained. Based on the values of critical heat flux ratio and Darcy number, a diagram is provided to determine the direction of heat transfer between the lower or upper walls while the fluid is flowing in the channel.     

Effect of an Inserted Porous Layer Located at a Wall of a Parallel Plate Channel on Forced Convection Heat Transfer

A theoretical study is performed on heat and fluid flow in partially porous medium filled parallel plate channel. A uniform symmetrical heat flux is imposed onto the boundaries of the channel partially filled with porous medium. The dimensional forms of the governing equations are solved numerically for different permeability and effective thermal conductivity ratios. Then, the governing equations are made dimensionless and solved analytically. The results of two approaches are compared and an excellent agreement is observed, indicating correctness of the both solutions. An overall Nusselt number is defined based on overall thermal conductivity and difference between the average temperature of walls and mean temperature to compare heat transfer in different channels with different porous layer thickness, Darcy number, and thermal conductivity ratio. Moreover, individual Nusselt numbers for upper and lower walls are also defined and obtained. The obtained results show that the maximum overall Nusselt number is achieved for thermal conductivity ratio of 1. At specific values of Darcy number and thermal conductivity ratio, individual Nusselt numbers approach to infinity since the value of wall temperatures approaches to mean temperature.     

Heat transfer enhancement for laminar natural convection along a vertical plate due to sub-millimeter-bubble injection

Sub-millimeter-bubble injection is one of the most promising techniques for enhancing heat transfer for the laminar natural convection of liquids. However, flow and heat transfer characteristics for laminar natural convection of water with sub-millimeter bubbles have not yet been fully understood. The purpose of this study is to experimentally clarify the effects of sub-millimeter-bubble injection on the laminar natural convection of water along a heated vertical plate. The use of thermocouples and a particle tracking velocimetry (PTV) technique are applied to temperature and velocity measurements, respectively. The temperature measurement shows that the ratio of the heat transfer coefficient with sub-millimeter-bubble injection to that without injection increases with an increase in the bubble flow rate or a decrease in the wall heat flux and that the ratio ranges from 1.35 to 1.85. Moreover, it is concluded from simultaneous measurement of temperature and velocity that the heat transfer enhancement is directly affected by flow modification due to bubbles rising near the heated vertical plate.     

Similarity conditions for heat transfer in incompressible two-dimensional laminar boundary layer flow

Summary Similarity conditions are presented for the solution of some problems of heat transfer in incompressible two-dimensional boundary layer flow. The treatment holds for forced convection as well as for free convection. For free convection no a priori restriction is made with respect to geometry or temperature distribution of the solid surface. For forced convection the treatment is restricted to uniform bulk flow parallel to a flat surface of non-uniform temperature or heat flux. The results are summarized in some tables that facilitate comparison with older work.     

Free convection flow over an uniform-heat-flux surface with temperature-dependent viscosity

The role of temperature-dependent viscosity is studied in laminar free convection flow adjacent to a vertical surface with uniform heat flux. The resulting non-similar equations are solved by using a suitable variable transformation and employing an implicit finite difference method. It is shown that the constant viscosity results evaluated at the ambient fluid temperature underestimate the Nusselt number and overestimate the drag coefficient. The heat transfer predictions for large values of the viscosity parameter may be two times the constant viscosity parameter prediction. The present analysis is in good agreement with the corresponding correlation of previous experimental investigation.     

Determination of thermal conductivities of Sn–Zn lead-free solder alloys with radial heat flow and Bridgman-type apparatus

The variations of thermal conductivities of solid phases versus temperature for pure Sn, pure Zn and Sn–9 wt.% Zn, Sn–14 wt.% Zn, Sn–50 wt.% Zn, Sn–80 wt.% Zn binary alloys were measured with a radial heat flow apparatus. The thermal conductivity ratios of liquid phase to solid phase for the pure Sn, pure Zn and eutectic Sn–9 wt.% Zn alloy at their melting temperature are found with a Bridgman-type directional solidification apparatus. Thus, the thermal conductivities of liquid phases for pure Sn, pure Zn and eutectic Sn–9 wt.% Zn binary alloy at their melting temperature were evaluated by using the values of solid phase thermal conductivities and the thermal conductivity ratios of liquid phase to solid phase.     

Radiation effects on combined convection over a vertical flat plate embedded in a porous medium of variable porosity

The present paper is concerned with the study of radiation effects on the combined (forced-free) convection flow of an optically dense viscous incompressible fluid over a vertical surface embedded in a fluid saturated porous medium of variable porosity with heat generation or absorption. The effects of radiation heat transfer from a porous wall on convection flow are very important in high temperature processes. The inclusion of radiation effects in the energy equation leads to a highly non-linear partial differential equations which are transformed to a system of ordinary differential equations using non-similarity transformation. These equations are then solved numerically using implicit finite-difference method subject to appropriate boundary and matching conditions. A parametric study of the physical parameters such as the particle diameter-based Reynolds number, the flow based Reynolds number, the Grashof number, the heat generation or absorption co-efficient and radiation parameter is conducted on temperature distribution. The effects of radiation and other physical parameters on the local skin friction and on local Nusselt number are shown graphically. It is interesting to observe that the momentum and thermal boundary layer thickness increases with the radiation and decrease with increase in the Prandtl number.     

Nonsimilar Boundary Layer Analysis of Free Convection Heat Transfer Over a Vertical Cylinder in Bidisperse Porous Media

This work presents a boundary layer analysis for the free convection heat transfer from a vertical cylinder in bidisperse porous media with constant wall temperature. A boundary layer analysis and the two-velocity two-temperature formulation are used to derive the nonsimilar governing equations. The transformed governing equations are solved by the cubic spline collocation method to yield computationally efficient numerical solutions. The effects of inter-phase heat transfer parameter, modified thermal conductivity ratio, and permeability ratio on the heat transfer and flow characteristics are studied. Results show that an increase in the modified thermal conductivity ratio and the permeability ratio can effectively enhance the free convection heat transfer of the vertical cylinder in a bidisperse porous medium. Moreover, the thermal nonequilibrium effects are strong for low values of the inter-phase heat transfer parameter.     

Heat and mass transfer of a water film falling down a tilted plate with radiant heating and evaporation

This paper has concerned the heat and mass transfer of a water film falling down a tilted plate with radiant heating and water evaporation. A cluster of physical models was developed for evaluating the properties of heat and mass transfer. A fully implicit control-volume finite-difference procedure was used to solve the coupling equations. The effects of various parameters on heat and mass transfer were investigated. The results showed that the mass fraction of water vapor in ambient atmosphere and the flow turbulence played key roles in the heat and mass transfer. The ambient atmospheric temperature dramatically affected the sensible heat flux. However its effect on the latent heat flux is negligibly small. The magnitude of solar incident flux had an intense influence on the water film temperature. Received on 29 January 1998     

Numerical analysis of transient laminar forced convection of nanofluids in circular ducts

In this study, forced convection heat transfer characteristics of nanofluids are investigated by numerical analysis of incompressible transient laminar flow in a circular duct under step change in wall temperature and wall heat flux. The thermal responses of the system are obtained by solving energy equation under both transient and steady-state conditions for hydro-dynamically fully-developed flow. In the analyses, temperature dependent thermo-physical properties are also considered. In the numerical analysis, Al₂O₃/water nanofluid is assumed as a homogenous single-phase fluid. For the effective thermal conductivity of nanofluids, Hamilton–Crosser model is used together with a model for Brownian motion in the analysis which takes the effects of temperature and the particle diameter into account. Temperature distributions across the tube for a step jump of wall temperature and also wall heat flux are obtained for various times during the transient calculations at a given location for a constant value of Peclet number and a particle diameter. Variations of thermal conductivity in turn, heat transfer enhancement is obtained at various times as a function of nanoparticle volume fractions, at a given nanoparticle diameter and Peclet number. The results are given under transient and steady-state conditions; steady-state conditions are obtained at larger times and enhancements are found by comparison to the base fluid heat transfer coefficient under the same conditions.