Free Convection in Confined Power-Law Fluids From Two Side-by-Side Cylinders in a Square Enclosure

Laminar free convection heat transfer in power-law fluids from two side-by-side cylinders (one hot and one cold) confined in a square duct has been studied numerically in the two-dimensional flow regime. For a fixed value of the ratio of cylinder radius to size of the enclosure, the effect of geometrical placement of the cylinders is studied on the resulting velocity and temperature fields in the laminar free convection regime by considering six asymmetric locations of the two cylinders. In particular, extensive results reported herein span the range of conditions of Grashof number, 10 to 10⁵; Prandtl number, 0.7 to 100, thereby yielding the range of the Rayleigh number as 7 to 10⁷; power-law index, 0.3 to 1.8; and the relative positions (dimensionless) of the cylinders with respect to the centerline, –0.25 to 0.25. The heat transfer characteristics are analyzed in terms of the local Nusselt number along the surfaces of the two cylinders and the enclosure walls. Overall, the average Nusselt number shows a positive dependence on both the Grashof number and the Prandtl number irrespective of the values of power-law index and relative positioning of the cylinders. Also, all else being equal, shear-thinning fluid behavior promotes heat transfer with reference to that in Newtonian fluids. When the two cylinders are situated close to the bottom wall, the rate of heat transfer is augmented with reference to that for the symmetric positioning of the cylinders along the horizontal mid-plane of the enclosure. Conversely, heat transfer deteriorates as the cylinders are located above the centerline of the enclosure. The present numerical results have been consolidated via the use of a modified Rayleigh number, thereby enabling the estimation of the average Nusselt number in a new application.     

Numerical Simulation of Steady Mixed Convection Around Two Heated Circular Cylinders in a Square Enclosure

In this paper, a numerical study has been carried out to investigate the steady-state mixed convection around two heated horizontal cylinders in a square two-dimensional enclosure. The cylinders are located at the middle of the enclosure height and the walls of the cavity are adiabatic. Streamlines and isotherms are produced and the effects of cylinder diameter, Reynolds number, and Richardson number on the heat transfer characteristics are numerically analyzed. The average Nusselt number over the surface of cylinders and average nondimensional temperature in the enclosure are also presented. The results show that both heat transfer rates from the heated cylinders and the dimensionless fluid temperature in the enclosure increase with increasing Richardson number and cylinder diameter. However, the trend of average Nusselt number and nondimensional temperature variation is completely opposite when Reynolds number increases. In addition, by increasing the cylinders diameter and Richardson number, the left cylinder is less affected by the inlet flow than right one.     

Effect of Fluid Yield Stress on Natural Convection From Horizontal Cylinders in a Square Enclosure

In this study, laminar natural convection heat transfer to Bingham plastic fluids from two differentially heated isothermal cylinders confined in a square enclosure (with isothermal walls) has been investigated numerically. The governing partial differential equations have been solved over the ranges of the dimensionless parameters, namely, Rayleigh number, 10² to 10⁶, Prandtl number, 10 to 100, and Bingham number, 0.01 to 100, for seven locations of inner cylinders as ±0.25, ±0.2, ±0.1 and 0. These values correspond to the range of Grashof number varying from 10 to 10⁵. The detailed flow and temperature fields are visualized in terms of the streamlines and isotherm contours. Further insights are developed by examining the iso-shear rate contours and the yield surfaces delineating the fluid-like and solid-like regions. The corresponding heat transfer results are analyzed in terms of the distribution of the local Nusselt number along the cylinder surface together with its surface averaged value as functions of the Rayleigh number, Prandtl number, Bingham number, and positions of the cylinders. It is found that the average Nusselt number increases with the increasing values of the Rayleigh number and decreases with the increasing Bingham number. For sufficiently large values of the Bingham number, the average Nusselt number reaches its asymptotic value wherein heat transfer takes place solely by conduction. Based on the present numerical results, simple correlations for the prediction of the average Nusselt number and the limiting Bingham number have been developed. Also, a dimensionless criterion denoting the cessation of convection regime is outlined for this configuration.     

冷热房间之间隔门通风性能的数值分析

应用数值模拟方法,对冷热房间之间隔门通风情况进行了分析。三维模拟区域主要运用稳态湍流和Boussinesq模型进行模拟,并着重对不同隔门高度和外部条件下流动换热情况作了对比研究。研究表明,两个墙壁间的温差及其引起的密度差是形成温度和速度分布的主要原因。隔门使两个房间之间形成对流,右侧高温房间下部有冷气流进入,而左侧低温房间有热气流由上方进入。当隔门高度降低时,零压差线的相对高度不断升高,对流效应减弱。     

Steady Mixed Convection in Power-Law Fluids from a Heated Triangular Cylinder

The steady mixed convective transport from a heated triangular cylinder immersed in power-law fluids in an unconfined vertical domain is investigated numerically. Two different configurations of the cylinder are chosen; one when the base of the cylinder is facing the flow and the other when the apex of the triangle is facing the flow. The simulation is performed for: Reynolds number (1 to 35), Richardson number (0 to 2), power law index (0.4 to 1.8) and Prandtl number, 50. The flow and thermal fields are visualized through the streamlines and isotherm contours at the close proximity of the heated object for various Reynolds numbers, Richardson numbers and power law indices. The distributions of the surface pressure coefficient and local Nusselt number provide further insight of the hydrodynamic and thermal characteristics. Finally, the total drag coefficient and average Nusselt numbers on the surface of the cylinder are computed to explore the overall macroscopic behavior of the involved thermo-hydrodynamics. The flow separation is observed to be more when the apex of the cylinder is facing the flow. The average heat transfer, measured in terms of the Nusselt number, and the total drag on the cylinder are also found higher for that configuration.     

Effect of Prandtl Number on Free Convection from Two Cylinders in a Square Enclosure

This study explores the effect of Prandtl number on the laminar natural convection heat transfer to Newtonian fluids in a square enclosure consisting of one hot circular cylinder and one cold circular cylinder. The walls of the square enclosure are maintained isothermal and at the same temperature as the cold cylinder and the fluid medium. The governing partial differential equations have been solved numerically over the following ranges of conditions: Grashof number, 10 to 10⁵; Prandtl number, 0.7 to 100 (or the range of Rayleigh numbers as 7 to 10⁷); and relative positioning of the cylinders, ?0.25 to 0.25. However, the ratio of the radius of the cylinder to the side of the enclosure is held fixed at 0.2. Extensive results on the streamline and isotherm contours, the local Nusselt number distribution, and the average Nusselt number are discussed to delineate the influence of Grashof and Prandtl numbers on them for a given location with respect to the horizontal center line. The surface-averaged Nusselt number shows a positive dependence on Grashof and Prandtl numbers for a fixed location of the two cylinders. The heat transfer results have been correlated as a function of the Rayleigh number and geometric parameters, thereby enabling its prediction in a new application.     

Coupled Natural Convection and Radiation in a Horizontal Rectangular Enclosure Discretely Heated from Below

A numerical study is carried out to investigate the interaction between natural convection and thermal radiation in a horizontal enclosure filled with air and heated discretely from below. The results are presented for a cavity having an aspect ratio A _r  = L′/H′ = 10, while the Rayleigh number and the emissivity of the walls are varied in the ranges 10³ ≤ Ra ≤ 10⁶ and 0 ≤ ε ≤ 1, respectively. The results of the study, presented in terms of flow and temperature patterns, average convective, radiative and total Nusselt numbers, evaluated on the cold wall, show that the problem has multiple solutions. Each of these solutions is characterized by a specific flow structure, and its appearance and range of existence depend strongly on the parameters Ra and ε. The amount of heat evacuated through the cold surface is dependent on the type of solution.     

Natural Convection from a Heated Elliptic Cylinder with a Different Axis Ratio in a Square Enclosure

A detailed study about the free convection over a heated elliptic cylinder, placed at the center of a square cavity having cooled walls, is performed. Simulations are carried out for three Rayleigh numbers (10⁴, 10⁵, and 10⁶) and two cavity aspect ratios (CR = 2.5 and 5.0) for different axis ratio (AR). The effect of AR on fluid flow and heat transfer characteristics for varying Rayleigh number and cavity aspect ratio are analyzed. The influence of AR is phenomenal at higher Ra and lower CR. At higher Ra, thermal plumes are observed above the cylinder for different ARs. Bicellular vortices are formed at low Ra by changing CR. The surface-averaged Nusselt number (Nu _avg ) increases with increasing AR and Ra. The value of Nu _avg increases with decreasing CR, and a correlation for Nu _avg in terms of AR is obtained for each CR.     

Lattice Boltzmann Simulation of Natural Convection in a Differentially Heated Square Enclosure Containing Heat Generating Low‐Prandtl Number Fluid

Lattice Boltzmann simulations were conducted for the free convective flow of a low‐Prandtl number (Pr = 0.0321) fluid with internal heat generation in a square enclosure having adiabatic top and bottom walls and isothermal side walls. The problem of free convection with volumetric heat source has represented itself in connection with advanced engineering applications, such as water‐cooled lithium–lead breeder blankets for nuclear fusion reactors and liquid metal sources of spallation neutrons for subcritical fission systems. A single relaxation time (SRT) thermal lattice Boltzmann method (LBM) was employed. While applying SRT, a D2Q9 model was used to simulate the flow field and temperature field. Results have been obtained for various Rayleigh numbers characterizing internal and external heating from 10³ to 10⁶. Flow and temperature fields in terms of stream function and isotherms in the enclosure were predicted for these cases. The temperature of the fluid in the enclosure was found higher than the heated wall temperature at high values of internal Rayleigh numbers. The internal heat generation affected the rate of heat transfer significantly as two convection loops are observed in the enclosure. The average Nusselt number at the heated and cold wall was determined for all the cases.     

The Effect of a Blockage on Forced Convection Heat Transfer From a Heated Square Cylinder to Power-Law Fluids

Forced convection heat transfer characteristics of a long, heated square cylinder blocking the flow of a power-law fluid in a channel is numerically investigated in this study. In particular, the role of the power-law index n, Reynolds number Re, Prandtl number Pr, and blockage ratio β(=B/H) on the rate of heat transfer from a square cylinder in a channel has been studied over the following ranges of conditions: 0.5 ≤ n ≤ 1.8, 60 ≤ Re ≤ 160, β = 1/4, 1/2, and 0.7 ≤ Pr ≤ 50. A semi-explicit finite-volume method is used on a nonuniform collocated grid arrangement. The third-order QUICK and the second-order central difference schemes are used to discretize the convective and diffusive terms, respectively, in the momentum and energy equations. Irrespective of the type of behavior of fluid (different values of n), the average Nusselt number increases as the blockage ratio increases. Similar to the unconfined flow configuration, the average Nusselt number increases monotonically with Reynolds and Prandtl numbers for both values of the blockage ratio and for all values of power-law index considered here. Further insights into the heat transfer phenomenon are provided by presenting isotherm contours in the vicinity of the cylinder for a range of values of the Reynolds number, Prandtl number, and power-law index for the two values of β considered in this work.     

Natural Convection Heat Transfer and Entropy Generation in a Porous Trapezoidal Enclosure Saturated with Power-Law Non-Newtonian Fluids

Abstract

In the present study, natural convection heat transfer and its associated entropy generation in a porous trapezoidal enclosure saturated with a power-law non-Newtonian fluid has been numerically investigated. Horizontal walls of the enclosure are assumed to be adiabatic while the side walls are considered to be kept at a constant temperature. A continuum-based approach is adapted here to model the fluid flow through porous media and the Darcy’s law is modified to account for non-Newtonian rheological behavior of the fluid. The obtained governing equations are discretized using the finite volume method and a detailed parametric study is undertaken to account for the effects of various relevant parameters of the problem on the heat transfer and entropy generation rates. It was shown that the impact of the power-law index on both entropy generation and heat transfer significantly intensifies in a convection-dominated flow regime inside the enclosure, especially for a shear thinning liquid. Moreover, heat transfer rate and entropy generation increase as the sidewall angle is elevated.     

Laminar Natural Convection of Bingham Fluids in Square Cross-Sectioned Cylindrical Annular Cavity with Differentially Heated Vertical Walls Subjected to Constant Heat Fluxes

Steady-state numerical simulations have been conducted to investigate natural convection of yield stress fluids obeying Bingham model in square cross-sectioned axisymmetric cylindrical annular enclosure with vertical walls subjected to constant heat fluxes for nominal Rayleigh number range of 10³ to 10⁶, nominal Prandtl number of 10 to 10³ for different values of internal cylinder radius. It is found that the mean Nusselt number on the inner periphery increases (decreases) with increasing nominal Rayleigh (Bingham) number due to strengthening (weakening) of thermal advection. However, the values of the mean Nusselt number on the inner periphery obtained for Bingham fluids are smaller than that obtained for Newtonian fluids for the same set of nominal Rayleigh and Prandtl numbers. The mean Nusselt number normalized by the corresponding value obtained for pure conductive transport increases with increasing internal radius before asymptotically approaching the mean Nusselt number for a square enclosure. This suggests that the ratio of the convective to the conductive transport strengthens with increasing cylinder radius in the cylindrical annular cavity. Detailed physical explanations have been provided for the effects of the aforementioned parameters on the mean Nusselt number on the inner periphery and correlations have been proposed for the mean Nusselt number on the inner periphery for both Newtonian and Bingham fluids.     

Influences of Boundary Condition on Laminar Natural Convection of Bingham Fluids in Square Cross-Sectioned Cylindrical Annular Enclosures with Differentially Heated Vertical Walls

Numerical simulations have been carried out to analyze steady-state laminar natural convection of yield stress fluids obeying Bingham model in square cross-sectioned cylindrical annular enclosures with differentially heated vertical walls for both constant wall temperature and constant wall heat flux boundary conditions for active walls. The simulations have been performed under the assumption of axisymmetry for a nominal Rayleigh number range of 10³ to 10⁶ and nominal Prandtl number range of 10 to 10³ for different ratio of internal cylinder radius to cylinder height range of 0.125 to 16. The mean Nusselt number on the inner periphery for the constant wall heat flux configuration has been found to be smaller than that in the case of constant wall temperature configuration for a given set of values of nominal Rayleigh and Prandtl numbers for both Newtonian and Bingham fluid cases. The mean Nusselt number normalized by the corresponding value obtained for pure conductive transport increases with increasing internal radius before approaching the corresponding mean Nusselt number for square enclosures regardless of the boundary conditions. Detailed physical explanations have been provided for the effects of the aforementioned parameters on the mean Nusselt number on the inner periphery. Finally, the new Nusselt number correlations have been proposed for laminar natural convection of both Newtonian and Bingham fluids in square cross-sectioned cylindrical annular enclosures for both constant wall temperature and constant wall heat flux boundary conditions.     

Natural Convection in a Differentially Heated Cavity with Parallel Heat-Generating Baffles

Natural convection in a horizontal differentially heated square cavity containing two vertical heat generating baffles is studied numerically. The baffles are assumed to generate heat uniformly at the same or different rates. Asymptotic steady-state results for the vorticity–stream function formulation are presented in the form of streamline and isotherm plots. The fluid flow, heat transfer, and average Nusselt number are investigated for different heat generation ratios and spacing between the baffles. Convection within the cavity gets augmented for increasing values of heat generation ratio. When the two baffles are located very near the cavity walls, an increase in heat generation ratio induces a strong buoyancy convective flow. When they are very close to each other an increase in heat generation ratio strengthens the innermost cell around the baffles, which in turn drives the global flow at a faster rate through a pair of intermediate inner cells. It is found that the blocking effect of the baffles strongly depends on heat generation ratio and spacing between the baffles. The heat transfer rate varies nonlinearly against spacing between the baffles, and the possible physical reason is given.     

The Effects of Magnetization Saturation on Thermomagnetic Convection in a Locally Heated Square Enclosure

Steady-state simulation study of thermomagnetic convection is presented for a square cavity with localized heat-sources. The external magnetic field conforms to Maxwell's equations. Effects of magnetization saturation of the ferrofluid medium on heat transfer enhancement are studied by invoking Langevin's law. Thermal interactions between the heaters and the fluid at convection-dominated regimes are visualized through streamline, heatline, and isotherm plots. The variation of Nu_avg,heater is depicted for increasing dipole strength of the magnetic field sources. Fluid-magnetization contours explain the layout of the Kelvin force-field. The positions of the magnets for maximum value of Nu_avg,heater are determined. For the same heat generation rate, effects of enclosure dimensions on heat transfer augmentation are studied.     

Entropy Generation Minimization Versus Thermal Mixing Due to Natural Convection in Differentially and Discretely Heated Square Cavities

Uniform temperature distribution is a key parameter in many thermal processing applications. A considerable amount of additional energy is used to enhance the fluid mixing in order to maintain the temperature uniformity, but that affects the overall efficiency of the process. In this article, an alternate approach is proposed for maintaining uniform temperature via various distributed/discrete heating strategies while maintaining the minimal entropy generation. The system of laminar natural convection in differentially and discretely heated square cavities filled with various materials (molten metals, air, aqueous solutions, oils) is considered, and finite element simulations are performed for a range of Rayleigh numbers (Ra = 10³–10⁵). Entropy generation is evaluated using finite-element basis sets for the first time in this work, and the derivatives at particular nodes are estimated based on the functions within adjacent elements. Analysis of entropy generation in each case is carried out and a detailed investigation of entropy production due to local heat transfer and fluid friction irreversibilities is presented. It is found that a high thermal mixing may not be the optimal situation for achieving uniform temperature distribution based on entropy production. A greater degree of temperature uniformity with moderate thermal mixing may correspond to minimum entropy generation with distributed heating. Further, based on entropy generation minimization approach, it has been thermodynamically established that the distributed heating methodology with multiple heat sources may be the energy efficient strategy for attaining adequate uniform temperature distribution with minimum entropy generation.     

Free-Forced Convection from a Heated Longitudinal Horizontal Cylinder

Abstract

Finite-difference solutions of a longitudinal three-dimensional boundary layer along a heated horizontal cylinder are presented for Pr = 1 and 10. The numerical results are compared with earlier asymptotic solutions. The comparison shows that the asymptotic solution is valid only for a narrow region close to the leading edge. Downstream from this narrow region, the asymptotic solution overpredicts the buoyancy effect along the bottom of the cylinder and underpredicts it along the top. The numerical solutions indicate that the flow becomes free-convection dominant far downstream from the leading edge even when Gr/Re² is small.     

Mixed Convection from a Heated Sphere in Bingham Plastic Fluids

In this work, the steady and laminar mixed-convection heat transfer from an isothermal sphere immersed in Bingham plastic fluids has been investigated in the aiding-buoyancy configuration. The pertinent coupled equations of motion and thermal energy have been solved numerically over the following ranges of conditions: Richardson number, 0 ≤ Ri ≤ 2, Bingham number, 0 ≤ Bn ≤ 10, Reynolds number, 0.1 ≤ Re ≤ 100 and Prandtl number, 10 ≤ Pr ≤ 100. Flow characteristics like streamlines, pressure coefficient, morphology of yielded/unyielded regions and drag coefficient are discussed extensively. Similarly, isotherms, local Nusselt number and average Nusselt number are thoroughly examined to develop an overall understanding of the corresponding heat transfer characteristics. All else being equal, in contrast to the positive role of the aiding-buoyancy free convection in Newtonian and power-law fluids, due to the fluid yield stress, heat transfer is impeded in viscoplastic fluids. While the average value of the Nusselt number is influenced by four dimensionless groups, namely, Reynolds number, Bingham number, Prandtl number and Richardson number, by using novel scaling, it has been possible to consolidate the present results via the use of the Colburn j-factor in a simple form. This is particularly suitable for predicting the value of the Nusselt number in a new application.     

含热源方腔的耦合自然对流换热的三维数值模拟

对具有内热源方腔的稳态层流耦合自然对流换热进行了三维的数值模拟，采用的模拟代码基于连续介质计算力学的开源库OpenFoam，解决了自然对流换热与固体传热的耦合问题。对外壁面为常温、方腔内充满含体积热源流体的自然对流计算结果表明，温度场、速度场与非耦合的工况有很大差异。     

Natural Convection in a Trapezoidal Enclosure Heated from the Side with a Baffle Mounted on Its Upper Inclined Surface

A numerical investigation examined the effects on heat transfer of mounting baffles to the upper inclined surfaces of trapezoidal cavities. Two thermal boundary conditions are considered. In the first, the left, short vertical wall is heated while the right, long vertical wall is cooled (buoyancy assisting mode along the upper inclined surface of the cavity). In the second, the right, long vertical wall is heated while the left, short vertical wall is cooled (buoyancy opposing mode along the upper inclined surface of the cavity). For each boundary condition, computations are performed for three baffle heights, two baffle locations, four Rayleigh number values, and three Prandtl number values. Results are displayed in terms of streamlines, isotherms, and local and average Nusselt number values. For both boundary conditions, predictions reveal a decrease in heat transfer in the presence of baffles, with its rate generally increasing with increasing baffle height and Prandtl number. For a given baffle height, a higher decrease in heat transfer is generally obtained with baffles located close to the short vertical wall. Average Nusselt number correlations for both boundary conditions are presented.