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.     

Laminar Natural Convection of Bingham Fluids in a Square Enclosure with Vertical Walls Subjected to Constant Heat Flux

In this study, two-dimensional steady-state simulations of laminar natural convection in square enclosures with vertical sidewalls subjected to constant heat flux have been carried out, where the enclosures are considered to be completely filled with a yield-stress fluid obeying the Bingham model. Yield stress effects on heat and momentum transport are investigated for nominal values of Rayleigh number (Ra) in the range 10³–10⁶ and a Prandtl number (Pr) range of 0.1–100. It is found that the mean Nusselt number Nu increases with increasing values of Rayleigh number for both Newtonian and Bingham fluids. However, Nu values obtained for Bingham fluids are smaller than that obtained in the case of Newtonian fluids with the same nominal value of Rayleigh number Ra due to weakening of convective transport. The mean Nusselt number Nu in the case of Bingham fluids is found to decrease with increasing Bingham number, and for large values of Bingham number Bn, the value settles to unity (Nu = 1.0) as heat transfer takes place principally due to thermal conduction. The Nu values for the vertical walls subjected to constant heat flux are smaller than the corresponding values in the same configuration with constant vertical wall temperatures (for identical values of nominal Rayleigh, Prandtl, and Bingham numbers). However, the value of Bingham number at which Nu approaches to unity remains the same for both constant wall temperature and constant wall heat flux configurations. It is demonstrated that for small values of Bingham number Nu increases with increasing Prandtl number, but the opposite behavior occurs for large values of Bingham number. New correlations are proposed for the mean Nusselt number Nu for both Newtonian and Bingham fluids for square enclosures with vertical walls subjected to constant heat flux, which are shown to satisfactorily capture the correct qualitative and quantitative behavior of Nu in response to changes in Ra, Pr, and Bn.     

Natural Convection in Power-Law Fluids in a Square Enclosure from Two Differentially Heated Horizontal Cylinders

Laminar natural convection has been numerically investigated from two differentially heated horizontal cylinders in a square enclosure filled with power-law fluids. Two basic configurations, namely, vertical- and diagonal-alignment of the cylinders at various locations have been considered. The coupled continuity, momentum and energy equations have been solved numerically to elucidate the effect of the Grashof number (10²–10⁴), Prandtl number (0.7–100) and power-law index (0.2–2) for a range of symmetric and asymmetric locations of the cylinders. The velocity and temperature fields are visualized in terms of streamlines, isothermal contours and plots of the local and average Nusselt number for different positions of the cylinders. The occurrence of the power-law index in the definitions of the Grashof and Prandtl numbers accentuates the interplay between the viscous, inertial and buoyancy forces thereby leading to nonlinearity in the observed trends. The presence of the dead zones coupled with the dominance of conduction under certain conditions strongly influences the overall heat transfer. All else being equal, it is possible to improve heat transfer for asymmetric positioning of the cylinders, especially at high values of the Prandtl number and Grashof number in shear-thinning fluids. A predictive correlation has been developed thereby enabling the estimation of the heat transfer coefficient in a new application in terms of the geometric and kinematic parameters.     

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.     

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.     

MHD Turbulent and Laminar Natural Convection in a Square Cavity utilizing Lattice Boltzmann Method

In this study, the lattice Boltzmann method was used to solve the turbulent and laminar natural convection in a square cavity. In this paper a fluid with Pr = 6.2 and different Rayleigh numbers (Ra = 10³, 10⁴,10⁵ for laminar flow and Ra = 10⁷, 10⁸,10⁹ for turbulent flow) in the presence of a magnetic field (Ha = 0, 25, 50, and 100) was investigated. (Results show that the magnetic field drops the heat transfer in the laminar flow as the heat transfer behaves erratically toward the presence of a magnetic field in a turbulent flow. Moreover, the effect of the magnetic field is marginal for a turbulent flow in contrast with a laminar flow.The greatest influence of the magnetic field is observed at Ra = 10⁵ from Ha = 0 to 100 as the heat transfer decreases significantly.     

Transient Natural Convection in Vertical Channels Symmetrically Heated at Uniform Heat Flux

A numerical transient analysis of natural convection in air between two vertical parallel plates, heated at uniform heat flux, is carried out. The problem is two-dimensional and laminar and the full Navier-Stokes and energy equations are employed. The control volume method is used to discretize the equations on a uniform grid. Results are given at different aspect ratio values and Rayleigh number values. The simulation allows detection of complex structures of the flow inside and outside the channel. Temperature profiles as a function of time show an overshoot and undershoot increase at the lowest aspect ratio and highest Rayleigh number. Inside the channel conductive and convective regimes as well as an inverse fluid motion are observed. Transient average Nusselt number presents oscillations before the steady-state.     

Heat Transfer of Bingham Fluids in an Annular Duct with Viscous Dissipation

ABSTRACT

Analytical expressions for the velocity and temperature profiles in a fully-developed laminar Poiseuille flow through a concentric annular duct of a Bingham fluid with constant wall heat flux at the inner and outer wall, in the presence of viscous dissipation are deduced and presented. It is found that the proportion of the heat generated by viscous dissipation near the outer wall increases with an increase of the dimensionless flow parameter, and a decrease of the duct radius ratio. The Nusselt numbers are first calculated based on a single bulk temperature for the entire duct cross section. The possibility of performing calculations of the relevant parameters discussed in this work is available via the Supplementary Material as an Excel file. Also in this work a new approach is employed, where two different bulk temperatures are used, one for each side of the radial location in the temperature profile whose derivative is zero. With this new approach the Nusselt number behavior is free of either unphysical discontinuities or negative values. As a consequence, the Nusselt number values better reflect the actual heat transfer coefficient at the walls and are more comparable with the heat transfer inside ducts when the temperature profile is symmetric.     

Numerical Study of Laminar Natural Convection in a Side-Heated Trapezoidal Cavity at Various Inclined Heated Sidewalls

Steady natural convection of air flow in a two-dimensional side-heated trapezoidal room was investigated numerically using a non-orthogonal, collocated finite-volume grid system. The considered geometry has an inclined left heated sidewall, a vertical right cooled sidewall, and two insulated horizontal upper and lower walls. Computations are performed for seven values of the heated sloping wall angle, three different values of aspect ratio, and five Rayleigh number values. Results are displayed in terms of streamlines, isotherms, and both local and average Nusselt number values. The principal result of this work is the great dependence of the flow fields and the heat transfer on the inclination angle, the aspect ratio, and the Rayleigh number. A correlation between the average Nusselt number, Rayleigh number, heated sloping wall angle, and aspect ratio is proposed.     

Lattice Boltzmann Simulation of Natural Convection in a Square Cavity with Linearly Heated Wall(s)

In this study, the lattice Boltzmann method is used in order to investigate the natural convection in a cavity with linearly heated wall(s). The bottom wall is heated uniformly and the vertical wall(s) are heated linearly, whereas the top wall is insulated. Investigation has been conducted for Rayleigh numbers of 10³ to 10⁵, while Prandtl number is varied from 0.7 to 10. The effects of an increase in Rayleigh number and Prandtl number on streamlines, isotherm counters, local Nusselt number and average Nusselt number are depicted. It has been observed that the average Nusselt number at the bottom wall augments with an increase in Prandtl number.     

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.     

Natural Convection Heat Transfer Enhancement in a Square Cavity Periodically Cooled from Above

The present article reports numerical results of natural convection within an air filled square cavity with its horizontal walls submitted to different heating models. The temperature of the bottom horizontal surface (hot temperature) is maintained constant, while that of the opposite surface (cold temperature) is varied sinusoidally with time. The remaining vertical walls are considered adiabatic. The parameters governing the problem are the amplitude (0 ≤ a ≤ 0.8) and the period (τ ≥ 0.001) of the variable temperature, the Rayleigh number (10³ ≤ Ra ≤ 7 × 10⁶), and the Prandtl number (Pr = 0.71). In constant cooling conditions (a = 0), up to three different solutions (monocellular flow MF, bicellular vertical flow BVF, and bicellular horizontal flow BHF) are obtained. Their existence ranges are delineated and, in the limits of the existence range of each solution, the transitions observed are identified and described. In the variable cooling conditions, the effect of the amplitude and the period of the exciting temperature on fluid flow and heat transfer is examined in the case of the MF, and BHF for specific values of Ra. Results are presented in terms of Ψ _max (t), Ψ _min (t), Nu(t) and streamlines, heatlines, and isotherms during the evolutions of selected flow cycles. In comparison with the constant heating conditions, it is found that the variable cooling temperature could lead to a drastic change in the flow structure and the corresponding heat transfer, especially at specific low periods of the cold variable temperature. This leads to a resonance phenomenon characterized by an important increase in heat transfer by about 46.1% compared to the case of a constant cold temperature boundary condition.     

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.     

Laminar Free Convection in Power-law Fluids in a Right Angle Triangular Duct with Heated Base

ABSTRACT

Laminar free convection in power-law fluids in a triangular duct is studied numerically to delineate the effects of the height-to-base ratio of the enclosure (0.2 to 2), power-law index (0.2 to 1.8), Grashof number (10 to 10⁴) and Prandtl number (0.7 to 100). The heat transfer is analyzed for the heated base with the other two walls being cold. Detailed kinematics is characterized by the formation of multiple recirculating zones ranging from two to four cells. Shear rate contours provide additional insights about the variation of the local viscosity in the fluid. Heatlines and the values of the Bejan number over the range of conditions are calculated to delineate the contributions of the entropy generation due to thermal effects and viscous dissipation. At low Grashof and/or Prandtl numbers, conduction dominates the overall heat transfer and this transition between the conduction and convection-dominated regimes is captured in terms of a modified Rayleigh number. The effect of aspect ratio on the Nusselt number is modulated by the values of Grashof and Prandtl numbers and power-law index. The present results have been consolidated via the use of a modified Rayleigh number for estimating the value of average Nusselt number in a new application.     

Mixed Convection of Nanofluid Flows in a Square Lid-Driven Cavity Heated Partially From Both the Bottom and Side Walls

This article presents the results of a numerical study on mixed convection within a square lid-driven which was heated simultaneously by two finite heat sources on the bottom and side walls, and also filled with nanofluids. The results were presented for different nanofluids. The governing equations were solved using a finite volume approach by the SIMPLE algorithm. The effects of the Rayleigh number, Reynolds number, the solid volume fraction, the dimensions of heaters, and their locations on the streamlines and isotherms contours were investigated accurately. Also, the effects of the above parameters on the average Nusselt number along two heat sources were precisely presented. Moreover, variations of the average Nusselt number of two heaters were considered whenever one heater was fixed and the location of the other heater was varied along the wall. In addition, variations of the length of one heater on the average Nusselt number were also studied whenever the length of the other heater was fixed.     

Numerical Investigation of the Entropy Generation Due to Natural Convection in a Partially Heated Square Cavity Filled With Nanofluids

A comprehensive numerical investigation has been carried out on the heat transfer performance and entropy generation within a rectangular cavity containing nanofluid. The cavity consists of two heat sources located on the bottom and a side wall. The effects of influential parameters including type and concentration of nanoparticles, radius of corner, width and thickness of heaters, heater distance from corners and aspect ratio of the enclosure were studied. The results showed that the Nusselt number enhanced by increasing the aspect ratio of the cavity, the distance of heaters from the corners, and concentration of nanoparticle and applying Cu as nanoparticle while it reduced by increasing the radius of the corner and the width and thickness of the heat sources. The entropy generation was found to be profoundly minimized by lowering the Rayleigh number. In addition, the entropy generation was attenuated by increasing the Eckert number, corner radius, the distance from the corner and concentration of nanoparticles and using Al₂O₃ as nanoparticle. On the other hand, increasing the aspect ratio of the cavity, width and thickness of the heaters augmented the entropy generation. Interestingly, the entropy generation of the system was lowered by just increasing the distance of one heater from the corner, whereas increasing the thickness and width of one heater resulted in larger entropy generation. This study provides valuable insight into the change in the amount of heat transfer and entropy by altering the geometry as well as fluid properties.     

Laminar Mixed-Convection Heat Transfer in a Lid-Driven Cavity with Modified Heated Wall

In this numerical study, steady laminar mixed-convection heat transfer in a two-dimensional square lid-driven cavity with a modified heated wall is investigated over a range of Richardson numbers, including 0.01, 1, and 10. The heated bottom wall of the cavity is characterized by rectangular, triangular, and sinusoidal wave shapes. The cooled top wall of the cavity is sliding with constant velocity, while the vertical walls are kept stationary and adiabatic. The governing equations are solved using a finite-volume technique. The results are presented in the form of streamlines, isotherms, and Nusselt number plots. The effects of the number of undulations and the amplitude on the flow field and heat transfer are also investigated. The predicted results demonstrate that the heat transfer enhancement is generally observed with the modification of the heated wall, while the improvement is found to be more profound for the case of rectangular wave and at low Richardson number.     

封闭方腔自然对流换热的研究

论述和分析了封闭方腔自然对流换热的研究进展,研究了采用商业软件FLUENT对此模拟的方法以及换热规律研究的可行性,所获得的数值模拟结果与研究文献做了对比分析,其模拟结果是正确的,由此表明:采用FLUENT软件通过数值模拟方法不仅能获得封闭腔自然对流换热的结果,还能研究其换热规律,是解决封闭腔自然对流换热的有效工具。     

封闭方腔内叉排管间熔融盐自然对流换热

以硝酸锂熔融盐为流体介质,对封闭方腔内交叉排布的四根热管间的介质自然对流换热进行了数值模拟,研究了热管不同位置和不同瑞利数对方腔内介质自然对流换热特性的影响。结果表明:管间距ε=0.5时,腔体内温度场和流场的对称性开始打破;ε=0.4和0.5时,下管及左右两管产生的热羽流冲刷上管热边界层,使上管局部Nu_φ分别在圆周角φ=180°、φ=280°和φ=80°处出现增强;ε=0.6时,上管不再受下管羽流影响。随着管间距增大,上管平均Nu_m越来越小,下管Nu_m越来越大;ε=0.4~0.6时,管间距对左右两管Nu_m影响很小,但ε=0.7时,左右两管平均换热均明显增强。     

Entropy Generation Due to Natural Convection in a Partially Open Cavity with a Thin Heat Source Subjected to a Nanofluid

This article presents a numerical study of natural convection cooling of a heat source mounted inside the cavity, with special attention being paid to entropy generation. The right vertical wall is partially open and is subjected to copper–water nanofluid at a constant low temperature and pressure, while the other boundaries are assumed to be adiabatic. The governing equations have been solved using the finite volume approach, using SIMPLE algorithm on the collocated arrangement. The study has been carried out for a Rayleigh number in the range 10³ < Ra < 10⁶, and for solid volume fraction 0 <? <0.05. In order to investigate the effect of the heat source and open boundary location, six different configurations are considered. The effects of Rayleigh numbers, heat source and open boundary locations on the streamlines, isotherms, local entropy generation, Nusselt number, and total entropy generation are investigated. The results indicate that when open boundary is located up, the fluid flow augments and hence the heat transfer and Nusselt number increase and total entropy generation decreases.