Laminar free convection from a horizontal semi-circular cylinder to power-law fluids

Extensive numerical results on the flow and thermal fields are presented for free convection from a semi-circular cylinder (flat base upward) immersed in quiescent power-law fluids for the following ranges of conditions: Grashof number, 10 ? Gr ? 10⁵, Prandtl number, 0.72 ? Pr ? 100, and power-law index, 0.2 ? n ? 1.8. The heat transfer characteristics are analyzed in terms of the isotherm patterns, local and average Nusselt number as functions of the pertinent dimensionless parameters. The flow field is visualized in terms of the streamline patterns adjacent to the surface of the cylinder for a range of values of the Grashof number, Prandtl number and power-law index. A separated flow region forms at as low values of the Prandtl number as Pr = 0.72 for n ? 1 (Newtonian and shear-thickening fluids); whereas for shear-thinning fluids (n < 1), the flow remains attached to the cylinder surface over the range of conditions encompassed here. The bubble size grows with Grashof number and it shrinks with Prandtl number. In order to quantify the deviation from the Newtonian behaviour, the normalized values of average Nusselt number are analyzed as a function of the power-law index. In addition, a correlation is proposed for average Nusselt number as a function of the Grashof number, Prandtl number and power-law index. In general terms, shear-thinning fluid behaviour enhances heat transfer whereas shear-thickening has adverse influence on it.     

Natural convection in a rectangular enclosure containing an oval-shaped heat source and filled with Fe3O4/water nanofluid

This work concerns with the study of natural convection heat transfer in rectangular cavities with an inside oval-shaped heat source filled with Fe₃O₄/water nanofluid. The finite element method is employed to solve the governing equations for this problem. Average Nusselt numbers are presented for a wide range of Rayleigh number (10³ ≤ Ra ≤ 10⁵), volume fraction of nanoparticles (0 ≤ ϕ ≤ 14%), and four different size and shapes of the heat source. Depending on concentration of the nanoparticle, geometry of the heat source, and the value of Rayleigh number different behaviors are monitored for average Nusselt numbers. Configuration of the heat source dictates a significant change on the behavior of the average Nusselt number, while addition of the nanoparticles has a negative effect on the magnitude of Nusselt number for this problem.     

Effects of viscous dissipation during forced convection of power-law fluids in microchannels

The effect of viscous dissipation in forced convection of power-law fluids through microchannels of different cross-sectional geometries is studied numerically over the ranges of power-law index, 0.8 ≤ n ≤ 1.2 and Brinkman number, 0.001 ≤ Br ≤ 0.1 while keeping Péclet number constant at Pe = 10. Two types of thermal boundary conditions, namely, uniform wall temperature (T₂) and uniform heat flux (H₂), have been employed at the microchannel wall and the results of the temperature fields are expressed in terms of Nusselt number. The interplay between the fluid rheology and viscous dissipation effect gives rise to significant alteration in the net convective transport and thus can be beneficial in the thermal design of biofluidic devices.     

Effect of aspect ratio on natural convection in power-law liquids from a heated horizontal elliptic cylinder

The influence of aspect ratio and shear-dependent viscosity on free convection heat transfer from a horizontal heated elliptic cylinder in power-law fluids has been investigated. In particular, the coupled momentum and energy equations have been solved numerically over the following ranges of conditions: Grashof number, 10 ? Gr ? 10⁵; Prandtl number, 0.72 ? Pr ? 100; power-law index, 0.3 ? n ? 1.5 and aspect ratio, 0.2 ? E ? 5. The new extensive results demonstrate the influence of the Grashof number (Gr), Prandtl number (Pr), power-law index (n) and aspect ratio (E) on the macroscopic heat and momentum transfer characteristics like local and average values of Nusselt number (Nu) and drag coefficients (C_D). Further insights are developed by examining the structure of the flow and temperature fields adjacent to the cylinder. Broadly speaking, all else being equal, shear- thinning fluid behaviour promotes heat transfer whereas shear-thickening viscosity has a deleterious effect on it with reference to that in Newtonian fluids. Also, the rate of heat transfer gradually increases as the cylinder shape passes from blunt to slender with respect to the direction of gravity. Finally, the present numerical values of the Nusselt number are correlated using a simple analytical form which facilitates interpolation of the present results for the intermediate values of the governing parameters. The paper is concluded by presenting detailed comparisons with the previous numerical and experimental results available in the literature, especially in Newtonian fluids.     

Effect of nanofluid variable properties on natural convection in enclosures filled with a CuO–EG–Water nanofluid

This work focuses on the study of natural convection heat transfer characteristics in a differentially-heated enclosure filled with a CuO–EG–Water nanofluid for different published variable thermal conductivity and variable viscosity models. The problem is given in terms of the vorticity–stream function formulation and the resulting governing equations are solved numerically using an efficient finite-volume method. Comparisons with previously published work are performed and the results are found to be in good agreement. Various results for the streamline and isotherm contours as well as the local and average Nusselt numbers are presented for a wide range of Rayleigh numbers (Ra = 10³–10⁵), volume fractions of nanoparticles (0 ≤ φ ≤ 6%), and enclosure aspect ratios (½ ≤ A ≤ 2). Different behaviors (enhancement or deterioration) are predicted in the average Nusselt number as the volume fraction of nanoparticles increases depending on the combination of CuO–EG–Water variable thermal conductivity and viscosity models employed. In general, the effects the viscosity models are predicted to be more predominant on the behavior of the average Nusselt number than the influence of the thermal conductivity models. The enclosure aspect ratio is predicted to have significant effects on the behavior of the average Nusselt number which decreases as the enclosure aspect ratio increases.     

Numerical modeling of natural convection in an open cavity with two vertical thin heat sources subjected to a nanofluid

This paper presents a numerical study of natural convection cooling of two heat sources vertically attached to horizontal walls of a cavity. The right opening boundary is subjected to the copper–water nanofluid at 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 the Rayleigh number in the range 10⁴ ≤ Ra ≤ 10⁷, and for solid volume fraction 0 ≤ φ ≤ 0.05. In order to investigate the effect of heat source location, three different placement configurations of heat sources are considered. The effects of both Rayleigh numbers and heat source locations on the streamlines, isotherms, Nusselt number are investigated. The results indicate that the flow field and temperature distributions inside the cavity are strongly dependent on the Rayleigh numbers and the position of the heat sources. The results also indicate that the Nusselt number is an increasing function of the Rayleigh number, the distance between two heat sources, and distance from the wall. In addition it is observed that the average Nusselt number increases linearly with the increase in the solid volume fraction of nanoparticles.     

Effects of severe blockage on heat transfer from a sphere in power-law fluids

The momentum and thermal energy equations describing the forced convection heat transfer from a heated sphere settling at the axis of a long cylindrical tube filled with a power-law fluid have been solved numerically. The extensive new results reported herein encompass wide ranges of conditions as: Reynolds number, 1 ≤ Re ≤ 100; Prandtl number, 5 ≤ Pr ≤ 100, power-law index, 0.2 ≤ n ≤ 2 and blockage ratio, 0.5 ≤ λ ≤ 0.95. The range of values of the power-law index (n) used here include both the shear-thinning (n < 1) and shear-thickening (n > 1) fluid behaviours. The overall heat transfer is strongly modulated by Re, n and λ depending upon whether the recirculation region is formed in the rear of the sphere and/or on the proximity of the tube wall. Furthermore, the results reported herein elucidate the effect of the type of thermal boundary condition (isothermal or isoflux) on the surface of the sphere as well as that of the velocity profile (uniform or fully developed Poiseuille profile) in the tube. Overall, the average Nusselt number bears a positive dependence on the Reynolds and Prandtl numbers and blockage ratio. The shear-thinning behaviour (n < 1) augments heat transfer over and above the corresponding Newtonian value whereas shear-thickening behaviour (n > 1) adversely influences it. The present numerical results (~ 4000 data) have been consolidated by incorporating the blockage factor into an existing expression valid for λ = 0 for Newtonian fluids.     

Pressure drop predictions for laminar flows of extended modified power law fluids in rectangular ducts

This paper reports on a numerical study of the friction factor-modified Reynolds number product, f × Re_M, for fully developed, laminar flows of pseudoplastic and dilatant fluids in rectangular ducts. Constitutive equations for the apparent viscosity that span from the low shear rate Newtonian region through the high shear rate Newtonian region were utilized, and a shear rate parameter was defined that determines the flow regime where the duct is operating. Numerical results for f × Re_M in all flow regimes are included along with correlation equations. Errors associated with applying power law solutions to flows of weakly non-Newtonian fluids are discussed.     

Steady natural convection flows in a square cavity with linearly heated side wall(s)

The present numerical study deals with natural convection flow in a closed square cavity when the bottom wall is uniformly heated and vertical wall(s) are linearly heated whereas the top wall is well insulated. Non-linear coupled PDEs governing the flow have been solved by penalty finite element method with bi-quadratic rectangular elements. Numerical results are obtained for various values of Rayleigh number (Ra) (10³ ⩽ Ra ⩽ 10⁵) and Prandtl number (Pr) (0.7 ⩽ Pr ⩽ 10). Results are presented in the form of streamlines, isotherm contours, local Nusselt number and the average Nusselt as a function of Rayleigh number.     

Non-Newtonian power-law flow and heat transfer across a pair of side-by-side circular cylinders

Flow and heat transfer of non-Newtonian power-law fluids across a pair of identical circular cylinders in side-by-side arrangement are investigated numerically by solving the continuity, momentum and energy equations along with the appropriate boundary conditions. The numerical calculations are performed in an unconfined computational domain for the following range of physical parameters: Reynolds number, Re = 1–40 and power-law index, n = 0.4–1.8 (covering shear-thinning, n < 1; Newtonian, n = 1 and shear-thickening, n > 1 behaviors) for gap ratio, T/D = 1.5–4.0 at a constant Prandtl number of 50. The global characteristics such as drag coefficients and average Nusselt number, etc. are calculated and the representative streamline and isotherm contours are presented for the above range of conditions. It has been found that the individual and overall drag coefficients decrease and the average Nusselt number increases with Reynolds number for all T/D and n considered here. The heat transfer is found higher in shear-thinning fluids than Newtonian fluids and followed by shear-thickening fluids for 1.5 ? T/D ? 4.0 and 1 ? Re ? 40.     

Momentum and heat transfer characteristics of a semi-circular cylinder immersed in power-law fluids in the steady flow regime

The continuity, momentum and energy equations describing the flow and heat transfer of power-law fluids over a semi-circular cylinder have been solved numerically in the two-dimensional steady flow regime. The influence of the Reynolds number (Re), Prandtl number (Pr) and power-law index (n) on the local and global flow and heat characteristics have been studied over wide ranges of conditions as follows: 0.01 ? Re ? 30, 1 ? Pr ? 100 and 0.2 ? n ? 1.8. The variation of drag coefficient and Nusselt number with the Reynolds number, Prandtl number and power-law index is shown over the aforementioned ranges of conditions. In addition, streamline and isotherm profiles along with the recirculation length and distribution of pressure coefficient and Nusselt number over the surface of the semi-circular cylinder are also presented to gain further insights into the nature of the underlying kinematics. The wake size (recirculation length) shows almost linear dependence on the Reynolds number (Re ? 1) for all values of power-law index studied herein. The drag values show the classical inverse variation with the Reynolds number, especially for shear-thinning fluids at low Reynolds numbers. The point of maximum pressure coefficient is found slightly displaced from the front stagnation point for highly shear-thinning fluids, whereas for shear-thickening and Newtonian fluids, it coincides with the front stagnation point. For fixed values of the Prandtl number and Reynolds number, the rate of heat transfer decreases with the gradual increase in power-law index; this effect is particularly striking at high Prandtl numbers due to the thinning of the thermal boundary layer. Conversely, as expected, shear-thinning behavior facilitates heat transfer and shear-thickening impedes it. The effect of power-law index on both momentum and heat-transfer characteristics is seen to be appreciable at low Reynolds numbers and it gradually diminishes with the increasing Reynolds number.     

Effects of thermal boundary conditions on natural convection flows within a square cavity

A numerical study to investigate the steady laminar natural convection flow in a square cavity with uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls has been performed. A penalty finite element method with bi-quadratic rectangular elements has been used to solve the governing mass, momentum and energy equations. The numerical procedure adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁵ and Prandtl number Pr, 0.7 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous Dirichlet boundary conditions. Non-uniform heating of the bottom wall produces greater heat transfer rates at the center of the bottom wall than the uniform heating case for all Rayleigh numbers; however, average Nusselt numbers show overall lower heat transfer rates for the non-uniform heating case. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes, power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Forced convection heat transfer from an elliptical cylinder to power-law fluids

Forced convection heat transfer to incompressible power-law fluids from a heated elliptical cylinder in the steady, laminar cross-flow regime has been studied numerically. In particular, the effects of the power-law index (0.2 ? n ? 1.8), Reynolds number (0.01 ? Re ? 40), Prandtl number (1 ? Pr ? 100) and the aspect ratio of the elliptic cylinder (0.2 ? E ? 5) on the average Nusselt number (Nu) have been studied. The average Nusselt number for an elliptic cylinder shows a dependence on the Reynolds and Prandtl numbers and power-law index, which is qualitatively similar to that for a circular cylinder. Thus, heat transfer is facilitated by the shear-thinning tendency of the fluid, while it is generally impeded in shear-thickening fluids. The average Nusselt number values have also been interpreted in terms of the usual Colburn heat transfer factor (j). The functional dependence of the average Nusselt number on the dimensionless parameters (Re, n, Pr, E) has been presented by empirically fitting the numerical results for their easy use in process design calculations.     

Comparative evaluation of the performance of sinuous symmetrically-heated natural convection channels

Buoyancy-driven flows established in vertical channels are investigated numerically. Two different sinuous morphologies for a vertical channel are proposed as alternatives to a typical (straight) vertical channel; one is a sinus-shaped channel, and the other is a three-segment channel. The objective is to carry out a comparative evaluation of the thermal and dynamic performance of the proposed configurations, comparing their performance with that corresponding to a typical vertical channel. The low-Reynolds k − ω turbulence model is employed to simulate the transitional or fully turbulent flow. The average Nusselt number and the dimensionless mass flow rate are obtained for Rayleigh numbers ranging 10² ≤ Ra_H ≤ 10¹⁵, with an aspect ratio of the channel equal to 0.1. Sizeable increases in the Nusselt number and the mass flow rate are found for the sinuous configurations, under given conditions.     

Laminar natural convection in a square cavity: Low Prandtl numbers and large density differences

Steady natural convection at low Prandtl numbers caused by large density differences in a square cavity heated through the side walls is investigated numerically and theoretically. An appropriate dimensionless parameter characterizing the density differences of the working fluid is identified by the Gay-Lussac number. The Boussinesq assumption is achieved when the Gay-Lussac number tends to zero. The Nusselt number is derived for the ranges in Rayleigh number 10 ? Ra ? 10⁸, in Prandtl number 0.0071 ? Pr ? 7.1 and in Gay-Lussac number 0 ? Ga < 2. The effects of the Rayleigh, Prandtl and Gay-Lussac numbers on the Nusselt number are discussed on physical grounds by means of a scale analysis. Finally, based on physical arguments, a heat transfer correlation is proposed, valid for all Prandtl and Gay-Lussac number ranges addressed.     

Numerical investigation for natural convection of a nanofluid in an inclined L-shaped cavity in the presence of an inclined magnetic field

The problem of natural convection in an inclined L-shaped enclosure filled with Cu/water nanofluid that operates under differentially heated walls in the presence of an inclined magnetic field is presented in this paper. The fully implicit finite difference method is used to solve the governing equations. A comparison with previously published results in special case of the present study is performed and a very good agreement is found. Heat transfer and fluid flow are examined for parameters of the Hartmann number (0 ≤ Ha ≤ 100), the nanoparticles volume fraction (0% ≤ ϕ ≤ 20%), the cavity inclination angle (0° ≤ ϑ ≤ 300°), the magnetic field inclination angle (0° ≤ γ ≤ 270°), the cavity aspect ratio (0.25 ≤ AR ≤ 0.6) and the Rayleigh number (10³ ≤ Ra ≤ 10⁶). It is found that, the presence of the magnetic field in the fluid region causes a significant reduction in the fluid flow and heat transfer characteristics. Also, a good enhancement in the heat transfer rate can be obtained by adding the copper nanoparticles to the base fluid.     

Natural convection in a nanofluid-filled eccentric annulus with constant heat flux wall: A lattice Boltzmann study with immersed boundary method

A numerical investigation of natural convection in a Cu–water nanofluid-filled eccentric annulus with constant heat flux wall is presented. The governing equations of the flow and temperature fields are solved by lattice Boltzmann method (LBM), and the Dirichlet and Neumann boundary conditions are treated using the immersed boundary method (IBM). Influences of the Rayleigh number (10³ ≤ Ra ≤ 10⁷), eccentricity (ε = −0.625,0 and 0.625), nanoparticles volume fraction (0 ≤ ϕ ≤ 0.03) and radial ratio (rr = 2.33,2.6 and 3) on the streamlines, isotherms and Nusselt number are studied. It is found that the inclusion of the nanoparticles into pure fluid changes the flow pattern. And the Nusselt number has a positive relationship with nanoparticle volume fraction, Rayleigh number and radial ratio. Also, it can be confirmed that Nusselt number in the case with negative eccentricity (ε = −0.625) is larger than the others.     

Natural convection in a square cavity filled with a porous medium: Effects of various thermal boundary conditions

Natural convection flows in a square cavity filled with a porous matrix has been studied numerically using penalty finite element method for uniformly and non-uniformly heated bottom wall, and adiabatic top wall maintaining constant temperature of cold vertical walls. Darcy–Forchheimer model is used to simulate the momentum transfer in the porous medium. The numerical procedure is adopted in the present study yields consistent performance over a wide range of parameters (Rayleigh number Ra, 10³ ⩽ Ra ⩽ 10⁶, Darcy number Da, 10⁻⁵ ⩽ Da ⩽ 10⁻³, and Prandtl number Pr, 0.71 ⩽ Pr ⩽ 10) with respect to continuous and discontinuous thermal boundary conditions. Numerical results are presented in terms of stream functions, temperature profiles and Nusselt numbers. Non-uniform heating of the bottom wall produces greater heat transfer rate at the center of the bottom wall than uniform heating case for all Rayleigh numbers but average Nusselt number shows overall lower heat transfer rate for non-uniform heating case. It has been found that the heat transfer is primarily due to conduction for Da ⩽ 10⁻⁵ irrespective of Ra and Pr. The conductive heat transfer regime as a function of Ra has also been reported for Da ⩾ 10⁻⁴. Critical Rayleigh numbers for conduction dominant heat transfer cases have been obtained and for convection dominated regimes the power law correlations between average Nusselt number and Rayleigh numbers are presented.     

Numerical thermal analysis and optimization of a water jet impingement cooling with VOF two-phase approach

The fluid flow and heat transfer characteristics of a free-surface liquid jet impingement cooling have been investigated numerically. The slot jet with water impinging normally on a flat plate is employed. To describe the turbulent structure, the turbulent governing equations are solved by a control-volume-based finite-difference method with a power-law scheme and the well-known turbulence model, which are associated with wall function. Numerical computations have been conducted with variations of jet exit Reynolds number (11,000 ≤ Re_d ≤ 17,000), dimensionless jet-to-surface distance (3 ≤ H/d₀ ≤ 12), dimensionless jet width (1 ≤ B/d₀ ≤ 2), and the heat flux (140 kW/m² ≤ q″ ≤ 280 kW/m²). The theoretical model developed is validated by comparing the numerical predictions with available experimental data in the literature. Under the studied ranges, the variations of local Nusselt numbers by hydraulic diameter Nu_d of the dimensionless jet-to-surface distance 3 ≤ H/d₀ ≤ 12 along the flat plate decrease monotonically from its maximum value at the stagnation point. In addition, the shape of the inlet area and jet-to-surface distance are optimized by using the response surface methodology (RSM) and the genetic algorithm (GA) method after solutions are carefully validated with available experimental results in the literature. Based on the optimal results, the optimum condition is in H/d₀ = 7.86 and B/d₀ = 2 for this physical model.     

Bénard convection from a circular cylinder in a packed bed

Bénard convection around a circular heated cylinder embedded in a packed bed of spheres is studied numerically. The Forchheimer–Brinkman–extended Darcy momentum model with the Local Thermal Non-Equilibrium energy model is used in the mathematical formulation for the porous layer. The governing parameters considered are the Rayleigh number (10³ ≤ Ra ≤ 5 × 10⁷) and the thermal conductivity ratio (0.1 ≤ k_r ≤ 10,000). The structural properties of the packed bed are kept constant as: cylinder-to-particle diameter ratio D/d = 20 and porosity ε = 0.5, while the Prandtl number is fixed at Pr = 0.71. It is found that the presence of the porous medium suppresses significantly the strong free convection produced in the empty enclosure, and reduces considerably the high intensity of the pair of vortices generated behind the cylinder. Also, the results show that the porous medium can play the role of insulator or enhancer of heat transfer from the heat source, depending mainly on their thermal conductivities regardless of the Rayleigh number.