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.     

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.     

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.     

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.     

Numerical Studies on Natural Convection in a Trapezoidal Enclosure With Discrete Heating

Abstract

A steady state laminar natural convection flow in a trapezoidal enclosure with discretely heated bottom wall, adiabatic top wall, and constant temperature cold inclined walls is performed. The finite volume based commercial code “ANSYS-FLUENT” is used to investigate the influence of discrete heating on natural convection flows in a trapezoidal cavity. The numerical solution of the problem covers various Rayleigh numbers ranging from 10³ to 10⁶, non-dimensional heating length ranging from 0.2 to 0.8 and Prandtl number is 0.7. The performance of the present numerical approach is represented in the form of streamfunction, temperature profile and Nusselt number. Heat transfer increases with increase of Rayleigh numbers at the corners of the cavity for same heating length from center of the bottom wall. However, the heat transfer rate is less and almost constant for the Rayleigh numbers considered. It is found that the average Nusselt number monotonically increases with increase of Rayleigh number and length of heat source. The variation of local and average Nusselt numbers is more significant for larger length of heating than smaller one. The heat transfer correlations useful for practical design problems have been predicted.     

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.     

Criteria for Predicting Transitions in Free Convection Heat Transfer From Isothermal Convex Bodies in Fluids With Any Prandtl Number: A New Analytical Model

This article presents criteria for predicting transitions in free convection from isothermal convex bodies in fluids with any Prandtl number (conduction–laminar and laminar–turbulent transitions). Laminar–turbulent transition results for vertical plates happening in the vicinity of Grashof number 10⁹ in fluids with any Prandtl number and other geometries show that this criterion based on Grashof number is a better choice compared to that of the Rayleigh number, as the Prandtl number yields a large change in the transition Rayleigh number. Besides, for convex geometries other than vertical plates, the laminar–turbulent transition occurs at values slightly less than Grashof number 10⁹ owing to longitudinal vortices bringing about separation over their surface and helping laminar–turbulent transition to arise sooner. On the other side, the results of the conduction–laminar transition criterion for different convex bodies disclose that the Rayleigh number is superior to the Grashof number. Likewise, the present results for five different geometries over a wide range of Prandtl numbers are compared with the available experimental data, and excellent agreement is found. This reveals that the proposed criteria is powerful enough to predict the two transition phenomena in free convection heat transfer from isothermal convex bodies for a wide range of convex geometries in fluids with any Prandtl number.     

Momentum and heat transfer from a square cylinder in power-law fluids

Numerical solutions are sought, using FLUENT, to the mass, momentum and thermal energy equations for the 2-D flow of power-law fluids over a cylinder of square cross-section. The major thrust of this work is to delineate the values of the Reynolds number denoting the onset of flow separation and the limits of the steady flow regime for both shear-thinning and shear-thickening type fluids. Extensive results are reported on streamline and vorticity contours over wide ranges of power-law index (0.2–1.4) corroborating the occurrence of these two transitions. Having established the limits of the steady flow regime, drag and Nusselt number results are obtained in this regime as functions of the Reynolds number (0.1–40), of Prandtl number (0.7–100) for highly shear-thinning fluids (power-law index < 0.5) thereby extending the range of currently available results to that encountered in practical applications. The Nusselt number shows positive dependence on both the Reynolds and Prandtl numbers. Also, shear-thinning characteristics can augment the rate of heat transfer by up to 100% under appropriate conditions.     

Buoyancy effects on laminar heat transfer in the thermal entrance region of horizontal rectangular channels with uniform wall heat flux for large prandtl number fluid

The Graetz problem for fully developed laminar flow in horizontal rectangular channels with uniform wall heat flux is extended by including buoyancy effects in the analysis for the case of large Prandtl number fluid. A general formulation valid for all Prandtl numbers is presented and the limiting case of large Prandtl number is approached by a numerical method. The typical developments of temperature profile, wall temperature and secondary flow in the thermal entrance region are presented for the case of square channel γ = 1. Local Nusselt number variations are presented for the aspect ratios γ = 0.2, 0.5, 1, 2 and 5 with Rayleigh number as parameter. Due to entry and secondary flow effects, a minimum Nusselt number occurs at some distance from the entrance, depending on the magnitude of Rayleigh number. This behavior is similar to that observed in the thermal entrance region where the transition from laminar to turbulent flow occurs. The effect of Rayleigh number is seen to decrease the thermal entrance length, and the Graetz solution, neglecting buoyancy effects, is found to be applicable only when Rayleigh number is less than about 10³. A study of the practical implications of large Prandtl number on heat transfer results for hydrodynamically and thermally fully developed case reveals that the present heat transfer results are valid for Prandtl number ranging from order 10 to infinity.     

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.     

WALL HEAT CONDUCTION EFFECT ON NATURAL CONVECTION IN AN ENCLOSURE FILLED WITH A NON-DARCIAN POROUS MEDIUM

ABSTRACT

A numerical analysis has been made of the conjugate natural convection in a rectangular enclosure filled with a fluid-saturated porous medium and surrounded with four solid walls. The conductance of the walls is assumed to be much greater than that of the cavity filled with a porous medium. The main objective was to investigate the influences of the ratio of thermal conductivity of the wall to that of the fluid-porous matrix composite, the Darcy-modified Rayleigh number, the Prandtl number, and the aspect ratio. The streamlines and isotherms are presented; also, the local and average Nusselt numbers are presented along the interface between walls and cavity. A non-Darcian model was employed and the numerical method was SIMPLE-C. The numerical results indicate that the wall heat conduction effects decrease the heat transfer rate. When the wall heat conduction is considered, the greater the conductance of the solid walls surrounding the cavity, the greater is the rate of heat transfer.     

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.     

Flows and heat transfer of the transition to an unsteady state in a finned cavity for different Prandtl numbers

Flows and heat transfer of the transition to an unsteady state in a finned cavity are studied for Prandtl numbers (Pr) from 0.1 to 100 and Rayleigh numbers (Ra) from 10⁷ to 10¹⁰. Transient flows are described in the finned cavity. Critical Rayleigh numbers of the transition to an unsteady state are obtained for different Prandtl numbers and the relation between two dimensionless parameters is given. The spectral analysis is applied for the oscillations of unsteady flows and the dominant frequency dependent on governing parameters is presented. Heat transfer of the transition to an unsteady flow is quantified and the corresponding relations dependent on the Prandtl number and Rayleigh number are gained. It is demonstrated that the flow rate and the Nusselt number of the finned cavity significantly increase due to the presentence of the fin, which depend on the Prandtl number and the Rayleigh number.     

Momentum and Heat Transfer from a Semi-Circular Cylinder to Power-Law Fluids in the Vortex Shedding Regime

Flow and heat transfer from a semi-circular cylinder to power-law fluids has been studied in the laminar vortex shedding regime for the range of conditions as follows: Reynolds number, 40 ≤ Re ≤ 140; Prandtl number, 0.7 ≤ Pr ≤ 50; and power-law index, 0.2 ≤ n ≤ 1.8. Extensive results are presented in terms of the streamline, vorticity and isotherm profiles, drag and lift coefficients, Strouhal number, and Nusselt number. The influence of Reynolds number, Prandtl number, and power-law index are seen to be more prominent in shear-thinning fluids (n < 1) than that in shear-thickening (n > 1) fluids. The shear-thinning fluid behavior enhances the rate of heat transfer, whereas shear-thickening fluid behavior impedes it.     

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.     

Laminar Flow and Heat Transfer to a Non-Newtonian Fluid in an Entrance Region of a Square Duct with Prescribed Constant Axial Wall Heat Flux

Abstract

Forced-convection heat transfer information as a function of the pertinent nondimensional numbers is obtained numerically for laminar incompressible non-Newtonian fluid flow in the entrance region of a square duct with simultaneously developing temperature and velocity profiles for constant axial wall heat flux with uniform peripheral wall temperature. The power-law model characterizes the non-Newtonian behavior.

Finite-difference representations are developed for the equations of the mathematical model, and numerical solutions are obtained assuming uniform inlet velocity and temperature distributions. Results are presented for local and mean Nusselt numbers as functions of the Graetz number and the Prandtl number in the entrance region. Comparisons are made with previous analytical work for Newtonian fluids. The results show a strong effect of the Prandtl number on the Nusselt numbers with fully developed and uniform velocity profiles representing the lower and upper limits, respectively. The results provide a new insight into the true three-dimensional character of the pseudoplastlc fluid flow in the entrance region of a square duct and are accurate.     

NUMERICAL EXPERIMENTS WITH FLUID CONVECTION IN TILTED NONRECTANGULAR ENCLOSURES

Numerical experiments were performed on an incompressible fluid contained in a tilted nonrectangular enclosure. Rayleigh numbers of l0²-l0⁵ and Prandtl numbers of 0.001-100 are considered. The wall angles are 22.5°, 45°, and 77.5° with aspect ratios of 3 and 6. Results indicate that the heat transfer and fluid motion within the enclosure are strong functions of Rayleigh number, Prandtl number, and orientation angle of the enclosure. For Rayleigh numbers greater than 1& and Prandtl numbers greater than 0.1, a minimum and a maximum mean Nusselt number occurred as the angle of orientation was increased from 0° to 360°. A transition in the mode of circulation occurred at the angles corresponding to the minimum or maximum rate of heat transfer.     

Three-Dimensional Numerical Study of Natural Convective Heat Transfer of Liquid in a Cubic Enclosure

ABSTRACT

Steady-state laminar natural convection in a cubic enclosure with a cold vertical wall and two hot square heaters with constant temperature on the opposite wall is studied numerically. The enclosure is filled with various liquids. Three-dimensional Navier–Stokes Equations are solved by employing the SIMPLE algorithm. Computations are performed for a range of Rayleigh number from 10³ to 10⁷ while enclosure aspect ratio varies from 0.05 to 1.6. The effects of Rayleigh number, enclosure aspect ratio, and Prandtl number on heat transfer characteristics are studied in detail. The results show that the flow field is very complex and heat transfer from the two heaters is not the same. The effects of Prandtl number are negligible in the range from 5 to 140 with other parameters kept constant. This allows the use of liquids such as water for studying other dielectric liquids, provided the flow geometry and other nondimensional parameters are similar. The overall Nusselt number increases markedly with Rayleigh number. It is also affected by enclosure aspect ratio. It attains the maximum value when aspect ratio is in the range of 0.1–0.2 and decreases as enclosure aspect ratio varies from 0.2 to 1.6. Also, various settings of cooling face and arrangement of heaters are investigated, and the results show that they have considerable effects on heat transfer of both heaters.     

Simulation of natural convection and entropy generation of MHD non-Newtonian nanofluid in a cavity using Buongiorno's mathematical model

In this paper, natural convection and entropy generation of non-Newtonian nanofluid, using the Buongiorno's mathematical model in a cavity in the presence of a uniform magnetic field has been analyzed by Finite Difference Lattice Boltzmann method (FDLBM). The cavity is filled with nanofluid which the mixture shows shear-thinning behavior. This study has been performed for the certain pertinent parameters of Rayleigh number (Ra = 10⁴ and 10⁵), Hartmann number (Ha = 0, 15, 30), buoyancy ratio number (N_r = 0.1, 1, and 4), power-law index (n = 0.4–1), Lewis number (Le = 1, 5, and 10), Thermophoresis parameter (N_t = 0.1, 0.5, 1), and Brownian motion parameter (N_b = 0.1, 1, 5). The Prandtl number is fixed at Pr = 1. The Results indicate that the augmentation of Hartmann number causes heat and mass transfer to drop. The increase in Rayleigh number enhances heat and mass transfer for various power-law indexes. The alteration of the power-law index changes heat and mass transfer. In addition, the rise of Hartmann number declines the shear-thinning behavior. The increase in the Lewis number augments mass transfer while it causes heat transfer to drop. The rise of the Thermophoresis and Brownian motion parameters ameliorate mass transfer and declines heat transfer significantly. The augmentation of buoyancy ratio number enhances heat and mass transfer. The augmentation of the power-law index declines various entropy generations in different Rayleigh numbers and Hartmann numbers. The increase in Hartmann number declines total entropy generation in different Rayleigh numbers. In addition, the rise of Rayleigh number and Hartmann number causes Bejan number to drop in various power-law indexes. The enhancement of the Lewis number provokes the total irreversibility to rise. Further, the total entropy generation increases as the buoyancy ratio number augments. It was shown that the increase in the Brownian motion and Thermophoresis parameters enhance the total irreversibility.     

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.