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.     

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.     

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.     

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.     

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.     

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.     

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.     

Numerical Investigation of Unsteady Natural Convection in an Inclined Square Enclosure With Heat-Generating Porous Medium

The unsteady laminar natural convection in an inclined square enclosure with heat-generating porous medium whose heat varies by a cosine function is investigated by a thermal equilibrium model and the Brinkman–Darcy–Forchheimer model numerically, with the four cooled walls of closure as isothermal. The numerical code based on the finite-volume method has been validated by reference data before it was adopted. Influence of dimensionless frequency and inclination angle on heat transfer characteristics in a square enclosure, such as flow distribution, isotherm, averaged Nusselt number on each wall, and time-averaged Nusselt number, are discussed, with specified value for Rayleigh number = 10⁸, Darcy number = 10^?4, Prandtl number = 7, porosity = 0.4, and specific heat ratio = 1. It is found that when the internal heat source varies by cosine, the Nusselt numbers of the four walls oscillate with the same frequency as the internal heat source; however, phase difference occurs. Moreover, frequency has little impact on time-averaged Nusselt number of the four walls, which is different from the phenomenon discovered in natural convection with suitable periodic varying wall temperature boundary condition. Moreover, inclination angle plays an important role in the heat transfer characteristics of the walls studied.     

Conjugate turbulent natural convection with surface radiation in air filled rectangular enclosures

Conjugate turbulent natural convection and surface radiation in rectangular enclosures heated from below and cooled from other walls, typically encountered in Liquid Metal Fast Breeder Reactor (LMFBR) subsystems, have been investigated by a finite volume method for various aspect ratios. The formulation comprises the standard two equation k–ε turbulence model with physical boundary conditions (no wall functions), along with the Boussinesq approximation, for the flow and heat transfer. As far as radiation is concerned, the radiosity – irradiation formulation for a transparent fluid of Prandtl number 0.7 has been employed. The conjugate coupling on the walls has been handled by using a fin type formulation. The Rayleigh number based on the width of the enclosure is varied from 10⁸ to 10¹² and the aspect ratio is varied from 0.5 to 2.0. Detailed results including stream lines, temperature profiles, and convective, radiative and overall Nusselt numbers are presented. A correlation for the mean convection Nusselt number in terms of Rayleigh number and aspect ratio is proposed for design purposes. The influence of the wall emissivity and the external heat transfer coefficient on the heat transfer from the enclosure has also been investigated.     

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.     

An analytical study of pulsating laminar heat convection in a circular tube with constant heat flux

Pulsating laminar convection heat transfer in a circular tube with constant wall heat flux is investigated analytically. The results show that both the temperature profile and the Nusselt number fluctuate periodically about the solution for steady laminar convection, with the fluctuation amplitude depending on the dimensionless pulsation frequency, ω*, the amplitude, γ, and the Prandtl number, Pr. It is also shown that pulsation has no effect on the time-average Nusselt numbers for pulsating convection heat transfer in a circular tube with constant wall heat flux.     

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.     

Critical condition for Rayleigh-Bénard convection of Bingham fluids in rectangular enclosures

Two-dimensional steady-state numerical simulations have been conducted for laminar Rayleigh-Bénard convection of Bingham fluids in rectangular enclosures to analyse the critical Rayleigh number Ra_crit for which convection ceases to influence the thermal transport and thermal conduction becomes the principal heat transfer mechanism. The influences of Bingham number Bn on the critical Rayleigh number Ra_crit have been investigated for different values of aspect ratio (height: length) AR (ranging from 1/4 to 4) and nominal Prandtl number Pr (ranging from 10 to 500) for both constant wall temperature (CWT) and constant wall heat flux (CWHF) boundary conditions for the horizontal walls. It has been found that Ra_crit increases with increasing values of Bn and AR, regardless of the boundary condition. The values of Ra_crit have been found to be greater in the case of CWT boundary condition than in the CWHF configuration for AR ≤ 1, whereas an opposite trend is obtained for AR > 1 for Bingham fluids. Additionally, Ra_crit has been found be insensitive to the change of Pr for Newtonian fluids (i.e. Bn = 0), whereas Ra_crit increases with increasing Pr for Bingham fluids irrespective of the boundary condition. A detailed scaling analysis has also been performed to elucidate the effects of Bn ,Pr , AR on Ra_crit for Bingham fluids. The results of scaling analysis and numerical findings have been utilised to propose a new correlation for Ra_crit for both Newtonian and Bingham fluids in the case of both CWT and CWHF boundary conditions.     

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.     

Viscous dissipation effects on the asymptotic behaviour of laminar forced convection for Bingham plastics in circular ducts

The present study concentrates on the effects of viscous dissipation and the yield shear stress on the asymptotic behaviour of the laminar forced convection in a circular duct for a Bingham fluid. It is supposed that the physical properties are constant and the axial conduction is negligible. The asymptotic temperature profile and the asymptotic Nusselt number are determined for various axial distributions of wall heat flux which yield a thermally developed region. It is shown that if the asymptotic value of wall heat flux distribution is vanishes, the asymptotic value of the Nusselt number is zero. The case of the asymptotic wall heat flux distribution non-vanishing giving a value of the Nusselt number dependent on the Brinkman number and on the dimensionless radius of the plug flow region was also analysed. For an infinite asymptotic value of wall heat flux distributions, the asymptotic value of the Nusselt number depends on the dimensionless radius of the plug flow region and on the dimensionless parameter which depends on the asymptotic behaviour of the wall heat flux. The condition of uniform wall temperature and convection with an external isothermal fluid were also considered. The comparison with other existing solutions in the literature in the Newtonian case is analysed.     

Numerical study of the laminar natural convection flow around horizontal isothermal cylinder

Natural convection around a horizontal circular cylinder under constant temperature or constant heat flux conditions in an infinite space has been the subject of numerous investigations in recent years. However, these studies use the inflow-outflow boundary in cylindrical coordinates that gives a sensible error, especially when the Rayleigh number is small. This investigation, that enters within the framework of general study dealing with natural convection from an array of cylinders, states the problem in cartesian coordinates system, involves the use of a control-volume method and resolves various apparent redundancies in boundary conditions. This problem was investigated numerically for laminar case by solving the full vorticity transport equation together with the stream function and energy equations. Results are obtained for 10¹ < Ra < 10⁶ and for Prandtl number equal to 0.7. Further, typical isotherms, local and mean Nusselt number, velocities and temperature distribution are clarified in detail. The numerical approach presented here appears to be sufficiently versatile to permit computation of a vertical array of cylinders.     

Exergy transfer characteristics of forced convective heat transfer through a duct with constant wall heat flux

The examination of exergy transfer characteristics caused by forced convective heat transfer through a duct with constant wall heat flux for thermally and hydrodynamic fully developed laminar and turbulent flows has been presented. The exergy transfer Nusselt number is put forward and the dependence relationships of the exergy transfer Nusselt number on the heat transfer Nusselt number, Reynolds number and Prandtl number are obtained. Expressions involving relevant variables for the local and mean convective exergy transfer coefficient, non-dimensional exergy flux and exergy transfer rate, etc. have been derived. By reference to a smooth duct, the numerical results of exergy transfer characteristics for fluids with different Prandtl number are obtained and the effect of the Reynolds number and non-dimensional cross-sectional position on exergy transfer characteristics is analyzed. In addition, the results corresponding to the exergy transfer and energy transfer are compared.     

The development of laminar natural-convective flow in a vertical uniform heat flux duct

An account of a theoretical and experimental study of laminar natural-convective flow in heated vertical ducts is presented. The ducts are open-ended and circular in cross-section and their internal surfaces dissipate heat uniformly.

Temperature and velocity fields and the relationship between Nusselt and Rayleigh numbers were obtained by solving the governing equations by a step-by-step numerical technique. Two Rayleigh numbers are introduced, one expressed in terms of the uniform heat flux and the other in terms of the mean wall temperature. The influence that the Prandtl number has on the relationship between the Nusselt and Rayleigh numbers is discussed. Three inlet conditions were examined; they all gave the same Nusselt relationship at small Rayleigh numbers and the differences between the Nusselt relationships obtained at large Rayleigh numbers were only small.

Experimentally determined Nusselt numbers, with air as the convected fluid, agreed satisfactorily with the theoretical relationship.     

Laminar natural convection in an inclined cavity with a wavy wall

In the present work, a numerical study of the effect of a hot wavy wall of a laminar natural convection in an inclined square cavity, differentially heated, was carried out. This problem is solved by using the partial differential equations, which are the vorticity transport, heat transfer and stream function in curvilinear co-ordinates. The tests were performed for different inclination angles, amplitudes and Rayleigh numbers while the Prandtl number was kept constant. Two geometrical configurations were used namely one and three undulations.The results obtained show that the hot wall undulation affects the flow and the heat transfer rate in the cavity. The mean Nusselt number decreases comparing with the square cavity. The trend of the local heat transfer is wavy. The frequency of the latter is different from the undulated wall frequency.     

HEAT AND FLUID FLOW ACROSS A SQUARE CYLINDER IN THE TWO-DIMENSIONAL LAMINAR FLOW REGIME

The flow structure and heat transfer characteristics of an isolated square cylinder in cross flow are investigated numerically for both steady and unsteady periodic laminar flow in the two-dimensional regime, for Reynolds numbers of 1 to 160 and a Prandtl number of 0.7. The effect of vortex shedding on the isotherm patterns and heat transfer from the cylinder is discussed. Heat transfer correlations between Nusselt number and Reynolds number are presented for uniform heat flux and constant cylinder temperature boundary conditions.