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.     

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.     

Prandtl and capillary effects on heat transfer performance within laminar liquid–gas slug flows

This paper investigates a two-phase non-boiling slug flow regime for the purposes of enhancing heat transfer in microchannel heat sinks or compact heat exchangers. The primary focus is upon understanding the mechanisms leading to enhanced heat transfer and the effect of using different Prandtl number fluids, leading to variations in Capillary number. Experimental work was conducted using Infrared thermography and results are presented in the form of Graetz solution, spanning both the thermal entrance and fully developed flow regions. Nusselt numbers enhancements were observed throughout when data was reduced to account for void fraction. However, the gaseous void was also noted to demonstrate an artificial increase with greater thicknesses of the liquid film, due to higher Capillary numbers. Up to 600% enhancement in heat transfer rates were observed over conventional Poiseuille flow. This was verified through Nusselt number measurements over inverse Graetz number ranges from 10^?4 to 1 and slug length to channel diameter ratios from 0.88 to 32. Varying Prandtl and Capillary numbers caused notable effects in the transition region between entrance and fully developed flows. Significant Nu oscillations were observed for low Pr fluids due to internal circulation within the slug. However, these oscillations are observed to be damped out when higher Prandtl number fluids are employed. The thickness of the liquid film surrounding the gas bubbles is shown to have a significant influence on heat transfer performance. Overall, this study provides a greater understanding of the mechanisms leading to significant enhancements in heat exchange devices employing two-phase gas–liquid flows without boiling.     

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.     

Steady mixed convective flow and heat transfer from tandem square cylinders in a horizontal channel

The two-dimensional laminar steady mixed convective flow and heat transfer around two identical tandem square cylinders confined in a horizontal channel are simulated by the high-accuracy multidomain pseudo-spectral method. The blockage ratio of the channel is chosen as 0.1, whereas the spacing between the cylinders is fixed with four widths of the cylinder. The Prandtl number is fixed at 0.7, the Reynolds number (Re) is studied in the range 5?≤?Re?≤?60, and the Richardson number (Ri) demonstrating the influence of thermal buoyancy ranges from 0 to 1. Numerical results reveal that, with the thermal buoyancy effect, the mixed convective flow remains steady. The variations of the overall drag and lift coefficients and the Nusselt numbers, are presented and discussed. Furthermore, the influence of thermal buoyancy on fluid flow and heat transfer is discussed and analyzed.     

Mixed convection about a rotating sphereConvection mixte autour d'une sphere tournanteMischkonvektion an einer rotierenden kugelCмeшaHHaя кoHвeкция oкoлo вpaщaющeйCя Cфepы

The paper presents a theoretical analysis of flow and heat transfer characteristics of the effects of buoyancy force on laminar boundary layer over a rotating sphere in forced flow under two kinds of heating conditions: uniform wall temperature and uniform surface heat flux. By applying appropriate coordinate transformations and using Merk's types of series, the governing momentum and energy equations are reduced to a set of coupled ordinary differential equations, which depend on wedge, rotation and buoyancy parameters. Numerical computations are carried out for Prandtl numbers 0.7,1.0 and for various values of buoyancy and rotation parameters. For aiding flow, it is found that both the friction factor and the local Nusselt number increase with increasing buoyancy force. The local free stream velocity increases with buoyancy which, in turn, affects the friction coefficient and Nusselt number. The coupling between rotation and buoyancy results in increased overshooting of the velocity profiles in the vicinity of the rotating sphere. For an equivalent buoyancy effect, heating by uniform surface heat flux yields larger local Nusselt number than heating by uniform wall temperature. The ratio Nu_UHF/Nu_UWT is higher for the rotating sphere (as compared to a nonrotating case) and further the ratio increases as the sphere spins faster. The effect of free stream, rotation and buoyancy on the eruption of flow is examined and also a suggestion for further investigation is made.     

A Parametric Study of Prandtl Number Effects on Laminar Natural Convection Heat Transfer From a Horizontal Circular Cylinder to Its Coaxial Triangular Enclosure

A parametric study of Prandtl number effects on laminar natural convection heat transfer in a horizontal equilateral triangular cylinder with a coaxial circular cylinder is conducted. The Prandtl number is varied over a wide range from 10^?2 to 10⁵, which corresponds to a variety of working fluids. The governing equations with the Boussinesq approximation for buoyancy are iteratively solved using the finite volume approach. It is shown that the flow patterns and temperature distributions are unique for low-Prandtl-number fluids (Pr ≤ 0.1), and are nearly independent of Prandtl number when Pr ≥ 0.7. In addition, the inclination angle of the triangular enclosure is found to noticeably affect the variations of the local Nusselt number, and to have insignificant influence on the average Nusselt numbers for low Rayleigh numbers when Pr ≥ 0.7.     

Axial heat conduction effects in the entrance region of laminar duct flows: Correlations for the local Nusselt number

Axial heat conduction effects within the fluid can be important for duct flows if the Prandtl number is low (liquid metals) or if the hydraulic diameter of the channel is very small (micro-channels). Exact analytical solutions for the extended Graetz problem for such situations are well known and documented in literature. In the present paper suitable correlations are developed for the local Nusselt number in the thermal entrance region in a parallel plate channel and in a pipe. Correlations are given for different thermal boundary conditions and for axial positions before (x < 0) and after (x > 0) the jump in wall temperature or wall heat flux.     

Effects of viscous dissipation on the heat transfer in a forced pipe flow. Part 2: Thermally developing flow

In this part of the study, consideration is given to thermally developing laminar forced convection in a pipe including viscous dissipation. The axial heat conduction in the fluid is neglected. Two different thermal boundary conditions are considered: the constant heat flux (CHF) and the constant wall temperature (CWT). Both the wall heating (the fluid is heated) case and the wall cooling (the fluid is cooled) case are considered. The distributions for the developing temperature and local Nusselt number in the entrance region are obtained. Results show that the temperature profiles and local Nusselt number are influenced by the Brinkman number (Br) and the thermal boundary condition used for the wall. Significant viscous dissipation effects have been observed for large Br.     

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.     

Heat and fluid flow across a rotating cylinder dissipating uniform heat flux in 2D laminar flow regime

Free-stream flow and forced convection heat transfer across a rotating cylinder, dissipating uniform heat flux, are investigated numerically for Reynolds numbers of 20–160 and a Prandtl number of 0.7. The non-dimensional rotational velocity (α) is varied from 0 to 6. Finite volume based transient heatline formulation is proposed. For Re = 100, the reasons for the onset/suppression of vortex shedding at a critical rotational velocity is investigated using vorticity dynamics. At higher rotational velocity, the Nusselt number is almost independent of Reynolds number and thermal boundary conditions. Finally, a heat transfer correlation is proposed in the 2D laminar flow regime. Cylinder rotation is an efficient Nusselt number reduction or cylinder-surface temperature enhancement technique.     

COMBINED RADIATION AND LAMINAR MIXED CONVECTION IN THE THERMAL ENTRANCE REGION OF HORIZONTAL ISOTHERMAL RECTANGULAR CHANNELS

The interaction of thermal radiation with laminar mixed convection for a gray fluid in the thermal entrance region of a horizontal isothermally heated rectangular channel is numerically investigated. The vorticity-velocity formulation of the Navier-Stokes equation and the integral formulation for radiation solved by finite-element nodal approximation are employed. The effects of radiation and convection on local Nusselt number, the development of bulk temperature, and the friction factor are examined. Secondary flow induced by the buoyancy effects leads to a significant enhancement in heat transfer in the entrance region. The result shows that the existence of secondary flow causes fluctuations in local Nusselt number and this phenomenon is reduced by the effect of thermal radiation and a large aspect ratio.     

Double-Diffusive Natural Convection in a Closed Annulus

A numerical study for steady laminar double-diffusive natural convection within a vertical closed annulus is examined with constant temperature and mass species (concentration) differences imposed across the vertical walls. The annulus has an aspect ratio of 1 and a curvature ratio of 2, while the fluid Prandtl number is 7. In this paper the problem is defined and the numerical solution procedure is validated. Moreover, the effect of buoyancy ratio on the flow structure and rite resulting heal and mass transfer rates is presented. It is determined that buoyancy ratio is the primary factor that defines flow structure, including concentration—dominated (buoyancy force) opposing flow, transitional flow, thermal-dominated flow, or concentration-dominated aiding flow. The relationship for buoyancy ratios, in the range -10 ≤ n ≤ 10, and the average NusseU and Sherwood numbers have been obtained for a thermal Rayleigh number of 50,000 and a Lewis number of 5. Future papers wilt include the effect of thermal Rayleigh number, Lewis number, and various geometric parameters on the flow structure and heat and nusi transfer.     

Natural convection heat transfer of water in a horizontal circular gap

An experimental study on the natural convection heat transfer on a horizontal downward facing heated surface in a water gap was carried out under atmospheric pressure conditions. A total of 700 experimental data points were correlated using Rayleigh versus Nusselt number in various forms, based on different independent variables. The effects of different characteristic lengths and film temperatures were discussed. The results show that the buoyancy force acts as a resistance force for natural convection heat transfer on a downward facing horizontal heated surface in a confined space. For the estimation of the natural convection heat transfer under the present conditions, empirical correlations in which Nusselt number is expressed as a function of the Rayleigh number, or both Rayleigh and Prandtl numbers, may be used. When it is accurately predicted, the Nusselt number is expressed as a function of the Rayleigh and Prandtl numbers, as well as the gap width-to-heated surface diameter ratio; and uses the temperature difference between the heated surface and the ambient fluid in the definition of Rayleigh number. The characteristic length is the gap size and the film temperature is the average fluid temperature.     

A universal entrance nusselt number for internal slip flow

A universal finite Nusselt number is found for laminar slip flow heat transfer at the entrance of a conduit. The Nusselt number expression is valid for both isothermal and isoflux thermal boundary conditions and for any conduit geometry. The value of the entrance Nusselt number is found to be dependent on two nondimensional parameters that include effects of rarefaction, fluid properties, and the fluid/wall interaction.     

Conjugate heat transfer with viscous dissipation in a microtube

Hydrodynamically developed, thermally developing, steady, laminar conjugate heat transfer of a liquid flow in the entrance region of a microtube is studied numerically. The finite volume method is used. The effects of the thermal conductivity ratio, the diameter ratio, the channel length and the viscous dissipation on the Nusselt number as well as on the temperature and the interface heat flux distribution are examined in detail. In the entrance region, large reductions are observed in the Nusselt number with increasing k_s/k_f and d_o/d_i. This is considered to be due to the axial conduction, which also extends to the exit region. The results also show that the Nusselt number decreases with increasing viscous dissipation for fixed values of k_s/k_f and d_o/d_i.     

Prandtl number dependence of laminar natural convection heat transfer in a horizontal cylindrical enclosure with an inner coaxial triangular cylinder

A numerical study of laminar convection heat transfer from a horizontal triangular cylinder to its concentric cylindrical enclosure is performed to investigate the Prandtl number effect on flow and heat transfer characteristics. The Prandtl number over several orders of magnitude (10^?2 < Pr < 10³) as well as different aspect ratios (AR = 1.2 and 2.0) and different Rayleigh numbers (Ra = 10³, 10⁴, 10⁵, and 10⁶) are considered. The finite volume approach is used to solve the governing equations, in which buoyancy is modeled via the Boussinesq approximation. The computed flow patterns and temperature fields are shown by means of streamlines and isotherms, respectively, and the local and average heat transfer coefficients are also presented. It is found that the flow and heat transfer characteristics for a low Prandtl number fluid (Pr = 0.03) are unique and they are almost independent of Prandtl number when Pr ? 0.7. The entire spectrum of Prandtl number investigated can be divided into three sections based on the variations of average heat transfer coefficients. In each section, correlating equations of the average Nusselt number to the Rayleigh number are proposed with the maximum deviation less than 3%.     

An Experimental Investigation of Heat Transfer Enhancement Mechanisms in Microencapsulated Phase-Change Material Slurry Flows

This article investigates laminar heat transfer characteristic of two-phase microencapsulated phase-change material (MPCM) suspension flows within minichannels under a constant wall heat flux boundary. Capsules containing paraffin wax with phase-change temperature between 36.1°C and 38.1°C are examined and found to be well suited for electronics cooling applications using liquid cold plate technologies. In particular, it is shown that the large thermal capacity of MPCM slurries around the phase-change temperature can lead toward greater isothermality of isoflux systems, a characteristic of significant interest to telecommunication, laser and biomedical applications. The principal focus of the study is to examine heat transfer characteristics within standard tube flow geometries, quantify the heat transfer augmentation/degradation observed, and finally, elucidate the mechanisms from which these result. Through the study volume concentrations of the MPCM slurry were varied between 0% and 30.2%. High-resolution local heat transfer measurements were obtained using infrared thermography and results presented in terms of local Nusselt number versus inverse Graetz parameter. These spanned both the thermal entrance and the fully developed flow regions with inverse Graetz number ranging from 10^?3 to 10⁰. Results show that significant heat transfer enhancements are attainable via the use of MPCM slurries over conventional single-phase coolants. Overall, the study highlights mechanisms that lead to significant heat transfer enhancements in heat exchange devices employing microencapsulated phase-change material slurries.     

A numerical study of laminar natural convective heat transfer around a horizontal cylinder inside a concentric air-filled triangular enclosure

A numerical investigation of steady-state laminar natural convective heat transfer around a horizontal cylinder to its concentric triangular enclosure was carried out. The enclosure was filled with air and both the inner and outer cylinders were maintained at uniform temperatures. The buoyancy effect was modeled by applying the Boussinesq approximation of density to the momentum equation and the governing equations were iteratively solved using the control volume approach. The effects of the Rayleigh number and the aspect ratio were examined. Flow and thermal fields were exhibited by means of streamlines and isotherms, respectively. Variations of the maximum value of the dimensionless stream function and the local and average Nusselt numbers were also presented. The average Nusselt number was correlated to the Rayleigh number based on curve-fitting for each aspect ratio. At the highest Rayleigh number studied, the effects of different inclination angles of the enclosure and various cross-section geometries of the inner cylinder were investigated. The computed results indicated that at constant aspect ratio, both the inclination angle and cross-section geometry have insignificant effects on the overall heat transfer rates though the flow patterns are significantly modified.     

Hydromagnetic double-diffusive convection in a rectangular enclosure with opposing temperature and concentration gradients

The finite-difference method is used to predict numerically the characteristics of hydromagnetic double-diffusive convective flow of a binary gas mixture in a rectangular enclosure with the upper and lower walls being insulated. Constant temperatures and concentrations are imposed along the left and right walls of the enclosure and a uniform magnetic field is applied in the x-direction. Consistent with what is reported by previous investigators, an oscillation in the flow is observed in the absence of the magnetic field for a specific range of buoyancy ratio values where the Prandtl number Pr=1, the Lewis number Le=2, the thermal Rayleigh number Ra_T=10⁵, and the aspect ratio A=2 for the enclosure. In the presence of the magnetic field, however, no oscillatory behavior is observed. Numerical results are reported for the effect of the heat generation or absorption coefficient and the Hartmann number on the contours of streamline, temperature, concentration and density. In addition, results for the average Nusselt and Sherwood numbers are presented and discussed for various parametric conditions. In this study, the thermal and compositional buoyancy forces are assumed to be opposite.