Heat Transfer Phenomena of Assemblages of Smooth Slip Spheres in Newtonian Fluids

Combined effects of slip velocity and volume fraction of slip spheres on the heat transfer characteristics of multiple slip spheres are numerically investigated within the framework of a free surface cell model combined with a linear slip velocity along the surface of the slip spheres. The governing conservation equations of the mass, momentum, and energy are solved by a segregated approach using a simplified marker and cell algorithm implemented on a staggered grid arrangement in spherical coordinates. The convection and diffusion terms of conservation equations are discretized using quadratic upstream interpolation for convective kinematics and second‐order central differencing schemes, respectively. Prior to obtaining new results, this numerical solver is validated by comparison of present results with the existing literature values. Further new results are obtained for a range of conditions as; Reynolds number, Re: 0.1–200; Prandtl number, Pr: 1–100; volume fraction of slip spheres, Φ: 0.1–0.5 and slip parameter, λ: 0.01–100. The effects of these dimensionless parameters on isotherm contours and local and average Nusselt numbers are thoroughly delineated. Finally, a new empirical correlation for the average Nusselt number of multiple smooth slip spheres is proposed on the basis of present numerical results.     

Forced convective heat transfer between assemblages of spherical particles and power‐law non‐Newtonian liquids with velocity slip at the interface

Heat transfer from spheres can be influenced by a varying degree of slip at the fluid‐particle interface along with the rheology of the surrounding continuous liquid and adjacent spheres. Thus in this study, the effects of dimensionless velocity slip parameter (λ) along with power‐law fluid rheology and other pertinent kinematic flow and heat transfer parameters on isotherm contours, local and average Nusselt numbers of assemblages of spherical slip particles are presented. This is done by adopting a segregated approach where dimensionless momentum and energy equations are solved by SMAC algorithm formulated in spherical coordinates within the finite difference formulation. Before obtaining new results, grid independence studies for either extreme values of power‐law consistency index of non‐Newtonian fluids are carried out. Finally, the major contribution of this study is the development of a correlative equation for the average Nusselt number of assemblages of spherical slip particles in power‐law fluids based on the present results (5880 data points) as a function of pertinent dimensionless parameters.     

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.     

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.     

Aspects of second-order velocity slip and radiation absorption on hydromagnetic stagnation point flow of Jeffrey fluid with chemical reaction

The present article describes the magnetohydrodynamic flow of a moving Jeffrey fluid along a convectively heated porous stretching surface with second-order velocity slip and radiation absorption effects. Furthermore, chemical reactions and viscous dissipation impacts are also taken into account. The governing equations are converted into dimensionless ordinary differential equations (ODEs) using appropriate similarity transformations. The highly nonlinear ODEs are solved numerically by employing a shooting technique based on the Runge–Kutta Cash–Karp formula. The figures are used to study the variations in temperature, velocity, and concentration profiles for several physical factors. The numerical values of the local skin friction, Sherwood number, and Nusselt number are explained and shown in tables. The analysis reveals that the velocity profile is enhanced for amplifying values of velocity ratio parameter and first-order velocity slip parameter. However, the temperature profile of Jeffrey nanofluid is highlighted w.r.t. Eckert number and radiation absorption parameter. This study may find significant applications in polymer production, food processing, instrumentation, combustion modeling, catalytic chemical reactors, and so on.     

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.     

Confined flow and heat transfer across a triangular cylinder in a channel

In this paper, fluid flow and heat transfer across a long equilateral triangular cylinder placed in a horizontal channel is studied for Reynolds number range 1–80 (in the steps of 5) and Prandtl number of 0.71 for a fixed blockage ratio of 0.25. The governing Navier-Stokes and energy equations along with appropriate boundary conditions are solved by using a commercial CFD solver FLUENT (6.3). The computational grid is created in a commercial grid generator GAMBIT. The flow and temperature fields are presented by stream-line and isotherm profiles, respectively. The wake/recirculation length, mean drag coefficient and average Nusselt number, etc. are calculated for the above range of conditions studied here. The critical value of the Reynolds number (i.e., transition to transient) is found to lie between Re = 58 and Re = 59. The average Nusselt number and the wake length increase with increasing value of the Reynolds number; however, the mean drag coefficient decreases with increasing value of the Reynolds number. Finally, simple correlations for wake length, mean drag coefficient and average Nusselt number are obtained for the range of conditions studied here.     

Application of heat transfer on MHD shear thickening fluid flow past a stretched surface with variable heat source/sink

The purpose of this study is to explore the viscous dissipation stimulus on the steady convective magnetohydrodynamic shear thickening liquid stream across a vertically stretched sheet. The impact of thermic heat, first-order velocity slip, and variable heat generation/absorption are considered and also ignored the effect of magnetic Reynold's number. We converted flow controlling equations into the set of dimensionless nonlinear ordinary differential equations by employing similarity variables to solve these coupled equations by R–K and shooting technique. The effect of different dimensionless variables on velocity, heat, friction factor, and local Nusselt numbers are presented through graphs and tables. Depreciation in velocity and growth in temperature distribution is detected when the Casson fluid parameter is increased. Temperature is the increasing function of the Eckert number.     

Natural Convection Flow with Surface Radiation Along a Vertical Wavy Surface

In this study, natural convection boundary layer flow of thermally radiating fluid along a heated vertical wavy surface is analyzed. Here, the radiative component of heat flux emulates the surface temperature. Governing equations are reduced to dimensionless form, subject to the appropriate transformation. Resulting dimensionless equations are transformed to a set of parabolic partial differential equations by using primitive variable formulation, which are then integrated numerically via iterative finite difference scheme. Emphasis has been given to low Prandtl number fluid. The numerical results obtained for the physical parameters, such as, surface radiation parameter, R, and radiative length parameter, ξ, are discussed in terms of local skin friction and Nusselt number coefficients. Comprehensive interpretation of velocity distribution is also given in the form of streamlines.     

Enhancement of Heat Transfer in Partially Heated Vertical Channel Under Mixed Convection by Using Al2O3 Nanoparticles

Laminar mixed convection in a two-dimensional symmetrically and partially heated vertical channel is investigated. The heaters are located on both walls and uniform temperature is applied on the heated sections. The number of heaters is considered as 1, 4, 8, and 10. Aluminum oxide/water nanofluid is considered as working fluid and the inlet velocity is uniform. The continuity, momentum and energy equations with appropriate boundary conditions are solved in dimensionless form, numerically. The study is performed for Richardson number of 0.01 and 10, Reynolds number of 100 and 500, and nanofluid volume fraction of 0% and 5%. Based on the obtained velocity and temperature distributions, the local and mean Nusselt number is calculated and plotted for different cases. The variation of the mean Nusselt number with the number of the heated portions is also discussed. It is found that the addition of nanoparticles into the base fluid increases mean Nusselt number but the rate of increase depends on Reynolds, Richardson numbers and number of heated portions. It is possible to increase mean Nusselt number 138% by increasing Reynolds number from 100 to 500, Richardson number from 0.01 to 10 and number of heated portions from 1 to 10 when volume fraction value is 5%.     

Nonsimilar boundary layer flow of Cross fluid induced by a heated stretched sheet

This paper deals with the nonisothermal boundary layer flow of Cross fluid due to a stretching sheet. Unlike previous studies on boundary layer flow of Cross fluid, a nonsimilar formulation is adopted to transform the boundary layer equations into nondimensional form. The problem is characterized by three dimensionless parameters, namely, the Deborah number, the Prandtl number, and dimensionless distance along the sheet. The transformed equations are simulated by a numerical scheme with the help of MAPLE software. The velocity and temperature profiles inside the boundary layer are calculated and shown graphically. The skin friction coefficient and Nusselt number at various axial stations are also tabulated for several values of Deborah number and Prandtl number.     

Numerical study of the heat sink with un-uniform fin width designs

The thermal performances of the heat sink with un-uniform fin width designs with an impingement cooling were investigated numerically. The governing equations are discretized by using a control-volume-based finite-difference method with a power-law scheme on an orthogonal non-uniform staggered grid. The coupling of the velocity and the pressure terms of momentum equations are solved by the SIMPLEC algorithm. The well-known k ? ε two-equations turbulence model is employed to describe the turbulent structure and behavior. The parameters include the five Reynolds number (Re = 5000–25000), three fin heights (H = 35, 40, 45 mm), and five fin width designs (Type-1–Type-5). The objective of this study is to examine the effects of the fin shape of the heat sink on the thermal performance. The results show that the Nusselt number increases with the Reynolds number. The increment of the Nusselt number decreases gradually with the increasing Reynolds number. Furthermore, the effects of fin dimensions on the Nusselt number at high Reynolds numbers are more significant than that at low Reynolds numbers. It is also found that there is potential for optimizing the un-uniform fin width design.     

Optimization of Mixed Convection in a Lid-Driven Enclosure with a Heat Generating Circular Body

The physical model considered here is a lid-driven enclosure with bottom heating and top cooling conditions, and a heat generating circular body is placed at the center. The vertical walls of the cavity are kept thermally insulated, and the top lid moves at a constant speed. The steady two-dimensional governing equations for the physical problem are transformed in a dimensionless form with dimensionless governing parameters that decide the fluid flow and heat transfer characteristics in the system. The solution of these transport equations is obtained numerically with the finite element approach using the Galerkin method of weighted residuals. The parametric study has been carried out for variation of the heat generation parameters, the Reynolds numbers, solid-fluid thermal conductivity ratios as well as the Richardson numbers. The working fluid is assigned as air with a Prandtl number of 0.71 throughout the simulation. Results are presented in the form of streamlines, isotherms, average Nusselt number, bulk temperature, and drag force for the afore mentioned parameters. The numerical results indicate the strong influence of the mentioned parameters on the flow structure and heat transfer as well as average Nusselt number, average bulk temperature, and drag force. An optimum combination of the governing parameters would result in higher heat transfer and lower drag force.     

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.     

Heat and mass transfer analysis of chemically reactive tangent hyperbolic fluid in a microchannel

This study addresses the fully developed magnetohydrodynamic flow of non-Newtonian fluid in a microchannel using tangent hyperbolic fluid model. The physical situation has been modeled by accessing boundary layer theory along with the physical aspects of thermophoresis and Brownian motion. The heat and mass transport phenomena are depicted through graphical interpretations. The modeled equations are nondimensionalized using dimensionless variables. The obtained corresponding equations are solved by employing Runge–Kutta–Fehlberg scheme accompanied with shooting technique. The fluctuations in distinct entities of physical connotations, like, the Nusselt number, friction factor and Sherwood number are explored in this examination. A notable reduction in the concentration field of the tangent hyperbolic fluid has been obtained for a larger chemical reaction parameter. The result shows that non-Newtonian fluids exhibit higher Nusselt number than Newtonian fluids. Furthermore, a significant enhancement in Nusselt number has been attained through a rise in the power-law index and thermophoresis aspect.     

Mixed Convection from a Heated Sphere in Bingham Plastic Fluids

In this work, the steady and laminar mixed-convection heat transfer from an isothermal sphere immersed in Bingham plastic fluids has been investigated in the aiding-buoyancy configuration. The pertinent coupled equations of motion and thermal energy have been solved numerically over the following ranges of conditions: Richardson number, 0 ≤ Ri ≤ 2, Bingham number, 0 ≤ Bn ≤ 10, Reynolds number, 0.1 ≤ Re ≤ 100 and Prandtl number, 10 ≤ Pr ≤ 100. Flow characteristics like streamlines, pressure coefficient, morphology of yielded/unyielded regions and drag coefficient are discussed extensively. Similarly, isotherms, local Nusselt number and average Nusselt number are thoroughly examined to develop an overall understanding of the corresponding heat transfer characteristics. All else being equal, in contrast to the positive role of the aiding-buoyancy free convection in Newtonian and power-law fluids, due to the fluid yield stress, heat transfer is impeded in viscoplastic fluids. While the average value of the Nusselt number is influenced by four dimensionless groups, namely, Reynolds number, Bingham number, Prandtl number and Richardson number, by using novel scaling, it has been possible to consolidate the present results via the use of the Colburn j-factor in a simple form. This is particularly suitable for predicting the value of the Nusselt number in a new application.     

Laminar forced convection heat transfer to ordered and disordered single rows of cylinders

We study laminar forced convection heat transfer to or from a single row of equidistantly and non-equidistantly spaced parallel cylindrical wires, perpendicular to the flow direction. We report average Nusselt numbers as a function of geometry and flow conditions, for open frontal area fractions between 0.04 and 0.95, Prandtl numbers between 0.7 and 10, and Reynolds numbers (based on the wire diameter and the free stream velocity) between 0.001 and 600. For equidistantly spaced rows of cylindrical wires we propose a general analytical expression for the average Nusselt number as a function of the Reynolds number, Prandtl number and the open frontal area fraction, as well as asymptotic scaling rules for small and large Reynolds. For all studied Prandtl numbers, equidistant rows exhibit decreasing average Nusselt numbers for decreasing open frontal area fractions at low Reynolds numbers. For high Reynolds numbers, the Nusselt number approaches that of a single cylinder in cross-flow, independent of the open frontal area fraction. For equal open frontal area fractions, the Nusselt number in non-equidistant rows is lower than in equidistant rows for intermediate Reynolds numbers. For very low and high Reynolds numbers, non-uniformity does not influence heat transfer.     

Heat Transfer of Power-Law Fluids in Plane Couette–Poiseuille Flows with Viscous Dissipation

Abstract

Analytical expressions for the velocity and temperature profiles, bulk temperature and Nusselt numbers, in a fully-developed laminar Couette–Poiseuille flow between parallel plates of a power-law fluid with constant, and distinct, wall heat fluxes, in the presence of viscous dissipation are deduced and presented. Both favorable and adverse pressure gradient cases were analyzed. The walls’ shear stresses ratio, which arises naturally when the dimensionless hydrodynamic solution is obtained, together with the fluid power-law index Brinkman number and the walls’ heat fluxes ratio are the independent variables in the heat transfer solutions. With the exception of Newtonian fluids, there are in general two distinct analytical solutions, one for positive and another for negative values of the walls’ shear stresses ratio. The existence of singular points are also observed, where for a given value of the power-law index, there are values of the walls’ shear stresses ratio for which the Nusselt number becomes independent of the Brinkman number. It was also found that in a Couette–Poiseuille flow, for each value of the power-law index there exists a certain negative value of the walls’ shear stresses ratio that makes the Nusselt numbers at both walls identically zero.     

A numerical study of convective heat transfer from a rotating cylinder with cross-flow oscillation

This article presents a numerical investigation of convective heat transfer from a rotating cylinder with cross-flow oscillation. A finite element analysis using Characteristic Based Split method (CBS) is developed to solve governing equations involving continuity, Navier–Stokes, and energy equations. Dynamic unstructured triangular grid is used employing improved lineal and torsional spring analogy which is coupled with the solver by the Arbitrary Lagrangian–Eulerian (ALE) formulation. After verifying the numerical code accuracy, simulations are conducted to study convective heat transfer past a rotating cylinder with cross-flow oscillation at Reynolds numbers of 50, 100, and 200. Different rotational speeds of the cylinder normalized by free stream velocity, in the range of 0–2.5 are considered at various oscillating amplitudes and frequencies and three different Prandtl numbers of 0.7, 6, and 20. Effects of oscillation and rotation of cylinder on the temperature and flow field, vortex lock-on, mean Nusselt number, and the pattern of vortex shedding are investigated in detail considering iso-temperature and iso-flux boundary conditions on the cylinder surface. It is found that similar to the fixed cylinder, beyond a critical rotating speed, vortex shedding is mainly suppressed. Also by increasing the non-dimensional rotational speed of the cylinder, both the Nusselt number and the drag coefficient decrease rapidly. However, in vortex lock-on region, the Nusselt number increases in a large amount.     

Mixed convection heat transfer from a horizontal plate to non-newtonian fluids

Steady laminar mixed convection of non-Newtonian fluids over a horizontal plate has been analyzed. After a suitable coordinate transformation to reduce the complexity of the governing boundary-layer equations, the resulting nonlinear coupled differential equations were solved with an implicit finite difference scheme. Of particular interest are the effects of the power-law viscosity index, the generalized Prandtl number and the buoyancy parameter on fluid flow and heat transfer characteristics. It was found that both the dimensionless skin friction group and the dimensionless heat transfer group increase with higher buoyancy effects for any non-Newtonian fluid. Dilatant fluids exhibit a distinctively different behavior with respect to dimensionless heat transfer group when compares to pseudoplastics in the leading edge of the flat plate. Furthermore, higher generalized Prandtl numbers generate lower skin friction and larger heat transfer coefficients.