Forced convection in cross flow of power law fluids over a tube bank

The forced convective heat transfer characteristics for incompressible power-law fluids past a bundle of circular cylinders have been investigated numerically. The cylinder-to-cylinder hydrodynamic interactions have been approximated via a simple cell model. The momentum and energy equations have been solved using a finite difference based numerical method for a range of physical and kinematic conditions. The role of the two commonly used thermal boundary conditions, namely, constant temperature or constant heat flux, on heat transfer characteristics has also been studied. Extensive numerical results elucidating the effect of shear-thinning viscosity on the values of Nusselt number have been obtained for Peclet numbers ranging from 1 to 5000, Reynolds number in the range 1-500, flow behaviour index 1?n?0.5 and three values of voidages, namely, 0.4, 0.5 and 0.6, typical of tubular heat exchangers and tube banks. Under all conditions, varying levels of enhancement in Nusselt number are observed due to shear-thinning behaviour. The surface averaged Nusselt number shows strong dependence on the values of voidage, power-law index, Reynolds and Peclet numbers. The paper is concluded by presenting comparisons with the scant experimental results available in the literature.     

Drag on chains and agglomerates of spheres in viscous newtonian and power law fluids

New experimental data on the free settling velocity of straight chains (up to twenty spheres) and planar clusters of touching spheres in Newionian and power law media are reported. The results embrace the following ranges of conditions: 0.65 ≤ n ≤ 1; Re < - 2.5 and 1.22 < m < 48.87 Pa·sⁿ. The straight chain drag measurements are in line with theoretical predictions for Newtonian fluids. The present results in power law fluids seem to suggest that it is possible to express the drag on a straight chain of spheres in terms of that on a single sphere of equal volume. Limited results with planar clusters are satisfactorily correlated using a volume equivalent sphere diameter.     

Effect of blockage on drag and heat transfer from a single sphere and an in-line array of three spheres

The effect of blockage ratio on the steady flow and heat transfer characteristics of incompressible fluid over a sphere and an in-line array of three spheres placed at the axis of a tube has been investigated numerically. The Navier-Stokes and thermal energy equations have been solved numerically using FLUENT for the following ranges of parameters: for a single sphere, 2 ≤ β ≤ 10; 1 ≤ Re ≤ 100; for the three-sphere system, for two values of sphere-to-sphere distance, namely s = 2 and 4. All computations were carried out for two values of the Prandtl number, i.e., 0.74 and 7, corresponding to the flow of air and water respectively. Extensive results on streamline patterns, wake characteristics (angle of separation and recirculation length), drag coefficient and Nusselt number are presented to elucidate the interplay between the blockage and the Reynolds number and their influence on drag and Nusselt number.     

Effect of blockage on heat transfer from a cylinder to power law liquids

The influence of planar confining walls on the steady forced convection heat transfer from a cylinder to power-law fluids has been investigated numerically by solving the field equations using FLUENT (version 6.2). Extensive results highlighting the effects of the Reynolds number (1?Re?40), power-law index (0.2?n?1.8), Prandtl number (1?Pr?100) and the blockage ratio (β=4 and 1.6) on the average Nusselt number have been presented. For a fixed value of the blockage ratio, the heat transfer is enhanced with the increasing degree of shear-thinning behaviour of the fluid, while an opposite trend was observed in shear-thickening fluids. Due to the modifications of the flow and temperature fields close to the cylinder, the closely placed walls (i.e., decreasing value of the blockage ratio) further enhance the rate of heat transfer as the fluid behaviour changes from Newtonian to shear-thickening fluids (n>1), the opposite influence is seen with the decreasing value of the flow behaviour index (n) in shear-thinning (n<1) fluids. Finally, the functional dependence of the present numerical results on the relevant dimensionless parameters has been presented in the form of closure relationships for their easy use in a new application.     

Forced convection heat transfer in tube banks in cross flow

The forced convection heat transfer characteristics for an incompressible, steady and Newtonian fluid flow over a bundle of circular cylinders has been investigated numerically. The inter-cylinder hydrodynamic interactions have been approximated by employing a simple cell model. The momentum and energy equations have been solved by using a finite difference based numerical solution procedure for a range of physical and kinematic conditions. Furthermore, the role of the type of thermal boundary condition, namely, a constant temperature or a constant heat flux, imposed on the surface of the cylinder has also been elucidated. Extensive results on the temperature fields, and on the variation of the Nusselt number on the surface of a typical cylinder in the assemblage have been obtained for two values of the Prandtl number (corresponding to air and water). The Reynolds number of flow was varied in the range 1-500 and the voidage of the assemblage ranged from 0.4 to 0.99 thereby covering the entire range of interest as encountered in tubular heat exchangers and in fibrous beds. The paper is concluded by presenting extensive comparisons with the limited analytical/numerical and/or experimental results available in the literature for the case of a single cylinder as well as that for tube bundles.     

Surface-to-bed heat transfer in fluidised beds of fine particles

This work reports experimental results on the heat transfer between a fluidised bed of fine particles and a submerged surface. Experiments have been carried out using different bed materials (polymers, ballotini, corundum, carborundum and quartz sand) with Archimedes number between 2 and 50. Dry air at ambient pressure and temperature has been used as fluidising gas. Three different exchange surfaces, namely a sphere and two cylinders with different base diameter and same height, have been used.Experimental results show that the heat transfer coefficient increases with particle Archimedes number and is almost independent from particle thermal conductivity for K_p/K_g > 30. Finally, the comparison of heat transfer coefficient for the different surfaces shows that the effect of the surface geometry may account for a 30% variation in the heat transfer coefficient, with higher differences occurring for coarser particles.     

Convective instability in packed beds with internal heat sources and throughflow

This article focuses on convective instabilities of throughflow in packed beds with internal heat sources. When a packed bed is heated with internal heat sources, the effects of throughflow on the onset conditions of convection have been examined numerically under the linear stability theory. The resulting conditions show that stationary instabilities occur at higher values of Darcy-Rayleigh number than the critical values as the amount of throughflow increases. The effects of free and rigid boundaries on the onset condition are also obtained for the Brinkman porous media with throughflow.     

Particle scale study of heat transfer in packed and bubbling fluidized beds

The approach of combined discrete particle simulation (DPS) and computational fluid dynamics (CFD), which has been increasingly applied to the modeling of particle‐fluid flow, is extended to study particle‐particle and particle‐fluid heat transfer in packed and bubbling fluidized beds at an individual particle scale. The development of this model is described first, involving three heat transfer mechanisms: fluid‐particle convection, particle‐particle conduction and particle radiation. The model is then validated by comparing the predicted results with those measured in the literature in terms of bed effective thermal conductivity and individual particle heat transfer characteristics. The contribution of each of the three heat transfer mechanisms is quantified and analyzed. The results confirm that under certain conditions, individual particle heat transfer coefficient (HTC) can be constant in a fluidized bed, independent of gas superficial velocities. However, the relationship between HTC and gas superficial velocity varies with flow conditions and material properties such as thermal conductivities. The effectiveness and possible limitation of the hot sphere approach recently used in the experimental studies of heat transfer in fluidized beds are discussed. The results show that the proposed model offers an effective method to elucidate the mechanisms governing the heat transfer in packed and bubbling fluidized beds at a particle scale. The need for further development in this area is also discussed. © 2009 American Institute of Chemical Engineers AIChE J, 2009     

Oscillatory flow and axial dispersion in packed beds of spheres

The effect of oscillations in the bulk flow on the axial dispersion coefficient in packed beds of spherical particles has been studied using the imperfect pulse tracer method with two probes located within the bed. Three bed sizes with diameters in the range 25-47.3 mm have been used with oscillation frequencies and amplitudes in the range 0-2.4 Hz and 0-3.5 mm, respectively. In the absence of oscillations, the axial dispersion coefficient increases linearly with interstitial velocity. For a given bulk velocity and oscillation frequency, the axial dispersion coefficient-amplitude relationship shows a minimum. Over the ranges of conditions studied, the best reduction (up to 50%) in the axial dispersion coefficient from the non-oscillation base case occurred at the highest frequency studied and when the wall effect was the greatest, i.e. when the column-to-particle size was the smallest. The axial dispersion coefficient was fitted to a mathematical model, which takes into account the diameters of both the column and the packing, the fluid velocity, and the oscillation intensity (frequency and amplitude). The model was adapted from those developed by Göebel et al. (1986) and Mak et al. (1991) so as to need no a priori assumptions about the relationship between oscillation parameters and the axial dispersion coefficient. The model provides near-perfect fits to the experimental data for the higher frequencies studied.     

The development of a mechanism for gas-particle heat transfer in shallow fluidised beds of large particles

A basic mechanism for gas-particle heat transfer in shallow fluidised beds of large particles has been developed on the basis of an experimental investigation of the fluidising behavior of such a system. The validity of the proposed mechanism was tested by comparison of experimentally measured outlet gas temperature fluctuations with those predicted by the mechanism. Whereas the predicted and measured fluctuations showed the same trends, the predicted fluctuation was usually a little higher than that measured. This is attributed to an over-simplification of the flow patterns within the bed.     

Predictions of heat transfer and pressure drop for a modified power law fluid flow in a square duct

Numerical Solutions for the Nusselt Numbers (CHF and CWT) and the Friction Factor times Reynolds Number have been obtained for fully developed laminar flow of a MPL (Modified Power Law) fluid within a square duct. The solutions are applicable to pseudoplastic fluids over a wide shear rate range from Newtonian at low shear rates through a transition region to power law behavior at higher shear rates. A shear rate parameter is identified, which allows the prediction of the shear rate range for a specified set of operating conditions. Numerical results of the Nusselt numbers (CHF and CWT) and the Friction factors times Reynolds number for the Newtonian and power law regions are compared with previous published results, showing agreement with 0.02% in Newtonian region and 4.0% in power law region.     

Performance of stress-transport models in the prediction of particle-to-fluid heat transfer in packed beds

Computational fluid dynamics (CFD) as a simulation tool allows obtaining a more complete view of the fluid flow and heat transfer mechanisms in packed bed reactors, through the resolution of 3D Reynolds averaged transport equations, together with a turbulence model when needed. This tool allows obtaining mean velocity and temperature values as well as their fluctuations at any point of the bed. An important problem when a CFD modeling is performed for turbulent flow in a packed bed reactor is to decide which turbulence model is the most accurate for this situation. Turbulence models based on the assumption of a scalar eddy viscosity for computing the turbulence stresses, so-called eddy viscosity models (EVM), seem insufficient in this case due to the big flow complexity. The use of models based on transport equations for the turbulence stresses, so-called second order closure modeling or Reynolds stress modeling (RSM), could be a better option in this case, because these models capture more of the involved physics in this kind of flow.To gain insight into this subject, a comparison between the performance in flow and heat transfer estimation of RSM and EVM turbulence models was conducted in a packed bed by solving the 3D Reynolds averaged momentum and energy equations. Several setups were defined and then computed. Thus, the numerical pressure drop, velocity, and thermal fields within the bed were obtained. In order to judge the capabilities of these turbulence models, the Nusselt number (Nu) was computed from numerical data as well as the pressure drop. Then, they were compared with commonly used correlations for parameter estimations in packed bed reactors. The numerical results obtained show that RSM give similar results as EVM for the cases checked, but with a considerably larger computational effort. This fact suggests that for this application, even though the RSM goes further into the flow physics, this does not lead to a relevant improvement in parameter estimation when compared to the performance of EVM models used.     

Using CFD to predict the behavior of power law fluids near axial-flow impellers operating in the transitional flow regime

A computational fluid dynamics (CFD) model of flow in a mixing tank with a single axial-flow impeller was developed with the Fluent^TM software. The model consists of an unstructured hexagonal mesh (158,000 total cells), dense in the region from the surface of the impeller. The flow was modeled as laminar and a multiple reference frame approach was used to solve the discretized equations of motion in one-quarter of a baffled tank. A solution of 0.1% Carbopol in water, a shear-thinning fluid, was found to be clear enough to measure impeller discharge angles using laser Doppler velocimetry. This is the first time that impeller discharge angles have been reported in the literature for a shear-thinning fluid with a hydrofoil impeller. Rheological measurements indicated that the Carbopol solution can be characterized by the power law (K=9,n=0.2) under the range of shear conditions (0.1- expected near the impeller in the mixing tank. The CFD model accurately predicted the dependence of power number and discharge angle on Reynolds number (as predicted by Metzner and Otto), for an A200 (pitched blade turbine or PBT) and an A315 (Hydrofoil) impeller operating in the transitional flow regime (Reynolds numbers: 25-400) with glycerin and 0.1% Carbopol solutions. Subsequently, the results of a systematic CFD study with power law fluids indicated that the power number and discharge angle of an axial-flow impeller in the transitional flow regime depends not only on the Reynolds number (as determined by Metzner and Otto's method) but also on the flow behavior index n. Consequently, an alternative to Metzner and Otto's method was pursued. The results of converged CFD simulations indicate that the near-impeller “average shear rate” increases not only with increasing RPM (as proposed by Metzner and Otto), but also with decreasing flow behavior index (n) and discharge angle in the transitional flow regime. Considering this result, an improved method of estimating the power number and discharge angle for power law fluids in the transitional flow regime is proposed.     

窄间隙矩形通道沸腾传热强化判定准则研究

为了研究以水为介质的窄间隙矩形通道内发生临界热流密度(CHF)时的传热强化,构建了一个判定准则。根据窄间隙矩形通道的流道结构特点,参考圆管环状流CHF预测解析模型,得到了可以预测通道间隙厚度不小于0.5 mm的窄间隙矩形通道内发生沸腾二相流环状流时的CHF解析模型。根据汽液二相介质的特点,推导出了在沸腾二相流系统中发生CHF时的传热强化判定准则,通过分析计算表明这个判定准则是合理的。这个判定准则适用于高Re数下窄间隙矩形通道内强迫流动时发生CHF的传热强化判定。     

End effects on interphase heat and mass transfer in a cubically packed bed of spheres

Heat (and mass) transfer data were obtained for the constant rate evaporation of water from two layers of porous spheres in simple cubic array (? = 0.4764) to a flowing air stream, over a particle Reynolds number range of 200-900. It was found that the cumulative effect of adding inert layers of non-porous spheres at the entrance and exit was to lower (rather than raise, as in the case of randomly packed beds) the extent of heat and mass transfer between the active spheres and the fluid, the minimum number of inert layers required to eliminate end effects being three at the entrance and two at the exit. The specific effect of adding only one inert layer to the entrance, with none at the exit, was to raise somewhat the extent of heat and mass transfer. Interpretations are offered for these trends. The simple cubic assemblage showed substantially lower values of the heat (and mass) transfer factors than those of randomly packed beds.     

Settling velocity of cubes in Newtonian and power law liquids

New extensive data on the free settling velocity of thirty cubes of various densities and sizes falling in scores of Newtonian and Power law liquids are reported herein to supplement the existing data, for there is very little prior data on cubes in power law liquids. The new data embrace the range of conditions as follows: sphericity of 0.805; power law index, 0.61 to 1 and consistency index, 0.0078-15.31 Pa sⁿ; Reynolds number, 0.0013 to 860. The new results are shown to be consistent with an existing drag correlation which has been tested extensively using the literature data for spherical and non-spherical particles falling in Newtonian and power law liquids with acceptable levels of accuracy.     

Influence of the turbulence model in CFD modeling of wall-to-fluid heat transfer in packed beds

Computational fluid dynamics as a simulation tool allows obtaining a more detailed view of the fluid flow and heat transfer mechanisms in fixed-bed reactors, through the resolution of 3D Reynolds averaged transport equations, together with a turbulence model when needed. In this way, this tool permits obtaining of mean and fluctuating flow and temperature values in any point of the bed. An important problem when modeling a turbulent flow fixed-bed reactor is to decide which turbulence model is the most accurate for this situation. To gain insight into this subject, this study presents a comparison between the performance in flow and heat transfer estimation of five different RANS turbulence models in a fixed bed composed of 44 homogeneous stacked spheres in a maximum space-occupying arrangement in a cylindrical container by solving the 3D Navier-Stokes and energy equations by means of a commercial finite volume code, Fluent 6.0^®. Air is chosen as flowing fluid. Numerical pressure drop, velocity and thermal fields within the bed are obtained. In order to judge the capabilities of these turbulence models, heat transfer parameters (Nu_w, k_r/k_f) are estimated from numerical data and together with the pressure drop are compared to commonly used correlations for parameter estimations in fixed-bed reactors.     

Computation of forced convection in slow flow through ducts and packed beds—IV. Convective boundary layers in cubic arrays of spheres

Detailed boundary-layer solutions are given for heat and mass transfer in forced convection through packed beds. The simple cubic and the dense cubic array of spheres are chosen as examples. Numerical results are given for uniform and catalytic surface conditions, and the uniform-accessibility assumption is quantitatively tested. Close agreement is found with the data of Karabelas et al.[1] for diffusion-controlled electrolysis in a dense cubic array.