Modelling the flow of power law fluids in a packed bed using a volume-averaged equation of motion

The development of a theoretical model for the prediction of velocity and pressure drop for the flow of a viscous power law fluid through a bed packed with uniform spherical particles is presented. The model is developed by volume averaging the equation of motion. A porous microstructure model based on a cell model is used. Numerical solution of the resulting equation is effected using a penalty Galerkin finite element method. Experimental pressure drop values for dilute solutions of carboxymethylcellulose flowing in narrow tubes packed with uniformly sized spherical particles are compared to theoretical predictions over a range of operating conditions. Overall agreement between experimental and theoretical values is within 15%. The extra pressure drop due to the presence of the wall is incorporated directly into the model through the application of the no-slip boundary condition at the container wall. The extra pressure drop reaches a maximum of about 10% of the bed pressure drop without wall effect. The wall effect increases as the ratio of tube diameter to particle diameter decreases, as the Reynolds number decreases and as the power law index increases.     

Upscaling the flow of generalised Newtonian fluids through anisotropic porous media

The homogenisation method with multiple scale expansions is used to investigate the slow and isothermal flow of generalised Newtonian fluids through anisotropic porous media. From this upscaling it is shown that the first-order macroscopic pressure gradient can be defined as the gradient of a macroscopic viscous dissipation potential, with respect to the first-order volume averaged fluid velocity. The macroscopic dissipation potential is the volume-averaged of local dissipation potential. Using this property, guidelines are proposed to build macroscopic tensorial permeation laws within the framework defined by the theory of anisotropic tensor functions and by using macroscopic isodissipation surfaces. A quantitative numerical study is then performed on a 3D fibrous medium and with a Carreau–Yasuda fluid in order to illustrate the theoretical results deduced from the upscaling.     

On the flow resistance of viscoelastic fluids through packed beds

Analytical solutions are obtained for the free surface cell model of packed beds using a third order fluid. Second order perturbed results indicate a substantial increase in resistance to the flow of a viscoelastic fluid through a packed bed. This predicted increase is in good agreement with experimental findings.     

Simulation of two generalised Newtonian fluids flow in micropore with dead end

This research deals with the numerical simulation of Carreau and power-law fluids flow in an open capillary of a reservoir. The capillary is connected to a dead end. The finite volume method (FVM) on a structured and co-located grid has been used. The numerical method has been validated through the comparison of numerical results against the analytical solutions of power-law fluid flow in a planar channel. The effects of fluids, the operating conditions and the aspect ratio of dead end at the low Reynolds (Re) numbers on the oil sweeping from the dead end are investigated. The simulation results show that by increasing the power-law exponent in the case of power-law fluids, the swept depth in the dead end increases. However, according to the results, the effect of Re number on the flow pattern and the oil sweeping from the dead end is insignificant at the investigated conditions. In the case of Carreau model, at the conditions investigated, the swept area increases as the power-law exponent increases, but the Reynolds number has still minor effects on the flow pattern. Also, as the aspect ratio of dead end increases, the sweep efficiency increases.     

Simulated configurational temperature of particles and a model of constitutive relations of rapid-intermediate-dense granular flow based on generalized granular temperature

In gas–solid flat-base spout bed with a jet, the flow of particles must go through an intermediate regime where both kinetic/collisional and frictional contributions play a role. In this paper, the statistical framework is proposed to define the generalized granular temperature which sums up the configurational temperature and translational granular temperature. The configurational temperature, translational and rotational granular temperatures of particles are simulated by means of CFD-DEM (discrete element method) in a 3D flat-base spout bed with a jet. The configurational temperatures of particles are calculated from instantaneous overlaps of particles. The translational and rotational granular temperatures of particles are calculated from instantaneous translational and angular velocities of particles. Roughly, the simulated translational and rotational granular temperatures increase, reach maximum, and then decrease with the increase of solids volume fractions. However, the configurational temperature increases with the increase of solids volume fractions. At high solid volume fraction, the predicted configurational temperatures are larger than the translational and rotational granular temperatures, indicating that the rate of energy dissipation do contributes by contact deformation of elastic particles. The generalized granular temperature is proposed to show the relation between the variance of the fluctuation velocity of deformation and the variance of the translational fluctuation velocity of particles. The constitutive relations of particle pressure, viscosity, granular conductivity of fluctuating energy and energy dissipation in rapid-intermediate-dense granular flows are correlated to the generalized granular temperature. The variations of particle pressure, shear viscosity, energy dissipation and granular conductivity are analyzed on the basis of generalized granular temperature in a flat-base spout bed with a jet. The axial velocities of particles predicted by a gas–solid two-fluid model of rapid-intermediate-dense granular flows agree with experimental results in a spout bed.     

Resistance to flow of molten polymers through granular beds

Summary The resistance to flow of molten polymers (exhibiting memory effects) through granular beds was investigated. The data were correlated using simple methods of rheological characteristics of polymer melts.

---

Zusammenfassung Es wird der Strömungswiderstand von Polymerschmelzen (welche Gedächtnis-Effekte zeigen) in Kornschüttungen untersucht. Bei der Korrelation der Meßwerte werden einfache Methoden zur rheologischen Charakterisierung der Schmelzen verwendet.

---

Notation a ^* regression coefficient - b constant in eq. [22] - b ^* regression coefficient - B swell ratio, defined by eq. [2] - B ₁ swell ratio, defined by eq. [17a] - B ₂ swell ratio, defined by eq. [18a] - B ₃ swell ratio, defined by eq. [19a] - B ₀ swell ratio during the extrusion of a Newtonian fluid from the capillary - B _exp swell ratio determined according to the method described byCogswell (22) - B _exp swell ratio determined according to the method described byMendelson andFinger (23) - C constant in eq. [22] - d _p particle diameter, m - D extrudate diameter, m - D ₀ capillary diameter, m - f _CM friction factor, defined by eq. [12] - F() function occurring in eq. [15] - k fluid consistency factor, Ns ⁿ /m² - l bed height, m - L ₁,L ₂ length scales, m - n flow behaviour index - p pressure, N/m² - p ₁₁ –p ₂₂ first normal-stress difference, N/m² - r correlation coefficient - S recoverable shear strain, defined by eq. [3] - S average value of recoverable shear strain - S _w recoverable shear strain at the wall - s bed permeability, defined by eq. [8], m² - v ₀ mean linear velocity of fluid, related to an empty cross-section of the column, m/s - w ₀ mass flow rate, kg/m² s - level of probability of the correlation - shear rate, s^–1 - shear rate in a porous bed, defined by eq. [14], s^–1 - porosity - coefficient of dynamic fluid viscosity, Ns/m² - relaxation time, defined by eq. [1], s - friction factor, defined by eq. [5] - density of molten polymer, kg/m³ - ₀ density of polymer at room temperature, kg/m³ - ₁₂ shear stress, N/m² - parameter, defined by eq. [7], Ns ⁿ /m¹⁺ⁿ - function occurring in eq. [4] - De Deborah number, defined by eq. [16] - Re_BK Reynolds number, defined by eq. [25] - Re_BK generalized Reynolds number, defined by eq. [6] - Re_CM generalized Reynolds number, defined by eq. [13] With 10 figures and 3 tables     

A new look at the laminar flow of power law fluids through granular beds

Summary The paper deals with laminar flow of power law fluids through granular beds. A critical review of the assumptions concerning the capillary model of the bed, applied by various authors, led us to the conclusion that the derivation of the correlation eq. [13] given byChristopher andMiddleman was based on a too simplified model of the granular bed. Taking advantage of the approach presented in the classical works ofKozeny andCarman (which seems to be partly overlooked by some authors, including our own previous works) a modified correlation equation for power law fluids [21], a corrected formula for shear rate in the bed [29] and for Deborah number [32], as well as corrected correlation equation for fluids exhibiting memory effects [34] were presented.

---

Zusammenfassung Diese Arbeit betrifft laminare Strömungen von Potenzgesetzflüssigkeiten durch Kornschüttungen. Eine kritische Prüfung der Annahmen, die von verschiedenen Autoren für das Kapillar-Modell der Schüttung gemacht worden sind, führt uns zu der Folgerung, daß die Herleitung der Korrelationsgleichung [13] nachChristopher undMiddleman auf einem übervereinfachten Modell der Kornschüttung basiert. Unter Nutzbarmachung der Annahmen, die in den klassischen Arbeiten vonKozeny undCarman dargestellt worden sind (sie wurden sowohl von manchen anderen Autoren als auch in unseren früheren Arbeiten nicht beachtet), werden nun eine modifizierte Korrelationsgleichung für die Potenzgesetzflüssigkeiten [21], eine korrigierte Formel für die Schergeschwindigkeit in der Schüttung [29], eine korrigierte Formel für die Deborah-Zahl [32] und eine korrigierte Korrelationsgleichung für Flüssigkeiten, die Gedächtnis-Effekte zeigen [34], angegeben.

---

Notation A constant in eq. [9] - d _p effective particle diameterd _p = 6/a (wherea is the specific surface of the bed), m - f _BK modified friction factor, defined by eq. [1] - k power law parameter, N s ⁿ /m² - K Kozeny constant, defined by eq. [8] - K ₀ constant depending on the shape of the channel cross-section - K ₁ constant, defined by eq. [5] - l bed height, m - l _e channel length, m - n power law parameter - p pressure drop due to friction, N/m² - r _h hydraulic radius, defined by eq. [6], m - s bed permeability, defined by eq. [16], m² - v ₀ mean linear velocity related to an empty crosssection of the column, m/s - v _e mean linear velocity in the channel, m/s - shear rate at the wall of the channel, s^–1 - shear rate at the wall of the channel calculated according to the formula [29], s^–1 - bed porosity - characteristic time of the fluid, s - friction factor, defined by eq. [25] - µ dynamic viscosity of the fluid, N s/m² - parameter, defined by eq. [15], N s ⁿ /m¹⁺ⁿ - De Deborah number, defined by eq. [33] - De ^* Deborah number, defined by eq. [32] - Re _BK modified Reynolds number, defined by eq. [2] - Re _BK modified Reynolds number, defined by eq. [26] - Re _BK ^* modified Reynolds number, defined by eq. [23] - Re _CM modified Reynolds number byChristopher andMiddleman, defined by eq. [14] - Re _CM modified Reynolds number, defined by eq. [17] With 3 figures and 1 table     

Three-dimensional flow of Newtonian and Boger fluids in square–square contractions

The flow of a Newtonian fluid and a Boger fluid through sudden square–square contractions was investigated experimentally aiming to characterize the flow and provide quantitative data for benchmarking in a complex three-dimensional flow. Visualizations of the flow patterns were undertaken using streak-line photography, detailed velocity field measurements were conducted using particle image velocimetry (PIV) and pressure drop measurements were performed in various geometries with different contraction ratios. For the Newtonian fluid, the experimental results are compared with numerical simulations performed using a finite volume method, and excellent agreement is found for the range of Reynolds number tested (Re₂ ≤ 23). For the viscoelastic case, recirculations are still present upstream of the contraction but we also observe other complex flow patterns that are dependent on contraction ratio (CR) and Deborah number (De₂) for the range of conditions studied: CR = 2.4, 4, 8, 12 and De₂ ≤ 150. For low contraction ratios strong divergent flow is observed upstream of the contraction, whereas for high contraction ratios there is no upstream divergent flow, except in the vicinity of the re-entrant corner where a localized atypical divergent flow is observed. For all contraction ratios studied, at sufficiently high Deborah numbers, strong elastic vortex enhancement upstream of the contraction is observed, which leads to the onset of a periodic complex flow at higher flow rates. The vortices observed under steady flow are not closed, and fluid elasticity was found to modify the flow direction within the recirculations as compared to that found for Newtonian fluids. The entry pressure drop, quantified using a Couette correction, was found to increase with the Deborah number for the higher contraction ratios.     

Boundary element analysis of three-dimensional mixing flow of Newtonian and viscoelastic fluids

The boundary element method (BEM) is implemented for the simulation of three-dimensional transient flows of typical relevance to mixing. Creeping Newtonian and viscoelastic fluids of the Maxwell type are examined. A boundary-only formulation in the time domain is proposed for linear viscoelastic flows. Special emphasis is placed on cavity flows involving simple- and multiple-connected moving domains. The BEM becomes particularly suited in multiple-connected flows, where part of the boundary (stirrer or rotor) is moving, and the remaining outer part (cavity or barrel) is at rest. In this case, conventional methods, such as the finite element method (FEM), generally require remeshing or mesh refinement of the three-dimensional fluid volume as the flow evolves and the domain of computation changes with time. The BEM is shown to be much easier to implement since the kinematics of the elements bounding the fluid is known (imposed). It is found that, for simple cavity flow induced by a rotating vane at constant angular velocity, the tractions at the vane tip and cavity face exhibit non-linear periodic dynamical behavior with time for fluids obeying linear constitutive equations. © 1998 John Wiley & Sons, Ltd.     

CFD modeling of pressure drop and drag coefficient in fixed beds: Wall effects

Simulations of fixed beds having column to particle diameter ratio (D/dp ) of 3, 5 and 10 were performed in the creeping, transition and turbulent flow regimes, where Reynolds number (dp VLL/L ) was varied from 0.1 to 10,000. The deviations from Ergun’s equation due to the wall effects, which are important in D/d p<15 beds were well explained by the CFD simulations. Thus, an increase in the pressure drop was observed due to the wall friction in the creeping flow, whereas, in turbulent regime a decrease in t...     

The flow behavior of fiber suspensions in Newtonian fluids and polymer solutions.

This is the second part of a study examining the mechanical properties and capillary flow of fiber suspensions in Newtonian fluids and in polymer solutions. In part I results for the viscous and elastic properties of the fiber suspensions were presented and it was shown that the fiber suspensions exhibited normal stresses in Newtonian as well as in viscoelastic suspending media. It was thus expected that circulating secondary flows would occur near the entrance to a capillary. Four types of fillers (glass, carbon, nylon and vinylon fibers) suspended in glycerin, HEC solutions and Separan solutions were investigated. The entrance flow patterns were visualized and the pressure fluctuations measured. The visualization enabled the eddies occurring in the fiber suspensions in Newtonian fluids to be analysed and classified into two tpyes. The results from the flow visualization were correlated with the pressure fluctuations. Empirical equations for the tube length correction factor due to the elasticity were obtained.     

Experimental study on filtration performance of the moving bed granular filter with axial flow

The filtration performance of the moving bed granular filter with axial flow (MBGF-AF) is investigated through a large cold experiment. The effect of different operation parameters on the filtration performance (collection efficiency, pressure drop) of the axial-flow moving bed filter is investigated in combination with the dust deposition effect and the mechanism of trapping dust by the capturing particles. The results show that the collection efficiency of MBGF-AF is enhanced by decreasing the superficial gas velocity, increasing the inlet dust concentration properly, or decreasing the moving velocity of the capturing particles. A model covering the above operation parameters is established to calculate the collection efficiency of the moving bed granular filter. It is used in a wide range of operating parameters for the MBGFs.     

Time periodic electroosmotic flow of micropolar fluids through microparallel channel

The time periodic electroosmotic flow of an incompressible micropolar fluid between two infinitely extended microparallel plates is studied.The analytical solutions of the velocity and microrotation are derived under the Debye-H(u|¨)ckel approximation.The effects of the related dimensionless parameters,e.g.,the micropolar parameter,the frequency,the electrokinetic width,and the wall zeta potential ratio of the upper plate to the lower plate,on the electroosmotic velocity and microrotation are investigated.The results show that the amplitudes of the velocity and the volume flow rate will drop to zero when the micropolar parameter increases from 0 to 1.The effects of the electrokinetic width and the frequency on the velocity of the micropolar fluid are similar to those of the Newtonian fluid.However,the dependence of the microrotation on the related parameters mentioned above is complex.In order to describe these effects clearly,the dimensionless microrotation strength and the penetration depth of the microrotation are defined,which are used to explain the variation of the microrotation.In addition,the effects of various parameters on the dimensionless stress tensor at the walls are studied.     

Effects of shear thinning on migration of neutrally buoyant particles in pressure driven flow of Newtonian and viscoelastic fluids

The pattern of cross stream migration of neutrally buoyant particles in a pressure driven flow depends strongly on the properties of the suspending fluid. These migration effects have been studied by direct numerical simulation in planar flow. Shear thinning has a large effect when the inertia or elasticity is large, but only a small effect when they are small. At moderate Reynolds numbers, shear thinning causes particles to migrate away from the centerline, creating a particle-free zone in the core of the channel, which increases with the amount of shear thinning. In a viscoelastic fluid with shear thinning, particles migrate either toward the centerline or toward the walls, creating an annular particle-free zone at intermediate radii. The simulations also give rise to precise determination of slip velocity distributions in the various cases studied.     

Numerical simulations of high pressure carbon dioxide fluid fluidized beds

Hydrodynamics of carbon dioxide fluid-particle mixtures are predicted using a low density ratio-based kinetic theory of granular flow in high pressure carbon dioxide fluid fluidized beds. A coexistence of particle waves and particle aggregates exists along bed height. The threshold to identify the occurrence of particle aggregates is suggested based on standard deviation of solid volume fractions in aggregative fluidization. The existence time fractions and frequencies of particle aggregates are predicted along axial direction. The effect of carbon dioxide fluid temperature and pressure on volume fraction and velocity distributions are analyzed at different inlet carbon dioxide velocities and particle densities in high pressure carbon dioxide fluidized beds. Simulated results indicate that the carbon dioxide-particles fluidization transits from particulate to aggregative states with the increase of inlet carbon dioxide velocities. The computed fluid volume fractions and heterogeneity indexes are close to the measurements in a high pressure carbon dioxide fluidized bed.     

煤仓内煤散料流动状态与力学行为影响因素

针对煤仓内煤散料流动问题及其力学行为,采用三维颗粒流模拟程序PFC3D建立了某型号煤仓与某种煤散料的离散元模型,简述了其力学模型与求解步骤,模拟分析了煤仓内煤散料卸料流动状态。通过分析水平向侧压力、颗粒速度场和接触力场,重点讨论了煤仓下部锥体内壁面摩擦系数、锥仓倾角和卸料口径等对煤散料颗粒流动状态和力学行为的影响。结果显示,深仓卸料流动为整体流动与中心流动混合状态,煤仓内壁摩擦系数、锥体倾角和卸料孔开口半径均对煤散料流动和水平侧压力有较大影响。     

Experimental investigation of heat transfer and pressure drop characteristics of flow through spirally fluted tubes

Spirally fluted tubes are used extensively in the design of tubular heat exchangers. In previous investigations, results for tubes with flute depths e/D_vi < 0.2 were reported, with most correlations applicable for Re ≥ 5000. This paper presents the results of an experimental investigation of the heat transfer and pressure drop characteristics of spirally fluted tubes with the following tube and flow parameter ranges: flute depth e/D_vi = 0.1−0.4, flute pitch p/D_vi = 0.4−7.3, helix angle θ/90° = 0.3−0.65, Re = 500−80,000, and Pr = 2−7. The heat transfer coefficients inside the fluted tube were obtained from measured values of the overall heat transfer coefficient using a nonlinear regression scheme. The friction factor data obtained consisted of 507 data points. The proposed correlation for the friction factor predicts 96% of the database within ±20%. The heat transfer correlation for the range 500 ≤ Re ≤ 5000 predicts 76% of the database (178 data points) within ±20%, and the correlation for the higher Re range predicts 97% of the 342 data points within ±20%. Comparison of heat transfer and friction data show that these tubes are most effective in the laminar and transition flow regimes. The present results show that the increase of flute depth in the range considered does not improve heat transfer.     

Numerical modelling of gas–solid fluidized beds using the two‐fluid approach

This paper details an approach to modelling gas–solid fluidized beds using the two‐fluid granular temperature model. Details concerning the difficulties associated with the boundary conditions, particularly for curved boundaries, are described along with a novel means of obtaining the internal stress of the solid‐phase, in part, by solving an implicit equation. This results in a scheme that is stable even when the solid volume fraction is close to maximum packing. A transient, mixed finite element discretization is used to solve the multi‐phase equations with a discontinuous finite element representation of the granular temperature and continuity equations. A new solution method is proposed to solve the coupled momentum and continuity equations based on Arnoldi iteration. Two fluidized beds are modelled, one in the bubbling regime and the other in the slugging regime. These simulations are compared with experiments. Copyright © 2001 John Wiley & Sons, Ltd.     

Stokes flow of micro-polar fluids by peristaltic pumping through tube with slip boundary condition

This paper studies the Stokes flow of micro-polar fluids by peristaltic pumping through the cylindrical tube under the effect of the slip boundary condition. The motion of the wall is governed by the sinusoidal wave equation. The analytical and numerical solutions for the axial velocity, the micro-polar vector, the stream function, the pressure gradient, the friction force, and the mechanical efficiency are obtained by using the lubrication theory under the low Reynolds number and long wavelength approximations. The impacts of the emerging parameters, such as the coupling number, the micro-polar parameter, the slip parameter on pumping characteristics, the friction force, the velocity profile, the mechanical efficiency, and the trapping phenomenon are depicted graphically. The numerical results infer that large pressure is required for peristaltic pumping when the coupling number is large, while opposite behaviors are found for the micro-polar parameter and the slip parameter. The size of the trapped bolus reduces with the increase in the coupling number and the micro-polar parameter, whereas it blows up with the increase in the slip parameter.