Drag on a large spherical aggregate with self-similar structure: An asymptotic analysis

A porous particle with radially varying permeability is used to model a large self-similar (fractal-like) aggregate in the continuum fluid flow regime. The Stokes flow in the viscous fluid around the aggregate is coupled with the solution of the flow inside the aggregate. This internal flow is shown to be composed of a central core governed by Darcy equation and a thin shell at the aggregate edge where the flow is described by Brinkman's equation. Solutions to the governing equations in each of these regions are sought as asymptotic expansions in terms of a suitably small permeability parameter k. The hydrodynamic jumps across the Brinkman layer are systematically established and used to compute the viscous drag experienced by the aggregate. These conditions, which generalize the widely used Saffman (tangential flow) boundary conditions, are more broadly applicable to flows past/through fine-grained deposits of arbitrary shape, and many other configurations of technological interest.     

An experimental study on fluid flow characteristics of superposed porous and fluid layers

In the present study, fluid flow characteristics of a porous layer overlaid by a fluid layer were investigated through experiments. The experimental results were analyzed in comparison with theoretical results of a porous medium bounded by impermeable walls. With spheres, the slip coefficient was found to be 0.0107 for Poiseuille flow over a porous layer. As the permeability decreased, the experimental results approached the values calculated by Darcy’s law and Forchheimer’s equation. In addition, the effects of the presence of a fluid layer over a porous medium were examined in terms of the friction factor. The present experimental data placed in the range of the Darcy to the non-Darcy region are shown to be in reasonable agreement with the proposed correlation.     

Creeping motions of a composite sphere in a concentric spherical cavity

An analytical study is presented for the quasisteady translation and steady rotation of a spherically symmetric composite particle composed of a solid core and a surrounding porous shell located at the center of a spherical cavity filled with an incompressible Newtonian fluid. In the fluid-permeable porous shell, idealized hydrodynamic frictional segments are assumed to distribute uniformly. In the limit of small Reynolds number, the Stokes and Brinkman equations are solved for the flow field of the system, and the hydrodynamic drag force and torque exerted by the fluid on the particle which is proportional to the translational and angular velocities, respectively, are obtained in closed forms. For a given geometry, the normalized wall-corrected translational and rotational mobilities of the particle decrease monotonically with a decrease in the permeability of its porous shell. The boundary effects of the cavity wall on the creeping motions of a composite sphere can be quite significant in appropriate situations. In the limiting cases, the analytical solutions describing the drag force and torque or mobilities for a composite sphere in the cavity reduce to those for a solid sphere and for a porous sphere.     

Gas−liquid multiphase computational fluid dynamics (CFD) of amine absorption column with structured-packing for CO2 capture

The gas−liquid multiphase Eulerian computational fluid dynamics (CFD) model was used to investigate hydrodynamics and CO₂ removal efficiency of a pilot-scale amine absorber with structured-packing. The structured-packing was represented by a porous media zone having porous resistance, gas−liquid interfacial drag force, and liquid dispersion force. This study aimed to find a reasonable way to identify four modification factors of the Ergun coefficient that determine the hydrodynamic characteristics of structured-packing. The two modification factors (a and b) for porous resistance were mainly related to the liquid holdup (h_L) with respect to the liquid load. The other two factors (c and d) for gas−liquid interfacial drag force depended on the specific wet pressure drop (ΔP_wet/L) versus the gas load factor. The h_L and ΔP_wet/L increased in parallel with the increase of a and c, respectively, while the slopes of h_L and ΔP_wet/L increased with b and d, respectively.     

Permeation through a bed on a vibrating medium: theory and experimental results

A dead-end filtration set-up with a vertically vibrated medium is used to study cake permeation. A key feature of these experiments is that a sudden increase in permeability at a certain critical vibration amplitude takes place, when the static loading is light. A theory to explain this phenomenon is put forward in terms of a relation to local fluidization near the medium, thus returning the clogged septum resistance to virtually its unclogged value. The fluidization is due to a particle stress induced by the vibration of the particle fluid mixture near the medium. This stress can be large enough to counteract the compressive stress that is caused by gravity and drag due to the fluid flow in the set-up.Estimates for the particle stresses are obtained; these are proportional to the amplitude decay inverse length λ. The latter is derived from the analysis of a vibrated particle-fluid mixture that is in a state of fluidization. It is argued that only in this state will the value of λ be large enough to generate the required particle stress. A much smaller value for λ is obtained when the particles in the medium make enduring contacts.The theory predicts a frequency dependence for the turnover point in the permeability according to the root of the applied frequency. This theoretical result is confirmed by the experiments. The theory also predicts that when the decay is too steep, so that the vibration amplitude vanishes at a distance of less than a particle diameter, no fluidization will occur. This is found to be true for larger cake masses.     

Dynamic Young's modulus of open-porosity titanium measured by the electromagnetic acoustic resonance method

As biological implants, porous titanium with adjustable mechanical properties can solve the stress-shielding effect. In this paper, porous titanium was prepared by the powder metallurgy method, where urea powders as the second phase were removed by heat treatment. Pore morphology (such as pore size and character) was controlled by the character of urea powders. The dynamic Young's moduli of such porous titanium with different morphology was measured by the electromagnetic acoustic resonance method. From the semi-log plots of Young's modulus versus the porosity, it was found that with increased porosity this modulus firstly decreases linearly, then decreases rapidly and goes to zero at certain porosity. However, the Young's modulus was independent of pore size. The relationship between Young's modulus and the porosity was explained by a parallel model based on the Minimum Solid Area method. The value of linear slop `b' and the percolation limit `P_C' were used for predicting the trend of Young's modulus varied with the porosity and pore size. So porous titanium with appropriate Young's modulus can be chosen as a candidate for bone substitutes.     

Mobility radius of fractal aggregates growing in the slip regime

A method is presented for the calculation of the mobility radius of fractal aggregates. The connection between the aggregate permeability and the monomer friction factor is derived, which makes it possible to convert the permeability in the continuum regime to that in the slip regime. The method elaborated here estimates the permeability of an aggregate treated either as impermeable sphere of the size equal to mobility radius or a permeable self-similar structure consisting of impermeable monomers. The internal permeability of a fractal aggregate growing in the slip regime is analyzed assuming that the aggregate consists of no more than twelve effective impermeable monomers, the number being a result of hydrodynamic considerations. The method makes it possible to estimate the aggregation number dependence of the mobility radius. A system of carbonaceous flame soot aerosol containing fractal aggregates with D=1.8 is analyzed. The mobility radius-aggregation number relation is found to be very close to that obtained experimentally. For large aggregation numbers this relation tends to that, which is valid for the dynamic radius.     

Explaining the abnormally high flow activation energy of thermoplastic polyurethanes

The rheological properties of a thermoplastic polyurethane (TPU) were studied at small and large deformation via three different types of rheometry: dynamic shear, capillary, and torque (an instrumented batch mixer). The effect of degradation during TPU processing on the melt viscosity was investigated and several factors, such as temperature, time, shear stress, and flow type that may affect the degradation were studied. Apparent activation energy of flow (E_a) was determined to be 328 kJ/mol, much larger than expected. A simple model was derived to describe the relationship of molecular weight and thermal dissociation of urethane linkages. Contributions of flow and the degradation reaction of TPU to overall activation energy were found to be additive: E_a=E_η+1.7ΔH_deg. True activation energy of flow (E_η) was estimated to be 144 kJ/mol. While the high apparent flow activation energies in dynamic shear and capillary rheometry can be explained by simple thermal degradation, melt viscosities interpreted from the instrumented batch mixer showed a much lower apparent activation energy (186 kJ/mol). This low value may be due to a combination of effects: errors in the relation between viscosity and mixer torque for TPU, side reactions resulting from air exposure, high stress level during the melting, and extensional stresses.     

Hydrodynamic drag forces on two porous spheres moving along their centerline

This paper numerically evaluates the hydrodynamic drag force exerted on two highly porous spheres moving steadily along their centerline (sphere #1 and sphere #2) through a quiescent Newtonian fluid over a Reynolds number ranging from 0.1 to 40. At creeping flow limit, the drag forces exerted on both spheres were identical. At higher Reynolds numbers the drag force on sphere #1 was higher than sphere #2, revealing the shading effects produced by sphere #1 on sphere #2. At dimensionless diameter (β, =d_f/2k^0.5, d_f and k are floc diameter and interior permeability, respectively) >20, the spheres can be regarded nonporous. At β<20, the drag forces dropped. At β<2, the drag forces approached “no-spheres” limit. An increased size ratio of two spheres (d_f1/d_f2) would increase the drag force on sphere #1 and reduce that on sphere #2. At increasing β for both spheres, the drag force on sphere #2 was increased because of the more difficult advective flow through its interior, and at the same time the drag was reduced owing to the stronger wake flow produced by the denser sphere #1. The competition between these two effects leads to complicated dependence of drag force on sphere #2 on β value. These effects were minimal when β became low. Two identical spheres could move steadily along their centerline. At higher Reynolds number, the two spheres would move closer because of the incorporation of inertia force. For spheres of different diameters, the sphere # 2 would move faster than sphere #1 regardless of their size ratio and β value. This occurrence yielded efficient coagulation when two porous spheres were moving in-line.     

ENHANCED TRANSFER FROM SUSPENSION TO POROUS BOUNDARIES

The flow of suspensions parallel to porous surfaces is typical to engineering and biological systems. In some cases the flow is associated with transfer of mass. This paper considers the effect of the presence of rigid particles near the porous boundary. The kinematic response of the particles to the viscous stresses imposed by the flowing fluid results in streamlines which penetrate the permeable boundary, thus providing a convective enhancement of the rate of transfer. This mechanism is totally absent with boundaries impermeable to flow. The general equations are developed and the important dimensionless parameters are defined. A particular example concerning linear shear flow past a catalytic boundary is evaluated.     

Effect of sample’s length on flow properties of open-cell metal foam and pressure-drop correlations

Many applications require fluid flow through the open pores of metal foam. The foam is usually treated as a porous medium for which the Darcy law and the Hazen-Dupuit-Darcy (or Forchheimer) equation are used to describe the pressure drop, and for obtaining the two important flow properties, i.e., the permeability and the form drag coefficient. Little or no attention is paid to the length (or thickness) of the porous medium in the flow direction. This paper establishes a minimum length necessary for the foam to have length-independent (or bulk) permeability and form drag coefficient. This minimum length is obtained experimentally for various types of open-cell aluminum foam subjected to airflow in the Forchheimer regime. Below this thickness values of the two key flow properties are not constant, and they include entrance/exit effects, which may explain some of the discrepancies in the reported values in the literature. The Forchheimer equation was recast in two different manners, which resulted in new non-dimensional numbers- one representing the form drag and the other the viscous drag. These numbers correlated very well with the thickness of the porous medium. The obtained correlations allow for determining the pressure drop given only the velocity and the thickness of an aluminum foam sample.     

Motions of a porous particle in stokes flow: Part 1. Unbounded single-fluid domain problem

Exact solutions in closed form have been found using the eigenfunction-expansion method for various linear and quadratic flows of anunbounded incompressible viscous fluid at low Reynolds number past aporous sphere with a uniform permeability distribution. The linear flows considered here are a simple shear and an axisymmetric uniaxial straining flows and the quadratic flows include a unidirectional paraboloidal and a stagnation-like flows as typical representations. The theoretical analysis determines a general motion of a freely suspended particle in the prescribed mean flow at infinity. Then the solutions are expressed in terms of fundamental singularity solutions for Slokes flow which will be applied to examine the motion of a porous sphere in the presence of a plane fluid-fluid interface in the forthcoming part of the present paper.     

Optimization of flow in additively manufactured porous columns with graded permeability

Chemical engineering systems often involve a functional porous medium, such as in catalyzed reactive flows, fluid purifiers, and chromatographic separations. Ideally, the flow rates throughout the porous medium are uniform, and all portions of the medium contribute efficiently to its function. The permeability is a property of a porous medium that depends on pore geometry and relates flow rate to pressure drop. Additive manufacturing techniques raise the possibilities that permeability can be arbitrarily specified in three dimensions, and that a broader range of permeabilities can be achieved than by traditional manufacturing methods. Using numerical optimization methods, we show that designs with spatially varying permeability can achieve greater flow uniformity than designs with uniform permeability. We consider geometries involving hemispherical regions that distribute flow, as in many glass chromatography columns. By several measures, significant improvements in flow uniformity can be obtained by modifying permeability only near the inlet and outlet.     

Fluid motion around and through a porous cylinder

The flow-field and solute transport through and around a porous cylinder is investigated numerically. The range of Reynolds number (based on the cylinder diameter and the uniform sinking rate of the cylinder) considered here is between 1 and 40 with Darcy number (Da) in the range 10^-6?Da?1.5 and porosity in the range 0.629?ε?0.999. The motivation of the present study is the application of flow through porous cylinder extensively applied in nuclear biological chemical filters as well as reduction of carbon fines in filtered water. The influence of Da on the drag coefficient, separation angle, recirculation length, streamline and vorticity pattern are investigated. The drag ratio, defined as the ratio of drag coefficient of porous cylinder to that of solid cylinder, is found to approach zero from unity as Da is increased from 10^-6 to 1.5. The separation point shifts towards the rear stagnation point as Da is increased. The time evolution of the solutal field at different Reynolds number and Darcy number is presented. A long slender concentration plume is found to evolve from the cylinder with decreasing concentration at the outer edge.     

On the apparent permeability of a porous layer backed by a perforated plate

Two-dimensional slow viscous flow from a fluid reservoir, through a porous layer and then through a perforated plate is studied assuming Stokes flow in the fluid reservoir and Darcy flow within the porous medium. It is first shown that the coupled Stokes/Darcy problem can be reduced to a Darcy problem when the various length scales are constrained such that Darcy's law is appropriate to describe flow in the porous layer in the vicinity of the perforations of the plate. The apparent permeability of the porous layer is studied as a function of the (uniform) thickness of the layer, and as a function of the size and spacing of the performations in the plate. The apparent permeability is shown to be significantly lower than the intrinsic permeability of the porous layer when the layer is sufficiently thin. Closed-form expressions for the apparent permeability are derived using conformal transformation techniques. We then present a model of particle deposition onto the perforated plate. The growth of the porous layer resulting from the deposition is studied, as is the evolution of its apparent permeability.     

Numerical study on the hydrodynamics of agglomerates at intermediate Reynolds numbers

The flow pattern and hydrodynamics of a heterogeneous permeable agglomerate in a uniform upward flow at intermediate Reynolds numbers(1–40) are analyzed from three-dimensional(3 D) computational fluid dynamics simulations. Different from the homogeneous or stepwise-varying permeability models used in previous papers, a continuously radially varying permeability model is used in the present study. The effects of two dimensionless parameters, the Reynolds number and the permeability ratio, on the flow field and the hydrodynamics were investigated in detail. The results reveal that unlike the solid sphere, a small recirculating wake initially forms inside the agglomerate. The critical Reynolds number for the formation of the recirculating wake is lower than that of the solid sphere and it decreases with the increase of permeability ratio. A correlation of drag coefficient as a function of the Reynolds number and permeability ratio is proposed. Comparisons of drag coefficients obtained by different permeability models show that at intermediate Reynolds numbers(1–40),the effect of radially varying permeability on the drag coefficient must be considered.     

Consideration of the dynamic forces during bubble growth in a capillary tube

Single bubbles were generated from a capillary tube in quiescent water and the bubble formation process was studied in detail using high-speed video at two pressures, 1.38 and 0.93 kPa. The bubble equivalent spherical radius, r, was derived from the data sets of 100 bubbles: (1) the a and b semi-axes values for an ellipsoidal model and (2) a (more accurate) cylindrical integration of the bubble image in 1-pixel layers. The two methods were compared and showed a significant improvement with the more accurate integration approach. Based on the time trends in the derivatives of a and b, three growth phases were identified and the different forces acting on the bubble were calculated. The analysis showed significant differences between the two cases, despite similar times from appearance to detachment. For the 0.93 kPa case, the bubble shape detachment is described by Cassini oval while for 1.38 kPa it is a lenmiscate. For the study conditions, the momentum force was negligible for both cases; however, the viscous drag force, added mass, and surface tension forces were not. The bubble eccentricity exhibited an oscillatory behavior, which we propose arose from highly non-linear wave-phenomena. Finally, only the low-pressure case was in agreement with the predictions of Oguz and Prosperetti (1993); for flows less than the critical flow, the high pressure was not in agreement, indicating a smaller value for the transition to super-critical flow for the study conditions.     

Stokes flow past periodic rows of porous cylinders

The flow of a viscous incompressible fluid at low Reynolds numbers in model filters—systems of porous permeable cylinders—is studied. A single row and hexagonal and square arrays of parallel cylinders perpendicular to the flow direction are considered. The flow field outside and inside a porous cylinder is determined by solving the Brinkman and Stokes equations. The drag force on a porous cylinder versus the cylinder permeability and the distance between the axes of neighboring cylinders is calculated. It is shown that rows of porous cylinders arranged into a square array have almost no mutual hydrodynamic effect at any array spacing. The cell model is shown to be applicable to the hexagonal array of porous cylinders over a wide range of packing densities.     

Terminal velocity of porous spheres

Terminal velocity of porous spheres was experimentally measured for a Reynolds number range of 0.2 to 120 for a normalized sphere radius, β = R/R of 15.6 to 33, where R and k are the sphere radius and permeability, respectively. The drag coefficient for 15 < β < 33 was found to be C_D = 24Ω/Re [1 + 0.1315 Re^{(0.82 - 0.05w)}] for 0.1 < Re ≤ 7 and C_D = 24Ω/Re [1 + 0.0853 Re^{(1.093 - 0.105w)}] for 7 < Re < 120 with w = log₁₀Re where Re is the sphere Reynolds number and Ω=2β² [1 - (tanh β/β)] / 2β² + 3[1 - tanh β/β)] At high Reynolds numbers, it was found that the porous sphere terminal velocity was less affected by the container walls than for the case of an impermeable sphere. However, at very low Reynolds numbers, the wall effects were found to be similar for both the permeable and the impermeable spheres.     

Flow Kinetics in Porous Ceramics: Understanding with Non-Uniform Capillary Models

The present work describes the development of a two-parameter non-uniform capillary model to describe kinetics of flow in porous solids with complex tortuous varying paths. Experimentally, the rate of fluid flow in such a non-uniform capillary is found to be orders of magnitude slower compared with a corresponding average uniform capillary. This slow rate is explained in terms of an extremely small 'effective' hydrodynamic radius. The origin of such an 'unphysical' radius is rationalized based on geometrical considerations and effective driving forces for flow through a stepped capillary. Infiltration rate parameters are derived from the geometry of the porous medium for both wetting and non-wetting conditions.