Mean Solid Fraction Research Articles

A two-phase pure slurry model for planetary cores: one-dimensional solutions and implications for Earth's F-layer

We develop and analyse a continuum model of two-phase slurry dynamics for planetary cores. Mixed solid–liquid slurry regions may be ubiquitous in the upper cores of small terrestrial bodies and have also been invoked to explain anomalous seismic structure in the F-layer at the base of Earth's liquid iron core. These layers are expected to influence the dynamics and evolution of planetary cores, including their capacity to generate global magnetic fields; however, to date, models of two-phase regions in planetary cores have largely ignored the complex fluid dynamics that arises from interactions between phases. As an initial application of our model, and to focus on fundamental fluid dynamical processes, we consider a non-rotating and non-magnetic slurry comprised of a single chemical component with a temperature that is tied to the liquidus. We study one-dimensional solutions in a configuration set up to mimic Earth's F-layer, varying gravitational strength$R$, the solid/liquid viscosity ratio$\lambda _{\mu }$and the interaction parameter$K$, which measures friction between the phases. We develop scalings describing behaviour in the limit$R \gg 1$and$\lambda _{\mu } \gg 1$, which are in excellent agreement with our numerical results. Application to Earth's core, where$R \sim 10^{28}$and$\lambda _{\mu } \sim 10^{22}$, suggests that a pure iron slurry F-layer would contain a mean solid fraction of at most 5 %.

Multi-mechanism theory of aerosol capture by fibrous filters, including fiber diameter/orientation dispersity and particle morphology effects. Preliminary tests vs. data for mobility-selected submicron particles

In Paper I (Sep. Purif. Technol. 257 (2021) 117676) we showed that a semi-analytic, multi-mechanism expression for the single-fiber capture fraction, ηcap,SF, (derived using asymptotically valid approximations: Ref<0.4, Pef≫1, R≪1, R⋅Pef1/3 arbitrary and Stkp≤Stkpcrit), facilitates a deterministic-, pseudo-continuum aerosol population-balance (PB-) approach to predicting fibrous filter performance. There we explicitly considered “deep” (Lf/df,g≫1), low solidity idealized fibrous filters (FFs) challenged by polydispersed aerosols—especially single-mode log-normal (LN) ASDs of modest spread captured by a spatially uniform array of fibers of a single diameter in crossflow. However, realistic fibrous filter media often possess a LN distribution of fiber diameters, as well as a near-Gaussian orientation distribution narrowly spread about normal incidence (θ=π/2). Moreover, even if this were not so, there would be meso-scale departures from a uniform average fiber solid fraction. We show here that our tractable aerosol PBE-approach to idealized FF performance (Paper I) can be generalized to incorporate these particular structural features of commercially available fibrous filter media. But, to clarify whether these generalizations are likely to be useful, if not fully sufficient, for practical circumstances, it is also necessary to compare such methods/predictions against selected sets of well-defined experimental results. We initiate this program here, having chosen the recent experiments of Kang et al. (2019) carried out using a commercially available fiberglass filter with Lf/df,g≃300, mean solid fraction of 0.039, and df,g=2.5μm, successively challenged by mobility-selected KCl(s) particles (with diameters between ca. 20 and 600nm) at the carrier gas velocities of 15 and 10 cm/s—capture conditions dominated by the transport mechanism of Brownian diffusion and convection, with “interception” (associated with non-negligible dp/df) becoming important above ca. dp=100nm. We conclude from these data that the effective interception diameter, dp,icpt,eff, of the particles studied is systematically larger than their stated mobility diameters—a situation which will deserve further attention in future studies. Encouraged by these preliminary but instructive comparisons, we expect that, for many current and future design purposes, our present class of semi-analytic/non-stochastic/multi-mechanism methods will provide a welcome complement, if not alternative, to much more computationally-intensive simulation methods for realistic fibrous media that have been described and implemented in the recent aerosol filtration literature. The consequences of including these structural features of fibrous filters in the presence of aerosol size- and shape polydispersity will be the subject of future studies, based on the generalized Population Balance Equation developed/proposed in Section 3.3.

Convective heat transfer in suspensions of prolate ellipsoids

Particle Resolved Simulations (PRS) using the Immersed Boundary Method (IBM) are performed for flow through suspensions of ellipsoids with aspect ratios of 2.5, 5, and 10 for solid volume fractions from 0.1 to 0.3 in the Reynolds number range from 10 to 200. The mean Nusselt number increases as Re1/2but shows only a weak dependence on the aspect ratio while increasing between 10-15% with an increase in solid fraction from 0.1 to 0.3. Two common practices of calculating Nusselt number in past literature are reconciled. It is shown that the suspension mean Nusselt number based on individual particles, by definition is always greater than or at the least equal to the Nusselt number based on the internal developing flow analogy. It is established that for Re≤50, the suspension heat transfer coefficient is very sensitive to the spatial distribution of particles or local-to-particle solid fractions. For the same mean solid fraction, suspensions dominated by particle clusters or high local solid fractions can exhibit Nusselt numbers which are lower than the minimum Nusselt number imposed by pure conduction on a single particle in isolation. This results from the dominant effect of thermal wakes at low Reynolds numbers. As the Reynolds number increases to 100 and beyond, the effect of particle clusters on heat transfer becomes less consequential. Unlike heat transfer, particle clustering has a less significant effect on mean fluid forces such as drag.

Influencing the solid fraction distribution in a circulating fluidized bed system using differently shaped internals

Solids distribution and gas–solid contact gain in importance for new and alternative energy conversion processes realized with fluidized bed systems. The so-called dual circulating fluidized bed (DCFB) system represents a reactor concept proposed for these novel processes. This concept is based on the interconnection of two circulating fluidized beds via two loop seals. In this work, a cold flow model, designed according to the DCFB system, is used to investigate the influence of constrictions placed along the height of the secondary reactor. Experiments were performed with differently shaped constrictions, aperture ratios, varying distance between the constrictions and different length of the horizontal section at the smallest cross section in the constriction. In general, the mean solid fraction in the upper part of the secondary reactor benefits from the constrictions (mean solid fraction up to +1800% compared to a straight reactor without any constriction). The mean solid fraction in the reactor increased with the solid circulation rate between the reactors and the superficial gas velocity in the secondary reactor. Further, investigations showed substantial deviations for the solid fraction in the upper part of the reactor for the investigated geometries of the constrictions. Fluid dynamic limitations restricting the operating range of the cold flow model could be observed. These limitations depend on combinations of solid circulation rate and superficial gas velocity in the secondary reactor and differ for the investigated geometries. Solid fraction distribution in the secondary reactor showed high solid fractions in the constricted sections (up to 60%) while it remained low between the constrictions.

Suspension flow through an asymmetric T-junction

The flow of a suspension through a bifurcating channel is studied experimentally and by computational methods. The geometry considered is an ‘asymmetric T’, as flow in the entering branch divides to either continue straight or to make a right angle turn. All branches are of the same square cross-section of side length $D$, with inlet and outlet section lengths $L$ yielding $L/D=58$ in the experiments. The suspensions are composed of neutrally buoyant spherical particles in a Newtonian liquid, with mean particle diameters of $d=250~\unicode[STIX]{x03BC}\text{m}$ and $480~\unicode[STIX]{x03BC}\text{m}$ resulting in $d/D\approx 0.1$ to $d/D\approx 0.2$ for $D=2.4~\text{mm}$. The flow rate ratio $\unicode[STIX]{x1D6FD}=Q_{\Vert }/Q_{0}$, defined for the bulk, fluid and particles, is used to characterize the flow behaviour; here $Q_{\Vert }$ and $Q_{0}$ are volumetric flow rates in the straight outlet branch and inlet branch, respectively. The channel Reynolds number $Re=(\unicode[STIX]{x1D70C}DU)/\unicode[STIX]{x1D702}$ was varied over $0&lt;Re&lt;900$, with $\unicode[STIX]{x1D70C}$ and $\unicode[STIX]{x1D702}$ the fluid density and viscosity, respectively, and $U$ the mean velocity in the inlet channel; the inlet particle volume fraction was $0.05\leqslant \unicode[STIX]{x1D719}_{0}\leqslant 0.30$. Experimental and numerical results for single-phase Newtonian fluid both show $\unicode[STIX]{x1D6FD}$ increasing with $Re$, implying more material tending toward the straight branch as the inertia of the flow increases. In suspension flow at small $\unicode[STIX]{x1D719}_{0}$, inertial migration of particles in the inlet branch affects the flow rate ratio for particles ($\unicode[STIX]{x1D6FD}_{\mathit{particle}}$) and suspension ($\unicode[STIX]{x1D6FD}_{\mathit{suspension}}$). The flow split for the bulk suspension satisfies $\unicode[STIX]{x1D6FD}&gt;0.5$ for $\unicode[STIX]{x1D719}_{0}&lt;0.16$ while $\unicode[STIX]{x1D719}_{0}=0.16$ crosses from $\unicode[STIX]{x1D6FD}\approx 0.5$ to $\unicode[STIX]{x1D6FD}&gt;0.5$ at $Re\approx 100$. For $\unicode[STIX]{x1D719}_{0}\geqslant 0.2$, $\unicode[STIX]{x1D6FD}&lt;0.5$ at all $Re$ studied. A complex dependence of the mean solid fraction in the downstream branches upon inlet fraction $\unicode[STIX]{x1D719}_{0}$ and $Re$ is observed: for $\unicode[STIX]{x1D719}_{0}&lt;0.1$, the solid fraction in the straight downstream branch initially decreases with $Re$, before increasing to surpass the inlet fraction at large $Re$ ($Re\approx 500$ for $\unicode[STIX]{x1D719}_{0}=0.05$). At $\unicode[STIX]{x1D719}_{0}&gt;0.1$, the solid fraction in the straight branch satisfies $\unicode[STIX]{x1D719}_{\Vert }/\unicode[STIX]{x1D719}_{0}&gt;1$, and this ratio grows with $Re$. Discrete-particle simulations employing immersed boundary and lattice-Boltzmann techniques are used to analyse these phenomena, allowing rationalization of aspects of this complex behaviour as being due to particle migration in the inlet branch.

Granular temperature in a liquid fluidized bed as revealed by diffusing wave spectroscopy

We report granular temperature and solid fraction fields for a thin rectangular bed (20×200 mm cross-section and 500 mm high) of glass particles (mean diameter of 165 μm and density of 2500 kg/m 3) fluidized by water for superficial velocities ranging from 0.05 U t , which is approximately double the minimum fluidization velocity, to 0.49 U t , where U t is the particle terminal velocity estimated by fitting the Richardson–Zaki correlation to the bed expansion data. At superficial velocities below 0.336 U t , the solid fraction and granular temperature are uniform throughout the bed. At higher superficial velocities, the solid fraction tends to decrease with height above the distributor, whilst the granular temperature first increases to a maximum before decaying towards the top of the bed. Correlation of the mean granular temperature with the mean solid fraction and the local granular temperature with the local solid fraction both suggest that the granular temperature in the liquid fluidized bed can be described solely in terms of the solid fraction. The granular temperature increases monotonically with solid fraction to a maximum at φ≈0.18 where it then decreases monotonically as φ approaches the close-packed limit.

Voidage waves in hydraulic conveying through narrow pipes

The nature of voidage waves in hydraulic conveying systems has been investigated computationally in this study. Voidage waves consisting of alternating regions of high and low solid fractions were observed to form spontaneously in hydraulic conveying through vertical pipes. The wavelengths of these voidage waves were found to be inversely related to mean solid fraction but independent of liquid conveying velocities for elastic solid particles. The wave structure in terms of the thickness and shape of the solid fraction profile over each dense phase was dependent on the mean solid fraction used. Pressure distributions along the longitudinal direction of the conveying pipe exhibited an interesting ‘step’ profile due to unequal pressure drops across the dense and dilute regions. With inelastic solid particles, a lower tendency for voidage wave formation was observed and a phase diagram was constructed to show combinations of operating conditions under which voidage waves would form. Finally, the study of voidage waves in hydraulic conveying systems was extended to the case of hydraulic conveying through horizontal channels. In addition to the various system parameters, voidage wave formation in this case was also affected by gravitational settling of the solid particles.

Effect of solid fraction on fluctuations and self-diffusion of sheared granular flows

The influences of solid fraction on the fluctuations and self-diffusion of sheared granular flows were experimentally studied in a shear cell device. Employing the image processing technology and particle tracking method, the local displacements and velocities of particles were measured. From the measured velocities, the fluctuations, granular temperatures and shear rates could be determined. The self-diffusion coefficients were determined from the history of particles’ diffusive displacements. The periodic cell concept was used for long-time tracking purpose. Because of the gravitation force, the flows consist of both a solid-like region with higher and more uniform velocities in the lower test section and a fluid-like region in the upper part. The velocity fluctuations and self-diffusions were anisotropic with greatest components in the streamwise direction. The flow rheology was very sensitive to the mean solid fraction of the granular flow. The dependence of the diffusion coefficients on the velocity fluctuations and the shear rate were discussed.

Measurements of velocity fluctuations of granular materials in a shear cell

This paper studies the flow behavior of granular materials in a shear cell. The average and fluctuating velocities in the streamwise and transverse directions were measured by the image processing technique and particle tracking method. The granular materials were put in a shear cell to simulate the Couette flow with a stationary upper wall. The flows consist of a low shear region with higher and more uniform velocities in the lower test section and of a high shear region in the upper part. The velocities were decreased with the height, and the fluctuations, the granular temperature and the shear rate were increased with the height. The velocity fluctuations were anisotropic and were greater in the streamwise direction. The influences of the lower wall velocity and the mean solid fraction on the flow behavior were investigated in this study.

Shape and extent of the void formed by a horizontal jet in a fluidized bed

AbstractThe introduction of reactant gas into a fluidized‐bed chemical reactor as a jet is a common design practice. However, the shape and extent of the void formed by a jet into a fluidized bed and the manner in which the void breaks up into bubbles have still not been described well.An experimental study of a horizontal jet into a 15 cm by 38 cm cross‐section fluidized bed was carried out using an X‐ray system. Instantaneous solid fraction averaged along 15‐cm chords across the bed were measured. The mean value and the fluctuating component of solid fraction were determined for two initial jet diameters, 0.64 and 1.27 cm, and three initial jet velocities, 23, 46 and 69 m/s. Maps of the mean solid fraction and of statistical properties of the fluctuating component show that there are three discernible regions in the jet‐influenced area of the bed. These are a coherent void, bubble trains, and a surrounding compaction zone.

Pressure Drop and Distribution of Moisture Content in Layers of Rayon Staple Dried by Throughflow of Air

The drying the falling-rate period of drying is known to be approximately pro-portional to the moisture content W.∂W/∂θ=-KWwhere W: moisture content (kg water/kg dried fiber)θ: time (min)K : proportionality constant (1/min)The author has found that K is a function of the temperature and humidity of drying air; and that ∂K/∂ω decreases at a rate proportional to R/G, where R is the drying rate, ∂W/∂θ and G is the mass velocity of dryingair (kg/m2). This relation is expressible thus by equation : ∂K/∂ω=J/G-∂W/∂θwhere J is a proportionality constant.The two partial differential equations were integrated numerically with results which a greed well with experimental data.The pressure drop across fiber layers cannot be represented by a single coefficient of resistance, becuause fiber layers are compressed in varying degrees by the applied air pressure.The mean solid fraction (1-e) in fiber layers compressed by air pressure was obtained from the compressional characteristics of fiber layers compressed by uniform pressure so as to modify the above-mentioned coefficient of resistance for the pressure drop. The coefficient of friction when divided by a certain power of (1-e) is a relatively constant value.