Mass transfer from a contaminated fluid sphere

The unsteady mass transfer from a contaminated fluid sphere moving in an unbounded fluid is examined numerically for unsteady‐state transfer. The effect of the interface contamination and the flow regime on the concentration profiles, inside and outside a fluid sphere, is investigated for different ranges of Reynolds number (0 < Re < 200) and Peclet number (0 < Pe < 10⁵), viscosity ratio between the dispersed phase and the continuous phase (0 < κ < 10), and the stagnant‐cap angle (0° < θ_cap < 180°). It was found that the stagnant‐cap angle significantly influences the mass transfer from the sphere to a surrounding medium. For all Peclet and Reynolds numbers and κ, the contamination reduces the mass transfer flux. The average Sherwood number increases with an increase of stagnant‐cap angle and reaches a maximum equal to the average one for a clean fluid sphere at low viscosity ratio and large Peclet numbers. A predictive equation for the Sherwood number is derived from these numerical results. © 2010 American Institute of Chemical Engineers AIChE J, 2011     

Mass transfer from a single fluid sphere to power-law liquids at moderate Reynolds numbers

The steady-state convective inter-phase mass transfer from a single Newtonian fluid sphere (free from surfactants) to a continuous phase with power-law viscosity has been studied at moderate Reynolds and Schmidt numbers under the conditions when the resistance to mass transfer in the dispersed phase is negligible. The species continuity equation, segregated from the momentum equations of both phases, has been numerically solved using a finite difference method. The effects of the Reynolds number (Re_o), power-law index (n_o), internal to external fluid characteristic viscosity ratio (k) and Schmidt number (Sc) on the local and average Sherwood number (Sh) have been analysed over the following ranges of conditions: 5?Re_o?200, 0.6?n_o?1.6, 0.1?k?50 and 1?Sc?1000. It has been observed that irrespective of the values of the Reynolds number and of the power-law index, as the value of k increases the average Sherwood number decreases for intermediate to large values of the Peclet number. As the value of the power-law index increases, the rate of mass transfer decreases for all values of the Reynolds number and the characteristic viscosity ratio thereby suggesting that shear-thinning behaviour facilitates mass transfer, whereas shear-thickening behaviour impedes it. Based on the present numerical results, a simple predictive correlation is proposed which can be used to estimate the rate of inter-phase mass transfer of a fluid sphere sedimenting in power-law liquids.     

Kinetic theory based computation of PSRI riser: Part I—Estimate of mass transfer coefficient

The PSRI benchmark challenge problem one is modeled using kinetic theory based CFD with the energy minimization multi-scale (EMMS) drag law. These computations give a better comparison than the previous models to measured solids mass flux, solids density and pressure drop.The computer model was also used to calculate axial and radial normal Reynolds stresses, energy spectra, power spectra, granular temperatures, the FCC viscosity and axial and radial dispersion coefficients. The computed cluster sizes agreed with the published empirical correlations. Then, the mass transfer coefficients and the Sherwood numbers are estimated based on particle cluster sizes. The conventional Sherwood number is scaled with the particle cluster diameter. The Sherwood number is the order of 10^-2 and the mass transfer coefficient is the order of . This Sherwood number is two orders of magnitude smaller than the diffusion controlled limit of two based on particle diameter, in agreement with the experimental data for fluidization of fine particles.     

Wall-to-liquid mass transfer in fixed beds at low flow rates

The wall-to-fluid mass transfer coefficient was obtained from coated wall dissolution experiments, for water flow through fixed beds of spheres with tube-to-particle diameter ratios of 2.9-11.6, and for particle Reynolds number (Re) in the range 3-200. The coefficients, in the form of dimensionless Sherwood numbers (Sh_wf), were shown to approach a nonzero limit as Re→0. The data were well-represented by the equation

     

Numerical study on steady and transient mass/heat transfer involving a liquid sphere in simple shear creeping flow

A numerical method is utilized to examine the steady and transient mass/heat transfer processes that involve a neutrally buoyant liquid sphere suspended in simple shear flow at low Reynolds numbers is described. By making use of the known Stokes velocity field, the convection‐diffusion equations are solved in the three‐dimensional spherical coordinates system. For the mass transfer either outside or inside a liquid sphere, Sherwood number Sh approaches an asymptotic value for a given viscosity ratio at sufficiently high Peclet number Pe. In terms of the numerical results obtained in this work, two new correlations are derived to predict Sh at finite Pe for various viscosity ratios. © 2013 American Institute of Chemical Engineers AIChE J, 60: 343–352, 2014     

The heat transfer coefficient between a particle and a bed (packed or fluidised) of much larger particles

The heat transfer coefficient has been measured for a heated phosphor-bronze sphere (diam. 2.0, 3.0 or 5.56 mm) added to a bed of larger particles, through which air at room temperature was passed. The bronze heat transfer sphere was attached to a very thin, flexible thermocouple and was heated in a flame to before being immersed in the bed. The cooling of the bronze sphere enabled the heat transfer coefficient, h, to be measured for a variety of U/U_mf, as well as diameters of both the particles in the bed and the heat transfer sphere. It was found that before the onset of fluidisation, h rose with U, but h reached a constant value for U?U_mf. These measurements indicate that in this situation (of a relatively small particle in a bed of larger particles) all the heat transfer is between the hot bronze sphere and the gas flowing over it. Consequently, a Nusselt number, based on the thermal conductivity of the gas, is easy to define and for U?U_mf (i.e. a packed bed), Nu is given by

     

Lift force in bubbly flow systems

From the significance of three-dimensional simulation of dispersed flow systems in many engineering fields, extensive study was conducted for lift force in a single particle system as well as a multiparticle system. In this study, the lift force in a single particle system was modeled by considering the effect of bubble deformation on the lift force. The model was finalized based on existing data obtained in the range of particle Reynolds number from 3.68 to 78.8, viscous number from 0.0435 to 0.203 and Eötvös number from 1.40 to 5.83. The viscous number is defined by where μ_f, ρ_f, σ, g and Δρ are, respectively, fluid viscosity, fluid density, surface tension, gravitational acceleration and density difference between phases. The applicability of the model to higher particle Reynolds number system such as an air-water system was qualitatively examined. The lift force model developed in a single particle system was extended to a multiparticle system. The applicability of the extended lift force model was qualitatively examined. The similarity between drag and lift forces were also discussed.     

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.     

TWO DROP MASS TRANSFER WITH CHEMICAL REACTION-ASYMPTOTIC CASE STUDIES

Mass transfer at very low and moderately high Peclet numbers has been analyzed for two interacting solid spheres and for drops in tandem. In the first case study, where the Peclet and Reynolds numbers approach zero, interactions between two drops with liquid phase chemical reaction affect the mass flux more drastically for gas resistance controlling cases than for liquid resistance controlling cases. The effects of drop size, spacing, and reaction rate on the Sherwood numbers have been considered and the various regimes of gas and liquid side control have been numerically established. The asymptotic value of the average Sherwood number as the interdrop distance approaches infinity is lower for the case of two drops than for two solid spheres, i.e. it was found that ¯Sh² _drops = ¯0.5Sh² _solidspheres, as (dAB/a)→∞

For the case of mass transfer at moderately high Peclet numbers potential flow, i.e. Re→∞, was assumed. This limited analysis indicates that there is no significant difference between the single drop and the two drop cases.     

Particle diameter prediction in supercritical nanoparticle synthesis using three-dimensional CFD simulations. Validation for anatase titanium dioxide production

A mathematical model is proposed and validated with experimental data for the estimation of the average diameter attained by the particles that are generated by decomposition of an organometallic precursor in supercritical CO₂ (scCO₂). Experiments have been performed for the synthesis of TiO₂ anatase nanoparticles, using diisopropoxititanium bis(acetilacetonate) (DIPBAT) as precursor, ethanol as reactant and scCO₂ as solvent. The model is solved by using computational fluid dynamics (CFD) for the experimental geometry: a tee piece as mixer followed by a cylindrical reactor. Peng-Robinson EOS with Huron-Vidal mixing rule has been used to predict density variations within hydrodynamic equations. A pseudo-first order kinetic (r_TiO2=kC_DIPBAT, with ) has been proposed for mass and population balances. The CFD simulations predict a reduction on particle diameter from 400 to 200 nm when the Reynolds number is increased from 280 to 1500. In this range, deviations with experimental data are lower than 15%. A parametric study of the kinetic constant reveals that for faster reactions (Da?10^-3) the trend of particle size with the Reynolds number is inverse, and particle diameter increases when the Reynolds number is increased.     

Mass transfer and kinetics of the three-phase hydrogenation of a dinitrile over a Raney-type nickel catalyst

The hydrogenation kinetics of a dinitrile over a Raney-type nickel catalyst was evaluated from experiments performed in a fed-batch operating autoclave at 320- and 2- hydrogen pressure. This complex catalytic reaction consists of two main parts: almost 100% selective hydrogenation of the dinitrile to the corresponding aminonitrile and consecutive hydrogenation to either the desired primary diamine or to pyrrolidine via ring formation. An extensive study has been made on the effects of mass transfer in the applied slurry-type reactor for this reaction. The gas-liquid mass transfer is enhanced by the presence of catalyst particles, and at typical hydrogenation conditions, k_La values up to can be reached. A Sherwood correlation for the three-phase reactor showed that important parameters in the gas-liquid mass transfer are stirrer speed and the density and viscosity of the solvent. The kinetic experiments were performed in absence of mass and heat transfer limitations. The kinetic data were modeled using two rate models based on Langmuir-Hinshelwood kinetics, assuming the reaction of dissociatively adsorbed hydrogen and nitrile compound as rate-limiting step. The first model involved competitive adsorption between hydrogen and organic compound and the second model was based on non-competitive adsorption. Both models successfully described both reaction parts. The reaction of dinitrile to aminonitrile is nearly 100% selective due to the relatively strong adsorption of the dinitriles as compared to the aminonitriles. By increasing the hydrogen partial pressure, higher yields of primary amine can be obtained. The models predict that operating in the mass-transfer regime at relatively high temperatures reduces the formation of the primary diamine.     

Structure and rate of growth of whey protein deposit from in situ electrical conductivity during fouling in a plate heat exchanger

The influences of calcium concentrations , Reynolds number (2000-5000) and temperature () upon the deposit structure and the rate of growth deposition have been investigated in a plate heat exchanger. This was done from in situ measurements of the deposit electrical conductivity via implementation of stainless steel electrodes in channels combined with assessments of deposit thickness. Calcium ions affect structures of deposits and increase the rate of deposit growth upon heated surfaces. This was attributed to the formation of weaker size aggregates at higher calcium concentrations and a higher number of calcium bindings, which reinforce adhesion forces between protein aggregates. Structures and appearances of deposits also were affected by flow rates whatever the calcium concentrations. Deposit growth rate was enhanced by increasing flow rate below a critical Reynolds number comprised between 3200 and 5000. On the contrary, above the critical Reynolds number, a limitation of the deposit and/or an escape of the deposit from the fouled layer into the core fluid occurred, caused by the predominance of particle breakage on the deposit formation. Fouling tended to be reduced at higher flow rate. It was noteworthy that rates of growth decrease during fouling experiments which may be explained by an increase in local shear stresses leading to particle breakage.     

Impact spreading and absorption of Newtonian droplets on topographically irregular porous materials

This paper concerns the combined influences of high impaction rates and surface geometries on the spreading-absorption behaviour of Newtonian fluid droplets under non-equilibrium conditions. A volume of fluid method is used to simulate the motion of up to diameter droplets onto surfaces comprising protuberances below in width. On both flat and systematically arranged pyramidal surfaces, spreading is an exponential function of the maximum depth of absorption. Surface spreading is furthermore a linear function of the droplet Reynolds number and a simple method of predicting spreading from the Reynolds number has been identified herein. At high impact velocities, kinetic energy driven spreading dominates over wetting driven flow; though, its relativity to absorption is a function of the droplet pressure pulse. Lateral (plan view) spreading on irregular surfaces is found to be inversely proportional to surface roughness. In this paper, surface geometry is shown to be a critical determinant of both spreading and absorption and thereby, the research herein provides details that could be useful in functionalising surfaces. Finally, new surface roughness equations derived herein make the comparison of surface spreading to surface roughness more realistic than commonly used roughness factors such as the Wenzel and R_a roughness.     

Mass and heat transfer from or to a single sphere in simple extensional creeping flow

The first detailed numerical investigation on the mass and heat transfer both outside and inside a solid or liquid sphere immersed in a simple extensional flow for a larger range of Peclet numbers (1–100,000) is presented. By making use of the known Stokes velocity field at small Reynolds numbers, a finite difference method with the control volume formulation is adopted to solve the convection‐diffusion transport equation. Simulation results show that the transport rate, which is represented by Sherwood number, is significantly affected by Peclet number and viscosity ratio. The flow direction, no matter a uniaxial extensional flow or a biaxial extensional flow, has no effect on the total transport rate but affects the concentration distribution a lot. Some comparisons between present simulations and previous studies are made to validate each other and confirm the reliability and applicable scopes of reported correlations. A few new correlations are put forward to predict the transfer rate at finite Peclet numbers for various values of viscosity ratios. © 2011 American Institute of Chemical Engineers AIChE J, 58: 3214–3223, 2012     

Estimate of solid flow rate from pressure measurement in circulating fluidized bed

A method is described to estimate solid mass flow rate based on measurement of pressure drop in horizontal section of circulating fluid bed (CFB). A theoretical model was derived based on momentum balance equation and used to predict the solids flow rate. Several approaches for formulating such models are compared and contrasted. A correlation was developed that predicts the solids flow rate as a function of pressure drop measured in the horizontal section of piping leading from the top of the riser to the cyclone, often referred to as the cross-over. Model validation data was taken from literature data and from steady state, cold flow, CFB tests results of five granular materials with various sizes and densities in which the riser was operated in core-annular and dilute flow regimes. Experimental data were taken from a 0.20 m ID cross-over piping and compared to literature data generated in a 0.10 m ID cross-over pipe. The solids mass flow rate data were taken from statistically designed experiments over a wide range of Froude number , load ratio , Euler number , density ratio , Reynolds number , and Archimedes number . Several correlations were developed and tested to predict the solids mass flux based on measuring pressure drop in the horizontal section of CFB. It was found that load ratio is a linear function of the Euler number and that each of these expressions all worked quite well (R² > 95%) for the data within the range of conditions from which the coefficients were estimated.     

Shape normalization for catalytic monoliths

We analyze the coupled convection, diffusion and reaction problem for laminar flow in a washcoated channel of uniform but arbitrary cross-sectional shape with nonuniform washcoat thickness. For the case of an isothermal first-order reaction, we obtain analytical solutions and show that the reactant exit mixing-cup conversion (χ_m) depends mainly on the transverse Peclet number (P) and the effective local Damköhler number (Φ_s²), which depends on relative effective diffusivity (δ), relative washcoat thickness (λ) and local Damköhler number (φ_s²), and is a weak function of the axial Peclet number Pe and the Schmidt number Sc (or local velocity profile). We generalize the results for nonlinear kinetics and show that the χ_m versus P curve is universal for all geometric shapes and washcoat profiles, provided P and Φ_s² are defined using the shape normalized length scales ( and for the fluid phase and washcoat, respectively) and the normalized reaction time scale presented in this work. Here, is defined as the ratio of flow cross-sectional area to the fluid-washcoat perimeter and is defined as the ratio of washcoat cross-sectional area to the fluid-washcoat perimeter. Furthermore, we show that the χ_m versus P curve in the kinetic regime (Φ_s²?1) is independent of the velocity profile and is given by the asymptotes χ_m=1 for P?Φ_s² and χ_m=Φ_s²/P for P?Φ_s². In the mass transfer controlled regime (Φ_s²?1), the conversion (χ_m), for the case of fully developed flow, is given by the asymptotes χ_m=1 for P?1 and χ_m≈P^−2/3 for P?1, with a transition around a P value of unity. For the practical case of long ducts (P?1 or ) we show that the high-conversion branch may be approximated by for all cases. We present both analytical and numerical results on the first eigenvalue (μ₁) and the Fourier weight (α₁) for some cases of practical interest and show that these asymptotic constants are insensitive to the channel geometric shapes and washcoat profile but depend on the numerical values of φ_s²,δ and λ. Finally, we use theoretical results to present criteria for optimal design of catalytic monoliths.     

The heat and mass transfer from a fluid sphere at large reynolds and peclet number

Analytic expressions have been derived for the steady rate of heat or mass transfer from a fluid sphere in uniform motion at large Reynolds and Peclet number. The solution is applicable to cases where both phase resistances are of the same order of magnitude, or when the resistance of one phase is negligible. The analysis has shown that if finite values of fluid viscosity and density are considered the transfer rates are considerably less than those obtained from the ideal fluid model. The maximum flux for a fluid sphere occurs near the equatorial plane and in this respect its behavior differs significantly from a solid sphere.