Disperse-phase mass transport

If the disperse phase has particles with different volumes and different masses (or material densities), at least two internal coordinates are necessary to describe a particle state. In order to derive a transport equation of the disperse-phase mass density pp, we will let 

For the case in which all particles have the same mass, Um = Un for the case in which all particles have the same material density, Um = Uy. Note that, like Un and Uy, the mass-average disperse-phase velocity will not usually be in closed form. In most applications, the mean particle velocity Up will be set to the mass-average disperse-phase velocity. Thus, unless noted otherwise, we will set Up = Um throughout the rest of this book. The particle-mass source terms are defined by 

Since mass is conserved during transport, the continuous contribution Gp is due to mass transfer from the fiuid to the solid phase. Likewise, the discontinuous term 5m might appear due to nucleation of particles with nonzero mass from the fluid phase. For systems with no mass transfer between the disperse and continuous phases, the right-hand side of Eq. (4.64) will be null. 

Since the total mass of the fluid-particle system is conserved, we can deflne the fluid-phase mass density by 

2 The particle mass can be written as Mp = PpVp, where pp is the material density of the particle and Vp is its volume. Note that, in addition to mass, either material density or volume could be included in the internal-coordinate vector. (In general, we will use mass and volume as the internal coordinates.) Thus, for example, fixing the particle masses to be equal does not imply that all particles have the same volume. 

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The transfer of analyte from one phase to another in liquid-liquid extraction is achieved at the interface between the two immiscible phases. With appropriate agitation of the phases, mass transport within the two phases is of minor importance for the transfer efficiency. Maximum exposure of the phase interface is therefore pursued in batch extractions by vigorous shaking, resulting in the formation of highly dispersed droplets of one phase in the other. However, such conditions are not achievable in continuous flow systems such as FIA. Nevertheless, in most cases phase transfer factors of... 

The parameter p (= 7(5 ) in gas-liquid sy.stems plays the same role as V/Aex in catalytic reactions. This parameter amounts to 10-40 for a gas and liquid in film contact, and increases to lO -lO" for gas bubbles dispersed in a liquid. If the Hatta number (see section 5.4.3) is low (below I) this indicates a slow reaction, and high values of p (e.g. bubble columns) should be chosen. For instantaneous reactions Ha > 100, enhancement factor E = 10-50) a low p should be selected with a high degree of gas-phase turbulence. The sulphonation of aromatics with gaseous SO3 is an instantaneous reaction and is controlled by gas-phase mass transfer. In commercial thin-film sulphonators, the liquid reactant flows down as a thin film (low p) in contact with a highly turbulent gas stream (high ka). A thin-film reactor was chosen instead of a liquid droplet system due to the desire to remove heat generated in the liquid phase as a result of the exothermic reaction. Similar considerations are valid for liquid-liquid systems. Sometimes, practical considerations prevail over the decisions dictated from a transport-reaction analysis. Corrosive liquids should always be in the dispersed phase to reduce contact with the reactor walls. Hazardous liquids are usually dispensed to reduce their hold-up, i.e. their inventory inside the reactor. 

A significant advance was made in this field by Watarai and Freiser [58], who developed a high-speed automatic system for solvent extraction kinetic studies. The extraction vessel was a 200 mL Morton flask fitted with a high speed stirrer (0-20,000 rpm) and a teflon phase separator. The mass transport rates generated with this approach were considered to be sufficiently high to effectively outrun the kinetics of the chemical processes of interest. With the aid of the separator, the bulk organic phase was cleanly separated from a fine dispersion of the two phases in the flask, circulated through a spectrophotometric flow cell, and returned to the reaction vessel. 

Reactions carried in aqueous multiphase catalysis are accompanied by mass transport steps at the L/L- as well as at the G/L-interface followed by chemical reaction, presumably within the bulk of the catalyst phase. Therefore an evaluation of mass transport rates in relation to the reaction rate is an essential task in order to gain a realistic mathematic expression for the overall reaction rate. Since the volume hold-ups of the liquid phases are the same and water exhibits a higher surface tension, it is obvious that the organic and gas phases are dispersed in the aqueous phase. In terms of the film model there are laminar boundary layers on both sides of an interphase where transport of the substrates takes place due to concentration gradients by diffusion. The overall transport coefficient /cl can then be calculated based on the resistances on both sides of the interphase (Eq. 1) ... 

The overall mass transport coefficient for mass transport from the dispersed organic phase into the continuous aqueous phase was then calculated according to the Calderbank equation (Eq. 4) ... 

The moments of the solutions thus obtained are then related to the individual mass transport diffusion mechanisms, dispersion mechanisms and the capacity of the adsorbent. The equation that results from this process is the model widely referred to as the three resistance model. It is written specifically for a gas phase driving force. Haynes and Sarma included axial diffusion, hence they were solving the equivalent of Eq. (9.10) with an axial diffusion term. Their results cast in the consistent nomenclature of Ruthven first for the actual coefficient responsible for sorption kinetics as ... 

Authors efforts in this part of the work have been concentrated on developing turbulence closures for the statistical description of two-phase turbulent flows. The primary emphasis is on development of models which are more rigorous, but can be more easily employed. The main subjects of the modeling are the Reynolds stresses (in both phases), the cross-correlation between the velocities of the two phases, and the turbulent fluxes of the void fraction. Transport of an incompressible fluid (the carrier gas) laden with monosize particles (the dispersed phase) is considered. The Stokes drag relation is used for phase interactions and there is no mass transfer between the two phases. The particle-particle interactions are neglected the dispersed phase viscosity and pressure do not appear in the particle momentum equation. 

Mass transfer rates are increased in the presence of eruptions because the interfacial fluid is transported away from the interface by the jets. For mass transfer from drops with the controlling resistance in the continuous phase, the maximum increase in the transfer rate is of the order of three to four times (S8), not greatly different from the estimate of Eq. (10-4) for cellular convection. This may indicate that equilibrium is attained in thin layers adjacent to the interface during the spreading and contraction. When the dispersed-phase resistance controls, on the other hand, interfacial turbulence may increase the mass transfer rate by more than an order of magnitude above the expected value. This is almost certainly due to vigorous mixing caused by eruptions within the drop. 

The presence of fine solid particles or a finely dispersed second liquid phase in the continuous absorbent phase can have a very strong effect on the mass transfer rate between the gas and the continuous phases. The mass transport into the solid particles or liquid drops can essentially alter the concentration gradient and, consequently, the absorption rate [27-36]. The qualitative explanation of this phenomenon is that the particle absorbs oxygen in the oxygen-rich hydro-dynamic mass transfer film, after which, desorption of oxygen takes place in the oxygen-poor bulk of the liquid. 

The essential difference between the homogeneous model and the heterogeneous one is that the latter model takes into account the fact that the diffusion of the absorbed component alternately occurs through continuous- and dispersed phases in the liquid boundary layer at the gas-hquid interface. The mass transport through this heterogeneous phase is a nonUnear process, one can get explicit mathematical expression for the absorption rate only after its simpHfica-tion. 

Reactions in which one reactant is gaseous, the other is in a liquid phase, and the catalyst is dispersed in the liquid phase, constitute a special but not unusual case, for example, the hydrogenation of a liquid alkene catalysed by platinum. A batch reactor is most commonly employed for laboratory scale studies of such reactions. Mass transport from the gaseous to the liquid phase may reduce the rate of such a catalytic reaction unless the contact between the gas and the liquid is excellent (see 1.6). 

Mechanistic Multiphase Model for Reactions and Transport of Phosphorus Applied to Soils. Mansell et al. (1977a) presented a mechanistic model for describing transformations and transport of applied phosphorus during water flow through soils. Phosphorus transformations were governed by reaction kinetics, whereas the convective-dispersive theory for mass transport was used to describe P transport in soil. Six of the kinetic reactions—adsorption, desorption, mobilization, immobilization, precipitation, and dissolution—were considered to control phosphorus transformations between solution, adsorbed, immobilized (chemisorbed), and precipitated phases. This mechanistic multistep model is shown in Fig. 9.2. 

The first two terms of the right side of Eq. (9.13) account for transport. These two terms are referred to as the dispersion and mass flow terms, respectively. The third and fourth terms represent the adsorption (forward) and desorption (backward) reactions between the exchangeable and solution phases. Selim et al. (1976a) assumed the backward reaction was a... 

A device consisting of an array of frustum-shaped cells that contain a drug dispersed in a permeable matrix is shown to obey zero-order release kinetics following an initial burst phase. Geometric shapes of dissolving solids or diffusion systems and the constraints of impermeable barriers influence mass transport and can be exploited as in the constant release wedge- or hemispheric-shaped devices. 

The two-phase theory of fluidization has been extensively used to describe fluidization (e.g., see Kunii and Levenspiel, Fluidization Engineering, 2d ed., Wiley, 1990). The fluidized bed is assumed to contain a bubble and an emulsion phase. The bubble phase may be modeled by a plug flow (or dispersion) model, and the emulsion phase is assumed to be well mixed and may be modeled as a CSTR. Correlations for the size of the bubbles and the heat and mass transport from the bubbles to the emulsion phase are available in Sec. 17 of this Handbook and in textbooks on the subject. Davidson and Harrison (Fluidization, 2d ed., Academic Press, 1985), Geldart (Gas Fluidization Technology, Wiley, 1986), Kunii and Levenspiel (Fluidization Engineering, Wiley, 1969), and Zenz (Fluidization and Fluid-Particle Systems, Pemm-Corp Publications, 1989) are good reference books. 

Particularly sophisticated models deal with reactive mass transport, including both the accurate description of the convective and dispersive transport of species, as well as the modeling of interactions of species in water, with solid and gaseous phases (precipitation, dissolution, ion exchange, sorption). 

Two impedance arcs, which correspond to two relaxation times (i.e., charge transfer plus mass transfer) often occur when the cell is operated at high current densities or overpotentials. The medium-frequency feature (kinetic arc) reflects the combination of an effective charge-transfer resistance associated with the ORR and a double-layer capacitance within the catalyst layer, and the low-fiequency arc (mass transfer arc), which mainly reflects the mass-transport limitations in the gas phase within the backing and the catalyst layer. Due to its appearance at low frequencies, it is often attributed to a hindrance by finite diffusion. However, other effects, such as constant dispersion due to inhomogeneities in the electrode surface and the adsorption, can also contribute to this second arc, complicating the analysis. Normally, the lower-frequency loop can be eliminated if the fuel cell cathode is operated on pure oxygen, as stated above [18],... 

Although the hydrogenation activity of metal sulfides is lower by several orders of magnitude than that of metal catalysts, sulfides allow operations under conditions that are impractical for metals. They are generally used as highly dispersed materials on a high surface area support, such as y-alumina, in fixed bed operation. Most important is catalyst design to minimize deactivation due to the deposition of metals (V, Ni) in the feed and of coke at the mouths of the pores. Metal sulfides can also be used as finely dispersed phases in continuous slurry reactors to reduce the mass transport limitations of heavy oils. 

Other models directly couple chemical reaction with mass transport and fluid flow. The UNSATCHEM model (Suarez and Simunek, 1996) describes the chemical evolution of solutes in soils and includes kinetic expressions for a limited number of silicate phases. The model mathematically combines one- and two-dimensional chemical transport with saturated and unsaturated pore-water flow based on optimization of water retention, pressure head, and saturated conductivity. Heat transport is also considered in the model. The IDREAT and GIMRT codes (Steefel and Lasaga, 1994) and Geochemist s Workbench (Bethke, 2001) also contain coupled chemical reaction and fluid transport with input parameters including diffusion, advection, and dispersivity. These models also consider the coupled effects of chemical reaction and changes in porosity and permeability due to mass transport. 

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