Mass transfer between phases concentration profiles

Equilibrium relations. Even when mass transfer is occurring equilibrium relations are important to determine concentration profiles for predicting rates of mass transfer. In Section 10.2 the equilibrium relation in a gas-liquid system and Henry s law were discussed. In Section 7.1C a discussion covered equilibrium distribution coefficients between two phases. These equilibrium relations will be used in discussion of mass transfer between phases in this section. 

Due to the consumption of reactants and the production or consumption of heat, concentration and temperature profiles can develop in the stagnant zone around and in the particle itself (Fig. 11). In the following paragraphs, criteria are derived to ensure that the effect of these gradients on the observed reaction rate is negligible [4, 27, 28]. In gas/liquid/solid slurry reactors, the mass transfer between the gas and liquid phase has to be considered, too (see Refs 9 and 29). 

All cases of practical importance in liquid chromatography deal with the separation of multicomponent feed mixtures. As shown in Chapter 2, the combination of the mass balance equations for the components of the feed, their isotherm equations, and a chromatography model that accounts for the kinetics of mass transfer between the two phases of the system permits the calculation of the individual band profiles of these compounds. To address this problem, we need first to understand, measure, and model the equilibrium isotherms of multicomponent mixtures. These equilibria are more complex than single-component ones, due to the competition between the different components for interaction with the stationary phase, a phenomenon that is imderstood but not yet predictable. We observe that the adsorption isotherms of the different compounds that are simultaneously present in a solution are almost always neither linear nor independent. In a finite-concentration solution, the amount of a component adsorbed at equilib-... 

Another two-dimensional, discrete element model was applied by Cartaxo and Rocha [43]. In this work, only the dynamic phenomena were investigated, that is, heat and mass transfer between the phases were not considered. Thns, the inflnence of the momentum coupling between the discrete particles and the conveying air on the air radial velocity and the mass concentration profiles was presented. An object-oriented numerical model was developed to simulate the conveying of large spherical particles (3 mm) through 9.14 m vertical tube with 7.62 cm bore size. 

By suitably choosing the solubility, the concentration of the reactant and the rate of reaction, either the mass transfer coefficients, or the interfacial area or both groups of parameters can be deduced from the overall rate of absorption (lA). Generally but not always, a steady flow of each phase through the reactor is assumed. Indeed the competition between the phsyical and chemical kinetics at the level of mass transfer between gas and liquid (the mass transfer reaction regime where the reaction belongs) may allow for the choice of the type of gas-liquid contactor (I). This is clearly shown in Fig. I that represents schematically the concentration profiles for A and B on each side of the interface. 

Actual concentration profiles (Fig. 1.28) in the very near vicinity of a mass transfer interface are complex, since they result from an interaction between the mass transfer process and the local hydrodynamic conditions, which change gradually from stagnant flow, close to the interface, to more turbulent flow within the bulk phases. 

The nature of mass transport in MEMED has been confirmed with both ampero-metric and potentiometric studies of bromine transfer from an aqueous phase to DCE [79]. Figure 17 shows typical amperometric data for this case, in which the DCE phase acts as a sink for Br2, and a depleted region of Br2 is measured adjacent to the droplet in the aqueous phase. Video images are also provided, which correspond to particular times during the amperometric transient at position (3) the edge of the developing concentration boundary layer, around the drop, reaches the electrode the concentration profile is then mapped out between points (3) and (4). The measured current, i, can be related to the local concentration, c, via... 

Ctl is the mass transfer term and arises because of the finite time taken for solute molecules to move between the two phases. Consequently, a true equilibrium situation is never established as the solute moves through the system, and spreading of the concentration profiles results. The effect is minimal for small particle size and thin coatings of stationary phase but increases with flow rate and length of column or surface. 

For a simple hydrogenation reaction between H2 transferred from the gas phase with the substrate S present in the bulk liquid phase (S+H2->P), considering no mass transfer resistance on the gas side and a gradientless concentration of the molecular catalyst in the liquid phase, various concentration profiles in the liquid boundary layer, or film, exist (Fig. 45.2). 

In this section, we have examined how the coupling between mass transfer and the chemical reaction defines the concentration profile of the limiting reagent (i.e., hydrogen), and how the mass or molar flow between the gas and the liquid phase can be computed. In the next section, the estimation of the overall rate of reaction (i.e., the reactor productivity) will be reviewed for different gas-liquid reactors. 

The equations describing the concentration and temperature within the catalyst particles and the reactor are usually non-linear coupled ordinary differential equations and have to be solved numerically. However, it is unusual for experimental data to be of sufficient precision and extent to justify the application of such sophisticated reactor models. Uncertainties in the knowledge of effective thermal conductivities and heat transfer between gas and solid make the calculation of temperature distribution in the catalyst bed susceptible to inaccuracies, particularly in view of the pronounced effect of temperature on reaction rate. A useful approach to the preliminary design of a non-isothermal fixed bed catalytic reactor is to assume that all the resistance to heat transfer is in a thin layer of gas near the tube wall. This is a fair approximation because radial temperature profiles in packed beds are parabolic with most of the resistance to heat transfer near the tube wall. With this assumption, a one-dimensional model, which becomes quite accurate for small diameter tubes, is satisfactory for the preliminary design of reactors. Provided the ratio of the catlayst particle radius to tube length is small, dispersion of mass in the longitudinal direction may also be neglected. Finally, if heat transfer between solid cmd gas phases is accounted for implicitly by the catalyst effectiveness factor, the mass and heat conservation equations for the reactor reduce to [eqn. (62)]... 

Figure 12-15 Sketch of concentration profiles between a spherical bubble and a solid spherical catalyst particle in a continuous liquid phase (upper) in a gas-liquid sluny reactor or between a bubble and a planar solid wall (lower) in a catalytic w bubble reactor, It is assmned that a reactant A must migrate from the bubble, tirough the drop, md to tiie solid catdyst smface to react. Concentration variations may occur because of mass transfer limitations around both bubble and solid phases.

FIGURE 3.1 Concentration profiles in a passive sampling device. The driving force of accumulation is the difference in chemical potentials of the analyte between the bulk water and the sorption phase. The mass transfer of an analyte is governed by the overall resistance along the whole diffusional path, including contributions from the individual barriers (e.g., aqueous boundary layer, biofilm layer, and membrane). 

In near-isothermal systems, it is assumed that the heat transfer between mobile and stationary phases is slow. This causes an additional band broadening contribution to appear [53]. Such a contribution can be especially important on the front of a sharp concentration profile. On the other hand, the heat transfer between the chromatographic column and the outside is fast enough to prevent the formation of a temperature front and of an associated secondary mass transfer zone. 

For single-component systems, the theoretical solutions obtained are easy to compare to experimental profiles. They differ only by the smoothing effect due to axial dispersion and to the finite kinetics of mass transfers in actual columns. In many cases, because of the qualities of the stationary phases currently available, these effects appear to be secondary compared to the major role of thermodynamics in controlling the band proffles in overloaded elution. Admittedly, the influence of the finite coliunn efficiency on the band profiles prevents a successful quantitative comparison between theoretical and experimental band profiles. However, these profiles are similar enough at high concentrations and the solutions of the ideal model indicate which are the trends to be expected. 

The analysis of equilibrium-stage operations is normally performed on the basis of counter-current flow between two phases. Because most separation processes, whether described in terms of equilibrium or mass transfer rates, operate in this flow scheme, it is useM to compare countercurrent to cocurrent flow. Figure 2.2 illustrates cocurrent and counter-current operation. Assuming mass transfer across a barrier between the two fluid phases, generic concentration profiles can be drawn for each case (Figure 2.3). 

First, the overall mass transfer coefRcient k a of the microreactor was estimated to be 3-8 s [43]. For intensified gas liquid contactors, kj a can reach 3 s while bubble columns and agitated tanks do not exceed 0.2 s Reducing the flow rate and, accordingly, the liquid film thickness is a means of further increasing kj a, which is limited, however, by liquid dry-out at very thin films. Despite such large mass transfer coefficients, gas-liquid microreactors such as the falling film device may still operate between mass transfer and kinetic control regimes, as fundamental simulation studies on the carbon dioxide absorption have demonstrated [44]. Distinct concentration profiles in the liquid, and even gas, phase are predicted. 

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