Mass transfer model, solution

The solution of Eqs. (9) is straightforward if the six parameters are known and the boundary conditions are specified. Two boundary conditions are necessary for each equation. Pavlica and Olson (PI) have discussed the applicability of the Wehner-Wilhelm boundary conditions (W3) to two-phase mass-transfer model equations, and have described a numerical method for solving these equations. In many cases this is not necessary, for the second-order differentials can be neglected. Methods for evaluating the dimensionless groups in Eqs. (9) are given in Section II,B,1. 

Dreybrodt W, Buhmann D. A mass transfer model for dissolution and precipitation of calcite from solutions in turbulent motion. Chem Geol 1991 107-122. 

Based on the cephalosporins concentration profiles in the BLM system (Fig. 2), the following mass transfer models describe the change in solute concentration in the respective phases ... 

In order to estimate resolution among peaks eluted from a chromatography column, those factors that affect N must first be elucidated. By definition, a low value of Hs will result in a large number of theoretical plates for a given column length. As discussed in Chapter 11, Equation 11.20 obtained by the rate model shows the effects of axial mixing of the mobile phase fluid and mass transfer of solutes on Hs. 

We first give a rather general mass-transfer model, which is useful for most processes of porous-solid extraction with dense gases. Two cases are possible [43] for a single particle loaded with solute. In (a), the solute is adsorbed over the internal surface of the particle, and is desorbed from the sites and diffuses out to the external surface, (b) The solute fills in the pore-cavities completely, and is dissolved from an inner core that moves progressively to the centre of the particle. 

Mass transfer models have been used to describe the leaching of soluble substances from porous particles into solution. Such models include a concentration difference that drives the concentration of soluble components in the solids and solution to equilibrate (14). Application of one such model (14) to track release from a solid into solution results in the following equations when applied to xylan conversion in a batch system ... 

This preliminary study suggests that mass transfer models could describe many features of xylan hydrolysis with accuracy similar to that of conventional first-order reaction-only models that have been long used to describe such systems. For example, a simple leaching model can describe release of xylan into solution as the product of a concentration gradient times a mass transfer coefficient. This model predicts that flowthrough operation could improve xylan release compared to a batch system by reducing the concentration in solution and thereby increasing the concen-... 

The enhancement factors are either obtained by fitting experimental results or are derived theoretically on the grounds of simplified model assumptions. They depend on reaction character (reversible or irreversible) and order, as well as on the assumptions of the particular mass transfer model chosen [19, 26]. For very simple cases, analytical solutions are obtained, for example, for a reaction of the first or pseudo-first order or for an instantaneous reaction of the first and second order. Frequently, the enhancement factors are expressed via Hatta-numbers [26, 28]. They can be used in combination with the HTU/NTU-method or with a more advanced mass transfer description method. However, it is generally not possible to derive the enhancement factors properly from binary experiments, and a theoretical description of reversible, parallel or consecutive reactions is based on rough simplifications. Thus, for many reactive absorption processes, this approach appears questionable. 

In this investigation we carried out experiments with simultaneous absorption of H2S and CO2 into aqueous 2.0 M diisopropanol-amine (DIPA) solutions at 25 °C. The results are evaluated by means of our mathematical mass transfer model both in penetration and film theory form. The latter version has been derived from the penetration theory mass transfer model [5],... 

If the amine depletion in the penetration zone in a simultaneous absorption situation is negligible, the mass transfer fluxes are independent of each other and the respective enhancement factors may be obtained easily from analytical solutions of single gas mass transfer models. 

The solute is disseminated in a solid matrix in the most of the supercritical extractions of natural products. If the interactions between solute and solid matrix are not important, the mass transfer models can be developed from the equations of microscope balances to a volume element of the extractor. If the mass transfer resistance is in its solid phases, the mathematical models must consider the solute transport within the solid particles or the surface phenomena. 

Bencala K. E. (1983) Simulation of solute transport in a mountain pool-and-riffle stream with a kinetic mass transfer model for sorption. Water Resour. Res. 19, 732—738. 

Mass transfer model that accounted for the mass transfer both inside and outside the emulsion globules, the reaction between the diffusing component and the internal reagent in the globules jointly. A perturbation solution to the resulting nonlinear equations contained the parameters Bi and Da. 

A field-flow fractionation (FFF) channel is normally ribbonlike. The ratio of its breadth b to width w is usually larger than 40. This was the reason to consider the 2D models adequate for the description of hydrodynamic and mass-transfer processes in FFF channels. The longitudinal flow was approximated by the equation for the flow between infinite parallel plates, and the influence of the side walls on mass-transfer of solute was neglected in the most of FFF models, starting with standard theory of Giddings and more complicated models based on the generalized dispersion theory [1]. The authors of Ref. 1 were probably the first to assume that the difference in the experimental peak widths and predictions of the theory may be due to the influence of the side walls. 

The development of mass transfer models require knowledge of three properties the diffusion coefficient of the solute, the viscosity of the SCF, and the density of the SCF phase. These properties can be used to correlate mass transfer coefficients. At 35 C and pressures lower than the critical pressure (72.83 atm for CO2) we use the diffusivity interpolated from literature diffusivity data (2,3). However, a linear relationship between log Dv and p at constant temperature has been presented by several researchers U>5) who correlated diffusivities in supercritical fluids. For pressures higher than the critical, we determined an analytical relationship using the diffusivity data obtained for the C02 naphthalene system by lomtev and Tsekhanskaya (6), at 35 C. 

Mass-Transfer Models Because the mass-transfer coefficient and interfacial area for mass transfer of solute are complex functions of fluid properties and the operational and geometric variables of a stirred-tank extractor or mixer, the approach to design normally involves scale-up of miniplant data. The mass-transfer coefficient and interfacial area are influenced by numerous factors that are difficult to precisely quantify. These include drop coalescence and breakage rates as well as complex flow patterns that exist within the vessel (a function of impeller type, vessel geometry, and power input). Nevertheless, it is instructive to review available mass-transfer coefficient and interfacial area models for the insights they can offer. 

Pulse analysis is a means to couple experimental measurements with a mass transfer model of the system to evaluate various parameters in the model. The experimental measurements are straightforward. A pulse, typically a square wave, of a solute enters the inlet of the system. The concentration protile of the solute at the system outlet is measured. This is shown graphically in Figure C.l. The mass transfer model is solved for the solute concentration using Laplace transforms. The solute concentration and various derivatives in the Laplace domain will be shown to be related to various moments of the concentration vs time data. 

Although the latter models assumed that the slowest step involves the incorporation of species at the periphery of expanding nucleus, other nucleation models proposed that mass transfer in solution controlled the new phase growth [30-32]. 

Limestone (CaCC>3) dissolution is an important phenomenon in stack gas desulfurization processes using limestone slurry to absorb SC>2 and produce CaSC>3/CaS04 waste solids (1). The rate of dissolution directly determines the need for excess limestone and interacts strongly with SC>2 removal and scale-free operation in the absorber. There is a need to know the dependence of dissolution rates on both solution composition and the type and grind of limestone. This paper presents a mass transfer model and... 

Modeling of experimental data requires integration of the dissolution rate over a particle size distribution. This is simplified by assuming that the dissolution rate per particle is directly proportional to the particle diameter. Because of the CO2 reaction, the dissolution rate of a particle in gmol/sec is not exactly proportional to the particle diameter. Therefore, the effect of the CO2 reaction is assumed to be the same for all particles as for a 10 pm (effective diameter) particle. The general mass transfer model is used to calculate rates for 10 Jim particles as a function of solution composition. 

The factor 1.88 is an adjustable constant which accounts for deviation of experimental results from the unadjusted model. Values of 8 and k are obtained from the general mass transfer model as a function of solution composition and temperature at a particle diameter of 10 Um. Use of k gives ... 

The enhancing and inhibiting effect of sulfite can be modeled as a shift in equilibrium at the calcite surface. Thus, the equilibrium [CaCOg0] is reduced by increased [CaS03°], much like a solid solution at the CaCOj surface. Using the general mass transfer model, the solution composition at the CaCO surface was calculated from the experimental rate data. 

Figure 8. Effect of sulfite with N2 sparging at 25 and 55°C. Curves calculated from mass transfer model. Including CaCOs°/CaSO,° solid solution (Figure 10). pH , 4.5 O and A, 5.0 0,5.5 , 5.75 and A, 6.0 and A, S5°C.

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