Mathematical models mass Transfer Coefficient

Ideal reactors can be classified in various ways, but for our purposes the most convenient method uses the mathematical description of the reactor, as listed in Table 14.1. Each of the reactor types in Table 14.1 can be expressed in terms of integral equations, differential equations, or difference equations. Not all real reactors can fit neatly into the classification in Table 14.1, however. The accuracy and precision of the mathematical description rest not only on the character of the mixing and the heat and mass transfer coefficients in the reactor, but also on the validity and analysis of the experimental data used to model the chemical reactions involved. 

For simplicity, this section discusses only the mass transfer of one component in a liquid-liquid system with negligible miscibility of both liquids and with one transitional component. On the other hand, calculations must consider mass transfer rates of several components and more or less strong variation in the mass flows along the column, where both complicate the equation considerably [21-23]. Chemical reactions may cause further complications. Their kinetics can enhance the mass transfer coefficients and, therefore, the reaction equations have to be part of the mathematical model of the extractor [24,25]. 

In order to solve the mathematical model for the emulsion hquid membrane, the model parameters, i. e., external mass transfer coefficient (Km), effective diffu-sivity (D ff), and rate constant of the forward reaction (kj) can be estimated by well known procedures reported in the Hterature [72 - 74]. The external phase mass transfer coefficient can be calculated by the correlation of Calderback and Moo-Young [72] with reasonable accuracy. The value of the solute diffusivity (Da) required in the correlation can be calculated by the well-known Wilke-Chang correlation [73]. The value of the diffusivity of the complex involved in the procedure can also be estimated by Wilke-Chang correlation [73] and the internal phase mass transfer co-efficient (surfactant resistance) by the method developed by Gu et al. [75]. 

According to their analysis, if c is zero (practically much lower than 1), then the fluid-film diffusion controls the process rate, while if ( is infinite (practically much higher than 1), then the solid diffusion controls the process rate. Essentially, the mechanical parameter represents the ratio of the diffusion resistances (solid and fluid-film). This equation can be used irrespective of the constant pattern assumption and only if safe data exist for the solid diffusion and the fluid mass transfer coefficients. In multicomponent solutions, the use of models is extremely difficult as numerous data are required, one of them being the equilibrium isotherms, which is a time-consuming experimental work. The mathematical complexity and/or the need to know multiparameters from separate experiments in all the diffusion models makes them rather inconvenient for practical use (Juang et al, 2003). 

The description of diffusion involves three complimentary mathematical models, often dignified as laws. The most fundamental, Fick s law of diffusion, uses a diffusion coefficient. In other cases, where convection is strong, the mixing will occur following the same mathematics as Fick s law but with a dispersion coefficient replacing the diffusion coefficient. In still others cases, where there is transport across some type of interface, the mixing is described as mass transfer and correlated with a mass transfer coefficient. Mass transfer coefficients... 

It should be noted that the local mass transfer coefficient can only be obtained experimentally and is case specific. An analytical relationship for the local mass transfer rate coefficient can be obtained if a mathematical expression describing the gradient of the dissolved concentration at the NAPL-water interface is known. Unfortunately, the local mass transfer coefficient usually is not an easy parameter to determine with precision. Thus, in mathematical modeling of contaminant transport originating from NAPL pool dissolution, k(t, x,y) is often replaced by the average mass transfer coefficient, k(t), applicable to the entire pool, expressed as [41]... 

The extraction of toluene and 1,2 dichlorobenzene from shallow packed beds of porous particles was studied both experimentally and theoretically at various operating conditions. Mathematical extraction models, based on the shrinking core concept, were developed for three different particle geometries. These models contain three adjustable parameters an effective diffusivity, a volumetric fluid-to-particle mass transfer coefficient, and an equilibrium solubility or partition coefficient. K as well as Kq were first determined from initial extraction rates. Then, by fitting experimental extraction data, values of the effective diffusivity were obtained. Model predictions compare well with experimental data and the respective value of the tortuosity factor around 2.5 is in excellent agreement with related literature data. 

During the extraction an unsteady process prevails. The present paper presents an unsteady state mathematical model for a fixed bed extractor (model 1). The overall mass transfer coefficients were calculated by matching the calculated and experimental values of oil loading in CO2. The results are compared with those obtained by the model developed by Catchpole et al, 1994 (model II). Good agreement between both models results and our experimental measurements were obtained, although the model II allows the best fit over the entire extraction curve. 

In this study, we focused our attention on investigating the adsorption dynamics in column packed with activated carbon fiber. By optimizing the breakthrough curve data with a mathematical model, effective overall mass transfer coefficient was obtained. And it can be given reasonable predictions compared with the experimental data of breakthrough curve. 

A new mathematical model was developed to predict TPA behaviors of hydrocarbons in an adsorber system of honeycomb shape. It was incorporated with additional adsorption model of extended Langmuir-Freundlich equation (ELF). LDFA approximation and external mass transfer coefficient proposed by Ullah, et. al. were used. In addition, rate expression of power law model was employed. The parameters used in the power model were obtained directly from the conversion data of hydrocarbons in adsorber systems. To get numerical solutions for the proposed model, orthogonal collocation method and DVODE package were employed. 

During mastication, nonvolatile flavor molecules must move from within the food, through the saliva to the taste receptors on the tongue, and the inside of the mouth, whereas volatile flavor molecules must move from the food, through the saliva and into the gas phase, where they are carried to the aroma receptors in the nasal cavity. The two major factors that determine the rate at which these processes occur are the equilibrium partition coefficient (because this determines the initial flavor concentration gradients at the various boundaries) and the mass transfer coefficient (because this determines the speed at which the molecules move from one location to another). A variety of mathematical models have been developed to describe the release of flavor molecules from oil-in-water emulsions. 

Cybulski and Moulijn [27] proposed an experimental method for simultaneous determination of kinetic parameters and mass transfer coefficients in washcoated square channels. The model parameters are estimated by nonlinear regression, where the objective function is calculated by numerical solution of balance equations. However, the method is applicable only if the structure of the mathematical model has been identified (e.g., based on literature data) and the model parameters to be estimated are not too numerous. Otherwise the estimates might have a limited physical meaning. The method was tested for the catalytic oxidation of CO. The estimate of effective diffusivity falls into the range that is typical for the washcoat material (y-alumina) and reacting species. The Sherwood number estimated was in between those theoretically predicted for square and circular ducts, and this clearly indicates the influence of rounding the comers on the external mass transfer. 

Next, a mathematical model that allows description of the separation and concentration of the components of a metallic mixture will be detailed the principal assumptions of the model are (1) convective mass transfer dominates diffusive mass transfer in the fluid flowing inside the HFs, (2) the resistance in the membrane dominates the overall mass transport resistance, therefore the overall mass transfer coefficient was set equal to the mass transfer coefficient across the membrane, and (3) chemical reactions between ionic species are sufficiently fast to ignore the contribution of the chemical reaction rates. Thus, the reacting species are present in equilibrium concentration at the interface everywhere [31,32,58,59]. For systems working under nonsteady state, it is also necessary to describe the change in the solute concentration with time both in the modules and in the reservoir tanks. The reservoir tanks will be modeled as ideal stirred tanks. 

By incorporating the film theory into the mathematical model for the batch slurry oxidation, a mass transfer coefficient of 0.015 cm/sec was obtained by matching the model to highly catalyzed (2000 ppm Mn added) slurry oxidation data. Saturation concentration of sulfite is most important in determining mass transfer coefficient(32). A correlation is given... 

At the lower temperature (783 K open symbols in Fig. 70) a substantially different behavior is observed. The imide band (A in Fig. 69 bottom) decreases quasi-linearly with the elapsed time (see Eq. 24). The aromatic band (V in Fig. 70 top) is complex, revealing two distinct decomposition patterns. At the beginning (first half) of the normalized time a slow linear decrease is observed, followed by a fast decrease. The decrease of the imide band and the change of the aromatic band in the second part of the curve are typical for a film diffusion-controlled reaction of shrinking particles in a gas flow in the Stokes regime. To confirm this observation a new mathematical model is used to fit the curves [321]. Starting from Eq. 20, the reaction velocity ks is substituted with kg=D Rf1 [321]. D is the diffusion velocity and kg the mass transfer coefficient between fluid and particle. The differential equation is solved and the time necessary to reduce a particle from a starting radius R0 to Rt is obtained [see Eq. (22)] [321],... 

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