Multicomponent transfer coefficient

Rate equations 28 and 30 combine the advantages of concentration-independent mass transfer coefficients, even in situations of multicomponent diffusion, and a familiar mathematical form involving concentration driving forces. The main inconvenience is the use of an effective diffusivity which may itself depend somewhat on the mixture composition and in certain cases even on the diffusion rates. This advantage can be eliminated by working with a different form of the MaxweU-Stefan equation (30—32). One thus obtains a set of rate equations of an unconventional form having concentration-independent mass transfer coefficients that are defined for each binary pair directiy based on the MaxweU-Stefan diffusivities. 

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 rate parameters of importance in the multicomponent rate model are the mass transfer coefficients and surface diffusion coefficients for each solute species. For accurate description of the multicomponent rate kinetics, it is necessary that accurate values are used for these parameters. It was shown by Mathews and Weber (14), that a deviation of 20% in mass transfer coefficients can have significant effects on the predicted adsorption rate profiles. Several mass transfer correlation studies were examined for estimating the mass transfer coefficients (15, jL6,17,18,19). The correlation of Calderbank and Moo-Young (16) based on Kolmogaroff s theory of local isotropic turbulence has a standard deviation of 66%. The slip velocity method of Harriott (17) provides correlation with an average deviation of 39%. Brian and Hales (15) could not obtain super-imposable curves from heat and mass transfer studies, and the mass transfer data was not in agreement with that of Harriott for high Schmidt number values. 

In order to calculate the multicomponent diffusion matrices [D], the binary diffusivities in both phases should be known. The him thickness representing an important model parameter is estimated via the mass transfer coefficients (57,83). The binary diffusivities and mass transfer coefficients were calculated from the correlations summarized in Table 3. 

Experimental studies were carried out to derive correlations for mass transfer coefficients, reaction kinetics, liquid holdup, and pressure drop for the packing MULTIPAK (35). Suitable correlations for ROMBOPAK 6M are taken from Refs. 90 and 196. The nonideal thermodynamic behavior of the investigated multicomponent system was described by the NRTL model for activity coefficients concerning nonidealities caused by the dimerisation (see Ref. 72). 

In Eq. (31c), Nxn and N n are the mass transfer rates. These are calculated from multicomponent mass transfer equations. The equations used take into account the mass transfer coefficients and interfacial areas generated in the specific contactor, reaction rates, heat effects, and any interactions among the above processes. 

V" kg Gas-phase mass transfer coefficient for multicomponent systems, same units as Icg... 

The multicomponent diffusion matrices [D] in both phases are determined via the binary diffusivities. The latter can be estimated using different correlations summarized in Tab. 9.3. The film thicknesses which represent important model parameters (cf. Section 9.4.4) are estimated via the mass transfer coefficients [16, 23]. 

First, we measured thermodynamic and mass transfer data of the multicomponent system olive 0U/CO2 (3,4). The phase equilibria was modulated by correlating the partition coefficients (Kj = y /x ) of each component present in the mixture as a function of the mole fraction of the FFA fraction in the liquid phase (3). Mass transfer studies were performed in a lab-scale countercurrent packed column. The experimental measured mass transfer coefficients were... 

The selection of adsorbents is critical for determining the overall separation performance of the above-described PSA processes for hydrogen purification. The separation of the impurities from hydrogen by the adsorbents used in these processes is generally based on their thermodynamic selectivities of adsorption over H2. Thus, the multicomponent adsorption equilibrium capacities and selectivities, the multi-component isosteric heats of adsorption, and the multicomponent equilibrium-controlled desorption characteristics of the feed gas impurities under the conditions of operation of the ad(de)sorption steps of the PSA processes are the key properties for the selection of the adsorbents. The adsorbents are generally chosen to have fast kinetics of adsorption. Nonetheless, the impact of improved mass transfer coefficients for adsorption cannot be ignored, especially for rapid PSA (RPSA) cycles. 

Table 5 shows examples of LDF mass transfer coefficients for adsorption of several binary gas mixtures on BPL activated carbon particles (6-16 mesh) at 23-30°C. The data show that the mass transfer coefficients are relatively large for these systems. There is a scarcity of multicomponent adsorption equilibria, kinetics, and heat data in the published literature. This often restricts extensive testing of theoretical models for prediction of multicomponent behavior. 

DA TA The matrix of multicomponent volumetric mass transfer coefficients [ AT ] is given in Example 5.6.1. as... 

In Chapter 7 we define mass transfer coefficients for binary and multicomponent systems. In subsequent chapters we develop mass transfer models to determine these coefficients. Many different models have been proposed over the years. The oldest and simplest model is the film model this is the most useful model for describing multicomponent mass transfer (Chapter 8). Empirical methods are also considered. Following our discussions of film theory, we describe the so-called surface renewal or penetration models of mass transfer (Chapter 9) and go on to develop turbulent eddy diffusivity based models (Chapter 10). Simultaneous mass and energy transport is considered in Chapter 11. 

Our knowledge of multicomponent mass transfer coefficients is improving, but this is a slow process. I still occasionally have to pray that my estimate of some coefficient will not be off by more than one order of magnitude. 

The development for multicomponent mixtures is best carried out by using n — l dimensional matrix notation. We, therefore, define a matrix of finite flux mass transfer coefficients [fc ] by... 

In Eqs. 7.1.9 we define n — 1 X n — 1 elements of the mass transfer coefficients with the help of n — 1 linear equations. It follows that the elements k - are not unique that is, another set of these coefficients can also lead to the same value of the fluxes N. Put another way, making mass transfer measurements in a multicomponent system for the fluxes and Ax does not uniquely determine the values of the mass transfer coefficients. A large... 

In many cases, a priori estimates of the film thickness f cannot be made, and we resort to empirical methods of estimating the mass transfer coefficients. Most published experimental works have concentrated on two component systems and there are no correlations for the multicomponent [/ ]. The need to estimate multicomponent mass transfer coefficients is very real, however. The question is How can we estimate multicomponent mass transfer coefficients when all we have to go on are binary correlations In this section we look at the various methods that have been proposed to answer this question. 

In their original development of the linearized theory Toor (1964) and Stewart and Prober (1964) proposed that correlations of the type given by Eqs. 8.8.5 and 8.8.7 could be generalized by replacing the Fick diffusivity D by the charactersitic diffusion coefficients of the multicomponent system that is, by the eigenvalues of the Fick matrix [ >]. The mass transfer coefficient calculated from such a substitution would be a characteristic mass transfer coefficient an eigenvalue of [/c]. For example, the Gilliland-Sherwood correlation (Eq. 8.8.5) would be modified as follows ... 

This approach is, in fact, equivalent to replacing the binary diffusivity D by the matrix of multicomponent diffusion coefficients [D] and the binary mass transfer coefficient with the... 

Estimation of Mass Transfer Coefficients for Nonideal Multicomponent Systems... 

Toor (1964) and Stewart and Prober (1964) did not use the method presented above they used the method described in Chapter 5. For the multicomponent penetration model, the following expression for the matrix of mass transfer coefficients is obtained (cf. Section 8.4.2) ... 

To calculate [A ], which is the value needed to compute mass transfer fluxes, we may avoid computing [Sh]. Equations 8.4.23 or 8.4.30 and 8.4.31 may be used directly as written to compute the multicomponent mass transfer coefficients with the eigenvalues of [A ] computed from the appropriate expression in Section 9.2 as described above. 

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