Mass transfer analysis coefficients

In the frontal analysis experiment described in Section 5.3.2, the transport model of chromatography was used to fit the experimental data [40]. Neglecting axial and eddy diffusion, band broadening was accounted for by one single mass transfer rate coefficient. The mass transfer rate coefficients estimated were small and strongly dependent on the temperature and solute concentration, particularly the rate coefficients corresponding to the imprinted L-enantiomer (Fig. 5.12). Above a concentration of ca. 0.1 g/L the mass transfer rate coefficients of the two enantiomers are similar. 

The presence of temperature gradients in a multicomponent system introduces an additional complication in the analysis of the mass transfer process such gradients influence the values of physical, thermodynamic, and transport properties, such as the diffusion coefficients. These property variations may be taken care of by introducing temperature dependent property functions or by using average values of the properties (as is done here). The consequence of this simplification is that the basic mass transfer analysis remains essentially unchanged from those in Chapters 8-10 and we need only consider the effect of mass transfer on the heat transfer process. 

The mass transfer film coefficients (kx and l y) are more directly related to physical properties, such as diffusivities and hydrodynamic conditions, than the overall mass transfer coefficients (Kx and Ky and are, therefore, easier to predict and correlate. On the other hand, since the interfacial compositions are difficult to measure or predict, it is more convenient to use overall mass transfer coefficients for process design and analysis. The relationship between the two sets of mass transfer coef-hcients may be derived by equating different groups in Equations 15.11 and 15.12 as follows ... 

In packed columns, it is conceptually incorrect to use the staged model even though it works if the correct height equivalent to a theoretical plate (HETP) is used. In this chapter we will develop a physically more realistic model for packed columns that is based on mass transfer between the phases. After developing the model for distillation, we will discuss mass transfer correlations that allow us to predict the required coefficients for common packings. Next, we will repeat the analysis for both dilute and concentrated absorbers and strippers and analyze cocurrent absorbers. A simple model for mass transfer on a stage will be developed for distillation, and the estimation of stage efficiency will be considered. After a mass transfer analysis of mixer-setder extractors. Section 16.8 and the appendix to Chapter 16 will develop the rate model for distillation. 

Thus adsorption asks a different kind of question than the questions asked in absorption, distillation, or extraction. In absorption or differential distillation, the basic question is how tall a tower is needed. This question is answered with a mass transfer analysis, including an operating line and an equilibrium line. The mass transfer analysis includes overall and individual coefficients summarized by dimensionless correlations. In staged distillation or extraction, the basic question is how many stages are needed. This question is resolved largely with operating and equilibrium lines, with the mass transfer aspects conveniently compressed into an efficiency. In absorption, distillation, and extraction, the analysis is sufficiently reliable to answer the questions without experiment. 

The main conclusion to be drawn from these studies is that for most practical purposes the linear rate model provides an adequate approximation and the use of the more cumbersome and computationally time consuming diffusing models is generally not necessary. The Glueckauf approximation provides the required estimate of the effective mass transfer coefficient for a diffusion controlled system. More detailed analysis shows that when more than one mass transfer resistance is significant the overall rate coefficient may be estimated simply from the sum of the resistances (7) ... 

According to this analysis one can see that for gas-absorption problems, which often exhibit unidirectional diffusion, the most appropriate driving-force expression is of the form y — y tyBM,. ud the most appropriate mass-transfer coefficient is therefore kc- This concept is to he found in all the key equations for the design of mass-transfer equipment. 

Gal-Or and Hoelscher (G5) have recently developed a fast and simple transient-response method for the measurement of concentration and volumetric mass-transfer coefficients in gas-liquid dispersions. The method involves the use of a transient response to a step change in the composition of the feed gas. The resulting change in the composition of the liquid phase of the dispersion is measured by means of a Clark electrode, which permits the rapid and accurate analysis of oxygen or carbon dioxide concentrations in a gas, in blood, or in any liquid mixture. 

Pavlushenko et al. (P4) in their dimensional analysis considered Ks, the volumetric mass transfer coefficient, to be a function of pc, pc, L, Dr, N, Vs, and g. They determined the following relationship for the dimensionless groupings ... 

In their analysis, however, they neglected the surface tension and the diffusivity. As has already been pointed out, the volumetric mass-transfer coefficient is a function of the interfacial area, which will be strongly affected by the surface tension. The mass-transfer coefficient per unit area will be a function of the diffusivity. The omission of these two important factors, surface tension and diffusivity, even though they were held constant in Pavlu-shenko s work, can result in changes in the values of the exponents in Eq. (48). For example, the omission of the surface tension would eliminate the Weber number, and the omission of the diffusivity eliminates the Schmidt number. Since these numbers include variables that already appear in Eq. (48), the groups in this equation that also contain these same variables could end up with different values for the exponents. 

Most studies on heat- and mass-transfer to or from bubbles in continuous media have primarily been limited to the transfer mechanism for a single moving bubble. Transfer to or from swarms of bubbles moving in an arbitrary fluid field is complex and has only been analyzed theoretically for certain simple cases. To achieve a useful analysis, the assumption is commonly made that the bubbles are of uniform size. This permits calculation of the total interfacial area of the dispersion, the contact time of the bubble, and the transfer coefficient based on the average size. However, it is well known that the bubble-size distribution is not uniform, and the assumption of uniformity may lead to error. Of particular importance is the effect of the coalescence and breakup of bubbles and the effect of these phenomena on the bubble-size distribution. In addition, the interaction between adjacent bubbles in the dispersion should be taken into account in the estimation of the transfer rates... 

GP 9] [R 16[ The extent of external transport limits was made in an approximate manner as for the internal transport limits (see above), as literature data on heat and mass transfer coefficients at low Peclet numbers are lacking [78]. Using a Pick s law analysis, negligible concentration differences from the bulk to the catalyst sur-... 

In Temkin s analysis, mass transfer is assumed to be absent in the two-component system. He found that an attenuation coefficient, A, is function of the relaxation times td and t, and the mass fration of suspension, x . 

Depending on the driving force we choose to employ in our analysis, there are several definitions of mass transfer coefficients that may be considered appropriate for use. If we consider an arbitrary interface between a fluid and the external surface of a catalyst particle, we might choose to define a mass transfer coefficient based on a concentration driving force (kc) as... 

In a typical pulse experiment, a pulse of known size, shape and composition is introduced to a reactor, preferably one with a simple flow pattern, either plug flow or well mixed. The response to the perturbation is then measured behind the reactor. A thermal conductivity detector can be used to compare the shape of the peaks before and after the reactor. This is usually done in the case of non-reacting systems, and moment analysis of the response curve can give information on diffusivities, mass transfer coefficients and adsorption constants. The typical pulse experiment in a reacting system traditionally uses GC analysis by leading the effluent from the reactor directly into a gas chromatographic column. This method yields conversions and selectivities for the total pulse, the time coordinate is lost. 

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. 

One difficulty in making measurements of transfer coefficients is that equilibrium is rapidly attained between particles and fluidising medium. This has in some cases been obviated by the use of very shallow beds. In addition, in measurements of mass transfer, the methods of analysis have been inaccurate, and the particles used have frequently been of such a nature that it has not been possible to obtain fluidisation of good quality. 

The low molecular diffusion coefficients of proteins and other biopolymers reduces the efficiency of mass transfer and compromises efficiency as flow rate is increased. Therefore, high-performance SEC columns are usually operated at modest flow rates, e.g., 1 ml/min or less. However, operation at very low flow rates is undesirable due to excessive analysis times, loss of efficiency due to axial analyte diffusion, and the risk of poor recovery due to analyte adsorption. 

Equations 6 and 7 are the well known Dankwerts boundary conditions (7). In Equation 5, Kj is the mass transfer coefficient around the gel, cuid cjn (t) the inlet solute concentration which is a function of analysis time. 

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