Mass transfer models Higbie penetration

The addition of various surfactants and micelle forming agents on the biphasic hydroformylation of olefins was also considered as a tool for enhancement of the reaction rate. The relation between the extent of emulsification of the reaction mixture and the performance of hydroformylation reaction was also investigated. Mass transfer effects in biphasic hydroformylation of 1-octene in the presence of cetyltri-methylammoniumbromide (CTAB), was studied by Lekhal etal. [33], A mass-transfer model based on the Higbie s penetration theory was proposed to predict the rate of hydroformylation in a heterogeneous gas-liquid-liquid system under... 

Given that, from the penetration theory for mass transfer across an interface, the instantaneous rale ol mass transfer is inversely proportional to the square root of the time of exposure, obtain a relationship between exposure lime in the Higbie mode and surface renewal rate in the Danckwerts model which will give the same average mass transfer rate. The age distribution function and average mass transfer rate from the Danckwerts theory must be deri ved from first principles. 

In calculating Ihe mass transfer rate from the penetration theory, two models for the age distribution of the surface elements are commonly used — those due to Higbie and to Danckwerts, Explain the difference between the two models and give examples of situations in which each of them would be appropriate. 

Penetration theory (Higbie, 1935)assumes that turbulent eddies travel from the bulk of the phase to the interface where they remain for a constant exposure time te. The solute is assumed to penetrate into a given eddy during its stay at the interface by a process of unsteady-state molecular diffusion. This model predicts that the mass-transfer coefficient is directly proportional to the square root of molecular diffusivity... 

The mass-transfer coefficient in each film is expected to depend upon molecular diffusivity, and this behavior often is represented by a power-law function k . For two-film theory, n = 1 as discussed above [(Eq. (15-62)]. Subsequent theories introduced by Higbie [Trans. AIChE, 31, p. 365 (1935)] and by Dankwerts [Ind. Eng. Chem., 43, pp. 1460-1467 (1951)] allow for surface renewal or penetration of the stagnant film. These theories indicate a 0.5 power-law relationship. Numerous models have been developed since then where 0.5 < n < 1.0 the results depend upon such things as whether the dispersed drop is treated as a rigid sphere, as a sphere with internal circulation, or as oscillating drops. These theories are discussed by Skelland [ Tnterphase Mass Transfer, Chap. 2 in Science and Practice of Liquid-Liquid Extraction, vol. 1, Thornton, ed. (Oxford, 1992)]. 

In Chapter 7 we discussed the basics of the theory concerned with the influence of diffusion on gas-liquid reactions via the Hatta theory for flrst-order irreversible reactions, the case for rapid second-order reactions, and the generalization of the second-order theory by Van Krevelen and Hofitjzer. Those results were presented in terms of classical two-film theory, employing an enhancement factor to account for reaction effects on diffusion via a simple multiple of the mass-transfer coefficient in the absence of reaction. By and large this approach will be continued here however, alternative and more descriptive mass transfer theories such as the penetration model of Higbie and the surface-renewal theory of Danckwerts merit some attention as was done in Chapter 7. 

Lekhal et al. [6] proposed a pseudo-homogeneous gas-liquid-liquid model based on the Higbie penetration theory to account for simultaneous absorption of two gases into the liquid phases. Because of the assumption of rapid liquid-liquid mass transfer of reactants leading to the equilibrium between two liquid phases, the model was simplified greatly and the detail of phase dispersion and distribution and multiphase flow was avoided. Reasonable success was achieved and the results of analysis suggested that the only limitation to the conversion of hydroformylation of 1-octene was the gas-liquid mass transfer of CO and H2. 

The Higbie penetration model for mass transfer compensates for transient behavior. It assumes that mass transfer occurs during brief phase contacts that do not allow enough time for steady-state conditions. In other words, the phases collide but do not have a definitive and continuous interface with respect to time. The mass transfer is prompted by turbulence that refreshes the interface, and the refresh rate is the limiting step in mass transfer. Eddies approach the surface at which point mass transfer by molecular diffiision is initiated and is described by Azbel (1981) ... 

However, the absence of any parameter related to the channel size limits the application of the model to different two-phase systems. Van Baten and Krishna (2004) and Irandoust and Andersson (1989) included in their models the contributions of both bubble caps and film (Eq. 2.2.34). Van Baten and Krishna (2004) evaluated the contribution of the caps according to the Higbie penetration theory (Eq. 2.2.35), whilst the transfer through the film was obtained based on mass transfer in a falling film in laminar flow (Eqs. 2.2.36 and 2.2.37). 

A relatively simple model to describe the gas-liquid mass transfer in circular channels with slug flow pattern was proposed by van Eaten and Krishna [47]. For their fundamental model the authors considered an idealized geometry of the Taylor bubbles as shown in Figure 7.12. The bubbles consist of two hemispherical caps and a cylindrical body. The Higbie penetration model was applied to describe the mass transfer process of a compound from the gas phase to the liquid (Equation 7.8). For a rising bubble, the liquid will flow along the bubble surface of the cap. The average distance... 

The gas-phase mass transfer is described by the stagnant film model whilst for the liquid phase Higbie s penetration model was used. The process of diffusion and simultaneous reaction in the liquid-phase penetration zone is given by the following balances [4] ... 

Penetration theory can also be applied to turbulent conditions by assuming the turbulence spectrum to consist of large eddies, capable of surface renewal, and small eddies responsible for the presence of eddy diffusivity The small eddies are damped when an element of liquid reaches the interface so that, during its residence time there, mass transfer occurs in accordance with the assumptions of the penetration theory If all the eddies stay at the interface for the same interval of time we talk about penetration theory with regular surface renewal or the Higbie model If there is random distribution of residence times with an age-independent fractional rate of surface renewal, s, the term penetration theory with random surface renewal, or the Danckwerts nK)del, is employed In the case of the Higbie model, the mass transfer coefficient is the same as that given by eqn (18). For the Danckwerts model it takes the form... 

Seminal work on segmented flow mass transfer was conducted by Higbie [38], who used experiments at < tp to prove the now well-established but then new penetration theory, i.e. that More recently, Irandoust et al. [39] modeled gas... 

More recently, Kawase and co-workers have presented semi-analytical framework for the estimation of the volumetric mass transfer coefflcient in bubble column reactors with and without an internal draft tube [51,62,67]. They have essentially combined the Higbie s penetration model with the isotropic turbulence to deduce the following expression for gas-liquid mass transfer with power law liquids ... 

Other models such as the penetration model (surface renewal theory) developed by Higbie and Danckwerts (Westerterp, van Swaaij, and Beenackers, 1998) consider the mass transfer process to be essentially non-stationary, and the surface is assumed to consist of elements of different age at the surface, returning into the bulk phase while new elements originating from the bulk phase take their place. Results of calculations by the two-film theory and the surface renewal theory are similar. Thus, only the two-film theory, which is easier to understand and which is therefore used most, is considered here. 

One of the traditional models for predicting the binary mass transfer coefficient is based on the penetration theory by Higbie [37]. It is used as our starting point. 

For mass transfer process, Zhang derived a model for the overall liquid-phase mass transfer coefficient XL,theo based on the Higbie penetration theory [28] ... 

Furthermore, van Baten and Krishna [55] formulated an equation for the description of kia, which is based on the fundamental approach. The authors combined the Higbie penetration model to describe mass transfer from the two hemispherical caps and the Pigford model to describe the mass transfer to the liquid film [56]. Furthermore, they used CFD simulations for studying the separate influence of different parameters on the volumetric mass transfer coefficient and compared these results with the predictions of Equation 12.29. 

The penetration theory can be viewed as the original surface-renewal model. This model was formulated by Higbie [51]. This model is based on the assumption that the liquid surface contains small fluid elements that contact the gas phase for a time that is equal for all elements. After this contact time they penetrate into the bulk liquid and each element is then replaced by another element from the bulk liquid phase. The basic mechanism captured in this concept is that at short contact times, the diffusion process will be unsteady. Considering that the fluid elements may diffuse to an infinite extend into the liquid phase, the model formulation developed earlier for diffusion into a semi-infinite slab can be applied describing this system. After some time the diffusion process will reach a steady state, thus the penetration theory predictions will then correspond to the limiting case described by the basic film theory. However, when the transient flux development is determining a notable amount of the total flux accumulated, the two models will give rise to different mass transfer coefficients. 

The investigations of Murrieta [268] tiiow that for calculation of the liquid-side controlled mass transfer coefficient in case of corrugated packii the Higbie penetration model (Chapter 1) can be used, when the erqmsure time 0g in it is calculated the equaticm... 

The so-called rate-based stage model presents a different way to the modeling of separation processes, by directly considering actual mass and heat transfer rates (Seader, 1989 Taylor and Krishna, 1993). A number of models fall into the general framework of the rate-based stage. In most cases, the film (Lewis and Whitman, 1924) or penetration and surface renewal (Higbie, 1935 Danckwerts, 1951) models find application, whereas the necessary model parameters are estimated by means of correlations. In this respect, the film model appears advantageous due to numerous correlation data available in the literature (see, e.g., Billet and Schultes, 1999). 

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