Mass transfer surface renewal theory

An apparent weakness of the film model is that it suggests that the mass transfer coefficient is directly proportional to the diffusion coefficient raised to the first power. This result is in conflict with most experimental data, as well as with more elaborate models of mass transfer [surface renewal theory considered in the next chapter, e.g., or boundary layer theory (Bird et al., I960)]. However, if we substitute the film theory expression for the mass transfer coefficient (Eq. 8.2.12) into Eq. 8.8.1 for the Sherwood number we find... 

The theory of the unsteady-state mass transfer (surface renewal theory) starts... 

Neither the penetration nor the surface renewal theory can be used to predict mass transfer coefficients directiy because T and s are not normally known. Each suggests, however, that mass transfer coefficients should vary as the square root of the molecular diffusivity, as opposed to the first power suggested by the film theory. 

Simplified Mass-Transfer Theories In certain simple situations, tne mass-transfer coefficients can be calculated from first principles. The film, penetration, and surface-renewal theories are attempts to extend tnese theoretical calculations to more complex sit-... 

Surface Renewal Theory. The film model for interphase mass transfer envisions a stagnant film of liquid adjacent to the interface. A similar film may also exist on the gas side. These h5q>othetical films act like membranes and cause diffu-sional resistances to mass transfer. The concentration on the gas side of the liquid film is a that on the bulk liquid side is af, and concentrations within the film are governed by one-dimensional, steady-state diffusion ... 

Janssen and Hoogland (J3, J4a) made an extensive study of mass transfer during gas evolution at vertical and horizontal electrodes. Hydrogen, oxygen, and chlorine evolution were visually recorded and mass-transfer rates measured. The mass-transfer rate and its dependence on the current density, that is, the gas evolution rate, were found to depend strongly on the nature of the gas evolved and the pH of the electrolytic solution, and only slightly on the position of the electrode. It was concluded that the rate of flow of solution in a thin layer near the electrode, much smaller than the bubble diameter, determines the mass-transfer rate. This flow is affected in turn by the incidence and frequency of bubble formation and detachment. However, in this study the mass-transfer rates could not be correlated with the square root of the free-bubble diameter as in the surface renewal theory proposed by Ibl (18). 

One other measurement technique that has been used to measure Kl over a shorter time period, and is thus more responsive to changes in wind velocity, is the controlled flux technique (Haupecker et al., 1995). This technique uses radiated energy that is turned into heat within a few microns under the water surface as a proxy tracer. The rate at which this heat diffuses into the water column is related to the liquid film coefficient for heat, and, through the Prandtl-Schmidt number analogy, for mass as well. One problem is that a theory for heat/mass transfer is required, and Danckwert s surface renewal theory may not apply to the low Prandtl numbers of heat transfer (Atmane et al., 2004). The controlled flux technique is close to being viable for short-period field measurements of the liquid film coefficient. 

The mass transfer rates for the case when d > d can easily be obtained from Eqs. 9 or 12 (see [48]). Using the surface renewal theory this case is not relevant because the boundary layer thickness is here considered to be infinite. 

The theories vary in the assumptions and boundary conditions used to integrate Fick s law, but all predict the film mass transfer coefficient is proportional to some power of the molecular diffusion coefficient D", with n varying from 0.5 to 1. In the film theory, the concentration gradient is assumed to be at steady state and linear, (Figure 3-2) (Nernst, 1904 Lewis and Whitman, 1924). However, the time of exposure of a fluid to mass transfer may be so short that the steady state gradient of the film theory does not have time to develop. The penetration theory was proposed to account for a limited, but constant time that fluid elements are exposed to mass transfer at the surface (Higbie, 1935). The surface renewal theory brings in a modification to allow the time of exposure to vary (Danckwerts, 1951). 

Several different mechanisms have been proposed to provide a basis for a theory of interphase mass transfer. The three best known are the two-film theory, the penetration theory, and the surface renewal theory. 

Developing correlations to describe mass transfer in rotating packed beds has proven to be a challenge. Penetration theory (31), film-flow theory (32), and modified surface-renewal theory (12) are some examples of leveraging previous work... 

Liquid-solid mass transfer has also been studied, on a limited basis. Application to systems with catalytic surfaces or electrodes would benefit from such studies. The theoretical equations have been proposed based on film-flow theory (32) and surface-renewal theory (39). Using an electrochemical cell with rotating screen disks, liquid-solid mass transfer was shown to increase with rotor speed and increased spacing between disks but to decrease with the addition of more disks (39). Water flow over naphthalene pellets provided 4-6 times higher volumetric mass transfer coefficients compared to gravity flow and similar superficial liquid velocities (17). 

Chang and Rochelle [12-15] investigated some aspects of SO2 absorption in an agitated cell filled with aqueous solution or suspension, and compared their measurements with those of other authors with simulated absorbed mass flows, based on enhancement factors from the film and surface renewal theory. The results showed that mass transfer, when corrected by an enhancement factor from the surface renewal theory, agreed clearly with the measurements. 

Several theories have been developed to describe the rate of interphase mass transfer. These include film theory, boundary layer theory, penetration theory, and surface renewal theory. In this chapter we will review the first two, along with an overview of empirical correlations that are used to describe mass transfer. A more thorough overview of mass transfer theories can be found in Bird, Stewart and Lightfoot [48], Clark [49], Logan [50], and Weber and DiGiano [51]. 

Ravetkar and Kale (1981) have shown that Higbie s surface-renewal theory can be successfully employed for the calculation of the liquid-side mass-transfer coefficients for Newtonian and non-Newtonian liquids as... 

Mass-Transfer Coefficient Denoted by kc, kx, Kx, and so on, the mass-transfer coefficient is the ratio of the flux to a concentration (or composition) difference. These coefficients generally represent rates of transfer that are much greater than those that occur by diffusion alone, as a result of convection or turbulence at the interface where mass transfer occurs. There exist several principles that relate that coefficient to the diffusivity and other fluid properties and to the intensity of motion and geometry. Examples that are outlined later are the film theory, the surface renewal theory and the penetration theory, all of which pertain to idealized cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flow through banks of tubes, in fixed beds of particles, and the like, correlations have been developed that follow the same forms as the above theories. Examples of these are provided in the subsequent section on mass-transfer coefficient correlations. 

When necessary, results of surface-renewal theories will be presented simultaneously, as these models may be applied directly to determine interfacial area and mass-transfer coefficients in the laboratory apparatus considered in Sections III,B and V. 

The objective of this entry is to introduce the readers to the fundamental principles of gas-to-liquid mass transfer, as well as its major applications. Therefore, the first section of the entry is on the three fundamental mechanisms of gas-to-liquid mass transfer the film theory, the penetration theory, and the surface renewal theory followed by the applications of gas-to-liquid mass transfer in unit operations that are widely used in various chemical processes. There is a vast pool of reported literature on different aspects of gas-to-liquid mass transfer processes, all of which is impossible to be included in this entry. Therefore, only typical gas-to-liquid mass transfer processes are presented here. 

The value of a varies with the system under consideration. For example, in equimolar counter diffusion, Na and Nb are of the same magnitude, but in opposite direction. As a result, a is equal to 1 and hence, Eq. (2) reduces to Eq. (1), where is equal to Convective mass transfer coefficients are used in the design of mass transfer equipment. However, in most cases, these coefficients are extracted from empirical correlations that are determined from experimental data. The theories, which are often used to describe the mechanism of convective mass transfer, are the film theory, the penetration theory, and the surface renewal theory. 

The fundamental principles of the gas-to-liquid mass transfer were concisely presented. The basic mass transfer mechanisms described in the three major mass transfer models the film theory, the penetration theory, and the surface renewal theory are of help in explaining the mass transport process between the gas phase and the liquid phase. Using these theories, the controlling factors of the mass transfer process can be identified and manipulated to improve the performance of the unit operations utilizing the gas-to-liquid mass transfer process. The relevant unit operations, namely gas absorption column, three-phase fluidized bed reactor, airlift reactor, liquid-gas bubble reactor, and trickled bed reactor were reviewed in this entry. 

The K j may be estimated using an empirical correlation or alternative physical model (e.g., surface renewal theory) with the Maxwell-Stefan diffusivity of the appropriate i-j pair D-j replacing the binary Fick D. Since most published correlations were developed with data obtained with nearly ideal or dilute systems where F is approximately unity, we expect this separation of diffusive and thermodynamic contributions to k to work quite well. We may formally define the Maxwell-Stefan mass transfer coefficient k - as (Krishna, 1979a)... 

The surface-renewal theory intends to provide a better representation of the physical mechanisms than the penetration theory, but it predicts the same dependency of the mass transfer coefficient upon the diffusion coefficient. The penetration theory can thus be looked upon as a special case of the surface renewal theory where the distribution function takes the form of (5.204). Moreover, both theories also contain an unknown fitting parameter and are thus in practice equivalent. For the quantitative determination of the transfer coefficient we need to relate s, t or tg to the measurable parameters of the system under consideration. For this reason these concepts have no predictive value. 

While the film and surface-renewal theories are based on a simplified physical model of the flow situation at the interface, the boundary layer methods couple the heat and mass transfer equation directly with the momentum balance. These theories thus result in anal3dical solutions that may be considered more accurate in comparison to the film or surface-renewal models. However, to be able to solve the governing equations analytically, only very idealized flow situations can be considered. Alternatively, more realistic functional forms of the local velocity, species concentration and temperature profiles can be postulated while the functions themselves are specified under certain constraints on integral conservation. Prom these integral relationships models for the shear stress (momentum transfer), the conductive heat flux (heat transfer) and the species diffusive flux (mass transfer) can be obtained. 

The integral method thus leads to a mass transfer coefficient which vary with the 2/3 power of the diffusion coefficient. This parameter dependency is between the linear one of the film theory and the square-root variation of the penetration and surface-renewal theories. 

The results obtained on nitrating chlorobenzene in a stirred cell with 0.56 mol dm nitric acid in 79-8 H2SO4 are consistent with the rate being fast reaction mass transfer controlled. Reasonable agreement is obtained between the experimentally determined rate and activation energy and the values of these quantities predicted by surface renewal theory. 

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