Mass-transfer coefficients film theory

The factor is known as the Stefan correction factor , [1.28]. In order to calculate the mass transferred using film theory, the mass transfer coefficient (3 has to be found. In cases where convection is negligible the mass transferred is calculated from equation (1.181), whilst where convection is significant the mass transferred is given by (1.183). 

Film Theory. Many theories have been put forth to explain and correlate experimentally measured mass transfer coefficients. The classical model has been the film theory (13,26) that proposes to approximate the real situation at the interface by hypothetical "effective" gas and Hquid films. The fluid is assumed to be essentially stagnant within these effective films making a sharp change to totally turbulent flow where the film is in contact with the bulk of the fluid. As a result, mass is transferred through the effective films only by steady-state molecular diffusion and it is possible to compute the concentration profile through the films by integrating Fick s law ... 

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. 

Mass-Transfer Coefficient Denoted by /c, K, 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 theoiy, the surface renewal theoiy, and the penetration the-oiy, all of which pertain to ideahzed cases. For many situations of practical interest like investigating the flow inside tubes and over flat surfaces as well as measuring external flowthrough 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. 

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-... 

In evaluating their results they assumed the film theory, and, because the oxygen is sparingly soluble and the chemical reaction rate high, they also assumed that the liquid film is the controlling resistance. The results were calculated as a volumetric mass-transfer coefficient based, however, on the gas film. They found that the volumetric mass-transfer coefficient increased with power input and superficial gas velocity. Their results can be expressed as follows ... 

Note that the transfer rate equation is based on an overall concentration driving force, (X-X ) and overall mass transfer coefficient, Kl. The two-film theory for interfacial mass transfer shows that the overall mass transfer coefficient, Kl, based on the L-phase is related to the individual film coefficients for the L and G-phase films, kL and ko, respectively by the relationship... 

In SSP, the boundaries for the mass balances are defined by the particle instead of the reactor or the reactor compartment dimensions and the process conditions are accounted for by a boundary condition. The mass transfer at the particle/gas interface is mostly described according to the film theory by using a mass-transfer coefficient. 

On the basis of the familiar two-film theory, the over-all mass transfer coefficients, as well as the over-all heights and times of transfer units,... 

This chapter will first provide some basics on ozone mass transfer, including theoretical background on the (two-) film theory of gas absorption and the definition of over-all mass transfer coefficients KLa (Section B 3.1) as well as an overview of the main parameters of influence (Section B 3.2). Empirical correction factors for mass transfer coefficients will also be presented in Section B 3.2. These basics will be followed by a description of the common methods for the determination of ozone mass transfer coefficients (Section B 3.3) including practical advice for the performance of the appropriate experiments. Emphasis is laid on the design of the experiments so that true mass transfer coefficients are obtained. 

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). 

Postulating that n is dependent on the turbulence in the system, Dobbins (1956) proposed that under sufficiently turbulent conditions, n approaches 0.5 (surface renewal or penetration theory), while under laminar or less turbulent conditions n approaches 1.0 (film theory). Thus, the selection of the value for n to predict the mass transfer coefficient should depend on the degree of turbulence in the system ... 

The connection between the film mass transfer coefficients and the over-all mass transfer coefficients is provided by the two-film theory from Lewis and Whitman (1924) the total resistance to mass transfer is the sum of the resistances in each phase. 

The two-film theory supposes that the entire resistance to transfer is contained in two fictitious films on either side of the interface, in which transfer occurs by molecular diffusion. This model leads to the conclusion that the mass-transfer coefficient kL is proportional to the diffusivity DAB and inversely proportional to the film thickness Zy as... 

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). 

As can be seen from Figure 8, if Fo < 0.02, the concentration changes within the film are confined largely to the surface layer and the local mass transfer coefficient is given by the Higbie penetration theory (9) as... 

Z.V.P. Murphy, S.K. Gupta, Estimation of mass transfer coefficient using a combined nonlinear membrane transport and film theory model, Desalination 109 (1997) 39-49. 

As is shown in Figure 2, in the two-phase model the fluid bed reactor is assumed to be divided into two phases with mass transfer across the phase boundary. The mass transfer between the two phases and the subsequent reaction in the suspension phase are described in analogy to gas/liquid reactors, i.e. as an absorption of the reactants from the bubble phase with pseudo-homogeneous reaction in the suspension phase. Mass transfer from the bubble surface into the bulk of the suspension phase is described by the film theory with 6 being the thickness of the film. D is the diffusion coefficient of the gas and a denotes the mass transfer coefficient based on unit of transfer area between the two phases. 6 is given by 6 = D/a. 

Like their random-packing efficiency model (above), the Bravo, Fair et al. structured-packing model is based on the two-film theory. The HTU is calculated from the mass transfer coefficients and interfacial areas using Eqs. (9.23) and (9.24). The HETP can be calculated from the HTU using Eqs. (9.12) and (9.13). The mass transfer coefficients are evaluated from... 

Film theory predicts that the mass transfer coefficient for a phase (or the overall mass transfer coefficient) is proportional to the diffusion coefficient and inversely proportional to the thickness of the stagnant zone. The diffusion coefficient can be calculated from either the Wilke-Chang or the FSG equations. However, 6 is difficult (if not impossible) to determine. Hence, mass transfer coefficients are often determined from empirical correlations. Also, Film theory is based on the assumption that the bulk fluid phases are perfectly mixed. While this might approach reality for well-mixed turbulent systems, this is certainly not the case for laminar systems. 

Gas-liquid interfacial areas, a, and volumetric liquid-side mass transfer coefficients, kLa, are measured in a high pressure trickle-bed reactor. Increase of a and kLa with pressure is explained by the formation of tiny bubbles in the trickling liquid film. By applying Taylor s theory, a model relating the increase in a with the increase in gas hold-up, is developed. The model accounts satisfactorily for the available experimental data. To estimate kLa, contribution due to bubbles in the liquid film has to be added to the corresponding value measured at atmospheric pressure. The mass transfer coefficient from the bubbles to the liquid is calculated as if the bubbles were in a stagnant medium. 

In practice, the contact time t is not known except in special cases in which the hydrodynamics are clearly defined. This is somewhat similar to the case of the stagnant-film theory in which the unknown quantity is the thickness of the stagnant layer 5 (in film theory, the liquid-phase mass-transfer coefficient is given by k L = DJ5). 

In the limiting case of mass transfer from a single sphere resting in an infinite stagnant liquid, a simple film-theory analysis122 indicates that the liquid-solid mass-transfer coefficient R s is equal to 2D/JV, where D is the molecular diffusivity of the solute in the liquid phase and d is the particle diameter. In dimensionless form, the Sherwood number... 

Although the second approach can be considered a simplification of the first many systems have been satisfactorily described, and it is often preferred due to its mathematical simphcity. Studies on mass transfer through aqueous-organic interfaces immobilized at the pore mouths of a microporous membrane have shown that for a HF device, the overall mass transfer coefficient obtained can be related to the individual phase mass transfer coefficients and the membrane resistance using simple film theory. However, many authors considered that this separation controlling, overall mass transfer resistance was dominated by the resistance of the membrane. This is because the permeability of the membrane is low and because the membrane is thick [4,19,20,23,52,53]. On the other hand, other authors reported for different systems and conditions that the kinetic control of... 

As the interface offers no resistance, mass transfer between phases can be regarded as the transfer of a component from one bulk phase to another through two films in contact, each characterized by a mass-transfer coefficient. This is the two-film theory and the simplest of the theories of interfacial mass transfer. For the transfer of a component from a gas to a liquid, the theory is described in Fig. 6B. Across the gas film, the concentration, expressed as partial pressure, falls from a bulk concentration Fas to an interfacial concentration Ai- In the liquid, the concentration falls from an interfacial value Cai to bulk value Cai-... 

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