Models for Transfer at a Gas-Liquid Interface

Several models have been proposed to describe the phenomena occurring when a gas phase is brought into contact with a liquid phase. The model that has been used most so far is the two-film theory proposed by Whitman [1] and by Lewis 

The resistance to transfer of A from the interface to the bulk liquid is supposed to be entirely located in the liquid film. Beyond that film the turbulence is sufficient to eliminate concentration gradients. This concept is illustrated in Fig. 6.2-1. 

The same concept has been applied to mass transfer in the gas and liquid phase, for which one can write, in the absence of reaction  

Again the absence of information on both and y leads to the introduction of mass transfer coefficients for the gas and liquid phase, feg and k, respectively. 

The two-film theory is an essentially steady-state theory. It assumes that the steady-state profiles corresponding to the given and are instantaneously realized. This requires that the capacity of the films be negligible. The two-film theory certainly lacks reality in postulating the existence of films near the interface between the gas and liquid, yet it contains the essential features of the phenomenon, that is, the dissolution and the diffusion of the gas. prior to transfer to the turbulent bulk of the liquid. Nevertheless the theory has enabled consistent correlation of data obtained in equipment in which the postulates are hard to accept completely. 

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Assume that a simple film model exists for the mass transfer, equilibrium is established at the gas-liquid interface, and the diffusion occurs at isobaric and isothermal conditions. Also assume that neither helium nor argon is absorbed so that N2=N3 = 0. Then, the Maxwell-Stefan equations for the diffusion of argon and helium are... 

The two-film model is a simple example of this approach. A system of two fluids exists, with a distinct interface between the two (gas/liquid or two immiscible liquids). For purposes of this example, we assume a gas/liquid interface Figure 3.33 illustrates the region near the interface. There will be a film (or boundary layer) on each side of the interface where, due to mass transfer from one phase to the second (gas to liquid in the figure), the concentration of A is changing from its value in the bulk phase, in gas and Ca.s in liquid. The thickness of the boundary layer on each side of the interface will typically be different and a function of the fluid and flow conditions in each phase. At steady-state, the flux of A can be described as ... 

The fllm theory is the simplest model for interfacial mass transfer. In this case it is assumed that a stagnant fllm exists near the interface and that all resistance to the mass transfer resides in this fllm. The concentration differences occur in this film region only, whereas the rest of the bulk phase is perfectly mixed. The concentration at the depth I from the interface is equal to the bulk concentration. The mass transfer flux is thus assumed to be caused by molecular diffusion through a stagnant fllm essentially in the direction normal to the interface. It is further assumed that the interface has reached a state of thermodynamic equilibrium. The mass transfer flux across the stagnant film can thus be described as a steady diffusion flux. It can be shown that within this steady-state process the mass flux will be constant as the concentration profile is linear and independent of the diffusion coefficient. Consider a gas-liquid interface, as sketched in Fig. 5.16. The mathematical problem is to formulate and solve the diffusion flux equations determining the fluxes on both sides of the interface within the two films. The resulting concentration profiles and flux equations can be expressed as ... 

Figure 9.7 shows concentration profiles schematically for A and B according to the two-film model. Initially, we ignore the presence of the gas film and consider material balances for A and B across a thin strip of width dx in the liquid film at a distance x from the gas-liquid interface. (Since the gas-film mass transfer is in series with combined diffusion and reaction in the liquid film, its effect can be added as a resistance in series.)... 

Considering homogeneous RSPs, mass transfer at the gas/vapor/liquid-liquid interface can be described using different theoretical concepts (57,59). Most often the two-film model (87) or the penetration/surface renewal model (27,88) is used, in which the model parameters are estimated via experimental correlations. In this respect the two-film model is advantageous since there is a broad spectrum of correlations available in the literature, for all types of internals and systems. For the penetration/surface renewal model, such a choice is limited. 

In most common separation processes, the main mass transfer is across an interface between a gas and a liquid or between two liquid phases. At fluid-fluid interfaces, turbulence may persist to the interface. A simple theoretical model for turbulent mass transfer to or from a fluid-phase boundary was suggested in 1904 by Nernst, who postulated that the entire resistance to mass transfer in a given turbulent phase lies in a thin, stagnant region of that phase at the interface, called a him, hence the name film theory.2 4,5 Other, more detailed, theories for describing the mass transfer through a fluid-fluid interface exist, such as the penetration theory.1,4... 

The result obtained from the film theory is that the mass transfer coefficient is directly proportional to the diffusion coefficient. However, the experimental mass transfer data available in the literature [6], for gas-liquid interfaces, indicate that the mass transfer coefficient should rather be proportional with the square root of the diffusion coefficient. Therefore, in many situations the film theory doesn t give a sufficient picture of the mass transfer processes at the interfaces. Furthermore, the mass transfer coefficient dependencies upon variables like fluid viscosity and velocity are not well understood. These dependencies are thus often lumped into the correlations for the film thickness, 1. The film theory is inaccurate for most physical systems, but it is still a simple and useful method that is widely used calculating the interfacial mass transfer fluxes. It is also very useful for analysis of mass transfer with chemical reaction, as the physical mechanisms involved are very complex and the more sophisticated theories do not provide significantly better estimates of the fluxes. Even for the description of many multicomponent systems, the simplicity of the model can be an important advantage. 

An analysis of chemical desorption has recently been published (Chem.Eng.Sci., 21 0980)), which is based on a number of simplifying assumptions the film theory model is assumed, the diffusivities of all species are taken to be equal to each other, and in the solution of the differential equations an approximation which is second order with respect to distance from the gas-liquid interface is used this approximation was introduced as early as 1948 by Van Krevelen and Hoftizer. However, the assumptions listed above are not at all drastic, and two crucial elements are kept in the analysis reversibility of the chemical reactions and arbitrary chemical mechanisms and stoichiometry.The result is a methodology for developing, for any given chemical mechanism, a highly nonlinear, implicit, but algebraic equation for the calculation of the rate enhancement factor as a function of temperature, bulk-liquid composition, interface gas partial pressure and physical mass transfer coefficient The method of solution is easily gene ralized to the case of unequal diffusivities and corrections for differences between the film theory and the penetration theory models can be calculated. 

The analytical model is shown in F ig. 1 for heat flow interaction with an environment consisting of Case I, heat transfer with an ambient through a convective heat transfer coefficient and. Case II, heat transfer with an ambient consisting of an imposed, constant heat flux. The system consists of a container (or, pipe as far as this analysis is concerned) initially filled with liquid at temperature T, Initially, both the wall and the liquid are at temperature Tj. At zero time, a pressurizing gas having temperature is introduced into the top of the container at x = 0. At this same time liquid discharge or flow is commenced such that the gas-liquid interface immediately starts to move downward at a velocity V. In addition, heat flow interaction of the nature of Case I or Case II between the outside of the container and the ambient starts. As a consequence of this, a transient process is introduced in the temperatures of both the wall and the gas. It is the purpose of this paper to present a solution for the thermal response of the gas and the wall. 

The mathematical model was constracted on the basis of a three-phase plug-flow reactor model developed by Korsten and Hoffmaim [63]. The model incorporates mass transport at the gas-liquid and liquid-solid interfaces and uses correlations to estimate mass-transfer coefficients and fluid properties at process conditions. The feedstock and products are represented by six chemical lumps (S, N, Ni, V, asphaltenes (Asph), and 538°C-r VR), defined by the overall elemental and physical analyses. Thus, the model accounts for the corresponding reactions HDS, HDN, HDM (nickel (HDNi) and vanadium (HDV) removals), HD As, and HCR of VR. The gas phase is considered to be constituted of hydrogen, hydrogen sulfide, and the cracking product (CH4). The reaction term in the mass balance equations is described by apparent kinetic expressions. The reactor model equations were built under the following assumptions ... 

A dynamic mathematical model of the three-phase reactor system with catalyst particles in static elements was derived, which consists of the following ingredients simultaneous reaction and diffusion in porous catalyst particles plug flow and axial dispersion in the bulk gas and liquid phases effective mass transport and turbulence at the boundary domain of the metal network and a mass transfer model for the gas-liquid interface. 

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