Film theory estimation

Example 9.5-1 Fast benzene evaporation Benzene is evaporating from a flat porous plate into pure flowing air. Using the film theory, estimate how much a concentrated solution increases the mass transfer rate beyond that expected for a simple theory. Then calculate the resulting change in the mass transfer coefficient defined by Eq. 9.5-2. In other words, find (Vi/k ci, and k/k as a function of the vapor concentration of benzene at the surface of the plate. 

The initial condition is c = cfO) at t = 0. The flux at the particle surface is obtained from the solution of Eq. (8.1) and the gas-liquid flux can be estimated from the film theory. 

The approach taken is semi-empirical. Point efficiencies are estimated making use of the two-film theory , and the Murphree efficiency estimated allowing for the degree of mixing likely to be obtained on real plates. 

Additional experiments in a loop reactor where a significant mass transport limitation was observed allowed us to investigate the interplay between hydrodynamics and mass transport rates as a function of mixer geometry, the ratio of the volume hold-up of the phases and the flow rate of the catalyst phase. From further kinetic studies on the influence of substrate and catalyst concentrations on the overall reaction rate, the Hatta number was estimated to be 0.3-3, based on film theory. 

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. 

The heat transfer coefficients estimated from correlations or analogies are the low flux coefficients and, therefore, need to be corrected for the effects of finite transfer rates before use in design calculations. We recommend the film theory correction factor given by Eq. 11.4.12. 

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. 

Bobrov et find for ethylene on silver films in UHV that an exposure of 1 L leads to a monolayer, based on a kinetic theory estimate, assuming a sticking factor of 1. They see Raman signals already at exposures of 0.03 L which saturate at 2 L. By comparison to the scattering from a crystal of naphthalene ( ) they estimate the enhancement as lO -lO (no further details are given). 

Various other ways of characterizing the role of mass transfer in PTC systems have been reported. For example, based on the two-film theory, Chen et al.(1991) derived algebraic expressions for the interphase flux of QY and QX. Nonlinear differential equations described the slow reaction in the organic phase, and coupled algebraic equations described the dissociation equilibria in the aqueous phase and the species mass balance. Model parameters were estimated... 

The electronic properties of monocrystalline and grained nanocrystalline CrSi2 films were estimated within the Effective Mass Theory. Inclusion of the grains inside the film increases the energy gap up to 60% compared to the monocrystalline film of the same effective thickness. 

The amount of additional information needed to be able directly to take into account heat and mass transfer in Model 4 is high. Using the two-film theory, information on the film thickness is needed, which is usually condensed into correlations for the Sherwood number. That information was not available for Katapak-S so that correlations for similar non-reactive packing had to be adopted for that purpose. Furthermore, information on diffusion coefficients is usually a bottleneck. Experimental data is lacking in most cases. Whereas diffusion coefficients can generally be estimated for gas phases with acceptable accuracy, this does unfortunately not hold for liquid multicomponent systems. For a discussion, see Reid et al. [8] and Taylor and Krishna [9]. These drawbacks, which are commonly encountered in applications of rate-based models to reactive separations, limit our ability to judge their value as deviations between model predictions and experimen-... 

It does not seem possible at this stage to single out any particular theory as the most acceptable. In any case, because the diffusivities do not differ greatly and also considering the error in their estimation, the exact value of p is not crucial. It is often adequate to use the film theory and then apply the necessary correction by using the appropriate value of p provided that the diffusivities are accurately known. Thus the methods and equations given in Chapter 14 should be applicable. 

Combined solution-diffusion-film-theory models have been presented already in several publications on aqueous systems, however, either 100% rejection of the solute is assumed [38], or detailed experimental flux and rejection results are required in order to find parameters by nonlinear parameter estimation [43, 44]. Consequently, it is difficult to apply these models for predictive purposes. In OSN, it is also important to account for the effect of different activities of the species on both sides of the membrane. We have proposed a set of equations [32], Eqs. (7) to (13), taking these factors into account We assume a binary system, although the equations could be generalized for a system of n components. In this analysis component 1 is the solute and component 2 is the solvent. The only parameters to be estimated, other than physical properties, are... 

The film thickness is estimated using Nusselt falling film theory given by Equation 8.10. 

The rate of transfer to the wall may be estimated from film theory as fe = D/6f for a given film thicknes s 5 F. We as sume here that 5 is the same within the slug, i. e. the liquid layer close to the wall is not partaking in the recirculation and the bubble, and obtain... 

For the bubble region, the mass transfer rate can be estimated from film theory ... 

The main transport parameters to be estimated are the mass transfer coefficients (gas-liquid (liquid side) fc , gas-liquid (gas side) kg, and liquid-solid fc )). Coupled to that is the estimation of the interfacial area per unit volume a, and often it is the combination (i.e., kia or kgO) that is estimated in a certain experimental procedure. Thermodynamic parameters, such as Henry s law constant (fZ) can be estimated in a simpler manner since their estimation on the flow or on any time-dependent phenomenon. Mass transfer coefticients may be evaluated in well-defined geometries with known flow fields using classical theories like film theory, penetration theory, surface renewal... 

The values for parameters in the mass balances have to be determined either theoretically or experimentally. These parameters are equilibrium ratio (Kj), mass transfer coefficients (fcLi and kci), diffusion coefficients in the gas and liquid phases (Du and Da), volume ratios (Uy and Op), and liquid holdup (bl). The equilibrium ratio, Kj, can be estimated using thermodynamic theories [4] for gases with a low solubility, the equilibrium ratio, Ki, can sometimes be replaced by Hemy s constant [5]. In Refs. [6,7], some methods for estimating ku and ka are discussed. The correlation equations for ku and ka contain the diffusion coefficients, D i and Dci, in the gas and liquid phases, respectively. The mass transfer coefficients, ku and ka, are formally related to the diffusion coefficients by the film theory ... 

In other words, we do not intend to include, except in a peripheral sense, the more profound aspects of transport theory. The mainstays here are Pick s law of diffusion, film theory, and the concept of the equilibrium stage. These have been, and continue to be, the preferred tools in everyday practice. What we bring to these topics compared to past treatments is a much wider, modem set of applications and a keener sense that students need to learn how to simplify complex problems (often an art), to make engineering estimates (an art as well as a science), and to avoid common pitfalls. Such exercises, often dismissed for lacking academic rigor, are in fact a constant necessity in the engineering world. 

An additional simple approximate analytical solution is available in the so-called thin film" theory near the diffusion limit when the Damkohler number (kb m/DA) representing the ratio of diffusion time to reaction time approaches zero. Smith et al. (1973) have obtained the following estimate of the facilitation ratio by a power series expansion ... 

Film theory employed for concentration polarization estimate. 

The outline of the presented material is as follows. The mass transfer that occurs at the NAPL-water interfaces at the pore scale is generally approximated using a linear model based on film theory. In extending this formulation to the representative elemental volume (REV) scale in porous media, it is necessary to define an overall mass transfer rate coefficient. The theoretical development of phenomenological models that are used to estimate these overall mass transfer coefficients is presented. Methods to up-scale these REV scale models to field scale are presented. A set of examples based on intermediate-scale laboratory tests is presented to demonstrate the use of these methods. 

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