Film resistance model

Multimedia models and field data indicate that this is a dominant process for the transfer of persistent organic pollutants to lakes (e.g., Bidleman McConnell, 1995 Mackay Wania, 1995). Fluxes of air-water gas exchange are typically calculated using the two-film resistance model (Schwartzenbach et al 1993). Fluxes can be in either direction and depend upon air and water concentrations, partition coefficients, and air and water resistivities. 

Isothermal Langmuir Kinetics and Film Resistance Models. 316... 

Our library contains FRFs for four simple isothermal mechanisms Langmuir kinetics, film resistance model, micropore diffusion, and pore-surface model. For each mechanism, a short description with the model equations is given, together with the expressions for the first-order FRF Fx p(w), and two second-order FRFs, F2,pp(w, w) and F2,pp(w, —[Pg.293]

If the overall mass transfer resistance is lumped in the fluid film surrounding the particle or in the thin skin at the particle surface, a simple, lumped parameter, film resistance model can be used ... 

FIGURE 11.7 First- and second-order FRFs for film resistance model. (From Petkovska, M. and Do, D.D., Nonlinear Dyn., 21, 353-376, 2000. With permission.)... 

Figure 11.19 shows the shape of the negative imaginary part of the first-order FRFs corresponding to the Langmuir kinetics and the film resistance models. If the product of the frequency and the characteristic time constant is used on the abscissa, the maximum is obtained for = 1, that is, for = 1 /t. In this way, the time constant can be estimated directly from the position of the maximum of — Imag(Fi p(u>)). 

FIGURE 11.19 The negative imaginary part of p(o)) for Langmuir kinetics and film resistance models. 

On the other hand, the time constant for the film resistance model was defined as (Equation (11.29)) ... 

The existing parameterizations of treat the exchange across the atmosphere-snow pack interface on the basis of the Whitman two-film resistance model, which assumes that a chemical that moves from snow pack to atmosphere has to first transfer from the bulk of the snow pack to the snow pack s surface, and then through the atmospheric boundary layer to the bulk atmosphere. 

As discussed in Chapter 7, this form can provide a good fit of the data if the reaction is not too close to equilibrium. However, most reaction engineers prefer a mechanistically based rate expression. This section describes how to obtain plausible functional forms for based on simple models of the surface reactions and on the observation that aU the rates in Steps 2 through 8 must be equal at steady state. Thus, the rate of transfer across the film resistance equals the rate of diffusion into a pore equals the rate of adsorption equals the rate of reaction equals the rate of desorption, and so on. This rate is the pseudohomo-geneous rate shown in Steps 1 and 9. 

The concentration of gas over the active catalyst surface at location / in a pore is ai [). The pore diffusion model of Section 10.4.1 linked concentrations within the pore to the concentration at the pore mouth, a. The film resistance between the external surface of the catalyst (i.e., at the mouths of the pore) and the concentration in the bulk gas phase is frequently small. Thus, a, and the effectiveness factor depends only on diffusion within the particle. However, situations exist where the film resistance also makes a contribution to rj so that Steps 2 and 8 must be considered. This contribution can be determined using the principle of equal rates i.e., the overall reaction rate equals the rate of mass transfer across the stagnant film at the external surface of the particle. Assume A is consumed by a first-order reaction. The results of the previous section give the overall reaction rate as a function of the concentration at the external surface, a. ... 

Schneider and Klein(5) have pointed out that the early stages of cross-flow microfiltration often follow such a pattern although the growth of the cake is limited by the cross-flow of the process liquid. There are a number of ways of accounting for the control of cake growth. A useful method is to rewrite the resistance model to allow for the dynamics of polarisation in the film layer as discussed by Fane 6. Equation 8.3 is then written as ... 

This model covers the case where we have combined resistances to diffusion (fluid-film and solid diffusion). In this case, the concentration in the main phase of the fluid (bulk concentration) is different from the one at the interface due to the effect of the fluid film resistance. The following equations can be used for Langmuir and Freundlich equilibrium equations (Miura and Hashimoto, 1977). The solutions of the fixed-bed model are the following ... 

Essentially, the above model is empirical as in real systems, both solid and fluid-film resistances play an important role in the adsorption process. An improved BDST model is found elsewhere (Ko et al., 2000, 2002). Finally, qm a and could be flow rate-dependent parameters (Walker and Weatherley, 1997). 

In the case of low resistance in fluid film or/and unfavorable equilibrium (equilibrium-limited system), a high contact time at the large bed would be beneficial for the equilibration step, whereas a high superficial velocity is not expected to lead to better results, since the fluid-film resistance is minimal. Furthermore, the same type of model can be used for different scales. 

A detailed transport model for resist dissolution has been developed (169). In conjunction with standard ellipsometric equations describing multilayer films, the model provides quantitative agreement with the observed traces from the in situ ellipsometer. Model parameters are thus extracted, and their significance in terms of molecular structures of the system can be established. This model can then be extended for predictive purposes in the design and selection of resist materials. 

Analytical equations for adsorbate uptake and radial adsorbent temperature profiles during a differential kinetic test are derived. The model assumes that the mass transfer into the adsorbent can be described by a linear driving force model or the surface barrier model. Heat transfer by Fourier conduction inside the adsorbent mass in conjunction with external film resistance is considered. 

The importance of adsorbent non-isothermality during the measurement of sorption kinetics has been recognized in recent years. Several mathematical models to describe the non-isothermal sorption kinetics have been formulated [1-9]. Of particular interest are the models describing the uptake during a differential sorption test because they provide relatively simple analytical solutions for data analysis [6-9]. These models assume that mass transfer can be described by the Fickian diffusion model and heat transfer from the solid is controlled by a film resistance outside the adsorbent particle. Diffusion of adsorbed molecules inside the adsorbent and gas diffusion in the interparticle voids have been considered as the controlling mechanism for mass transfer. 

Since liquid does not completely wet the packing and since film thickness varies with radial position, classical film-flow theory does not explain liquid flow behavior, nor does it predict liquid holdup (30). Electrical resistance measurements have been used for liquid holdup, assuming liquid flows as rivulets in the radial direction with little or no axial and transverse movement. These data can then be empirically fit to film-flow, pore-flow, or droplet-flow models (14,19). The real flow behavior is likely a complex combination of these different flow models, that is, a function of the packing used, the operating parameters, and fluid properties. Incorporating calculations for wetted surface area with the film-flow model allows prediction of liquid holdup within 20% of experimental values (18). 

The model based on the concept of pure limiting film resistance involves the steady-state concept of the heat transfer process and omits the essential unsteady nature of the heat transfer phenomena observed in many gas-solid suspension systems. To take into account the unsteady heat transfer behavior and particle convection in fluidized beds, a surface renewal model can be used. The model accounts for the film resistance adjacent to the heat transfer... 

Figure 7.3 shows the sequence of steps converting phase resistances into a tray efficiency. Gas and liquid film resistances are added to give the point efficiency (Sec. 7.1,2), Had both vapor and liquid on the tray been perfectly mixed, the Murphree tray efficiency would have equaled the point efficiency (see Sec. 7,1.1). Since the phases are not perfectly mixed, a model of the vapor- and liquid-mixing patterns is... 

Characteristic Time Model This model was developed by Reverchon and Osseo in 1994. As in model I it is assumed that the extraction is uniform along the bed (and in the form used) that the external film resistance can be neglected (see Equations 8 and 9 in reference 3). 

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