Lumped parameter mass transfer

One very common sinplification is to assume that the film diffusion and diffusion in the particles can be lumped together in a lumped parameter mass transfer expression. In this form the total of all the mass transfer is assumed to be proportional to the driving force caused by the concentration difference or by the driving force caused by the difference in amount adsorbed The value is the... 

Possibilities for a single resistance include a linear rate expression with a lumped parameter mass transfer coefficient based either on the external fluid film or on a hypothetical solid film, depending on which film is controlling the rate of uptake of adsorbate. A quadratic driving force expression, again with a lumped parameter mass transfer coefficient, may be used instead. Alternatively, intraparticle diffusion, if the dominant form of mass transfer, may be described by the general diffusion equation (Pick s second law) with its appropriate boundary conditions, as described in Chapter 4. 

The mass-transfer coefficients depend on complex functions of diffii-sivity, viscosity, density, interfacial tension, and turbulence. Similarly, the mass-transfer area of the droplets depends on complex functions of viscosity, interfacial tension, density difference, extractor geometry, agitation intensity, agitator design, flow rates, and interfacial rag deposits. Only limited success has been achieved in correlating extractor performance with these basic principles. The lumped parameter deals directly with the ultimate design criterion, which is the height of an extraction tower. 

The connection of the overall mass transfer coefficient of the lumped kinetic and the parameters of the general rate model is... 

The transfer interface produced by most of the mass transfer apparatus considered in this book is in the form of bubbles. Measuring the surface area of swarms of irregular bubbles is very difficult. This difficulty in determining the interfacial area is overcome by not measuring it separately, but rather lumping it together with the mass transfer coefficient and measuring kLa as one parameter. 

Due to the complexity of most waste waters and unknown oxidation products, differences in lumped parameters such as COD or preferably DOC are used to quantify treatment success. A model to describe the oxidation process, including physical and chemical processes, based on a lumped parameter has been tried (Beltran et al., 1995). COD was used as a global parameter for all reactions of ozone with organic compounds in the chemical model. The physical model included the Henry s law constant, the kLa, mass transfer enhancement (i. e. the determination of the kinetic regime of ozone absorption) as well as the... 

Process-scale models represent the behavior of reaction, separation and mass, heat, and momentum transfer at the process flowsheet level, or for a network of process flowsheets. Whether based on first-principles or empirical relations, the model equations for these systems typically consist of conservation laws (based on mass, heat, and momentum), physical and chemical equilibrium among species and phases, and additional constitutive equations that describe the rates of chemical transformation or transport of mass and energy. These process models are often represented by a collection of individual unit models (the so-called unit operations) that usually correspond to major pieces of process equipment, which, in turn, are captured by device-level models. These unit models are assembled within a process flowsheet that describes the interaction of equipment either for steady state or dynamic behavior. As a result, models can be described by algebraic or differential equations. As illustrated in Figure 3 for a PEFC-base power plant, steady-state process flowsheets are usually described by lumped parameter models described by algebraic equations. Similarly, dynamic process flowsheets are described by lumped parameter models comprising differential-algebraic equations. Models that deal with spatially distributed models are frequently considered at the device... 

Ma et al. [104] attributed a decrease in diffusivity with an increase in initial concentration to pore diffusion effects. Because zeolites are bi-dispersed sorbents, both surface and pore diffusions may dominate different regions. In micropores, surface diffusion may be dominant, while pore diffusion may be dominant in macropores. This, therefore, supports the use of a lumped parameter (De). To explore further the relative importance of external mass transfer vis-a-vis internal diffusion, Biot number (NBl — kf r0/De) was considered. Table 9 summarizes the NBi values for the four initial concentrations. The NBi values are significantly larger than 100 indicating that film diffusion resistance was negligible. 

Both mass transfer resistances - as can be seen from the analytical solutions -act in a different way and cannot be combined to a so-called overall transfer resistance . The introduction of such a lumped parameter will hide essential physical effects, evoked by these two resistances separately. 

Many working groups have modeled the performance of diesel particulate traps during the past few decades. Concentrated parameter models (CSTR assumption) have been applied for the evaluation of formal kinetic models and model parameters. The formal kinetic parameters lump the heat and mass transfer effects with the reaction kinetics of the complicated reaction network of diesel soot combustion. Those models and model parameters were used for the characterization of the performance of different filter geometries and filter materials, as well as of the performance of a variety of catalytically active coatings and fuel additives [58],... 

The simplified lumped parameter model (M2) can also be used to match the data of Figure 6. This suggests the following correlations for the overall mass transfer coefficients in presence of reaction. 

Because of their structural and conformational complexity, polypeptides, proteins, and their feedstock contaminants thus represent an especially challenging case for the development of reliable adsorption models. Iterative simulation approaches, involving the application of several different isothermal representations8,367 369 enable an efficient strategy to be developed in terms of computational time and cost. Utilizing these iterative strategies, more reliable values of the relevant adsorption parameters, such as q, Kd, or the mass transfer coefficients (the latter often lumped into an apparent axial dispersion coefficient), can be derived, enabling the model simulations to more closely approximate the physical reality of the actual adsorption process. 

Today most hollow fiber BOs are designed such that the gas phase flows inside the fibers and the blood flows outside and across the fibers (Figure 23.8). Like flat sheet BOs, several investigators have used a lumped parameter approach to determine mass-transfer and friction factor correlations for BOs. Different investigators have developed slightly different correlations. The differences in the correlations are probably indicative of the level of accuracy of this approach. 

As shown in the preceding parts, kinetic parameters cannot be directly calculated when internal heat transfer limits pyrolysis. A model taking into account both kinetic scheme and heat- mass transfers becomes necessary, A one-dimension model has already been implemented and solved. It features a classical set of equations for heat and mass transfers in porous media, i.c. heat transfer through convection, conduction, radiation and mass transfer due to pressure gradient (Darcy s law) and binary diffusion. Different kinetic schemes from e literature arc and will be tested mass-loss as lumped first order reaction, formation of volatiles, tars and char from decomposition of cellulose, hcmicellulose and lignin [26], the Broido-Shafi2adeh model [30] and the one proposed by Di Blasi [31]. None of them can describe the composition of the volatiles and supplementary schemes have to be found. 

As already mentioned, the effects of several parameters are often lumped into one (see also Section 6.5.3.1). In this case, all band broadening effects are included in a dispersion coefficient. The so-called apparent dispersion coefficient Dapp is used here to distinguish from the axial dispersion coefficient, Dm, which is assumed to be independent of concentration and only influenced by the quality of the packing. The lumped parameter Dapp includes peak broadening effects caused by the fluid dynamics of the packing (axial dispersion), as well as by all other mass transfer effects that might occur, and was first introduced by van Deemter et al. (1956). 

The other subgroup of the lumped rate approach consists of the reaction dispersive model where the adsorption kinetic is the rate-limiting step. It is an extension of the reaction model (Section 6.2.4.3). Like the mass transfer coefficient in the transport dispersive model, the adsorption and desorption rate constants are considered as effective lumped parameters, kads,eff and kdes.eff- Since no film transfer resistance exists (Cpi = q), the model can be described by Eq. 6.79 ... 

The drawback of this approach compared with that described in Section 6.5.2 is that all errors are lumped into the isotherm parameters rather than the effective mass transfer coefficient, because either the wrong column or isotherm model is chosen. This approach is thus recommended to get a quick first idea of system behavior using only little amounts of sample, and not for a complete analysis, especially if binary mixtures with component interactions are investigated. The significance of the results decreases even further if some plant and packing parameters are only guessed or even neglected. 

In the framework of the TDM model, the transport coefficient is the last parameter to be determined according to Fig. 6.9. All prior experimental errors and model inaccuracies are lumped into this parameter. In addition it cannot be excluded that the mass transfer depends on concentration because of surface diffusion or adsorption kinetics. However, in many cases, e.g. for the target solutes discussed in this book, the transfer coefficient can be assumed to be independent of operating conditions (especially flow rate) with reasonable accuracy. 

The GRM is the most comprehensive model of chromatography. In principle, it is the most realistic model since it takes into account all the phenomena that may have any influence on the band profiles. However, it is the most complicated model and its use is warranted only when the mass transfer kinetics is slow. Its application requires the independent determination of many parameters that are often not accessible by independent methods. Deriving them by parameter identification may be acceptable in practical cases but is not easy since it requires the acquisition of accurate band profile data in a wide range of experimental conditions. This explains why the GRM is not as popular as the equilibrium-dispersive or the lumped kinetic models. 

In the equilibrium-dispersive model, we assume that the mobile and the stationary phases are constantly in equilibrium. We recognize, however, that band dispersion takes place in the column through axial dispersion and nonequilibrium effects e.g., mass transfer resistances, finite kinetics of adsorption-desorption). We assume that their contributions can be lumped together in an apparent dispersion coefficient. This coefficient is related to the experimental parameters by... 

Henry s law (i.e., c = po/H) relates the equilibrium oxygen solubility in the liquid (c ) to oxygen partial pressure in the gas (po) by the corresponding Henry s law constant (H). Because the liquid film mass transfer coefficient, l, is difficult to measure independently, kj a (i.e., times a) is used. It is a lumped parameter known as the volumetric mass transfer coefficient to characterize the overall mass transfer rate. 

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