Lumped parameter model mass transfer

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... 

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. 

This chapter discusses the two common models used to describe diffusion and suggests how you can choose between these models. For fundamental studies where you want to know concentration versus position and time, use diffusion coefficients. For practical problems where you want to use one experiment to tell how a similar one will behave, use mass transfer coefficients. The former approach is the distributed-parameter model used in chemistry, and the latter is the lumped-parameter model used in engineering. Both approaches are used in medicine and biology, but not always explicitly. 

This chapter presents an alternative model for diffusion, one using mass transfer coefficients rather than diffusion coefficients. The model is most useful for mass transfer across phase boundaries. It assumes that large changes in the concentration occur only very near these boundaries and that the solutions far from the boundaries are well mixed. Such a description is called a lumped-parameter model. 

Mass transfer coefficients provide especially useful descriptions of diffusion in complex multiphase systems. They are basic to the analysis and design of industrial processes like absorption, extraction, and distillation. Mass transfer coefficients are not useful in chemistry when the focus is on chemical kinetics or chemical change. They are not useful in studies of the solid state, where concentrations vary with both position and time, and lumped-parameter models do not help much. 

Simultaneous heat and mass transfer, the subject of this chapter, is a complicated process. Analyzing this process to And simple but useful results depends on making effective approximations. The approximations exploit both the similar mathematics used for the processes and the similar numerical values of the transport coefficients. This can be true for both distributed and lumped-parameter models. More specifically, for gases, D and a are nearly equal, and k and h/pCp are very similar. 

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

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... 

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],... 

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. 

Considering the models in Table I, it follows that the response of model III-T will be more close to reality due to (i) the correct way the transfer phenomena in and between phases is set up, and (ii) radial gradients are taken into account. Therefore, the responses of the different models will be compared to that one. It is obvious that the different models can be derived from model III-T under certain assumptions. If the mass and heat transfer interfacial resistances are negligible, model I-T will be obtained and its response will be correct under these conditions. If the radial heat transfer is lumped into the fluid phase, model II-T will be obtained. This introduces an error in the set up of the heat balances, and the deviations of type II models responses will become larger when the radial heat flux across the solid phase becomes more important. On the other hand, the one-dimensional models are obtained from the integration on a cross section of the respective two-dimensional versions. In order to adequately compare the different models, the transfer parameters of the simplified models must be calculated from the basic transfer... 

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. 

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... 

In choosing between these two models, one needs to consider the specific process. The use of mass transfer coefficients represents a lumped, more global view of the many process parameters that contribute to the rate of transfer of a species from one phase to another, while diffusion coefficients are part of a more detailed model. The first gives a macroscopic view, while the latter gives a more microscopic view of a specific part of a process. For this reason, the second flux equation is a more engineering representation of a system. In addition, most separation processes involve complicated flow patterns, limiting the use of Pick s Law. A description of correlations to estimate values of k for various systems is contained in Appendix B. 

Analogous to the experimental approaches discussed in the previous section, mathematical models have been developed to describe mass transfer at all three levels—cellular, multi-cellular (spheroid), and tissue levels. For each level two approaches have been used—the lumped parameter and distributed parameter models. In the former approach, the region of interest is considered to be a perfectly mixed reactor or compartment. As a result, the concentration of each region has no spatial dependence. In the latter approach, a more detailed analysis of the mass transfer process leads to information on the spatial and/or temporal changes in concentrations. Models for single cells and spheroids were reviewed in Section III,A and are part of the tissue-level models (Jain, 1984) hence, we will focus here only on tissue-level models. 

The estimated model parameters are given in table 1. Note that the estimated model parameters cannot be considered to represent intrinsic kinetic constants. They represent lumped parameters which can be disguised by possible heat and mass transfer effects which are not accounted for in the model. 

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