Separator model equations

The advantage of the MTC model, as opposed to the CRK model, is that the exchange rate constant kp is no longer an empirical constant, but is now defined in terms of more fundamental processes that can be separately modeled. Equations 3.1 and 3.12 are linked via the equality... 

Unfortunately, there are a few problems with this type of model. First, if there is more than one desired constituent in the calibration samples, each individual constituent requires solving a separate model equation. More important, a separate peak in the spectrum must be identified for each constituent. Remember that each peak must be completely isolated from interference from other constituents. This is not always practical, and in mixture samples this can be nearly impossible. Therefore, the use of least squares regression models tends to be restricted to pure or simple mixture samples. 

The grain- or multigrain model consists of separate model equations for characteristic grains or micro-particles and the macro-scale pellet. The characteristic micrograin is described by the following diffusion-reaction partial- differential equation for the product species s ... 

Continuation methods, also called imbedding and path-fallowing methods, were first applied to the solution of separation models involving large numbers of nonhnear equations by Salgovic, Hlavacek, and llavsky Eng. ScL, 36, 1599 (1981)] and by Byrne and... 

Kittrell et al. (1965a) also performed two types of estimation. First the data at each isotherm were used separately and subsequently all data were regressed simultaneously. The regression of the isothermal data was also done with linear least squares by linearizing the model equation. In Tables 16.7 and 16.8 the reported parameter estimates are given together with the reported standard error. Ayen and Peters (1962) have also reported values for the unknown parameters and they are given here in Table 16.9. 

The initial structure of the program is then followed by statements reflecting the dynamic model equations. These are also provided with comment lines with surrounding braces to distinguish them from the executable program lines. Note that the kinetic rate equations are expressed separately apart from the balance equations, to provide additional simplicity and additional flexibility. The kinetic rates are now additional variables in the simulation and the rates can... 

Note that, by construction, ( >2 = 0. For cases where no reactions occur in environment 2, )i is constant, and the transport equation for (5)2 is not needed. A separate model must be provided for the scalar dissipation rate e. ... 

Besides the fluorescence coefficient, the optical absorption and scattering coefficients of the sample are the most important parameters in quantative fluorescence spectroscopy of turbid media. In principle two or, if the anisotropy parameter has to be determined, three independent measurements are sufficient to separate the coefficients that appear in all equations as sums or proportions. However, for better accuracy, one of the geometrical parameters (sample thickness, angle of incidence, distance from the irradiated spot) as well as the wavelength of irradiation should be varied over a wide range, and then the data should be fitted with the help of the corresponding model equation. 

A thorough study on the ion-exchange mechanism and the effect of distinct counterions in this PO mode was recently presented by Gyimesi-Forras et al. [41]. A large variety of distinct acid additives to methanol, acetonitrile, and tetrahydrofuran (Table 1.1) (without any base added) was investigated in view of the stoichiometric displacement model and their effect on the enantiomer separation of 2-methoxy-2-(l-naphthyl)propionic acid. The stoichiometric displacement model (Equation 1.1) was obeyed also in the PO mode, as revealed by linear plots of log k vs. acid concentration. The slopes and intercepts along with the concentration ranges used with the distinct competitor acids are summarized in Table 1.1. 

The phase separation model for nonionic micelle formation has been modified for ionic micelle formation to give an equation close to Equation 12 for Ai (26.). ... 

One way to include the effect of gas-phase pressure-driven flow is to use eq 44 as a separate momentum equation.The models that do this are primarily CFD ones. Another way to include pressure-driven flow is to incorporate eq 44 into the Stefan—Maxwell equations, as per tihe dusty-gas... 

Indicator Variables Indicator variables are binary variables having a value of either 0 or 1, indicating if a particular structural or substructural characteristic is present or is missing in the molecular graph. Indicator variables are used by some authors in QSPR equations presenting correlations between a property and descriptors, including indicator variables. However, indicator variables are redundant. Instead of using an indicator, a QSPR model can be split into two separate models, one for all those compounds for which the indicator variable is zero, and one for all those compounds for which the indicator variable is 1. 

In this study the numerical simulations were performed with a 3-D mechanistic global Cologne Model of the Middle Atmosphere (COMMA) based on the primitive equations expressed in spherical coordinates for the horizontal and log-pressure coordinates in the vertical direction. The model equations are solved on the basis of an explicit numerical scheme (leapfrog) with a fixed time step of 450 sec. To avoid separate evolution at even and odd time steps, a Robert time filter is used. 

A Hammett equation was also established for substituted benzenes. A separate model was established for halogenated benzenes. It has been found that halogen substituents behave differently than substituents such as -CH3 and N02 (Hansch and Leo, 1995). For this reason, an accurate comparison of halogenated substituents and other substituents could not be made. 

Powers MF, Vickery DJ, Arehole A, Taylor R. A nonequilibrium-stage model of multicomponent separation processes—V. Computational methods for solving the model equations. Computers Chem Eng 1988 12 1229-1241. 

The phase separation model follows exactly the description of a two-phase equilibrium, i.e., equating the respective chemical potentials of the particular surfactant in both phases (i.e., monomers in the nonpolar solvent and the micelles) at the critical concentration (CMC). Thus, (assuming ideal condition)... 

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... 

In rate-based multistage separation models, separate balance equations are written for each distinct phase, and mass and heat transfer resistances are considered according to the two-film theory with explicit calculation of interfacial fluxes and film discretization for non-homogeneous film layer. The film model equations are combined with relevant diffusion and reaction kinetics and account for the specific features of electrolyte solution chemistry, electrolyte thermodynamics, and electroneutrality in the liquid phase. 

Mujtaba (1997) used the minimum time to evaluate the performance of continuous column operation under multiple separation duties. However, time does not explicitly appear in continuous column model equations but the feed rate is a measure of the batch time (t = BJF). Note, maximisation of the feed rate will therefore ensure minimisation of the batch time. 

The hydrocracker simulator was also converted to subroutine form for inclusion in the nonlinear programming model of the Toledo process complex. The subroutine was considerably simplified, however, to save computer time and memory. The major differences are (1) the fractionation section is represented by correlations instead of by a multi-stage separation model, (2) high pressure flash calculations use fixed equilibrium K-values instead of re-evaluating them as a function of composition, and (3) the beds in each reactor are treated as one isothermal bed, eliminating the need for heat balance equations. 

In this section we replace the CSTR by a plug-flow reactor and consider the conventional control structure. Section 4.5 presents the model equations. The energy balance equations can be discarded when the heat of reaction is negligible or when a control loop keeps constant reactor temperature manipulating, for example, the coolant flow rate. The model of the reactor/separation/recycle system can be solved analytically to obtain (the reader is encouraged to prove this) ... 

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