Models with Dispersive Variability

RD models have certain limitations in applications to the real world. For example, members of a population do not necessarily disperse in the same way, there is always some variability. To take this fact into account. Cook considered that the population explicitly consists of distinct subpopulations, namely dispersers with density py and nondispersers with density P2- These subpopulations are allowed to have different birth rates. They interbreed fully, and all newborns have the same probability of being a disperser [309]. The evolution equations for the RD system are given by 

We have assumed logistic population growth, with K being the carrying capacity of the environment. The probability of a newborn being a disperser is /u. = rj/(rj +r2 . Note that if rj 0, the whole population disperses and (7.17) becomes the standard RD equation. Since the kinetics satisfy the KPP criteria, we can calculate the front velocity from the Hamilton-Jacobi formalism. The corresponding Hamiltonian is 

Consider the evolution equations for both populations in the general form [292] 

To calculate the front velocity for (7.21) we reduce the system to a Hamilton-Jacobi equation. To do so, we employ the hyperbolic scaling (4.33), the exponential transformation (4.35), the large scale limit e 0, and the definitions in (4.38) to obtain the Hamilton-Jacobi equation 

The hyperbolic extension to Cook s models was proposed in [181]. The system of RT equations is 

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In particular cases simplified reactor models can be obtained neglecting the insignificant terms in the governing microscopic equations (without averaging in space) [9]. For axisymmetrical tubular reactors, the species mass and heat balances are written in cylindrical coordinates. Himelblau and Bischoff [9] give a list of simplified models that might be used to describe tubular reactors with steady-state turbulent flow. A representative model, with radially variable velocity profile, and axial- and radial dispersion coefficients, is given below ... 

Liquid-liquid extraction is carried out either (1) in a series of well-mixed vessels or stages (well-mixed tanks or in plate column), or (2) in a continuous process, such as a spray column, packed column, or rotating disk column. If the process model is to be represented with integer variables, as in a staged process, MILNP (Glanz and Stichlmair, 1997) or one of the methods described in Chapters 9 and 10 can be employed. This example focuses on optimization in which the model is composed of two first-order, steady-state differential equations (a plug flow model). A similar treatment can be applied to an axial dispersion model. 

For adsorption rate, LeVan considered four models axial dispersion (this is not really a rate model but rather a flow model), external mass transfer, linear driving force approximation (LDF) and reaction kinetics. The purpose of this development was to restore these very compact equations with the variables of Wheeler equation for comparison. 

The model is referred to as a dispersion model, and the value of the dispersion coefficient De is determined empirically based on correlations or experimental data. In a case where Eq. (19-21) is converted to dimensionless variables, the coefficient of the second derivative is referred to as the Peclet number (Pe = uL/De), where L is the reactor length and u is the linear velocity. For plug flow, De = 0 (Pe ) while for a CSTR, De = oo (Pe = 0). To solve Eq. (19-21), one initial condition and two boundary conditions are needed. The closed-ends boundary conditions are uC0 = (uC — DedC/dL)L=o and (dC/BL)i = i = 0 (e.g., see Wen and Fan, Models for Flow Systems in Chemical Reactors, Marcel Dekker, 1975). Figure 19-2 shows the performance of a tubular reactor with dispersion compared to that of a plug flow reactor. 

In summary, the microscale description provides two important pieces of information needed for the development of mesoscale models. First, the mathematical formulation of the microscale model, which includes all of the relevant physics needed to completely describe a disperse multiphase flow, provides valuable insights into what mesoscale variables are needed and how these variables interact with each other at the mesoscale. These insights are used to formulate a mesoscale model. Second, the detailed numerical solutions from the microscale model are directly used for validation of a proposed mesoscale model. When significant deviations between the mesoscale model predictions and the microscale simulations are observed, these differences lead to a reformulation of the mesoscale model in order to improve the physical description. Note that it is important to remember that this validation step should be done by comparing exact solutions to the mesoscale model with the microscale results, not approximate solutions that result... 

Natural systems rarely contain perfectly uniform, regular voids. A regular shape model may be unrealistic for macropores and fractures with irregular walls. Thus, it is useful to examine the impact of systematic variations in channel diameter on solute dispersion. Variable shape models are attractive for simulating pore scale dispersion because a single unit cell is often able to capture a wide range of transport processes, from convection in the center of the channel to diffusion in backwater zones near the apex (see Fig. 3-2B). Furthermore, the macroscopic behavior of such models can be predicted from well-defined geometric parameters. 

These are systems where the state variables are varying in one or more directions of the space coordinates. The simplest chemical reaction engineering example is the plug flow reactor. These systems are described at steady state either by an ordinary differential equation (where the variation of the state variables is only in one direction of the space coordinates, i.e. one dimensional models, and the independent variable is this space direction), or partial differential equations (when the variation of the state variables is in more than one direction of the space coordinates, i.e. two dimensional models, and the independent variables are these space directions). The ordinary differential equations of the steady state of the one-dimensional distributed model can be either initial value differential equations (e.g. plug flow models) or two-point boundary value differential equations (e.g. models with superimposed axial dispersion). The equations describing the unsteady state of distributed models are invariably partial difierential equations. 

It is desirable at this point to try to draw together a few of the threads that have permeated the previous discussion, in order to give them the prominence they deserve. We have seen that the product distributions and rates of reaction of the higher alkanes with hydrogen are dependent upon operational variables, especially temperature, reactant pressures, time-on-stream and the state of the surface, as well as on the nature of the metal, its support (if any) and its dispersion. Unfortunately the variables that are controllable, namely temperature and reactant pressures, do not give results that are immediately suitable for modelling, because these variables, and others, also affect the coverage of the surface by unreactive... 

For the dynamic simulation of the SMB-SFC process a plug-flow model with axial dispersion and linear mass-transfer resistance was used. The solution of the resulting mass-balance equations was performed with a finite difference method first developed by Rouchon et al. [69] and adapted to the conditions of the SMB process by Kniep et al. [70]. The pressure drop in the columns is calculated with the Darcy equation. The equation of state from Span and Wagner [60] is used to calculate the mobile phase density. The density of the mobile phase is considered variable. 

Until now, most evaluation studies and model intercomparisons relied on comparison of the outputs of models with observations or other model results. However, there is a limited amount of observations of sufficient quality compared to the complexity of the phenomena, not to mention the inherent variability of atmospheric dispersion, which makes it difficult to measure. Therefore, such validation studies alone do not provide sufficient proof of a model s quality and its capabilities to address problems outside the range of the validation data sets. A scientific assessment of the model is an important means to obtain information about the capabilities, reliability, and quality of a model. The scientific assessment seeks ... 

Very recently, Sorrentino et proposed a model that predicts effective diffusivity in heterogeneous systems with dispersed impermeable domains of variable orientation and distribution. In addition to the sinuous pathways... 

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