Well mixed cell model

Well-Mixed Cell Model. A conceptually simple approach is based on the representation of the airshed by a three-dimensional array of well-mixed vessels (34, 35, 36). As before, we assume that the airshed has been divided into an array of L cells. Instead of using the array simply as a tool in the finite-difference solution of the continuity equations, let us now assume that each of these cells is actually a well-mixed reactor with inflows and outflows between adjacent cells. If we neglect diffusive transport across the boundaries of the cells and consider only convective transport among cells, a mass balance on species i in cell k is given by... 

Therefore, the well-mixed cell model can also be described as the result of the finite difference approximation of the spatial derivatives of (7)— i.e., of the conservation equations in which diffusion has been neglected. 

MacCracken et al. (36) have applied the well-mixed cell model in describing pollutant transport and dispersion in the San Francisco Bay Area. 

The plate models assume that the column is divided into a series of an arbitrary number of identical equilibrium stages, or theoretical plates, and that the mobile and the stationary phases in each of these successive plates are in equihbrimn. The plate models are in essence approximate, empirical models because they depict a continuous column of length I by a discrete number of well-mixed cells. Although any mixing mechanism is dearly absent from the actual physical system, plate models have been used successfully to characterize the column operation physically and mathematically. Therefore, by nature, plate models are empirical ones, which cannot be related to first principles. 

The coefficient representing axial dispersion, E, is measured using a tracer by pulse, sinusoidal, or step-change residence time distribution tests, or by measuring backflow. Sometimes the phenomenon is represented by a cefl model, in which the number of well-mixed cells fits dispersion. The coefficient representing radial dispersion, is determined by measuring the radial spread of a tracer from the centerline toward the wall. 

The approximation of the river as a series of discrete well-mixed cells introduces additional dispersion into the model. Even if a value of = 0 is input, some dispersion wUl still be predicted by the model. Banks (1974) developed a mixed cell model which may be used to quantify this numerical dispersion ... 

Fig 10 I Concept of the theoielical plate model Well mixed m each cell and mobile phase IS in equilibrium with stationary phase... 

The horizontal dispersion of a plume has been modeled by the use of expanding cells well mixed vertically, with the chemistry calculated for each cell (31). The resulting simulation of transformation of NO to NO2 in a power plant plume by infusion of atmospheric ozone is a peaked distribution of NO2 that resembles a plume of the primary pollutants, SO2 and NO. The ozone distribution shows depletion across the plume, with maximum depletion in the center at 20 min travel time from the source, but relatively uniform ozone concentrations back to initial levels at travel distances 1 h from the source. 

Mass and energy transport occur throughout all of the various sandwich layers. These processes, along with electrochemical kinetics, are key in describing how fuel cells function. In this section, thermal transport is not considered, and all of the models discussed are isothermal and at steady state. Some other assumptions include local equilibrium, well-mixed gas channels, and ideal-gas behavior. The section is outlined as follows. First, the general fundamental equations are presented. This is followed by an examination of the various models for the fuel-cell sandwich in terms of the layers shown in Figure 5. Finally, the interplay between the various layers and the results of sandwich models are discussed. 

Here we focus on the issue of how to build computational models of biochemical reaction systems. The two foci of the chapter are on modeling chemical kinetics in well mixed systems using ordinary differential equations and on introducing the basic mathematics of the processes that transport material into and out of (and within) cells and tissues. The tools of chemical kinetics and mass transport are essential components in the toolbox for simulation and analysis of living biochemical systems. 

When biochemical systems are studied in vitro, it is typically under well mixed conditions. Yet the contents of living cells are not necessarily well mixed and the biochemical workings within cells are inseparably coupled to the processes that transport material into, out of, and within cells. The three processes responsible for mass transport in living systems are advection, diffusion, and drift. Characterizing which, if any, of these processes is active in a given system is an important component of building differential equation-based models of living biochemical systems. 

We divide the airshed models discussed here into two basic categories, moving cell models and fixed coordinate models. In the moving cell approach a hypothetical column of air, which may or may not be well mixed vertically, is followed through the airshed as it is advected by the wind. Pollutants are injected into the column at its base, and chemical reactions may take place within the column. In the fixed coordinate approach the airshed is divided into a three-dimensional grid. 

The authors have used this intermediate approach to treat power plants in the Los Angeles Basin modeling study (50). Some of these plants are situated along the coastline, and their emissions are advected across the Basin under prevailing wind conditions. Typically emissions from these plants travel a horizontal distance of 2-5 miles before they are considered well-mixed in the vertical. Since an individual cell is 2 miles by 2 miles and horizontal dispersion of the plume under low winds extends about 2 miles after a 2-5 mile traverse, the assumption of approximately uniform distribution immediately downwind of the source is reasonable. A computational scheme for apportioning emissions among cells downwind of the source under these circumstances is described by Roberts et al. (50). 

The equivalent TMB model can also be considered as a series of well-mixing tanks [108-113]. The bed volume, V, is considered as equivalent to a certain number of theoretical stages (N), each stage being considered as an ideal mixing cell of volume V/N distributed between the fluid and the stationary phases, in accordance with the bed phase ratio, with... 

For models of the phytoplankton populations in coastal oceanic waters and in lakes, the sinking rate of phytoplankton cells is an important contribution to the mortality of the population. The cells have a net downward velocity, and they eventually sink out of the euphotic zone to the bottom of the water body. This mechanism has been investigated and included in phytoplankton population models (5,12). However, for the application of these equations to a relatively shallow vertically well mixed river or estuary, the degree of vertical turbulence is sufficient to eliminate the effect of sinking on the vertical distribution of phytoplankton. 

To connect the two markedly different scenarios observed in the static and the well-mixed environments, it is natural to analyze the role of increasing mobility (Reichenbach et al., 2007). Karolyi et al. (2005) studied the above competition model combined with dispersion by a chaotic map that represents advection of fluid elements in the alternating sine-flow. By continuously changing the frequency of the chaotic dispersion as a control parameter, it is possible to follow the transitions between the two limiting situations. When the chaotic mixing is much faster than the local population dynamics, the killer and resistant cells gradually disappear from the population and only the sensitive cells survive. This is because the killer cells... 

Since nonisothermality in tubular reactors often leads to radial as well as axial gradients, and since the mixing-cell model in its one-dimensional form is not very convenient anyway, it seems logical to see what a two-dimensional mixing-cell model might entail. 

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