Distributed model, simplest

THE SIMPLEST DISTRIBUTED MODEL Example 1 The Tubular Reactor... 

Therefore, we must make assumptions to be able to arrive at simple models which are useful for fermenter design and performance predictions. Various models can be developed based on the assumptions concerning cell components and population as shown in Table 6.1 (Tsuchiya et al., 1966). The simplest model is the unstructured, distributed model which is based on the following two assumptions ... 

In the continuum solvent distribution models, Vei is evaluated by resorting to the description of the solvent as a dielectric medium. This medium may be modeled in many different ways, being the continuous methods quite flexible. We shall consider the simplest model only, i.e. an infinite linear isotropic dielectric, characterized by a scalar dielectric constant e. The interested reader can refer to a recent review (Tomasi and Persico, 1994) for the literature regarding more detailed and more specialistic models. However, the basic model we are considering here is sufficient to treat almost all chemical reactions occurring in bulk homogeneous solutions. 

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. 

With regard to the unsteady state behaviour, only the simplest distributed model based on Fickian diffusion with constant effective diffusivities will be considered in this section. However, two important phenomena which are usually neglected in the literature will be included in the unsteady state modelling because of their importance. These are the adsorption mass capacity of the porous catalyst surface and the heat of chemisorption accompanying the steps of the CSD process. 

Binomial distribution. The simplest common probability model used for binary outcomes from a series of independent trials. (Where tried is to be understood in the sense of the first meaning of the definition.) If n is the number of trials, 0 is the probability of success in an individual trial and X is the number of successes, then the probability mass function of the binomial distribution is... 

Poisson distribution. The simplest probability distribution which may be used to model the number of events occurring during a given time interval. (Named after the... 

Example 2.9 Probability Calculation from Exponential Distribution. The simplest distribution used to model the times to failure (lifetime) of items or survival times of patients is the exponential distribution. The p.d.f. of the exponential distribution is given by... 

Geometric Distribution The geometric distribution models the number of i.i.d. Bernoulli trials needed to obtain the first success. It is the simplest of the waiting time distributions, that is, the distribution of discrete time to an event, and is a special case of the negative binomial distribution (we will show this distribution in the next section). Here the number of required trials is the r.v. of interest. As an... 

In simple situations, queueing models can be applied to obtain preUminary results quickly. Ap-pUcation of queueing models to a situation requires that the assumptions of the model be met. These assumptions relate to the interarrival time and service time probabihty distributions. The simplest model assumes Poisson arrivals and exponential service times. These assumptions may be approxi-... 

The competition between reaction and diffusion can be represented by the lEM-model. t is identified with a diffusion time t = yL /33 (see Sec. 3.2 above) where different diffusivities 3j and hence different micromixing times t- may be used for each species. This simple lumped parameter model gives results comparable to those of more sophisticated distributed models, at least for reaction systems which are not too "stiff". An interesting property is revealed by numerical simulations. The simplest way to represent partial segregation in a fluid is to consider that it consists of a mixture of macrofluid (fraction 3) and microfluid (fraction 1-3). It turns out that the ratio (1 - 3)/3 is always close to that of two characteristic times [28 QlSj. In the case of erosive mixing of two reactants in a CSTR (erosion controls mixing and the product of erosion is a microfluid), one finds... 

Several distribution models have been developed to describe a toxicant s movement through the body. These models help estimate the concentration and duration of a toxicant in a living individual. The simplest distribution models treat the body as one compartment. This assumes that the chemical is spread evenly through the body and eliminated at a steady rate proportional to the amount left in the systan. In reality, few toxicants follow a one-compartment model. A two-compartment model is slightly more complex and usually more realistic. In this estimation, the first compartment clears the toxin from the body steadily with time, similarly to the one-compartment model. The toxicant concentration in a second compartment, however, rises as the first declines. This represents the toxicant s movonent fiom one area of the body to a second compartment, for example, from the blood to adipose tissue (body fat). The concentration in the second compartment also dechnes with time, but at a different rate. This concept can be expanded to multicompartment models that consider additional regions of the body. 

The simplest way of introducing Che pore size distribution into the model is to permit just two possible sizes--Tnlcropores and macropotes--and this simple pore size distribution is not wholly unrealistic, since pelleted materials are prepared by compressing powder particles which are themselves porous on a much smaller scale. The small pores within the powder grains are then the micropores, while the interstices between adjacent grains form the macropores. An early and well known model due to Wakao and Smith [32] represents such a material by the Idealized structure shown in Figure 8,2,... 

When experimental data is to be fit with a mathematical model, it is necessary to allow for the facd that the data has errors. The engineer is interested in finding the parameters in the model as well as the uncertainty in their determination. In the simplest case, the model is a hn-ear equation with only two parameters, and they are found by a least-squares minimization of the errors in fitting the data. Multiple regression is just hnear least squares applied with more terms. Nonlinear regression allows the parameters of the model to enter in a nonlinear fashion. The following description of maximum likehhood apphes to both linear and nonlinear least squares (Ref. 231). If each measurement point Uj has a measurement error Ayi that is independently random and distributed with a normal distribution about the true model y x) with standard deviation <7, then the probability of a data set is... 

Often in stress analysis we may be required to make simplified assumptions, and as a result, uneertainties or loss of aeeuraey are introdueed (Bury, 1975). The aeeuraey of ealeulation deereases as the eomplexity inereases from the simple ease, but ultimately the eomponent part will still break at its weakest seetion. Theoretieal failure formulae are devised under assumptions of ideal material homogeneity and isotropie behaviour. Homogeneous means that the materials properties are uniform throughout isotropie means that the material properties are independent of orientation or direetion. Only in the simplest of eases ean they furnish us with the eomplete solution of the stress distribution problem. In the majority of eases, engineers have to use approximate solutions and any of the real situations that arise are so eomplieated that they eannot be fully represented by a single mathematieal model (Gordon, 1991). 

In the CHS model only nearest neighbors interact, and the interactions between amphiphiles in the simplest version of the model are neglected. In the case of the oil-water symmetry only two parameters characterize the interactions b is the strength of the water-water (oil-oil) interaction, and c describes the interaction between water (oil) and an amphiphile. The interaction between amphiphiles and ordinary molecules is proportional to a scalar product between the orientation of the amphiphile and the distance between the particles. In Ref. 15 the CHS model is generalized, and M orientations of amphiphiles uniformly distributed over the sphere are considered, with M oo. Every lattice site is occupied either by an oil, water, or surfactant particle in an orientation ujf, there are thus 2 + M microscopic states at every lattice site. The microscopic density of the state i is p.(r) = 1(0) if the site r is (is not) occupied by the state i. We denote the sum and the difference of microscopic oil and water densities by and 2 respectively and the density of surfactant at a point r and an orientation by p (r) = p r,U(). The microscopic densities assume the values = 1,0, = 1,0 and 2 = ill 0- In close-packing case the total density of surfactant ps(r) is related to by p = Ylf Pi = 1 - i i. The Hamiltonian of this model has the following form [15]... 

Exponential distribution The exponential distribution is the simplest component life distribution. It is suited to model chance failures when the rate at which events occur remains constant over time. It is often suitable for the time between failures for repairable equipment. 

Tire simplest model for describing binary copolyinerization of two monomers, Ma and Mr, is the terminal model. The model has been applied to a vast number of systems and, in most cases, appears to give an adequate description of the overall copolymer composition at least for low conversions. The limitations of the terminal model generally only become obvious when attempting to describe the monomer sequence distribution or the polymerization kinetics. Even though the terminal model does not always provide an accurate description of the copolymerization process, it remains useful for making qualitative predictions, as a starting point for parameter estimation and it is simple to apply. 

The simplest kind of gridpoint model is one where only one spatial dimension is considered, most often the vertical. Such one-dimensional models are particularly useful when the conditions are horizontally homogeneous and the main transport occurs in the vertical direction. Examples of such situations are the vertical distribution of CO2 within the ocean (except for the downwelling regions in high latitudes, Sie-genthaler, 1983) and the vertical distribution of... 

Axial Dispersion. Rigorous models for residence time distributions require use of the convective diffusion equation. Equation (14.19). Such solutions, either analytical or numerical, are rather difficult. Example 15.4 solved the simplest possible version of the convective diffusion equation to determine the residence time distribution of a piston flow reactor. The derivation of W t) for parabolic flow was actually equivalent to solving... 

The conceptually simplest model, which for reasons explained later is called UNEQ, is based on the multivariate normal distribution. Suppose we have carried... 

The simplest model for the ionic distribution at liquid-liquid interfaces is the Verwey-Niessen model [10], which consists of two Gouy-Chapman space-charge layers back to... 

The distribution of distance h is characterized either by the most probable value /imax, defined by the condition dw(/i)/d/ = 0, or by the mean square of distance h, defined in the usual manner, h2 = J/i2w(/i) dh/ w(h) dh. The ratio (hmax)2/h2 gives the width of the distribution. The closer this value is to unity, the narrower the distribution. This ratio is equal to for the simplest model in statistics, the random walk . 

In its simplest form a partitioning model evaluates the distribution of a chemical between environmental compartments based on the thermodynamics of the system. The chemical will interact with its environment and tend to reach an equilibrium state among compartments. Hamaker(l) first used such an approach in attempting to calculate the percent of a chemical in the soil air in an air, water, solids soil system. The relationships between compartments were chemical equilibrium constants between the water and soil (soil partition coefficient) and between the water and air (Henry s Law constant). This model, as is true with all models of this type, assumes that all compartments are well mixed, at equilibrium, and are homogeneous. At this level the rates of movement between compartments and degradation rates within compartments are not considered. 

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