Model simplification

In order to be able to obtain analytical solutions we must first simplify the balance equations. Although the balance equations are fundamental and rigorous, they are nonlinear, nonunique, complex and difficult to solve. In other words, they do not have a general solution and so far, only particular solutions for special problems have been found. 

Using the above relation, the original variable can be expressed in terms of the dimensionless variable and its characteristic value as, 

By substituting the new variables into the original equations we will acquire information that allows the simplification of a specific model. Length and time scales, for example, can lead to geometrical simplifications such as a reduction in dimensionality. 

Object submerged in a fluid. Consider an object with a characteristic length L and a thermal conductivity k that is submerged in a fluid of constant temperature Too and convection coefficient h (see Fig.5.9). If a heat balance is made on the surface of the object, it must be equivalent to the heat by conduction, i.e., 

The maximum value possible for the temperature gradient must be the difference between the central temperature, Tc, and the surface temperature, Ts, 

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The parameter term (k x) which is called Damkohler Number I, is dimensionless and is now the single governing parameter in the model. This results in a model simplification because originally the three parameters, x, k and Cao. all appeared in the model equation. 

A problem with the solution of initial-value differential equations is that they always have to be solved iteratively from the defined initial conditions. Each time a parameter value is changed, the solution has to be recalculated from scratch. When simulations involve uptake by root systems with different root orders and hence many different root radii, the calculations become prohibitive. An alternative approach is to try to solve the equations analytically, allowing the calculation of uptake at any time directly. This has proved difficult becau.se of the nonlinearity in the boundary condition, where the uptake depends on the solute concentration at the root-soil interface. Another approach is to seek relevant model simplifications that allow approximate analytical solutions to be obtained. 

As both phases occupy the full flow field concurrently, two sets of conservation equations correspond to these two phases and must be complemented by the set of interfacial jump conditions (discontinuities). A further topological law, relating the void fraction, a, to the phase variables, was needed to compensate for the loss of information due to model simplification (Boure, 1976). One assumption that is often used is the equality of the mean pressures of the two phases, ... 

Another potential model simplification involves assuming negligible energy accumulation in the gas phase as compared to that in the solid, which is equivalent to the earlier approximation [Eq. (66)] based on the relative magnitude of the energy accumulation in the gas and solid. For our system, the accumulation of energy in the solid is approximately 250 to 300 times that in the gas phase due to the relative thermal capacitance of the gas [Eq. (65)] and the similarity of the temporal behavior of the gas and catalyst temperatures (e.g., Fig. 19). Thus the accumulation term in the energy balance... 

As a further model simplification, a linearization of the film concentration profiles has been studied. This causes no significant changes in the simulation results and at the same time reduces the total number of equations by half and stabilizes the numerical solution (142). The assumption of chemical equilibrium in... 

Today, the most widely used model simplification in polymer processing simulation is the Hele-Shaw model [5], It applies to flows in "narrow" gaps such as injection mold filling, compression molding, some extrusion dies, extruders, bearings, etc. The major assumptions for the lubrication approximation are that the gap is small, such that h < . L, and that the gaps vary slowly such that... 

Although this methodology has the advantage of not requiring model simplifications, it is extremely time-consuming and cannot be used in a realtime optimization scheme. 

Fig. 3.40(a), top]. In situation II ( AG/ = AGj > Xc) the effect is completely lost since both reaction zones have exactly the same shape [Fig. 3.40(b), top]. Thus the initial ion distribution, even when it coincides with one of them, cannot be inside the other. As a consequence, only the descending (diffusion-controlled) branch of this dependence is seen in Figure 3.40(a) (bottom). Such a high sensitivity of the results to the shape and relative location of the ionization and recombination zones makes any model simplifications of these zones undesirable. 

A large number of investigations do not even consider the formation of the industrially important acrylic acid (Models I -III). The most detailed Model V, on the other hand, is too complex for a practical application. Investigations of model simplifications for industrially relevant catalysts are either nonexis-tant or lead to differing results (Models I-IV). 

Permeability is observed to vary enormously in what looks like homogeneous rock bodies, and it varies much more between the different rock types that constitute a studied system. Disregarding this large range of different prevailing permeabilities, mathematical models accommodate only single permeability values for entire case studies or for large sections of studied systems. Thus representative values are selected in the frame of a whole list of model simplifications. Such simplifications can lead to... 

To develop three-phase, HLM system model [1,2], the transport model simplification analysis, developed by Shagxu Hu [73] for the two-phase system, is used. 

A good understanding of the physical-chemical phenomena taking place in a process can lead to significant model simplifications for control purposes. Such simplifications can be done by excluding from the balances (model) those terms that have small contributions. 

The QSSA is a useful tool in reaction analysis. Material balances for hatch and plug-flow reactors are ordinary differential equations. By applying Equation 5.81 to the components that are QSSA species, their material balances become algebraic equations. These algebraic equations can be used to simplify the reaction expressions and reduce the number of equations that must be solved simultaneously. In addition, appropriate use of the QSSA can eliminate the need to know several difficult-to-measure rate constants. The required information can be reduced to ratios of certain rate constants, which can be more easily estimated from data. In the next section we show how the QSSA is used to develop a rate expression for the production of a component from a statement of the elementary reactions, and illustrate the kinetic model simplification that results from the QSSA model reduction.. 

In order to establish a kinetic model, a number of assumptions regarding the operation of the photocatalytic reactor should apply (Ibrahim, 2001). Consideration of the applicability of model assumptions is relevant for any type of photocatalytic reactor model. In the specific case of Photo-CREC-Air, and given the special design of this unit, the following model simplifications can be adopted ... 

Kreul et al. [16] used an NEQ model of homogeneous RD and, via a series of case studies, studied the importance of various model simplifications. They found little difference between the full MS description of multi-component mass transfer and... 

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