Full scalar model

In the full scalar model, the full light intensity inside the resist is calculated, subject to the standard scalar approximation, involving the requirement that the three components of the electric field vector be treated separately as scalar quantities, with each scalar electric field component satisfying the wave equation. In addition, when two fields of light (for example, two plane waves) are added together, the scalar approximation dictates that the sum of the field would simply be the sum of the scalar amplitudes of the two fields. Implicit in the scalar approximation is... 

The effect of turbulence on scalars in the flow (c, T, reaction kinetics) is strong, and is sensitive to the details of the velocity and turbulence fields. Models that have been formulated to solve the combination of velocity and scalar fields have not yet accounted for the multiplicity of interactions between the fields, especially when complex reaction kinetics exist. Steady progress continues in the application of full PDF models to these problems. 

For instantaneous reactions the problem is thus reduced to the calculation of the presumed PDF of a passive scalar or tracer. A large number of alternative presumed PDFs have been listed and discussed by [2, 60, 67]. Each presumed PDF has its advantages and disadvantages, but none of them are generally applicable. The concept of the full PDF approaches is to formulate and solve additional transport equations for the PDFs determining the evolution of turbulent flows with chemical reactions. These models thus require modeling and solution of additional balance equations for the one-point joint velocity-composition PDF. The full PDF models are thus much more CPU intensive than the moment closures and hardly tractable for process engineering calculations. These theories are comprehensive and well covered by others (e.g., [2, 8, 26]), thus these closures are not examined further in this book. For Unite rate chemical reaction processes neither of the asymptotic simplifications explained above are applicable and appropriate elosures for 5c (w) are very difficult to achieve. 

However, if the correlation matrix p is rank-deficient, but the scalar dissipation matrix is full rank, the IEM model cannot predict the increase in rank of p due to molecular diffusion. In other words, the last term on the right-hand side of (6.105), p. 278, due to the diffusion term in the FP model will not be present in the IEM model. The GIEM model violates the strong independence condition proposed by Pope (1983). However, since in binary mixing the scalar fields are correlated with the mixture fraction, it does satisfy the weak independence condition. The expected value on the left-hand side is with respect to the joint PDF (c, f x, t). 

For most numerically solved models, a control-volume approach is used. This approach is based on dividing the modeling domain into a mesh. Between mesh points, there are finite elements or boxes. Using Taylor series expansions, the governing equations are cast in finite-difference form. Next, the equations for the two half-boxes on either side of a mesh point are set equal to each other hence, mass is rigorously conserved. This approach requires that all vectors be defined at half-mesh points, all scalars at full-mesh points, and all reaction rates at quarter-mesh points. The exact details of the numerical methods can be found elsewhere (for example, see ref 273) and are not the purview of this review article. The above approach is essentially the same as that used in CFD packages (e.g.. Fluent) or discussed in Appendix C of ref 139 and is related to other numerical methods applied to fuel-cell modeling. ... 

In full-scale Are modeling, a diffusion flame structure is usually assumed. However, in many fire situations, such as underventilated fires, premixed or partially premixed flame theory may be more appropriate. The Burke-Schumann description of the diffusion flames can be used to conveniently represent the transport of gaseous species by a single scalar quantity called mixture fraction. For a simple one-step reaction ... 

However, as discussed in chap 1.2.7, the gradient-diffusion models can fail because counter-gradient (or up>-gradient) transport may occur in certain occasions [15, 85], hence a full second-order closure for the scalar flux (1.468) can be a more accurate but costly alternative (e.g., [2, 78]). 

We regard Eq.(7.1.1) as a set of constraints the variable z has to obey, thus as (a part of) the model of some physical (technological) process. So if the condition (7.1.2) is not obeyed then the model is simply wrong, having no solution. Let thus the condition be satisfied. If we have AT < M then certain (M-M ) rows of matrix (C, c), thus certain scalar equations are linear combinations of the remaining M ones, and can be deleted. This done, the matrix of the new system is of full row rank. In what follows, let us suppose that linearly dependent equations have been deleted a priori, hence assume... 

Using Eq. (11) or (12) as controllability measures has three drawbacks First, Eq. (11) results from assuming a full block input imcertainty with a scalar weighting function w,. This description can be very conservative. To reduce this conservativeness, the system should be scaled before the determination of the weighting function w,. This is to some extent equivalent to substituting y(G) by y ((3), the minimized condition number, in (11). Secondly, the desired performance is indirectly represented in Eq. (11) by the weighting function Wj. Finally, it can be applied only for minimum phase models. 

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