First-order scalar model

Independently proposed hy Mack and Bernard, this model accounts for the effect of a changing aerial image focus (i.e., defocus effect) as it propagates 

Optical lithography modeling, in Microlithography Science and Technology, J.R. Sheats B.W. Smith, Eds., p. 120, Marcel Dekker, New York (1998). 

The expression for the intensity of the standing waves for the normal incidence condition is provided in Section 12.7. 

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Similar techniques have been employed to derive ASM models for turbulent reacting flows (Adumitroaie et al. 1997). It can be noted from (3.102) on p. 84 that the chemical source term will affect the scalar flux. For example, for a scalar involved in a first-order... 

Figure 33 The temperature response function r of the first-order decomposition rate constants ki and ky.. The r-function is a scalar that relates k at any given temperature to its maximal rate at a reference temperature (Tref = 30 °C, see text for details). Also shown is the modeled fit of the Lloyd and Taylor equation (Equation (12),...

We illustrate the behavior for a first order transition between a vapor and a dense liquid in the framework of a simple Lennard-Jones model. The condensation of a vapor into a dense liquid upon cooling is a prototype of a phase transition that is characterized by a single scalar order parameter - the density, p. The thermodynamically conjugated field is the chemical potential, p. The qualitative features, however, are general and carry over to other types of phase coexistence, e.g., Sect. 3.4. 

Direct measurement of turbulent fluctuations in canopies was not possible until relatively recently so their first recourse was to simple, first-order eddy diffusivity or X -theory models of turbulent fluxes. Typically the vertical eddy flux of a scalar, C, or of mean momentum, that is windspeed, U, was written as3... 

The scalar models for calculating aerial image intensity neglects the effects of polarization of light. This is a legitimate approximation for numerical apertures less than 0.7. Scalar models are the simplest of the aerial models and come in three main types the zero-order, first-order, and high-NA scalar models. 

As seen in Chapter 2 for turbulent flow, the length-scale information needed to describe a homogeneous scalar field is contained in the scalar energy spectrum E k, t), which we will look at in some detail in Section 3.2. However, in order to gain valuable intuition into the essential physics of scalar mixing, we will look first at the relevant length scales of a turbulent scalar field, and we develop a simple phenomenological model valid for fully developed, statistically stationary turbulent flow. Readers interested in the detailed structure of the scalar fields in turbulent flow should have a look at the remarkable experimental data reported in Dahm et al. (1991), Buch and Dahm (1996) and Buch and Dahm (1998). 

The last term on the right-hand side is unclosed and represents scalar transport due to velocity fluctuations. The turbulent scalar flux ( , varies on length scales on the order of the turbulence integral scales Lu, and hence is independent of molecular properties (i.e., v and T).17 In a CFD calculation, this implies that the grid size needed to resolve (4.70) must be proportional to the integral scale, and not the Batchelor scale as required in DNS. In this section, we look at two types of models for the scalar flux. The first is an extension of turbulent-viscosity-based models to describe the scalar field, while the second is a second-order model that is used in conjunction with Reynolds-stress models. 

In order to understand the physical basis for turbulent-diffusivity-based models for the scalar flux, we first consider a homogeneous turbulent flow with zero mean velocity gradient18 and a uniform mean scalar gradient (Taylor 1921). In this flow, velocity fluctuations of characteristic size... 

The final expression needed to complete the SR model is a closure for eaD appearing in (A.20). In order to develop a model starting from (A.2), we will first introduce the scalar... 

Among experimental studies of chemical reactions in turbulent media, fast reactions in tubular reactors with multijet injection of reactants are very popular, since the first experiments of Mao and Toor (34) and Vassiliatos and Toor (35). Their data have been (and are still) extensively exploited for testing theoretical models, although one may ask if homogeneous isotropic turbulence was perfectly controlled in these experiments. In order to rule out this objection, a new series of experiments was recently performed by Bennani et al. (28, 29, 30, 36) in a 0.29 m i.d. tube eliminating the influence of boundary layers. Turbulence was created by a grid and carefully controlled by velocity fluctuation measurements. Previous studies (2) had confirmed that the decrease of Ig with a non-reacting species (passive scalar) obeys Corrsin s equation ... 

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