The lumped parameter model

In deriving the lumped parameter model, it is customary to let E be the nominal exposure energy (i.e., intensity times the exposure time), 7(x) be the normalized image intensity, and I z) be the relative intensity variation with depth into the resist. The exposure energy as a function of position within the resist E z is 

Equation (12.126) can be integrated over the development time. If s So the remaining thickness is by definition zero, such that 

Accordingly, the effective resist thickness for the case of absorption only causing a variation in intensity with depth in the resist [i.e., the case where I(z) decays exponentially] can he calculated from Eq. (12.128) as 

If the resist is only slightly absorbing, such that a-yD g 1, the exponential can be approximated by the first few terms in its Taylor series expansion as 

And from Eq. (12.130), it can be seen that the effect of absorption is to make the resist appear thicker than it actually is during the development process. 

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The lumped parameter model of Example 13.9 takes no account of hydrodynamics and predicts stable operation in regions where the velocity profile is elongated to the point of instability. It also overestimates conversion in the stable regions. The next example illustrates the computations that are needed... 

During the flight of droplets in the spray, the forced convective and radiative heat exchanges with the atomization gas lead to a rapid heat extraction from the droplets. A droplet undergoing cooling and phase change may experience three states (a) fully liquid, (b) semisolid, and (c) fully solid. If the Biot number of a droplet in all three states is smaller than 0.1, the lumped parameter model 1561 can be used for the calculation of droplet temperature. Otherwise, the distributed parameter model 1541 should be used. 

In the lumped parameter model, the transient temperature of a single droplet during flight in a high speed atomization gas is calculated using the modified Newton s law of cooling, 1561 considering the frictional heat produced by the violent gas-droplet interactions due... 

The lumped-parameter model approach becomes particularly useful when dealing with the plasticating extrusion process discussed in the next subsection, where, in addition to melt flow, we are faced with the elementary steps of solids handling and melting. 

Consider the dynamic behavior of a process that can be considered perfectly mixed. The lumped parameter model has the following form ... 

Within the framework of the lumped-parameter models, it is relatively simple to derive information about the mixing regime from multi-tracer data. As discussed above for the H- Kr tracer pair, the predicted concentrations for any tracer can be calculated for different lumped-parameter models with various parameter values, and the model that best fits the data can be found. This inverse modeling procedure can be illustrated by plotting the predicted output concentrations for any pair of tracers versus each other (Fig. 20). By plotting the measured concentrations in such a figure containing a series of curves that represent different models and parameter values, the curve that best fits the... 

The most widely used development rate models are the kinetic development rate model, enhanced kinetic development rate model,and the lumped parameter model ° proposed hy Mack. We briefly outline their derivation here. ... 

The integral form of the lumped parameter model can be derived by applying the definition of the development rate to Eq. (12.136) or solving for the slope in Eq. (12.138) to yield... 

Equation (12.140) is the integral form of the lumped parameter model. With this equation, it is possible to generate a normalized CD versus exposure curve, as long... 

A good measure of the resist profile is the sidewall angle, which can be predicted with the lumped parameter model. To derive an expression for the sidewall slope, it is customary to rewrite Eq. (12.135) in terms of development rate as... 

The lumped parameter models are useful because they simplify the theory and the coefficients can often be estimated with reasonable accuracy. The most common way to determine 1 q is with a sum of resistances approach fRuthven et al.. 19941 that is similar to the approach used in Section 15.1. 

The advantage of a lumped parameter model over a distributed parameter model, is that the lumped parameter model is much easier to solve. The modeler should ensure, however, that the lumped parameter representation is an adequate approximation of the trae process behaviour. It may be difficult to determine a-priori whether a Imnped parameter description is valid or not. There are, however, some criteria that can help the modeler, sueh as the Peclet number in systems with mass transport (Westerterp et al, 1984). 

The models mentioned earlier are limited to single cell. McKay et al. [29] has developed a two-phase isothermal ID model of reactant and water dynamics. It is validated nsing a multicell stack. The lumped parameter model depends on six tunable parameters associated with the estimation of voltage, the membrane water v or transportation, and the accumulation of liquid water in the gas channels. The water flooding fault is embedded in this model by the assumption of liquid water layer of uniform thickness at the GDL channel interface. This water layer spreads across the GDL surfece as the liquid water volume in the channel increases, thus, reducing the surface area. This increases the calculated current density that will reduce the cell voltage at a fixed total stack current. 

This chapter discusses the two common models used to describe diffusion and suggests how you can choose between these models. For fundamental studies where you want to know concentration versus position and time, use diffusion coefficients. For practical problems where you want to use one experiment to tell how a similar one will behave, use mass transfer coefficients. The former approach is the distributed-parameter model used in chemistry, and the latter is the lumped-parameter model used in engineering. Both approaches are used in medicine and biology, but not always explicitly. 

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