Profile optimum

Depending on the process conditions, different profiles of the active phase over the particle will be obtained. A completely uniform distribution of the active material over the particle is not always the optimum profile for impregnated catalysts. It is possible to purposely generate profiles in order to improve the catalyst performance. Fig. 3.28 shows four major types of active phase distribution in catalyst spheres. 

Once the optimum profile(s) has been established, its practicality for implementation must be assessed. For a continuous process, the equipment must be able to be designed such that the profile can be followed through space by adjusting rates of reaction, mass transfer, heat transfer, and so on. In a dynamic problem, a control system must be designed that will allow the profile to be followed through time. If the profile is not practical, then the optimization must be repeated with additional constraints added to avoid the impractical features. 

Table 3 shows the responses to be optimized, among which AUC was considered the most important. A constant release between 1 and 12 h (passing through 60% dissolved at 8 h) and a complete dissolution after 14 h were chosen for an optimum profile (AUC=713% h). 

Nonisothermal systems are accounted for by the introduction of temperature-control units into the generic reactor unit representation. These units consist of elements associated with the manipulation of temperature changes and constitute temperature profiles (profile-based approach) and heaters/coolers (unit-based approach). The assumption of thermal equilibrium between the contacting phases reduces the need for a single temperature per shadow reactor compartment. The profile-based system (PBS) finds the optimum profiles without considering the details of heat transfer mechanisms. Because the profiles are imposed rather than... 

As seen in the previous chapter, all the approaches used to solve the dynamie optimization problem integrate, at some point, the dynamical system of the ehemieal proeess. In order to obtain more effieiently the values of the optimum profile of the control variable, a suitable model of the system should be developed. That means that the complexity of the model should be limited, but, in the same time, the model should represent the plant behaviour with good accuracy. The best way to obtain such a model is by using the model reduction techniques. However, the use of a classical model reduction approach is not always able to lead to a solution [6]. And very often, the physical structure of the problem is destroyed. Thus, the procedure has to be performed taking into account the process knowledge (units, components, species etc.). 

The optimum profiles of the control variables (Figure 3) are obtained after several time-consuming, trial-and-error iterations. The solution was obtained after a number of about 150 manual iterations, not taking into aeeount the iterations performed by the solver. In each manual iteration, the initial profile was modified by the user, while the solver is trying to optimize this profile. The advantage of having a reduced model at this point is obvious. 

Further, the optimum profiles were implemented into Aspen Dynamics. The agreement between the responses of the nonlinear and reduced model is excellent (Figure 4). The difference between the reduced and the nonlinear model response is less than 2.3% at the end of the time span. 

Given an objective function to be maximized or minimized, it is possible, in principle, to compute an optimum profile of reflux ratio and distillate rate for the entire distillation cycle. The principles of such approaches are introduced in Section 17.2.4. 

Analogously, if peak profiles C are used as an input, iterations start with the calculations of the conjugated spectra SF In any case, the iterative calculations of C and are repeated until reaching the optimum profiles. Three stopping criteria have been defined as follows (i) reaching a convergence... 

A conical shape is by no means the optimum profile for the piston. The profile shown in Fig. 1.15(b) has been found to provide better support for the... 

Figure 9.19. Diffusion decay profiles. When the signal attenuation is too fast (traee (a)), the later data points do not contribute to the profile, when it is too slow (trace (c)) there is insufficient attenuation to provide an accurate determination of D. Traee (b) shows an optimum profile where all data points would eontribute to characterisation of the decay.

As a reinforcement, Kevlar 49 is in intense competition with carbon fibre. They are both much more costty than E-glass, and are considered only where their outstanding mechanical properties are really needed. Not surprisingly, there is currently increasing interest in hybrid composites. A combination of two or more types of reinforcement is used to produce an optimum profile of cost and properties for a given application. 

Figure 26.6 Simulated temperature profile in the reaction channel (solid line) compared with the optimum profile (dashed line) [29],...

Figures 3, 4 and 5 present selected optimum profiles of the pellet size R (r), te[0,T, for fixed value of R ax and various R and selected profiles of conversion degree and relating to them profiles of overall rate of process, r T).

The optimum profile of the characteristic pellet size is a non decreasing function of the space time. 

For small deformations, we expand P e - -u — ( ) out to second order in (u — C)- The zero-order term gives a constant. The first-order term drops out since the spatial average of (u — C) is zero if the volume of the liquid is conserved. The second-order term yields the following equation for the optimum profile ... 

While the fabrication of the optimum profiles described above requires the use of isotropic wet etching techniques, dry etching methods may also be used to produce geometries that minimize band broadening in pressure-driven flow systems. In this case, a shallow etch in the side-regions by an amount Ad = ad followed by a deep etch of the entire channel by a distance d yields a cross sec-... 

The optimum profile exponent p, which minimizes the modal dispersion and the difference in the delay of all the modes and maximizes the bandwidth, is expressed as follows on the basis of an analysis of scalar wave equation ... 

Toxvaerd minimizes the free energy of a system with a planar surface, with respect to the parameters of a trial function for p(z), and subject to the constraints of fixed gas and liquid densities, p( o°), and of fixed number of molecules, which is achieved by a suitable condition on the position of the equimolar dividing surface. His trial function for p(z) comprises a constant term, Kp +P ). md two tanh functions with arguments [2A, g(z-5)], where A is used for zb, and where i = (Ar -A, )ln2. Clearly there is a discontinuity in the derivative p (z) at z h, but this is imperceptible in his published graphs. From the optimum profile and the minimized free energy he calculates the surface tension as the surface excess free energy with respect to the equimolar dividing surface. 

Cherry BW, Harrison NL (1970) The optimum profile for a lap joint. J Adhes 2 125... 

The approximations in these equations are for weakly guiding fibers and paraxial rays. Hence the optimum profile is close to parabolic. The pulse width is a factor of A/8, or dl/16, times that for the step profile, in Eq. (3-3), and is therefore considerably reduced. Since l/tj is one measure of the informationcarrying capacity of a waveguide, we deduce that capacity is increased by a factor of 8/A, or 16/0. We plot tj of Eq. (3-7) as the normalized time ctjzn o against q, corresponding to the solid curve in Fig. 3-3, for A = 0.01 or 0c = 0.14. There is a cusp at q pt, which means that ray dispersion is very sensitive to small variations about qopt- For example, when q = opt the pulse width increases by a factor of nearly 10. The normalized pulse width for a step profile with the same value of A is included for comparison. 

Fig. 3-3 The solid curve is the normalized pulse width ctj/2 co of E<1-(3-7) for clad power-law profiles as a function of q, and the horizontal line is the corresponding value of Eq. (3-2) for the step profile. When the fiber materials are dispersive, the dashed curve plots the normalized pulse width cti/zn, which shifts the optimum profile exponent from g p, s 1.98 to gopt -I- gopt = 2.26, assuming weak guidance [4]. For all three plots A = 0.01, or 6c = 0.14.

We showed in Section 2-13 that the transit time for a step-profile fiber is independent of the cross-sectional geometry. Consequently Eqs. (3-2) and (3-3) give the ray dispersion for step-profile fibers of arbitrary cross-section. We also found in Section 2-13 that the ray transit time for the noncircular, clad power-law profiles of Eq. (2-55) is identical to the transit time for the symmetric, clad power-law profiles in Table 2-1, page 40, i.e. dependent on only. Thus Eqs. (3-8) and (3-9) also give the optimum profile and minimum pulse spread for those noncircular profiles [5], which includes the clad parabolic-profile fiber of elliptical cross-section. In other words, ray dispersion on step-profilefibers of arbitrary cross-section and clad power-law profilefibers of noncircular cross-section is also given by the corresponding solutions for planar waveguides. 

---