Dynamic behavior

Calculations were made for a reactor containing a bundle of 2,500 1-in. tubes packed with catalyst pellets and surrounded by molten salt to absorb the heat of reaction. The calculations showed that even with tubes of only- 1 -i n. diam5reT7 raiitat remp erature gradrentS -are-severe-for-thia-extTemely-exothermic system. 

A thorough study of a nonisothermal fixed-bed reactor for the reaction 

Thus far we have considered only steady-state operation of fixed-bed reactors. The response to variations in feed composition, temperature, or flow rate is also of significance. The dynamic response of the reactor to these involuntary disturbances determines the control instrumentation to be used. Also, if the system is to be put on closed-loop computer control, a knowledge of the response characteristics is vital for developing a control policy. It is beyond the scope of this book to treat the dynamic behavior of reactors, but it is necessary to draw attention to the availability of information on the subject. 

The solution of the steady-state form of the mass- and energy-conservation equations for fixed-bed reactors has been found to be complex. When transient conditions are considered the situation becomes rather intractable. For the special case of isothermal operation only the mass balance is involved, and it is possible for many types of kinetics to 

Beek has reviewed the stability criterion of Barkelew, which is applicable for several rate expressions, including second-order, auto-catalytic, and product-inhibited ones. A simpler, but, unnecessarily conservative, proposal is that of Wilson, developed in connection with the study of the hydrogenation of nitrobenzene (Example 13-4). This criterion states that instability cannot occur if 

The strong parametric sensitivity resulting from this coupling largely determines the start-up procedure, steady-state and transient operations (Ruiz et al., 1995). Moreover, even small variations in the input variables might lead to unstable conditions (Alejski and Duprat, 1996). This fact implies that a feasible operation of a RD column requires several control loops to mitigate the effects of disturbances. 

The content of this rather classical control approach sets the stage for a next novel contribution to be introduced in chapter 8, involving the trade-off between design, responsiveness and sustainability aspects. Thus, the outcome of this section serves as a reference case to be compared with that of the life-span inspired design methodology (c/. section 1.4). 

As in previous chapters, the synthesis of MTBE is used as case study. As the focus of this section is the d3mamic behavior of the unit around a desired steady state, we choose appropriate processing conditions and design parameters to avoid multiplicity as much as possible. The steady state of interest is the high-purity one as given elsewhere Hauan et al. (1995a,6) Jacobs and Krishna (1993) Schrans et al. (1996). 

2 Model Description, Control Structure and Disturbance Scenarios 

The RD model consists of sets of algebraic and differential equations, which are obtained from the mass, energy and momentum balances performed on each tray, reboiler, condenser, reflux drum and PI controller instances. Additionally, algebraic expressions are included to account for constitutive relations and to estimate physical properties of the components, plate hydraulics and column sizing. Moreover, initial values are included for each state variable. A detailed description of the mathematical model can be found in appendix A. The model is implemented in gPROMS /gOPT and solved using for the DAE a variable time step/variable order Backward Differentiation Formulae (BDF). 

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Maler L, Widmalm G and Kowalewski J 1996 Dynamical behavior of carbohydrates as studied by carbon-13 and proton nuclear spin relaxation J. Phys. Chem. 100 17 103-10... 

Pieranski P, Brochard F and Guyon E 1973 Static and dynamic behavior of a nematic liquid crystal in a magnetic field. Part II Dynamics J.Physique 34 35-48... 

Based on observations concerning the dynamical behavior we already conjectured that there exist seven almost invariant sets - a conjecture that we now want to check numerically. We employ the subdivision algorithm for subtrajectories of length mr = 0.1. The final box-collection corresponding to the total energy E = 4.5 after 18 subdivision steps consists of 18963 boxes. 

M. Dellnitz and O. Junge. On the approximation of complicated dynamical behavior. To appear in SIAM J. Num. Anal., 1998. Also available as Preprint SC 96-35, Konrad Zuse Zentrum, Berlin (1996)... 

A detailed examination of LN behavior is available [88] for the blocked alanine model, the proteins BPTI and lysozyme, and a large water system, compared to reference Langevin trajectories, in terms of energetic, geometric, and dynamic behavior. The middle timestep in LN can be considered an adjustable quantity (when force splitting is used), whose value does not significantly affect performance but does affect accuracy with respect to the reference trajectories. For example, we have used Atm = 3 fs for the proteins in vacuum, but 1 fs for the water system, where librational motions are rapid. 

To begin with, we compare the stepsizes used in the simulations (Fig. 3). As pointed out before, it seems to be unreasonable to equip the Pickaback scheme with a stepsize control, because, as we indeed observe in Fig. 3, the stepsize never increases above a given level. This level depends solely on the eigenvalues of the quantum Hamiltonian. When analyzing the other integrators, we observe that the stepsize control just adapts to the dynamical behavior of the classical subsystem. The internal (quantal) dynamics of the Hydrogen-Chlorine subsystem does not lead to stepsize reductions. 

To understand the importance of investigating the dynamical behavior of molecules... 

The Universal Modeling Language is used to describe a software system [4, 5], Several kinds of diagrams exist to model the diverse properties of the system. Thus a description of the system can be developed that enables the systematic and uniform documentation of the system. The class diagram, for example, represents the classes and their relationships. But also interacting diagrams exist, to describe the dynamic behavior of the system and its objects. 

Despite the very restricted circumstances In which these equations properly describe the dynamical behavior, they are the starting point for almost all the extensive literature on the stability of steady states in catalyst pellets. It is therefore Interesting to examine the case of a binary mixture at the opposite limit, where bulk diffusion controls, to see what form the dynamical equations should take in a coarsely porous pellet. 

Equilibrium Theory. The general features of the dynamic behavior may be understood without recourse to detailed calculations since the overall pattern of the response is governed by the form of the equiUbrium relationship rather than by kinetics. Kinetic limitations may modify the form of the concentration profile but they do not change the general pattern. To illustrate the different types of transition, consider the simplest case an isothermal system with plug flow involving a single adsorbable species present at low concentration in an inert carrier, for which equation 30 reduces to... 

SPACEEIL has been used to study polymer dynamics caused by Brownian motion (60). In another computer animation study, a modified ORTREPII program was used to model normal molecular vibrations (70). An energy optimization technique was coupled with graphic molecular representations to produce animations demonstrating the behavior of a system as it approaches configurational equiHbrium (71). In a similar animation study, the dynamic behavior of nonadiabatic transitions in the lithium—hydrogen system was modeled (72). 

Whereas there is extensive Hterature on design methods for azeotropic and extractive distillation, much less has been pubUshed on operabiUty and control. It is, however, widely recognized that azeotropic distillation columns are difficult to operate and control because these columns exhibit complex dynamic behavior and parametric sensitivity (2—11). In contrast, extractive distillations do not exhibit such complex behavior and even highly optimized columns are no more difficult to control than ordinary distillation columns producing high purity products (12). 

It can easily accommodate difficult or unusual dynamic behavior such as large time delays and inverse responses. 

Pressure Drop Methods for estimating fluid-dynamic behavior of crossflow plates are analogous, whether the plates be bubble-cap, sieve, or valve. The total pressure drop across a plate is defined by the general equation (see Fig. 14-29)... 

From the standpoint of collector design and performance, the most important size-related property of a dust particfe is its dynamic behavior. Particles larger than 100 [Lm are readily collectible by simple inertial or gravitational methods. For particles under 100 Im, the range of principal difficulty in dust collection, the resistance to motion in a gas is viscous (see Sec. 6, Thud and Particle Mechanics ), and for such particles, the most useful size specification is commonly the Stokes settling diameter, which is the diameter of the spherical particle of the same density that has the same terminal velocity in viscous flow as the particle in question. It is yet more convenient in many circumstances to use the aerodynamic diameter, which is the diameter of the particle of unit density (1 g/cm ) that has the same terminal settling velocity. Use of the aerodynamic diameter permits direct comparisons of the dynamic behavior of particles that are actually of different sizes, shapes, and densities [Raabe, J. Air Pollut. Control As.soc., 26, 856 (1976)]. 

G.T. Gray III and P.S. Follansbee, Influence of Peak Pressure and Pulse Duration on the Substructure Development and Threshold Stress Measurements in Shock-loaded Copper, in Impact Loading and Dynamic Behavior of Materials (edited by C.Y. Chiem, H.-D. Kunze, and L.W. Meyer), Deutsche Gesellschaft fuer Metall-kunde, Germany, 1988, 541 pp. 

A molecular dynamics force field is a convenient compilation of these data (see Chapter 2). The data may be used in a much simplified fonn (e.g., in the case of metric matrix distance geometry, all data are converted into lower and upper bounds on interatomic distances, which all have the same weight). Similar to the use of energy parameters in X-ray crystallography, the parameters need not reflect the dynamic behavior of the molecule. The force constants are chosen to avoid distortions of the molecule when experimental restraints are applied. Thus, the force constants on bond angle and planarity are a factor of 10-100 higher than in standard molecular dynamics force fields. Likewise, a detailed description of electrostatic and van der Waals interactions is not necessary and may not even be beneficial in calculating NMR strucmres. 

The model that best describes the mechanism is usually very complicated. For dynamic studies that require much more computation (and on a more limited domain) a simplified model may give enough information as long as the formalities of the Arrhenius expression and power law kinetics are incorporated. To study the dynamic behavior of the ethylene oxide reactor. 

Flowever, with CFD, configurations with mostly known or at least steady-state boundary conditions and surface temperatures are calculated. In cases where the dynamic behavior of the building masses and the changing driving forces for the natural ventilation are of importance, thermal modeling and combined thermal and ventilation modeling mu.st be applied (see Section 11..5). 

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