Dynamic Specifications

The dynamic performance of a system can be deduced by merely observing the location of the roots of the system characteristic equation in the s plane. The time-domain specifications of time constants and damping coefficients for a closedloop system can be used directly in the Laplace domain. 

If all tbe roots lie on the negative real axis, we know the system is overdamped or critically damped (all real roots). 

The farther out on tbe negative axis the roots lie, the faster will be the dynamics of the system (the smaller the time constants). 

The roots that lie close to the imaginary axis will dominate the dynamic response since the ones farther out will die out quickly. 

The farther any complex conjugate roots are from the real axis the more underdamped the system will be. 

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There are a number of time-domain specifications. A few of the most frequently used dynamic specifications are listed below (see also Prob. 6.11). The traditional test input signal is a step change in setpoint. 

The steadystate error is another time-domain specification. It is not a dynamic specification, but it is an important performance criterion. In many loops (but not all) a steadystate error of zero is desired, i.e, the value of the controlled variable should eventually level out at the setpoint. 

Subsequently, we used Aspen Dynamics for time-domain simulations. A basic control system was implemented with the sole purpose of stabilizing the (open-loop unstable) column dynamics. Specifically, the liquid levels in the reboiler and condenser are controlled using, respectively, the bottoms product flow rate and the distillate flow rate and two proportional controllers, while the total pressure in the column is controlled with the condenser heat duty and a PI controller (Figure 7.4). A controller for product purity was not implemented. 

Flow cytometry is a very versatile technique [223] which allows the analysis of more than 104 cells per second [369,370]. This high number results in statistically significant data and distributions of cell properties. Therefore, flow cytometry is a key technique to segregate biomass (into distinct cell classes) and to study microbial populations and their dynamics, specifically the cell cycle [76, 87, 116, 200, 214, 221, 295, 329, 330, 409, 418]. Individual cells are aligned by means of controlled hydrodynamic flow patterns and pass the measuring cell one by one. One or more light sources, typically laser(s), are focused onto the stream of cells and a detection unit(s) measure(s) the scattered and/or fluorescent light (Fig. 24). Properties of whole cells such as size and shape can be... 

Dynamic specific modulus E /y, the ratio of dynamic Young s modulus to specific gravity, and loss tangent tan 8 can be used to study the viscoelastic nature of wood. E /y is related to sound velocity and tan 8 to sound absorption or damping within the wood. A large E /y and small tan 8 characterize the acoustic quality of soundboard wood [3]. 

The Wigner function has the valuable property that the time evolution equation for the quantum dynamics in the Wigner representation resembles that for the classical Liouville dynamics. Specifically, the Schrodinger equation can be transformed to [70]... 

Whatever the limiting mechanism, ultimately the current becomes limited by concentration polarization, i.e., by the transport of redox species from the bulk electrolyte to the semiconductor surface. The situation in this regard is no different from that at metal electrode-electrolyte interfaces. As in the latter case, hydro-dynamic (specifically RDE, Table 2) voltammetry is best suited to study mass transport. AC impedance spectroscopy can be another useful tool in this regard [82]. 

We therefore begin our discussion by examining the validity of the nearly integrable picture since, as mentioned, it would be the most legitimate approach to treat a class of systems that can generate slow dynamics. Specifically, we take the FPU model as a representative model of nearly integrable Hamiltonian systems. Its explicit form is given as... 

The second term in the spin Hamiltonian relates to collision dynamics. Specifically, is expressed as... 

In gas-phase dynamics, the discussion is focused on the TD quantum wave packet treatment for tetraatomic systems. This is further divided into two different but closed related areas molecular photofragmentation or half-collision dynamics and bimolecular reactive collision dynamics. Specific methods and examples for treating the dynamics of direct photodissociation of tetraatomic molecules and of vibrational predissociation of weakly bound dimers are given based on different dynamical characters of these two processes. TD methods such as the direct projection method for direct photodissociation, TD golden rule method and the flux method for predissociation are presented. For bimolecular reactive scattering, the use of nondirect product basis and the computation of the initial state-selected total reaction probabilities by flux calculation are discussed. The descriptions of these methods are supported by concrete numerical examples and results of their applications. 

This chapter summarizes many of the contributions that the recoil technique of generating excited radiotracer atoms in the presence of a thermal environment is making to the field of chemical dynamics. Specific topics discussed critically include characterization of the generation and behavior of excited molecules including fragmentation kinetics and energy transfer, measurement of thermal and hot kinetic parameters, and studies of reaction mechanisms and stereochemistry as a function of reaction energy. Distinctive features that provide unique approaches to dynamical problems are evaluated in detail and the complementarity with more conventional techniques is addressed. Prospects for future applications are also presented. 

To compute two physical parameters, the tortuosity factor r and the dynamic specific surface area flv[Pg.381]

The simplest ionic solution is a mixture of a single solvent (e.g., water) and a single ionic solute (e.g., sodium chloride). We may represent such a solution by a model, which consists of large numbers of molecules of species w (solvent), c (cations), and a (anions), which interact according to specified prescriptions and which obey certain laws of dynamics (specifically, classical mechanics). Numbering all the particles of a given species from 1 to N, and denoting species type by a subscript, we may write the Hamiltonian for such a model as... 

Fletcher, C.A.J. (1991) Computational Techniques for Fluid Dynamics, Specific Techniques for Different Flow Categories, vol. 2 Springer, Berlin. 

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