Time-varying behavior

ANS Yes, the time varying behavior of elastance will mathematically result in an inverse force-velocity relationship of muscle. However, as I have just shown there is an additional dependence of pressure on flow that is independent of volume and it is this additional pressure loss that must be accounted for by a resistance term. Furthermore, Dr. Suga recently published the results of a study which indicated a correction term had to be added to his time varying elastance model in order for the isovolumetric and ejecting pressure-volume relationships to coincide. This correction term was of the same magnitude as our resistance term. So you cannot just use a time-varying elastance to describe the dynamics of the left ventricle. 

Various mathematical models were suggested for the analysis of left ventricular diastole (Ghista et al.y 1969 Mirsky, 1973 Rabkin and HSU 1975, Ghista and Hamid 1977, Ghista and Ray 1980, a, b Moskowitz, 1980). These models are directed toward either a part of diastole or toward analytical expressions for the cardiac elastic properties. Different parameters are used to describe the time varying behavior discussed earlier. They can be the general stiffness parameters, or parameters defined only for the need of the analysis, as for example the left ventricular medium s strain energy density (Mirsky, 1973) or others. 

A static system has structure but no associated activity (e g., bridge, building) and a dynamic system involves time-varying behavior for complex systems (e.g., machine, U.S. economy). It deals with internal feedback loops and time delays that affect the behavior of the entire system [12,13]. 

Moreover, one important phenomenon to be analyzed in SCs is the so-called bullwhip effect. The bullwhip effect is the magnification of demand fluctuations as one moves up the SC from markets to suppliers. The basic phenomenon is not new and has been known to scientists for some time. Forrester (1961) illustrated the effect in a series of case studies, and pointed out that it is a consequence of industrial dynamics or time varying behaviors of industrial organizations. Here, the variation of the quotient given in Eq. (7.58) is proposed as a measure of the bullwhip effect. is the quotient of the production generated and the real demand received by the SC for product i. In the ideal case in which information is certain, amount of production... 

The medium resistance R, which theoretically should be constant, often varies with time. This behavior results when some of the soHds penetrate the medium or when it compresses under appHed pressure. For convenience, the resistance of the piping and the feed and outlet ports is sometimes included in R. 

When Sn2 reactions are carried out on these substrates, rates are greatly increased for certain nucleophiles (e.g., halide or halide-like ions), but decreased or essentially unaffected by others. For example, a-Chloroaceto-phenone (PhCOCH2Cl) reacts with KI in acetone at 75°C 32,000 times faster than l-Chlorobutane, ° but a-bromoacetophenone reacts with the nucleophile triethylamine 0.14 times as fast as iodomethane. The reasons for this varying behavior are not clear, but those nucleophiles that form a tight transition state (one in which bond making and bond breaking have proceeded to about the same extent) are more likely to accelerate the reac-tion. ... 

In the distillation column example, the manipulated variables correspond to all the process parameters that affect its dynamic behavior and they are normally set by the operator, for example, reflux ratio, column pressure, feed rate, etc. These variables could be constant or time varying. In both cases however, it is assumed that their values are known precisely. 

The third period corresponds to the last five data points where it is obvious that the assumption of a nearly constant qM is not valid as the slope changes essentially from point to point. Such a segment can still be used and the integral method will provide an average qM, however, it would not be representative of the behavior of the culture during this time interval. It would simply be a mathematical average of a time varying quantity. 

Propagation problems. These problems are concerned with predicting the subsequent behavior of a system from a knowledge of the initial state. For this reason they are often called the transient (time-varying) or unsteady-state phenomena. Chemical engineering examples include the transient state of chemical reactions (kinetics), the propagation of pressure waves in a fluid, transient behavior of an adsorption column, and the rate of approach to equilibrium of a packed distillation column. 

The time-dependent behaviors of metal ions adsorption were measured by varying the equilibrium time between the adsorbate and adsorbent in the range of 30-300 min. The concentration of Pb(ll) and Zn(ll) were kept as 50 ttg/mL while the amount of resin added was 0.5 g. The experiments were performed at pH 4 for Pb L... 

Various adsorption parameters for the effective removal of Pb + and ions by using new synthesized resin as an adsorbent from aqueous solutions were studied and optimized. Time-dependent behavior of Pb + and ions adsorption was measured by varying the equilibrium time between in the range of 30-300 min. The percentage adsorption of Pb + plotted in Fig. 26.2 as a function of contact time... 

The time-dependent behavior of ions adsorption was measured by varying the equilibrium time between the adsorbent (ground sumac leaves) and adsorbent (Cu " ions) in the range of 30 min and 24 h. The concentration of was kept 40 pg ttiL , particle size 710 pm, and amount of adsorbent 0.1 g. 

Time-depended behavior of Cu + ion adsorption was measured by varying the equilibrium time between in the range of 0.5-72 h. The percentage adsorption of Cu + ions plotted in Fig. 28.2 as a function of contact time. The percentage adsorption of Cu + indicates that the equilibrium between the Cu + ions and sumac leaves was attained 4 h. Therefore, 4 h stirring time was found to be appropriate for maximum adsorption and was used in all subsequent measurement. The effect of temperature and pH the adsorption equilibrium of Cu + on sumac leaves was investigated by varying the solution temperature from 283 to 303 and pH from 6 to 10. The results are presented in Fig. 28.3. The results indicated that the best adsorption results were obtained at pH 8 at 293 K. 

In the control literature and control applications, regulation is often addressed as forcing the output of a dynamical system to reach a desirable constant value. While for many physical systems this is the case due to the proper nature of the system, for other interesting systems, time varying reference signals are imposed to obtain a suitable behavior of the system. In this section, a review of some results relative to the regulator problem, for the linear and non linear case is presented. Extension of these results to the case of discretetime systems will be also introduced. 

The formulation of combustion dynamics can be constructed using the same approach as that employed in the previous work for state-feedback control with distributed actuators [1, 4]. In brief, the medium in the chamber is treated as a two-phase mixture. The gas phase contains inert species, reactants, and combustion products. The liquid phase is comprised of fuel and/or oxidizer droplets, and its unsteady behavior can be correctly modeled as a distribution of time-varying mass, momentum, and energy perturbations to the gas-phase flowfield. If the droplets are taken to be dispersed, the conservation equations for a two-phase mixture can be written in the following form, involving the mass-averaged properties of the flow ... 

The second important point is that the dependence of the polymerization rate on the dose rate varies from 0.90 to 0.48, going from lower to high relative rates at the same time. This behavior seems to be related to the exhaustiveness of the drying method employed. 

Let us count the number of periods of si2(t) in Figure 4.57, once the periodic oscillatory steady-state loop has been reached. For t > 20 the system has stabilized in its varying behavior there are 9 separate periods in the top graph of Figure 4.57 in each complete limit cycle of length about 11 time units. This can be verified by counting the number of local maxima between subsequent large amplitude drops in the top profile curve of... 

As we have discussed in depth elsewhere, despite the similarities in the structures of hypericin and hypocrellin, which are centered about the perylene quinone nucleus, their excited-state photophysics exhibit rich and varied behavior. The H-atom transfer is characterized by a wide range of time constants, which in certain cases exhibit deuterium isotope effects and solvent dependence. Of particular interest is that the shortest time constant we have observed for the H-atom transfer is 10 ps. This is exceptionally long for such a process, 100 fs being expected when the solute H atom does not hydrogen bond to the solvent [62]. That the transfer time is so long in the perylene quinones has been attributed to the identification of the reaction coordinate with skeletal motions of the molecule [48, 50]. 

From a holistic point of view, the time-varying parameters V (/,) and k (/,) fitting the observed data could represent the dynamic behavior of a com-... 

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