Order parameter response time

We shall consider the following two limiting cases of thermal nonlinearities (1) steady-state regime, where Tr and (Tj is the order parameter response time), and (2) transient regime, corresponding to and x. ... 

The response time of the nematic to an absorbed energy pulse is determined by the order parameter response. Here a connection can be made to experiments on ultrasonic propagation. The order parameter cannot change faster than a molecular vibration period. Since S is determined by the interactions of many molecules, the characteristic time is much longer than the molecular time scale. Moreover, as the phase transition point is approached, there is a slowing of order parameter response typical of critical phenomena. This has been verified by ultrasonic experiments. Therefore, the thermal grating response time should be comparable with the ultrasonic data. ... 

The practical goal of EPR is to measure a stationary or time-dependent EPR signal of the species under scrutiny and subsequently to detemiine magnetic interactions that govern the shape and dynamics of the EPR response of the spin system. The infomiation obtained from a thorough analysis of the EPR signal, however, may comprise not only the parameters enlisted in the previous chapter but also a wide range of other physical parameters, for example reaction rates or orientation order parameters. 

Cohen and Coon observed that the response of most uncontrolled (controller disconnected) processes to a step change in the manipulated variable is a sigmoidally shaped curve. This can be modelled approximately by a first-order system with time lag Tl, as given by the intersection of the tangent through the inflection point with the time axis (Fig. 2.34). The theoretical values of the controller settings obtained by the analysis of this system are summarised in Table 2.2. The model parameters for a step change A to be used with this table are calculated as follows... 

There are different time scales associated with the various emissions and uptake processes. Two terms that are frequently used are turnover time and response or adjustment) time. The turnover time is defined as the ratio of the mass of the gas in the atmosphere to its total rate of removal from the atmosphere. The response or adjustment time, on the other hand, is the decay time for a compound emitted into the atmosphere as an instantaneous pulse. If the removal can be described as a first-order process, i.e., the rate of removal is proportional to the concentration and the constant of proportionality remains the same, the turnover and the response times are approximately equal. However, this is not the case if the parameter relating the removal rate and the concentration is not constant. They are also not equal if the gas exchanges between several different reservoirs, as is the case for C02. For example, the turnover time for C02 in the atmosphere is about 4 years because of the rapid uptake by the oceans and terrestrial biosphere, but the response time is about 100 years because of the time it takes for C02 in the ocean surface layer to be taken up into the deep ocean. A pulse of C02 emitted into the atmosphere is expected to decay more rapidly over the first decade or so and then more gradually over the next century. 

If a system is disturbed by periodical variation of an external parameter such as temperature (92), pressure, concentration of a reactant (41,48,65), or the absolute configuration of a probe molecule (54,59), then all the species in the system that are affected by this parameter will also change periodically at the same frequency as the stimulation, or harmonics thereof (91). Figure 24 shows schematically the relationship between stimulation and response. A phase lag <)) between stimulation and response occurs if the time constant of the process giving rise to some signal is of the order of the time constant Inim of the excitation. The shape of the response may be different from the one of the stimulation if the system response is non-linear. At the beginning of the modulation, the system relaxes to a new quasi-stationary state, about which it oscillates at frequency cu, as depicted in Fig. 24. In this quasi-stationary state, the absorbance variations A(v, t) are followed by measuring spectra... 

Fitting Dynamic Models to Experimental Data In developing empiricaTtransfer functions, it is necessary to identify model parameters from experimental data. There are a number of approaches to process identification that have been published. The simplest approach involves introducing a step test into the process and recording the response of the process, as illustrated in Fig. 8-21. The xs in the figure represent the recorded data. For purposes of illustration, the process under study will be assumed to be first-order with dead time and have the transfer function... 

A related approach which has been used successfully in industrial applications occurs in discrete-time control. Both Dahlin (43) and Higham (44) have developed a digital control algorithm which in essence specifies the closed loop response to be first order plus dead time. The effective time constant of the closed loop response is a tuning parameter. If z-transforms are used in place of s-transforms in equation (11), we arrive at a digital feedback controller which includes dead time compensation. This dead time predictor, however, is sensitive to errors in the assumed dead time. Note that in the digital approach the closed loop response is explicitly specified, which removes some of the uncertainties occurring in the traditional root locus technique. 

Typical response times are of the order 15-30 minutes (i), days (ii), and several days (iii). The net microorganism growth rate is a key design parameter and its reciprocal is termed sludge age or mean cell residence time. An adequate control has to take into consideration, that the growth rate is time-varying (5). 

Therefore, switch-off times are independent of the field strength and directly dependent on material parameters, such as viscosity coefficients and elastic constants, and the cell configuration. Therefore, they are often three or four orders of magnitude larger than the switch-on times. However, sophisticated addressing techniques can produce much shorter combined response times ( on + off The nematic director should be inclined, e.g. 1° pretilt,... 

Here Qm and Q can be separate order parameters or order parameter components. In combination, Equations (4) and (5) provide a means of predicting the elastic constant variations associated with any phase transition in which the relaxation of Q in response to an applied stress is rapid relative to the time scale of the experimental measurement. A classic example of the success that this approach can have for describing the elastic behaviour of real materials is provided by the work of Errandonea (1980) for the orthorhombic monoclinic transition in LaPsOi4 (Fig. 4). 

The Fluid Flame energy systems are designed to run themselves without the need for full-time operators. The system is fully automated and is equipped with annunciators to alert operators when problems arise. A number of electrical process controllers are used to maintain constant temperatures in the bed and in the vapor space of the combustion cell in order to provide steam at constant pressure. Signals from these controllers dictate the feed rate from the metering bin and the amount of excess air fed into the system. The temperature is controlled in order to maintain constant parameters throughout the system and to keep temperatures below the slagging temperatures of the ash. Normal temperature variation is less than +5C° (9F°), and the response time to process variations is rapid. 

The quadratic response function is obtained as the third derivative of the time-averaged quasi-energy. The program is then to expand the energy to third order in the first-order parameters ... 

In order to select a particular experimental technique to measure x , it is very important to keep in mind which parameter of the third-order nonlinear response has to be characterized. For example, if one wants to determine the time-response due to molecular reorientation, one cannot choose Third-Harmonic Generation or Electric-Field-Induced Second-Harmonic Generation, since none of these techniques provide time-response information. Depending on the parameter of interest, a specific technique must be chosen. The following physical mechanisms can contribute to the third-order nonlinear response [54] ... 

The quadratic model (Eq.3.3) allowed the generation of the 3-D response surface image (Fig. 3.5) for the main interaction between injection time and voltage. The quadratic terms in this equation models the curvature in the true response function. The shape and orientation of the curvature results from the eigenvalue decomposition of the matrix of second-order parameter estimates. After the parameters are estimated, critical values for the factors in the estimated surface can be found. For this study, a post hoc review of our model... 

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