Population transient behavior

The dynamic model used in predicting the transient behavior of isothermal batch crystallizers is well developed. Randolph and Larson (5) and Hulburt and Katz (6) offer a complete discussion of the theoretical development of the population balance approach. A summary of the set of equations used in this analysis is given below. 

In one respect, the variable-yield model has been a disappointment in the sense that it was hoped that the transient behavior of its solutions would better fit the transient behavior seen in experiments with certain algae [CNIJ. The experiments, described in [CM], involved the growth of a Chlamydomonas reinhardii population on a nitrogen substrate. Following a step increase in the dilution rate, damped oscillations were observed in cell numbers. Cunningham and Nisbet [CNl] note that the singlepopulation variable-yield model could not reproduce these oscillations without the introduction of time delays into the equations. See also the monograph [NG]. 

Transient behavior is determined by calculating how the pore population changes... 

The parameters appearing in the present stochastic model can be easily estimated from some correlations and determined by eonducting simple batch experiments. The transient behavior of molecules can be evaluated by analytically solving a system of the governing ordinary differential equations of the present stochastic model. The stochastic modeling, therefore, provides an easier approach than the deterministic population balance modeling in which a partial differential equation must be solved. These facts are very attractive from the practical point of view. 

When the radiation field is switched on at time t = 0, one may observe at first a transient behavior of the population densities. The probability amplitudes a(t) and b(t) show a damped oscillation (see Sect.2.9.5) and ap-... 

A suggestion for the existence of at least three populations of adsorbed Ru(II) comes from the time evolution of the transient UV-vis absorption spectra. These spectra show that the recovery of the initial Ru(II) spectra occurs with two parallel (fast and slow) second-order components. The rate constants for these two components show remarkably little dependence on the nature of the coordinating ligands. Both of these components are attributed to recombination of the adsorbed Ru(III) with the injected electrons. Thus there is a small luminescent population of Ru(II) that does not engage in electron injection, a non-luminescent population that injects and recombines rapidly, and a third population that injects rapidly and recombines slowly. A detailed picture of the nature of the ligand/semiconductor interaction and how it affects the behavior of these systems awaits further study. 

Similar transient signals were obtained from time-dependent quantum mechanical calculations performed by Meier and Engel, which well reproduce the observed behavior [49]. They show that for different laser field strengths the electronic states involved in the multiphoton ionization (MPI) are differently populated in Rabi-type processes. In Fig. 13 the population in the neutral electronic states is calculated during interaction of the molecule with 60-fs pulses at 618 nm. For lower intensities the A state is preferentially populated by the pump pulse, and the A state wavepacket dominates the transient Na2+ signal. However, for the higher intensities used in the... 

Fisher 1998 demonstrates that the more that is learned about the biomarker (half-life, time course in blood or urine, and development of PBPK model) and the exposed population (age, body weight, pharmacoge-netic traits, behavioral factors that affect exposure, and time between exposure and sample measurement), the more refined dose estimates can become. Without such information, a highly transient metabolite like TCE is not a reliable marker of exposure, unless exposure is nearly continuous and uniform. That may not be the case in the general population, so TCE in blood may not be a good biomarker for assessment of general-population exposure, although PBPK models are available to extrapolate from biomarker concentration to external dose in both animals and humans (Clewell et al. 2000). 

Figure 6e, f show the experimentally relevant case in which a mixture of mechanisms occurs, i. e., E 0 and Wetu 0- The parameters have been chosen such that under steady-state excitation conditions 40% of the upconversion is generated by GSA/ESA, and 60% by GSA/ETU. Panel e shows that following a short pulse the properties of both panels a and c can be identified. Specifically, a nonzero N2 is observed at time = 0, but a delayed maximum and a long decay time are also observed. This provides a way to identify intensity involving both GSA/ESA and GSA/ETU contributions. This transient curve is triexponential, involving the decay of the GSA/ESA population, and the rise and decay of the GSA/ETU population (dashed lines). The analogous square-wave transient is shown in Fig. 6f. Termination of the square pulse leads to a simple biexponential decay curve, with a fast component corresponding to the natural decay rate of the upper state, and a slow component corresponding to twice the decay rate of the intermediate state (dashed lines). Again, a small deviation from pure biexponential behavior is observed at short times due to the effect of k2- The relative contributions of each mechanism, in this case 40 60, can be determined from the decay curve as shown in Fig. 6f. This information can be introduced directly into Eq. (10) for data simulation.

Only the case of steady coinjection of surfactant solution and gas into a one-dimensional core initially filled with surfactant solution is addressed. Calculated transient foam displacement well represents both the measured wetting liquid saturations and pressure profiles with physically meaningful parameter values. It is predicted and experimentally verified that foam moves in a piston-like fashion through a linear porous medium presaturated with surfactant solution. Moreover, the proposed population-balance predicts the entire spectrum of unique steady foam-flow behavior in the capillary-pressure regime. 

When the system is prepared in the higher of the coupled electronic states, the electronic population typically decays on a time scale of several fs, followed by transient partial recuiTences which evolve into more or less erratic fluctuations around a finite long-time limit. Examples of this type of behavior will be discussed in Section 3. The calculations have revealed that the electronic population decay is intimately related to the multi-mode nature of the problem. The strong coupling between the nuclear degrees of freedom affects not only electronic, but also nuclear observables, such as the expectation values of the position and momentum operators of the various modes ... 

This setup allows one to obtain two types of transient data (i) at fixed tu, the monochromator grating is scanned or (ii) at fixed monochromator frequency, the time delay is changed. Figure 4.13 shows the transient absorption spectra of CO molecules adsorbed on a Pf(lll) surface and obtained for differ-enf fime delays. A distinct shift to lower frequencies and a spectral broadening are observed at small fp as well as the rapid return of the spectrum to the equilibrium lineshape. Such temporal behavior can be described as originating from a one-phonon band of fhe adlayer which shiffs with the increasing excited-state population. 

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