Phase accumulation model

Wave Function Matching and Phase-Accumulation Model... 

In the literature, this scattering picture is referred to as phase-accumulation model [28, 29]. 

A breakthrough curve with the nonretained compound was carried out to estimate the axial dispersion in the SMB column. A Peclet number of Pe = 000 was found by comparing experimental and simulated results from a model which includes axial dispersion in the interparticle fluid phase, accumulation in both interparticle and intraparticle fluid phases, and assuming that the average pore concentration is equal to the bulk fluid concentration this assumption is justified by the fact that the ratio of time constant for pore diffusion and space time in the column is of the order of 10. ... 

We present experimental results on photophysical deactivation pathways of uracil and thymine bases in the gas phase and in solvent/solute complexes. After photoexcitation to the S2 state, a bare molecule is tunneled into and trapped in a dark state with a lifetime of tens to hundreds of nanoseconds. The nature of this dark state is most likely a low lying nn state. Solvent molecules affect the decay pathways by increasing IC from the S2 to the dark state and then further to the ground state, or directly from S2 to S0. The lifetimes of the S2 state and the dark state are both decreased with the addition of only one or two water molecules. When more than four water molecules are attached, the photophysics of these hydrated clusters rapidly approaches that in the condensed phase. This model is now confirmed from other gas phase and liquid phase experiments, as well as from theoretical calculations. This result offers a new interpretation on the origin of the photostability of nucleic acid bases. Although we believe photochemical stability is a major natural selective force, the reason that the nucleic acid bases have been chosen is not because of their intrinsic stability. Rather, it is the stability of the overall system, with a significant contribution from the environment, that has allowed the carriers of the genetic code to survive, accumulate, and eventually evolve into life s complicated form. 

Equation (41) is modelled qualitatively in Fig. 4 for the simplest MQMAS experiment involving two pulses (which can be easily adapted to STMAS experiment also). Four subspaces of the density matrix describe the phase accumulated by two arbitrary crystallites. From left to right, the first subspace, Fig. 4a, corresponds to the MQC m ( — m at time = 0 (or I — m >< ml, depending on Iml and /). When perfect excitation is assumed the coherence of all crystallites will have the same phase and amplitude. In Fig. 4b the same subspace is shown for time t = r, with a coherence phase accumulated according to Eq. (41). The two crystallites accumulate a different phase depending on their values. In Fig. 4c 1 the projection of the density matrix on to the CT subspace is shown following a conversion pulse. Ideally, this is similar in phase and amplitude to its predecessor and corresponds to time t2 = 0. Finally, in Fig. 4dl the phase accumulation at t2 = —i/l tx is shown. All crystallites refocus at the same k... 

Its application to the case of the LNT lean-phase accumulation demonstrated its nimierical viability and prediction superiority over other existing methods. Gimparisons between CAT-PP and ideal PFR models highlight the substantial ideality of the system analyzed, while the first approach might provide further information on the dynamic and spatial behavior of active siuface coverage during the catalyst operation. 

The penetration theory can be viewed as the original surface-renewal model. This model was formulated by Higbie [51]. This model is based on the assumption that the liquid surface contains small fluid elements that contact the gas phase for a time that is equal for all elements. After this contact time they penetrate into the bulk liquid and each element is then replaced by another element from the bulk liquid phase. The basic mechanism captured in this concept is that at short contact times, the diffusion process will be unsteady. Considering that the fluid elements may diffuse to an infinite extend into the liquid phase, the model formulation developed earlier for diffusion into a semi-infinite slab can be applied describing this system. After some time the diffusion process will reach a steady state, thus the penetration theory predictions will then correspond to the limiting case described by the basic film theory. However, when the transient flux development is determining a notable amount of the total flux accumulated, the two models will give rise to different mass transfer coefficients. 

The stagnant-film model discussed previously assumes a steady state in which the local flux across each element of area is constant i.e., there is no accumulation of the diffusing species within the film. Higbie [Trans. Am. Jn.st. Chem. Eng., 31,365 (1935)] pointed out that industrial contactors often operate with repeated brief contacts between phases in which the contact times are too short for the steady state to be achieved. For example, Higbie advanced the theory that in a packed tower the liquid flows across each packing piece in laminar flow and is remixed at the points of discontinuity between the packing elements. Thus, a fresh liquid surface is formed at the top of each piece, and as it moves downward, it absorbs gas at a decreasing rate until it is mixed at the next discontinuity. This is the basis of penetration theoiy. 

Example 8 Calculation of Rate-Based Distillation The separation of 655 lb mol/h of a bubble-point mixture of 16 mol % toluene, 9.5 mol % methanol, 53.3 mol % styrene, and 21.2 mol % ethylbenzene is to be earned out in a 9.84-ft diameter sieve-tray column having 40 sieve trays with 2-inch high weirs and on 24-inch tray spacing. The column is equipped with a total condenser and a partial reboiler. The feed wiU enter the column on the 21st tray from the top, where the column pressure will be 93 kPa, The bottom-tray pressure is 101 kPa and the top-tray pressure is 86 kPa. The distillate rate wiU be set at 167 lb mol/h in an attempt to obtain a sharp separation between toluene-methanol, which will tend to accumulate in the distillate, and styrene and ethylbenzene. A reflux ratio of 4.8 wiU be used. Plug flow of vapor and complete mixing of liquid wiU be assumed on each tray. K values will be computed from the UNIFAC activity-coefficient method and the Chan-Fair correlation will be used to estimate mass-transfer coefficients. Predict, with a rate-based model, the separation that will be achieved and back-calciilate from the computed tray compositions, the component vapor-phase Miirphree-tray efficiencies. 

The theory of quenched-annealed fluids is a rapidly developing area. In this chapter we have attempted to present some of the issues already solved and to discuss only some of the problems that need further study. Undoubtedly there remains much room for theoretical developments. On the other hand, accumulation of the theoretical and simulation results is required for further progress. Of particular importance are the data for thermodynamics and phase transitions in partly quenched, even quite simple systems. The studies of the models with more sophisticated interactions and model complex fluids, closer to the systems of experimental focus and of practical interest, are of much interest and seem likely to be developed in future. 

The global design equations for packed beds—e.g.. Equations (10.1), (10.9), (10.39), and (10.40)—all have a similar limitation to that of the axial dispersion model treated in Chapter 9. They all assume steady-state operation. Adding an accumulation term, da/dt accounts for the change in the gas-phase inventory of component A but not for the surface inventory of A in the adsorbed form. The adsorbed inventory can be a large multiple of the gas-phase inventory. 

This model does not predict a stationary phase in a batch fermentation if kd is constant. A nearly stationary phase can be modeled if kd is made to depend on the accumulation of a toxic product. 

This example is based on the model description of Sec. 3.3.4, and involves a multicomponent, semi-batch system, with both heating and boiling periods. The compositions and boiling point temperatures will change with time. The water phase will accumulate in the boiler. The system simulated is based on a mixture of n-octane and n-decane, which for simplicity will be assumed to be ideal but which has been simulated using detailed activity coefficient relations by Prenosil (1976). 

As mentioned at the beginning of Section 3.2.3 the separation process can be modeled by mass balances. Two mass balances have to be made as the liquid appears in two phases dispersed in the gas phase (the droplets) and continuous in the accumulated state. Figure 3.2.12 describes the process and the conditions in addition to what was explained at the very beginning in Section 3.2.1. 

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