Transition criterion

Furthermore, when d is smaller than the right-hand side of the above equation, the flow transition criterion is, according to Barnea et al. (1983) ... 

In the upper portion of the bed, trickling flow occurs, but when the gas velocity has increased enough, it tends to blow away liquid held up between the packing particles and pulsing flow is observed. This transition criterion is given in Equation (5). 

Equation (61) is the transition criterion provided the conditions given by equation (52) are satisfied. From Table I it can be seen that these conditions are satisfied only in the case of gas-solid fluidized beds and in some cases of solid-liquid fluidized beds where ps Pl- Therefore, for other multiphase dispersions [such as gas-liquid (bubble columns) and solid-liquid fluidized beds (where pl is not negligible)] the comparison of dynamic wave velocity with continuity wave velocity is not valid for deciding the bed stability. Further, the above analysis holds for transition from region I to II (point P in Fig. 1) and not for III to II (point Q). Therefore, the criterion does not hold for bubble columns and dilute dispersions. 

In the previous section, it was pointed out that the critical gas hold-up is very sensitive to the d -v oo relationship and the values of m, Cy, and a. It was thought desirable to calculate the value of a for all the experimental points shown in Table V. For each point, four values of m (1, 1.4, 1.9 and 2.4) and three levels of Cy were considered. The values of a are given in Table VI. The average value works out to be 3.14. Table V also gives the comparison between the experimental critical gas hold-up and the critical gas hold-up predicted using the transition criterion [Eq. (25)] with dispersion coefficient equal to 3. The comparison between predicted and experimental gas hold-up for bubble columns is shown in Fig. 30. It can be seen... 

Within this framework, one can imagine a two dimensional ductile-brittle transition criterion which would be composed of two perpendicular boundaries in the graph la-Mw (Fig. 7). The vertical boundary would correspond to a critical molar mass M c- In polymers of low crystallinity or fully amorphous, M c would be sharply linked to the entanglement molar mass M c 5Me. In polymers of medium to high crystallinity, M c would be the molar mass below which it is impossible to have values higher than lac- The horizontal boundary would correspond to U = lac-... 

Using the same transition criterion as for Newtonian fluids, namely, Re, < 10, the flow in the whole of the boundary layer is likely to be laminar. 

Transition criterion and turbulent boundary layer flow... 

Brutin et al. [9] studied the transition from steady to unsteady flow boiling and developed a transition criterion. This study was performed experimentally in minichannels but the theory also applies to confined geometries such as microchannels. This criterion is based on observations of the unsteady two-phase flow the two-phase flow stops at a location in the microchannel, then all the fluid after this location is expelled toward the exit, whereas all the fluid before this location is sent back to the entrance. This two-phase backflow has been evidenced previously [7] however, the location of this split remained unexplained. 

It is further of interest to note that in the particular case of thin laminar liquid layers in channel flow, the liquid layer thickness (as appears in Fr, Equation 27), can be obtained explicitly by solving the steady-state two-fluid momentum equations. In this case, the transitional criterion for wind generated waves, > 1 reduces to a critical superficial gas phase Reynolds number, Re > 1.113 x 10 [103]. 

For sufficient long vertical surface, fhe condensate film may become furbulent. The transition criterion may be expressed in terms of Reynolds number defined as... 

Let us now look at the influence of R on the solubility at fixed T, Co, and other parameters. As R increases, the maximum of the fluctuation probability (Equation 13.13) appears and shifts right and up, and the minimum of/(r) shifts down and to the left (Figure 13.4b). By means of the same reasoning as above, one deduces the existence of a separation transition criterion at the size of the nanosystem i tr- This value J tr is a function of the degree of supersaturation (temperature and initial composition). 

The condition that the Gibbs free energy of the total system for a new (two-phase) configuration (Figure 13.3b-d) is smaller than for the starting (single-phase) one (Figure 13.3a) is taken as the phase transition criterion. 

It foUows that, when one deals with the phase diagram, one must determine and plot the solubility curves in one T-C nanophase diagram (in the authors definitions) as well as the final equilibrium compositions, the boundaries of the phases of the already transformed system found by the transition criterion... 

The next step is carried out by considering the second mechanism. The transition criterion between the first and second mechanisms should be evaluated if the bifurcation is possible. Also the coexistence of the first and second mechanisms should be considered. In this case the relative importance between them is evaluated using the scaling parameters. It is also important to obtain the characteristic time constant for each mechanism. It will give an estimate of the time required to complete the transfer process within the subsystem by that particular mechanism. 

When the liquid starts to flow as a film, three different regimes are possible as discussed previously. The stratified to slug flow transition criterion gives the required gas velocity of 300 m/s, which is about twice the expected gas velocity. However, the more important transition is that to the annular flow, which is determined by the entrainment process. As shown below, for the case of the sample calculation the onset of entrainment velocity is exceeded by the expected gas... 

Proof Given an input data sequence, the execution delays for all operations in G, are known. Therefore, the latency of G, can be computed as the longest weighted path in G,. The precise transition criterion ensures that transferring control from any control element to its successors incurs no penalty in tains of cycles for state vertices. Recall that the done signals of stateless vertices are immediately asserted upon being enabled, and hence stateless vertices do not affect the execution delay and control delay. 

Lemma 8.U The precise adaptive control implementation of G, satisfies the precise transition criterion. 

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