The closed-flow case

We consider the time evolution of a localized perturbation of the form C(x,y,t = 0) = Co exp [— (x2 +y2) /2a2], with Co and a small, on top of the unstable C = 0 solution. In terms of the reaction A + B — 2B, this represents a diluted localized patch of the activating substance B, added to the A component. C = 1 corresponds to pure B and C = 0 to pure A. The numerical simulations show that there is a critical value of Da, in this particular case Dac 2, at which the dynamics changes qualitatively. 

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In the closed flow case we obtain again two different behaviors above and below a critical value Dac. When Da > Dac and the localized perturbation added to the (7 = 0 state is large enough the phenomenology is similar to the autocat alytic case and configurations similar to those in Fig. 7.2 are obtained the reaction product... 

Again, there is a main transition at a critical Damkohler number, Da = Dac. For smaller values of Da the initial perturbation is quickly diluted and the activator decays to the C state, as in the bistable case. The same behavior is observed when the initial perturbation is not sufficiently large. For Da > Dac the perturbation grows as in the bistable case, forming a growing filament that eventually fills the whole system (in the closed flow case), or covers the unstable manifold of the chaotic saddle (in open flows). The filament consists now of a pulse of the C concentration, with a maximum close to the excited state, and accompanied by a smaller pulse of C2. In the closed... 

J d) is plotted against d/d in Fig. 6-7 also shown are the curves for the two ideal reactors, taken from Fig. 6-5. The comparison brings out pertinent points about reactor behavior. Although the plug-flow reactor might be expected to be a better representation of the laminar case than the stirred-tank reactor, the RTD for the latter more closely follows the laminar-reactor curve for 6/6 from about 0.6 to 1.5. However, there is no possibility for 6 to be less than 0.5 in the laminar-flow case. Hence the stirred-tank form is not applicable at all in the low 6 region. At high 6 the three curves approach coincidence. Conversions for these reactors are compared in Sec. 6-7. 

The effect of molecular diffusion is to decrease the radial concentration differences and increase the conversion. Cleland and Wilhelm [23] showed that the conversion depends on a diffusion parameter a = D /kR and a kinetic parameter kL/u or kt. The values in Table 6.1 correspond to a = 0, and the plug-flow case is a = oo. The conversions for a < 0.01 are so close to those for a = 0, that radial diffusion can be neglected. For a> 1.0, radial diffusion makes the conversion almost the same as for plug flow. The conversions for O = 0.1 fall about midway between these extremes, as shown in Figure 6.6. 

In Figure 8, the experimental results from the (4 m/day frontal advance rate, oil free) short core flood are compared to the simulated pressure drops which were based on the limiting capillary pressure principle. In this particular case was chosen at 0.35 over a range of water fractional flows from 0.01 to 0.15 to closely match the experimental data. For Sw > a fractional flow curve was chosen which matched the experimental data closely by appropriately adjusting the gas phase relative permeability curve. The water relative permeability curve remains the same as defined in the Appendix under gas/water relative permeabilities. The composite foam fractional flow curve can be seen in Figure 9. Notice the vertical section in the curve for the foam flow case lies at = 0.35. 

To determine how the IADs scale with flow rates, in order to compare performance with the vane PMD, simulations were also run at multiple flow rates between 1 x 10 and 2.76 x 10 kg/s for a 325 x 2300 screen in LH2 at a fixed liquid and pressurant gas temperature of 20.3 K each and a tank pressure of 101 kPa. Table 14.4 presents the resultant expulsion efficiencies and PMD mass for the screen channel LAD cases at different demand flow rates. The results show that up to a mass flow rate of 0.0049 kg/s, the LAD can achieve a maximum expulsion efficiency of 98.1% of the small-scale LH2 tank. This upper limit is due to the fact that the gallery arm cannot access the small residual propellant pool in between consecutive arms at very low fill levels due to the closed flow... 

Since acrylic polymerizations liberate considerable heat, violent or mnaway reactions are avoided by gradual addition of the reactants to the kettie. Usually the monomers are added by a gravity feed from weighing or measuring tanks situated close to the kettie. The rate of monomer addition is adjusted to permit removal of heat with full flow of water in the condenser and a partial flow in the cooling jacket. Flow in the jacket can be increased to control the polymerization in cases of erroneous feed rates or other unexpected circumstances. A supply of inhibitor is kept on hand to stop the polymerization if the cooling becomes inadequate. 

Open-Loop versus Closed-Loop Dynamics It is common in industry to manipulate coolant in a jacketed reacdor in order to control conditions in the reacdor itself. A simplified schematic diagram of such a reactor control system is shown in Fig. 8-2. Assume that the reacdor temperature is adjusted by a controller that increases the coolant flow in proportion to the difference between the desired reactor temperature and the temperature that is measured. The proportionality constant is K. If a small change in the temperature of the inlet stream occurs, then depending on the value or K, one might observe the reactor temperature responses shown in Fig. 8-3. The top plot shows the case for no control (K = 0), which is called the open loop, or the normal dynamic response of the process by itself. As increases, several effects can be noted. First, the reactor temperature responds faster and faster. Second, for the initial increases in K, the maximum deviation in the reactor temperature becomes smaller. Both of these effects are desirable so that disturbances from normal operation have... 

Close-Clearance Stirrers For some pseiidoplastic fluid systems stagnant fluid may be found next to the -essel walls in parts remote from propeller or turbine impellers. In such cases, an anchor impeller maybe used (Fig, 18-6), The fluid flow is principally circular or helical (see Fig, 18-7) in the direction of rotation of the anchor. Whether substantial axial or radial fluid motion also occurs depends on the fluid iscosity and the design of the upper blade-supporting spokes. Anchor agitators are used particularly to obtain irnpro ed heat transfer in high-consistency fluids,... 

Jet interaction should not be taken into account when the jets are closely adjacent to each other, are propagated in confined conditions, and entrainment of the ambient air is restricted. This may be the case for concentrated air supply when air diffusers are uniformly positioned across the wall and the jets are replenished by the reverse flow, which decreases the jet velocity. This effect should be taken into consideration using the confinement coefficient discussed in Section 7.4.5. For the same reason, jet interaction should not be taken into consideration when air is supplied through the ceiling-mounted air diffusers and they are uniformly distributed across the ceiling. 

The largest disturbances can occur when the supply jet is disturbed or deflected by the process. In this case the jet can blow in unwanted directions, such as to the side or backward into the surroundings. The operator can disturb the horizontal jet by standing too close to the jet inlet and also by preventing the induced flow from passing the contaminant source. 

Figure 9-13 shows the Ynryn P ° obtained for the first case where four regions are defined a region of complete separation, two regions where only one outlet stream is 100 % pure and a last region where neither of them is 100 % pure. The closed circles are numerical results based on the equivalence between the TMB and the SMB the thick lines connect those results. The thin line in Fig. 9-13 has two branches. The diagonal 7 -7 corresponds to zero feed flow rate therefore, 7 must be higher than Yn- The horizontal branch Ym corresponds to zero raffinate flow rate in this case, the extract flow rate is 25.09 mL min k... 

The above results show close agreement between the experimental and theoretical friction factor (solid line) in the limiting case of the continuum flow regime. The Knudsen number was varied to determine the influence of rarefaction on the friction factor with ks/H and Ma kept low. The data shows that for Kn < 0.01, the measured friction factor is accurately predicted by the incompressible value. As Kn increased above 0.01, the friction factor was seen to decrease (up to a 50% X as Kn approached 0.15). The experimental friction factor showed agreement within 5% with the first-order slip velocity model. 

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