Condensation limiting cases

As the feed-to-steam ratio is increased in the flow sheet of Fig. 11-125 7, a point is reached where all the vapor is needed to preheat the feed and none is available for the evaporator tubes. This limiting case is the multistage flash evaporator, shown in its simplest form in Fig. 11-125 7. Seawater is treated as before and then pumped through a number of feed heaters in series. It is given a final boost in temperature with prime steam in a brine heater before it is flashed down in series to provide the vapor needed by the feed heaters. The amount of steam required depends on the approach-temperature difference in the feed heaters and the flash range per stage. Condensate from the feed heaters is flashed down in the same manner as the brine. 

Two limiting cases for gasification at the fuel surface were considered. In case 1, the fuel concentration was assumed constant and independent of time, i.e., f(Cf) = Cf and in case 2, it was assumed that the fuel mass flux was constant and independent of time or pressure, i.e.,/(Cy) = — D 8Cf/ dx = rfi. Case 1 was identified with a condensed phase behaving as a boiling liquid or subliming solid, and case 2 with a polymer undergoing irreversible decomposition at constant temperature. 

Equation (6.87) is a condensed mathematical statement of the second law the inequality applies to any real process, which is necessarily irreversible, and the equality applies to the limiting case of the reversible process. 

A cornerstone of condensed phase reaction theory is the Kramers-Grote-Hynes theory. In a seminal paper Kramers solved the Fokker-Plank equation in two limiting cases, for high and low friction, by assiuning Markovian dynamics y(t) 5(t). He foimd that the rate is a non-monotonic function of the friction ( Kramers turnover .) Further progress was made by Grote and Hynes - who... 

The instanton method takes into account only the dynamics of the lowest energy doublet. This is a valid description at low temperature or for high barriers. What happens when excitations to higher states in the double well are possible And more importantly, the equivalent of this question in the condensed phase case, what is the effect of a symmetrically coupled vibration on the quantum Kramers problem The new physical feature introduced in the quantum Kramers problem is that in addition to the two frequencies shown in Eq. (28) there is a new time scale the decay time of the flux-flux correlation function, as discussed in the previous Section after Eq. (14). We expect that this new time scale makes the distinction between the comer cutting and the adiabatic limit in Eq. (29) to be of less relevance to the dynamics of reactions in condensed phases compared to the gas phase case. 

The Usadel equation can be solved in some limiting cases [15]. In Fig. 2 we present the spatial dependence of the condensate function (singlet and triplet). One can see that the SC penetrates the F layer over a short distance of the order whereas the TC penetrates over a long distance = y/Df/2itT. The amplitude of the long-range part of the TC has a maximum at a = 7r/4. 

This problem of performance defects due to heat transfer and velocity lag when condensation is completed in the chamber is now treated. Very interesting limiting cases which have been detailed by Altman and Carter (5) are reviewed first. Then, the more general case of the effect of simultaneous heat transfer and velocity lag on nozzle performance is developed. 

Particularly extractions of fatty substances are carried out with propane in high pressure liquid/liquid mode. Although this extraction medium may be used in supercritical condition, this limiting case of liquid/liquid extraction was considered for design, because the required vaporization heat is maximized. The extraction is operated at a pressure of merely 80 bar and a temperature of 55°C. The further steps are the first decompression, evaporation and overheating to vapour conditions at 15 bar, 65°C followed by the second precipitation at 10 bar, 65°C. Afterwards the vapour will be liquified through condensation, recompressed and - in this case - reheated to extraction conditions. 

The foregoing discussion presents a contrast of the two limiting ideal cases. Both the hydrolysis and condensation rates are affected by parameters other than pH, such as solvent, type of alkoxide, and concentration. Only the tendencies of the solution behavior are discussed in this chapter. The ideal limiting-case behavior may not always be observed. 

Equation 6.3-1 is a limiting case of Raoult s law, which will be introduced in a more general context in Section 6.4. It is the fundamental relation used in the analysis of equilibrated gas-liquid systems containing one condensable component. A wide variety of problems occur in... 

The atmosphere of Mars has several features that are distinct from that of the Earth and require a somewhat different planetary history. At likely nebular temperatures and pressures at its radial distance. Mars is too small to have condensed a dense early atmosphere from the nebula even in the limiting case of isothermal capture (Hunten, 1979 Pepin, 1991). Therefore, regardless of the plausibility of gravitational capture as a noble-gas source for primary atmospheres on Venus and Earth, some other way is needed to supply Mars. This may include solar-wind implantation or comets. An important feature is that, in contrast to Earth, martian xenon apparently did not evolve from a U-Xe progenitor, but rather from SW-Xe. This requires that accreting SW-Xe-rich materials that account for martian atmospheric xenon are from sources more localized in space or time and so have not dominated the terrestrial-atmospheric xenon budget. There are insufficient data to determineif the martian C/N ratio is like the terrestrial value, but it appears that the initial C/H2O ratio may have been. Further constraints on the sources of the major volatUes are required. 

Both limiting cases are presented in Fig. 4.22. Actual condensation rates lie between the two extremes. 

When one of the components in a mixture is a condensable vapor and the pores are small enough, the condensate can block gas-phase diffusion through the pores. This is the limiting case of surface diffusion where the adsorbed layer fills the pore. This condensate will evaporate on the low partial pressure side of the membrane. The Kelvin equation predicts that condensation can occur in small pores even through the partial pressure of that component is below its vapor pressure. The Kelvin equation represents thermodynamic equilibrium between the gas phase and fluid in the pore ... 

In this chapter, we develop efficient methods for the calculation of 4WM and SRF processes of large polyatomic molecules in condensed phases (e.g., solution, solid matrices, and glasses). The key quantity in the present formulation is the nonlinear response function R(t3,t2,ti)> which contains all the microscopic information relevant for any type of 4WM and SRF.6,11 12>19-2o>57 In Section II we introduce the nonlinear response function and derive the general formal expression for 4WM. The two ideal limiting cases of time-... 

There are two limiting cases. One is the dew point, where L/V = 0 and the product is entirely vapor but at the threshold of condensation. The relationship between pressure and temperature at the dew point is obtained by a summation of mole fractions ... 

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