Fluid dynamic critical pressure

The velocity in the narrowest cross section in a safety valve may reach the sound velocity at maximum. Simultaneously, the pressure decreases to its minimum value, the fluid dynamic critical pressure, and the mass flow rate is maximized. 

The fluid dynamic critical pressure and the critical pressure ratio r) = Pait/po in a nozzle throat may be calculated by differentiating the sizing coefficient, Eq. (15.21), with respect to the pressure ratio and setting the result equal to zero. Enuther mathematical transformations lead to the critical pressure ratio of a real gas ... 

Fluid dynamic critical pressure and temperature in the throat area of the nozzle and the narrowest flow cross section of the safety valve ... 

Iterative procedure to calculate the fluid dynamic critical pressure and temperature. 

A much more complex behaviour is observed for the process of penetration of various proteins into phospholipid monolayers. This behaviour depends strongly on the protein and the solution properties although some common features are observed. Fainerman et al. [116] studied the P-lactoglobulin penetration dynamics into DPPC monolayers. For a (i-lactoglobulin bulk concentration of 510 mol/l and molar areas of the lipid larger than the critical value, A > A, first order phase transitions are observed. Thus, two-dimensional condensed phase are formed although at these molar area values the pure DPPC monolayer exists only in the fluid-like state and does not form any domains. The first-order phase transition in the DPPC monolayer becomes visible by the characteristic break point in the dynamic surface pressure curve Fl(t) (see Fig. 4.50). 

As a result of the significant variation in thermodynamic properties near and at the critical point, it is difficult to use Computational Fluid Dynamics (CFD) when modeling supercritical flows. Also, since small changes in temperature and pressure can have large effects on the structure of a fluid near the critical point, local property values are very important. 

In this Chapter we have made an attempt to describe mixing and heat and mass transfer in supercritical fluids. Using Laser Doppler Anemometry, Computational Fluid Dynamics and High-Pressure Calorimetry, some basic guidelines have been derived. When compared to the behavior of ordinary liquids, the behavior of SCCO2 is quite different, especially near the critical point. However, when the thermodynamic behavior of CO2 is taken into account (in terms of constant-pressure heat capacity, viscosity, and thermal conductivity), its behavior is consistent with that of other liquids. 

A critical aspect, especially in the design of cold-wall reactors is the fluid dynamics of the gas flow. Even though the pressure in the reactor is reduced, the gas density is still sufficient to apply conventional fluid mechanics to its flow. Fluid mechanics tells us that a gas flowing down a smooth tube will move without turbulence as long as the Reynolds number, R , is less than 2000. The Reynolds number in a tube is given by ... 

SEE is an instrumental approach not unlike PLE except that a supercritical fluid rather than a liquid is used as the extraction solvent. SFE and PLE employ the same procedures for preparing samples and loading extraction vessels, and the same concepts of static and dynamic extractions are also pertinent. SFE typically requires higher pressure than PLE to maintain supercritical conditions and, for this reason, SFE usually requires a restrictor to control better the flow and pressure of the extraction fluid. CO2 is by far the most common solvent used in SFE owing to its relatively low critical point (78 atm and 31 °C), extraction properties, availability, gaseous natural state, and safety. 

The phenomenon of critical flow is well known for the case of single-phase compressible flow through nozzles or orifices. When the differential pressure over the restriction is increased beyond a certain critical value, the mass flow rate ceases to increase. At that point it has reached its maximum possible value, called the critical flow rate, and the flow is characterized by the attainment of the critical state of the fluid at the throat of the restriction. This state is readily calculable for an isen-tropic expansion from gas dynamics. Since a two-phase gas-liquid mixture is a compressible fluid, a similar phenomenon may be expected to occur for such flows. In fact, two-phase critical flows have been observed, but they are more complicated than single-phase flows because of the liquid flashing as the pressure decreases along the flow path. The phase change may cause the flow pattern transition, and departure from phase equilibrium can be anticipated when the expansion is rapid. Interest in critical two-phase flow arises from the importance of predicting dis-... 

For chemical applications, vibrational spectroscopy of high-pressure fluid phases, including liquids and compressed gases, is of special importance (Buback, 1991). The fluid, i.e., the non-solid region of a substance, is illustrated in Fig. 6.7-2. The packing density of the circles is approximately proportional to the density of a substance. The bottom left part of Fig. 6.7-2 shows the vapor pressure curve which, up to the critical point, separates the liquid phase from the gas phase. Above the critical temperature (7 ), the density of a substance may change continuously between gaseous and liquid like states vibrational spectroscopic methods make it possible to study the structure and dynamics... 

Gas flow processes through microporous materials are important to many industrial applications involving membrane gas separations. Permeability measurements through mesoporous media have been published exhibiting a maximum at some relative pressure, a fact that has been attributed to the occurrence of capillary condensation and the menisci formed at the gas-liquid interface [1,2]. Although, similar results, implying a transition in the adsorbed phase, have been reported for microporous media [3] and several theoretical studies [4-6] have been carried out, a comprehensive explanation of the static and dynamic behavior of fluids in micropores is yet to be given, especially when supercritical conditions are considered. Supercritical fluids attract, nowadays, both industrial and scientific interest, due to their unique thermodynamic properties at the vicinity of the critical point. For example supercritical CO2 is widely used in industry as an extraction solvent as well as for chemical... 

Dynamic light scattering has been utilized to directly measure droplet diffusion coefficients in near-critical and supercritical fluids. Blitz et al.f and Fulton et measured the effect of pressure on... 

CCT, critical cracking thickness Boltzmann constant (1.381x10 local permeability [m ] fracture resistance [N m ] average permeability in/of compact [m ] particle shape factor compact thickness [m] initial particle number concentration [m refractive index of particle material refractive index of dispersion material number density of ion i dimensionless number dimensionless number Stokes number Peclet number capillary pressure [N-m ] dynamic pressure [N m ] local liquid pressure in the compact [N-m local solid pressure in the compact [N-m ] superficial fluid velocity [m-s q gas constant [J K ] centre to centre distance [m]... 

To extract TMB from TMB-methanol mixtures it is necessary to find a solvent that is relatively immiscible in methanol yet is miscible with TMB at the same conditions. TMB is very soluble in benzene, hexane, heptane, nonane, and carbon tetrachloride indicating that it exhibits very lipophilic characteristics (Plank and Christopher, 1976 Niswonger, Plank, and Laukhuaf, 1985 Schmidt, Plank, and Laukhuf, 1985 Munster et al., 1984). Hence, TMB should be soluble in the more common supercritical fluid solvents such as ethane and carbon dioxide. Methanol is moderately miscible with xenon, ethane, ethylene, and carbon dioxide since a single phase is obtained at pressures of less than —200 bar at temperatures between the respective critical temperatures of the binary components (Brunner, 1985). To obtain quickly an estimate of the distribution coefficient for TMB in carbon dioxide, ethane, and ethylene, rapid screening experiments were performed with a dynamic flow apparatus at temperatures ranging from 0 to 55°C at a number of pressures. From this preliminary study it was found that carbon dioxide does not... 

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