Equilibrium phase-space dispersion

Figure 2. The temperature dependence of the equilibrium phase space dispersions, reported in relation to their zeroth order values. Note that for the time-local QDT expansion formulation, (p )2 = (p )o [cf. Eq. (5.2)] and the lowest order correction to the momentum dispersion is of the 4 order. The system-bath interaction strength is rjB = 0.5MUh and the cut-off frequency cUc = Uh- Included as inserts are also the memory-kernel COP-CS-QDT results.

Figure 2 depicts the equilibrium phase space dispersions q ) and (p ) as the functions of temperature, resulting from the exact (solid), the fourth-order cumulant (dash), and the second-order cumulant (dot) expansion formulations. The latter amounts to the POP-CS-QDT or the CODDE as they share the same equilibrium properties. The dissipation-free values,... 

E. General theory of equilibrium phase-space dispersions... 

The problem to be solved in this paragraph is to determine the rate of spread of the chromatogram under the following conditions. The gas and liquid phases flow in the annular space between two coaxial cylinders of radii ro and r2, the interface being a cylinder with the same axis and radius rx (0 r0 < r < r2). Both phases may be in motion with linear velocity a function of radial distance from the axis, r, and the solute diffuses in both phases with a diffusion coefficient which may also be a function of r. At equilibrium the concentration of solute in the liquid, c2, is a constant multiple of that in the gas, ci(c2 = acj) and at any instant the rate of transfer across the interface is proportional to the distance from equilibrium there, i.e. the value of (c2 - aci). The dispersion of the solute is due to three processes (i) the combined effect of diffusion and convection in the gas phase, (ii) the finite rate of transfer at the interface, (iii) the combined effect of diffusion and convection in the liquid phase. In what follows the equations will often be in sets of five, labelled (a),..., (e) the differential equations expression the three processes (i), (ii) (iii) above are always (b), (c) and (d), respectively equations (a) and (e) represent the condition that there is no flow over the boundaries at r = r0 and r = r2. 

The preceding condition of thermodynamic equilibrium implies that the curvature of the dispersed phase becomes zero at the transition from two to three phases. In the middle-phase microemulsion, the pressures p and p2 fluctuate in time and space because of the instability of the interface between the two media (see below), and intuition suggests that Eqs. (37) and (40) be replaced with their average, hence that the condition of zero curvature be replaced with the condition of mean (with respect to time) average curvature. 

In this equation, c is the concentration in the fluid phase and q is the quantity in the solid phase. The column porosity e (expressed as phase ratio f = (1 -e)/e) defines the fraction of the fluid phase in the column. Furthermore, u stands for the linear velocity and t and x are the time and space coordinates, respectively. All contributions leading to band-broadening are lumped in a simplifying manner into an apparent dispersion coefficient, D p. In equation (21-2), it is assumed that the two phases are constantly in equilibrium expressed by the adsorption isotherms. Due to the nonlinear character of the isotherm equations, the solution of equation (21-2) requires the use of numeri-... 

A computational procedure that marches forward in space must necessarily start from an initial condition that represents a deviation from equilibrium. For a partly dispersed shock wave, the difference in the vapour and liquid phase flow variables just downstream of the frozen shock discontinuity constitute the required initial departure from equilibrium. For a fully dispersed shock wave an initial, arbitrary perturbation of the flow must be specified. Step-by-... 

Measure the amount of water, e.g. 1 litre, with a graduated cylinder and pour into the zinc or brass vessel. Seal the vessel with a rubber bung and then shake it for 1 minute. This produces the vapour-space equilibrium between the liquid and gas phases. Carefully discharge any gauge pressure and then fit the dispersion cylinder and measuring head in position in the vessel. 

A computer analysis was performed of the loss-of-load event with delayed reactor trip, similar to that used in safety valve capacity evaluation, except that a conservative 20% safety valve blowdown and initial conditions biased to maximize pressurizer liquid level were assumed. The purpose of this analysis was to determine the pressurizer liquid level response and the RCS subcooling under these conservative conditions. For additional conservatism, adjustments were made to the computer-calculated pressurizer level on the basis of a very conservative pressurizer model. This model assumed that the initial saturated pressurizer liquid did not mix with the cooler insurge liquid, that the initial liquid remained in equilibrium with the pressurizer steam space, and that the steam which flashed during blowdown remained dispersed in the liquid phase and caused the liquid level to swell. The adjusted pressurizer water level vs time curve showed a maximum level of 78%, Reference 2, (1874 ft" ), below the safety valve nozzle elevation which is greater than 100% level, so that dry saturated steam flow to the safety valves is assured throughout the blowdown. The computer analysis also showed that adequate subcooling was maintained in the RCS during the blowdown, so that steam bubble formation is precluded. 

When gas phase adsorption takes place in a large column, heat generated due to adsorption cannot be removed from the bed wall and accumulated in the bed because of poor beat transfer characteristics in packed beds of particles. A typical model of this situations is an adiabatic adsorption. The fundamental relations for this case are Eqs. (8-22), (8-38), (8-39) and (8-40), which are essentially similar to those employed by Pan and Basmadjian (1970). Thermal equilibrium between particle and fluid is assumed and oidy axial dispersion of heat is taken into account while mass transfer resistance between fluid phase and particle as well as axial dispersion is considered. This situation is identical with the model employed in the previous section. For further simpliHcation, axial dispersion effect may be involved in the overall mass transfer coefficient of the linear driving force model as discussed in Chapter S. In this case, after further justifiable simplifications such as negligible heat capacity and accumulation of adsorbate in void spaces, a set of basic equations to describe heat and mass balances can be ven as follows. 

Sieve tray columns have also fotmd an application in staged countercurrent liquid extraction operations. The perforated plates, arranged in much the same way as in gas-liquid contact, act to break up accumulations of the dispersed phase and provide fresh surfaces for renewed mass transfer. Each of the plates, or rather the space between them, is a potential equilibrium stage, but the efficiency E is generally quite low. 

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