Equations Rayleigh

This equation was first given by Lord Rayleigh and is called the Rayleigh equation. Integration between the initial number of moles n o in the still with composition over any time yields the following ... 

By using vapor-liquid equilibrium data the above integral can be evaluated numerically. A graphical method is also possible, where a plot of l/(y - xj versus Xr is prepared and the area under the curve over the limits between the initial and fmal mole fraction is determined. However, for special cases the integration can be done analytically. If pressure is constant, the temperature change in the still is small, and the vapor-liquid equilibrium values (K-values, defined as K=y/x for each component) are independent from composition, integration of the Rayleigh equation yields ... 

Fischer A, J Bauer, RU Meckenstock, W Stickler, C Griebler, P Maloszewski, M Kastner, HH Richnow (2006) A multitracer test proving the reliability of Rayleigh equation-based approach for assessing hiodegra-datyion in a BTEX contaminated aquifer. Environ Sci Technol 40 4245-4252. 

Compare the computer predictions with the Rayleigh equation prediction, where ... 

Equation 14.7 is known as the Rayleigh Equation and describes the material balance around the distillation pot. 

Figure 14.8 Integration of the Rayleigh Equation for constant reflux ratio. 

The vapor pressure in the bubble is related to the liquid pressure at the bubble interface and the surface tension force by Eq. (2-3). Introducing this result into Eq. (2-28), the Rayleigh equation (Rayleigh, 1917) for isothermal bubble dynamics is obtained as... 

The Rayleigh equation (9.3.3) rests on the assumption of constant D/s. This problem was discussed by Allegre et al. (1977). The mineralogy of the cumulate during fractional crystallization varies nearly step-wise provided phase boundaries remain nearly linear... 

Figure 9.5 Evolution of the concentration with the fraction crystallized (from right to left) for the fractional crystallization model [Rayleigh equation (9.3.3), heavy line] and two models of magma chamber with periodic recharge, periodic eruption and continuous fractionation [equations (9.4.7) and (9.4.8)].

In the case of reactions where the products do not continue to exchange with other phases in the system, as might be the case during precipitation of a mineral from solution, Rayleigh fractionation may best describe the changes in 5 E values for the individual components. The well-known Rayleigh equation (Rayleigh 1902) is ... 

Equations (3.1) and (3.2) yield the following relationship, the so-called Rayleigh equation ... 

The starting point for the Forster-Zuber theory (F4, F5, F6) is the Rayleigh equation (Rl) for a bubble growing in a liquid medium. In this... 

Since Rayleigh scattering does not apply to particles in the colloidal size range, we do not present the derivation in detail instead, Table 5.1 summarizes some key steps in the development of the Rayleigh equation ... 

TABLE 5.1 Steps Involved in the Derivation of the Rayleigh Equation... 

Rayleigh X Rs, particle size Applicable for (R/X) < 1/20 extension of the Rayleigh equation to solutions allows the measurement of osmotic pressure, molecular weight, and turbidity of colloidal or polymer solutions see Section 5.3... 

Equation 13 reduces to the Rayeigh equation (3) when the ratio of the gas-phase diffusivities, , is unity. Since gas-phase diffusivity is inversely proportional to the square root of the reduced mass, in the case of fission product-sodium systems where sodium has the smallest molecular weight, the above diffusivity ratio is less than unity. Therefore, the Rayleigh equation, which was derived on the basis of equilibrium vaporization, in fact represents an upper limit for the fractional fission-... 

Equilibrium Vaporization. The cesium release results presented in this chapter may also be used to demonstrate our earlier conclusion that equilbirium vaporization represents the upper limit for the fractional fission-product release as a function of sodium vaporization. Figure 6 shows three cesium release curves. Curve A was calculated from the Rayleigh Equation in conjunction with the partial molar excess free energy of mixing of infinitely dilute cesium—sodium solutions reported... 

We shall call this a quasilinear Fokker-Planck equation, to indicate that it has the form (1.1) with constant B but nonlinear It is clear that this equation can only be correct if F(X) varies so slowly that it is practically constant over a distance in which the velocity is damped. On the other hand, the Rayleigh equation (4.6) involves only the velocity and cannot accommodate a spatial inhomogeneity. It is therefore necessary, if F does not vary sufficiently slowly for (7.1) to hold, to describe the particle by the joint probability distribution P(X, V, t). We construct the bivariate Fokker-Planck equation for it. 

A one-dimensional Fokker-Planck equation was used by Smoluchowski [19], and the bivariate Fokker-Planck equation in phase space was investigated by Klein [21] and Kramers [22], Note that, in essence, the Rayleigh equation [23] is a monovariate Fokker-Planck equation in velocity space. Physically, the Fokker-Planck equation describes the temporal change of the pdf of a particle subjected to diffusive motion and an external drift, manifest in the second- and first-order spatial derivatives, respectively. Mathematically, it is a linear second-order parabolic partial differential equation, and it is also referred to as a forward Kolmogorov equation. The most comprehensive reference for Fokker-Planck equations is probably Risken s monograph [14]. 

Equation (58) is equivalent to the fractional Rayleigh equation [75, 77], and therefore we refer to Eq. (58) as the fractional Ornstein-Uhlenbeck process. For the sharp initial condition Wo(x) = <5(x — xo), the solution to this process is, according to Eq. (46), given by... 

The integration of the fractional Klein-Kramers equation (69) over the position coordinate leads in, the force-free limit, to the fractional Rayleigh equation... 

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