Equation Eotvos

Homogeneous Nucleation (a) Using Eq. 8.6-2, calculate the rate of homogeneous nucleation of styrene as a function of temperature at atmospheric pressure and a temperature range from 145°C to 325°C. In calculating the pressure in the bubble, assume that it equals the vapor pressure (extrapolate it from lower temperature values). Use the Eotvos equation a = 2.1 (p/M)1 (Tc — T — 6), where the surface tension is in erg/cm3, temperature is in °C, and density in g/cm3, to evaluate the surface tension as a function of temperature. The critical temperature... 

Kapitza and Milner s equation for effect of magnetic field on boilffig-point, 376 Katayama s modification of Eotvos equation, 142... 

Later, Katayama replaced the density of the liquid by the difference in density between the liquid and saturated vapor in the Eotvos equation ... 

Sirk7 assumed that % is proportional to l/Vm2 (Fm=molar volume), hence %=XoVmo2 /Vm2, where Xo> V0 are the values at the absolute zero. The EotvOs equation ( 7.VIII G), aV2,3=K Tc—T)> with 7 =0 (when x=a) gives ... 

Moreover, for many liquids, knowing the temperature dependence of the density of the liquid, the Eotvos equation allows us to calculate the temperature coefficient of the surface... 

Figure 5.12 Estimated normal boiling point temperatures, of [C Cjim][Nty ionic liquids as a function of alkyl side chain length, n. O, 3" using the Eotvos equation with the data under discussion [26] , using the Guggenheim equation with the data under discussion [26] A, using the Guggenheim equation with data from [1] , using the Clausius-Clapeyron relation with experimental vapor pressure values from [33-35].

Figure 3.6 Decrease in water surface tension with temperature. This dependency can be described by the Ramsay-Shields or the Eotvos equations fTc is the critical temperature) ...

Equation (12-46) implies that the Eotvos number cannot exceed a value of about 16. Since the spherical-cap regime requires Eo > 40 (see Fig. 2.5), stability considerations explain why drops falling in gases and drops in many liquid-liquid systems never attain the spherical-cap regime. Moreover, since We = 4Eo/3Cd and is nearly constant for large drops in air, it is also possible to... 

For values of the Eotvos number higher than 40 (for the air-water system, this corresponds to dbub > 17 mm), the above equation simplifies to the well-known Davies-Taylor equation ... 

Eotvos having previously given a rather less accurate equation in which the constant 6° was not subtracted from the critical temperature. Mv is the molecular volume and Tc the critical temperature. 

Eotvos deduced his equation theoretically from considerations of corresponding states of liquids of similar molecular constitution, which are rather difficult to follow. The central point of the theory is, however, that surfaces should be compared on the basis of the number of molecules per unit area, which is, if the molecules are similar in shape and symmetrically packed, proportional to (Jfv)1. 

The accuracy of (5) is within the limits of experimental error for many liquids. As the linear relation does not usually hold near the critical temperature, (6) is less accurate, though it is preferable theoretically as containing one less arbitrary constant. Refinements of the linear equation have usually been made along one of two lines the first is to follow Eotvos s plan of introducing the twro-thirds power of the molecular volume, as in equation (1) of Ramsay and Shields, or Katayama s modification2... 

The temperature dependence of the surface tension of pure liquid is given by the Eotvos-Ramsay equation ... 

Esmaeeli et al [43] solved the full Navier-Stokes equations for a bubble rising in a quiescent liquid, or in a liquid with a linear velocity profile. The calculations were performed in 2-dimensional flow, but similar results have also been reported for 3-dimensional calculations. The surface tension forces were included, and the interface was allowed to deform. R was shown that deformation plays a major role in the lift on bubbles. Bubbles with a low surface tension have a larger Eotvos number, and are more prone to deform. 

If we further increase the temperature towards the critical temperature, Tc, the restraining force on the surface molecules diminishes, and the vapor pressure increases, and when Tc is reached, the surface tension vanishes altogether (y= 0). There are several empirical approaches using critical properties and molar volume to predict the surface tension of pure liquids. By comparing the surfaces on the basis of the number of similarly shaped and symmetrically packed molecules per unit area, Eotvos derived an equation in 1886,... 

It should be noted that Eotvos, Ramsay-Shields and Sugden s parachor equations are empirical in nature and their theoretical foundations are rather obscure. There have been several attempts to associate these equations with strict thermodynamical terms, but none have been successful. 

If the excess components of y and dueto orientational entropy, are, respectively, 61 and i.e., excess surface entropy over that corresponding to a normal Eotvos constant [17]—Equation 1 becomes... 

The above equation holds quite accurately for normal liquids at temperatures much nearer to the critical vedue than does the original form of Eotvos. In equation (4), according to Ramsay and Shields, the surface tension of the liquid becomes zero when the temperature reaches a value which is lesser than the critical temperature t, by 6°C. It follows that at critical temperature the surface tension of a normal liquid will become negative. The above prediction, however, is not universal. Several liquids have shown a value of zero only at the critical temperature and not earlier. For such liquids, Katayama s equation Is found to be the most suitable. It is expressed... 

The model species, total mass, momentum, and energy continuity equations are similar to those presented in Section 13.7 on fluidized bed reactors. Constant values of the gas and liquid phase densities, viscosities, and diffusivities were assumed, as well as constant values of the interphase mass transfer coefficient and the reaction rate coefficient. The interphase momentum transfer was modelled in terms of the Eotvos number as in Clift et al. [1978]. The Reynolds-Averaged Navier-Stokes approach was taken and a standard Computational Fluid Dynamics solver was used. In the continuous liquid phase, turbulence, that is, fluctuations in the flow field at the micro-scale, was accounted for using a standard single phase k-e model (see Chapter 12). Its applicability has been considered in detail by Sokolichin and Eigenberger [1999]. No turbulence model was used for the dispersed gas phase. Meso-scale fluctuations around the statistically stationary state occur and were explicitly calculated. This requires a transient simulation and sufficiently fine spatial and temporal grids. 

The question of the dependence of the size has been discussed a lot, and some equations of state have been proposed in the literature such as that of Eotvos in which the product ya depends only on temperature. 

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