Hard sphere fluids real gases

FIGURE 2.2 Radial distribution functions for (a) a hard sphere fluid, (A) a real gas, (c) a liquid, (li) a crystal. 

Just as the ideal gas forms a convenient point of reference in discussing the properties of real gases, so does the hard-sphere fluid in discussing the properties of liquids. This is especially true at low densities, where the role of intermolecular forces in real systems is not so important. In this limit, the hard-sphere model is useful in developing the theory of solutions, as will be seen in chapter 3. 

As the density of a fluid is decreased, the effects of forces between molecules weaken, and the fluid behaves more like an ideal gas that is, the behavior of real fluids may simplify under extreme conditions. Another extreme occurs by making the temperature high, for then many simple fluids behave as if they were composed of hard spheres ... 

Real fluids are neither ideal gases nor are they composed of hard spheres. But if the density is low, a gas might be nearly ideal, or if the temperature is high, a gas might behave somewhat like a fluid of hard spheres. In such cases the ideal-gas or hard-sphere models may serve as references in expansions that approximate real behavior. In this section we consider Taylor expansions (see Appendix A) of the compressibility factor Z about that for the ideal gas. The expansions may be done using either density or pressure as the independent variable we introduce the density expansion first. 

Recall that the virial equations originate from Taylor expansions about the ideal gas. Alternatives can be obtained by expanding, not about ideal gases, but about hard spheres. Real fluids approach hard-sphere behavior in the isochoric high-temperature limit (4.5.1), so we use the parameter P = 1/RT as the independent variable. Then on truncating the expansion at first-order, we have... 

As has been made clear in these earlier sections, the Enskog theory, even for hard-sphere systems, is an ad hoc extension of the dilute-gas theory to a dense system. The theory does not, therefore, properly account for the correlations in velocity space discussed above. Furthermore, while a number of methods have been suggested whereby the Enskog theory may be applied to real fluids, all of them require further steps away from a rigorous theory, and, for that reason, none of them is particularly satisfactory. Here just one of the approaches suggested is considered, which has the virtue of self-consistency and is not significantly less accurate than other proposals. 

The essence of the application of these equations to real fluids consists of three parts first the replacement of the hard-sphere results for the pure gas viscosity and the interaction viscosity by the values for the real fluid system second, the evaluation of a pseudo-radial distribution function for each binary interaction to replace the hard-sphere equivalent at contact and, finally, the selection of a molecular size parameter for each binary interaction to account for the mean free-path shortening in the dense gas. 

In the first coarse-grained approach, the discrete phase is treated as an Eulerian continuum, interpenetrating with the real continuous phase. The particle-particle interactions are then captured by an effective particle phase rheology obtained from kinetic theory of granular flows. These so-called two-fluid (Euler-Euler) models have been very successful at predicting the dynamic properties of, e.g., gas-solid fluidized beds (see Van derHoefet al, 2008 Verma et al, 2013). Despite their success, two-fluid models also have their limitations they are usually limited to idealized cases ofmonodisperse hard sphere particles, while extensions to polydisperse mixtures (e.g., in size or in contact properties) are difficult to make. Also, because no particles are explicitly tracked, it is difficult to include particle properties which may vary from particle to particle, such as particle temperature, surface moisture concentration, or chemical surface species concentrations. 

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