Pure real fluids

Equation 153 is vaUd only for pure species / in the ideal gas state. For a real fluid, an analogous equation is as follows ... 

The authors noted that when their friction parameter M= (pG/,) 8/G is real, it is equivalent to the real slip parameter s = fe used by McHale et al. [14]. From this analysis, a real interfacial energy G /8 is related to the slip length b, for a purely viscous fluid, by... 

The nature of imaginary (or ficticious or hypothetical) standard states is most easily explained by reference to Fig. 12.18. The two dashed lines shown both conform to ideal-solution behavior as prescribed by Eq. (12.45). The points labeled fl(LR) and f°(HL) are both fugacities of pure i, but only Jf(LR) is the fugacity of pure i as it actually exists at the given T and P. The other point P(HL) represents an imaginary state of pure i in which its imaginary properties are fixed at values other than those of the real fluid. Either choice of value for f° fixes the entire line which represents ff = xj°. 

It is known that incompressible fluids represent a useful model for real fluids in fluid mechanics and heat and mass transfer. Their thermal equation of state is v = v0 = const. For pure substances and also for mixtures, isobaric and isochoric specific heat capacities agree with each other, cp = cv = c. 

The ideal gas equation of state cannot describe real fluids in most situations because the fluid molecules themselves occupy a finite volume and because they exert forces of attraction and repulsion on each other. As the gas is cooled, and assuming its pressure is below the critical point, a temperature is reached where the intermolecular interactions result in a transition from the gas phase to the liquid phase. The ultimate fluid model would be one that could describe this transition as well as the fluid behavior over the entire range of temperature and pressure. Such a model would also be capable of representing mixtures as well as pure components. 

Given the apparent arbitrariness of the assumptions in a purely continuum-mechanics-based theory and the desire to obtain results that apply to at least some real fluids, there has been a historical tendency to either relax the Newtonian fluid assumptions one at a time (for example, to seek a constitutive equation that allows quadratic as well as linear dependence on strain rate, but to retain the other assumptions) or to make assumptions of such generality that they must apply to some real materials (for example, we might suppose that stress is a functional over past times of the strain rate, but without specifying any particular form). The former approach tends to produce very specific and reasonable-appearing constitutive models that, unfortunately, do not appear to correspond to any real fluids. The best-known example is the so-called Stokesian fluid. If it is assumed that the stress is a nonlinear function of the strain rate E, but otherwise satisfies the Newtonian fluid assumptions of isotropy and dependence on E only at the same point and at the same moment in time, it can be shown (see, e.g., Leigh29) that the most general form allowed for the constitutive model is... 

The ratio of fugacity to partial pressure,// / ), called the fugacity coefficient, ([) , is in common use as a measure of departure of a real fluid/ from its ideal-gas value. For a pure fluid, the fugacity coefficient is simply < >=f/p. For an ideal-gas mixture, ([) = 1 for all i for a pure ideal gas, ([) = 1. For a real fluid, by rearranging Equation (4.305),... 

In fact, since we know that fluids exist in thermodynamically stable states (experimental observation 7 of Sec. 1.7), we will take as an empirical fact that drS < 0 for all real fluids at equilibrium,.and instead establish the restrictions placed on the equations of state of fluids by this stability condition. We first study the problem of the intrinsic stability of the equilibrium state in a pure single-phase fluid, and then the mutual stability of two interacting systems or phases. 

Equations 9.3-3 to 9.3-5 resemble those obtained in Sec. 9.1 for the ideal gas mixture. There is an important difference, however. In the present case we are considering an ideal mixture of fluids that are not ideal gases, so each of the pure-component properties here will not be an ideal gas property, but rather a real fluid property that must either be measured or computed using the techniques described in Chapter 6. Thus, the molar volume Vj is not equal to RT/P, and the fugacity of each species is not equal to the pressure. 

We will now discuss the problem of determining effective or optimal diameters for use with the HSE theory for real fluids when both the form of the intermolecular potential and its parameters are unknown but accurate equations of state which represent the PVT behavior over an extensive range are available for the pure components. 

Similarly as in preceding models (cf. Sect. 1.1, Rems. 6, 9, 8, 42 in Chaps. 2, 3, respectively) we exclude unusual situations by regularity conditions. Even though some exclusions are similar to those for pure materials and possible in (especially non-reacting) mixtures (e.g. disintegration of real fluid mixture to more phases which is outside of our models), the situation is much more complicated in chemical reacting mixtures because of non-linearity of chemical reaction rates in our model (transport phenomena are linear as in pure fluid of Sect. 3.7). 

As described in Section 2.4.3, the behavior of a real fluid can also be expressed by means of the fugacity, which is applicable to mixtures as well The fugadty of a species in a mixture is defined in the same way as the fugacity of a pure component... 

Most of the results of this initial paper are comparisons with simulation data for chains with various parameters, although pure-component parameters for six hydrocarbons and two associating fluids were fitted. No results for mixtures of real fluids are presented. 

One of the reasons that the Huang and Radosz version of SAFT has been adopted (and is widely referred to as original SAFT) is that they undertook an extensive pure-component parameterization for over 100 pure fluids. This meant that their equation of state could be used immediately for real fluids of industrial interest without any intermediate steps. Both Huang and Radosz were employed by Exxon during the development of SAFT. The fact that the model had the backing of a major oil company may also help explain its rapid adoption and use as an engineering tool. 

At this step, the concept of residual molar enihalpy needs to be introduced /i is a difference measure for how a substance deviates from the behaviour of a p>erfect gas having the same temperature T as the real substance. The molar enthalpy of a pure fluid can thus be written as the summation of the molar enthalpy of a p>erfect gas having the same temperature as the real fluid plus a residual term ... 

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. 

Most of the information about purely 2D fluids has been obtained from computer simulations or from the application of theories commonly used in the three-dimensional case. In this subsection, we will summarize the most important studies and results about the properties of the 2D Lermard-Jones system, which is the most widely used model. The 2D L-J system is defined through the intermolecular potential given in Eq. (12), where now the distance is considered only in two dimensions, the L-J parameters a and real fluid. In both computer simulations and theories, the thermodynamic properties are commonly expressed in reduced (adimensional) units, marked with a superscript, which are related to the real imits through the L-J parameters as follows ... 

Equations of State. Equations of state having adjustable parameters are often used to model the pressure—volume—temperature (PVT) behavior of pure fluids and mixtures (1,2). Equations that are cubic in specific volume, such as a van der Waals equation having two adjustable parameters, are the mathematically simplest forms capable of representing the two real volume roots associated with phase equiUbrium, or the three roots (vapor, Hquid, sohd) characteristic of the triple point. 

Affinity chromatography (12) has become an important tool in the isolation of purified fractions of such substances as enzymes. Advantage is taken of specific interactions such as antigen-antibody interactions. One substance of the pair (e.g. antigen) is bonded to a support. When a mixture is passed through the column, the specific interaction retains the corresponding antibody relative to other substances. A change of mobile phase conditions then elutes the pure antibody. This method has a real potential for analysis of specific proteins in body fluids. 

The fact that the velocity of a fluid changes from layer to layer is evidence of a kind of friction between these layers. The layers are mathematical constructs, but the velocity gradient is real and a characteristic of the fluid. The property of a fluid that describes the internal friction or resistance to flow is the viscosity of the material. Chapter 4 is devoted to a discussion of the measurement and interpretation of viscosity. For now, it is enough for us to recall that this property is quantified by the coefficient of viscosity 77 of a material. The coefficient of viscosity has dimensions of mass length-1 time-1, kg m-ls-1 in SI units. In actual practice, the cgs unit of viscosity, the poise (P), is widely used. Note that pure water at 20°C has a viscosity of about 0.01 P = 10-3kgm-ls-1... 

Mixtures of equisized charged spheres were also treated by the MSA. Such a system is then uniquely characterized by the ratio of the critical temperatures of the pure components. Harvey [235] found that a continuous critical curve from the dipolar solvent to the molten salt is maintained until the critical temperature of the ionic component exceeds that of the dipolar component by a factor of about 3.6. This ratio is much higher than theoretically predicted for nonionic model fluids. We recall that for NaCl the critical line is still continuous at a critical temperature ratio of about 5. Thus, the MSA of the charged-hard-sphere-dipolar-hard-sphere system captures, at least in part, some unusual features of real salt-water systems with regard to their critical curves. 

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