Homogeneous fluids

As pointed out previously, the separation of homogeneous fluid mixtures requires the creation or addition of another phase. The most common method is by repeated vaporization and condensation— distillation. The three principal advantages of distillation are... 

Distillation is by far the most commonly used method for the separation of homogeneous fluid mixtures. The cost of distillation varies with operating pressure, which, in turn, is mainly determined by the molecular weight of the materials being separated. Its widespread use can be attributed to its ability to... 

The direct correlation fimction c(r) of a homogeneous fluid is related to the pair correlation fimction tiirough the Omstein-Zemike relation... 

In this section we discuss the frequency spectrum of excitations on a liquid surface. Wliile we used linearized equations of hydrodynamics in tire last section to obtain the density fluctuation spectrum in the bulk of a homogeneous fluid, here we use linear fluctuating hydrodynamics to derive an equation of motion for the instantaneous position of the interface. We tlien use this equation to analyse the fluctuations in such an inliomogeneous system, around equilibrium and around a NESS characterized by a small temperature gradient. More details can be found in [9, 10]. 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

Antontsev S.N., Kazhikhov A.V., Monakhov V.N. (1990) Boundary value problems in mechanics of homogeneous fluid. Studies in Math. Their Appl. 22, North-Holland. 

Fluid mixing is a unit operation carried out to homogenize fluids in terms of concentration of components, physical properties, and temperature, and create dispersions of mutually insoluble phases. It is frequently encountered in the process industry using various physical operations and mass-transfer/reaction systems (Table 1). These industries include petroleum (qv), chemical, food, pharmaceutical, paper (qv), and mining. The fundamental mechanism of this most common industrial operation involves physical movement of material between various parts of the whole mass (see Supplement). This is achieved by transmitting mechanical energy to force the fluid motion. 

The systems of interest in chemical technology are usually comprised of fluids not appreciably influenced by surface, gravitational, electrical, or magnetic effects. For such homogeneous fluids, molar or specific volume, V, is observed to be a function of temperature, T, pressure, P, and composition. This observation leads to the basic postulate that macroscopic properties of homogeneous PPIT systems at internal equiUbrium can be expressed as functions of temperature, pressure, and composition only. Thus the internal energy and the entropy are functions of temperature, pressure, and composition. These molar or unit mass properties, represented by the symbols U, and S, are independent of system size and are intensive. Total system properties, J and S do depend on system size and are extensive. Thus, if the system contains n moles of fluid, = nAf, where Af is a molar property. Temperature... 

Consider a closed, nonreacting PTT system containing n moles of a homogeneous fluid mixture. The mole numbers of the individual chemical species sum to... 

For the more general case of a homogeneous fluid in which the vary for whatever reason, it is assumed that n J is a function, F, of the as well as of nS and nV ... 

Equations 54 and 58 through 60 are equivalent forms of the fundamental property relation apphcable to changes between equihbtium states in any homogeneous fluid system, either open or closed. Equation 58 shows that ff is a function of 5" and P. Similarly, Pi is a function of T and C, and G is a function of T and P The choice of which equation to use in a particular apphcation is dictated by convenience. Elowever, the Gibbs energy, G, is of particular importance because of its unique functional relation to T, P, and the the variables of primary interest in chemical technology. Thus, by equation 60,... 

When equations 54 and 58 through 60 are appHed to the special case of one mole of a homogeneous fluid of constant composition, n = 1, dn = 0, and these equations reduce to the following ... 

Because the macroscopic-intensive properties of homogeneous fluids in equilibrium states ate functions of T, P, and composition, it follows that the total property of a phase fiM can be expressed functionally as in equation 113 ... 

The definitions of enthalpy, H, Helmholtz free energy. A, and Gibbs free energy, G, also give equivalent forms of the fundamental relation (3) which apply to changes between equiUbrium states in any homogeneous fluid system ... 

Equations (4-34) and (4-35) are general expressions for the enthalpy and entropy of homogeneous fluids at constant composition as functions of T and P. The coefficients of dT and dP are expressed in terms of measurable quantities. 

Partial Molar Properties Consider a homogeneous fluid solution comprised of any number of chemical species. For such a PVT system let the symbol M represent the molar (or unit-mass) value of any extensive thermodynamic property of the solution, where M may stand in turn for U, H, S, and so on. A total-system property is then nM, where n = Xi/i, and i is the index identifying chemical species. One might expect the solution propei fy M to be related solely to the properties M, of the pure chemical species which comprise the solution. However, no such generally vahd relation is known, and the connection must be establi ed experimentally for eveiy specific system. 

If chemical reactions occur only over the catalyst and none on the walls or in the homogeneous fluid stream in the recycle loop, then conservation laws require that the two balances should be equal. 

The discussion so far has related to homogenous fluids where the density is the same everywhere in the fluid. In the presence of a surface, the density p r) is a function of position and the fluid is inhomogeneous. 

For homogeneous fluids, it is desirable to have a systematic method of going beyond the closures that have been discussed above. Equation (21) can be generalized... 

It is desirable to have a systematic procedure for going beyond these approximations. Attard [110] has suggested using Eq. (80) for this purpose. In an application to a homogeneous fluid, we use Eq. (80) but regard the source of the inhomogeneity as one of the molecules. When Eq. (80) is used in this manner, we shall call it the OZ2 equation. 

The equilibrium theory of homogeneous fluids may be constructed by using the hierarchy of the direct correlation functions [48]. This approach has been of much utility for the development of the theory of inhomogeneous simple fluids. The hierarchy of the direct correlation functions is defined by the following relation... 

Expanding /g around the global equilibrium solution /eq at u = 0 in available scalar products using the vectors cg and u, we have, formally, in the homogeneous fluid approximation (Vm = 0),... 

It follows that 1/T is the integrating factor of SQ. Now since SQ is a function of two variables (in the simple case of a homogeneous fluid), and since the integrating factor of such a magni-" tude is usually also a function of the same two variables, we must regard the proposition that the integrating factor of SQ is a function of one variable only as expressing a physical, not a mathematical, truth. 

Taking the simple case of a homogeneous fluid of unchanging composition we see that its state may be defined in terms of any pair of the three variables temperature T, specific volume v, and pressure p. If the state is to be normally defined, T must be taken as one variable, and v must be taken as the other, because there is the condition to be satisfied that no external work is done when the temperature changes whilst the variable remains constant. This condition is satisfied by v, but not by/>. 

If we have unit mass of a homogeneous fluid having a uniform temperature T, and with its surface exposed to a uniform normal pressure p, its state can be defined (if we regard the chemical nature as fixed) in terms of the specific volume r, and the absolute temperature T. 

An equation in which the entropy of a homogeneous fluid is expressed as a function of its energy and volume is called by Planck (1909) a canonical equation. 

This notation admits of generalisation (Gibbs, 1876). The total energy of a homogeneous fluid is a continuous and single-valued function of the masses mi, m2, m3,. . mh of its constituents, of the total volume Y, and the total entropy S ... 

The conditions which lead a homogeneous fluid mixture to split into two separate fluid phases can be described by classical thermodynamic stability analysis as discussed in numerous textbooks.9 Such analysis has often been... 

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