Real fluids

It is detemrined experimentally an early study was the work of Andrews on carbon dioxide [1], The exact fonn of the equation of state is unknown for most substances except in rather simple cases, e.g. a ID gas of hard rods. However, the ideal gas law P = pkT, where /r is Boltzmaim s constant, is obeyed even by real fluids at high temperature and low densities, and systematic deviations from this are expressed in tenns of the virial series ... 

Unlike the pressure where p = 0 has physical meaning, the zero of free energy is arbitrary, so, instead of the ideal gas volume, we can use as a reference the molar volume of the real fluid at its critical point. A reduced Helmlioltz free energy in tenns of the reduced variables and F can be obtained by replacing a and b by their values m tenns of the critical constants... 

Sengers and coworkers (1999) have made calculations for the coexistence curve and the heat capacity of the real fluid SF and the real mixture 3-methylpentane + nitroethane and the agreement with experiment is excellent their comparison for the mixture [28] is shown in figure A2.5.28. 

Cubic equations, although simple and able to provide semiquantitative descriptions of real fluid behavior, are not generally useful for accurate representation of volumetric data over wide ranges of T and P. For such appHcations, more comprehensive expressions with large numbers of adjustable parameters are needed. 7h.e simplest of these are the extended virial equations, exemplified by the eight-constant Benedict-Webb-Rubin (BWR) equation of state (13) ... 

Property changes are readily determined for fluids in the ideal gas state, and these in combination with residual properties are used to compute property changes of real fluids. The computational scheme is suggested in Figure 5, and is based on the following identity ... 

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

Limiting L ws. Simple laws that tend to describe a narrow range of behavior of real fluids and substances, and which contain few, if any, adjustable parameters are called limiting laws. Models of this type include the ideal gas law equation of state and the Lewis-RandaH fugacity rule (10). 

A.ssessmentofUNIFy C. UNIFAC is a method to predict the activity of binary Hquid solutions in the absence of all data except stmctural information. Because state-of-the-art real fluid estimation methods are empirical or semi-empirical, the use of more data results in improved activity estimation. 

A real fluid will have a velocity gradient when flowing due to the viscosity of the fluid. 

Viscosity of the flowing fluid is constant. In general, this is not true for most real fluids. However, the effect is negligible if p at the average pressure is used. 

Lattice gases are micro-level rule-based simulations of macro-level fluid behavior. Lattice-gas models provide a powerful new tool in modeling real fluid behavior ([doolenQO], [doolenQl]). The idea is to reproduce the desired macroscopic behavior of a fluid by modeling the underlying microscopic dynamics. 

This chapter is organized into two main parts. To give the reader an appreciation of real fluids, and the kinds of behaviors that it is hoped can be captured by CA models, the first part provides a mostly physical discussion of continuum fluid dynamics. The basic equations of fluid dynamics, the so-called Navier-Stokes equations, are derived, the Reynolds Number is defined and the different routes to turbulence are described. Part I also includes an important discussion of the role that conservation laws play in the kinetic theory approach to fluid dynamics, a role that will be exploited by the CA models introduced in Part II. 

Since in real fluids, some of the energy of fluid flow is typically converted into heat by viscous forces, it is convenient to generalize equation 9.7 so that it allows for dissipation. Consider the momentum of fluid flowing through the volume dT (= pv). Since its time rate of change is given by d pv)/dt = dp/dt)v -f p dv/dt), we can use equations 9.3 and 9.7 to rewrite this expression as follows ... 

The negative sign indicates that all real fluids contract when the pressure increases. Since Bv/v is a mere number, the elasticity has the same dimensions as pressure. 

The relation between shear stress and shear rate for the Newtonian fluid is defined by a single parameter /z, the viscosity of the fluid. No single parameter model will describe non-Newtonian behaviour and models involving two or even more parameters only approximate to the characteristics of real fluids, and can be used only over a limited range of shear rates. 

When the fluid behaviour can be described by a power-law, the apparent viscosity for a shear-thinning fluid will be a minimum at the wall where the shear stress is a maximum, and will rise to a theoretical value of infinity at the pipe axis where the shear stress is zero. On the other hand, for a shear-thickening fluid the apparent viscosity will fall to zero at the pipe axis. It is apparent, therefore, that there will be some error in applying the power-law near the pipe axis since all real fluids have a limiting viscosity po at zero shear stress. The procedure is exactly analogous to that used for the Newtonian fluid, except that the power-law relation is used to relate shear stress to shear rate, as opposed to the simple Newtonian equation. 

The mass rate of flow of the real fluid between y = 0 and y — oo... 

An equation of state is an algebraic expression which relates temperature, pressure and molar volume, for a real fluid. 

In many ways these problems are more taxing and certainly longer than those in Volume 4, which gives the solutions to problems in Volume 1, and yet they have considerable merit in that they are concerned with real fluids and, more importantly, with industrial equipment and conditions. For this reason we hope that our efforts will be of interest to the professional engineer in industry as well as to the student, who must surely take some delight in the number of tutorial and examination questions which are attempted here. 

Figure 3.9 shows velocity profiles for various values of n. The profiles for the limiting values n = 0 and = are ol interest but it should be remembered that the behaviour of real fluids lies in the approximate range 0.2[Pg.120]

An alternative approach is possible. Just at the coefficients B, C, D, etc. define the thermodynamic properties of the real fluid so coefficients B°, C°, D°, etc. define thermodynamic properties for a hypothetical fluid which we will call the primary fluid. The primary fluid can be regarded as having the properties which the real fluid might have in the absence of association. It is assumed that when secondary interactions such as hydrogen bonding are imposed on the primary fluid the real fluid will be simulated. This assumption is an acceptable approximation at low densities, but is unlikely to hold at high densities where the addition of hydrogen bonds may produce new structural features. 

The primary fluid is replaced by the real fluid methyl fluoride. This removes the need to evaluate f(B0,C0,D°,etc.), as the thermodynamic function X required is simply that for methyl fluoride. 

In practice all real fluids have nonzero viscosity so that the concept of an inviscid fluid is an idealization. However, the development of hydrodynamics proceeded for centuries neglecting the effects of viscosity. Moreover, many features (but by no means all) of certain high Reynolds number flows can be treated in a satisfactory manner ignoring viscous effects. 

A more sophisticated approach is to avoid the postulate of a shock and instead to state the differential equations of conservation of mass, momentum, and energy to include more properties of a real fluid. Including the effects of viscosity, heat conditions, and diffusion along with chem reaction gives eqs with a unique solution for given boundary conditions and so solves the determinacy problem. The boundary conditions are restricted by the assumption that the reaction begins and is completed with the region considered. 

M-W. Evans C.M. Ablow, Chem Revs 61 (1961), p 147 (Definition of term detonation wave) p 152 (Steady detonation waves in real fluids) p 157 (Cylindrically symmetric flow in the steady zone of detonation wave) p 159 (Spherically symmetric flow in the steady zone of detonation wave) p 166 (Stability of detonation waves in which reaction is not complete) p 167 (One- dimensional transient reaction waves) 172,... 

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