Liquids as incompressible

Properties near the critical point are quite different compared to states at lower temperatures and pressures. As the difference between vapor and liquid becomes less clear near the critical point, the liquid becomes substantially more compressible than typical liquids. This is indicated on the PVgraph by the gentle slope of the isotherm as it approaches the critical point. Isotherms below but near the critical temperature (not shown in Figure 2-2) show similar behavior. The usual approximation that treats liquids as incompressible is acceptable only at temperatures well below the critical. In the supercritical region, the behavior of a fluid is somewhere between that of a liquid and a gas. The gentle slope of the isotherms indicates that the fluid is quite compressible, even at high, liquid-like densities Qow molar volumes). 

Equations [5.16] and [5.18] constitute Euler s equations for the flow of a compressible but inviscid liquid. They are not sufficient to describe the behaviour of an inviscid liquid, but need also a functional relationship between temperature, pressure and density of the liquid, often referred to as its equation of state. A common approximation is to treat the liquid as incompressible, when the equation of state, at constant temperature, becomes p = constant. The solution of a simple one-dimensional problem is given in the following example. 

The condition of irrotational flow is very important and needs further consideration. First, the equation for general irrotational flow will be derived and then the simplification introduced by treating the liquid as incompressible will be considered. 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

In fluid mechanics the principles of conservation of mass, conservation of momentum, the first and second laws of thermodynamics, and empirically developed correlations are used to predict the behavior of gases and liquids at rest or in motion. The field is generally divided into fluid statics and fluid dynamics and further subdivided on the basis of compressibility. Liquids can usually be considered as incompressible, while gases are usually assumed to be compressible. 

Depart from Geometric Similarity. Adding length to a tubular reactor while keeping the diameter constant allows both volume and external area to scale as S if the liquid is incompressible. Scaling in this manner gives poor results for gas-phase reactions. The quantitative aspects of such scaleups are discussed... 

In this equation, liquid has been assumed as incompressible. In the following Keller and Herring equations, the liquid compressibility has been taken into account to the first order of R jc-x. where c x is the sound velocity in the liquid far from a bubble [41]. 

This says that the sum of the local pressure (P) and static head (pgz), which we call the potential (4>), is constant at all points within a given isochoric (incompressible) fluid. This is an important result for such fluids, and it can be applied directly to determine how the pressure varies with elevation in a static liquid, as illustrated by the following example. 

The average specific volume V is a function of the gas and liquid specific volumes and the mass fraction of gas. The liquid may be treated as incompressible but, in general, the specific volume of the gas and the mass fraction will change along the length of the pipe. Differentiating equation 7.51... 

C From right to left, the line curves up initially and then becomes horizontal as it crosses the Tiquid/vapour curve. Once all vapour has been liquidized, the line climbs almost vertically as liquid is incompressible, leading to a rapid increase in pressure for a small decrease in volume. 

Solids are usually not used in cycles. Liquids can be approximated as incompressible (dv = 0) substances since their specific volumes remain near constant during a process. 

Consider the vacuum forming of a polymer sheet into a conical mold as shown in Figure 7.84. We want to derive an expression for the thickness distribution of the final, conical-shaped product. The sheet has an initial uniform thickness of ho and is isothermal. It is assumed that the polymer is incompressible, and it deforms as an elastic solid (rather than a viscous liquid as in previous analyses) the free bubble is uniform in thickness and has a spherical shape the free bubble remains isothermal, but the sheet solidifies upon confacf wifh fhe mold wall fhere is no slip on fhe walls, and fhe bubble fhickness is very small compared fo ifs size. The presenf analysis holds for fhermoforming processes when fhe free bubble is less than hemispherical, since beyond this point the thickness cannot be assumed as constant. 

Liquids can be considered as incompressible in many cases. For small differences in height, a gas might be regarded as incompressible. For an incompressible fluid, with constant g. Equation (2) becomes ... 

Taking into account the aforementioned effects of ice formation in porous materials, a macroscopic quintuple model within the framework of the Theory of Porous Media (TPM) for the numerical simulation of initial and boundary value problems of freezing and thawing processes in saturated porous materials will be investigated. The porous solid is made up of a granular or structured porous matrix (a = S) and ice (a = I), where it will be assumed that both phases have the same motion. Due to the different freezing points of water in the macro and micro pores, the liquid will be distinguished into bulk water ( a = L) in the macro pores and gel water (a = P, pore solution) in the micro pores. With exception of the gas phase (a = G), all constituents will be considered as incompressible. 

For liquids—and even for gases and vapors when the change in pressure is small—the fluid may be considered as incompressible for all practical purposes, and thus we may take wx = w2 = w as a constant. In turbulent flow the value of a is only a little more than unity, and, as a simplifying assumption, it will now be omitted. Then Eq. (10.5) becomes ... 

General case. In practice many flow problems approach either the isothermal or the adiabatic process, but in the event that neither applies it is possible to use Eq. (10.5) or Eq. (10.10). The same procedure may be used for vapors when they depart too widely from the perfect gas laws. If the pressure ratio p2/pi is of the order of 0.95 or more, the error will be negligible if the fluid is treated as incompressible and the equations for liquids are used. 

Explain in your own words and without the use of jargon (a) the three ways of obtaining values of physical properties (b) why some fluids are referred to as incompressible (c) the liquid volume additivity assumption and the species for which it is most likely to be valid (d) the term equation of state (e) what it means to assume ideal gas behavior (f) what it means to say that the specific volume of an ideal gas at standard temperature and pressure is 22.4 L/mol (g) the meaning of partial pressure (h) why volume fraction and mole fraction for ideal gases are identical (i) what the compressibility factor, z, represents, and what its value indicates about the validity of the ideal gas equation of state (j) why certain equations of state are referred to as cubic and (k) the physical meaning of critical temperature and pressure (explain them in terms of what happens when a vapor either below or above its critical temperature is compressed). 

A substance whose specific volume (or density) does not change with temperature or pressure is called an incompressible substance. The specific volumes of solids and liquids essentially remain constant during a process, and thus they can be approximated as incompressible substances without sacrificing much in accuracy. 

The densities of liquids are essentially constant, and thus the flow of liquids is typically incompressible. Therefore, liquids are usually referred to as incompressible substances. A pressure of 210 atm, for example, causes the density of liquid water at 1 atm to change by just 1 percent. Gases, on the other hand, arc highly compressible. A pressure change of Just 0.01 atm, for example, causes a change of 1 percent in the density of atmospheric air. 

Liquid flows are incompressible to a high level of accuracy, but the level of variation in density in gas flows and the consequent level of approximation made when modeling gas flows as incompressible depends on die Mach number defined as Ma - V/c, where c is the speed of sound whose value is 346 m/s in air at room lemperatute at sea level. Gas flows can often be approximated as incompressible if the density changes are under about 5 percent, which is usually the case when Ma < 0.3. Therefore, the comprc-ssibility effects of air can be neglected at speeds under about 100 m/s. Nole that the flow of a gas is not necessarily a compressible flow. 

Ideal liquids are incompressible and their flow is by definition frictionless. Real liquids are characterized by cohesion forces operating between the molecules, which bring about frictional forces, whose action is known as internal friction. 

In this section, we consider these problems in some detail, although with the major simplifications of assuming that the processes are isothermal and that the liquid is incompressible. As we shall see, the governing equations for even this simplified ID problem are nonlinear, and thus most features can be exposed only by either numerical or asymptotic techniques. In fact, the problem of single-bubble motion in a time-dependent pressure field turns out to be not only practically important, but also an ideal vehicle for illustrating a number of different asymptotic techniques, as well as introducing some concepts of stability theory. It is for this reason that the problem appears in this chapter. 

To the extent that a given liquid may be portrayed as incompressible, its availability can be obtained from the expression... 

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