Incompressible liquid

As an example of a simple application of Bernoulli s equation, consider the case of steady, fully developed flow of a liquid (incompressible) through an inclined pipe of constant diameter with no pump in the section considered. Bernoulli s equation for the section between planes 1 and 2 shown in Figure 1.5 can be written as... 

Liquid Incompressible. If this is the case, the Poynting correction (PC) becomes... 

The fugacity of a pure liquid component is close, but not identical, to its vapour pressure. By assuming the liquid incompressible, by combining the equations (5.43) and (5.79), and by integrating at constant temperature between the saturation pressure and the system pressure P, gives the relation ... 

At low to moderate pressures, assuming liquid incompressibility, the pure species molal volumes in (4-78) can be estimated by the method of Cavett using the empirical equation... 

The total fugacity, if the liquid is considered to be incompressible, isj calculated as a function of the vagor pressure by the expression ... 

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... 

Equation (3.20) is conventionally termed the Kelvin equation. The tacit assumption is made at the integration stage that K is independent of pressure, i.e. that the liquid is incompressible. 

Flows are typically considered compressible when the density varies by more than 5 to 10 percent. In practice compressible flows are normally limited to gases, supercritical fluids, and multiphase flows containing gases. Liquid flows are normally considerea incompressible, except for certain calculations involved in hydraulie transient analysis (see following) where compressibility effects are important even for nearly incompressible hquids with extremely small density variations. Textbooks on compressible gas flow include Shapiro Dynamics and Thermodynamics of Compre.ssible Fluid Flow, vol. 1 and 11, Ronald Press, New York [1953]) and Zucrow and Hofmann (G .s Dynamics, vol. 1 and 11, Wiley, New York [1976]). 

For fully developed incompressible horizontal gas/hquid flow, a quick estimate for Ri may be obtained from Fig. 6-27, as a function of the Lockhart-MartineUi parameter X defined by Eq. (6-131). Indications are that liquid volume fractious may be overpredicled for liquids more viscous than water (Alves, Chem. Eng. Prog., 50, 449-4.56 [19.54]), and uuderpredicted for pipes larger than 25 mm diameter (Baker, Oil Gas]., 53[12], 185-190, 192-195 [1954]). 

For fully developed incompressible cocurrent upflow of gases and liquids in vertical pipes, a variety of flow pattern terminologies and descriptions have appeared in the hterature some of these have been summarized and compared by Govier, Radford, and Dunn Can. J. Chem. Eng., 35, 58-70 [1957]). One reasonable classification of patterns is illustrated in Fig. 6-28. 

Many transient flows of liquids may be analyzed by using the full time-dependent equations of motion for incompressible flow. However, there are some phenomena that are controlled by the small compressibility of liquids. These phenomena are generally called hydraulic transients. 

For pressure, slow and fast systems are compared. An example of a slow system is control of the pressure of compressible gas in a large process volume. A fast system could be the pressure of fuel oil (incompressible liquid) supply to a burner. For flow, a single set of settings is recommended. 

The molecules of liquids are separated by relatively small distances so the attractive forces between molecules tend to hold firm within a definite volume at fixed temperature. Molecular forces also result in tlie phenomenon of interfacial tension. The repulsive forces between molecules exert a sufficiently powerful influence that volume changes caused by pressure changes can be neglected i.e. liquids are incompressible. 

If the mobile phase is a liquid, and can be considered incompressible, then the volume of the mobile phase eluted from the column, between the injection and the peak maximum, can be easily obtained from the product of the flow rate and the retention time. For more precise measurements, the volume of eluent can be directly measured volumetrically by means of a burette or other suitable volume measuring vessel that is placed at the end of the column. If the mobile phase is compressible, however, the volume of mobile phase that passes through the column, measured at the exit, will no longer represent the true retention volume, as the volume flow will increase continuously along the column as the pressure falls. This problem was solved by James and Martin [3], who derived a correction factor that allowed the actual retention volume to be calculated from the retention volume measured at the column outlet at atmospheric pressure, and a function of the inlet/outlet pressure ratio. This correction factor can be derived as follows. 

D. E. Martire, Unified Approach to the Theory of Chromatography Incompressible Binary Mobile Phase (Liquid Chromatography) in Theoretical Advancement in Chromatography and Related Separation Techniques (Ed. F. Dondi, G. Guiochon, IGuwer, Academic Publishers, Dordrecht, The Netherlands,(l993)261. 

The column length, as well as providing the required efficiency, is also defined by the D Arcy equation. The D Arcy equation describes the flow of a liquid through a packed bed in terms of the particle diameter, the pressure applied across the bed, the viscosity of the fluid and the linear velocity of the fluid. The D Arcy equation for an incompressible fluid is given as follows. 

Permeability is normally determined using linear flow in the incompressible or compressible form, depending on whether a liquid or gas is used as the flowing fluid. The volumetric flowrate Q (or Q ,) is determined at several pressure drops. Q (or Q ,) is plotted versus the average pressure p . The slope of this line will yield the fluid conductivity K or, if the fluid density and viscosity are known, it provides the intrinsic permeability k. For gases, the fluid conductivity depends on pressure, so that... 

Filter aids as well as flocculants are employed to improve the filtration characteristics of hard-to-filter suspensions. A filter aid is a finely divided solid material, consisting of hard, strong particles that are, en masse, incompressible. The most common filter aids are applied as an admix to the suspension. These include diatomaceous earth, expanded perlite, Solkafloc, fly ash, or carbon. Filter aids build up a porous, permeable, and rigid lattice structure that retains solid particles and allows the liquid to pass through. These materials are applied in small quantities in clarification or in cases where compressible solids have the potential to foul the filter medium. 

When the cake structure is composed of particles that are readily deformed or become rearranged under pressure, the resulting cake is characterized as being compressible. Those that are not readily deformed are referred to as sem-compressible, and those that deform only slightly are considered incompressible. Porosity (defined as the ratio of pore volume to the volume of cake) does not decrease with increasing pressure drop. The porosity of a compressible cake decreases under pressure, and its hydraulic resistance to the flow of the liquid phase increases with an increase in the pressure differential across the filter media. 

An incompressible fluid is a fluid whose density remains constant during flow. Liquids are normally treated as being incompressible, as a gas can be when only slight pressure variation occurs. 

A fluid can be considered as being liquid, which is incompressible, or a gas, which is easily compressible. When a force of sufficient magnitude is applied to a fluid, motion will occur provided the frictional resistance within an open system is overcome. 

In this section incompressible liquid flow is dealt with, and the effect of compressibility is ignored. 

Themtal. Thermal relief is needed in a vessel or piping run that is liquid-packed and can be isolated, for example pig launchers and meter provers. Liquid is subject to thermal expansion if it is heated. It is also incompressible. The thermal expansion due to heating by the sun from a nighttime temperature of 80°F to a sun-heated temperature of 120 F can be enough to rupture piping or a vessel. The required capacity of thermal relief valves is very small. 

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. 

Pumps are a mechanical device that forces a fluid to move from one position to another. Usually a pump refers to the mechanical means to move incompressible (or nearly incompressible) fluid or liquid. Pumps are our earliest machine and are to this day one of our most numerous mechanical devices. 

Air or gas compressors are very similar in design and operation to liquid pumps discussed earlier. The air and gas compressor is a mover of compressed fluids the pumps are movers of basically incompressible fluids (i.e., liquids). 

Liquids are almost incompressible. For example, if a pressure of 100 pounds per square inch, psi, is applied to a given volume of water that is at atmospheric pressure, the volume will decrease by only 0.03 per cent. It would take a force of approximately 32 tons to reduce its volume by 10 per cent however, when this force is removed, the water immediately returns to its original volume. Other liquids behave in about the same manner as water. 

If fluidity, the physical property of a substance that enables it to flow and incompressibility were the only properties required, any liquid that is not too thick might be used in a hydraulic system. However, a satisfactory liquid for a particular system must possess a number of other properties. The most important properties and some characteristics are discussed in the following paragraphs. 

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