Incompressibility pressure effects

It is clear that sound, meaning pressure waves, travels at finite speed. Thus some of the hyperbolic—wavelike-characteristics associated with pressure are in accord with everyday experience. As a fluid becomes more incompressible (e.g., water relative to air), the sound speed increases. In a truly incompressible fluid, pressure travels at infinite speed. When the wave speed is infinite, the pressure effects become parabolic or elliptic, rather than hyperbolic. The pressure terms in the Navier-Stokes equations do not change in the transition from hyperbolic to elliptic. Instead, the equation of state changes. That is, the relationship between pressure and density change and the time derivative is lost from the continuity equation. Therefore the situation does not permit a simple characterization by inspection of first and second derivatives. 

Liquids and solids are almost incompressible. Therefore, changes of atmospheric pressure have little effect on the entropy of substances in liquid or solid states. Ordinary changes in pressure have essentially no effect on melting and freezing. Although the elevation is high and atmospheric pressure is very low, water on Pike s Peak still freezes at 273.15 K. You will learn more about pressure effects on state changes in the next section. 

Because liquids and solids are almost incompressible, pressure has little effect on their solubility, but it has a major effect on gas solubility. Consider the piston-cylinder assembly in Figure 13.10, with a gas above a saturated aqueous solution of the gas. At a given pressure, the same number of gas molecules enter and leave the solution per unit time that is, the system is at equilibrium ... 

Changes in pressure have significant effects only on equilibrium systems with gaseous components. Aside from phase changes, a change in pressure has a negligible effect on liquids and solids because they are nearly incompressible. Pressure changes can occur in three ways ... 

Tabular and calculated standard equibbrium constants characterize interrelations between components of individual reactions under standard conditions (at temperature 25 °C and pressure 1 bar). Water is practically incompressible, so the effect of pressure not important. With pressure increase from 100 to 10 kPa (1-1,000 bar) values pFC in water solutions change by about 0.1- 0.2. Temperature has much greater effect. With temperature increase by 100 °C the values of equibbrium constant can change by 1-2, sometimes by 3 orders of magnitude. For this reason under conditions close to surficial, the pressure effect may be disregarded. 

Because Uquids and solids are nearly incompressible, pressure has little effect on the rates of melting and freezing a plot of pressure vs. temperature for a solid-liquid phase change is typically a nearly vertical straight Une. 

The same as classic thermodynamics, polymer thermodynamics is function of pressure, temperature and composite. But in many cases, pressure effects on polymer thermodynamics was neglected, because polymer thermodynamics were often studied under atmosphere. The classic theory of polymer thermodynamics is Flory-Huggins hard lattice theory. In this theory, the hard lattice is incompressible. A rigorously incompressible system should be unaffected by pressure. However, since experimental results show that the critical temperature for polymer demixing system is strongly affected by pressure, it is clear that polymer containing systems show significant departures from this ideal limit. We wish... 

By far, the most often cited work is the early paper by Outmans (1963), which applied differential-equation-based filtration methods developed in chemical engineering to static and dynamic invasion in the borehole. In this single-phase flow study, where lineal flow was assumed and the applied differential pressure was completely supported by the mudcake, Outmans derived the well known Vt law, subject to the further proviso of cake incompressibility. (The effects of cake nonlinearity and compaction can be important over time e.g., see Figure 14-7.) Thus, the Vt law cannot be used when the net flow resistance offered by the formation is comparable to that of the mudcake (e.g., thin muds in permeable formations, or thick muds in very impermeable rocks). Also, the law does not apply to slimholes, where the radial geometry is important. Finally, the Vt law does not generally apply to reservoirs with two-phase, immiscible flow, or miscible flow, or both. Only under these restrictive assumptions does Outmans correctly derived law hold. 

This was previously interpreted as a pressure effect because with the larger L/D, a higher pressure is required to extrude the rubber. However, this interpretation may be incorrect, because gum rubber is practically incompressible and therefore, viscosity is not expected to depend upon pressure. This is demonstrated in the following experiments. 

As for any incompressible single-phase flow, the equivalent pressure P = p + pgz where g = acceleration of gravity z = elevation, may be used in place of p to account for gravitational effects in flows with vertical components. 

When the continmty equation and the Navier-Stokes equations for incompressible flow are time averaged, equations for the time-averaged velocities and pressures are obtained which appear identical to the original equations (6-18 through 6-28), except for the appearance of additional terms in the Navier-Stokes equations. Called Reynolds stress terms, they result from the nonlinear effects of momentum transport by the velocity fluctuations. In each i-component (i = X, y, z) Navier-Stokes equation, the following additional terms appear on the right-hand side ... 

The effect of pressure on the properties of an incompressible fluid, an ideal gas. and a non-ideal gas is now considered. 

For an incompressible fluid flowing in a horizontal pipe of constant cross-section, in the absence of work being done by the fluid on the surroundings, the pressure change due to frictional effects is given by ... 

As will be outlined below, the computation of compressible flow is significantly more challenging than the corresponding problem for incompressible flow. In order to reduce the computational effort, within a CED model a fluid medium should be treated as incompressible whenever possible. A rule of thumb often found in the literature and used as a criterion for the incompressibility assumption to be valid is based on the Mach number of the flow. The Mach number is defined as the ratio of the local flow velocity and the speed of sound. The rule states that if the Mach number is below 0.3 in the whole flow domain, the flow may be treated as incompressible [84], In practice, this rule has to be supplemented by a few additional criteria [3], Especially for micro flows it is important to consider also the total pressure drop as a criterion for incompressibility. In a long micro channel the Mach number may be well below 0.3, but owing to the small hydraulic diameter of the channel a large pressure drop may be obtained. A pressure drop of a few atmospheres for a gas flow clearly indicates that compressibility effects should be taken into account. 

Lockhart and Martinelli divided gas-liquid flows into four cases (1) laminar gas-laminar liquid (2) turbulent gas-laminar liquid (3) laminar gas-turbulent liquid and (4) turbulent gas-turbulent liquid. They measured two-phase pressure drops and correlated the value of 0g with parameter % for each case. The authors presented a plot of acceleration effects, incompressible flow (3) no interaction at the interface and (4) the pressure drop in the gas phase equals the pressure drop in the liquid phase. 

Martinelli and Nelson (M7) developed a procedure for calculating the pressure drop in tubular systems with forced-circulation boiling. The procedure, which includes the accelerative effects due to phase change while assuming each phase is an incompressible fluid, is an extrapolation of the Lockhart and Martinelli x parameter correlation. Other pressure drop calculation procedures have been proposed for forced-circulation phase-change systems however, these suffer severe shortcomings, and have not proved more accurate than the Martinelli and Nelson method. 

Two common types of one-dimensional flow regimes examined in interfacial studies Poiseuille and Couette flow [37]. Poiseuille flow is a pressure-driven process commonly used to model flow through pipes. It involves the flow of an incompressible fluid between two infinite stationary plates, where the pressure gradient, Sp/Sx, is constant. At steady state, ignoring gravitational effects, we have... 

The first method, which is the more flexible, is to use an activity coefficient model, which is common at moderate or low pressures where the liquid phase is incompressible. At high pressures or when any component is close to or above the critical point (above which the liquid and gas phases become indistinguishable), one can use an equation of state that takes into account the effect of pressure. Two phases, denoted a and P, are in equilibrium when the fugacity / (for an ideal gas the fungacity is equal to the pressure) is the same for each component i in both phases ... 

Spiering et al. (1982) have developed a model where the high-spin and low-spin states of the complex are treated as hard spheres of volume and respectively and the crystal is taken as an isotropic elastic medium characterized by bulk modulus and Poisson constant. The complex is regarded as an inelastic inclusion embedded in spherical volume V. The decrease in the elastic self-energy of the incompressible sphere in an expanding crystal leads to a deviation of the high-spin fraction from the Boltzmann population. Pressure and temperature effects on spin-state transitions in Fe(II) complexes have been explained based on such models (Usha et al., 1985). 

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