Compressibility, isothermal

Pressure depletion in the reservoir can normally be assumed to be isothermal, such that the isothermal compressibility is defined as the fractional change in volume per unit change in pressure, or... 

Reservoir fluids (oil, water, gas) and the rock matrix are contained under high temperatures and pressures they are compressed relative to their densities at standard temperature and pressure. Any reduction in pressure on the fluids or rock will result in an increase in the volume, according to the definition of compressibility. As discussed in Section 5.2, isothermal conditions are assumed in the reservoir. Isothermal compressibility is defined as ... 

Two other important quantities are the isobaric expansivity ( coefficient of themial expansion ) and the isothermal compressibility k, defined as... 

Figure 4.14 Behavior of thermodynamic variables at Tg for a second-order phase transition (a) volume and fb) coefficient of thermal expansion a and isothermal compressibility p.

By an assortment of thermodynamic manipulations, the quantities dn/dp and [N (d G/dp )o] can be eliminated from Eq. (10.48) and replaced by the measurable quantities a, /3, and dn/dT the coefficients of thermal expansion, isothermal compressibility, and the temperature coefficient of refractive index, respectively. With these substitutions, Eq. (10.48) becomes... 

Many of the unusual properties of the perfluorinated inert fluids are the result of the extremely low intermolecular interactions. This is manifested in, for example, the very low surface tensions of the perfluorinated materials (on the order of 9-19 mN jm. = dyn/cm) at 25°C which enables these Hquids to wet any surface including polytetrafluoroethene. Their refractive indexes are lower than those of any other organic Hquids, as are theh acoustic velocities. They have isothermal compressibilities almost twice as high as water. Densities range from 1.7 to 1.9 g/cm (l )-... 

Other Refrigeration Methods. Cryocoolers provide low temperature refrigeration on a smaller scale by a variety of thermodynamic cycles. The Stirling cycle foUows a path of isothermal compression, heat transfer to a regenerator matrix at constant volume, isothermal expansion with heat transfer from the external load at the refrigerator temperature, and finally heat transfer to the fluid from the regenerator at constant volume. 

Isothermal Gas Flow in Pipes and Channels Isothermal compressible flow is often encountered in long transport lines, where there is sufficient heat transfer to maintain constant temperature. Velocities and Mach numbers are usually small, yet compressibihty effects are important when the total pressure drop is a large fraction of the absolute pressure. For an ideal gas with p = pM. JKT, integration of the differential form of the momentum or mechanical energy balance equations, assuming a constant fric tion factor/over a length L of a channel of constant cross section and hydraulic diameter D, yields,... 

For isothermal compressible flow of a gas with constant compressibility factor Z through a packed bed of granular solids, an equation similar to Eq. (6-114) for pipe flow may be derived ... 

The energy required to reversibly separate gas mixtures is the same as that necessary to isothermally compress each component in the mixture from the partial pressure of the gas in the mixture to the final pressure of the mixture. This reversible isothermal work is given by the familiar relation... 

The Intercooled Regenerative Reheat Cycle The Carnot cycle is the optimum cycle between two temperatures, and all cycles try to approach this optimum. Maximum thermal efficiency is achieved by approaching the isothermal compression and expansion of the Carnot cycle or by intercoohng in compression and reheating in the expansion process. The intercooled regenerative reheat cycle approaches this optimum cycle in a practical fashion. This cycle achieves the maximum efficiency and work output of any of the cycles described to this point. With the insertion of an intercooler in the compressor, the pressure ratio for maximum efficiency moves to a much higher ratio, as indicated in Fig. 29-36. 

The theoretical required (isothermal) compression work in the compressor, which is assumed to operate isothermally at To, is... 

The modulus indicates that heat is absorbed (+), during die isodrermal expansion, but released (—) during die isothermal compression. In the adiabatic processes no heat is supplied or removed from die working gas, and so... 

The chapter on equation-of-state properties provides the basic approaches used for describing the high-pressure shock-compression response of materials. These theories provide the basis for separating the elastic compression components from the thermal contributions in shock compression, which is necessary for comparing shock-compression results with those obtained from other techniques such as isothermal compression. A basic understanding of the simple theories of shock compression, such as the Mie-Gruneisen equation of state, are prerequisite to understanding more advanced theories that will be discussed in subsequent volumes. 

Isothermal compression is presented here to represent the upper limits of cooling and horsepower savings. It is the equivalent of an infinite number of intercoolers and is not achievable in the practical types of compressors described in this book. For an isothermal process. 

Calculate the theoretical power for each case (1) no intercooling, (2) one intercooler, (3) two intercoolers, (4) isothermal compression. 

Using Equation 2.84 to establish the theoretical limit of isothermal compression,... 

For theoretical cycle work performed in an isothermal compression cycle (For ideal fluid case)... 

The experiments result in an explicit measure of the change in the shock-wave compressibility which occurs at 2.5 GPa. For the small compressions involved (2% at 2.5 GPa), the shock-wave compression is adiabatic to a very close approximation. Thus, the isothermal compressibility Akj- can be computed from the thermodynamic relation between adiabatic and isothermal compressibilities. Furthermore, from the pressure and temperature of the transition, the coefficient dO/dP can be computed. The evaluation of both Akj-and dO/dP allow the change in thermal expansion and specific heat to be computed from Eq. (5.8) and (5.9), and a complete description of the properties of the transition is then obtained. 

The discussion of the last section is then useful in considering the evaporative cycles. We shall see that the effect of water injection downstream of the compressor (and possibly in the cold side of the heat exchanger) may lead towards the [CBTJiXr type of plant, with increased cold side effective specific heat and hence increased heat exchanger effectiveness. Water injection in the compressor may lead to a plant with isothermal compression. 

A MC study of adsorption of living polymers [28] at hard walls has been carried out in a grand canonical ensemble for semiflexible o- 0 polymer chains and adsorbing interaction e < 0 at the walls of a box of size C. A number of thermodynamic quantities, such as internal energy (per lattice site) U, bulk density (f), surface coverage (the fraction of the wall that is directly covered with segments) 9, specific heat C = C /[k T ]) U ) — U) ), bulk isothermal compressibility... 

The predictions of the theory are obtained for a one-phase system in which the isothermal compressibility for the uncharged system is finite dss > 0). In order to investigate a system with dss = 0, a more sophisticated Hamiltonian has to be considered in order to stabilize the system. 

A wide variety of physical properties are important in the evaluation of ionic liquids (ILs) for potential use in industrial processes. These include pure component properties such as density, isothermal compressibility, volume expansivity, viscosity, heat capacity, and thermal conductivity. However, a wide variety of mixture properties are also important, the most vital of these being the phase behavior of ionic liquids with other compounds. Knowledge of the phase behavior of ionic liquids with gases, liquids, and solids is necessary to assess the feasibility of their use for reactions, separations, and materials processing. Even from the limited data currently available, it is clear that the cation, the substituents on the cation, and the anion can be chosen to enhance or suppress the solubility of ionic liquids in other compounds and the solubility of other compounds in the ionic liquids. For instance, an increase in allcyl chain length decreases the mutual solubility with water, but some anions ([BFJ , for example) can increase mutual solubility with water (compared to [PFg] , for instance) [1-3]. While many mixture properties and many types of phase behavior are important, we focus here on the solubility of gases in room temperature IFs. 

Isothermal compression takes place when the heat of compression is removed during compression and when the temperature of the gas stays constant. The characteristic equation is... 

Corollary.—A fluid emits or absorbs heat on isothermal compression according as it expands or contracts, respectively, with rise of temperature at constant pressure. 

An isothermal compression of the fluid is made at 91 from C3 to V4. Work is added to the system, and, to maintain isothermal conditions, a quantity of heat q is removed from the system and absorbed in a heat reservoir maintained at the temperature 9. ... 

The form of equations (8.11) and (8.12) turns out to be general for properties near a critical point. In the vicinity of this point, the value of many thermodynamic properties at T becomes proportional to some power of (Tc - T). The exponents which appear in equations such as (8.11) and (8.12) are referred to as critical exponents. The exponent 6 = 0.32 0.01 describes the temperature behavior of molar volume and density as well as other properties, while other properties such as heat capacity and isothermal compressibility are described by other critical exponents. A significant scientific achievement of the 20th century was the observation of the nonanalytic behavior of thermodynamic properties near the critical point and the recognition that the various critical exponents are related to one another ... 

The total work of compression from a pressure P to a pressure Pi is found by integrating equation 8.27. For an ideal gas undergoing an isothermal compression ... 

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