Isotherm compression, isothermal

It is expressed as a ratio of the isothermal work input to the actual work input An air compressor is required to raise the pressure of the air with the minimum possible work input, which occurs with isothermal compression. Isothermal compression is an ideal case since none of the work input is absorbed in raising the temperature of the compressed air, i.e. in... 

Starting at condition A with the ethane in the liquid phase, and assuming isothermal depletion, then as the pressure is reduced so the specific volume increases as the molecules move further apart. The relationship between pressure and volume is governed by the compressibility of the liquid ethane. 

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 ... 

Neumann has adapted the pendant drop experiment (see Section II-7) to measure the surface pressure of insoluble monolayers [70]. By varying the droplet volume with a motor-driven syringe, they measure the surface pressure as a function of area in both expansion and compression. In tests with octadecanol monolayers, they found excellent agreement between axisymmetric drop shape analysis and a conventional film balance. Unlike the Wilhelmy plate and film balance, the pendant drop experiment can be readily adapted to studies in a pressure cell [70]. In studies of the rate dependence of the molecular area at collapse, Neumann and co-workers found more consistent and reproducible results with the actual area at collapse rather than that determined by conventional extrapolation to zero surface pressure [71]. The collapse pressure and shape of the pressure-area isotherm change with the compression rate [72]. 

The limiting compression (or maximum v value) is, theoretically, the one that places the film in equilibrium with the bulk material. Compression beyond this point should force film material into patches of bulk solid or liquid, but in practice one may sometimes compress past this point. Thus in the case of stearic acid, with slow compression collapse occurred at about 15 dyn/cm [81] that is, film material began to go over to a three-dimensional state. With faster rates of compression, the v-a isotherm could be followed up to 50 dyn/cm, or well into a metastable region. The mechanism of collapse may involve folding of the film into a bilayer (note Fig. IV-18). 

A rather different approach is to investigate possible adsorption isotherm forms for use with Eq. X-43. As is discussed more fully in Section XVII-7, in about 1914 Polanyi proposed that adsorption be treated as a compression of a vapor in the potential held U x) of the solid with sufficient compression, condensation to liquid adsorbate would occur. If Uq(x) denotes the held necessary for this, then... 

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

Fig. 45 Adsorption isotherms of nitrogen at 77 K on zirconia powder and its compacts. (A) uncompressed (B) compressed at 3 tonin (C) lOtonin (D) 30tonin" (E) 70tonin" (F) lOOtonin". Open symbob denote adsorption, solid symbols desoprtion. (Courtesy Avery and...

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 )-... 

Several types of fluids are used as refrigerants in mechanical compression systems ammonia, halocarbon compounds, hydrocarbons, carbon dioxide, sulfur dioxide, and cryogenic fluids. A wide temperature range therefore is afforded. These fluids boil and condense isotherm ally. The optimum temperature or pressure at which each can be used can be deterrnined from the economics of the system. The optimum refrigerant can be deterrnined only... 

Sometimes for compression is expressed for the isothermal case, which is always lower than that for the adiabatic case. The difference defines... 

The monolayer resulting when amphiphilic molecules are introduced to the water—air interface was traditionally called a two-dimensional gas owing to what were the expected large distances between the molecules. However, it has become quite clear that amphiphiles self-organize at the air—water interface even at relatively low surface pressures (7—10). For example, x-ray diffraction data from a monolayer of heneicosanoic acid spread on a 0.5-mM CaCl2 solution at zero pressure (11) showed that once the barrier starts moving and compresses the molecules, the surface pressure, 7T, increases and the area per molecule, M, decreases. The surface pressure, ie, the force per unit length of the barrier (in N/m) is the difference between CJq, the surface tension of pure water, and O, that of the water covered with a monolayer. Where the total number of molecules and the total area that the monolayer occupies is known, the area per molecules can be calculated and a 7T-M isotherm constmcted. This isotherm (Fig. 2), which describes surface pressure as a function of the area per molecule (3,4), is rich in information on stabiUty of the monolayer at the water—air interface, the reorientation of molecules in the two-dimensional system, phase transitions, and conformational transformations. 

As the barrier moves, the molecules are compressed, the intermolecular distance decreases, the surface pressure increases, and a phase transition may be observed in the isotherm. These phase transitions, characterized by a break in the isotherm, may vary with the subphase pH, and temperature. The first-phase transition, in Figure 2, is assigned to a transition from the gas to the Hquid state, also known as the Hquid-expanded, LE, state. In the Hquid... 

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. 

If the power requirement of the gaseous diffusion process were no greater than the power required to recompress the stage upflow from the pressure on the low-pressure side of the barrier to that on the high-pressure side, then the power requirement of the stage would be Z RTLq (1 /r) for the case where the compression is performed isotherm ally. The power requirement per unit of separative capacity would then be given simply by the ratio... 

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,... 

Adiabatic Frictionless Nozzle Flow In process plant pipelines, compressible flows are usually more nearly adiabatic than isothermal. Solutions for adiabatic flows through frictionless nozzles and in channels with constant cross section and constant friction factor are readily available. 

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 ... 

If the compression cycle approaches the isothermal condition, pV = constant, as is the case when several stages with intercoolers are used, a simple approximation of the power is obtained from the following formula ... 

The Carnot refrigeratiou cycle is reversible and consists of adiabatic (iseutropic due to reversible character) compression (1-2), isothermal rejection of heat (2-3), adiabatic expansion (3-4) and isothermal addition of heat (4-1). The temperature-entropy diagram is shown in Fig. 11-70. The Carnot cycle is an unattainable ideal which serves as a standard of comparison and it provides a convenient guide to the temperatures that should be maintained to achieve maximum effectiveness. 

Vapor-Compression Cycles The most widely used refrigeration principle is vapor compression. Isothermal processes are realized through isobaric evaporation and condensation in the tubes. Standard vapor compression refrigeration cycle (counterclockwise Ranldne cycle) is marked in Fig. ll-72<7) by I, 2, 3, 4. 

To reduce the work of compression in this cycle a two-stage or dualpressure process may be usedwhereby the pressure is reduced by two successive isenthalpic expansions. Since the isothermal work of compression is approximately proportional to the logarithm of the pressure ratio, and the Joule-Tnomson cooling is roughly proportional to... 

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 thermal efficiency of the process (QE) should be compared with a thermodynamically ideal Carnot cycle, which can be done by comparing the respective indicator diagrams. These show the variation of temperamre, volume and pressure in the combustion chamber during the operating cycle. In the Carnot cycle one mole of gas is subjected to alternate isothermal and adiabatic compression or expansion at two temperatures. By die first law of thermodynamics the isothermal work done on (compression) or by the gas (expansion) is accompanied by the absorption or evolution of heat (Figure 2.2). 

The adiabatic expansion and compression serve only to change the temperature of tire gas widrout heat being absorbed or evolved, i.e. iso-entropic changes. The heat changes are therefore only related to the work which is done during the isothermal stages, which is given by... 

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... 

It follows that the efficiency of the Carnot engine is entirely determined by the temperatures of the two isothermal processes. The Otto cycle, being a real process, does not have ideal isothermal or adiabatic expansion and contraction of the gas phase due to the finite thermal losses of the combustion chamber and resistance to the movement of the piston, and because the product gases are not at tlrermodynamic equilibrium. Furthermore the heat of combustion is mainly evolved during a short time, after the gas has been compressed by the piston. This gives rise to an additional increase in temperature which is not accompanied by a large change in volume due to the constraint applied by tire piston. The efficiency, QE, expressed as a function of the compression ratio (r) can only be assumed therefore to be an approximation to the ideal gas Carnot cycle. 

Figure 4.2. Pressure-volume compression curves. For isentrope and isotherm, the thermodynamic path coincides with the locus of states, whereas for shock, the thermodynamic path is a straight line to point Pj, V, on the Hugoniot curve, which is the locus of shock states.

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. 

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