Compressed gases thermodynamic functions

We now briefly discuss how thermodynamics can work for us or, better, how thermodynamics functions to solve a problem where it can help to provide the answer. We wish to illustrate this for a relatively simple problem how much work is required to compress a unit of gas per unit time (Figure 2.4) from a low to a high pressure. Figure 2.5 schematically gives the path to the answer and the structure of the solution. In fact, the same steps will have to be taken to apply thermodynamics to problems such as the calculation of the heat released from or required for a process, of the position of the chemical or phase equilibrium, or of the thermodynamic efficiency of a process. 

The solvent strength of a supercritical fluid (compressed gas) may be adjusted continuously from gas-like to liquid-like values, as described qualitatively by the solubility parameter. The solubility parameter, 8 (square root of the cohesive energy density) (2), is shown for gaseous, liquid, and SCF carbon dioxide as a function of pressure in Figure 1. It is a thermodynamic property which can be calculated rigorously as... 

The Fig. 1 phase diagram is for orientation and the regions indicated have been selected for thermodynamic computations p]. Thermodynamic functions may be calculated for the ideal-gas state from spectroscopic data [ ]. Density dependence of the functions on isotherms then may be computed from P-U-T data and used to establish values over that part of the P-V-T surface which is below and to the right of the coexistence region of Fig. 1. However, additional, detailed properties are required, to establish related values for compressed liquid states. These are described below. 

Note that thermodynamic interpretation of e is essentially different for compressible and incompressible fluid. Compressible fluid can be considered as a two-parametric system. According to the second law of thermodynamics, there is a state function - entropy s, playing the role of a thermodynamic potential. If e and specific volume 1 jp are taken as independent parameters, then the equation of state of compressible gas will be s = s(e, 1/p), and the perfect differential of the entropy is... 

By allowing the (volume) fraction of holes to be temperature and pressure dependent, lattice gas models are able to interpret thermal expansion and compression coefficients in thermodynamic functions for polymer solutions and mixtures. Onodera has developed such a model with which he examines changes of volume on mixing and pressure dependence of critical temperatures. 

Note 3.6 (Notation of pressure and volume change for fluid and thermodynamic functions). Most textbooks on thermodynamics (de Groot and Mazur 1962 Kestin 1979 Kondepudi and Prigogine 1998) contain treatments of the perfect gas, therefore we have [Pg.100]

In many industrial processes one of the thermodynamic functions is constant or nearly constant. Examples include compression in a highspeed compressor (near-constant entropy) and a gas flowing through an orifice (constant enthalpy). These concepts are discussed below. This type of process typically is described in... 

Because of these reasons, when calculating thermodynamic tables for compressed Freon-22, preference should be given to the function C T) found in Barho s work [0.37]. It is important that the reference data in [0.28, 3.1] were constructed using exactly these values of thermodynamic functions of Freon-22 in the ideal gas state. Contained in Ref. [0.21] are the coefficients of Eqs. (1.12)-(1.14) for the calculation of C, Hj — Tfg) and Sj from data by Barho at T... 

Close to the gas-liquid critical point one can show that the quantities Sp, Sv, and St, which are small deviations from the critical point in terms of the variables p, V, and t, are simply related to each other as well as to thermodynamic functions like the isobaric thermal expansion coefficient, ap, and the isothermal compressibility, Kt (as well as all others ). ... 

The adiabatic efficiency is a function of the pressure ratio, and thus, dependent on the thermodynamic state of the gas undergoing compression. ... 

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 thermodynamic 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 the 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. 

Returning to 3D lattice models, one may note that sine-Gordon field theory of the Coulomb gas should enable an RG (e — 4 — D) expansion [15], but this path has obviously not yet followed up. An attempt to establish the universality class of the RPM by a sine-Gordon-based field theory was made by Khodolenko and Beyerlein [105]. However, these authors did not present a scheme for calculating the critical exponents. Rather they argued that the grand partition function can be mapped onto that of the spherical model of Kac and Berlin [106, 297] which predicts a parabolic coexistence curve, i.e. fi — 1/2. This analysis was severely criticized by Fisher [298]. Actually, the spherical model has some unpleasant thermodynamic features, never observed in real fluids. In particular, it is associated with a divergence of the compressibility KTas the coexistence curve (rather than the spinodal line) is approached. By a determination of the exponent y, this possibility could also be ruled out experimentally [95, 97]. 

As shown in Chap. 6, ideal-gas heat capacities, rather than the actual heat capacities of gases, are used in the evaluation of thermodynamic properties such as internal energy and enthalpy. The reason is that thermodynamic-property evaluation is conveniently accomplished in two steps first, calculation of ideal-gas values from ideal-gas heat capacities second, calculation from PVT data of the differences between real-gas and ideal-gas values. A real gas becomes ideal in the limit as P - 0 if it were to remain ideal when compressed to a finite pressure, its state would remain that of an ideal-gas. Gases in these hypothetical ideal-gas states have properties that reflect their individuality just as do real gases. Ideal-gas heat capacities (designated by Cf and Cy) are therefore different for different gases although functions of temperature, they are independent of pressure. 

The first term on the left-hand side describes the variation of the fluid momentum in time and the second term describes the transport of the momentum in the flow (convective transport). The first term on the right-hand side describes the effect of gradients in the pressure p the second term, the transport of momentum due to the molecular viscosity p (diffusive transport) the third term, the effect of gravity g and in the last term, F lumps together all the other forces acting on the fluid. Techniques for solving the set of four equations (one continuity and three momentum equations) are discussed in a later section of this entry. When the flow is compressible, it is usually necessary to close the system of equations listed above using a thermodynamic equation of state (such as the ideal gas law) that calculates the density as a function of temperature and pressure. 

Both the rarefaction and compressibility effects are a function of the Kn that determines the degree of nonequilibrium of the gas flow. For the continuum fiow regime, thermodynamic equilibrium has to be maintained which means that the mean free path must be smaller than the characteristic length (/sTn = j 1). Typically, in order to obtain a stable estimation of the macroscopic properties, the ratio, Ub > 100, should be satisfied. For gas microflows, the effects of compressibility and rarefaction are linked and tend to craiflict with each other. The Kn is also related to the Reynolds number. Re, and the Mach number, Ma, with the relation... 

P). Note the expression for (C) is also a function of the particle diameter (dp) and includes known thermodynamic and physical properties of the chromatographic system. Consequently, with the aid of a computer, the optimum particle diameter (dp(opt)) can be calculated as that value that will meet the equality defined in equation (18). However, it will be seen in due course that these equations can be simplified. The equation for a flow of liquid though a packed bed will, however, differ for a compressible fluid, i.e., a gas. Due to the compressibility of a gas, the flow rate can not be described by the simple D Arcy law for liquids. From chapter 2, it is seen that... 

There are statistical, and perhaps systematic, uncertainties associated with the primary measured quantities used to define the thermodynamic state these should be considered in the reporting of experimental uncertainties of the measured quantity and accounted for in the regressions used to determine a correlation. All the uncertainties are likely to have state point dependence, so that the assessment in the dilute gas regime will differ markedly firom that in the critical region or in the compressed liquid (Perkins et al. 1991a). The uncertainty reported for a correlation should be a function of the fluid state point. The coverage factor (based on, perhaps, two standard deviations in a normal distribution) should be applied to the appropriate distribution associated with the combined standard uncertainty of the correlation. 

Equation (2.4-21) can be used for a reversible adiabatic compression as well as for an expansion. It is an example of an important fact that holds for any system, not just an ideal gas For a reversible adiabatic process in a simple system the final temperature is a function of the final volume for a given initial state. All of the possible final state points for reversible adiabatic processes starting at a given initial state lie on a single curve in the state space, called a reversible adiabat. This fact will be important in our discussion of the second law of thermodynamics in Chapter 3. 

---