Vapor isochore

Several techniques are available for measuring values of interaction second virial coefficients. The primary methods are reduction of mixture virial coefficients determined from PpT data reduction of vapor-liquid equilibrium data the differential pressure technique of Knobler et al.(1959) the Bumett-isochoric method of Hall and Eubank (1973) and reduction of gas chromatography data as originally proposed by Desty et al.(1962). The latter procedure is by far the most rapid, although it is probably the least accurate. 

FIGURE 3.20 Successive cooling curves for hydrate formation with successive runs listed as Sj < S2 < S3. Gas and liquid water were isochorically cooled into the metastable region until hydrates formed in the portion of the curve labeled Sj. The container was then heated and hydrates dissociated along the vapor-liquid water-hydrate (V-Lyy-H) line until point H was reached, where dissociation of the last hydrate crystal was visually observed. (Reproduced from Schroeter, J.R, Kobayashi, R., Hildebrand, M.A., Ind. Eng. Chem. Fundam. 22, 361 (1983). With permission from the American Chemical Society.)... 

Cp = isobaric specific heat c = isochoric specific heat e = specific internal energy h = enthalpy k = thermal conductivity p = pressure s = specific entropy t = temperature T = absolute temperature u = specific internal energy 4 = viscosity V = specific volume / = subscript denoting saturated liquid g = subscript denoting saturated vapor... 

Typical uncertainties in density are 0.02% in the liquid phase, 0.05% in the vapor phase and at supercritical temperatures, and 0.1% in the critical region, except very near the critical point, where the uncertainty in pressure is 0.1%. For vapor pressures, the uncertainty is 0.02% above 180 K, 0.05% above 1 Pa (115 K), and dropping to 0.001 mPa at the triple point. The uncertainty in heat capacity (isobaric, isochoric, and saturated) is 0.5% at temperatures above 125 K and 2% at temperatures below 125 K for the liquid, and is 0.5% for all vapor states. The uncertainty in the liquid-phase speed of sound is 0.5%, and that for the vapor phase is 0.05%. The uncertainties are higher for all properties very near the critical point except pressure (saturated vapor/liquid and single pliase). The uncertainty in viscosity varies from 0.4% in the dilute gas between room temperature and 600 K, to about 2.5% from 100 to 475 K up to about 30 MPa, and to about 4% outside this range. Uncertainty in thermal conductivity is 3%, except in the critical region and dilute gas which have an uncertainty of 5%. 

Figure 5. Previous experimental results and the P-T conditions for in situ observations in this study for the system CH4-H2O. The experimental data for the univariant P-T relations of the assemblage methane hydrate-water-methane vapor were taken from Marshall et al. (circles) [10], Dyadin et al. (squares) [11], and Nakano et al. (triangles) [12]. All symbols are for si methane hydrate, except those squares branching out at higher P-T conditions. Points A, B, C and D (solid squares) indicate the P-T conditions for four invariant points, and they are for the following assemblages, respectively si methane hydrate-liquid water (lw)-ice Ih-methane vapor (v), si and sll methane hydrates-lw-ice Ih, si and sll methane hydrates-lw-v, and si and sH methane hydrates-lw-ice VI. Points E, F and G (dots) are P-T points along the isochore of pure water for 1,220 kg/m [14], and point H (dot) is a P-T point along the isochore of pure water for 1,047 kg/m . The former isochore was defined by the melting P-T condition of ice VI at point D (16.6 °C and 0.84 MPa) [15], and the latter isochore by that of ice Ih at point B (28.7 °C and 99 MPa Chou et al., unpublished results). The latter isochore is also the univariant P-T conditions for the assemblage si and sll methane hydrates-liquid water (Chou et al., unpublished results).

Figure 3. Pressure-temperature-density diagram for liquid water. LV represents the liquid-vapor boundary CP represents the critical point. The isochors are labeled by density in g/cm. ...

A large number of other cycles and variations to the standard cycles considered above have been proposed. We consider only a few additional cycles here. The Stirling cycle, shown in Fig. 5.2-5, operates with a vapor-phase working fluid, rather than a two-phase mixture as considered above. In this process the compressor and turbine, which are on the same shaft, are cooled and heated, respectively, in order to operate isothermally. The heat exchanger operates isochorically (that is, at constant volume). The P-V and T-S traces of-this cycle are shown in Fig. 5.2-6. The properties and path are shown in the table. 

An isochoric equation of state, applicable to pure components, is proposed based upon values of pressure and temperature taken at the vapor-liquid coexistence curve. Its validity, especially in the critical region, depends upon correlation of the two leading terms the isochoric slope and the isochoric curvature. The proposed equation of state utilizes power law behavior for the difference between vapor and liquid isochoric slopes issuing from the same point on the coexistence cruve, and rectilinear behavior for the mean values. The curvature is a skewed sinusoidal curve as a function of density which approaches zero at zero density and twice the critical density and becomes zero slightly below the critical density. Values of properties for ethylene and water calculated from this equation of state compare favorably with data. 

In 1976, Hall and Eubank (12,13) published two papers which have direct bearing upon the present equation of state. In the first paper, they noted the rectilinear behavior for the mean of the vapor and liquid isochoric slopes issuing from the same point on the vapor pressure curve near the critical point and the power law behavior for the difference in these slopes. The second paper presented an empirical description of the critical region which generally agreed with the scaling model but differed in one significant way—the curvature of the vapor pressure curve. 

The basic function of an isochoric equation of state is to describe isochores as they issue from the vapor pressure curve. Figure 1 illustrates... 

An isochoric equation has been developed for computing thermodynamic functions of pure fluids. It has its origin on a given liquid-vapor coexistence boundary, and it is structured to be consistent with the known behavior of specific heats, especially about the critical point. The number of adjustable, least-squares coefficients has been minimized to avoid irregularities in the calculated P(p,T) surface by using selected, temperature-dependent functions which are qualitatively consistent with isochores and specific heats over the entire surface. Several nonlinear parameters appear in these functions. Approximately fourteen additional constants appear in auxiliary equations, namely the vapor-pressure and orthobaric-densities equations, which provide the boundary for the P(p,T) equation-of-state surface. 

In our experience, a necessary but insufficient condition for a well-behaved critical isotherm is that, at the critical point, the slope of the critical isochore from the equation of state be equal to the slope of the vapor-pressure equation, 6P/6T = dP /dT. This constraint always is applied in the following work via the least-squares program (7). 

Iteration for Coexisting Densities. Orthobaric densities near the critical point generally cannot be obtained accurately from isochoric PpT data by extrapolation to the vapor-pressure curve because the isochore curvatures become extremely large near the critical point. The present, nonanalytic equation of state, however, can be used to estimate these densities by a simple, iterative procedure. Assume that nonlinear parameters in the equation of state have been estimated in preliminary work. For data along a given experimental isochore (density), it is necessary merely to find the coexistence temperature, Ta(p), by trial (iteration) for a best, least-squares fit of these data. 

These tables summarize the thermophysical properties of air in the liquid and gaseous states as calculated from the pseudo-pure fluid equation of state of Lemmon et al. (2000). The first table refers to liquid and gaseous air at equilibrium as a function of temperature. The tabulated properties are the bubble-point pressure (i.e., pressure at which boiling begins as the pressure of the liquid is lowered) the dew-point pressure (i.e., pressure at which condensation begins as the pressure of the gas is raised) density (/ ) enthalpy (H) entropy (S) isochoric heat capacity (CJ isobaric heat capacity (C ) speed of sound (u) viscosity (rj) and thermal conductivity (A). The first line of identical temperatures is the bubble-point (liquid) and the second line is the dewpoint (vapor). The normal boiling point of air, i.e., the temperature at which the bubble-point pressure reaches 1 standard atmosphere (1.01325 bar), is 78.90 K (-194.25 °C). 

When molar volume data are not available for at least one high pressure value, as for liquid oxygen, the assumption of the linearity of the p T isochores may be used in order to make reasonable estimates. In this case the entire p-F-T diagram can be described by a series of straight lines determined each by the V value at saturated vapor pressure and by the slope yV. At high pressure the assumed linearity is not valid the error on />, however, remains smaller than 1 % and the accuracy on is then of the same order of magnitude. 

The first coefficient describes the most common case, namely how much entropy AS flows in if the temperature outside and (also inside as a result of entropy flowing in) is raised by AT and the pressure p and extent of the reaction are kept constant. In the case of the secmid coefficient, volume is maintained instead of pressure (this only works well if there is a gas in the system). In the case of J = 0, the third coefficient characterizes the increase of entropy during equilibrium, for example when heating nitrogen dioxide (NO2) (see also Experiment 9.3) or acetic acid vapor (CH3COOH) (both are gases where a portion of the molecules are dimers). Multiplied by T, the coefficients represent heat capacities (the isobaric Cp at constant pressure, the isochoric Cy at constant volume, etc.). It is customary to relate the coefficients to the size of the system, possibly the mass or the amount of substance. The corresponding values are then presented in tables. In the case above, they would be tabulated as specific (mass related) or molar (related to amount of substance) heat capacities. The qualifier isobaric and the index p will... 

H. Mansoorian, K. R. Hall, and P. T. Eubank, "Vapor Pressure and PVT Measurements Using the Bumett-Isochoric Method," Proc. Seventh Symp. Thermophysical Properties, ASME, New York, 1977, p. 456. 

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