The Pressure Series

It is often convenient to use (T, p) as the independent variables in place of (r, pf) and, for this purpose, an expansion of Z in is used which may be written as  

The coefficients of this series are uniquely related to the virial coefficients as may be shown by eliminating p from the right hand side of eq 3.3 using eq 3.1 and collecting terms, leading to B = BjRT and C = C — B )j RTf-, Table 3.1 compares the coefficients of the density (eq 3.2) and pressure (eq 3.4) explicit expansions of Z. The composition dependence of the coefficients B, C, - rnsL mixture may be determined by combining these relations with eq 3.3. 

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Higher virial coefficients are defined analogously. AH virial coefficients depend on temperature and composition only. The pressure series and density series coefficients are related to one another ... 

Equation 24 foi the pressure series expansion can therefore be written as follows ... 

Equation (4-187) is the virial equation in pressure, and B, C, D, . . . , are the pressure-series virial coefficients. Like the density-series coefficients, they depend on temperature and composition only. Moreover, the two sets of coefficients are related ... 

FIG. 50. Internal stress as a function of fmax at three temperatures, for the pressure series in Figure 46 in the SiH4-H2 discharge running at an excitation frequency of 50 MHz and a power of 10 W. (From E. A. G. Hamers, Ph.D. Thesis. Universiteit Utrecht. Utrecht, the Netherlands. 1998, with permission.)... 

The pressure-series coefficients and density-series coefficients ate related ... 

Multiple sets of Burnett data were obtained for each isotherm—three sets for ethylene and two sets for helium. Each set consisted of data from a series of four consecutive expansions from the highest to the lowest pressure compatible with our optimum accuracy and precision. The initial pressure for each set was selected so as to intersperse the data from all of the sets over the entire pressure range of interest, 0.3 MPa to 3.7 MPa. Consistent with the extent of the nonideal behavior of the gas, the density-series generalized equation was applied to the ethylene data and the pressure-series generalized equation was applied to the helium data. The parameters in the resulting overdetermined sets of equations then were evaluated using the least-squares constraint. 

B and C are called the pressure-series virial coefficients, B is the second virial coefficient and C the third virial coefficient. The fugacity coefficient can be obtained from the compressibility factor according to... 

The residual part of the Gibbs energy may be obtained in terms of the pressure series, eq 3.4, in a manner analogous to that used to obtain Table 3.2 gives the expansions of Gm, IfSt, Cptm, and in terms of... 

A comparison of the two expansions for the compressibility factor indicates, however, that when truncated after C, the volume series provides better results than the pressure series but when truncated after B, they provide comparable results, with the pressure series actually doing better at higher densities due to compensation of errors (Chueh and Prausnitz, 1967). 

The virial equation of state is a power series in the reciprocal molar volume or in the pressure ... 

The vapor pressure of a crude oil at the wellhead can reach 20 bar. If it were necessary to store and transport it under these conditions, heavy walled equipment would be required. For that, the pressure is reduced (< 1 bar) by separating the high vapor pressure components using a series of pressure reductions (from one to four flash stages) in equipment called separators , which are in fact simple vessels that allow the separation of the two liquid and vapor phases formed downstream of the pressure reduction point. The different components distribute themselves in the two phases in accordance with equilibrium relationships. 

Under standard conditions of temperature and pressure (STP), the first four members of the alkane series (methane, ethane, propane, and butane) are gases. As length of the carbon increases the density of the compound increases (pentane) to C yHgg are liquids, and from C.,gH3g, the compounds exist as wax-like solids at STP. 

From the above plot, it can be seen that the recovery factor for gas reservoirs depends upon how low an abandonment pressure can be achieved. To produce at a specified delivery pressure, the reservoir pressure has to overcome a series of pressure drops the drawdown pressure (refer to Figure 9.2), and the pressure drops in the tubing, processing facility and export pipeline (refer to Figure 9.12). To improve recovery of gas, compression facilities are often provided on surface to boost the pressure to overcome the pressure drops in the export line and meet the delivery pressure specified. 

The pressure equipment directive was adopted by the European Parliament and the European Council in May 1997. It harmonises the national laws of the 15 Member States of the European Union relating to equipment subject to the pressure risk. That directive is one of the series of technical harmonisation directives such as for machinery, medical devices, simple pressure vessels, gas appliances and so on, which were foreseen by the Communities programme for the elimination of technical barriers to trade. It therefore aims to ensure the free placing on the market and putting into service of the equipment concerned within the European Union and the European Economic Area. At the same time it permits a flexible regulatory environment, allowing European industry to develop new techniques increasing thereby its international competitiveness. 

The pressure coefficients can be related to the volume coefficients by reverting the series and one finds that... 

The nth virial coefficient = < is independent of the temperature. It is tempting to assume that the pressure of hard spheres in tln-ee dimensions is given by a similar expression, with d replaced by the excluded volume b, but this is clearly an approximation as shown by our previous discussion of the virial series for hard spheres. This is the excluded volume correction used in van der Waals equation, which is discussed next. Other ID models have been solved exactly in [14, 15 and 16]. ... 

Figure A2.5.6 shows a series of typical p, Fisothemis calculated using equation (A2.5.1). (The temperature, pressure and volume are in reduced units to be explained below.) At sufficiently high temperatures the pressure decreases monotonically with increasing volume, but below a critical temperature the isothemi shows a maximum and a minimum.

Although Eq. (10.50) is still plagued by remnants of the Taylor series expansion about the equilibrium point in the form of the factor (dn/dc2)o, we are now in a position to evaluate the latter quantity explicitly. Equation (8.87) gives an expression for the equilibrium osmotic pressure as a function of concentration n = RT(c2/M + Bc2 + ) Therefore... 

La.mina.r Flow Elements. Each of the previously discussed differential-pressure meters exhibits a square root relationship between differential pressure and flow there is one type that does not. Laminar flow meters use a series of capillary tubes, roUed metal, or sintered elements to divide the flow conduit into innumerable small passages. These passages are made small enough that the Reynolds number in each is kept below 2000 for all operating conditions. Under these conditions, the pressure drop is a measure of the viscous drag and is linear with flow rate as shown by the PoiseuiHe equation for capilary flow ... 

In the paraffin series, methane, CH, to / -butane, C H q, are gases at ambient conditions. Propane, C Hg, and butanes are sometimes considered in a special category because they can be fiquefied at reasonable pressures. These compounds are commonly referred to as fiquefied petroleum gases (qv) (LPG). The pentanes, to pentadecane [629-62-9], fiquids, commonly called distillates, which include gasoline [8006-61-9], kerosene... 

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