Isothermal Compressibilities

The isothermal compressibility of a fluid is defined as follows:

c f = 1 V ( V ρ ) T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(18.15)
 

This expression can be also given in term of fluid density, as follows:

c f = 1 ρ ( ρ P ) T This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(18.16)
 

For liquids, the value of isothermal compressibility is very small because a unitary change in pressure causes a very small change in volume for a liquid. In fact, for slightly compressible liquid, the value of compressibility (co) is usually assumed independent of pressure. Therefore, for small ranges of pressure across which co is nearly constant, Equation (18.16) can be integrated to get:

c o ( p p b )=ln( ρ o ρ ob ) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(18.17)
 

In such a case, the following expression can be derived to relate two different liquid densities ( ρ o ρob) This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. at two different pressures (p, pb):

ρ o = ρ ob [ 1+ c o (p p b ) ] This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(18.18)
 

The Vasquez-Beggs correlation is the most commonly used relationship for co.

For natural gases, isothermal compressibility varies significantly with pressure. By introducing the real gas law into Equation (18.16), it is easy to prove that, for gases:

c g = 1 P 1 Z ( Z P ) r This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(18.19)
 

Note that for an ideal gas, cg is just the reciprocal of the pressure. “cg” can be readily calculated by graphical means (chart of Z versus P) or by introducing an equation of state into Equation (18.19).

Surface Tension

Surface tension is a measure of the surface free energy of liquids, i.e., the extent of energy stored at the surface of liquids. Although it is also known as interfacial force or interfacial tension, the name surface tension is usually used in systems where the liquid is in contact with gas.

Qualitatively, it is described as the force acting on the surface of a liquid that tends to minimize the area of its surface, resulting in liquids forming droplets with spherical shape, for instance. Quantitatively, since its dimension is of force over length (lbf/ft in English units), it is expressed as the force (in lbf) required to break a film of 1 ft of length. Equivalently, it may be restated as being the amount of surface energy (in lbf-ft) per square feet.

Katz et al. (1959) presented the Macleod-Sudgen equation for surface tension ( σ This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers. ) calculations in dynes/cm for hydrocarbon mixtures:

σ 1/4 = i=1 n Pc h i [ ρ i 62.4(M W l ) x i ρ g 62.4(M W g ) y i ] This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(18.20)
 

where:

Pchi = Parachor of component “i”,

xi = mole fraction of component “i” in the liquid phase,

yi = mole fraction of component “i” in the gas phase.

In order to express surface tension in field units (lbf/ft), multiply the surface tension in (dynes/cm) by 6.852177x10-3. The parachor is a temperature independent parameter that is calculated experimentally. Parachors for pure substances have been presented by Weinaug and Katz (1943) and are listed in Table 18.1.

Table 18.1. Parachors for pure substances (Weinaug and Katz, 1943)
Component Parachor
CO2 78.0
N2 41.0
C1 77.0
C2 108.0
C3 150.3
iC4 181.5
nC4 189.9
iC5 225.0
nC5 231.5
nC6 271.0
nC7 312.5
nC8 351.5

Weinaug and Katz (1943) also presented the following empirical relationship for the parachor of hydrocarbons in terms of their molecular weight:

Pc h i =4.6148734+2.558855M W i +3.404065 10 4 M W i 2 + 3.767396 10 3 M W i This equation is not rendering properly due to an incompatible browser. See Technical Requirements in the Orientation for a list of compatible browsers.
(18.21)
 
  • This correlation may be used for pseudo-components or for those hydrocarbons not shown in Table 18.1.

Action Item

Answer the following problem, and submit your answer to the drop box in Canvas that has been created for this module.

Please note:

  • Your answer must be submitted in the form of a Microsoft Word document.
  • Include your Penn State Access Account user ID in the name of your file (for example, "module2_abc123.doc").
  • The due date for this assignment will be sent to the class by e-mail in Canvas.
  • Your grade for the assignment will appear in the drop box approximately one week after the due date.
  • You can access the drop box for this module in Canvas by clicking on the Lessons tab, and then locating the drop box on the list that appears.

Problem Set

  1. Which of all the properties we have studied so far would you use to compare and distinguish among wet natural gas, dry natural gas, and gas-condensate systems? Explain why.