# Isothermal Compressibilities

The isothermal compressibility of a fluid is defined as follows:

${c}_{f}=-\frac{1}{V}{\left(\frac{\partial V}{\partial \rho }\right)}_{T}$
(18.15)

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

${c}_{f}=-\frac{1}{\rho }{\left(\frac{\partial \rho }{\partial P}\right)}_{T}$
(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:

${c}_{o}\left(p-{p}_{b}\right)=\mathrm{ln}\left(\frac{{\rho }_{o}}{{\rho }_{ob}}\right)$
(18.17)

In such a case, the following expression can be derived to relate two different liquid densities at two different pressures (p, pb):

${\rho }_{o}={\rho }_{ob}\left[1+{c}_{o}\left(p-{p}_{b}\right)\right]$
(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:

${c}_{g}=\frac{1}{P}-\frac{1}{Z}{\left(\frac{\partial Z}{\partial P}\right)}_{r}$
(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 ($\sigma$) calculations in dynes/cm for hydrocarbon mixtures:

${\sigma }^{1/4}=\sum _{i=1}^{n}Pc{h}_{i}\left[\frac{{\rho }_{i}}{62.4\left(M{W}_{l}\right)}{x}_{i}-\frac{{\rho }_{g}}{62.4\left(M{W}_{g}\right)}{y}_{i}\right]$
(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:

$Pc{h}_{i}=-4.6148734+2.558855M{W}_{i}+3.404065\cdot {10}^{-4}M{W}_{i}^{2}+\frac{3.767396\cdot {10}^{3}}{M{W}_{i}}$
(18.21)

• This correlation may be used for pseudo-components or for those hydrocarbons not shown in Table 18.1.

# Action Item

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