Gas Properties: Specific Volume, Viscosity, and Compressibility
Specific Volume (v) is the volume occupied by a unit mass of material. It is the inverse of density and does not depend on the amount of matter. For example, two pieces of iron with different sizes have different weights and volumes, but their specific gravities are equal. This is independent of the material amount. Examples of intensive properties include boiling point, brightness, color, hardness, and melting point.
Where: V is volume, m is mass, and ρ is the density of the material.
It is expressed in units of volume per unit mass. For example:
or
Specific Volume for an Ideal Gas For an ideal gas, the following equation also applies:
Where R is the gas constant (R = 0.082), M is the molar mass of the gas, T is temperature, and P is gas pressure.
Measuring the Viscosity of a Gas Through a Capillary Poiseuille’s Law for Gases
Consider a capillary with radius r and length L, through which a gas flows with a pressure difference of P – P0 at its ends.
Poiseuille’s law, derived for an incompressible viscous fluid, states that the volumetric flow rate G = dV/dt (volume of fluid passing through the capillary’s cross-section per unit time) is directly proportional to the pressure gradient along the tube, i.e., the ratio (P – P0)/L.
For a gas flowing through the capillary, the volume of gas entering per unit time at pressure P is not equal to the volume leaving at pressure P0 (ambient) due to the compressibility of gases. However, the mass of gas entering per unit time equals the mass leaving per unit time.
We can write Poiseuille’s law as:
dV/dt is the volume of gas passing through the capillary’s cross-section at a distance x from the tube’s end per unit time. dP/dx is the pressure gradient at that point.
Given the ideal gas law PV = nRT:
• n is the number of moles, n = m/M
• m is the mass of gas in volume V
• M is the molecular weight
• R = 8.3143 J/(K·mol) is the gas constant
• T is the absolute temperature.
Poiseuille’s law can be written as:
The negative sign indicates that gas pressure P decreases along the capillary.
Integrating this equation, considering that dm/dt is constant along the capillary, with pressure P at x = 0 and P0 (atmospheric) at x = L:
Correlation of Gas Z Factor Many mathematical models exist, generalizable by:
Z is the compressibility factor, indicating the deviation from the ideal gas model. If Z = 1, the ideal model applies. However, Z values range from 0 to 1 and can exceed 1. The deviation Z can be calculated using various mathematical models. From the above equation:
Solving this for different models yields Z. Z expresses the deviation from the model relative to the ideal model. At a given pressure and temperature, we have a specific volume. With an experimental measurement of specific volume, we can determine Z, where Vi is the ideal molar specific volume and Vr is the real molar specific volume (measured). This expression is also used to find Z using specific volumes calculated with the ideal model and other mathematical models.
Isothermal Compressibility of Gas (Cg)
The variation of fluid compressibility with pressure and temperature is crucial for reservoir engineering calculations. For liquids, compressibility is small and often assumed constant, but this is not the case for gases. The isothermal compressibility of a gas is the change in volume per unit pressure change. For an ideal gas, Z = 1 (constant), and the compressibility is Cg = 1/P.
Multicomponent and Single-Component Systems Systems can be simple or complex, homogeneous or heterogeneous, and single- or multi-component. A gas in a cylinder is a simple system. A system may consist of subsystems or phases, distinguished by discontinuities in physical properties. A phase is a homogeneous system or part of a system where each intensive property has the same value at every point. A system with several phases is heterogeneous. A phase can be a chemically pure, single-component system or a multi-component system (binary, ternary, etc.). Do not confuse phase with state of aggregation. For example, a water-oil mixture (immiscible) is a single binary liquid phase; at certain concentrations and temperatures, it separates into two liquid phases, each with alcohol and water at different concentrations, remaining in a single state of aggregation. A water-ice mixture is a complex system with two phases in different states of aggregation.
Multicomponent technology is rapidly evolving, enabling material combinations with advantages such as reduced cycle times compared to two-component machines; simpler and faster machine operation; simplified work steps; reduced assembly effort; improved quality and repeatability; reduced space requirements; and lower power consumption. This text was repeated in the original document.
The number of phases that can coexist in a system is not arbitrary. For a single-component system, the chemical potential is a function of
and
If two phases coexist,
and
must be met.
This yields a relationship,
called the coexistence curve. These two phases can coexist with another phase,
and three phases can coexist simultaneously when
This is a system of two equations with two unknowns:
and
When the system has a solution, we have a triple point determined by these equations.
For a system with
chemical components, the Gibbs potential in each phase will be a function of the variables, or equivalently, the molar Gibbs potential will be
The variables
are not all independent, subject to the condition
Stability criteria indicate that
and
must be concave in
and
and convex in
or
When the stability criterion fails in a multicomponent system, phase transitions occur. Each phase generally has a different composition: for example, saltwater boiling at atmospheric pressure with coexisting steam is always much more diluted in salt; this is the basis of distillation, as repeatedly condensing the steam yields pure water. Just as the maximum number of phases that can coexist in a single-component system is three, in a system with
chemical components,
phases can coexist. This is known as the Gibbs phase rule. To prove this, assume that
phases coexist. The coexistence condition implies that the chemical potential of component 1 should be equal in the
phases:
This is a system of
independent equations relating
,
, and
mole fractions
of each phase
The same condition must be satisfied for the chemical potentials of each of the
components, resulting in a system of
equations to determine
unknowns (
). The number of equations cannot exceed the number of unknowns.
For a single-component system, this rule is met, so
, meaning at most three phases can coexist. In binary systems,
.
Gas Viscosity The viscosity of a fluid (gas or liquid) is its resistance to flow. Gas viscosity is often determined by the rate of momentum transfer from faster-moving to slower-moving layers. It is the relationship between shear stress and shear rate, which applies to a portion of fluid acquiring motion (dynamic viscosity). Different types of viscosity exist, with the most studied being dynamic and kinematic viscosity, the latter being the resistance generated when a fluid flows under gravity. Gas viscosity exhibits the following behavior:
– At low pressures (less than 1500 psf), increasing temperature increases viscosity.
– At high pressures (greater than 1500 psf), increasing temperature decreases viscosity.
– At any temperature, increasing pressure increases viscosity.
– Viscosity is higher for gases with heavier components.