Why should process engineers know what is compressibility, Z? How to find the compressibility of an unknown gas? Concept of the corresponding state

Two things right in the beginning

Why gas compressibility, Z is important?

1. Process Design: Pipelines, storage tanks, and distillation columns are just a few examples of the various industrial processes involving gases that compressibility is crucial for designing and optimizing.

2. Equipment Sizing: The compressibility factor of the gas must be taken into consideration when sizing compressors, blowers, and other gas-handling equipment.

3. Gas Transport: It's important to comprehend the compressibility factor when moving gases through pipelines, including natural gas.

4. Safety Analysis: Safety considerations for high-pressure systems take compressibility into account. For evaluating the blowdown characteristics, relief valve sizing, and overall system response during emergency scenarios, accurate knowledge of the compressibility factor is necessary.

5. Phase Equilibrium: Compressibility is related to how gases behave in different phases (vapour, liquid, and dense phases). It aids in determining the circumstances under which a gas turns into a liquid or vice versa.?

How to find the compressibility of an unknown gas?

See the graph of Z vs. Pr.

To determine the compressibility factor (Z), you must know the gas's reduced temperature (Tr) and reduced pressure (Pr) values.

How to do it:

1. On the graph's x-axis, find the reduced pressure (Pr) value of the unknown gas.

2. From Pr move up vertically until you cross the curve representing the unidentified gas's known reduced temperature (Tr) value.

3. The y-coordinate of the intersection point provides the compressibility factor (Z) for the unidentified gas at those particular Pr and Tr.

At the point where the Pr and Tr curves converge, all gases have the same Z value.]

Background

The simplest definition of corresponding state in one sentence is "For most gases when the ratio of actual temperature to critical temperature or the ratio of actual pressure to critical pressure is the same, they exhibit similar thermodynamic properties."

What is the corresponding state theory?

Van der Walls discovered through experiments that most gases have comparable properties close to the critical point. Following that, he put forth the "Theory of Corresponding State."

According to the theory of corresponding states, the thermodynamic properties of most gases become similar or "correspond" to one another when the temperature is expressed as a percentage of the critical temperature (reduced temperature, Tr). According to the theory of corresponding states, the various thermodynamic properties, including enthalpy, specific heat, compressibility factor, and phase behaviour, approach similar values when gases are compared at the same reduced temperature (Tr). The theory of corresponding states makes use of the idea of reduced variables, such as reduced temperature and reduced pressure. It is possible to compare various substances and analyse their behaviour close to the critical point by expressing the temperature and pressure in terms of these condensed variables.

The theory of corresponding states is primarily applicable to simple, non-polar gases where intermolecular forces are primarily due to London dispersion forces. For complex molecules with additional intermolecular forces like H bonds, the theory may not accurately predict their behaviour.

Explanation

What are reduced properties

In thermodynamics, the reduced properties of a fluid are a set of state variables scaled by the fluid's state properties at its critical point. Reduced properties provide a measure of the “departure” of the conditions of the substance from its own critical conditions and are defined as follows:

Pr = P/Pc, Tr = T/Tc and Vr = V/Vc P, T and V are temperature, pressure and volume. Subscript c stands for critical point and r stands for reduced temperature and pressure. If Pr = Tr = Vr = 1, the substance is at its critical condition. If we are beyond critical conditions, Tr > 1, Pr > 1 and Vr > 1. By the same token, if all the conditions are subcritical, Tr < 1, Pr < 1 and Vr < 1. Critical conditions become the scaling factor by which substances can be compared among each other in terms of their “departure from criticality” or reduced properties.

How to find reduced specific volume

The reduced specific volume of a fluid is computed from the ideal gas law at the substance's critical pressure and temperature:

Vr = V Pc/R Tc

This property is useful when the specific volume and either temperature or pressure are known, in which case the missing third property can be computed directly.

How and why corresponding state work?

Why do all gases have the same compressibility, Z at reduced temperature, Tr

van der Walls discovery

The ratio of a gas's actual temperature to its critical temperature is known as the reduced temperature or Tr.?The ratio of actual temperature to its critical pressure is defined as critical pressure or Pr.

Van der Walls discovered that, when plotting the compressibility factors of various gases against reduced temperature and reduced pressure, all the curves condense onto a single curve known as the "compressibility chart" or "Z chart" when expressed in terms of reduced temperature and pressure.

It can be seen above that all of the curves overlap and get closer to the value of 1 as the reduced pressure gets closer to 1, showing the compressibility of all gases as a function of reduced pressure. This indicates that gases behave similarly, and their compressibility tends to be close to one at low temperatures.

Due to the dominance of intermolecular forces over kinetic energy at lower temperatures, the compressibility of all gases equalises and becomes independent of the particular gas under consideration.

An indicator of a gas's departure from ideal gas behaviour is its compressibility factor (Z), where Z=1 denotes ideal gas behaviour. Most gases behave differently than they would under ideal gas conditions at low temperatures and moderate pressures, and at these conditions, different gases have different compressibility factors.

This confirms what van der Walls wrote in his theory of the corresponding state

“The principle of corresponding states that gases may behave differently at different temperatures and pressures, but at the same reduced temperature (Tr) and reduced pressure (Pr), the compressibility factor of all gases tends to converge to the same value, regardless of their individual chemical properties. Therefore, since the reduced pressure Pr is the same for all gases, their compressibility factor Z will also be the same.

Fundamentals

All gases have the same compressibility at reduced temperatures because at low temperatures, all gases tend to behave similarly and follow similar physical laws. At lower temperatures, the particles of gases move slower, and their kinetic energy decreases. Due to this reduction in kinetic energy, the intermolecular forces between the molecules become more dominant, causing the gases to condense and become more compressible. As the temperature is reduced, the effect of these intermolecular forces becomes more pronounced, and the compressibility of gases becomes more similar regardless of the specific gas being considered. Therefore, all gases have similar compressibility at reduced temperatures.

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