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FLUID MECHANICS - THEORY

   

The ideal gas law will be introduced in this section. The conditions in which the ideal gas law is valid will be examined through the compressibility factor.

     
    Ideal Gas Law

Table: Gas Constant for
Common Gases*
  Gas Constant, R
(kJ/kg-K)
Air 0.2870
Ammonia 0.4882
Carbon Dioxide 0.1889
Helium 2.0771
Hydrogen 4.1243
Nitrogen 0.2968
Oxygen 0.2598
R-12 0.06876
R-134a 0.08149


Universal Gas Constant Versus
Gas Constant

 

The relationship between pressure and temperature for most gases can be approximated by the ideal gas law,

 
pV = mRT
 

where p is the absolute pressure, V is the volume, m is the mass, T is the absolute temperature (units in Kelvin or Rankine) and R is the gas constant. Kelvin is related to Celsius by TK = TC + 273.15 and Rankine is related to Fahrenheit byTR = TF + 459.67.

This equation is also referred to as the perfect gas law or the equation of state for an ideal gas. The gas constant R for some common gases is given in the table. Note that the density ρ is given by m/V, hence the ideal gas law can be written in terms of the density as

     p = ρRT

The ideal gas law can also be written in per mole basis as follows:

     
where n is the number of moles and is the universal gas constant. The number of moles is given by n = m/M where M is the molecular weight of the gas. The universal gas constant is 8.314 kJ/kmol-K for all gases, and it is related to the gas constant by:

     

   
    Compressibility Factor


Compressibility Chart for
Various Gases**

Table: Critical Pressure and
Temperature for Various Gases*
  Critical
Temp.
Tc (K)
Critical
Pressure
pc (MPa)
Air 133.0 3.77
Ammonia 405.5 11.3
Carbon
Dioxide
304.1 7.38
Helium 5.19 0.227
Hydrogen 33.2 1.30
Nitrogen 126.2 3.39
Oxygen 154.6 5.04
R-12 385.0 4.14
R-134a 374.2 4.06


Validity of the Ideal Gas Law**

 

Having introduced the ideal gas law, the next important question is: Under what conditions will the ideal gas law be a good approximation for real gases? To address this issue, the compressibility factor Z is introduced, and it is defined as:

     Z = pV/mRT

Obviously, when Z approaches unity, then the ideal gas law holds.

The compressibility factor for various gases is plotted as a function of the reduced pressure (pR) and reduced temperature (TR) in the compressibility chart as shown in the figure. The reduced pressure and temperature are defined as follows:

     pR = p/pc
     TR = T/Tc

where pc and Tc are the critical pressure and critical temperature, respectively. The critical temperature is the maximum temperature that liquid and vapor phases can coexist in equilibrium. The corresponding pressure is the critical pressure. Values of critical pressure and temperature for some common gases are summarized in the table.

From the compressibility chart, it is observed that Z approaches unity when pR < 0.1 or TR > 2. Hence, these are the conditions when the behavior of real gases can best be approximated using the ideal gas law.


* Source: Sonntag, R. E. and Borgnakke, C., "Introduction to Engineering Thermodynamics," John Wiley, Inc., New York, 2001.

** Reference: Su, G. J., "Modified Law of Corresponding States," Ind. Eng. Chem. (international ed.), Vol. 38, pp. 803, 1946.

     
   
 
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