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4.3.3.1: Gas Phase under Hydrostatic Pressure

Ideal Gas under Hydrostatic Pressure

 The gas density vary gradually with the pressure. As first approximation, the ideal gas model can be employed to describe the density. Thus equation (11) becomes
\[
    \dfrac{\partial P } {\partial z} = - \dfrac{g\, P}{ R\, T}     
    \label{static:eq:dzRhoGas} \tag{50}
\]
Separating the variables and changing the partial derivatives to full derivative (just a notation for this case) results in
\[
    \dfrac{dP} {P} = - \dfrac{g\, dz}{ R\, T}  
    \label{static:eq:sepDzRhoGas} \tag{51}
\]
Equation (51) can be integrated from point "0'' to any point to yield
\[
    \ln \dfrac{P} {P_0} = - \dfrac{g }{R\,T} \left(z - z_0\right)  
    \label{static:eq:dzRhoGasSolutionA} \tag{52}
\]
It is convenient to rearrange equation (52) to the following
\[
     \dfrac{P} {P_0} = {e}^    {- \left( \dfrac{g (z -z_o) }{R\,T} \right)}
    \label{static:eq:dzRhosolutionT} \tag{53}
\]
Here the pressure ratio is related to the height exponentially. Equation (53) can be expanded to show the difference to standard assumption of constant pressure as
\[
    \dfrac{P} {P_0} = 1 -  
    \overbrace{\dfrac{\left(z - z_0\right) g }{R\,T}}
            ^{-\dfrac{h\,\rho_0\, g}{P_0}} +
    \dfrac{\left(z - z_0\right)^2 g }{6\,R\,T} +
  \cdots
    \label{static:eq:dzRhoGasSolutionExpantion} \tag{54}
\]
Or in a simplified form where the transformation of \(h = (z − z_0)\) to be
\[
    \dfrac{P} {P_0} = 1 + \dfrac{\rho_0\, g}{P_0}
        \left( h - \overbrace{\dfrac{h^2}{6} + \cdots}^{\text{correction factor}}
        \right)
    \label{static:eq:dzRhoGasSolutionExpantionSimpleC} \tag{55}
\]
Equation (55) is useful in mathematical derivations but should be ignored for practical use.

Real Gas under Hydrostatic Pressure

The mathematical derivations for ideal gas can be reused as a foundation for the real gas model (\(P = Z \rho R T\)). For a large range of \(P/P_c\) and \(T/T_c\), the value of the compressibility factor, \(Z\), can be assumed constant and therefore can be swallowed into equations (53) and (54). The compressibility is defined in Thermodyanimcs Chapter. The modified equation is  
\[
     \dfrac{P} {P_0} = {e}^{- \left( \dfrac{g\, (z -z_o) }{Z\,R\,T} \right)}
    \label{static:eq:dzRhosolutionTz} \tag{56}
\]
Or in a series form which is
\[
    \dfrac{P} {P_0} = 1 -  
    \dfrac{\left(z - z_0\right) g }{Z\, R\,T} +
    \dfrac{\left(z - z_0\right)^2 g }{6\,Z\,R\,T} +
  \cdots
    \label{static:eq:dzRhoGasSolutionExpantionA} \tag{57}
\]
Without going through the mathematics, the first approximation should be noticed that the compressibility factor, \(Z\) enter the equation as \(h/Z\) and not just \(h\). Another point that is worth discussing is the relationship of Z to other gas properties. In general, the relationship is very complicated and in some ranges \(Z\) cannot be assumed constant. In these cases, a numerical integration must be carried out.

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.