REAL Gas Mixtures EXPLAINED
Don’t Let This Cost You Points
on Your Exam! Part 2

DOFPro Team

Kay’s Rule Review

  1. Calculate the pseudocritical temperature.
  2. Calculate the pseudocritical pressure.
  3. Calculate the pseudoreduced temperature.
  4. Calculate the pseudoreduced pressure.
  5. Use the generalized compressibility charts to calculate \(z_m\).
  6. Use \(z_m\) to calculate anything else needed.

Kay’s Rule (cont.)

\[ P_r' =\frac{P}{P_c'} \]

\[ T_r' = \frac{T}{T_c'} \]

\[ z_m = z_m(T_r', P_r') \]

DWSIM and Equation of State

Real Gas Example

A stream of

20% methane,

30% ethane, and

50% propane

at \(T = 423\ \mathrm{K}\) and \(P = 70\ \mathrm{atm}\) flows at \(100.0\ \mathrm{kmol/h}\).

Estimate the volumetric flow rate using Kay’s rule and compare with the ideal gas law and the Peng-Robinson, Soave-Redlich-Kwong, and Lee-Kesler-Plöcker equations of state.

Data from Wikipedia

methane: \(T_c = 190.56\ \mathrm{K}\) \(P_c = 4.5992\ \mathrm{MPa}\)
ethane: \(T_c = 305.32\ \mathrm{K}\) \(P_c = 4.8714\ \mathrm{MPa}\)
propane: \(T_c = 370\ \mathrm{K}\) \(P_c = 4.23\ \mathrm{MPa}\)

Example (cont.)

\[70\ \mathrm{atm} \boldsymbol{\cdot} 0.101325\ \mathrm{\frac{MPa}{atm}}= 7.093\ \mathrm{MPa} = 7,093,750\ \mathrm{Pa}\]

Ideal Gas

\[ \hat{V}=\frac{RT}{P}=\frac{8.314\mathrm{\frac{J}{mol \cdot K}}\boldsymbol{\cdot}423\ \mathrm{K}}{7.093\times10^{6}\ \mathrm{Pa}}=4.959\times10^{-4}\ \mathrm{\frac{m^{3}}{mol}}=0.4959\ \mathrm{\frac{L}{mol}} \]

\[ \dot{V}=\dot{n}\hat{V}=100\ \mathrm{\frac{kmol}{h}}\boldsymbol{\cdot}\frac{1000\ \mathrm{mol}}{1\ \mathrm{kmol}}\boldsymbol{\cdot}4.959\times10^{-4}\ \mathrm{\frac{m^{3}}{mol}}=49.59\ \mathrm{\frac{m^{3}}{h}} \]

Kay’s Rule Calculation

\[ T_c' = 0.200(4.23) + 0.300(305.32) + 0.500(370) = 314.71\ \mathrm{K} \]

\[ P_c' = 0.200(4.5992) + 0.300(4.8714) + 0.500(4.23) = 4.496\ \mathrm{MPa} \]

\[ T_r' = \frac{T}{T_c'} = \frac{423}{314.7} = 1.344 \]

\[ P_r' = \frac{P}{P_c'} = \frac{7.093}{4.496} = 1.577 \]

Example (cont.)

Using Kay’s rule

\[ P_r' = \frac{P}{P_c'} = \frac{7.093}{4.496} = 1.577 \]

\[ T_r' = \frac{T}{T_c'} = \frac{423}{314.7} = 1.344 \]

\[ z_{m}=0.7833 \]

\[ \dot{V}_{KR}=z_{m}\dot{V}_{IG}=38.84\ \mathrm{\frac{m^{3}}{h}} \]

DWSIM Video

Example Spreadsheet

The Takeaways

  1. Kay’s rule is the simplest method for calculating properties of mixtures of real gases by hand, and it is very good for aliphatic hydrocarbons, but it is still a fair amount of work
  2. DWSIM or some other software application is the easiest way to calculate properties of mixtures of real gases using equations of state.
  3. Life gets much more interesting than the ideal gas law when you have mixtures of real gases.






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