Thermodynamic Analysis Of Gas Behavior In Explosive Venting And Work Calculation: A Case Study On Nitrogen And Carbon Dioxide
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Thermodynamic Analysis of Gas Behavior in Explosive Venting and Work
Calculation: A Case Study on Nitrogen and Carbon Dioxide
#1
A tank containing carbon dioxide at 400 K and 50 bar is vented until the temperature in the tank falls to
300 K. Assuming there is no heat transfer between the gas and the tank, find the pressure in the tank at the
end of the venting process and the fraction of the initial mass of gas remaining in the tank if carbon
dioxide obeys the ideal gas and PR equations of state. Constant pressure heat capacity of carbon dioxide:
πΆπ(π½ πππβ1 πΎβ1)=22.243+(5.977Γ10β2)π+(β3.499Γ10β5)π2+(7.464Γ10β9)π3
You need to first show that the PR entropy departure function is given by
(π β ππΌπΊ)π,π
ππ = π ππππ(π β π΅) π€βπππ π΅ = ππ
π π
Answer: The content you've provided is part of a larger thermodynamic problem involving COβ in a tank
that is vented adiabatically, and it asks you to derive the Peng-Robinson (PR) entropy departure function
and then solve for the final pressure and mass of the gas remaining in the tank.
Answer to the First Part: Derivation of the PR Entropy Departure Function
The first part of your question asks you to derive the PR entropy departure function, which
involves comparing the entropy of a real gas (using the Peng-Robinson equation of state) to the
entropy of an ideal gas.
Starting point: Ideal Gas Entropy and Real Gas Entropy
For a real gas, the entropy change SβSIGS - S^{IG} can be calculated using the following
general thermodynamic relation:
dS=CPdTTβRln(Z)dS = C_P \frac{dT}{T} - R \ln(Z)
Where:
β’ SS is the entropy of the real gas
β’ SIGS^{IG} is the entropy of the ideal gas
β’ ZZ is the compressibility factor
β’ CPC_P is the heat capacity at constant pressure
β’ RR is the universal gas constant
For the ideal gas, the entropy change is:
dSIG=CPdTTdS^{IG} = C_P \frac{dT}{T}
Therefore, the difference in entropy between the real gas and the ideal gas (i.e., the entropy
departure) is:
SβSIG=β«T1T2CPdTTβRβ«T1T2ln(Z)dTS - S^{IG} = \int_{T_1}^{T_2} C_P \frac{dT}{T} - R
\int_{T_1}^{T_2} \ln(Z) dT
This leads to the general expression for entropy departure:
(SβSIG)=βRβ«T1T2ln(Z)dT(S - S^{IG}) = -R \int_{T_1}^{T_2} \ln(Z) dT
Peng-Robinson Equation of State and Compressibility Factor
The Peng-Robinson equation of state (PR EOS) is used to describe the behavior of real gases:
P=RTVβbβaV(V+b)+b(Vβb)P = \frac{RT}{V - b} - \frac{a}{V(V + b) + b(V - b)}
Calculation: A Case Study on Nitrogen and Carbon Dioxide
#1
A tank containing carbon dioxide at 400 K and 50 bar is vented until the temperature in the tank falls to
300 K. Assuming there is no heat transfer between the gas and the tank, find the pressure in the tank at the
end of the venting process and the fraction of the initial mass of gas remaining in the tank if carbon
dioxide obeys the ideal gas and PR equations of state. Constant pressure heat capacity of carbon dioxide:
πΆπ(π½ πππβ1 πΎβ1)=22.243+(5.977Γ10β2)π+(β3.499Γ10β5)π2+(7.464Γ10β9)π3
You need to first show that the PR entropy departure function is given by
(π β ππΌπΊ)π,π
ππ = π ππππ(π β π΅) π€βπππ π΅ = ππ
π π
Answer: The content you've provided is part of a larger thermodynamic problem involving COβ in a tank
that is vented adiabatically, and it asks you to derive the Peng-Robinson (PR) entropy departure function
and then solve for the final pressure and mass of the gas remaining in the tank.
Answer to the First Part: Derivation of the PR Entropy Departure Function
The first part of your question asks you to derive the PR entropy departure function, which
involves comparing the entropy of a real gas (using the Peng-Robinson equation of state) to the
entropy of an ideal gas.
Starting point: Ideal Gas Entropy and Real Gas Entropy
For a real gas, the entropy change SβSIGS - S^{IG} can be calculated using the following
general thermodynamic relation:
dS=CPdTTβRln(Z)dS = C_P \frac{dT}{T} - R \ln(Z)
Where:
β’ SS is the entropy of the real gas
β’ SIGS^{IG} is the entropy of the ideal gas
β’ ZZ is the compressibility factor
β’ CPC_P is the heat capacity at constant pressure
β’ RR is the universal gas constant
For the ideal gas, the entropy change is:
dSIG=CPdTTdS^{IG} = C_P \frac{dT}{T}
Therefore, the difference in entropy between the real gas and the ideal gas (i.e., the entropy
departure) is:
SβSIG=β«T1T2CPdTTβRβ«T1T2ln(Z)dTS - S^{IG} = \int_{T_1}^{T_2} C_P \frac{dT}{T} - R
\int_{T_1}^{T_2} \ln(Z) dT
This leads to the general expression for entropy departure:
(SβSIG)=βRβ«T1T2ln(Z)dT(S - S^{IG}) = -R \int_{T_1}^{T_2} \ln(Z) dT
Peng-Robinson Equation of State and Compressibility Factor
The Peng-Robinson equation of state (PR EOS) is used to describe the behavior of real gases:
P=RTVβbβaV(V+b)+b(Vβb)P = \frac{RT}{V - b} - \frac{a}{V(V + b) + b(V - b)}
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Subject
Physics