Intermediate Thermodynamics Assignment 6 — Non-Ideal Models
 Instructions
$\xi$ is a parameter related to your student ID, with $\xi_1$ corresponding to the last digit, $\xi_2$ to the last two digits, $\xi_3$ to the last three digits, etc. For instance, if your ID is 199225962, then $\xi_1=2$, $\xi_2=62$, $\xi_3=962$, $\xi_4=5962$, etc. Keep a copy of the assignment — the assignment will not be handed back to you. You must be capable of remembering the solutions you hand in.
 05.04.14
 Question #1
Starting from the Van der Waals equation of state for a non-ideal gas: $$P=\frac{\overline{R}T}{\overline{v}-b}-\frac{a}{\overline{v}^2}$$ prove that: $$a=\frac{27}{64} \frac{\overline{R}^2 T_{\rm c}^2}{P_{\rm c}} ~~~{\rm and}~~~ b=\frac{1}{8} \frac{\overline{R} T_{\rm c}}{P_{\rm c}}$$ with $T_{\rm c}$ and $P_{\rm c}$ the critical temperature and pressure respectively.
 Question #2
Refrigerant-12 has a specific volume of $\rm 0.02638~m^3/kg$ at a pressure of 800 kPa. Estimate the temperature of the gas by the use of (a) the ideal-gas equation of state, (b) the van der Waals equation of state, (c) the Redlich-Kwong equation of state, and (d) the generalized compressibility chart.
 Question #3
Can the following fluids in the specified states be treated as ideal gases? Assume that a gas can be approximated as ideal if the generalized compressibility factor differs from that of an ideal gas by less than 10%.
 (a) Air at 0.1 MPa, 20$^\circ$C (b) Air at 13 MPa, 900$^\circ$C (c) Methane at 2 MPa, 1000$^\circ$C (d) Water at 0.1 MPa, 20$^\circ$C (e) Water at 0.01 MPa, 30$^\circ$C (f) Refrigerant 12 at 1 MPa, 50$^\circ$C
 Question #4
 (a) Calculate the specific volume of nitrogen for a pressure of 3 MPa and a temperature of $(165+\xi_2)$ K (i) assuming ideal gas behaviour, and (ii) using compressibility charts (b) Nitrogen has a density of $\rm 0.14~g/cm^3$ and a temperature of 150 K. Estimate the compressibility factor and the pressure of the nitrogen. (c) Calculate the specific volume of refrigerant-12 at a pressure of $(1.0+0.1\times\xi_1)$ MPa and a temperature of $\rm 60^\circ C$ using the generalized compressibility chart. Assume that the molecular weight is 120.91 kg/kgmol.
 Question #5
Consider $\rm CO_2$ in a $\rm 1~m^3$ tank initially at a pressure 10.35 MPa and at a temperature of $\rm 61.6^\circ C$. A valve is attached to the tank and lets $\rm CO_2$ escape such that when heat is added, the pressure within the tank always remains 10.35 MPa. After the heat addition process, the temperature of the $\rm CO_2$ left in the tank is of $\rm 90^\circ C$. Calculate the heat added to the tank and the mass of $\rm CO_2$ that left the tank. Take $c_p=0.862$ kJ/kgK. Hint: you can not assume either thermally or calorically perfect gas.
 Question #6
Octane (C$_{8}$H$_{18}$) is expanded isentropically from a pressure of 124.5 bars and a temperature of 683 K to a pressure of 49.8 bars. Determine the temperature after the expansion process using (i) an ideal model, and (ii) the generalized correction charts.
 05.01.19
 $\pi$