Question 8.4: Methane is to be compressed from atmospheric pressure to 30 ...
Methane is to be compressed from atmospheric pressure to 30 MN/m² in four stages. Calculate the ideal intermediate pressures and the work required per kilogram of gas. Assume compression to be isentropic and the gas to behave as an ideal gas. Indicate on a temperature–entropy diagram the effect of imperfect intercooling on the work done at each stage.
The ideal intermediate pressures are obtained when the compression ratios in each stage are equal. If the initial, intermediate, and final pressures from this compressor are P_{1}, P_{2}, P_{3}, P_{4}, and P_{5}, then:
P_{2}/P_{1}=P_{3}/P_{2}=P_{4}/P_{3}=P_{5}/P_{1}
as in problem 8.1.
P_{5}/P_{1} = (30,000/101.3) = 296.2
and: (P_{5}/P_{1})^{0.25} = 4.148
Hence: P_{2} = 4.148 P_{1} = (4.148 × 101.3) = 420 kN/m²
P_{3} = 4.148 P_{2} = 1.74 MN/m²
P_{4} = 4.148 P_{3} =7.23 MN/m²
The work required per kilogram of gas is:
W=nP_{1}V_{1}\frac{\gamma }{\gamma -1} \left[(\frac{P_{5}}{P_{1}} )^{(\gamma -1)/n\gamma }-1\right] (equation 8.46)
For methane, the molecular mass = 16 kg/kmol and the specific volume at STP =(22.4/16) = 1.40 m³/kg.
If \gamma = 1.4, the work per kilogram is:
W=(4\times 101,300\times 1.40)(1.4/0.4)[(296.2)^{0.4/(4\times 1.4)}-1]
= 710,940 J/kg or 711 kJ/kg
The effect of imperfect cooling is shown in Figs 8a and 8b
