Hyperloop: Compressor work

Classical Mechanics Level 4

The Hyperloop is a hypothetical new fast transport system between cities, which works by launching pods that carry people through a very low air pressure tunnel. The Hyperloop reduces friction between the pods and the tunnel by supporting the pod on a cushion of air. This air is gathered through the front of the pod and compressed from the initial ambient pressure and temperature of \(99~\mbox{Pa}\) and \(293~\mbox{K}\) to \(2.1~\mbox{kPa}\) and \(857~\mbox{K}\). The rate of air compressed in this way is \(0.49~\mbox{kg/s}\) and the total compressor input power is \(276~\mbox{kW}\). We define the efficiency \(\eta\) of the compressor as the (work done on the gas)/(input work to the compressor). If the pressure as a function of volume during the compression is \(P= AV^{-\alpha}\), where \(A\) and \(\alpha\) are constants, what is the efficiency of the compressor?

Details and assumptions

  • The molar mass of air is \(29~\mbox{g/mol}\).
  • Ignore any change in the bulk kinetic energy of the air.

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