6 Due 9/24

1. Two cubic meters of a gas at 200 kPa expands polytropically with $n=5/3$ to standard atmospheric pressure. What is the work done by the gas in the process?

2. A piston cylinder combination contains 2 kg of air at the ambient temperature of 25$\POW9,{\circ}$C and at the unknown ambient pressure. To provide a source of cooling on this hot Florida day, the piston is now pulled up until the air reaches a temperature of -100$\POW9,{\circ}$C. It can be assumed that the process is a polytropic one with $n=1.4$. What is the work that must be done on the air in the cylinder to pull up the piston and produce the cold air? The piston must also do work on the air outside the cylinder, which remains at the ambient pressure. What is the expression for that work?

3. Air in a piston/cylinder combination with an initial volume of 1 m$\POW9,{3}$ is initially kept at a pressure of 150 kPa by the weight of the piston and the atmospheric pressure. It expands to 2 m$\POW9,{3}$, at which time the piston hits against a linear spring. The air then continues to expand to 4 m$\POW9,{3}$ while the pressure increases to 300 kPa due to the increasing spring force. Find the work done by the air in the complete process. Draw the $PV$ diagram very neatly and graphically show the two parts of the work done by the air.

4. One type of a multistep process is a cycle. Consider the idealized car engine cycle, called Otto cycle. Assume that in state 1, the air-fuel mixture has been taken in and is still at 100 kPa and 27$\POW9,{\circ}$C ambient conditions. Then the air is compressed until state 2, where the volume is one tenth of the intake volume. Assume that this compression stroke is polytropic with $n=1.4$. Then from state 2 to state 3, an isochoric 2,500$\POW9,{\circ}$C increase in temperature takes place. Finally, from state 3 to state 4, the power stroke, the air expands polytropically with $n=1.4$ until the volume returns to the original intake volume value. Finally, for the process from 4 back to 1, a true car engine would dump the hot air to the exhaust and take in fresh cool air. However, the Otto cycle assumes that the air at step 4 is isochorically cooled down from 4 to 1 to the original intake temperature.

   Model the air-fuel mixture as pure air. Find the pressure, specific volume, and temperature at each of the 4 stages of the cycle. Then find the net specific work produced in the complete cycle.

   Also show the cycle and the work graphically in a very neat $Pv$ diagram.