The Arctic Research Station's Ice Cellar · AI Generator #GEN-AF2D8060
Physics · thermodynamics
Problem
An Arctic research station uses a clever passive cooling system for food storage. They dig a cylindrical pit of radius R=1.5m and depth h=2.0m into the permafrost and fill it with air at atmospheric pressure. The permafrost walls maintain a constant temperature of Tc=250K. During the brief Arctic summer, the air temperature above ground reaches Th=290K. The researchers notice that as the day warms up, the air pressure inside the sealed pit increases, and when they open a valve to the atmosphere, cold air rushes out. They decide to use this as a heat engine by connecting the pit to an insulated cylinder with a movable piston that can extract work during the daily temperature cycle. Assuming the air in the pit undergoes a quasi-static cycle where it: (1) heats at constant volume from Tc to Th, (2) expands isothermally at Th back to atmospheric pressure, (3) cools at constant pressure back to Tc, and (4) compresses isothermally at Tc back to the original state. If the pit contains n=850mol of air (treated as an ideal diatomic gas), what is the net work output per cycle?