MAB120 - MAN120 - MAB125 semester one 2010 Problem Solving Assignment see Blackboard site for additional details, including submission instructions...

2. Work and Heat Engines Thermodynamics allowsus to study ideal models for a heat engine that con-verts heat energy to mechanical energy. For an ide-alised engine, we can assume that gases obey idealgas laws such as PV = nRT, or for adiabatic expan-sions, PV = constant ( 1:4 for air). Note thatP = pressure, V = volume and T = temperatureof the gas, and n and R can be taken to be con-stants. The term adiabatic simply implies that inthe process of compressing (or expanding) a gas bychanging its pressure and volume, the temperatureof the gas also changes.The work done by the system in one complete cyclein an ideal heat engine is given by the area enclosedin the cycle when illustrated on a P ô€€€ V (pressureversus volume) diagram. By denition, as the gaschanges from volume Va to Vb the work done by thegas isWork =Z VbVaP dV :An example of a heat engine is the Rankine cycle,which is an idealised version of a steam engine. De-tails of such a cycle are shown in the diagram.VP0:05 0:15 0:251020c = (0:01; 20:0)d = (0:03; 20:0)b = (0:01; 1:03)a = (0:25; 1:03)In this cycle, liquid water at low temperature andpressure is heated at constant volume within aboiler. This occurs along path bc. The water is con-verted to steam and expands along path cd. The ex-pansion continues adiabatically along path da. Thewater is then cooled and condensed back to liquidalong path ab. There is no work done along path bcbecause the volume remains constant.(a) Using the ideal gas law PV = nRT, show thatthe appropriate expression for work along anisothermal (constant temperature) path from to isWork = nRT ln(V ô€€€ V) :(b) Show that the appropriate expression for workalong an adiabatic path from to isWork = (PV ô€€€ PV)=(1 ô€€€ ) :(c) Determine the total work done in single cycle ofthe Rankine cycle, using the information pro-vided in the gure above.