The ideal thermodynamic cycle
Key FactsGyroscopic Couple: The rate of change of angular momentum () = (In the limit).
- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.
OverviewThe Carnot Cycle is an entirely theoretical thermodynamic cycle utilising reversible processes. The thermal efficiency of the cycle (and in general of any reversible cycle) represents the highest possible thermal efficiency (this statement is also known as Carnot's theorem - for a more detailed discussion see also Second Law of Thermodynamics ). This ultimate thermal efficiency can then be used to compare the efficiencies of other cycles operating between the same two temperatures. The thermal efficiency of any engine working between the temperatures of T1 and T2 is: 2) it can be seen that in order to improve the thermal efficiency of an engine, we should basically increase the value of (T2 - T1), i.e. increase the temperature difference under which the engine works.
Thermal EfficiencyThe thermal efficiency of a cycle, also denoted by , is a measure of the ability to convert heat energy into work. Therefore, the thermal efficiency can be defined as: 3) becomes:
- a reversible adiabatic (i.e. isentropic) compression of the gas from temperature T1 to T2 (step 1-2),
- followed by an isothermal heating with expansion (step 2-3),
- then a reversible adiabatic (isentropic) expansion of the gas from T2 to T1 (step 3-4),
- and ending with an isothermal cooling with compression which reverts the system back to its initial state (step 4-1).
Example - Heat supplied to a steam engine
Consider a steam engine for which the steam is supplied at and condensed at . If the thermal efficiency of the steam engine is of the Carnot efficiency, find the heat required (expressed in ) to produce a work output of 1 horsepower () for 1 minute.
We know that the Carnot efficiency of an engine working between temperatures and is given by: 13) and (8) in equation (11), and also considering that , we get the heat required (expressed in ) to produce for as: