Carnot's Heat Engine
Carnot's Heat Engine:
Carnot's heat engine is an ideal heat engine operating between two temparatures. A carnot's engine consists of an ideal gas as working substance, enclosed in a cylinder whose walls and piston are perfectly adiabatic and whose base is perfectly diathermic. Any heat engine works in a cyclic process. The cyclic process in carnot's heat engine is called as Carnot's cycle.
The cycle has four steps, an isothermal expansion as heat is absorbed, followed by an adiabatic expansion, then an isothermal contraction as heat is shed, finally an adiabatic contraction to the original configuration.
Step 1: Isothermal Expansion :
Taking the temperature of the heat reservoir to be TH (H for hot), the expanding gas
follows the isothermal path PV = nRTH in the (P, V) plane.
The work done by the gas in a small volume expansion ΔV is just PΔV , the area under
the curve (as we proved in the last lecture).
Hence the work done in expanding isothermally from volume Va to Vb is the total area
under the curve between those values,
Vb V b
work done isothermally = ∫ PdV = ∫(nRTH/V) dV = nRTH ln(Vb / Va) .
Since the gas is at constant temperature TH, there is no change in its internal energy
during this expansion, so the total heat supplied must be nRTH ln (Vb / Va) , the same as the
external work the gas has done.
Step 2: Adiabatic Expansion :
To find the work the gas does in expanding adiabatically from Vb to Vc, say, the above
analysis is repeated with the isotherm PV = nRTH replaced by the adiabat PV γ = PbVbγ ,
work done adiabatically Wadiabat = ∫ PdV =PbVbγ ∫ (dV /Vγ) = PbVbγ (Vc1−γ − Vb1−γ /(1-γ) ).
Again, this is the area under the curve, in this case under the adiabat, from b to c in the
(P, V) plane.
Since points b, c are on the same adiabat, PcVc γ = PbVbγ , and the expression can be written
Wadiabat = ( PcVc − PbVb )/(1− γ) .
Steps 3 and 4: Completing the Cycle :
The above two are the initial steps in a heat engine, but it is equally necessary for the engine to get back to where it began, for the next cycle.
The general idea is that the piston drives a wheel which continues to turn and pushes the gas back to the original volume.
To ensure that the engine is as efficient as possible, this return path to the starting point
( Pa ,Va ) must also be reversible. We can’t just retrace the path taken in the first two legs,
that would take all the work the engine did along those legs, and leave us with no net
output. Now the gas cooled during the adiabatic expansion from b to c, from TH to TC,
say, so we can go some distance back along the reversible colder isotherm TC. But this
won’t get us back to ( Pa , Va ) , because that’s on the TH isotherm. The simplest option—
the one chosen by Carnot—is to proceed back along the cold isotherm to the point where
it intersects the adiabat through a, then follow that isotherm back to a.
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