PhysicsLAB Resource Lesson
Heat Cycles

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Heat Engine
A heat engine is a device that converts thermal energy into useable, mechanical work. Engines complete a cycle in which they receive heat from a high-temperature, "hot reservoir," convert part of it to useable work, and exhaust the remainder to a low-temperature, "cold reservoir."
According to the law of conservation of energy, this process can be written mathematically as
Qin = Wdone + Qout
Qhot = Wdone + Qcold
You can remember this formula if you model a car's operation. 
    • Qin is in the chemical energy found in the gasoline you buy
    • work done = ΔKE of the car as it takes you from place to place
    • Qout is the wasted heat and raw gasoline that escape from the car’s tailpipe
Efficiency of a heat engine
We define the efficiency of a heat engine as the ratio of the work done to the amount of energy put into the system to make it run.
e = Workdone / Qin
Since Qin = Wdone + Qout
e = (Qin - Qout) / Qin
e = 1 - (Qout / Qin)
Carnot Cycle and Ideal Efficiencies
On a PV-diagram, a heat engine is represented by a heat cycle, in which the effective work done by the engine is associated with the interior area of the cycle. The larger the area and the greater the temperature differential between the hot and cold sinks, the greater the efficiency of the engine. This equates to saying that without a ΔT no net work could be accomplished during the cycle - that is, the work done by the gas in expanding would be equal to the amount of work done by the piston during compression.
Each stroke of the cycle is generally composed of one of the four gas processes: isothermal, isobaric, isochroic, and adiabatic. In 1824, Sadi Carnot proposed an ideal, reversible heat engine comprised of these four operations:
  • an isothermal expansion (AB) followed by an adiabatic expansion (BC)
  • an isothermal compression (CD) followed by an adiabatic compression (DA)
Carnot's analysis showed that this ideal heat engine's efficiency depended solely on the temperature differential between the hot and cold reservoirs.
e = 1 - (Qout / Qin )
Qin only occurs during AB (at Thot ) and Qout only during CD (at Tcold ).
e = 1 - (Tcold / Thot )
In fact, his calculations show that a 100% efficient engine can only be obtained if the temperature of the cold reservoir was 0 K.
e = 1 - (0 / Thot ) = 100%
Kelvin-Planck later used this concept to restate the second law of thermodynamics as "no device exists that can completely transform a given amount of heat into useful work." 
Heat Pump/Refrigerator
A heat pump removes heat from a system with the assistance of an mechanical or electrical input.
According to the law of conservation of energy, this process can be written mathematically as
Qin + Win = Qout
Qcold + Win = Qhot
You can remember this formula by reminding yourself about the operation of the refrigerator in your kitchen.
    • Qin or Qcold  is the heat present in the food which you want to cool down
    • Win is the electric energy [Pt = energy where P = I2R = IV] does the work in running the refrigerator’s motor/compressor
    • Qout or Qhot is the heat exhausted into the room which raises the temperature in your kitchen
Efficiency of a heat pump
We define the efficiency of a heat pump as its coefficient of performance, COP.
COP = Qcold / Win

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Catharine H. Colwell
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