The purpose of building this kind of heat engine is that it is possible
to show that it is the most efficient engine for given temperatures,
, at which heat is provided and extracted,
respectively. You can imagine it as an engine consisting of a piston moving
inside a cylinder filled with a working gas and closed off at the end opposite
the piston, or a cylinder with a movable piston on either end - your fantasy
is the limit. The gas is not allowed to escape though. In a very specific
sequence we will alternately heat or cool the gas and compress or expand it.
The term reversible basically means that we build the engine such that
any type of friction or turbulent gas motion is completely eliminated.
This can be achieved
as closely - without ever reaching it - as one desires given enough
money is available.
A simple example for a reversible process could be the compression
an elastic spring.
When you compress it you put a certain amount of energy into it which you can
gain back as you relax the spring to its original position.
You have to watch out though that the mechanism used to compress and
relax the spring incurs zero friction.
Instead of a spring you could use of course a gas confined in a piston/cylinder
arrangement. This is now more complicated. To make the process reversible you
avoid any type of friction between piston and cylinder and, at the same time,
you have to have a perfect seal between piston and the cylinder wall so that
no gas can escape.
Additionally, when you compress a gas it will warm up. So, if you wish
to avoid the gas loosing even
the slightest amount of heat you have to make sure that the cylinder walls and
piston head do not absorb any heat even if they and the gas have different
temperature. Again, the more money available the closer you can get to
the ideal state.
The process of compressing/expanding gas without any heat transfer to/from
the environment is called an adiabatic process and we will use this
process in our engine.
Another process of interest here is the isothermal
of a gas. Isothermal means that the temperature of the gas does not
change during the process. If you compress a gas isothermally you have to put
in work (pushing the piston) but you simultaneously have to remove heat to
avoid a rise in the temperature of the gas. If you consequently let the
gas expand back to its original state you a) get back the work you previously
put in but b) you also must put heat (exactly the same amount you removed
during compression ) into the gas to avoid it getting cooler. To achieve
complete reversibility the heat transfer has to take place at vanishingly small
temperature difference which of course can be achieved only in a thought
|Figure 3 : The Carnot Cycle||
In Figure 3 we finally putting our heat engine
together showing in a p-v
(pressure - volume) diagram the processes the working gas has to undergo,
see diagram in Figure 3a).
- 1 → 2 : isothermal compression at cold temperatur Tc
work Wc in , heat Qc removed
- 2 → 3 : adiabatic compression, temperature rises to Th
work W23 in
- 3 → 4 : isothermal expansion at hot temperature Th
work Wh out, heat Qh in
- 4 → 1 : adiabatic expansion, temperature falls back to Tc
work W41 out
At the moment numbers don't play a role, but some items of interest are :
Wnet = Wh + W41 - Wc - W23
Equivalently because of conservation of energy :
Wnet = Qh - Qc
Efficiency is in general defined as the ratio between benefit, here
Wnet, and the cost associated with providing
Qh by burning some type of fuel.
η = Wnet / Qh = ( Qh - Qc ) /
Qh = 1 - Qc / Qh
The sketch in Figure 3b) is exactly equal to that in Figure 3a) except for the
direction of all arrows. As Carnot engine provides useful work by absorbing
heat at a higher temperature, Th, and rejects heat at a cooler
temperature , Tc, the cycle in Figure 3b) needs work to be
Wsupplied = Wh + W41 - Wc - W23
and in return transfers heat from the cold side, temperature Tc
to the hot side, temperature Th
. Actually, Stirling engines can
be used that way and your refrigerator or thermal heat pump does it all
the time. The important point is here that if the values for pressure,
volume and temperatures are exactly the same in Figure 3a) and b) and every
process is exactly reversible then :
Wnet = Wsupplied