Caption: A annotated, colorized version of Sadi Carnot's (1796--1832) 1824 schematic diagram of the Carnot engine.
Features Continued:
For a heat engine/refrigerator the quantities are all positive/negative or zero.
If all the quantities are zero, the heat engine/refrigerator is a useless trivial case.
where we just define heat engine/refrigerator to have (Q_H > 0)/(Q_H < 0) and NOT worry about the ambiguous case of Q_H = 0.
we which just the ordinary heat engine efficiency. A perfect heat engine has η_h = 1 and perfectly bad one has η_h = 0.
If Q_H < 0,
we which a non-standard figure of merit for refrigerators. A perfect refrigerator has η_r = 1 and perfectly bad one has η_r = 0.
The standard figure of merit for refrigerators is the coefficent of performance.
Note that η_h = 1 - η_r for a reversible engine in general.
We allow reversible engine 1 to be scaled by a factor S: i.e., when scaled, reversible engine 1 quantities are S*Q_H_1, S*Q_C_1, and S*W_1.
S can be negative, and so can effect the change from heat engine mode to refrigerator mode.
The upper/lower case is a heat engine/refrigerator.
Nothing forbids us from considering the general engine so big that Q_H_2 + S*Q_H_1 > 0 which implies -1/(S*Q_H_1) > 1/Q_H_2, and thus we find
Thus, η_h_2 > η_h_1 , and engine 2 is more efficient than reversible engine 1 in heat engine mode.
Note since Q_C_2 + S*Q_C_1 = 0, the combined engine takes heat energy from the hot bath and turns it all into work without rejecting any heat energy to the cold bath.
Since this is NEVER observed, following Sadi Carnot, we conclude NO heat engine can be more efficient than a reversible engine in heat engine mode.
Now
Nothing forbids us from considering the general engine so big that -Q_H_2 + S*Q_H_1 < 0 which implies 1/Q_H_2 > 1/(S*Q_H_1)????, and so
Thus, η_r_2 > η_r_1 , and engine 2 is more efficient than reversible engine 1 in refrigerator mode.
Since -W_2 + S*W_1 = 0 , the combined engine takes heat energy from the cold bath and rejects it to the hot bath without any work done.
Since this is NEVER observed, following Sadi Carnot, we conclude NO refrigerator can be more efficient than a reversible engine in refrigerator mode.
But is there a reversible engine?
Yes, the Carnot engine which we will NOT describe here.
So the maximum efficiencies for a heat engine and refrigerator are, respectively,
Its efficiency is high, its power (energy per unit time) output is low.
In fact, the maximum efficiency is only achieved exactly in the limit of zero power.
Also they suggest---and the suggestion is true---that usually it is best to make T_H as high as possible for a heat engine even though it is NOT a Carnot engine.
Of course, there are practical limitations. You can't have your heat engine melt or become dangerously hot.
Usually, you can't lower T_C to improve efficiency since T_C is usually just the ambient medium.