Caption: A annotated, colorized version of Sadi Carnot's (1796--1832) 1824 schematic diagram of the Carnot engine.

UNDER CONSTRUCTION till whenever

1. For an engine cycle (AKA thermodynamic cycle) of a heat engine/refrigerator, let Q_H be the heat energy removed from the hot bath by the working fluid, Q_C be the heat energy rejected to the cold bath by the working fluid, and W be the work output.

   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.

2. By energy conservation (which in this context is also the 1st law of thermodynamics), we have

         Q_H = Q_C + W .

3. A general figure of merit for heat engines and refrigerators is

         η = | W/Q_H + (1/2)(Q_H/|Q_H|-1) | ,

   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.

4. If Q_H > 0,

         η_h = W/Q_H = 1 - Q_C/Q_H ,

   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,

         η_r = | W/Q_H -1 | = Q_C/Q_H ,

   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.

5. Now assume we have reversible engine 1. By reversible, we mean it can operate as a heat engine with Q_H_1, Q_C_1, and W_1 or as a refrigerator with -Q_H_1, -Q_C_1, and -W_1.

   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.

6. Consider a general engine 2 with quantities with ±Q_H_2, ±Q_C_2, and ±W_2.

   The upper/lower case is a heat engine/refrigerator.

7. We now combine reversible engine 1 and engine 2 to make a combined engine which obeys

         ±Q_H_2 + S*Q_H_1 = (±Q_C_2 + S*Q_C_1) + (±W_2 + S*W_1) .

8. Consider the upper case of the combined engine and scale reversible engine 1 so that Q_C_2 + S*Q_C_1 = 0.

   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

         η_h_2 = 1 - Q_C_2/Q_H_2 = 1 - |SQ_C_1|/Q_H_2 > 1 - |SQ_C_1|/|S*Q_H_1| = 1- Q_C_1/Q_H_1 = η_h_1 .

   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.

9. Consider the lower case of the combined engine and scale reversible engine 1 so that -W_2 + S*W_1 = 0.

   Now

       η_r_2 = |W_2/Q_H_2 - 1| = |S*W_1/Q_H_2 - 1| ,

   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

       η_r_2 = |W_2/Q_H_2 - 1| = |S*W_1/Q_H_2 - 1| > |W_1/Q_H_1 - 1| = η_r_1 ,

   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.

10. Sadi Carnot concluded that a reversible engine must be the most efficient heat engine/refrigerator possible.

    But is there a reversible engine?

    Yes, the Carnot engine which we will NOT describe here.

11. It can be shown that the Carnot engine has Q_C/Q_H = T_C/T_H, where T_C is the Kelvin temperature of the hot bath, T_C is the Kelvin temperature of the cold bath.

    So the maximum efficiencies for a heat engine and refrigerator are, respectively,

    η_h = 1 - T_C/T_H     and     η_r = T_C/T_H .

12. If the Carnot engine is the most efficient heat engine/refrigerator why isn't it widely used?

    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.

13. Carnot engines do have special uses in experimentation.

14. The maximim efficiencies give the upper limits to practical heat engines/refrigerators.

    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.