Clausius’ Equivalence of Transformations — Part III — Energy Viewpoint
A fresh look at Clausius’ transformations in terms of gravitational potential energy
Clausius defines two transformations:
- transformation of heat from a high temperature to a low temperature
- conversion of heat to work and vice-versa
But what do these transformations actually mean? What do they represent?
Why is there a transformation from ‘hight to low’ temperature?
We gain some insight by looking not at a heat engine but instead at a standard energy diagram showing water falling through a gravitational field.
Energy diagram — water falling through a gravitational field
The water starts at a height h=H1 in a gravitational field. As the water falls, it does work, turning a flywheel. After doing work, the water joins a pool of water at height h=H2<H1.
The water at height H1 has a potential energy (to within a constant) given by
- E1=mgH1
As the water falls, some amount of that energy will be expended as work. The flywheel cannot extract more energy from the falling water than is available. In the lower pool, the water has potential energy given by
- E2 = mgH2
The maximum amount of energy the flywheel can extract is then just the difference between these two potential energies is given by
- diff E=mg(H1-H2)
Hence, the maximum amount of work that can be done by the falling water in this example is
- W=mg(H1-H2)=E1-E2
Energy picture in two parts
Now redefine the energy picture into two distinct parts, to align with Clausius’ two transformations.
Part 1 — Work done
- Heat equal to W is transferred from the hot bath and totally converted to work by the engine
Part 2 — Energy Conservation
- Energy equal to E2 is transferred from the higher pool at height H1 to the lower pool at height H2.
the meaning of the 2nd part is clear from the diagram of water falling through a gravitational field — each particle of water starts with an energy E1 at the top bath. It gives up W energy to turn the flywheel. The same particle of water then joins the lower bath at an energy level of H2.
Clausius Transformations and the Energy Picture
Clausius’ two transformations now align directly with the reformulated energy version defined above.
Transformation Type 1— heat to work
Clausius representation
- Heat equal to W is transferred from the hot bath and totally converted to work by the engine
Energy representation
- Energy equal to W is transferred from the higher pool at height H1 to the flywheel and totally converted to work
Transformation Type 2 — transfer of heat from high T to low T
Clausius representation
- Heat Qc is transferred from a high temperature Th to a low temperature Tc
Energy representation
- Energy equal to E2 is transferred from the higher pool at height H1 to the lower pool at height H2.
In the energy picture, the energy is E2. In the thermodynamic picture, the energy is Tc.
Of course, energy E2 or heat Qc is not transferred per se from the hot bath to the cold bath.
A particle traverses a path from the hot bath to the cold bath. The particle starts out with an energy of E1 in the gravitational picture or a temperature of Th in the thermodynamic picture. Recall, that temperature is directly proportional to total internal energy — take a simple gas:
- U(T)= (3/2)NKT
The particle gives up energy W=Qh-Qc=E1-E2 energy to the flywheel, doing work. The particle is then at energy E2 and enters the cold bath at temperature Tc.
Heat and Entropy Flows
Here is a diagram that shows the heat flows and corresponding entropy flows for a generic heat engine.
The entropy flows are shown with respect to the engine e.g. +Qc/Th indicates a positive flow of entropy into the engine at a temperature of Th.
Summary
Clausius’ two transformations can be seen as being consistent with energy conservation rules. In this case, energy conservation dictates that the maximum amount of energy available to do work is the energy difference between the two heat baths or, as in the example above, the two pools of water at different heights.
The energy that is not used to do work must match the energy of the lower pool, since that is the energy of the water as it transits past the flywheel and stops falling.