How do you convert heat into electricity?

Question

There are two neighboring water tanks each with 1 cubic meter of water, both at 15 degrees C.

A heat pump transfers heat from one to the other, yielding one tank with 12 C and the other 21 C. Due to 50% efficiency, one tank goes up 6 C and the other down 3 C. The total energy used would be (100^3)(21 C-15 C) + (100^3)(12 C-15 C) = 3000 kcals = 3.487 KWh.

Now that you have a heat differential, use a geothermal generator to convert the heat into energy.

The 21 C water boils a liquid, turns a turbine, and condenses in the 12 C tank, creating electricity in the process. In this process, if the 21 degree water decreases to 14 C and the 12 degree water increases to become 14 C (55% efficiency), then this would result in a decrease in average temperature of 16.5 C to 14 C, and producing 5.81 KWh, which means that the system would net 2.323 KWh each cycle, therefore generating electricity by converting heat to electricity. I don’t think this can work, but why not?

Answer 1

The hole in this particular perpetual motion machine is in the impossibly high efficiency with which the heat is turned into electricity.  The theoretically best that can be done to convert a temperature difference to another form of energy is the Carnot efficiency, which is

η = (Thot − Tcold) / Thot

where Thot is the absolute temperature of the hot reservoir and Tcold is the temperature of the cold reservoir.  Any higher efficiency would necessarily violate the second law of thermodynamics, the one about entropy always increasing.

In your particular example, the temperature reservoirs are at 21 deg C and 12 deg C, which convert to 294 deg K and 285 deg K.  This results in a Carnot efficiency of

η = (294 − 285) / 294

= 0.031

The actual efficiency will necessarily be less than this.

In other words, such a machine would work but you would be putting in a lot more energy into the process than you would be getting out.  On the other hand, if the energy were free, such as the naturally-occurring temperature difference between the top and bottom of the ocean, then it might make economic sense to extract the energy.  That is, if you can solve the practical problems and come up with the necessary capital, both of which would be difficult.

Answer 2

The tanks start off at thermal equilibrium thus no heat exchange can take place between them.