# Geothermal heat pump or heat engine power requirements

1. Jul 11, 2013

### bkraabel

1. The problem statement, all variables and given/known data
In winter, you like to keep your house interior at 21.0 degrees C. Your geothermal
heating system, which was advertised as being reversible, draws thermal energy from an
underground reservoir at 347 K. In a cold winter, with the average outdoor temperature
being 0.0 degrees C, thermal energy escapes from the house at a rate of 1000 W +-5 percent.
Distributing the thermal energy supplied by the heating system throughout the house
requires 180 W (provided by the heating system). Is your heating system reversible
within your 5% margin of error, or was the advertisement false? Ignore any inefficiency
in converting electrical energy to mechanical energy

2. Relevant equations
For reversible heat pump, coefficient of performance is the maximum possible:
$COP_{max} =\frac{T_{out}}{T_{out}-T_{in}}$
where $T_{in}=273$ K and $T_{out}=294$ K (I think).

For reversible heat engine (Carnot engine), maximum efficiency is
$\eta_{max} =1-\frac{T_{out}}{T_{in}}$
where $T_{in}=347$ K and $T_{out}=294$ K (I think).

3. The attempt at a solution
I don't see how to relate these efficiencies (or anything else) to the power requirements of 1180 W. Also, I don't see how the outdoor temperature of 0 Celsius is relevant; it seems like we're just transferring heat from the ground to the house.

Last edited: Jul 11, 2013
2. Jul 11, 2013

### Andrew Mason

The problem is poorly drafted because this is not a heat pump. A heat pump causes heat flow from a cooler reservoir to a warmer reservoir. Mechancial work is needed to cause this heat flow. But in this case, the heat will flow without work. You just have to cause air to circulate through the hot reservoir and return to the house.

You could run a heat engine between the geothermal reservoir and the house and heat the house that way. Reversing this would cause heat to flow from the house to the geothermal reservoir - that would be a heat pump.

AM

3. Jul 11, 2013

### haruspex

I agree with AM that a heat pump pumps heat, usually against the gradient. It could pump it with the gradient, it's true, but that is not useful. A reversible one can pump against the gradient in either direction, i.e. whichever way the gradient runs.
The description seems to be that of a heat engine, generating power (180W+) from the gradient. You know the source and sink temperatures and the heat flow, so you can calculate the theoretical power output. You can then compare that with the 180W. But what any of that has to do with 'is it reversible' I have no idea.

4. Jul 11, 2013

### Andrew Mason

I agree with Haruspex that this does not appear to have anything to do with reversibility. The work output of the heat engine is not used to drive a thermodynamic process. It just facilitates natural heat flow. Furthermore, the heat is escaping from the 21° C house to a 0° C reservoir which is obviously not reversible. You don't need to do any calculation to see that even if you could save that output of the heat engine, there is no way to reverse the heat flow - more work would be needed due to the extra temperature difference.

AM

5. Jul 13, 2013

### bkraabel

reversible heat engine

Ok, after further reflection, I think the device is working as a heat engine between the ground and the interior of the house. The "waste" heat is dumped into the house to keep it warm (1000 W) and the work (180 W) is used to distribute this heat. For a reversible (Carnot) engine, the efficiency is
$\eta=1-\frac{T_{out}}{T_{in}}=15\%$
From the data given, we know that the heat engine draws 1180 W from the ground, so its real efficiency would be
$\eta_{real}=W/Q_{in}=15\%$
which would suggest that this is, indeed, a reversible engine.

6. Jul 13, 2013

### haruspex

That looks right.

7. Jul 13, 2013

### Andrew Mason

Does that make the heating system reversible? That was the question.

AM

8. Jul 13, 2013

### bkraabel

I think it makes the engine a reversible engine because only a reversible engine can have the maximum theoretical Carnot efficiency.

9. Jul 13, 2013

### haruspex

If the original question has been quoted accurately, either it is a trick question or the questioner does not understand what is meant by a reversible heat pump. I would answer it with words like: The information provided gives no evidence either way on whether it is a reversible heat pump; it does however indicate that it is a reversible heat engine.