Heat pump efficiency

Here's something that surprised me...I guess I didn't pay close enough attention in thermodynamics class....

What's the most efficient way to heat a house, a) burn natural gas, or b) run an electrically powered "heat pump" system?

I would have answered a), thinking that nothing could be more efficient than to burn an energy source directly into heat. But this is totally wrong. It is actually far more efficient to let the power company burn that gas to generate electricity, and then use the electricity to run your heat pump.

The outside air may be colder than the inside, but it still stores plenty of heat - the only trick is how to get it from the outside to the inside. Heat doesn't naturally move from a colder place to a hotter place, so it takes energy to pump it, but there is a multiplier factor: a given amount of energy can transfer several times that amount of heat.

Heat pumps really seem like a case of "something for nothing". How can energy E magically pump 3E or 4E of heat from the freezing outdoors to the inside of your house?

The first key is that the outside and inside temperatures are actually not that different, when measured in the Kelvin scale, i.e., starting from -460 Farenheit. The difference between 68 degrees indoors and 32 outdoors is only about 6% on this scale.

The second key is that the obstacle to transferring heat from outdoors to in is not energy, but entropy. After all, we are not talking about creating any energy - just moving it around. Energy ordinarily doesn't move from cold to hot places because it has lower entropy in the hot place ¹; however, the entropy difference is not that great for normal temperatures because it depends on the temperature difference in Kelvin.

The third key is that the energy we use to run the heat pump has to be in a very low-entropy form, such as natural gas, rather than a high-entropy form such as the air inside the house. (We could not use the energy in that air to power the heat pump!)

To pump energy from outside to inside, then, all we have to do is make up the relatively small difference in entropy, which we can do by taking a little bit of low-entropy energy and converting it to high entropy.

Of course this doesn't tell us how to make a heat pump, it just tells us something about the performance we can expect. But a heat pump is not complicated - it is just a refrigerator or air conditioner turned backwards.

Added 8/4/2013: Since the heat pump is just a standard heat engine running in reverse, its efficiency is the inverse of that of a heat engine. The efficiency of an ideal heat engine is W/Q=(H-C)/H where H is the hot temperature and C is the cold, and W is the work done while Q is the heat transferred.  The "efficiency" of a heat pump is just the inverse of this, Q/W, and it will always be greater than one, at least for a reasonably well constructed engine.

I learned this from the book "Sustainable Energy - Without the Hot Air", by David MacKay, a book I highly recommend.

¹ Hot energy has lower entropy than cold energy because the cold energy spreads over more "degrees of freedom". For example, one Joule at a cold temperature may be shared over N particles, while at a hot temperature it is only shared over N/2 particles, because the particles are all moving around faster on average.