# Using heat energy to increase temperature

It's been a long time since I studied thermodynamics in college years ago, but I was just wondering this:

I know that a large bathtub of 10 C water has more heat energy than a small cup of 20 C water, and that heat naturally flows from objects of higher temperature to objects of lower temperature, but is there any way to use the heat energy of the bathtub to get the temperature of the cup water higher on a small scale level like this?

I purposefully chose number values that would make it hard to use the heat energy of the bathtub water to do something conventional like drive a turbine to generate electricity to heat the cup water because I'm specifically curious about possibility and efficiencies without huge amounts of energy.

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People use a similar scheme to warm their homes from heat stored in the ground outside their house. Expensive to install I guess but 'free' heating! Like here: energysavingtrust.org.uk/Generate-your-own-energy/… –  Nic Jul 28 '11 at 18:16

In this case, you are moving heat from 283K to 293K. To move one Joule of heat across that temperature gradient requires $1 - \frac{283}{293} = \frac{10}{293}$ Joules of work at minimum. This is for a theoretical ideal heat engine. To do it in real life, you will need more work than that because you cannot be perfectly efficient.