Why there is no conflict between "heat always flows from hot object to cold object" and the concept of entropy

DS = Q/T where S is entropy, Q is heat, and T is temperature

When heat flows between a steak and a plate there is no violation of energy conservation; the energy lost by the steak is gained by the plate. However, conservation of energy does not explain why the heat always flows from the hot steak to the cold plate; this is where the second law of thermodynamics comes in. Suppose the steak is at temperature T, the plate is at a slightly lower temperature 0.95 T, and that a small amount of heat Q is transferred from the steak to the plate.

Then the entropy of the steak decreases by Q/T while the entropy of the plate increases by Q/0.95 T. It can be seen then that the entropy lost by the steak is smaller than the entropy gained by the plate, so the total entropy of the Universe has increased; this process is therefore consistent with the second law of thermodynamics.

If, on the other hand, hea Q had flowed from the cold plate to the hot steak, the entropy lost by the plate (Q/0.95 T) would have been greater than the entropy gained by the steak (Q/T), and the total entropy of the Universe would have decreased. This violates the second law of thermodynamics, so we can be sure that the process is impossible. Heat flow is said to be an irreversible process - you will never see heat flowing spontaneously from a cold body to a hotter one.

A quote from John von Neumann - talking to Claude Shannon about one of Shannon's new ideas in information theory

"You should call it entropy, for two reasons. In the first place your uncertainty function has been used in statistical mechanics under that name, so it already has a name. In the second place, and more important, no one really knows what entropy really is, so in a debate you will always have the advantage."