Our hands are the most common source for measuring heat transfer$^{\dagger}$ in our day-to-day life (they do not measure temperature, though they do indicate relative temperature). Now consider the following:
- When we rub our hands their temperature rises due to the friction. Now if there is equal increment in the temperature of both the hands then there should be no flow of heat (zeroth law of thermodynamics).
Hence we should feel no difference than usual as there is no heat, rather due to there being a temperature gradient heat should flow out of our hands to surrounding and our hands should feel cooler (assuming that surrounding has same temperature as our normal body temperature).
So
- why do our hands feel warm when we rub them?
One possible explanation is that one of the hand gets slightly warmer than the other and hence there is a heat flow from one to other and hence we feel hot. But there is a problem that if that were the case then we would get an alternating hot and cold sensation, which isn't what is observed.
$\dagger$ It is a common misconception that our hands measure the temperature. Rather they measure the heat flow (which is an indicator of relative temperature). Related: Veritasium: Misconceptions About Heat.
Also note that I know that the increment in temperature is due to the friction so no explanations needed for that.
Best Answer
The frictional heat generated in the contact area between our hands flows roughly equally to each hand. The interface gets hotter, and there is a heat flow by conduction away from the interface in both directions. At the interface itself, there is a discontinuity in the heat flux, with opposite signs to the two fluxes. The difference between the two heat fluxes is the frictional heat generation rate per unit area. So we have $$\left[-k\frac{dT}{dx}\right]^+-\left[-k\frac{dT}{dx}\right]^-=q_{gen}$$So, $$\left[-k\frac{dT}{dx}\right]^+=\frac{q_{gen}}{2}$$and$$\left[k\frac{dT}{dx}\right]^-=\frac{q_{gen}}{2}$$
As a result of all this, the temperature is highest at the interface (higher than the bulk of your hands), and this is what you sense as your skin temperature.