Pan size affecting evaporation time

The amount of energy required to evaporate the water stays the same, no matter what the size of the pan is, but a bigger pan could potentially collect more energy since the area in contact with the cooktop would be larger. There is also a larger portion of the water in contact with the pot, making it easier for more energy to transfer to it.

This question on the physics stackexchange basically ask the same question you do here, and states that the surface area of the water will have a significant effect on the rate of evaporation if the stove transfers enough energy. Given that there is no change in the temperature of the sauce during the reduction, the rate of evaporation will be proportional to the surface area of the sauce (as stated in the final formula there).

Yes, it can. The issue is the amount of surface area available for evaporation.

This will result in evaporative cooling which will result in additional heat being needed to maintain that same temperatre (and thus a simmer), but it does mean that you can reduce things faster by putting them in a wider vessel.

You may also be able to speed the evaporation by improving air flow in the kitchen -- moving the moist air away from the cooking vessel.

If you'd like to test this experimentally, you'll need two similar glasses, a sheet pan, and some rubbing alcohol:

• pour some alcohol into the two glasses, as equal as you can easily get them.
• dump the more full shot on the sheet pan.
• wait an hour.
• pour what remains (if any) from the sheet pan back into the empty glass.
• compare

If there's any left will be a function of the ambient temperature and concentration of alcohol.

In theory, you could speed the evaporation up by using a broader pan. But first, as erikadude noted, it depends on the energy pumped into your sauce. If you are on an electric burner with a fixed size, and put on it a pan much wider than the burner, the sauce in the outer parts of the pan will not get hot enough, and your evaporation rate will go up, but not proportionally to the pan surface.

But the more important point is that there is a reason for the recipe to require that much time. If it prescribes to be cooked for such a long time, then it is a slow cooking recipe which intentionally blends the taste of the ingredients by having them spend a long time on low heat. If you change the times needed, you will change the finished taste. If it is not convenient for you to make a 4h recipe, the better alternative would be to use a different recipe. You could either use recipes which start from a concentrate (tomato paste, ajwar, kyoopoloo would all fit) or choose a sauce which is thickened by a roux or by melting grated cheese into the sauce. This is usually much more energy and time efficient than speeding up a traditional slowcook recipe.