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when we blow out air its kinda adiabatic process(the time is so short that air is unable to exchange energy wid surroundings)
let us, for once, streamline this question and only keep it specific t othe case in which surrounding temp. is lower than 37 C. applying elementary fluid mechanics AV is a constant ( equation of continuity) as the aperture (mouth) has different cross-sections for both cases the velocity will change accordingly. hence, in a wide opening, the amount of air per unit volume will be lesser hence, causing little scattering of the original coat of air present around the hair, hence, a small increase in temp. but, a uniform blow from a small opening will intially interact wid only a small section og the hand and then cause movement in the rest of the cover of air. due to the stream of fast CO2 moving outside da rate of evaporation is greater for the moisture hence, cooling is caused.
note: in both cases the initial relaxation of the lungs is taken as the same for application of eqn of continuity.
another point, the application of da adiabatic process cud b used for the sheath of air around the hand. as the air from da narrow opening comes out it has more k.e. hence, vit displaces the air around da hand over a larger area hence, lowering da temp, which isnt done in da bigger opening. this case can b seen as one of perfect overlap in da case of the wide opening and of irregularity in da case of da narrow 1.













Your hands are surrounded by a thin layer of air that has been somewhat warmed by your hands, and it acts as an insulating blanket between your hands and the cold surrounding air.
The simple explanation here is that when you blow with a wide open mouth, the speed and pressure of the air coming out are necessarily low, and you are in effect dumping warm moist air from your core (lungs) into your hands, replacing that less warm insulating layer. So your hands feel warmer.
Since your lung power is the same, when you narrow your mouth, you blow with much higher pressure and speed due to the smaller opening. This air stream disturbs and blows away the thin layer of insulating air around your hands, replacing it with the cooler surrounding air, and so your hands feel colder. This is forced convection, and also a little bit of cooling by evaporation of the tiny amount of moisture on the skin of your hands.
If you want a more scientific explanation, involving precise relationship between pressure, air speed, volume, specific heats and a host of other items, you will have to look up Venturi and Bernoulli somewhere else.