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There is a device which works as heat of pump

There is a device which works as heat of pump. This device is able to heat the air at 1 degree per second. The velocity of hot air is 0.5 m/sec.
The professor is placing an alluminium cup full of water. The problem is, how long its going to take the hot air from heating pump to rise the water from 25 deg to 34 deg.
You are totally free to make any assumption.

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Problem:

There is a device which work as heat pump. This device is able to heat the air at 1 degree per second. The velocity of hot air is 0.5 m/sec.
The professor is placing an aluminum cup full of water. The problem is , how long it's going to take the hot air from heating pump to rise the water T from 25 deg to 34 deg.
Solution:

Assumptions:

1) The air temperature rises by 1C/sec from T0 = 80C till Tf = 200C, then it remains constant.

Therefore, if denote Ta = air temperature, then we will have:
(1)
2) The heat absorbed by the aluminum cup is negligible.

Thus, the heat from the hot air is used for raising the internal energy of the water:
(2)
where
mass of water in the cup [kg]
specific heat of liquid water,
Q = heat exchanged between hot air and water [J]

3) The heat transfer from the inner wall of the cup to the water is by natural convection and water temperature rises uniformly all over:
(Newton's law) (3)
where
coefficient of heat transfer by natural convection, w  100 W/m2.K
S = area of the cup washed by the hot air [m2]
temperature of inner side of cup [C]
temperature of water [C]

4) The ...

Solution Summary

There is a device which works as heat of pump. This device is able to heat the air at 1 degree per second. The velocity of hot air is 0.5 m/sec.

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