# 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.

https://brainmass.com/physics/convection/device-works-heat-pump-157402

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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 1C/sec from T0 = 80C till Tf = 200C, 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 thickness of the cup wall is small comparing to its diameter, so that the conduction through the aluminum wall occurs like in the case of a flat wall:

(Fourier's law) (4)

where

coefficient of thermal conductivity for Aluminum, 237 W/m.K

= thickness of cup wall [m]

temperature of outer side of cup [C]

5) The heat transfer from hot air to the aluminum cup is by forced convection and is governed by Newton's law:

(5)

where

coefficient of heat transfer by forced convection [W/m2.K]

temperature of the air stream [C], according to (1)

By eliminating from equations (2), (3), (4) and (5), we will get the following equations of thermal equilibrium:

(6)

(7)

(8)

Unknowns: , and .

We need to find out how changes in time, so that we will eliminate the other 2 unknowns and we will get an ordinary differential equation in :

(8) (9)

(10)

(11)

We replace that is (7) and one yields:

(12)

(13)

If we denote

(14)

then we will get

(15)

(16)

By replacing that in (6), one yields:

(17)

(18)

Let's denote:

(19)

(20)

From (14), we have

(21)

(18) (22)

The general solution of this equation is

(23)

Initial condition:

(24)

According to (1), we will have:

for t < 120 (25)

for t 120 (26)

Computing the integrals, one yields:

for t < 120 (27)

(26)

for t 120 (28)

In order to compute the parameter (B) defined by (19), we need first to determine the coefficient of heat transfer air.

We assume that convection occurs in the same way like on a flat plate, for which we have the formula:

(29)

where

Nu = Nusselt number, defined by

(30)

Re = Reynolds number:

(31)

Pr = Prandtl number, Pr (air) = 0.75

air = 1.225 kg/m3, air = 0.026 W/m.K, air = 1.710-5 Ns/m2, D = 0.058 m (average diameter of cup), Vair = 0.5 m/s

(14)

The cup area is

and its volume is given by

(32)

r = 0.022 m

Using (27) and (28) with the computed value of B, we can determine the time when water temperature reaches 34C.

Because we are dealing with a transcendent equation, an easy way to solve it is by using EXCEL.

https://brainmass.com/physics/convection/device-works-heat-pump-157402