A Heat Engine with few moving parts can be constructed to extract energy from th

Question

A Heat Engine with few moving parts can be constructed to extract energy from the simple cycle of day and night. An aluminum cable 25 meters long and 25 cm in diameter hangs from a fixed point. During the day, when it is 80*F outside, the cable is just long enough that is can be attached, by means of a ratchet to a 55 ton mass that rests on the ground. At night, when the temperature drops to 50*F, the cable contracts, lifting the mass off the ground.The mass is then released from the cable and work is done. When it warms back up to 80*F the next day the cable will once again expand to the point where it can be attached to the mass, and the cycle repeats. (Day and night cycle, from 80*f to 50*f. Determine the work (in J) released when the mass falls.

Explanation / Answer

Hi,

In this case, the work released is equal to the gravitational potential energy stored in the system (cable - mass).

Said energy can be calculated as:

U = mgh ; where m is the mass (in this case, the one that was initially on the ground) and h is the height above the ground.

According to the problem, the height is equal to:

h = L ; so we have to find the change in the length of the cable in order to calculate the released work.

The change in length is calculated as follows:

L = L T ; where is the coefficient of linear expansion (which is equal to 23*10-6 1/°C in the case of the aluminum)

T1 = 80°F = 26.7 °C ; T2 = 50°F = 10 °C

L = (25 m)(23*10-6 1/°C)(26.7 - 10)°C = 9.603*10-3 m

This means that the work released would be:

W = (55*103 kg)(9.8 m/s2)(9.603*10-3 m) = 5180 J

If we compare this value with the heat received by the cable:

V = R2H = (12.5*10-2 m)2 (25 m) = 1.227 m3

M = pV = (2698.4 kg/m3)(1.227 m3) = 3311 kg (density of aluminium = 2698.4 kg/m3)

Q = MCT = (3311 kg)(900 J/kgK)(16.7 K) = 4.976*107 J (heat capacity of aluminum = 900 J/kgK)

We can see that the heat is much bigger than the work (as it should be, because there is no heat engine that can transform the heat entirely into work).

Note: in this case the possible resistance of the mass to be lifted was not considered. If we would to consider said effect, the work would be much lower.

I hope it helps.

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