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There is a minimum energy needed to drive a heat pump or refrigerator. An ideal or reversible heat pump would consume energy and this would be the minimum amount of energy possible in perfectly ideal conditions. What is an ideal or "reversible" process?

Bouncing Ball

A Reversible Process
When Barry lets his golf ball fall to the ground, it bounces back. (try your mouse over the image) If there were no friction it would bounce back to his hand - exactly where it started. This would be a reversible process. As the golf ball falls, its potential energy is converted into kinetic energy. On reaching the ground the kinetic energy is converted into compression of the golf ball material. This then re-expands converting the stored compression energy back to exactly the same amount of kinetic energy, but in the reverse direction. This kinetic energy propels the ball upwards back to its starting point. However due to air friction, and losses in compression it never comes back to the same point. Barry can improve the situation by using a more "bouncy" ball, or checking out what happens on the moon where there is no air (more difficult!) but always the ball never quite makes it back. The reversible process is an "ideal" one which would be obtained in a perfect world.

Piston moving in cylinder

A Reversible Heat Pump Cycle - A Carnot Cycle
Carnot was the first person to realize that temperature difference is the key to understanding the power available from a steam engine, and his name is used to describe a reversible heat pump or engine cycle.

Imagine a cylinder barrel containing a pure fluid with a trapped by a piston. The barrel is perfectly insulated but the bottom is thin and a good conductor.

Start at point 1. The piston is near the top of the cylinder and the vapour is at high pressure and in contact with a hot reservoir H at the same temperature. Force the piston down very slowly, compressing the vapour and condensing it. (try your mouse over the image) The latent heat of condensation is transferred to the Hot Body, H, which is at the same temperature. At point 2 condensation is complete, H is removed and the piston starts to move very slowly upwards. The fluid pressure falls to point 3 but no heat transfer takes place.The temperature falls due to the expansion of the fluid until it reaches exactly that of C, the cold body. This is point 3. When point 3 is reached, the cold body C is brought into contact with the cylinder and heat is transferred from the cold body to the fluid, which expands to point 4. Vapour is formed, pushing the piston upwards - the fluid is boiling at temperature C. The cold body is removed at point 4 and the piston starts to move downward again, raising the pressure, but without heat transfer taking place, until point 1 is reached again.

The overall effect of the complete cycle is to transfer heat from the fluid from C at the lower temperature to the fluid at H, the higher temperature. Work is done on the piston because it has to be forced down against the higher pressure, but when it is moving upwards the pressure is lower. The quantity of work is represented by the area of the diagram 1-2-3-4.

This is a reversible cycle. It can in principle be equally used as a heat engine to produce power. In this case the cycle would be 4-3-2-1 and heat would flow from H to C, and at the same time the piston provides power. Like the bouncing ball, this cycle is an idealisation because in practice heat will only flow in measurable quantities if there is a temperature difference, but for the sake of argument we have supposed the temperature differences to be infinitely small. And friction is of course absent.

We know that this is the best efficiency which can be obtained because if an engine of greater efficiency producing more power could be devised, it could be used to drive the heat pump with extra work to spare. This extra work would be, in effect, coming from nothing. We would have created a perpetual motion machine which is impossible.

W = Q1 - Q2

The reversible engine E draws heat Q1 from the hot reservoir and deposits heat Q2 at a lower temperature to the cold reservoir. It generates work W. Because energy cannot be created or destroyed, the amount of work is the difference between Q1 and Q2. This work can be used to drive an identical heat pump P as shown. In this situation nothing is changed because the amount of heat entering and leaving each reservoir is exactly matched. The net heat flow is zero. This is analogous to the perfect bouncing ball. Never achievable, but we can see that it is the "perfection" which could never be bettered.


An Impossible Case
If all the heat could be converted to work by the engine E, as shown, the result would be that a quantity of heat Q2 flows from the cold to the hot reservoir with no other change occurring. We know that in our universe this does not happen. Heat only flows to lower temperatures. The question of where all the heat in the universe will end up is a matter for astronomers and philosophers.

W = Q1 - Q2

So its back to the reversible cycle for the concept of lowest power input. The relationship between Q1, Q2 and W depends only on the temperatures of the hot and cold reservoirs, just as Carnot predicted. But what do we mean by temperature, how can it be defined?

There are several practical reasons why a the cycle shown above is not suitable for a practical refrigeration machine. The most practical solution is a vapour compression cycle. This cycle, even if brought to perfection cannot achieve Carnot efficiency.

Learn more about Sadi Carnot

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