On the Conversion of Heat into Mechanical Work


By Robert M. Hazen, Ph.D., George Mason University

The second law of thermodynamics places limits on how energy can transfer from one form to another. Heat tends to flow from hot to cold. Similarly, the conversion of heat into work in an engine cannot be performed with 100 percent efficiency because some of the heat must escape into the environment and some has to be used to reinitialize the engine.

An image of a steam engine on a white background.
Nicolas Carnot’s equation for efficiency is now used to calculate the efficiency of any engine. (Image: Martin Bergsma/Shutterstock)

Waterwheel and Steam Engine Analogy

An analogy can be drawn between heat energy and the gravitational energy of water passing over a waterwheel. Heat is a more abstract concept, whereas water can be held in the hand. Hence, imagine designing a waterwheel with maximum possible efficiency. The elevation of water above sea level is analogous to fuel temperature, and the level at the bottom of the waterwheel is analogous to the low-temperature reservoir where the water flows out. In this design, sea level is considered zero energy or absolute zero.

First, some water is going to be leaking away, it is going to evaporate into the air, or it may soak into the material of which the waterwheel is made. The waterwheel will be designed with extreme efficiency toward minimizing evaporation levels, loss of water, and leaking by sealing all cracks very carefully. But still, a little bit of water will be lost, and there’s no way around that.

An image of a waterwheel in a village
An analogy can be drawn between heat energy and the gravitational energy of water passing over a waterwheel. (Image: IMG Stock Studio/Shutterstock)

Second, some of the water’s energy has to be used to turn the wheel by pulling down the buckets to go around. So some of the water’s energy has to be utilized just to raise up the buckets.

The waterwheel can be made of extremely lightweight materials and high-quality lubricants toward smoother operation of the wheel, but there is always going to be some water loss as the wheel goes around. And this loss is exactly like the energy used to draw the piston of a steam engine back.

But no matter how carefully the waterwheel is designed, some of the potential energy of the water is always squandered, because the waterwheel is always going to be at some point above sea level, and its efficiency is going to be limited. That’s the intrinsic loss that Nicolas Carnot recognized.

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Mathematical Law for Maximum Efficiency

Carnot’s great contribution was the derivation of the exact mathematical law that stipulates the maximum possible efficiency for any engine. That is, the percentage of heat energy that can be utilized for performing work, as opposed to the percentage always squandered (due to design considerations). It turns out that the maximum possible efficiency of any engine depends entirely on two easily measurable figures.

First, there is ‘T’ for the hot end (heat energy flowing into the system to do work). And at the other end of the system, there is T of the cold reservoir, usually called ‘Tcold’. That’s the temperature of the material that flows out of the engine, the exhaust. Every power plant/engine has an exhaust of some sort, and it’s the temperature of that exhaust.

An engine is just a mechanical device imposed between the hot reservoir and cold reservoir, just like a water wheel is a mechanical device imposed between high gravitational energy and lower gravitational energy.

Efficiency then can be defined mathematically as the difference between these two temperatures, the difference between Thot and Tcold, divided by the temperature of the hot reservoir.

Percentage-wise, it has to be multiplied times a hundred. The following equation is used to calculate all sorts of efficiencies, the maximum possible efficiency for different kinds of power plants, engines, and so forth. This is highly significant for understanding possible efficiency for modern engines.

E= 1-TCold/THot×100

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The Conversion of Heat into Work in Today’s Society

An image of a man with coal in his hands.
Fuels are graded according to the heat they produce. (Image: Parilov/Shutterstock),

In Carnot’s day, before these principles were understood, typical steam engines had an efficiency of only six percent. But when people saw Carnot’s reasoning, they realized they could get much higher efficiencies. Therefore, they began to design improvements, better insulation, better materials, tighter-fitting steam hoses, and so forth. And today, with these improvements, efficiency has been raised to about 40 percent, for example, in coal-burning power plants.

That’s close to 90 percent of the theoretical limit, so it is getting very close to the maximum theoretical efficiency of a coal-burning power plant. Here’s a clear case where theoretical ideas and understanding have led to dramatic benefits for society because energy is used more efficiently today.

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Kinds of Fuel According to the Heat They Contain

Fuels can be graded according to the heat they contain. Carnot’s explanation illustrates why some hot-burning fuels like coal, gas, and oil, for example, are all very much valued. It’s because they burn at such high temperatures, and therefore can make more efficient plants. About 90 percent of all the industrialized world’s energy comes from fossil fuels.

It’s still possible to scrape coal off the ground in some places and to drill shallow wells, so for the time being, these fossil fuels are going to be very valuable and play a very important part in the world’s energy supply. These fossil fuels represent ancient life forms that have been buried and transformed by the Earth’s heat and temperature. They are by definition limited resources.

Common Questions about the Conversion of Heat into Mechanical Work

Q: Why are fossil fuels more valuable than other fuel sources?

When fossil fuels burn, they reach much higher temperatures than other fuels. So, a greater conversion of heat into work occurs.

Q: What is the intrinsic loss according to Nicolas Carnot?

To increase the conversion of heat into work in any engine, Nicolas Carnot realized that they must be designed perfectly with lightweight pistons, good lubricant, better insulation, etc. He also realized that no matter how perfectly the engines are designed, there is always going to be energy loss in the system.

Q: What was Nicolas Carnot’s efficiency equation?

Nicolas Carnot derived a mathematical equation in which the efficiency of any engine could be calculated. In this equation, The difference between hot and cold ends is divided by the temperature of the hot reservoir. The better the conversion of heat into work, the higher the engine efficiency.

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