Heat engines. Carnot engine, Carnot cycle. Gasoline engine. Otto cycle. Second law of thermodynamics.

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James Watt invented the steam engine in 1763. It brought about the great industrial revolution. Before the days of the steam engine men had to depend on their own muscles or on animal, wind and water power for work they wished to do. The steam engine made possible power-driven machines that could do the work of many men and brought a new era of machines, factories and automation. The modern farmer produces more with his tractor and farm equipment than hundreds of farmers could have produced before the machine age. The modern factory produces goods in vast quantities, bringing great economies and great advances in the general standard of living.

Heat engines. Today the steam engine, steam turbine, and internal combustion engine utilize the energy of fossil fuels (coal, oil and natural gas) to power the machinery of our modern society. They are examples of what are called heat engines. Heat engines can be viewed simply as devices that transform heat into mechanical energy. They do this in a two step process: 1. They first transform the fuel into heat through a combustion process. 2. They transform the heat energy into mechanical energy. In general, they perform step 2 by employing a working fluid (a gas) that takes on a quantity of heat Q, expands in a cylinder pushing a piston and doing work, thus converting part of Q into work, and then discarding the rest of Q to the surroundings (i.e. to the atmosphere).

Efficiency of heat engines. One pound of coal produces about 13,000 Btu of heat. We know that 1 Btu = 778 ft∙lb of mechanical energy so we might expect a steam engine to get 13,000×778 ft∙lb of mechanical energy from every pound of coal. In fact, a steam engine provides only from about 5% to about 30% of this value. Where does the rest of the energy go? Stack and friction losses account for only a small part of what is lost. By far the largest part appears as heat rejected in the exhaust. The efficiency of the modern gasoline engine is 15 - 20%. Again, the largest part of the heat lost is in the exhaust. That sounds like very poor economy. Can’t we devise engines that do better than that?

Carnot engine. A Carnot engine is an idealized engine operating under a Carnot cycle. The Carnot cycle is a theoretical thermodynamic cycle proposed by Sadi Carnot in 1824 and modified later by Clapeyron. It can be shown that the Carnot cycle is the most efficient cycle for converting a given amount of thermal energy into work.

In the Carnot cycle the working fluid (a gas) takes in heat at some fixed high temperature T₁, converts part of it to mechanical energy in the form of work, and discards the rest to some low temperature reservoir (i.e. the surroundings, the atmosphere) at temperature T₂. The amount of heat taken in is Q₁, the amount discarded at temperature T₂ is Q₂, and the difference Q₁ - Q₂, is the amount converted to work.

The cycle consists of four steps. The steps are detailed in Fig. 1 and 2. The cylinder in Fig. 1 is assumed to be well insulated so that no heat will pass through its walls.

Step 1. a → b. An isothermal expansion of the gas at temperature T₁. The gas is heated and allowed to expand isothermally. It does work on its surroundings and takes in a quantity of heat Q₁. See Fig. 1 and 2.

Step 2. b → c. An adiabatic expansion of the gas until the temperature drops to T₂.

Step 3. c → d. An isothermal compression of the gas at temperature T₂. Here work is done on the gas and a quantity of heat Q₂ is discarded to the reservoir.

Step 4. d → a. An adiabatic compression of the gas until its temperature reaches T₁. During this step compression work is done on the gas by the surroundings. On completion of this step, the gas is back at the same state as it was at Step 1 and the cycle has been completed.

It is seen that the Carnot cycle consists of two isothermal processes and two adiabatic processes. The net amount of work W done during the entire cycle is that shown enclosed by the curves, abcda, in Fig. 2 where W = Q₁ - Q₂.

Efficiency of a heat engine. If a heat engine takes in a quantity of heat Q₁ at a high temperature T₁, transforms a portion of the heat W to work, and discards the rest Q₂ to a low temperature reservoir at temperature T₁, its efficiency is given by the fraction of the heat taken in that is converted into work i.e.

Efficiency of a Carnot engine. It can be shown that the efficiency of a Carnot engine is given by

where T₁ is the absolute (Kelvin) temperature at which heat is taken in by the working fluid and T₂ is the absolute temperature of the reservoir to which the heat is exhausted.

No engine can be more efficient than a Carnot engine. It has been proven that no engine operating between two given temperatures can be more efficient than a Carnot engine operating between the same two temperatures. It has also been proven that all Carnot engines operating between the same two temperatures have the same efficiency, irrespective of the nature of the working substance.

The gasoline engine. A cycle of the gasoline engine consists of four strokes:

Stroke 1. Intake stroke. Starting with the piston at the top of the cylinder, the piston moves down drawing in an explosive mixture of air and gasoline vapor from the carburetor, the inlet valve being open and the exhaust valve being closed.. See Fig. 3.

Stroke 2. Compression stroke. The intake valve closes and the piston moves up, performing an approximately adiabatic compression of the air-gasoline mixture to one-seventh or one-eight its original volume.

Stroke 3. Power stroke. Just before the piston reaches the top of the compression stroke, an electric spark is produced between the points of a spark plug set in the cylinder head. The spark ignites the air-gas mixture in the cylinder and the resulting explosion produces hot gases that force the piston down in an approximately adiabatic expansion.

Stroke 4. Exhaust stroke. The exhaust valve opens and the piston moves up, forcing the exhaust gases out of the cylinder.

Otto cycle. The behavior of a gasoline engine can be approximated by an idealized cycle called the Otto cycle. An assumption is made that there is no friction, no loss of heat through the cylinder walls, and that the working fluid is air, which behaves like an ideal gas with constant heat capacities.

Starting at point a in Fig. 4, air at atmospheric pressure is:

1) compressed adiabatically in a cylinder to point b

2) heated at constant volume to point c

3) allowed to expand adiabatically to point d

4) cooled at constant volume to point a, after which the cycle is repeated.

Line ab corresponds to the compression stroke, bc to the explosion, cd to the power stroke, and da to the exhaust stroke of a gasoline engine.

Efficiency of the Otto cycle. The thermal efficiency of the Otto cycle is given by

where V₁ and V₂ are respectively the maximum and minimum volumes of air in the cylinder and γ is the ratio of the specific heats of air, i.e. γ = c_p/c_v,. The ratio V₁ / V₂ is called the compression ratio, and is about 7 for a gasoline engine. For a compression ratio of 7 and a value of γ = 1.4, the efficiency is about 0.54. Because of friction, loss of heat through the cylinder walls, etc. the efficiency of an actual engine is much less.

Second law of thermodynamics. The second law of thermodynamics has been stated in a number of ways. Some common statements are:

1. Clausius: It is impossible to construct a machine that, while operating in a cycle, will produce no other effect than the transfer of heat from a cold body to a body of higher temperature.

2. Kelvin-Planck: It is impossible to construct an engine that, while operating in a cycle, will absorb heat from a single reservoir and produce an equivalent amount of heat.

Mechanical energy can be converted completely into heat, but heat cannot be converted completely into mechanical energy. This is an example of a one-sidedness of nature that is revealed in many ways. Heat always flows from a hotter to a cooler body and never the other way; rocks weather, iron rusts, people grow old. These are all examples of irreversible processes that take place in one direction only and express the second law of thermodynamics.

References

Dull, Metcalfe, Brooks. Modern Physics.

Sears, Zemansky. University Physics.

Schaum. College Physics.

Semat, Katz. Physics.

Kittsley. Physical Chemistry.

Warner. Thermodynamic Fundamentals.

Zemansky. Heat and Thermodynamics.