Thermal efficiency

In thermodynamics, the thermal efficiency ($eta\_\{th\}\; ,$) is a dimensionless performance measure of a thermal device such as an internal combustion engine, a boiler, or a furnace, for example. The input, $Q\_\{in\}\; ,$, to the device is heat, or the heat-content of a fuel that is consumed. The desired output is mechanical work, $W\_\{out\}\; ,$, or heat, $Q\_\{out\}\; ,$, or possibly both. Because the input heat normally has a real financial cost, a memorable, generic definition of thermal efficiency is ["Fundamentals of Engineering Thermodynamics", by Howell and Buckius, McGraw-Hill, New York, 1987]

:$eta\_\{th\}\; equiv\; frac\{\; ext\{What\; you\; get\{\; ext\{What\; you\; pay\; for.$

From the first law of thermodynamics, the output can't exceed what is input, so

:$0\; le\; eta\_\{th\}\; le\; 1.0.$

When expressed as a percentage, the thermal efficiency must be between 0% and 100%. Due to inefficiencies such as friction, heat loss, and other factors, thermal efficiencies are typically much less than 100%. For example, a typical gasoline automobile engine operates at around 25% thermal efficiency, and a large coal-fueled electrical generating plant peaks at about 46%. The largest diesel engine in the world peaks at 51.7%. In a combined cycle plant, thermal efficiencies are approaching 60%. [ [http://www.ge-energy.com/prod_serv/products/gas_turbines_cc/en/h_system/index.htm GE Power’s H Series Turbine] ]

Heat engines

When transforming thermal energy into mechanical energy, the thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. Thermal efficiency is defined as

:$eta\_\{th\}\; equiv\; frac\{W\_\{out\{Q\_\{in\; =\; 1\; -\; frac\{Q\_\{out\{Q\_\{in$

Carnot efficiency

The second law of thermodynamics puts a fundamental limit on the thermal efficiency of heat engines. Surprisingly, even an ideal, frictionless engine can't convert anywhere near 100% of its input heat into work. The limiting factors are the temperature at which the heat enters the engine, $T\_H,$, and the temperature of the environment into which the engine exhausts its waste heat,$T\_C,$, measured in the absolute Kelvin or Rankine scale. From Carnot's theorem, for any engine working between these two temperatures:

:$eta\_\{th\}\; le\; 1\; -\; frac\{T\_C\}\{T\_H\},$

This limiting value is called the Carnot cycle efficiency because it is the efficiency of an unattainable, ideal, lossless (reversible) engine cycle called the Carnot cycle. No heat engine, regardless of its construction, can exceed this efficiency.

Examples of $T\_H,$ are the temperature of hot steam entering the turbine of a steam power plant, or the temperature at which the fuel burns in an internal combustion engine. $T\_C,$ is usually the ambient temperature where the engine is located, or the temperature of a lake or river that waste heat is discharged into. For example, if an automobile engine burns gasoline at a temperature of $T\_H\; =\; 1500^circ\; F\; =\; 1089\; K,$ and the ambient temperature is $T\_C\; =\; 70^circ\; F\; =\; 294\; K,$, then its maximum possible efficiency is given by:

:$eta\_\{th\}\; le\; 1\; -\; frac\{294\; K\}\{1089\; K\}\; =\; 73.0\%,$

In practice, because the operating cycles of real engines are nowhere near as efficient as the Carnot cycle, coupled with other irreversibilities such as the combustion process itself and friction, real engines fall far short of the Carnot efficiency. Real automobile engines are only around 25% efficient. Combined cycle power stations efficiencies are higher, approaching 46%, but still fall at least 15 points short of the Carnot value. As Carnot's theorem only applies to heat engines, devices that convert the fuel's energy directly into work without burning it, such as fuel cells, can exceed the Carnot efficiency.

It can be seen that since $T\_C,$ is fixed by the environment, the only way for a designer to increase the theoretical efficiency of an engine is to increase $T\_H,$, the operating temperature of the engine. For this reason the operating temperatures of engines have increased greatly over the long term, and new materials such as ceramics to enable engines to stand higher temperatures are an active area of research.

Energy conversion

For an energy conversion device like a boiler or furnace, the thermal efficiency is

:$eta\_\{th\}\; equiv\; frac\{Q\_\{out\{Q\_\{in$.

So, for a boiler that produces 210 kW (or 700,000 BTU/h) output for each 300 kW (or 1,000,000 BTU/h) heat-equivalent input, its thermal efficiency is 210/300 = 0.70, or 70%. This means that the 30% of the energy is lost to the environment.

An electric resistance heater has a thermal efficiency of at or very near 100%, so, for example, 1500W of heat are produced for 1500W of electrical input. When comparing heating units, such as a 100% efficient electric resistance heater to an 80% efficient natural gas-fueled furnace, an economic analysis is needed to determine the most cost-effective choice.

Heat pumps and refrigerators

Heat pumps, refrigerators and air conditioners use work to move heat from a colder to a warmer place, so their function is the opposite of a heat engine. Their efficiency is measured by a coefficient of performance (COP). Heat pumps are measured by the efficiency with which they add heat to the hot reservoir, COP_heating; refrigerators and air conditioners by the efficiency with which they remove heat from the cold interior, COP_cooling:

:$\{COP\}\_\{heating\}\; equiv\; frac\{Q\_H\}\{W\},$

:$\{COP\}\_\{cooling\}\; equiv\; frac\{Q\_L\}\{W\},$

The reason for not using the term 'efficiency' is that the coefficient of performance can often be greater than 100%. Since these devices are moving heat, not creating it, the amount of heat they move can be greater than the input work. Therefore, heat pumps can be a more efficient way of heating than simply converting the input work into heat, as in an electric heater or furnace.

Since they are heat engines, these devices are also limited by Carnot's theorem. The limiting value of the Carnot 'efficiency' for these processes, with the equality theoretically achievable only with an ideal 'reversible' cycle, is:

:$\{COP\}\_\{heating\}\; le\; frac\{T\_H\}\{T\_H\; -\; T\_C\},$

:$\{COP\}\_\{cooling\}\; le\; frac\{T\_C\}\{T\_H\; -\; T\_C\},$

The same device used between the same temperatures is more efficient when considered as a heat pump than when considered as a refrigerator:

:$\{COP\}\_\{heating\}\; -\; \{COP\}\_\{cooling\}\; =\; 1,$

This is because when heating, the work used to run the device is converted to heat and adds to the desired effect, whereas if the desired effect is cooling the heat resulting from the input work is just an unwanted byproduct.

Energy efficiency

The 'thermal efficiency' is sometimes called the energy efficiency. In the United States, in everyday usage the SEER is the more common measure of energy efficiency for cooling devices, as well as for heat pumps when in their heating mode. For energy-conversion heating devices their peak steady-state thermal efficiency is often stated, e.g., 'this furnace is 90% efficient', but a more detailed measure of seasonal energy effectiveness is the Annual Fuel Utilization Efficiency (AFUE). [HVAC Systems and Equipment volume of the "ASHRAE Handbook", ASHRAE, Inc., Atlanta, GA, USA, 2004]

Example of energy efficiency

ee also

*Electrical efficiency
*Mechanical efficiency
*Figure of merit

References