# Thermal efficiency: Wikis

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# Encyclopedia

In thermodynamics, the thermal efficiency ($\eta_{th} \,$) is a dimensionless performance measure of a device that uses thermal energy, such as an internal combustion engine, a boiler, a furnace, or a refrigerator 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[1]

$\eta_{th} \equiv \frac{\text{What you get}}{\text{What you pay for}}.$

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

$0 \le \eta_{th} \le 1$

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 engines' efficiencies are typically much less than 100%. For example, a typical gasoline automobile engine operates at around 25% 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%.[2]

There are two types of thermal efficiency- 1.Indicated thermal efficiency 2.Brake thermal efficiency

## Heat engines

Heat engines transform thermal energy, or heat, Qin into mechanical energy, or work, Wout. They cannot do this task perfectly, so some of the input heat energy is not converted into work, but is dissipated as waste heat Qout into the environment

$Q_{in} = W_{out} + Q_{out} \,$

The thermal efficiency of a heat engine is the percentage of heat energy that is transformed into work. Thermal efficiency is defined as

$\eta_{th} \equiv \frac{W_{out}}{Q_{in}} = 1 - \frac{Q_{out}}{Q_{in}}$

The efficiency of even the best heat engines is low; usually below 50% and often far below. So the energy lost to the environment by heat engines is a major waste of energy resources, although modern cogeneration, combined cycle and energy recycling schemes are beginning to use this heat for other purposes. Since a large fraction of the fuels produced worldwide go to powering heat engines, perhaps up to half of the useful energy produced worldwide is wasted in engine inefficiency. This inefficiency can be attributed to three causes. There is an overall theoretical limit to the efficiency of any heat engine due to temperature, called the Carnot efficiency. Second, specific types of engines have lower limits on their efficiency due to the inherent irreversibility of the engine cycle they use. Thirdly, the nonideal behavior of real engines, such as mechanical friction and losses in the combustion process causes further efficiency losses.

### HCV and Gross CV or LCV, and Net CV

To complicate matters, there are at least two different definitions of Calorific Value in wide use, and wich one is being used significantly affects any quoted efficiency. Not stating whether an efficiency is HCV or LCv renders such numbers very misleading.[3]

### Carnot efficiency

The second law of thermodynamics puts a fundamental limit on the thermal efficiency of all 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 an absolute scale, such as the Kelvin or Rankine scale. From Carnot's theorem, for any engine working between these two temperatures:[4]

$\eta_{th} \le 1 - \frac{T_C}{T_H}\,$

This limiting value is called the Carnot cycle efficiency because it is the efficiency of an unattainable, ideal, reversible engine cycle called the Carnot cycle. No device converting heat into mechanical energy, 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:

$\eta_{th} \le 1 - \frac{294 K}{1089 K} = 73.0%\,$

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 Carnot efficiency of an engine is to increase $T_H\,$, the operating temperature of the engine. This is a general principle that applies to all heat engines. 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.

### Engine cycle efficiency

The Carnot cycle is reversible and thus represents the upper limit on efficiency of an engine cycle. Practical engine cycles are irreversible and thus have inherently lower efficiency than the Carnot efficiency when operated between the same temperatures $T_H\,$ and $T_C\,$. One of the factors determining efficiency is how heat is added to the working fluid in the cycle, and how it is removed. The Carnot cycle achieves maximum efficiency because all the heat is added to the working fluid at the maximum temperature $T_H\,$, and removed at the minimum temperature $T_C\,$. In contrast, in an internal combustion engine, the temperature of the fuel-air mixture in the cylinder is nowhere near its peak temperature as the fuel starts to burn, and only reaches the peak temperature as all the fuel is consumed, so the average temperature at which heat is added is lower, reducing efficiency.

$\eta_{th} = 1 - r^{1-\gamma}\,$
The higher the compression ratio, the higher the temperature in the cylinder as the fuel burns and so the higher the efficiency. However the maximum compression ratio usable is limited by the need to prevent preignition (knocking), where the fuel ignites by compression before the spark plug fires. The specific heat ratio of the air-fuel mixture γ varies somewhat with the fuel, but is generally close to the air value of 1.4. This standard value is usually used in all the engine cycle equations below, and when this approximation is used the cycle is called an air-standard cycle.
• Trucks: Diesel cycle In the Diesel cycle used in diesel truck and train engines, the fuel is ignited by compression in the cylinder. The efficiency of the Diesel cycle is dependent on r and γ like the Otto cycle, and also by the cutoff ratio, rc, which is the ratio of the cylinder volume at the beginning and end of the combustion process:[4]
$\eta_{th} = \frac{1 - r^{1-\gamma}(r_c^\gamma - 1)}{\gamma(r_c - 1)} \,$
The Diesel cycle is less efficient than the Otto cycle when using the same compression ratio. However, practical Diesel engines are 30% - 35% more efficient than gasoline engines.[5] This is because, since the fuel is not introduced to the combustion chamber until it required to ignite, the compression ratio is not limited by the need to avoid knocking, so higher ratios are used than in spark ignition engines.
• Power plants: Rankine cycle The Rankine cycle is the cycle used in steam turbine power plants. The overwhelming majority of the world's electric power is produced with this cycle. Since the cycle's working fluid, water, changes from liquid to vapor and back during the cycle, their efficiencies depend on the thermodynamic properties of water. The thermal efficiency of modern steam turbine plants with reheat cycles can reach 47%, and in combined cycle plants it can approach 60%.[4]
• Gas turbines: Brayton cycle The Brayton cycle is the cycle used in gas turbines and jet engines. It consists of a compressor turbine that increases pressure of the incoming air, then fuel is continuously added to the flow and burned, and the hot exhaust gasses are expanded in a turbine. The efficiency depends largely on the ratio of the pressure inside the combustion chamber p2 to the pressure outside p1[4]
$\eta_{th} = 1 - \bigg(\frac{p_2}{p_1}\bigg)^\frac{1-\gamma}{\gamma} \,$

### Other inefficiencies

The above efficiency formulas are based on simple idealized mathematical models of engines, with no friction and working fluids that obey simple thermodynamic rules called the ideal gas law. Real engines have many departures from ideal behavior that waste energy, reducing actual efficiencies far below the theoretical values given above. Examples are:

• friction of moving parts
• inefficient combustion
• heat loss from the combustion chamber
• departure of the working fluid from the thermodynamic properties of an ideal gas
• aerodynamic drag of air moving throgh the engine
• energy used by ancillary equipment like oil and water pumps
• inefficient compressors and turbines
• imperfect valve timing

Another source of inefficiency is that engines must be optimized for other goals besides efficiency, such as low pollution. The requirements for vehicle engines are particularly stringent: they must be designed for low emissions, adequate acceleration, fast starting, light weight, low noise, etc. These require compromises in design (such as altered valve timing) that reduce efficiency. The average automobile engine is only about 35% efficient, and must also be kept idling at stoplights, wasting an additional 17% of the energy, resulting in an overall efficiency of 18%.[5] Large stationary electric generating plants have fewer of these competing requirements as well as more efficient Rankine cycles, so they are significantly more efficient than vehicle engines, around 50% Therefore, replacing internal combustion vehicles with electric vehicles, which run on a battery that is charged with electricity generated by burning fuel in a power plant, can greatly increase the thermal efficiency of energy use in transportation, thus decreasing the demand for fossil fuels.

## 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. The work energy (Win) that is applied to them is converted into heat, and the sum of this energy and the heat energy that is moved from the cold reservoir (QC) is equal to the total heat energy added to the hot reservoir (QH)

$Q_H = Q_C + W_{in} \,$

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, COPheating; refrigerators and air conditioners by the efficiency with which they remove heat from the cold interior, COPcooling:

$\mathrm{COP}_{\mathrm{heating}} \equiv \frac{Q_H}{W_{in}}\,$
$\mathrm{COP}_{\mathrm{cooling}} \equiv \frac{Q_C}{W_{in}}\,$

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:

$\mathrm{COP}_{\mathrm{heating}} \le \frac{T_H}{T_H - T_C}\,$
$\mathrm{COP}_{\mathrm{cooling}} \le \frac{T_C}{T_H - T_C}\,$

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

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

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).[6]

## References

1. ^ Fundamentals of Engineering Thermodynamics, by Howell and Buckius, McGraw-Hill, New York, 1987
2. ^ GE Power’s H Series Turbine
3. ^ http://www.claverton-energy.com/the-difference-between-lcv-and-hcv-or-lower-and-higher-heating-value-or-net-and-gross-is-clearly-understood-by-all-energy-engineers-there-is-no-right-or-wrong-definition.html
4. ^ a b c d e Holman, Jack P. (1980). Thermodynamics. New York: McGraw-Hill. pp. 217. ISBN 0-07-029625-1.
5. ^ a b "Where does the energy go?". Advanced technologies and energy efficiency, Fuel Economy Guide. US Dept. of Energy. 2009. Retrieved 2009-12-02.
6. ^ HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, USA, 2004

There are two types of thermal efficiency- 1.Indicated thermal efficiency 2.Brake thermal efficiency

# Wikibooks

Up to date as of January 23, 2010
(Redirected to Jet Propulsion/Thermal efficiency article)

### From Wikibooks, the open-content textbooks collection

For heat engines the thermal efficiency is the mechanical work output for each unit of heat input. For the Carnot cycle it is

$\eta_c = 1 - \frac {T_0}{T_{max}}$

where

T0 = ambient absolute temperature

Tmax = maximum cycle absolute temperature

Actual engines using other cycles will have lower efficiencies than the Carnot cycle since they have irreversible processes such as heat flow across finite temperature differences and non-adiabatic compression and expansions, as well as friction.

Heat engines use as high a temperature as possible to maximize the efficiency and are limited by temperature capabilities of real materials used to make the engines.

# Simple English

The thermal efficiency ($\eta_\left\{th\right\} \,$) is a dimensionless performance measure of a thermal device such as an internal combustion engine, a boiler, or a furnace, for example.

The input, $Q_\left\{in\right\} \,$, to the device is heat, or the heat-content of a fuel that is consumed. The desired output is mechanical work, $W_\left\{out\right\} \,$, or heat, $Q_\left\{out\right\} \,$, or possibly both. Because the input heat normally has a real financial cost, a memorable, generic definition of thermal efficiency is[1]

$\eta_\left\{th\right\} \equiv \frac\left\{\text\left\{What you get\right\}\right\}\left\{\text\left\{What you pay for\right\}\right\}.$

From the first and second law of thermodynamics, the output can not exceed what is input, so

$0 \le \eta_\left\{th\right\} \le 1.0.$

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 36%. In a combined cycle plant thermal efficiencies are approaching 60%.

## Heat engines

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

$\eta_\left\{th\right\} \equiv \frac\left\{W_\left\{out\right\}\right\}\left\{Q_\left\{in\right\}\right\}$,

or via the first law of thermodynamics to substitute waste heat rejection for the work produced,

$\eta_\left\{th\right\} = 1 - \frac\left\{Q_\left\{out\right\}\right\}\left\{Q_\left\{in\right\}\right\}$.

For example, when 1000 joules of thermal energy is transformed into 300 joules of mechanical energy (with the remaining 700 joules dissipated as waste heat), the thermal efficiency is 30%.

## Energy conversion

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

$\eta_\left\{th\right\} \equiv \frac\left\{Q_\left\{out\right\}\right\}\left\{Q_\left\{in\right\}\right\}$.

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, for example, move heat, rather than convert it, so other measures are needed to describe their thermal performance. The common measures are the coefficient-of-performance (COP), energy-efficiency ratio (EER), and seasonal-energy-efficiency ratio (SEER).

The Efficiency of a Heat pump (HP) and Refrigerators (R)*:
$E_\left\{HP\right\}=\frac\left\{|Q_H|\right\}\left\{|W|\right\}$

$E_\left\{R\right\}=\frac\left\{|Q_L|\right\}\left\{|W|\right\}$

$\displaystyle E_\left\{HP\right\} - E_\left\{R\right\} = 1$

If temperatures at both ends of the Heat Pump or Refrigerator are constant and their processes reversible:

$E_\left\{HP\right\}=\frac\left\{T_H\right\}\left\{T_H - T_L\right\}$

$E_\left\{R\right\}=\frac\left\{T_L\right\}\left\{T_H - T_L\right\}$

*H=high (temperature/heat source), L=low (temperature/heat source)


## 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).[2]

## References

1. Fundamentals of Engineering Thermodynamics, by Howell and Buckius, McGraw-Hill, New York, 1987
2. HVAC Systems and Equipment volume of the ASHRAE Handbook, ASHRAE, Inc., Atlanta, GA, USA, 2004