Thermal Efficiency Formula | Calculation

As a result of this statement, we define the thermal efficiency, η_th, of any heat engine as the ratio of the work it does, W, to the heat input at the high temperature, Q_H. The thermal efficiency formula is then:

The thermal efficiency, η_th, represents the fraction of heat, Q_H, converted to work.

The air-standard Otto cycle thermal efficiency is a function of compression ratio and κ = c_p/c_v.

Thermal efficiency for Diesel cycle:

The thermal efficiency of the Brayton cycle in terms of the compressor pressure ratio (PR = p₂/p₁), which is the parameter commonly used:

The thermal efficiency of the simple Rankine cycle and in terms of specific enthalpies would be:

The thermal efficiency, η_th, represents the fraction of heat, Q_H, converted to work. It is a dimensionless performance measure of a heat engine that uses thermal energy, such as a steam turbine, an internal combustion engine, or a refrigerator. For refrigeration or heat pumps, thermal efficiency indicates the extent to which the energy added by work is converted to net heat output. Since it is a dimensionless number, we must always express W, Q_H, and Q_C in the same units.

Since energy is conserved according to the first law of thermodynamics and energy cannot be converted to work completely, the heat input, Q_H, must equal the work done, W, plus the heat that must be dissipated as waste heat Q_C into the environment. Therefore we can rewrite the formula for thermal efficiency as:

To give the efficiency as a percent, we multiply the previous formula by 100. Note that η_th could be 100% only if the waste heat Q_Cis zero.

In general, the efficiency of even the best heat engines is quite low. In short, it is very difficult to convert thermal energy to mechanical energy. The thermal efficiencies are usually below 50% and often far below. Be careful when comparing it with efficiencies of wind or hydropower (wind turbines are not heat engines). There is no energy conversion between thermal and mechanical energy.[/lgc_column]

Carnot's Efficiency

In 1824, a French engineer and physicist, Nicolas Léonard Sadi Carnot advanced the study of the second law by forming a principle (also called Carnot’s rule) that specifies limits on the maximum efficiency any heat engine can obtain. In short, this principle states that the efficiency of a thermodynamic cycle depends solely on the difference between the hot and cold temperature reservoirs.

Carnot’s principle states:

No engine can be more efficient than a reversible engine (Carnot heat engine) operating between the same high-temperature and low-temperature reservoirs.
The efficiencies of all reversible engines (Carnot heat engines) operating between the same constant temperature reservoirs are the same, regardless of the working substance employed or the operation details.

Carnot Efficiency

The formula for this maximum efficiency is:

where:

is the efficiency of the Carnot cycle, i.e., it is the ratio = W/Q_H of the work done by the engine to the heat energy entering the system from the hot reservoir.
T_C is the absolute temperature (Kelvins) of the cold reservoir,
T_H is the absolute temperature (Kelvins) of the hot reservoir.

The formula for Brayton Cycle

first law - example - brayton cycle — The ideal Brayton cycle consists of four thermodynamic processes. Two isentropic processes and two isobaric processes.

The thermal efficiency of simple Brayton cycle, for ideal gas and in terms of specific enthalpies can be expressed in terms of the temperatures:

Thermal Efficiency of Rankine Cycle

Example of Heat Engine — The Rankine cycle closely describes the processes in steam-operated heat engines commonly found in most thermal power plants.

The thermal efficiency of the simple Rankine cycle and in terms of specific enthalpies are:

It is very simple equation and for determination of the thermal efficiency you can use data from steam tables.

References:

Nuclear and Reactor Physics:

J. R. Lamarsh, Introduction to Nuclear Reactor Theory, 2nd ed., Addison-Wesley, Reading, MA (1983).
J. R. Lamarsh, A. J. Baratta, Introduction to Nuclear Engineering, 3d ed., Prentice-Hall, 2001, ISBN: 0-201-82498-1.
W. M. Stacey, Nuclear Reactor Physics, John Wiley & Sons, 2001, ISBN: 0- 471-39127-1.
Glasstone, Sesonske. Nuclear Reactor Engineering: Reactor Systems Engineering, Springer; 4th edition, 1994, ISBN: 978-0412985317
W.S.C. Williams. Nuclear and Particle Physics. Clarendon Press; 1 edition, 1991, ISBN: 978-0198520467
Kenneth S. Krane. Introductory Nuclear Physics, 3rd Edition, Wiley, 1987, ISBN: 978-0471805533
G.R.Keepin. Physics of Nuclear Kinetics. Addison-Wesley Pub. Co; 1st edition, 1965
Robert Reed Burn, Introduction to Nuclear Reactor Operation, 1988.
U.S. Department of Energy, Nuclear Physics and Reactor Theory. DOE Fundamentals Handbook, Volume 1 and 2. January 1993.

Advanced Reactor Physics:

K. O. Ott, W. A. Bezella, Introductory Nuclear Reactor Statics, American Nuclear Society, Revised edition (1989), 1989, ISBN: 0-894-48033-2.
K. O. Ott, R. J. Neuhold, Introductory Nuclear Reactor Dynamics, American Nuclear Society, 1985, ISBN: 0-894-48029-4.
D. L. Hetrick, Dynamics of Nuclear Reactors, American Nuclear Society, 1993, ISBN: 0-894-48453-2.
E. E. Lewis, W. F. Miller, Computational Methods of Neutron Transport, American Nuclear Society, 1993, ISBN: 0-894-48452-4.

See above:

Thermal Efficiency