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PowerPedia:Carnot Heat Engine

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A Carnot heat engine is a engine that operates on the reversible Carnot cycle. The basic model for this engine was developed by Nicolas Léonard Sadi Carnot in the 1824. The Carnot engine model was graphically expanded upon by Benoit Paul Émile Clapeyron in 1834 and mathematically elaborated upon by Rudolf Clausius in the 1850s and 60s from which the concept of entropy emerged.

Contents

Description

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Carnot Cycle (Image from http://hyperphysics.phy-astr.gsu.edu/)

Every thermodynamic system exists in a particular state. A thermodynamic cycle occurs when a system is taken through a series of different states, and finally returned to its initial state. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine. A heat engine acts by transferring energy from a warm region to a cool region of space and, in the process, converting some of that energy to mechanical work. The cycle may also be reversed. The system may be worked upon by an external force, and in the process, it can transfer thermal energy from a cooler system to a warmer one, thereby acting as a refrigerator rather than a heat engine.

In the cycle diagram,s from the original 1824 paper by Sadi Carnot entitled "On the Motive Power of Fire", we are told to “imagine two bodies A and B, kept each at a constant temperature, that of A being higher than that of B. These two bodies, to which we can give or from which we can remove the heat without causing their temperatures to vary, exercise the functions of two unlimited reservoirs of caloric. We shall call the first the furnace and the second the refrigerator.� Carnot then explains how we can obtain motive power, i.e. “work�, by carrying a certain quantity of heat from the body A to the body B.

The “working body� (system), a term introduced by Clausius in 1850, can be any fluid or vapor body through which heat Q can be introduced or transmitted through to produce work. In 1824, Sadi Carnot, in his famous paper Reflections on the Motive Power of Fire, had postulated that the fluid body could be any substance capable of expansion, such as vapor of water, vapor of alcohol, vapor of mercury, a permanent gas, or air, etc. Although, in these early years, engines came in a number of configurations, typically QH was supplied by a boiler, wherein water was boiled over a furnace; QC was typically a stream of cold flowing water in the form of a condenser located on a separate part of the engine. The output work W here is the movement of the piston as it is used to turn a crank-arm, which was then typically used to turn a pulley so to lift water out of flooded salt mines. Carnot defined work as “weight lifted through a height�.

For any cycle operating between temperatures TH and TC, none can exceed the efficiency of a Carnot cycle. Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. Thus, Equation 3 gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient.

In other words, maximum efficiency is achieved if and only if no new entropy is created in the cycle. Otherwise, since entropy is a state function, the required dumping of heat into the environment to dispose of excess entropy leads to a reduction in efficiency. So Equation 3 gives the efficiency of any reversible heat engine.

Carnot cycle

The Carnot cycle is a particular thermodynamic cycle, modeled on the Carnot heat engine, studied by Nicolas Léonard Sadi Carnot in the 1820s and expanded upon by Benoit Paul Émile Clapeyron in the 1830s and 40s. Every thermodynamic system exists in a particular state. A thermodynamic cycle occurs when a system is taken through a series of different states, and finally returned to its initial state. In the process of going through this cycle, the system may perform work on its surroundings, thereby acting as a heat engine.

A heat engine acts by transferring energy from a warm region to a cool region of space and, in the process, converting some of that energy to mechanical work. The cycle may also be reversed. The system may be worked upon by an external force, and in the process, it can transfer thermal energy from a cooler system to a warmer one, thereby acting as a refrigerator rather than a heat engine. The Carnot cycle is a special type of thermodynamic cycle. It is special because it is the most efficient cycle possible for converting a given amount of thermal energy into work or, conversely, for using a given amount of work for refrigeration purposes.

Carnot cycle steps

The Carnot cycle when acting as a heat engine consists of the following steps:

  1. Reversible isothermal expansion of the gas at the "hot" temperature, TH (isothermal heat addition). During this step (A to B on diagram) the expanding gas causes the piston to do work on the surroundings. The gas expansion is propelled by absorption of heat from the high temperature reservoir.
  2. Reversible adiabatic expansion of the gas. For this step (B to C on diagram) we assume the piston and cylinder are thermally insulated, so that no heat is gained or lost. The gas continues to expand, doing work on the surroundings. The gas expansion causes it to cool to the "cold" temperature, TC.
  3. Reversible isothermal compression of the gas at the "cold" temperature, TC. (isothermal heat rejection) (C to D on diagram) Now the surroundings do work on the gas, causing heat to flow out of the gas to the low temperature reservoir.
  4. Reversible adiabatic compression of the gas. (D to A on diagram) Once again we assume the piston and cylinder are thermally insulated. During this step, the surroundings do work on the gas, compressing it and causing the temperature to rise to TH. At this point the gas is in the same state as at the start of step 1.


The temperature-entropy diagram

The behavior of a Carnot engine or refrigerator is best understood by using a temperature-entropy (TE) diagram, in which the thermodynamic state is specified by a point on a graph with entropy (S) as the horizontal axis and temperature (T) as the vertical axis. For a simple system with a fixed number of particles, any point on the graph will represent a particular state of the system. A thermodynamic process will consist of a curve connecting an initial state (A) and a final state (B). The area under the curve will be:

\Delta Q=\int_A^B T\,dS \quad\quad(1)

which is the amount of thermal energy transferred in the process. If the process moves to greater entropy, the area under the curve will be the amount of heat absorbed by the system in that process. If the process moves towards lesser entropy, it will be the amount of heat removed. For any cyclic process, there will be an upper portion of the cycle and a lower portion. For a clockwise cycle, the area under the upper portion will be the thermal energy absorbed during the cycle, while the area under the lower portion will be the thermal energy removed during the cycle. The area inside the cycle will then be the difference between the two, but since the internal energy of the system must have returned to its initial value, this difference must be the amount of work done by the system over the cycle. Mathematically, for a reversible process we may write the amount of work done over a cyclic process as:

\Delta W = \oint PdV = \oint (TdS-dU) \quad\quad\quad\quad(2)

Since dU is an exact differential, its integral over any closed loop is zero and it follows that the area inside the loop on a T-S diagram is equal to the total work performed if the loop is traversed in a clockwise direction, and is equal to the total work done on the system as the loop is traversed in a counterclockwise direction.

The Carnot cycle

Evaluation of the above integral is particularly simple for the Carnot cycle. The amount of energy transferred as work is

\Delta W = \oint PdV = (T_H-T_C)(S_B-S_A)

The total amount of thermal energy transferred between the hot reservoir and the system will be

\Delta Q_H=T_H(S_B-S_A)\,

and the total amount of thermal energy transferred between the system and the cold reservoir will be

\Delta Q_C=T_C(S_B-S_A)\,.

The efficiency η is defined to be:

\eta=\frac{\Delta W}{\Delta Q_H}=1-\frac{T_C}{T_H} \quad\quad\quad\quad\quad\quad\quad\quad\quad(3)

where

ΔW is the work done by the system (energy exiting the system as work),
ΔQH is the heat put into the system (heat energy entering the system),
TC is the absolute temperature of the cold reservoir, and
TH is the temperature of the hot reservoir.

This efficiency makes sense for a heat engine, since it is the fraction of the heat energy extracted from the hot reservoir and converted to mechanical work. It also makes sense for a refrigeration cycle, since it is the ratio of energy input to the refrigerator divided by the amount of energy extracted from the hot reservoir.

Carnot's theorem

It can be seen from the above diagram, that for any cycle operating between temperatures TH and TC, none can exceed the efficiency of a Carnot cycle. Carnot's theorem is a formal statement of this fact: No engine operating between two heat reservoirs can be more efficient than a Carnot engine operating between the same reservoirs. Thus, Equation 3 gives the maximum efficiency possible for any engine using the corresponding temperatures. A corollary to Carnot's theorem states that: All reversible engines operating between the same heat reservoirs are equally efficient. In other words, maximum efficiency is achieved if and only if no new entropy is created in the cycle. Otherwise, since entropy is a state function, the required dumping of heat into the environment to dispose of excess entropy leads to a reduction in efficiency. So Equation 3 gives the efficiency of any reversible heat engine.

Efficiency of real heat engines

Carnot realised that in reality it is not possible to build a thermodynamically reversible engine, so real heat engines are less efficient than indicated by Equation 3. Nevertheless, Equation 3 is extremely useful for determining the maximum efficiency that could ever be expected for a given set of thermal reservoirs.

Although Carnot's cycle is an idealisation, the expression of Carnot efficiency is still useful. Consider the average temperatures,

<T_H> = \frac{1}{\Delta S} \int_{Q_{in}} TdS
<T_C> = \frac{1}{\Delta S} \int_{Q_{out}} TdS

at which heat is input and output, respectively. Replace TH and TC in Equation (3) by <TH> and <TC> respectively.

For the Carnot cycle, or its equivalent, <TH> is the highest temperature available and <TC> the lowest. For other less efficient cycles, <TH> will be lower than TH , and <TC> will be higher than TC. This can help illustrate, for example, why a reheater or a regenerator can improve thermal efficiency.

Efficiency

The efficiency of a heat engine relates how much useful power is output for a given amount of heat energy input.

From the laws of thermodynamics:

dW \ = \ dQ_c \ - \ (-dQ_h)
where
dW = − PdV is the work extracted from the engine. (It is negative since work is done by the engine.)
dQh = ThdSh is the heat energy taken from the high temperature system .(It is negative since heat is extracted from the source, hence ( − dQh) is positive.)
dQc = TcdSc is the heat energy delivered to the cold temperature system. (It is positive since heat is added to the sink.)

In other words, a heat engine absorbs heat energy from the high temperature heat source, converting part of it to useful work and delivering the rest to the cold temperature heat sink.

In general, the efficiency of a given heat transfer process (whether it be a refrigerator, a heat pump or an engine) is defined informally by the ratio of "what you get" to "what you put in."

In the case of an engine, one desires to extract work and puts in a heat transfer.

\eta = \frac{-dW}{-dQ_h} = \frac{-dQ_h - dQ_c}{-dQ_h} = 1 - \frac{dQ_c}{-dQ_h}

The theoretical maximum efficiency of any heat engine depends only on the temperatures it operates between. This efficiency is usually derived using an ideal imaginary heat engine such as the Carnot heat engine, although other engines using different cycles can also attain maximum efficiency. Mathematically, this is due to the fact that in thermodynamic reversibilr processes, the change in entropy of the cold reservoir is the negative of that of the hot reservoir (i.e., dSc = − dSh), keeping the overall change of entropy zero. Thus:

\eta_{max} = 1 - \frac{T_cdS_c}{-T_hdS_h} \equiv 1 - \frac{T_c}{T_h}

where Th is the absolute temperature of the hot source and Tc that of the cold sink, usually measured in kelvins. Note that dSc is positive while dSh is negative; in any reversible work-extracting process, entropy is overall not increased, but rather is moved from a hot (high-entropy) system to a cold (low-entropy one), decreasing the entropy of the heat source and increasing that of the heat sink.

The reasoning behind this being the maximal efficiency goes as follows. It is first assumed that if a more efficient heat engine than a Carnot engine is possible, then it could be driven in reverse as a heat pump. Mathematical analysis can be used to show that this assumed combination would result in a net decrease in entropy. Since, by the second law of thermodynamics, this is forbidden, the Carnot efficiency is a theoretical upper bound on the efficiency of any process.

Empirically, no engine has ever been shown to run at a greater efficiency than a Carnot cycle heat engine.

Other criteria of heat engine performance

One problem with the ideal Carnot efficiency as a criterion of heat engine performance is the fact that by its nature, any maximally-efficient Carnot cycle must operate at an infinitesimal temperature gradient. This is due to the fact that any transfer of heat between two bodies at differing temperatures is irreversible, and therefore the Carnot efficiency expression only applies in the infinitesimal limit. The major problem with that is that the object of most heat engines is to output some sort of power, and infinitesimal power is usually not what is being sought.

A much more accurate measure of heat engine efficiency is given by the endoreversible process, which is identical to the Carnot cycle except in that the two processes of heat transfer are not treated as reversible. As derived in Callen (1985), the efficiency for such a process is given by:

\eta = 1 - \sqrt{\frac{T_c}{T_h}}

The endoreversible efficiency much more closely models the observed data.

References and resources

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General
Patents
  • U.S. Patent 5027602 (G.patent; PDF) Engine Carnot; Glen Edwards Heat engine, refrigeration and heat pump cycles approximating the Carnot cycle and apparatus therefor
  • U.S. Patent 5819554 (G.patent; PDF) Engine Carnot; Glen Rotating vane compressor with energy recovery section, operating on a cycle approximating the ideal reversed Carnot cycle
  • U.S. Patent 4691523 (G.patent; PDF) Engine Carnot; Rosado Thermodynamic process for a practical approach to the Carnot cycle
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