Carnot's theorem (thermodynamics)

Carnot 's theorem is a coefficient of efficiency (efficiency) theorem for heat engines . According to this theorem, the efficiency of the Carnot cycle does not depend on the nature of the working fluid and the design of the heat engine and is a function of the temperature of the heater and the refrigerator ^[1] .

Content

History

In 1824, Sadi Carnot came to the conclusion: “The driving force of heat does not depend on the agents taken for its development; its quantity is exclusively determined by the temperatures of the bodies, between which, ultimately, the transfer of the calorific is carried out ”

Carnot’s reasoning logic was as follows: “... one can reasonably compare the driving force of heat with the power of falling water: both have a maximum that cannot be surpassed, whatever the machine for using the action of water would be in one case, and the substance used in the other to develop heat

The driving force of the falling water depends on the height of the fall and the amount of water; the driving force of heat also depends on the amount of caloric consumed and depends on what can be called and what we really will call the height of its fall, that is, on the temperature difference of the bodies between which the calorific exchange takes place. When water falls, the driving force is strictly proportional to the difference in levels in the upper and lower reservoir. When the calorific value falls, the driving force undoubtedly increases with the temperature difference between the hot and cold bodies ....

Wording

Some modern authors (K. V. Glagolev, A. N. Morozov from MSTU named after N. E. Bauman) already speak of two Carnot theorems, cited: “The above arguments allow us to proceed to the formulation of the first and second Carnot theorems . They can be formulated in the form of the following two statements:

1. The efficiency of any reversible heat engine operating on the Carnot cycle does not depend on the nature of the working fluid and the device of the machine, but is a function of only the temperature of the heater and refrigerator: $\eta =1-F(T_{H},T_{X}).$ ${\ displaystyle \ eta = 1-F (T_ {H}, T_ {X}).}$

2. The efficiency of any heat engine operating in an irreversible cycle is less than the efficiency of a machine with a reversible Carnot cycle, provided that the temperatures of their heaters and refrigerators are equal: $\eta _{n}<\!\eta _{o}.$ ${\ displaystyle \ eta _ {n} <\! \ eta _ {o}.}$

Other authors (for example, B. M. Yavorsky and Yu. A. Seleznev) point to three aspects of one Carnot theorem, cited (see pages 151–152.):

3 °. Thermal efficiency the reversible Carnot cycle is independent of the nature of the working fluid and is determined only by the temperature of the heater $T_{H}$ ${\ displaystyle T_ {H}}$ and fridge $T_{X}$ ${\ displaystyle T_ {X}}$ :

\eta _{k}={\frac {T_{H}-T_{X}}{T_{H}}}=1-{\frac {T_{X}}{T_{H}}}.

{\ displaystyle \ eta _ {k} = {\ frac {T_ {H} -T_ {X}} {T_ {H}}} = 1 - {\ frac {T_ {X}} {T_ {H}}} .}

$\eta _{k}<1$ ${\ displaystyle \ eta _ {k} <1}$ , because it is almost impossible to fulfill the condition $T_{H}\rightarrow \infty$ ${\ displaystyle T_ {H} \ rightarrow \ infty}$ and it is theoretically impossible to implement a refrigerator in which: $T_{X}=0$ ${\ displaystyle T_ {X} = 0}$ .

4 °. Thermal efficiency $\eta _{o}$ ${\ displaystyle \ eta _ {o}}$ an arbitrary reversible cycle cannot exceed the thermal efficiency reversible Carnot cycle carried out between the same temperatures $T_{H}$ ${\ displaystyle T_ {H}}$ and $T_{X}$ ${\ displaystyle T_ {X}}$ heater and refrigerator:

\eta _{o}<{\frac {T_{H}-T_{X}}{T_{H}}}.

{\ displaystyle \ eta _ {o} <{\ frac {T_ {H} -T_ {X}} {T_ {H}}}.}

5 °. Thermal efficiency $\eta _{n}$ ${\ displaystyle \ eta _ {n}}$ arbitrary irreversible cycle is always less than thermal efficiency reversible Carnot cycle conducted between temperatures $T_{H}$ ${\ displaystyle T_ {H}}$ and $T_{X}$ ${\ displaystyle T_ {X}}$ :

\eta _{n}<{\frac {T_{H}-T_{X}}{T_{H}}}.

{\ displaystyle \ eta _ {n} <{\ frac {T_ {H} -T_ {X}} {T_ {H}}}.}

Points 3 ° - 5 ° constitute the content of Carnot's theorem.

Proofs of Carnot's theorem

There are several different proofs of this theorem.

Proof of Sadi Carnot

... In various positions, the piston experiences pressures more or less significant from the air in the cylinder; the elastic force of air varies both from a change in volume and from a change in temperature, but it should be noted that with equal volumes, that is, for similar positions of the piston, the temperature will be higher during rarefaction than under compression. Therefore, in the first case, the elastic force of the air will be greater, and hence the driving force produced by the movement from the expansion will be greater than the force needed for compression. Thus, there will be an excess of driving force, an excess that can be used for something. Air will serve us as a heat engine; we used it even in the most advantageous way, since there was not a single useless restoration of the caloric balance.

Modern Proof for Perfect Gas

One of the evidence is presented in the book of D. ter Haar and G. Vergeland, “Elementary Thermodynamics” (see Fig.).

One of the possible options for the theoretical Carnot cycle

DE process:

Since the gas is perfect, $(dU/dV)_{T}=0$ ${\ displaystyle (dU / dV) _ {T} = 0}$ and internal energy remains constant. All heat received from the tank at temperature $T_{H}$ ${\ displaystyle T_ {H}}$ turns into external work:

Q_{D-E}=\int \limits _{ik}^{cd}p{dV}=RT_{H}\ln {\frac {V_{cd}}{V_{ik}}}.\qquad

{\ displaystyle Q_ {DE} = \ int \ limits _ {ik} ^ {cd} p {dV} = RT_ {H} \ ln {\ frac {V_ {cd}} {V_ {ik}}}. \ qquad }

[one]

Process B-C:

Similarly, work performed during compression is converted to heat, which is transferred to a cold tank:

Q_{B-C}=\int \limits _{gh}^{ef}p{dV}=RT_{X}\ln {\frac {V_{ef}}{V_{gh}}}.\qquad

{\ displaystyle Q_ {BC} = \ int \ limits _ {gh} ^ {ef} p {dV} = RT_ {X} \ ln {\ frac {V_ {ef}} {V_ {gh}}}. \ qquad }

[2]

EB and CD processes:

Because the gas is perfect and $U$ ${\ displaystyle U}$ depends only on temperature $T$ ${\ displaystyle T}$ from the equation $Q=U_{2}-U_{1}+A$ ${\ displaystyle Q = U_ {2} -U_ {1} + A}$ it follows that the work done in one of these two adiabatic processes completely compensates for the work done in the other process. Indeed, using the adiabatic condition $C_{V}dT+pdV=0$ ${\ displaystyle C_ {V} dT + pdV = 0}$ we get:

C_{V}(T_{H}-T_{X})=\int \limits _{cd}^{gh}pdV=-\int \limits _{ef}^{ik}pdV.

{\ displaystyle C_ {V} (T_ {H} -T_ {X}) = \ int \ limits _ {cd} ^ {gh} pdV = - \ int \ limits _ {ef} ^ {ik} pdV.}

To find a connection between $V_{ik}$ ${\ displaystyle V_ {ik}}$ , $V_{cd}$ ${\ displaystyle V_ {cd}}$ , $V_{gh}$ ${\ displaystyle V_ {gh}}$ and $V_{ef}$ ${\ displaystyle V_ {ef}}$ , note that, according to the Poisson equation $TV^{R/C_{V}}=const$ ${\ displaystyle TV ^ {R / C_ {V}} = const}$ , in adiabatic processes:

(E → B): $T_{H}V_{cd}^{x-1}=T_{X}V_{gh}^{x-1},$ ${\ displaystyle T_ {H} V_ {cd} ^ {x-1} = T_ {X} V_ {gh} ^ {x-1},}$

(C → D): $T_{X}V_{ef}^{x-1}=T_{H}V_{ik}^{x-1},$ ${\ displaystyle T_ {X} V_ {ef} ^ {x-1} = T_ {H} V_ {ik} ^ {x-1},}$

and therefore

{\frac {V_{cd}}{V_{ik}}}={\frac {V_{gh}}{V_{ef}}}.

{\ displaystyle {\ frac {V_ {cd}} {V_ {ik}}} = {\ frac {V_ {gh}} {V_ {ef}}}.}

Substituting this relation into equations [1] and [2], we obtain:

{\frac {Q_{B-C}}{Q_{D-E}}}={\frac {T_{H}}{T_{X}}}.

{\ displaystyle {\ frac {Q_ {BC}} {Q_ {DE}}} = {\ frac {T_ {H}} {T_ {X}}}.}

At the same time, we arrive at the result ... that the efficiency of the optimal cycle is

\eta _{max}={\frac {T_{H}-T_{X}}{T_{H}}}.

{\ displaystyle \ eta _ {max} = {\ frac {T_ {H} -T_ {X}} {T_ {H}}}.}

Literature

S. Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. - Paris, Gautier-Villars, Imprimeur-Libraire, 1878.
Carnot Nicolas Leonard Sadi, Translation by V.R. Bursiana and Yu.A. Krutkova. Reflections on the driving force of fire and on machines capable of developing this force.
D. Ter Haar, G. Vergeland. Elementary thermodynamics. Translation from English by I. B. Vikhansky. Edited by N.M. Placids. (D. TER HAAR, Oxford University, H. WERGELAND, Norwegian Institute of Technology, Trondheim. ELEMENTS OF THERMODYNAMICS. Addison-Wesley Publishing Company). - M.: Mir Publishing House, 1968.
Yavorsky B.M., Detlaf A.A. Handbook of Physics. For students and engineers of universities. Seventh edition, revised. - M .: Publishing house "Science", 1979.
Glagolev K.V., Morozov A.N. Physical thermodynamics. - M .: Publishing house of MSTU named after N.E.Bauman, 2004.
Yavorsky B.M., Seleznev Yu.A. Physics. Reference Guide: For applicants to universities. - 5th ed., Revised .. - M.: Fizmatlit, 2004.

Notes

↑ Carnot Theorem // Physical Encyclopedia. In 5 volumes. - M .: Soviet Encyclopedia. Editor-in-chief A.M. Prokhorov. 1988.

Links

http://nature.web.ru/db/msg.html?mid=1165074&uri=page1.html

[1] Carnot Theorem // Physical Encyclopedia. In 5 volumes. - M .: Soviet Encyclopedia. Editor-in-chief A.M. Prokhorov. 1988.