Page 1

Some observations on

Carnot cycle as the genesis

of the heat pipe and

thermosyphon

K. C. CHENG, Department of Mechanical Engineering, University of

Alberta, Edmonton, Alberta, Canada T6G 2G8

〈kwocheng@hotmail.com〉

Received 29th June 1998

Revised 12th November 1998

The purpose of this article is to show that the limiting case of the Carnot vapor cycle with a

temperature difference of only a few degrees between hot and cold reservoirs, can be

interpreted as the idealized heat pipe (Carnot heat pipe) for transporting heat, utilizing the

boiling condensation processes. The observation of the basic relationship (definition of

absolute temperature) Q Q

T T

1212

=

for Carnot cycle reveals that the Carnot heat engine

becomes a heat transfer device when the temperature difference is very small. The

Clapeyron–Clausius equation for the infinitesimal (differential) Carnot vapour cycle

becomes the genesis for the operating principle of the heat pipe and two-phase

thermosyphon. The Carnot heat pipe is independent of the system and working fluid. Within

the context of the idealized Carnot heat pipe, it is shown that thermodynamics, fluid

mechanics and heat transfer (thermal sciences) are closely coupled in heat pipe operation.

The present observation is based on various expositions of classical thermodynamics

available in the historical literature. It is of interest to note that the Carnot vapor cycle

provides the basic principle for heat pipe and two-phase thermosyphon in addition to the

classical cases of heat engine, refrigeration machine and heat pump.

1. INTRODUCTION

‘A theory is the more impressive the greater the simplicity of its premises, the more

different kinds of things it relates, and the more extended its area of applicability.

Hence the deep impression that classical thermodynamics made upon me. It is the

only physical theory of universal content concerning which I am convinced that,

within the framework of the applicability of its basic concepts, it will never be

overthrown’.

Albert Einstein (1946)

The first atmospheric steam engine that performed useful work was invented by Thomas

Newcomen in 1712. James Watt patented his design for a steam engine with separate

International Journal of Mechanical Engineering Education Vol 28 No 1

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70

K. C. Cheng

condenser in 1769 and attained his first commercial success in 1775. Steam engines utilized

a change in phase of the working fluid. A Carnot steam engine used a liquid-vapor cycle

(1824). Carnot was the first to study the theory of heat engines. His investigation on the

efficiency of steam engines led to the beginning of thermodynamics. Carnot was interested

in the amount of work that could be obtained from a heat engine. A steam engine operating

on a Carnot cycle is impractical and the practical steam cycle is referred to as the Rankine

cycle (1859).

The Carnot heat engine efficiency ηc? 0 30

ever, it is useful as a heat transfer device, such as heat pipe or thermosyphon. Carnot appears

to be the first to have shown the limitations associated with converting heat energy into

work.

The importance of Carnot’s concepts were not fully realized until later in the nineteenth

century. William Thomson (Lord Kelvin) (1824–1907) and Rudolph Clausius (1822–1888)

recognized the importance of Carnot’s work and are credited with completing the theory of

heat engines and the establishment of the fundamental laws of classical thermodynamics.

The Carnot or maximum efficiency for a heat engine is based on the concept of reversibility.

The development of the science of thermodynamics in the nineteenth century has been

studied by many historians of science. It is interesting to note that Carnot’s conclusions were

published before the equivalence of heat and work (the First Law of Thermodynamics) was

established. This undoubtedly accounts for the limited impact of Carnot’s work during the

nineteenth century in which the steam engine had its greatest impact on civilization. In 1824,

Sadi Carnot at the age of 28 years, published a monograph entitled ‘Reflections on the

Motive Power of Fire’, dealing with the efficiency of heat engines. The principle expounded

is now recognized as the ‘Second Law of Thermodynamics’.

The objective of this paper is to point out the connection between the Carnot (steam)

cycle and the operating principle or thermodynamic aspects of the heat pipe and two-phase

thermosyphon. A heat pipe is basically a two-phase natural convection device with a self-

contained evaporator and condenser, designed as a heat transfer device and utilizing boiling

and condensation phenomena. The heat pipe is closely related to the two-phase

thermosyphon. In the thermosyphon the condensate returns to the evaporator under the

action of gravitational forces rather than under the action of capillary forces as in the heat

pipe. For this reason, in the thermosyphon the evaporator must always be below the con-

denser. In the heat pipe it is possible to transfer heat downward.

Provided that the pressure gradient in the vapor is kept small, the axial temperature

gradient along the heat pipe can be small, resulting in a device of very high equivalent

thermal conductivity. The performance of the Carnot cycle is independent of the system used

and the working substance. Heat transport depends solely on isothermal expansion or com-

pression of the working fluid and the resulting temperature change caused solely by adiabatic

expansion or compression of the fluid.

Heat pipes were invented independently by R. S. Gaugler in 1942 and by G. M. Grover in

1963. The invention of the heat pipe was completely independent of Carnot’s theory of heat

engine. Heat pipes were developed because of technological needs. The situation is some-

what similar to the invention of the steam engine.

Various thermodynamic aspects of heat pipe operation were discussed by the following

investigators; Tien [1, 2], Casarosa and Latrofa [3, 4], Vasiliev and Konev [5], Pryputniewicz

and Haapala [6], Kobayashi [7], Richter and Gottschlich [8]. Within the scope of this paper,

no attempt is made to review recent works on thermodynamic aspects of the heat pipe or the

thermosyphon. The expositions of the operating principles and theory of heat pipes can be

found in books by Dunn and Reay [9], Ivanovskii et al. [10, Faghri [11], Peterson [12] and

., for example, may not be practical. How-

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Some observations on Carnot cycle as the genesis of the heat pipe

71

others. Fig. 1 shows the schematic diagrams of the conventional capillary-driven heat pipe,

closed two-phase thermosyphon, closed loop two-phase thermosyphon and Carnot heat en-

gine with a small temperature difference between the heat source and sink. One may note

that in heat pipe operation, thermodynamics, fluid mechanics and heat transfer are closely

coupled. The portraits of Carnot, Clapeyron, Clausius and Kelvin are shown in Fig. 2.

2. THE CARNOT VAPOR CYCLE AS THE GENESIS OF THE HEAT PIPE AND

THERMOSYPHON

Every engineer knows the Carnot cycle. Carnot cycle is well discussed in every textbook of

thermodynamics. It forms the basic principle for the steam engine, refrigeration machine and

heat pump. The ability to transport very large quantities of heat with very small temperature

differences is the main feature characterizing the heat pipe. When the temperature difference

between hot and cold reservoirs is very small, the heat engine cycle is not practical and the

limiting case apparently leads to the Carnot heat pipe cycle as a heat transfer device.

For the Carnot cycle, the system absorbs heat Q1 from the reservoir at absolute tempera-

ture T1 and rejects heat Q2 to the reservoir at absolute temperature T2. From Kelvin’s

absolute temperature (or Carnot’s reversible heat engine), one obtains,

Q

T

Q

T

Q

T

Q

T

W

T

1

1

2

2

12

12

==

−

−

=∆

(1)

The net heat absorbed by the system is Q1– Q2. Since the system is carried through a cycle,

there is no change in the internal energy. Thus from the first law, the net work done by the

system is,

W = Q1 – Q2

(2)

The efficiency of a heat engine is defined as,

ηc==

−

Q

=

−

T

W

Q

QQTT

1

12

1

12

1

(3)

Equation (3) represents the maximum fraction of the heat energy that could be converted into

work. The efficiency of a cyclic heat engine other than a Carnot engine (η), will be no

greater than that for a Carnot engine (ηc) operating between the same temperature difference.

ηη

?

c

(4)

Thus one obtains,

Q

T

Q

T

Q QT

1

T

1

1

2

2

2

2

1

0

−

??

or

(5)

The Carnot or maximum efficiency of a heat engine is based on the concept of reversibil-

ity. The maximum efficiency depends only upon the absolute temperature of the high

temperature source (T1) and the low temperature sink (T2) and is independent of the system

or the working fluid. Carnot conceptually invented the mechanical refrigeration cycle with

continuous power input. A Carnot (reversible) heat engine, when used as a heat pump,

produces the greatest heat transfer for a given amount of work.

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International Journal of Mechanical Engineering Education Vol 28 No 1

Fig. 1(a). Schematic diagrams of a typical heat pipe.

Fig. 1(b). Closed two-phase thermosyphon.

One may consider an ideal heat pipe operating on a Carnot vapor cycle. A steady circula-

tion of the working fluid in the heat pipe is maintained by the work produced by the gas or

vapor. Consequently, the circulation of the fluid causes transport of the heat from the warm

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Some observations on Carnot cycle as the genesis of the heat pipe

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International Journal of Mechanical Engineering Education Vol 28 No 1

Fig. 1(c). Closed-loop two-phase thermosyphon.

to the cold region. The heat of evaporation is removed in the boiling process from the hot

region and nearly the same amount of heat is released when the fluid is condensed and is

transferred to the cold region. When the temperature difference (T1– T2) is kept as small as

possible, a large quantity of heat Q2 can be transported to the condenser section. For Carnot

heat pipe, no distinction is made between the capillary action and the gravity since the

Carnot cycle is independent of the system and working fluid. Within the scope of Carnot

heat pipe, one does not have to consider the distinction between the operating principles of

the heat pipe and thermosyphon.

In Carnot heat pipe (see Fig. 1(d)), the heat input Q1 results in a vaporization of liquid

and, hence, a volume increase. An adiabatic expansion results in a new liquid-vapour

mixture at lower temperature T2. Heat, Q2 is then extracted from the mixture by condensing

the vapor at temperature T2. An adiabatic compression brings the process to the initial state

of the cycle.

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K. C. Cheng

A steam engine operating on a Carnot cycle is impractical. In practice a Rankine cycle or

its variation must be used. Similarly a heat pipe operating on a Carnot cycle is impractical

and a Rankine cycle must be employed. From the above brief account, it is obvious that the

Carnot vapor cycle can also be regarded as the basic principle for heat pipe operation. For

Carnot vapor cycle, the heat absorption or rejection is caused by isothermal expansion or

compression and the temperature change (decrease or increase) is caused solely by adiabatic

expansion or compression. For a heat pipe, the Carnot efficiency may be regarded to be less

International Journal of Mechanical Engineering Education Vol 28 No 1

Fig. 1(d). Carnot heat engine (pipe) with small temperature difference.

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Some observations on Carnot cycle as the genesis of the heat pipe

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International Journal of Mechanical Engineering Education Vol 28 No 1

Fig. 2. The portraits of four pioneers in classical thermodynamics.

than 0.25 (i.e. T T

between the heat pipe and Carnot heat engine is of special interest. It is seen that the Carnot

vapor cycle provides the upper limit for heat pipe thermal performance. The Carnot

efficiency shows that only a fraction of the heat supplied is transformed into mechanical

work. The Carnot cycle consists of two isothermal processes and two adiabatic processes

21

3 4

=

), a relatively low figure for a heat engine. The connection

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K. C. Cheng

International Journal of Mechanical Engineering Education Vol 28 No 1

Fig. 3. Carnot’s thought experiments for the Carnot cycle (heat engine) and Clapeyron’s

p–v diagram.

working with a constant quantity of fluid. A simple and clear exposition of the work of

Carnot and Clausius is also given by Ackeret [13]. Carnot’s thought experiments for the

Carnot cycle and Clapeyron’s original pressure–volume (p–v) diagram are shown in Fig. 3.

If one considers the entropy generated in heat engine cycle Gs, one obtains from entropy

balance,

Q

T

G

Q

T

s

1

1

2

2

+=

(6)

Solving equations (2) and (6), one has:

WQ

T

T

T Gs

2

=−

−

1

2

1

1

(7)

The meaning of the first term is clear and the second term represents the reduction of the net

work caused by the irreversible heat engine. In principle equation (7) is also applicable to

heat pipe operation. The heat quantity Q2 can be found from equation (2) (energy balance).

The connection between heat pipe and heat engine is of special interest.

The steam engine invented by Watt, although far better than any previous model, was

quite inefficient. In Carnot’s time an efficiency of 5–7% was all that could be expected; this

means that 93–95% of the heat energy of the burning fuel was wasted. Carnot was interested

in determining how far the steam engine might be improved further.

3. THE CLAPEYRON–CLAUSIUS EQUATION (1834, 1850)

From the infinitesimal (differential) Carnot cycle for a liquid-vapour mixture, Clapeyron

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Some observations on Carnot cycle as the genesis of the heat pipe

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International Journal of Mechanical Engineering Education Vol 28 No 1

Note dW represents the net work done in the infinitesimal cycle and L is the latent heat

absorbed at the higher temperature. One also notes that the small volume changes in the

adiabatic (isentropic) processes are neglected. Equation (8) is one form of the Clapeyron–

Clausius equation. It expresses the slope of the vapor pressure curve at any temperature in

terms of the three commonly measured quantities, namely, the pressure–temperature relation

for two phases in equilibrium, the latent heat, and the volumetric expansion corresponding to

the change of phase. For very small temperature differences in the Carnot cycle, one may

write equation (9) in the following form.

∆ = ∆ •∆ =∆

PW

T

T

L

v

(10)

Equation (10) may be regarded as the operating principle of the ideal Carnot heat pipe in its

simplest form. A fraction of the absorbed latent heat (∆T?T)L is transformed into mechanical

work for the circulation of fluid in the Carnot cycle. The latent heat L represents the heat

needed to change v by ∆v at constant temperature T. ∆P represents change in P when T

changes by ∆T at constant volume v. ∆P · ∆v is the work done by the vapor. Thus a substan-

tial amount of heat is released when the fluid is condensed and is transferred to the cold

reservoir. It is also seen that for a given small temperature difference ∆T, the heat transport is

greater at higher operating temperature T for a given working fluid since the Carnot

efficiency ∆T?T is small. This observation agrees with the actual operating characteristics of

the heat pipe. Equation (10) shows that the pressure drop ∆P and the temperature drop ∆T

are closely coupled. The pressure drop ∆P is related to the hydrodynamics in the heat pipe.

The Clapeyron–Clausius equation reveals the important operating parameters of the heat

pipe such as vapor pressure, density of vapor and liquid at the temperature of interest, a well

as the latent heat of vaporization.

For the heat pipe some other important properties are thermal conductivity, viscosity,

surface tension, wetting ability and others. The vapor–pressure curve dictates the tempera-

ture range of applicability for a given fluid. At a specified operating temperature, the vapor

pressure should be reasonably high for a small temperature difference within the heat pipe.

In a heat pipe, the conversion of thermal energy to kinetic energy is required for the circula-

tion of the working fluid. For a small temperature difference, the average temperature of the

hot and cold reservoirs can be regarded as the operating temperature of the heat pipe. The

Clapeyron–Clausius equation provides the basic principle of the heat pipe and some physical

insight into its operation.

The infinitesimal Carnot vapor cycle for the derivation of the Clapeyron–Clausius

equation is shown in Fig. 4.

(1834) and Clausius (1850) deduced the well-known Clapeyron–Clausius equation for a two-

phase (saturated) mixture of liquid (f) and vapor (g) given by,

where the saturation pressure P and temperature T are for a liquid and vapor in equilibrium

state in the Carnot engine, L the latent heat per unit mass and vg and vf the specific volumes

of the vapor and liquid phases, respectively. Equation (8) contains the essence of the second

law of thermodynamics. It is more instructive to write equation (8) in the following form,

d

d

T

or d

d

T

gf

P

T

L

W

L

T

()

vv

−==

(9)

d

d

gf

P

T

L

T

=

−

()

vv

(8)

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International Journal of Mechanical Engineering Education Vol 28 No 1

Fig. 4. Derivation of the Clapeyron–Clausius equation from an infinitesimal (differential)

Carnot cycle.

4. SOME OBSERVATIONS ON THERMODYNAMIC TEMPERATURE

DIFFERENCE AS THE DRIVING FORCE FOR MOTIVE POWER

Equations (3) and (10) show that temperature difference is the driving force for the produc-

tion of motive power. Without a temperature difference between the hot and cold reservoirs,

the cyclic processes consisting of isothermal expansion and compression, along with

adiabatic expansion and compression for the Carnot cycle, cannot be performed. For Carnot

cycle, heat is absorbed by working fluid due to an isothermal expansion and is rejected due

to an isothermal compression utilizing the special nature of heat. The temperature drop or

rise is caused by adiabatic expansion or compression. Thus, the Carnot cycle is based on the

special nature (principle) of heat leading to the Second Law of Thermodynamics.

In Fourier’s Analytical Theory of Heat (1822), an infinitesimal temperature difference

between molecules or a temperature gradient is the driving force for heat conduction. This is

in sharp contrast to the infinitesimal difference for Carnot cycle in the derivation of the

Clapeyron–Clausius equation. It is of interest to observe that Newton’s cooling law (1701)

was based on temperature difference (potential) as the driving force for convective heat

transfer and was proposed long before the fundamental laws of classical thermodynamics

had been firmly established. It is seen that the temperature difference has completely

different physical meanings in the works of Newton (1701), Fourier (1822) and Carnot

(1824). The Carnot cycle can be run in the opposite direction in a reverse power cycle mode.

Carnot’s temperature difference represents thermodynamic potential. Carnot’s discovery of

the fundamental concepts of refrigeration is indeed remarkable. It is noted that Carnot’s

infinitesimal temperature difference for Carnot cycle does not lead to Fourier’s heat conduc-

tion problem. The difference between Carnot’s theory of heat and Fourier’s heat conduction

theory is clearly seen.

Sadi Carnot was concerned with the conversion of heat into work. He was the first to

consider quantitatively the manner in which heat and work are interconverted. He was thus

the founder of the science of thermodynamics (heat movement). Carnot’s thought or concep-

tion was influenced to a large extent by the water wheel analogy of a heat engine assigning

to temperature the role of the water level. Carnot adhered to the caloric theory of heat. He

noticed that the caloric could do work only if there was a drop in temperature in the heat

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Some observations on Carnot cycle as the genesis of the heat pipe

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International Journal of Mechanical Engineering Education Vol 28 No 1

engine. Carnot’s various concepts and principles remain completely valid today in spite of

his wrong caloric hypothesis. His erroneous conclusions and methods of proof can be

corrected by using the energy principle. Although Carnot was not correct in his views as to

the nature of heat flow, he did recognize the conversion of heat into work (pressure times

volume change, PdV).

5. CARNOT’S EPOCH-MAKING MEMOIRE ON ‘REFLECTIONS ON THE

MOTIVE POWER OF FIRE’

At present the historical literature relating to Carnot’s work on the Second Law of thermody-

namics is very extensive. An excellent historical overview on the development of classical

thermodynamics in the nineteenth century is presented by Mach [14] an Cardwell [15].

Review articles are presented by Barnett [16], Koenig [17, 18], Thurston [19], Mendoza

[20], Challey [21], Lervig [22], Klein [23] and Cropper [24] among many others. Useful

historical notes in thermodynamics can be found in Soumerai’s monograph [25].

Carnot’s famous memoire consists of three distinct parts (Koenig [17, 18]). In the first

part Carnot describes the reversible cycle and demonstrates the fundamental theorem. In the

second part, Carnot deduces seven further theorems relating to the thermal properties of

gases, and illustrates these theorems to some extent by numerical data and calculations.

Carnot derives the expression for the work done by the infinitesimal cycle in the famous long

footnote using calculus. Application of his theoretical work for the infinitesimal cycle to the

case of vapor cycle led to the famous Clapeyron–Clausius equation. In the third part, Carnot

discusses the practical advantages of different working fluids and the efficiency of steam

engines in operation in his time compared to the theoretical maximum (technology). Carnot’s

important contributions to science, namely the Carnot cycle and Carnot’s theorem on revers-

ible engines are contained in the first part of the Reflections. The ideas led Clausius (1850)

and William Thomson (1851) to the final establishment of the Second Law of thermo-

dynamics. Effectively, the mathematization of Carnot’s original theory of heat was

performed and completed by Clapeyron (1834), Thomson (1851), and Clausius (1850).

The development of the ‘Second Law’ for the present purpose is well reviewed by

Koenig [17, 18] and further details will be omitted.

From the viewpoint of the operating principles of the heat pipe, the following theorems or

propositions from Carnot’s monograph are of special interest [20]:

(1) The motive power of heat is independent of the agents employed to realize it; its

quantity is determined solely by the temperatures of the bodies between which is

effected, finally, the transfer of the caloric (Carnot’s theorem).

Wherever there exists a difference of temperature, motive power can be produced.

The maximum of motive power resulting from the employment of steam is also the

maximum of motive power realizable by any means whatever.

The necessary condition of the maximum, is that in the bodies employed to realize the

motive power of heat, there should not occur any change of temperature which may

not be due to a change in volume.

The descent of caloric produces more motive power at lower degrees of temperature

than at higher.

The quantity of heat, due to the change of volume of a gas at constant temperature,

becomes greater as its temperature is raised.

(2)

(3)

(4)

(5)

(6)

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International Journal of Mechanical Engineering Education Vol 28 No 1

Carnot’s reasoning on the general laws for the theory of the heat engine was based on his

ingenious use of ‘thought experiments’ based on the concept of limit. He employed a gas,

enclosed in a cylindrical vessel (a b c d) with a moveable piston (c d) as shown in Fig. 3.

Carnot describes three of the four steps of a particular steam cycle, namely: (1) iso-

thermal conversion of a quantity of water into steam; (2) adiabatic expansion of this steam;

(3) isothermal condensation of the product of (2). Carnot did not specify the last step,

adiabatic compression to return to the original state of the closed cycle. The four steps noted

are applicable to an ideal Carnot heat pipe operation. In discussing the production of motive

power in the second part of the ‘Reflections’; the following statement is quite suggestive.

‘We shall see shortly that this principle is applicable to any machine set in motion by

heat. According to this principle, the production of heat alone is not sufficient to give

birth to the impelling power: it is necessary that there should also be cold; without it,

the heat would be useless.’

The motion of working fluid in an idealized Carnot heat pipe is certainly set by the action

of heat. When the temperature difference between the hot and cold reservoirs is small, one

has a Carnot heat pipe as a limiting case of the Carnot heat engine.

The concept of absolute temperature was implicit in the long footnote (second part of the

‘Reflections’), showing the numerical calculation on ‘a relation between the motive power

and the thermodynamic degree’. Carnot uses the temperature different ∆T = 0.001°C and

1°C in his numerical calculation.

The Reflections are devoted also to an attempt to show, by computation from experimen-

tal data from other investigators, that ‘the quantity of motive power produced is really

independent of the agents used (Carnot theorem)’. Carnot’s experimental verification is of

special interest. Table 1 summarizes Carnot’s results and compares them with the true values

computed by taking the theoretical efficiency equal to (

absolute temperatures of source and sink respectively (Koenig [18]). Clapeyron’s numerical

results (Mendoza, [20]) corresponding to Carnot’s result in Table 1 are shown in Table 2 for

comparison.

It is of interest to note that Carnot effectively derived the following analytical expression

for the Carnot theorem,

),

TTT

121

−

where T1 and T2 are the

Table 1. Carnot’s numerical results. Number of units of motive power produced

by descent of 1000 units of heat.

Agent From 1°C

to 0°C

From 78.7°C

to 77.7°C

From 100°C

to 99°C

Air

Water vapor

Alcohol vapor

Any agent, true value

Carnot efficiency ηc

1.395

1.290

—

1.556

3.65 × 10–3

—

1.212

1.230

1.213

2.84 × 10–3

—

1.112

—

1.144

2.68 × 10–3

Note: Carnot’s unit of motive power is the quantity of work required to lift 1 cubic m of

water through 1 m (about 9810 joules) and his unit of heat is the quantity required to raise

the temperature of 1 kg of water by 1°C (about 4185 joules).

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International Journal of Mechanical Engineering Education Vol 28 No 1

Table 2. Clapeyron’s numerical results.

Names of liquids Temperature of boiling, °CWork developed/1000 units of

heat/1°C temperature fall

Sulfuric ether

Alcohol

Water

Oil of turpentine

35.5

78.8

100

156.8

1.365

1.208

1.115

1.076

∆

= − ∆ =

t

∆

( )

W

Q

t t

t

C t

Ψ( ,)

(11)

where C(t) is the universal function known as the Carnot function and is a function of the

temperature t independent of the nature of the working fluid, ∆W the work produced and Q

heat input. In 1834 Clapeyron applied Carnot’s results to vapor–liquid equilibrium and

arrived at the relationship now called the Clapeyron–Clausius equation (10) valid for an

idealized heat pipe operation. The equation contains an unknown temperature function later

identified as the absolute temperature scale by Clausius.

The experimental determination of the Carnot function is shown in Fig. 5 [26] using the

experimental data computed by Carnot and Clapeyron; the straight line represents Kelvin’s

absolute temperature scale.

Carnot points out the possible causes for the source of errors, in various available experi-

mental data on thermal properties of fluids, and notes that the fundamental law is verified in

a particular case. The numerical results in Tables 1 and 2 generally agree with the known

linear relationship between Carnot’s original function and temperature (t). The Carnot func-

tion C(t) is equivalent to T t J

( )

(J = proportional factor (unit) for mechanical equivalent of

heat) in equation (8). Table 1 shows that for the temperature difference ∆t = 1°C, the Carnot

efficiency is lower at a higher temperature. One may also conclude that for Carnot heat pipe

(∆t = small), the heat transport to the cold reservoir is larger at a higher operating tempera-

ture. This observation agrees with the known heat pipe operating characteristics for a given

working fluid.

Carnot’s numerical calculation deals with small temperature difference T1– T2= 1°C and

()

TTT

121

0

−≈ or T T Q Q

1212

1

≈≈ (an infinitesimal cycle) in his memoir. The limiting

case can be considered as the Carnot heat pipe cycle. It is believed that this observation is of

historical interest in classical thermodynamics. It is instructive to show the Carnot steam

cycle and the Rankine cycle (Fig. 6) in three different plots (Sears, [27]) for reference here.

The thermodynamic cycle of heat pipe similar to the Rankine cycle can be found in

Richter and Gottschlich [8] and Pryputniewicz and Haapala [6], for example.

The thermodynamic characteristics of some working fluids at evaporation temperature

t2= 30°C and condensation temperature tc= 10°C [28] are shown in Table 3. The Rankine

cycle efficiency is seen to be larger when the absolute pressure ratio P P

and condensation is larger. It is seen that for small temperature difference such as ∆t = 20°C,

the Carnot efficiency and Rankine efficiency agree within a few percent. Thus for very small

temperature difference ∆t, the Clapeyron–Clausius equation reveals quite simply the basic

second law thermodynamic relationship for Carnot heat pipe operation.

ec for evaporation

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Table 3. Thermodynamic characteristics of some working fluids at evaporation

temperature te = 30°C and condensation temperature tc = 10°C

Working fluidPressure Pe at

evaporation,

kgf?cm2 a

Pressure Pc at

condensation,

kgf?cm2 a

Rankine cycle

efficiency,

ηR

ηη

RC

Freon R-22

Freon R-113

Propane

Ammonia

Water

12.15

0.55

10.98

11.91

0.04

6.94

0.24

6.48

6.28

0.01

0.0641

0.0627

0.0634

0.0642

0.0633

97.1

95.0

96.1

97.3

95.9

Note: Carnot efficiency ηC = 0.0660.

Fig. 5. The experimental determination of the Carnot function showing numerical values

computed by Carnot, Clapeyron and Kelvin (steam, t = 0 ~ 230°C) [26].

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Fig. 6. The Carnot team cycle and the Rankine cycle in: (a) p–v (pressure–volume)

diagram, (b) T–s (temperature–entropy) diagram, (c) h–s (enthalpy–entropy) diagram

(Sears, [27]).

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6. CHRONOLOGY (HISTORICAL NOTES)

The historical beginnings of classical thermodynamics lie in the study of the efficiency of

heat engines by Sadi Carnot. A brief historical summary is presented in the form of notes on

some aspects of thermodynamics relating to the origins of heat pipe as one understands them

today. The exposition of classical thermodynamics has been made by many authors in the

past.

where k is the latent heat of evaporation per unit volume of the liquid, ρ the density of

the liquid, δ the density of the saturated vapor, P the saturation vapor pressure at the

temperature t and C a function of the temperature, t independent of the nature of the

substance. C(t) is the Carnot function and contains the mechanical equivalent of heat

implicitly.

1836 The first use of the Perkins Tube, a heat transfer device operating like a closed-loop

two-phase thermosyphon.

1848 W. Thomson (Kelvin) defines an absolute temperature scale (independent of the

thermometric substance) based on Carnot’s work.

1850 Clausius, in essence, combines Carnot’s theory with the energy principle (First Law)

and thus he creates the basis of the Second Law of thermodynamics. His work marks

an epoch in the history of physics and the beginning of thermodynamics as a science.

1854 Clausius introduces the concept of entropy which leads to a new formulation of the

Second Law.

1887 Planck divides changes of state into two classes: i.e. reversible and irreversible

processes.

1939 E. Schmidt conducts experiments demonstrating a heat transfer mechanism similar to

the heat pipe transporting heat by using the working fluid close to its critical point.

1942 R. S. Gaugler filed the first US heat pipe patent (heat transfer device) for application

to the cooling of the interior of an ice box.

1953 M. J. Lighthill’s paper on ‘Theoretical Considerations on Free Convection in Tubes’

(Thermosyphon).

1963 G. M. Grover, US Patent on evaporation–condensation heat transfer device.

1964 Grover, Cotter and Erickson publish the first technical paper on the heat pipe.

(Structures of very high thermal conductance.)

1965 T. P. Cotter publishes the first heat pipe analysis (theory of heat pipes).

1824 Publication of Carnot’s monograph ‘Reflections on the Motive Power of Fire’ which

includes the theorem which states that the maximum efficiency of a heat engine

depends only on the temperature difference (T1– T2) and the hottest temperature, T1,

()

TTT

121

−

and not on the working fluid of the system. Carnot uses the caloric

theory of Lavoisier and suggests that the heat absorbed from a hot reservoir is equal to

the heat rejected by a cold reservoir in his first study of reversible cycles; ironically

this is approximately true for the heat pipe operation, but the reasoning is wrong.

1834 Clapeyron applies Carnot’s theorem to the study of vapor–liquid equilibrium and

obtains the formula:

k

P

tC

d

=−

1

δ

ρ

d

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7. CONCLUDING REMARKS

The influence of Carnot’s conceptualization of the ideal heat-engine was so far-reaching in

classical thermodynamics that it would be difficult to follow up all the implications related to

engineering and engineering sciences. Apparently the invention of the heat pipe was not

based on Carnot’s theory of heat.

An attempt is made to trace the origins of the thermodynamic aspects of the heat pipe

operations to the limiting case of the infinitesimal (differential) Carnot vapor cycle (Carnot

heat pipe). Since the Carnot cycle is impractical, the actual heat pipe operation is based on a

variation of the Rankine cycle. For a very small temperature difference, the Carnot cycle and

the Rankine cycle are nearly identical. The Clapeyron–Clausius equation shows that

thermodynamcs, fluid mechanics and heat transfer are closely coupled in heat pipe operation.

In a heat pipe the generation of internal work from heat is used to circulate the working fluid

in two-phases. The expansive property of water and other working substances by boiling is

important for both heat-engines and heat pipe.

The limiting case treated by Carnot clearly corresponds to the idealized operation of the

heat pipe. This observation is believed to be of historical interest for the heat pipe theory and

principle. The case of an expansive heat engine working within an infinitely small range of

temperatures corresponds to the heat pipe operation for heat transport. The close relationship

between Carnot heat-engine and heat pipe is of special interest. The Carnot vapor cycle

provides the basic principle for heat pipe and two-phase thermosyphon in addition to the

classical cases of heat engine, refrigeration machine and heat pump.

The conversion of heat into motion can also be seen in various natural convection

phenomena. Some examples of heat engine in natural convection phenomena are:

The application of Carnot cycle also led to the theoretical derivation of Stefan-Boltmann’s

law for blackbody radiation in 1884, confirmation of the measurement of the depression of

the freezing point in 1850 and the construction of happy water-drinking birds (toy) operating

between warm body at temperature T1 and cool head at temperature T2 with liquid ether

being driven up by vapor pressure.

The Carnot heat–heat pipe is considered to be the most efficient idealized heat pipe

device since it neglects the irreversible process entirely. Its performance provides the upper

limit for the heat pipe. The present observation on the close relationship (or connection)

between the Carnot vapor cycle and the operation of the heat pipe and two-phase

thermosyphon is based on various available expositions of classical thermodynamics in the

historical literature. The practical heat pipe operation belongs to the domain of non-

equilibrium thermodynamics.

It is of special interest to observe that the development of heat pipe science and tech-

nology proceeded rather rapidly based on the scientific methods of mathematization (theory)

and experimentation known since the times of Galileo and Newton.

It is noted that Carnot’s discovery of the fundamental concepts of refrigeration, the

reverse Carnot cycle, anticipated the ‘art’ of refrigeration by some 10 year (Perkin’s

mechanical refrigeration system in 1834). It is quite remarkable that Sadi Carnot’s discovery

of an infinitesimal (differential) reversible cycle in 1824 preceded the invention of the

modern heat pipe (1942, 1963) by about 118 years.

(1)

(2)

(3)

Bénard’s cells,

global atmospheric heat engine (heat cosmology), and

natural convection in closed space.

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ACKNOWLEDGEMENT

The preparation of this article was supported by an operating grant from the Natural Sciences

and Engineering Research Council of Canada.

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