# What Work Does the Earth Do as a Heat Engine?

2013-12-12

A heat engine is a device that repeatedly converts thermal energy into kinetic energy. It needs the difference of temperature and a working substance with a high rate of expansion to do useful work for us. It can be done in two ways: utilizing the change of volume of the working substance that hot and cold reservoirs make alternately expand or compress, and utilizing convection that the change of density resulting from the change of volume causes. A heat engine is not confined to an artificial one. The Earth also does useful work for living systems as a heat engine. The heat engine whose hot reservoir is solar radiation heat maintains the circulation of air and water, while the heat engine whose hot reservoir is geothermal heat maintains the circulation of mantle and mineral nutrition.

## 1. The history of heat engines and theories of their principles

In thermodynamics, a heat engine is a system that does a certain amount of net positive work repeatedly through the conversion of thermal energy to kinetic energy. A heat engine was first devised by an engineer in Ancient Greek, Heron of Alexandria (Ήρων ο Αλεξανδρεύς)[1], but this chapter takes up the Modern history of the practical application of heat engines and the theoretical reflection upon their principles.

### 1.1. The improvement of steam engines

In 1712 Thomas Newcomen (1664 – 1729) first succeeded in commercializing a heat engine. The animation (Fig. 1) below shows how the Newcomen engine works. When the valve is opened, steam is let out of the boiler to fill the space in the cylinder and lift the piston upward. The valve is then closed and another valve sends a spray of cold water into the cylinder, creating a partial vacuum under the piston. Pressure difference drives the piston down, raising the pump gear.

More than 100 Newcomen steam engines had been employed principally to pump water out of mines until 1733 when his patent for it expired. Despite this practical success people at that time including Newcomen himself did not understand the cause of the work rightly. The pump equipment was heavier than the steam piston and they wondered what drove the piston down or raised the pump gear. Most of them believed the vacuum in the cylinder rather than atmospheric pressure did the work[3], much less did they realize the difference of temperature was essential to heat engines.

Then James Watt (1736 – 1819) improved the Newcomen steam engine and devised a double-acting engine, utilizing steam pressure alternately above and below the piston. His engine converted the vertical movement of the piston into circular motion, for which there were more industrial applications. Thus he became a leading figure in the British Industrial Revolution. The following illustration (Fig. 2) was drawn in 19 century. Let me explain its mechanism using this illustration.

The pipe (v) injects steam into the upper part of the cylinder (J), propelling the piston downward and evacuating the steam from under the cylinder to a separate condensation chamber (H) immersed in a cold water tank (R). Thus the condensation makes the pressure of the lower part of the cylinder low. When the piston reaches the bottom, steam is injected into the lower part of the cylinder and the steam in the upper part of the cylinder is evacuated to the condensation chamber. The low pressure of the upper part of the cylinder and the weight of the pump raise the piston. This cycle is repeated automatically.

Watt’s steam engine was different from Newcomen’s in that a steam condenser was separated from the cylinder and steam pressure instead of atmospheric pressure pushed the piston down to increase the efficiency of the engine[5]. Watt also noticed that injection of steam to the end of the stroke is unnecessary and adoption of adiabatic expansion or compression can increase the efficiency of the engine[6].

Watt was not a mere engineer. He had an insight into the essential principle of a heat engine. But, generally speaking, because of their empiricist tradition the British engineers made technological improvements on heat engines by trial and error without theoretical reflection. In contrast to Britain, the Continent had a rationalistic tradition and produced a theoretical genius, Nicolas Léonard Sadi Carnot (1796 – 1832). He published Reflections on the Motive Power of Fire and on Machines Fitted to Develop that Power (Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance) in 1824 and established the general principles of a heat engine.

### 1.2. Carnot’s theory of heat engines in general

At the beginning of the Reflections, he wrote what motivated him to write this treatise, “Notwithstanding the work of all kinds done by steam-engines, notwithstanding the satisfactory condition to which they have been brought to-day, their theory is very little understood, and the attempts to improve them are still directed almost by chance[7]“. Although he analyzed steam-engines in those days, his theory can be applied to heat engines in general.

According to Carnot, there are two conditions for a heat engine to produce motive power. The first one is the difference in temperature.

The production of motive power in steam-engines is then due not to an actual consumption of caloric, but to its transportation from a warm body to a cold body, that is, to its re-establishment of equilibrium, an equilibrium considered as destroyed by any cause whatever, by chemical action such as combustion, or by any other. 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.[8]

Watt must also have noticed that not only a hot reservoir but also a cold reservoir is necessary for a heat engine because he separated the cold condenser from the hot cylinder. Carnot compared the motive power of a heat engine to that of a water wheel.

The motive power of a waterfall depends on its height and on the quantity of the liquid; the motive power of heat depends also on the quantity of caloric used, and on what may be termed, on what in fact we will call, the height of its fall, that is to say, the difference of temperature of the bodies between which the exchange of caloric is made.[9]

Carnot thought “the quantity of caloric absorbed or relinquished is always the same[10]" just as the quantity of water flowing from and into a water wheel is always the same, while both of them are producing power. As the term “caloric (calorique)" he used suggests, he believed in the caloric theory that was in fashion at that time and assumed conservation of heat. After he wrote the Reflections, however, he, inspired by the paper[11] by Count Rumford (Sir Benjamin Thompson; 1753 – 1814) in 1798 that reported the frictional heat generated by boring cannon at the arsenal was seemingly inexhaustible, abandoned the law of heat conservation, and admitted that part of heat could be converted into work[12]. That is to say, he stated what is today called the first law of thermodynamics[13].

The first law of thermodynamics is the topic I will bring back in the next section and let us proceed to the second condition for a heat engine to produce motive power. Carnot thought the difference in temperature was not sufficient. In fact, the mere flow of heat from a hot reservoir to a cold reservoir without working substances does not result in the motive power that we expect a heat engine to produce. A heat engine needs working substances susceptible to changes in volume through the alternation of heat and cold.

Of course, as the direct contact of a hot reservoir with a cold reservoir contracts the hot reservoir and expands the cold reservoir, heat can do work without the third working substance, however little the change in volume might be. Furthermore, the third working substance is not something different from a hot or cold reservoir. It expands in contact with a hot reservoir and functions as a cold reservoir. It contracts in contact with a cold reservoir and functions as a hot reservoir. The reason the working substance (usually gas) does useful work lies in its higher coefficient of thermal expansion than the cold reservoir which is usually water. So, I will use the term “working substance" in this sense. While the first condition is necessary for producing work in general, the second condition is necessary for producing useful work.

Carnot said, “Heat can evidently be a cause of motion only by virtue of the changes of volume or of form which it produces in bodies[14]" . It indicates that in order to increase the efficiency of a heat engine we should avoid the heat transfer that does not contribute to changes in volume.

Since every re-establishment of equilibrium in the caloric may be the cause of the production of motive power, every re-establishment of equilibrium that shall be accomplished without production of this power should be considered as an actual loss. Now, very little reflection would show that all change of temperature which is not due to a change of volume of the bodies can be only a useless reestablishment of equilibrium in the caloric. The necessary condition of the maximum is, then, 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 of volume. Reciprocally, every time that this condition is fulfilled the maximum will be attained.[15]

The heat engine that fulfills the maximum is called an ideal engine. It is different from a real heat engine that does not necessarily convert all of the change of temperature into a change of volume, but still, the ideal engine is scientifically significant just as the ideal gas is, though it is different from a real gas. The ideal gas expands the gas in a cylinder slowly and gradually, namely through a quasi-static process, so that a sequence of states is infinitesimally close to equilibrium. A real engine operates its piston faster and it does not maintain the equilibrium so that convection or vortex motion occurs in the gas of the cylinder. Because a real engine does this excessive work, its heat efficiency is lower than that of the ideal engine.

While the dynamic process of a real engine is irreversible, the quasi-static process of an ideal engine is reversible, that is to say, you can transfer heat from a cold reservoir to a hot one using the work produced by the heat flow from a hot reservoir to a cold reservoir. Carnot proved by reductio ad absurdum that a heat engine cannot do more work than an ideal engine. If it could, it could produce work without changing the original difference of temperature, but it is absurd[16]. To use the current terms, such a heat engine is a perpetual motion machine of the first kind which violates the first law of thermodynamics, the law of conservation of energy.

The bottom line is that a heat engine needs two conditions, a difference in temperature and a working substance (especially fluid with a high expansion rate). Carnot, however, regarded the former as more important. Though all the heat engines at that time were steam engines, Carnot realized that the working substances for a heat engine did not need to be steam. His abstraction was free from historical constraints at this point. The following is the bold hypothesis that is today called Carnot’s theorem.

The motive power of heat is independent of the agents employed to realize it; its quantity is fixed solely by the temperatures of the bodies between which is effected, finally, the transfer of the caloric.[17]

This theorem holds true even today. The efficiency of a heat engine is the function of only temperatures.

### 1.3. Development of thermodynamics after Carnot

On account of rather than despite the revolutionary theory of Reflections (Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance) it was not estimated at all. Some assume it was because his explanation was not mathematical, but the paper[18] by Benoît Paul Émile Clapeyron (1799 – 1864) in 1834, 2 years later after Carnot’s death, which mathematically formulated Carnot’s theorem, elicited no responses from the academic establishment except William Thomson (Lord Kelvin ; 1824 – 1907) who read the paper during his study in France and developed the theory of Carnot.

Another scientist who was unknown until Thomson discovered was James Prescott Joule (1818 – 1889). He showed the mechanical work can be converted into heat by the well-known experiment, where the mechanical work of spinning a paddle-wheel in an insulated barrel of water increased the temperature. He then jumped to the reverse proposition that heat can be converted into mechanical work.

You see, therefore, that living force [energy] may be converted into heat, and that heat may be converted into living force, or its equivalent attraction through space. All three, therefore – namely, heat, living force, and attraction through space (to which I might also add light, were it consistent with the scope of the present lecture) – are mutually convertible into one another. In these conversions nothing is ever lost. The same quantity of heat will always be converted into the same quantity of living force. We can therefore express the equivalency in definite language applicable at all times and under all circumstances.[19]

This idea of Joule is incompatible with the law of heat conservation Carnot presumed. Joule tried to convince Thomson out of Carnot’s theory, but he hesitated to decide which was right[20]. It was Rudolf Julius Emmanuel Clausius (1822 – 1888) who solved this problem.

Clausius concluded that the partial consumption of heat to generate the work of a piston is compatible with the heat flow from a hot reservoir to a cold reservoir[21]. This was also the conclusion that Carnot reached in the manuscript published posthumously. All of the work can be converted into heat but not all of the heat can be converted into work. This is the current view and Joule’s hypothesis turned out to be false. The irreversibility led Clausius and Thompson to discover the second law of thermodynamics.

Clapeyron, Thompson, and Clausius contributed to establishing Carnot’s theory as classic thermodynamics but in the end, it was a theory of an ideal engine that neglects convection and vortex motion as valueless turbulence. They were the subjects of fluid mechanics and heat transfer physics. In 1858 Hermann Ludwig Ferdinand von Helmholtz (1821–1894) established three laws of vortex motion[22] and early in the 20th century at the beginning Baron Rayleigh (John William Strutt; 1842 – 1919) and Henri Claude Bénard (1874–1939) studied the typical natural convection (Rayleigh-Bénard convection). This article pays attention to the similarity between two sorts of work produced by an ideal heat engine and convection that have been studied separately.

## 2. The structure and operation of Carnot heat engine

First, let’s analyze the Carnot heat engine as an ideal heat engine. In this chapter, we will survey four steps of the Carnot cycle, work produced by the Carnot heat engine, and its efficiency. Then we will recognize how the current standard explanation of thermodynamics reflects the theory of Carnot.

### 2.1. Four steps of the Carnot Cycle

Carnot drew the following piston-and-cylinder diagram (0 of Fig. 3) to explain the operation of an ideal heat engine in Reflections (Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance). As this diagram is overcrowded, I analyze it into four steps (1-4).

The cycle of an ideal heat engine consists of the following four steps:

1. Isothermal Expansion: The gas in the cylinder absorbs heat from the high-temperature reservoir, which increases the entropy of the gas, but, as the gas expands to push the piston, the temperature of the gas remains constant.
2. Adiabatic Expansion: The cylinder is thermally insulated from the high-temperature reservoir so that entropy remains constant. The gas continues to expand, which causes the gas to cool.
3. Isothermal Compression: The heat flows from the gas in the cylinder into the low-temperature reservoir, which decreases the entropy of the gas, but, as the piston compresses the gas, the temperature of the gas remains constant.
4. Adiabatic Compression: The cylinder is again thermally insulated from the low-temperature reservoir so that entropy remains constant. The piston continues to compress the gas, which causes the gas to warm.

At the last step, the gas comes back to the same state as the start of the first step, thus forming a cycle, called the Carnot cycle.

### 2.2. The Work produced by the Carnot heat engine

The figure below (Fig. 4) is the Carnot cycle plotted on a pressure-volume graph. Two isothermal stages follow the isotherm lines and two adiabatic stages move between isotherms.

V [m3] of the horizontal line multiplied by P [Pa=N/m2] of the vertical line is work W [J=Nm=m3×N/m2]. So, the yellow area bounded by the cycle path represents the total work that can be done during one cycle. The total area is the integral of VA-VB-B-A plus VB-VC-C-B minus VA-VD-D-A minus VD-VC-C-D.

The equation to integrate is the first law of thermodynamics. Heat energy Q [J] put into a gas system in the cylinder increases the temperature of the gas system, namely the internal energy (U), or does work (W), pushing the piston to the outside. Meanwhile, the total amount of energy is preserved.

$Q=\Delta U+W=nC_{v}\Delta T+P\Delta V$

As it is a reversible quasi-static process, the following differential equation can be applied to the infinitesimal change of internal energy.

$\delta Q=dU+\delta W=nC_{v}dT+PdV$

where n is the amount of a substance, Cv is specific heat at constant volume per mole, T is the absolute temperature, P is the pressure and V is the volume of the gas system. To use the ideal gas law PV=nRT, we get an equation:

$\delta Q=nC_{v}dT+\frac{nRT}{V}dV=0$

Let us calculate the areas of the 4 steps of the Carnot cycle using this equation.

1. Isothermal Expansion (A→B)

As this step is isothermal, dT=0. Its work is equivalent to Q2 or the positive work done by the expansion.

$W_{AB}=Q_{2}=nRT_{2}\int_{V_{A}}^{V_{B}}\frac{1}{V}dV=nRT_{2}(lnV_{B}-lnV_{A})=nRT_{2} ln \frac{V_{B}}{V_{A}}$

As this step has no heat transfer, δQ=0.

$nC_{v}dT+PdV=0$

If we use this equation and replace the increase in volume with the increase in internal energy, we will get its positive work.

$W_{BC}=\int_{V_{B}}^{V_{C}}PdV=-nC_{v}\int_{T_{2}}^{T_{1}}dT=-nC_{v}(T_{1}-T_{2})=nC_{v}(T_{2}-T_{1})$

3. Isothermal Compression (C→D)

As this step is isothermal, dT=0. Its work is equivalent to Q1 or the negative work done by the compression.

$W_{CD}=Q_{1}=nRT_{1}\int_{V_{C}}^{V_{D}}\frac{1}{V}dV=-nRT_{1}(lnV_{C}-lnV_{D})=-nRT_{1} ln \frac{V_{C}}{V_{D}}$

As this step has no heat transfer, δQ=0.

$W_{DA}=\int_{V_{D}}^{V_{A}}PdV=nC_{v}(T_{2}-T_{1})-nC_{v}\int_{T_{1}}^{T_{2}}dT=-nC_{v}(T_{2}-T_{1})$

The total net work is the aggregate of these.

$W=W_{AB}+W_{BC}+W_{CD}+W_{DA}$

Eq.0. $=nRT_{2} ln \frac{V_{B}}{V_{A}}+nC_{v}(T_{2}-T_{1})-nRT_{1} ln \frac{V_{C}}{V_{D}}-nC_{v}(T_{2}-T_{1})=nRT_{2} ln \frac{V_{B}}{V_{A}}-nRT_{1} ln \frac{V_{C}}{V_{D}}$

Because TVγ-1 is constant (γ=Cp/Cv) in the quasi-static adiabatic process, we can formulate the equation

$T_{2}V_{B}^{\gamma -1}=T_{1}V_{C}^{\gamma -1}$

$T_{2}V_{A}^{\gamma -1}=T_{1}V_{D}^{\gamma -1}$

as to adiabatic compression. The equation (10) divided by the equation (11) is

$\left (\frac{T_{2}V_{B}}{T_{2}V_{A}} \right )^{\gamma -1}=\left (\frac{T_{1}V_{C}}{T_{1}V_{D}} \right )^{\gamma -1}$

that is to say,

$\frac{V_{B}}{V_{A}}=\frac{V_{C}}{V_{D}}$

From the equation (09) and (13), we get

$W=nR(T_{2}-T_{1}) ln \frac{V_{B}}{V_{A}}$

Now we can recognize two conditions for a heat engine to do work in this equation (14). The first condition was the difference in temperature between two heat reservoirs. If T2-T1=0, W=0. No work can be done. The other condition was an expandable and compressible substance. If there is no substance, that is to say, n=0 or no change of volume, that is to say, VA=VB, W=0. It follows that we must increase the difference of temperature between two heat reservoirs and the amount of substance that has a high rate of expansion to increase the work of a heat engine.

### 2.3. The thermal efficiency of the Carnot heat engine

Even if the work by a heat engine gets big, it would not be desirable if the efficiency is low. Let’s consider what we should do to improve the thermal efficiency, defining the thermal efficiency as the rate of work to the heat from the hot reservoir,

$\eta =\frac{W}{Q_{2}}=\frac{Q_{2}-Q_{1}}{Q_{2}}=1-\frac{Q_{1}}{Q_{2}}$

From the equations (04), (07) and (13),

$\frac{Q_{1}}{Q_{2}}=\frac{nRT_{2} ln \frac{V_{B}}{V_{A}}}{nRT_{1} ln \frac{V_{C}}{V_{D}}}=\frac{T_{2} ln \frac{V_{B}}{V_{A}}}{T_{1} ln \frac{V_{B}}{V_{A}}}=\frac{T_{2}}{T_{1}}$

So, from the equations (15) and (16) we get

$\eta =1-\frac{Q_{1}}{Q_{2}}=1-\frac{T_{1}}{T_{2}}$

It tells us that thermal efficiency is entirely determined by the ratio of the low temperature to the high temperature. We can recognize that Carnot was right in that he considered the difference in temperature to be decisive. The formula also tells us that we must keep the quasi-static process and use the cold reservoir at absolute zero (T1=0) to make the thermal efficiency 1. As it is impossible to make the absolute temperature zero, we cannot make the efficiency 1.

Carnot knew that the efficiency of real steam-engines is far less than the theoretical maximum[25]. The real heat engine whose thermal efficiency is the closest to that of the Carnot engine is Stirling engine invented by Robert Stirling (1790 – 1878）in 1816. Many types of Stirling engines have been conceived and the following (Fig. 4) is one of those.

Despite the high thermal efficiency, Stirling engines are employed only for some limited purposes, because their equipment is large and heavy and their capital cost per unit power is high. The heat engine that is most widely used is the internal combustion engine, typically the Otto engine. The animation below (Fig. 5) shows its four-stroke cycle: air and vaporized fuel are drawn in at the 1st stroke, fuel vapor and air are compressed and ignited at the 2nd stroke, fuel combusts and the piston is pushed downwards at the 3rd stroke and exhaust is driven out at the 4th stroke.

Compared to a Stirling engine of the same power rating, an internal combustion engine currently has lower thermal efficiency but lower capital cost and is usually smaller and lighter. So, it is used for transportation and many other purposes. Thermal efficiency is not the only criterion to choose a heat engine.

Carnot did not know an internal combustion engine, but his principle can be applied to it. It is applicable also to what traditional thermodynamics does not treat as a heat engine. The next chapter focuses on more complex but more beneficial heat engines than artificial heat engines.

## 3. The structure and operation of global heat engines

Not only humans but also all living things have used two natural heat engines far before artificial heat engines were invented. One is a heat engine whose hot reservoir is the thermal energy caused by solar radiation, whose cold reservoir is outer space, and whose working substance is the atmosphere. The other is a heat engine whose hot reservoir is the geothermal energy, whose cold reservoir is the crust of the Earth and outer space in the end, and whose working substance is the mantle.

The former causes the convection of the atmosphere and the latter causes that of the mantle. All of the work of convection is converted into heat in the end and emitted into outer space. So the amount of heat that two global heat engines receive (Qin) is equal to the amount of heat that two global heat engines emit (Qout). Seen from the outside, the thermal efficiency of the Earth as a heat engine is zero.

$\eta =1-\frac{Q_{in}}{Q_{out}}=1-1=0$

Yet the work two global heat engines do is so important as to decide our existence on the surface of the Earth. This chapter elucidates this mechanism.

### 3.1. The atmospheric convection and the circulation of water

As for the artificial heat engines that we have developed since the 18th century, expansion and compression of gas produce useful work, while convection or vortex motion of gas is regarded as a waste. As for the heat engines of the atmosphere, on the other hand, the useful work is done not by expansion and compression of the atmosphere but by convection.

There are two kinds of convection: natural convection, where density differences in the fluid generate the fluid motion, or forced convection, where an external source generates the fluid motion. The atmospheric circulation is natural convection. The vertical temperature gradient of the troposphere is -6.5K/km. It means the higher the cooler. When the bottom of the fluid is heated and the surface is cooler than the bottom, the form of natural convection is decided by Rayleigh number (Ra).

$\mathrm{Ra}_{L} = \frac{g \beta} {\nu \alpha} (T_2 - T_1) L^3= \frac{C_p} {\nu k} g \rho \beta(T_2 - T_1) L^3$

where L=scale of convection, g=acceleration due to gravity, T2=surface temperature, T1=fluid temperature far from the surface, ν=kinematic viscosity, α=thermal diffusivity, β=thermal expansion coefficient, k=thermal conductivity, ρ=density, Cp=specific heat capacity.

When the Rayleigh number is below 1700, heat transfer is primarily in the form of conduction and when it exceeds 1700, heat transfer is primarily in the form of convection (Fig. 6).

When the Rayleigh number exceeds 5×104, the regular pattern of convection cells fluctuates and when it exceeds 106, turbulence occurs.

The factors gρβ(T2-T1) in the Rayleigh number represent buoyancy exerted by a fluid whose volume is L3. The difference of temperature plays no less important role here than in the equation (14) that defines the work by the Carnot engine. On the other hand, there are some differences. While the Carnot engine does work just by expanding and compressing the working fluid, in natural convection fluid arising from resistance to gravity does work. This is why the equation of the Rayleigh number depends on the properties of the working fluid unlike the equation (14).

With all these differences, the atmospheric circulation by natural convection has a cycle with four steps similar to that of the Carnot cycle (Fig. 7).

The atmospheric circulation consists of the following four steps:

1. Isothermal Expansion: Solar radiation heats the air on the Earth’s surface. The air expands, becomes less dense, and ascends, with the temperature of the air constant.
2. Adiabatic Expansion: The air, thermally insulated from the surroundings, continues to expand, which causes the air to cool.
3. Isothermal Compression: When the air is lifted aloft to the tropopause, the air radiates heat to the stratosphere. Then the air is compressed, becomes denser, and descends, with the temperature of the air constant.
4. Adiabatic Compression: The air, again thermally insulated from the surroundings, continues to be compressed, which causes the air to warm.

At the last step, the air comes back to the same state as the start of the first step, thus forming a cycle.

The troposphere has not only a vertical temperature gradient but also a horizontal temperature gradient. Owing to the difference in incidence angles of solar radiation, low latitudes are hot and high latitudes are cold. Another difference comes from that of specific heat between sea and land. Thanks to many factors the actual atmosphere makes complicated motion, but roughly speaking, it consists of three representative cells: the Hadley cell, the Ferrel cell, and the Polar cell.

The waste living systems discharge can be converted to heat and, so long as the waste heat is carried through the atmospheric circulation to outer space, we do not have to worry about the environmental problems. The atmospheric circulation is also important to living things in minimizing the difference in temperature of the Earth’s surface. Humans have utilized this heat engine employing windmills, windjammers, and so on before we began wind power generation. What is more important is that the circulation of air causes the circulation of water.

Although there is much water on the surface of the Earth, about 97% of it is mixed with salt. Desalination is difficult, but solar radiation vaporizes and desalts seawater. As moist air rises, the adiabatic expansion cools air and water vapor begins to condense, forming clouds and then precipitation so that fresh water returns to the earth’s surface. It not only desalts seawater but also distributes fresh water widely. The atmospheric circulation, therefore, redresses an imbalance in water as well as an imbalance in temperature.

Ocean circulation is another important circulation of water. Besides the wind-driven horizontal circulation of surface water, there is a density-driven vertical circulation of the ocean water, called thermohaline circulation. It circulates this way. The wind-driven surface currents in the Atlantic Ocean head poleward, get cold, salty, and dense, then flow downhill into the deep water basins at high latitudes, resurface in the Pacific Ocean and come back to the Atlantic Ocean. The thermohaline circulation results from the gap of temperature widened by solar radiation but results in narrowing it.

The wind-driven circulation and the thermohaline circulation are classified as forced convection. The troposphere has natural convection because its bottom is hot and its surface is cold. The ocean has no natural convection because its surface is hot and its bottom is cold unless a submarine volcano erupts. It is still possible to make a heat engine utilizing the vertical temperature gradient of the ocean.

The ocean thermal energy conversion (OTEC) is such a heat engine. The closed-cycle OTEC adopts ammonia or R-134a as working fluids. As they have low boiling points, warm surface seawater can vaporize the fluid and the expanding vapor turns the turbo-generator. Cold water from deep-ocean water condenses the vapor into a liquid, which is then recycled through the closed-cycle system. The open-cycle OTEC directly vaporizes the warm surface seawater at low pressure. OTEC is one of the candidates for renewable energy, but it does not reach the stage of practical application because of low energy efficiency.

The energy efficiency of OTEC is low because we must make the artificial circulation of working fluids. Why then don’t we generate electricity using the existing natural circulation of water? Yes, it has already been put to practical use. Hydroelectric power generation is it. It generates electricity consuming the potential energy of water, which the heat engine of solar radiation brings about through the circulation of water.

### 3.2. The mantle convection and the circulation of mineral nutrition

Another hot reservoir of the Earth as a heat engine is geothermal energy under the crust of the Earth. Its energy has two origins: heat from the decay of radiogenic isotopes, in particular uranium, thorium, and potassium, and heat from the original formation of the planet, especially the giant impact, the collision between the young smaller Earth and a Mars-sized body about 4.5 billion years ago.

According to the measurements of the geoneutrino flux from the Kamioka Liquid-Scintillator Antineutrino Detector, Japan, and from the Borexino detector, Italy, heat from the decay of uranium-238 and thorium-232 amounts to 21TW (plus 4 TW from the decay of potassium-40), about half of the current total heat flux, 44TW [29], while the other half is assumed to be the Earth’s primordial heat supply.

The heat of this sort from the interior of the Earth is the hot reservoir and the Earth’s crust and its outside is the cold reservoir of the heat engine under the ground. The figure below (Fig. 8) depicts the natural heat convection of the mantle.

The Earth’s tectonic plates moving due to the mantle convection trigger volcanism and earthquake. While the useful work the troposphere does for us is not the expansion and compression of the total atmosphere but its internal convection, the useful work the mantle does for us is not its internal convection but the expansion and compression of the mantle, the crustal upheaval, and depression.

You might think that the work of the mantle is rather harmful because they account for volcanism and earthquakes, but in fact, geothermal energy brings us more benefit than damage – more than the benefit of hot springs and geothermal power generation. Were it not for the mantle convection, it would be difficult for living systems to get most of the essential elements, above all, phosphorus and they would be far poorer than they actually are.

The bacterial isolate GFAJ-1 was once thought to incorporate arsenic into their basic biological structures in place of phosphorus, but later it was found that GFAJ-1 lacks the ability to grow in phosphorus-depleted, arsenate-containing medium, that is to say, GFAJ-1 is an arsenate-resistant, but still a phosphate-dependent, bacterium[31]. Hence, we can say no living systems can lack phosphorus. Of course, there are other elements essential to life, nitrogen, potassium, and so on. So, I will use a comprehensive term, mineral nutrition.

As their ions are soluble in water, their solution easily flows from land to sea according to gravity and stays at the bottom of the sea. The Earth’s crust, however, repeats the upheaval and depression, mineral nutrition at the bottom of the sea can be raised above sea level or barren land can be depressed to the bottom of the sea. In this way, the mantle convection has circulated mineral nutrition and prevented the land from drying it up all through geological time.

Even if mantle convection should stop, the circulation of air and water would continue to erode the land and fill up the submarine trench until it would make the Earth’s surface smooth. Suppose the Earth had stopped diastrophism long years ago. In this case, the entire surface would have sunk under the sea and no terrestrial living things could have evolved. Could we expect instead a rich evolution of aquatic life? The answer is no.

The amount of water on the Earth is 1.4×109 km3. Dividing it by the area of the Earth’s surface, 5.1×108 km2, we get the average depth of water 2.7km. If the Earth’s surface were to become smooth, the entire surface would be below the compensation depth (200m) of photosynthesis. Today about 90% of all marine life lives in the photic zone above the depth of 200m. Aquatic life in the euphotic zone can prosper if sufficient mineral nutrition is available there.

Without the coastal upwelling or the eruption of submarine volcanoes, however, the euphotic zone would be lacking it. As a result, the ocean would have two zones, both undesirable for life: the surface zone with sufficient sunlight and insufficient mineral nutrition and the deep zone with sufficient mineral nutrition and insufficient sunlight. Thus the number and the variety of life would be far poorer than they are today.

### 3.3. Two global heat engines that enable living systems

Let me summarize my conclusion. The difference in temperature and a working substance are conditions for a heat engine to do work. The work useful for us can be done in two ways. One is the change of volume of the working substance that hot and cold reservoirs make alternately expand or compress. The other is convection that the change of density resulting from the change of volume causes. The circulation of mineral nutrition is the result of the former work and the circulation of water is the result of the latter work.

The mantle convection and the circulation of mineral nutrition bring us more benefit than damage, even if the convection causes the disaster such as earthquakes and volcanic eruptions, just as the atmospheric convection and the circulation of water bring us more benefit than damage, even if the convection causes the disaster such as hurricanes and torrential rains. The heat inside the earth and heat by solar radiation are two important hot reservoirs of the earth as a heat engine that enables the survival of living systems.

Mars has not evolved as rich life as the Earth has. One of the reasons is that Mars is a poorer heat engine than the Earth. Mars is a tenth as heavy as the Earth. The atmosphere of Mars is very thin because of the weak gravity. It has convection but its scope and effects are limited. Although Mars has mantles, no plate movements due to the mantle convection can be recognized. It used to have volcanic eruptions, but the small planet has gotten cold faster than the Earth. A planet that does not sufficiently function as a heat engine is a dead planet.

## 4. References

Related Work
Annotations
1. Wilhelm Schmidt. Herons von Alexandria Druckwerke und Automatentheater. Literaricon (January 27, 2014). “Pneumatika" Book ΙI, Chapter XI.
2. Emoscopes. “Animation showing the operation of a Newcomen atmospheric engine.” Licensed under CC-BY-SA.
3. 山本義隆. 『熱学思想の史的展開―熱とエントロピー』現代数学社 (January 1, 1987). p. 262.
4. Amédée Guillemin. La vapeur. First published: 1876. Republished: HardPress (June 14, 2018).
5. “Letter from Boulton to Erasmus Darwin 4 Jan. 1790." in The Selected Papers of Boulton and Watt, Vol. 1: The Engine Partnership, 1775-1825. ed. Jennifer Tann. The MIT Press (October 28, 1981). p. 72.
6. James Watt. “1782 Specification of Patent." in James Watt and the steam revolution. ed. Eric Robinson. A. M. Kelley; 1st edition (1969). p. 96ff.
7. “Malgré les travaux de tous genres entrepris sur les machines à feu, malgré l’état satisfaisant où elles sont aujourd’hui parvenues, leur théorie est fort peu avancée, et les essais d’amélioration tentés sur elles sont encore dirigés presque au hasard." Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 6.
8. “La production de la puissance motrice est donc due, dans les machines à vapeur, non à une consommation réelle du calorique, mais à son transport d’un corps chaud à un corps froid, c’est-à-dire à son rétablissement d’équilibre, équilibre supposé rompu par quelque cause que ce soit, par une action chimique, telle que la combustion, ou par toute autre. Nous verrons bientôt que ce principe est applicable à toute machine mise en mouvement par la chaleur. D’après ce principe, il ne suffit pas, pour donner naissance à la puissance motrice, de produire de la chaleur : il faut encore se procurer du froid ; sans lui la chaleur serait inutile." Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 10-11.
9. “La puissance motrice d’une chute d’eau dépend de sa hauteur et de la quantité du liquide ; la puissance motrice de la chaleur dépend aussi de la quantité de calorique employé, et de ce qu’on pourrait nommer, de ce que nous appellerons en effet la hauteur de sa chute, c’est-à-dire de la différence de température des corps entre lesquels se fait l’échange du calorique." Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 28.
10. “la quantité de calorique absorbée ou abandonnée est toujours la même" Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 42.
11. Benjamin Thompson. “An Experimental Enquiry Concerning the Source of the Heat which is Excited by Friction." in The Collected Works of Count Rumford, Volume I: The Nature of Heat. ed. Sanborn C. Brown. p. 22.
12. Nicolas Léonard Sadi Carnot. “Extrait de notes inédites de Sadi Carnot." in Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 92.
13. Nicolas Léonard Sadi Carnot. “Extrait de notes inédites de Sadi Carnot." in Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 93.
14. “La chaleur ne peut évidemment être une cause de mouvement qu’en vertu des changements de volume ou de forme qu’elle fait subir aux corps" Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 14.
15. “Puisque tout rétablissement d’équilibre dans le calorique peut être la cause de la production de la puissance motrice, tout rétablissement d’équilibre qui se fera sans production de cette puissance devra être considéré comme une véritable perte : or, pour peu qu’on y réfléchisse, on s’apercevra que tout changement de température qui n’est pas dû à un changement de volume des corps ne peut être qu’un rétablissement inutile d’équilibre dans le calorique. La condition nécessaire du maximum est donc qu’il ne se fasse dans les corps employés à réaliser la puissance motrice de la chaleur aucun changement de température qui ne soit dû à un changement de volume. Réciproquement, toutes les fois que cette condition sera remplie, le maximum sera atteint." Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 23-24.
16. Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 20-22.
17. “La puissance motrice de la chaleur est indépendante des agents mis en œuvre pour la réaliser ; sa quantité est fixée uniquement par les températures des corps entre lesquels se fait en dernier résultat le transport du calorique." Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 38.
18. Émile Clapeyron. Mémoire sur la puissance motrice de la chaleur. JACQUES GABAY. First published: 1834.
19. James Prescott Joule. “On Matter, Living Force, and Heat." in The Scientific Papers of James Prescott Joule, Volume 1. ed. William Scoresby, Baron William Thomson Kelvin, Baron Lyon Playfair Playfair.
20. William Thomson Baron Kelvin. “An Account of Carnot’s Theory of the Motive Power of Heat – with Numerical Results Deduced from Regnault’s Experiments on Steam." in Mathematical and Physical Papers, Volume 1. p. 119.
21. Rudolf Clausius. Über die Anwendung der mechanischen Wärmetheorie auf die Dampfmaschine. p. 7.
22. Hermann Ludwig Ferdinand von Helmholtz. “Über Integrale der hydrodynamischen Gleichungen, welche den Wirbelbewegungen entsprechen." in Journal für die reine und angewandte Mathematik. Zeitschriftenband 1858. p. 25-55.
23. Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 17. + Dake. “Piston diagrams for thermodynamics.” Licensed under CC-BY-SA and modified by me.
24. Keta. “Carnot cycle." Licensed under CC-BY-SA and modified by me.
25. “Nous avons cependant choisi notre exemple parmi les meilleures machines à vapeur connues. La plupart des autres leur sont bien inférieures. L’ancienne machine de Chaillot, par exemple, élève 20 mètres cubes d’eau à 33 mètres pour 30 kilogrammes de charbon brûlé, ce qui revient à 23 unités de puissance motrice par kilogramme, résultat neuf fois moindre que celui cité ci-dessus, et 180 fois moindre que le maximum théorique." Nicolas Léonard Sadi Carnot. Réflexions sur la puissance motrice du feu et sur les machines propres à développer cette puissance. First published: 1824. p. 117.
26. Van helsing. “Beta Stirling engine animation." Licensed under CC-BY-SA.
27. Zephyris. “Animated scheme of a four stroke internal combustion engine, Otto principle." Licensed under CC-BY-SA.
28. LBMethod.org/jonas. “Simulation of 3D Rayleigh-Bèrnard convection with Rayleigh number 10^4 and Prandtl number 1. Temperature is mapped onto the colors of the spectrum, and streamlines are shown in white." Licensed under CC-BY-SA.
29. Gando, A., et al. “Partial radiogenic heat model for Earth revealed by geoneutrino measurements." Nature geoscience 4.9 (2011): 647.
30. “Below a depth of about 700 km, the descending slab begins to soften and flow, losing its form. Below: Sketch showing convection cells commonly seen in boiling water or soup. This analogy, however, does not take into account the huge differences in the size and the flow rates of these cells.” U.S. Geological Survey. “What drives the plates?.”
31. Erb, Tobias J., et al. “GFAJ-1 is an arsenate-resistant, phosphate-dependent organism." Science 337.6093 (2012): 467-470.