Theory for Closed Systems

Abstract

This chapter describes the fundamentals of thermodynamics focusing on a closed system, through which no matter comes in and out. The discussion starts from the concepts of heat, temperature, and work, and then moves on to the two concepts of energy and entropy. An emphasis is placed on the essential characteristics of energy as “to be conserved” and that of entropy as “the measure quantifying dispersion”. The concept of entropy is introduced with a manner different from most of thermodynamic textbooks so that the readers who are totally new to the science of thermodynamics can grasp the essential image of entropy together with that of energy as easily as possible. After establishing the relationships between heat, work, energy, entropy, Carnot engine, absolute temperature, and ideal-gas temperature, the fundamental equation for exergy balance is given and then how we can proceed exergy calculation is described in terms of heat transfer by long-wavelength radiation, convection, conduction both at steady state and at unsteady state, solar radiation, and finally long-wavelength effective sky radiation. All of them are discussed with numerical examples.

Appendices

Column 4A: Quantifying Things

We use a variety of quantities not only in scientific and engineering discussions but also in everyday life. What is basic and universal in any quantities is to systematize comparing things. Here we confirm how such systematization proceeds.

Let us suppose that there are two small children, whose mother has just prepared two pieces of cake for them as afternoon snack. These children are so small that they do not yet know how to measure the amounts of cake as adults do. The mother passed one piece of cake to the elder son and the other to the younger. She was a bit busy and her mind was still very much occupied by the work she was doing during the afternoon period so that she just happened to cut the pieces of cake not very evenly. The elder son did not look happy with the piece of cake given and tries to take a bit from the piece that his younger brother had. Then a quarrel started····such a situation could easily be observed in an average family.

This indicates that such small children who have not yet learned how to measure the size of things do know how to compare things as an unconscious behavior. Otherwise, they would not start quarreling. Of the most importance is that these children had grown to be able to judge the sizes of cakes by comparison.

One of the most fundamental quantities is length. So, let us discuss further with length as an example. Suppose that we measure the height of two persons who are in the same place. If we are asked to do so, we ask two persons to stand next to each other and take a look at both heads and judge which is taller. This is actually the same as what the elder brother did almost unconsciously in the short story above. Let us call this first step for quantification “direct” comparison.

Suppose next that these two persons are in different rooms and not allowed to move from one room to the other. Then, how can we compare their heights? In this case, if we prepare one string of rope, a wooden stick, or something else to make it as a measure, then we can make a comparison.

Let us suppose that we have a stick clearly longer than both heights. First, we compare one’s height and the length of the stick and thereby mark his height on the stick. Second, we go to the other room where the other person is standing and then compare his height with the mark on the stick. This is called “indirect” comparison, since we use a stick as a measure existing in between the two persons whose heights are to be compared.

If a stick that we find and use is much shorter than the heights of two persons, how can we compare? This can be done as an extension of the idea of “indirect” comparison. What we must do first this time is to mark the length of the short stick in series on the wall surface and count the number of the marks. After the last mark, there is necessarily a distance left shorter than the stick so that we can have the ratio of this shorter distance over the length of the stick. As a whole, we can measure the height of one person by adding the sum of the ratio finally obtained to the product of the number of marks counted and the unit size of the stick used. Now that we have the unit with a short stick, theoretically speaking, we can measure any sizes, long or short.

The process described above, from direct comparison via indirect comparison with a long stick to indirect comparison with a short stick, is universal. In such a manner, all human societies in the past must have determined their own units, respectively. For example, in ancient Egypt, they used the unit called “Stadion”, which is considered to be equivalent to 158–185 m. In Japan, “Shaku” equivalent to 10/33 m was used and in the U.K., “Feet” equivalent to 0.3048 m was used.

Globalization of human societies required the universal unit of length and also other units for easier quantitative communications among the societies. At present, the unit of “metre” defined in the international system of units, which is usually called SI, is used worldwide, as we are familiar with it.

The determination of the unit of “metre” originates from the commemorating work performed by two French scientists, J. B. J. Delambre (1749–1822) and P. F. A. Méchain (1744–1804) in the field of geodetics by the order of then French authority in the late eighteenth century [1]. The idea was to determine the unit of meter with the size of the Earth, as one ten-millionth of the distance between the equator and the Arctic. This is really universal, since all of we humans live on the Earth surface and it is fair to make use of the Earth in order to determine the unit for measuring various lengths of things.

In the contemporary SI units, one meter is defined to be the distance for which the light travels in 1/299792458 s.

Column 4B: Three Dipolar Characteristics of Quantities: Extensive Versus Intensive, Flow Versus State, and Continuous versus Discontinuous

Not only in the field of thermodynamics, but also in other fields of science and engineering, there are a lot of quantities that we have to use. When we encounter with a new quantity, it requires us having some time to digest its meaning by forging a firm link between its qualitative aspect and the quantity itself. Recognition of three kinds of dipolar characteristics described below may help you digest more easily the meaning of whatever new quantities you encounter with.

The first is whether the quantity in question is “extensive” or “intensive”. Most of the fundamental quantities, such as length, area, volume, mass, time, heat, work, energy, entropy, and exergy are all extensive. “Extensive” here implies additive or subtractive. We can easily confirm this characteristic in the quantity of area. For example, suppose that there are two rooms, 20 m² and 15 m² each. We all know that their sum is 35 m² by simply adding each other.

Intensive quantities, on the other hand, are always in the form of ratio with respect to at least two extensive quantities. Let us think about the characteristic of density as an example. The density of air at ordinary room temperature is 1.2 kg/m³. This indicates that each of one cubic meters of space contains 1.2 kg mass of air. This is exactly the ratio of two extensive quantities, mass and volume. In the case of liquid water, its density is 1,000 kg/m³. This means that the molecules of liquid water are much more densely populated in the space of 1 m³ than those of air are.

Temperature is also one of the intensive quantities. You may wonder why, since it does not look intensive for no symbol of division, “/”, in the unit of temperature as either  °C or K. But it is exactly intensive. We can easily confirm this fact as follows. Suppose that there are two bottles of the same size, whose capacity, maximum volume, is 1,000 ml and they contain some amounts of water: one is 250 ml at 50 °C and the other 750 ml at 20 °C. Mixing these two volumes of water pouring water from one bottle to the other results in the water temperature at 27.5 °C, that is exactly calculated as (50 °C × 250 + 20 °C × 750)/(250 + 750). If the water temperature is additive as an extensive quantity, then the result should have been 70 °C, by simply adding 50 °C to 20 °C. Such a result, a higher value of temperature than those of temperature before mixing, never happens spontaneously. Therefore, temperature is one of the intensive quantities. What was described in Sect. 4.1.2 proves formally that absolute temperature is one of the intensive quantities.

As demonstrated in the above paragraph, intensive quantities are featured by their characteristic of multiplication and division, which is in good contrast to that of addition and subtraction in extensive quantities. These characteristics are in fact related to four rules of arithmetic: addition; subtraction; multiplication; and division, which we all learn in primary schools.

Digesting the essence of classical thermodynamics is believed to be not easy and I must confess that it was so, but it must become much easier, and even more fun, if what has been described above, that is the characteristics of extensive and intensive quantities, is kept in mind. This is one of the things that I learned through my exergy research.

The second is whether the quantity in question represents either “flow” or “state”. This is especially helpful for understanding how a system works. It is essential for those who make their own analyses by setting up balance equations of momentum, mass, energy, entropy, and exergy for a system that they focus on.

“Exergy-entropy process” discussed in Chaps. 1 and 2 let us recognize the importance of flow. This is equivalent to what was pointed out by Plato (423–348BC), an ancient Greek philosopher, quoting Heraclitus (535–475BC) by a famous statement: “all things move and nothing remains still” and also by Kamo no Chomei (1153–1216), a Japanese novelist and poet, by a sentence: “The flowing river never stops and yet the water never stays the same”.

Most of the textbooks on thermodynamics deal with various quantities of state such as temperature, pressure, energy, volume, entropy, but work and heat as the quantities of flow are used only in the very beginning just to introduce a closed system and then they tend not to be at focus. The reason for this tendency is that it is recognized that work and heat cannot be dealt with as a kind of quantities of state, in which exact differential forms necessarily exist. This is right in the sense of mathematical formality, but if this let us overlook the importance of work and heat as the quantity of flow, it is wrong.

The tendency that work and heat are not explained well as the quantity of flow in most of the textbooks on thermodynamics may have been due to the fact that the science of thermodynamics was developed very much aiming at clarifying the thermo-physical and -chemical characteristics of a variety of matters, but not at describing how a variety of systems including living systems sustain their respective specific states by the inflow and outflow of work and heat for a closed system, and by those of mass carrying energy and entropy in addition to work and heat for an open system.

All equations to be set up with respect to momentum, mass, and energy become easier to understand if these balance equations are in the form of the quantities of inflow being equal to the sum of a change in the quantity of state and the quantities of outflow. That is expressed in an abbreviated symbolic form as [in] = [stored] + [out].

In addition to distinguishing the quantities of state and flow, there are two more quantities to be considered distinctly. One is the quantity of “generation” in the case of entropy balance and the other the quantity of “consumption” in the case of exergy balance.

Recognition of the quantities of flow, generation, and consumption together with the quantity of state is the key for a better understanding of thermodynamics itself and also its wider application to the built environment and its associated systems.

The third is whether the quantity in question is continuous or discontinuous. The length of something can be at any values in the unit of metre, for example, exactly at 2 m, 567.8 m, 3.452 m, 0.0028739 m, and so on. For this reason the quantity of length is continuous. On the other hand, the number of objects including living creatures is discontinuous. We can count the number of people for example as 3 persons or 4 persons, but not as 3.2 persons in reality, which is possible only in calculation. Similar to this is the currency used in our societies. The amount of money paid or received in reality cannot be any values. For example, in Japan we use Yen and its smallest amount in the coins used at present is 1 Yen. Therefore, for example, we can pay or receive either 450 Yen or 451 Yen, but not 450.5 Yen. The same applies to Euro. In the currency of Euro, the smallest amount is assigned to 1/100 of 1 Euro called 1 cent. Therefore, we can pay or receive 1.55 Euro or 1.56 Euro, but not 1.555 Euro. In this sense, the quantity of currency is discontinuous.

Whether something is regarded to be continuous or discontinuous is, I think, also related to the question whether the name of an object in question is called as a corresponding countable noun or uncountable noun. For example, bread is uncountable so that we need to say a piece of bread, while on the other hand, for example, spoon is countable so that we can say two spoons. For countable objects, we ask how many, but for uncountable, we ask how much. There are few or many in the case of countable objects, while on the other hand, little or much in the case of uncountable objects. The number of countable objects can be of course counted, but the amount of uncountable objects cannot, but has to be measured by their respective appropriate measures.

Since the ancient Greek period till the early twentieth century, there were continual discussions made by philosophers and scientists on the existence of tiny building blocks that compose of all things in nature including living creatures. This is exactly the question whether things are continuous or discontinuous. Quantities such as length and time are continuous. So must be mass. We can measure mass of water almost at any values, for example, 200 g, 95.34 g, 0.031805 g, and so on. It looks self-evident that water is continuous. But, in fact, water is composed of tiny building blocks, each of which is called water molecule that is the assembly of two hydrogen atoms and one oxygen atom.

This was found by a series of theoretical considerations and also careful experiments made by eminent scientists such as Planck, Perrin, Einstein, and others who finally proved that all things are made of molecules or atoms as tiny building blocks. Atomic and molecular sizes are so tiny that no visible error is induced by measuring the amounts of objects with the quantity of mass or volume instead of counting the number of atoms and molecules. In 1 g of water, there exist an enormous number of molecules that is about 3.33 × 10²².

It may not be so hard to accept that things having mass consist of such tiny building blocks, but we also have to accept that energy has a similar feature to mass. There was no doubt that energy is one of the continuous quantities until the turn of the century from nineteenth to twentieth, when Planck happened to discover the quantum nature of energy carried by radiation in the course of his theoretical consideration on the measured spectral distribution of high-temperature radiation available from the window of a furnace. Since the radiation was theoretically and experimentally confirmed to be a kind of wave very clearly until then, it was not easy for even capable scientists including Planck himself to be convinced that radiation has a particle-like characteristic. Einstein then gave revolutionary proof that radiation is composed of tiny building block to be called photons, each of which carry energy, whose amount is exactly proportional to the frequency of radiation. It should be noted that his proof was made by applying thermodynamic consideration to photon gas in an enclosure as a closed system.

For ordinary temperature range, discontinuous nature does not look obvious and there is no problem to deal with the quantity of energy as continuous, but for extremely high and low temperature ranges, it becomes dominant so that we need to take it into consideration.

Why we are suntanned at beach or at high up in mountain areas is due to such a particle-like characteristic of ultra-violet range of solar radiation, which amounts to only 4 % of the whole energy carried by solar radiation. Why we can see stars at dark night in spite of the fact that the amount of light available from those stars is so scant is due also to the particle-like characteristic of visible range of radiation coming from those stars existing so far away in Universe. The heat capacity of matters in ordinary temperature range is rather constant, but toward the temperature of matter approaching to the absolute zero, the lower the temperature of matters is, the lower also their heat capacity toward zero. This can also be explained with the consideration of discontinuous nature of energy.

The above discussion on whether it is continuous or discontinuous may also be extended to other varieties of thoughts in science and art. Some examples are as follows: the relationships between exact differential and finite differential equations; infinite decimals and corresponding fractions; algebra and geometry; musical sound and musical note; circulation and blood vascular system; mind and nervous system; and function (kata) and structure (katachi), the last of which was introduced in the beginning of Chap. 3. All of them must be related to how human brains work.

Column 4C: A Short Story on the Science of Thermal Radiation

The set of three concepts, energy, entropy, and absolute temperature forms the foundation of thermodynamics. At first glance, thermodynamics may not look having a relation to electrodynamics, but, their combination led to having the concept of radiant entropy and furthermore the quantum nature of energy. The fourth-power law of radiant energy together with the third-power law of radiant entropy were developed by merging electrodynamics and thermodynamics together.

This fact is not described in most textbooks on heat transfer, although the fourth-power law of energy is briefly described to let the readers perform the calculation of radiant heat transfer. Such a way of description is not enough, I thought, especially in order to derive the concept of radiant exergy, with a sufficient confidence, in relations not only to long-wavelength thermal radiation but also to short-wavelength thermal radiation. Therefore, I decided to come back once to the very basic of explanation given by Planck (1912) on radiant energy and entropy, although it took quite a while and required a lot of patience to digest the essence due to its theoretical formality.

What follows here is a brief introduction of what I learned from that study. In fact it happened to give me a chance to learn the history of thermal science and also to confirm the importance of application well woven with the threads of fundamentals.

As the first step, we must review the energy balance equation of a closed system, in which there are the input and output of energy by work and heat, but not those of matter. Focusing on the fact that energy is basically a quantity of state, while on the other hand, work and heat are quantities of flow, we try to change the way of expression of energy balance equation, using the following relationships: the work can be rewritten as the product of pressure and an infinitesimal increase of volume, and the heat as the product of absolute temperature and an infinitesimal increase of entropy. Their substitution to the energy balance equation of a closed system yields the energy balance equation in a very beautiful form that all terms are expressed only by the quantities of state (See Eq. (4.36)).

This closed-system energy balance equation is combined with another equation representing the relationship between pressure, volume, and temperature, that is the characteristic equation of, for example, a rather low-density gas in the closed system, and thereby we reach the equation for the entropy of the closed system as a function of temperature and pressure. The equation so far we have reached allows us to know that the entropy as a quantity of state increases as the temperature increases or as the volume increases. This suggests that the entropy disposal is essential in realizing the dynamic equilibrium of a system.

The next step is to apply the above-mentioned procedure in terms of a closed system to the derivation of the fourth-power law of radiant energy, the third-power law of radiant entropy, and thereby also to the derivation of radiant exergy.

Suppose a closed system, in which no matter is contained, that is in vacuum. Such a system may look nonsense to be discussed, but it is to be discussed in relation to electromagnetic radiation, applying the set of four equations on electromagnetism established successfully and beautifully by Maxwell (1865) based on the experimental evidence found by Coulomb, Ørsted, Faraday, Ampere, and others.

Maxwell’s equations are in fact based on the conservation laws of electric charge, momentum, and energy, so that even in the vacuum closed system, there are a certain electromagnetic pressure exerted on the internal surface of the closed system and also a certain amount of electromagnetic energy held by the vacuum space of the closed system.

From that point on, assuming that the internal surface of the closed system is perfectly white, 100 % of reflectance, the momentum balance equation at the internal surface can be set up together with the relationships between solid angular spectral characteristics of electromagnetic radiation and the speed of light. This brings about the final result that the electromagnetic pressure equals one-third of the density of electromagnetic energy contained by the closed system.

Substitution of this relationship into the energy balance equation of the closed system expressed only by the quantities of state mentioned above results in having the formulae to be expected, that is, the whole of radiant energy proportional to the fourth-power of absolute temperature, the whole of radiant entropy proportional to the third-power of absolute temperature, and the proportional constant for entropy to be 4/3 of that for energy. This theoretical work was done first by Boltzmann in 1884.

The purpose of Boltzmann’s work was to explain the experimental evidence found by Stephan et al. by the end of 1879 with a then precise measurement, which showed that the radiant energy emitted from a high-temperature furnace was proportional to the fourth-power of absolute temperature of the matter inside the furnace. Those days, the value of the proportional constant had to be empirically determined, since Planck’s equation was not yet discovered by then, and it was estimated to be 5.2–5.5 W/(m²K⁴).

The technology for measuring the characteristics of thermal radiation developed further very much and Weber et al. (1888) found the following characteristics of spectral distribution of radiant energy: the wavelength, at which the spectral rate of energy becomes the maximum, shifts from long to short, as the source temperature increases; the product of the wavelength, at which the maximum spectral energy is available, and the corresponding absolute temperature of the source is always constant.

Wien (1893) made an effort to explain this experimental facts by combining the theory of electrodynamics and optics together with thermodynamics and reached a conclusion that the spectral rate of energy should be expressed as a function of the above-mentioned product of the wavelength and the absolute temperature as a constant and the integration of this function must include the fourth-power and the absolute temperature of the source, although the exact form of this function was not found. Commemorating the theoretical work done by Wien, the product of the wavelength and the absolute temperature of the radiant source giving the maximum spectral rate of energy is called Wien’s displacement law.

Late nineteenth century, quite a few leading scientists including Kelvin were skeptical about combining thermodynamics and electrodynamics. This view must have been caused by the difficulty of imagining a vacuum closed system containing energy and entropy carried solely by electromagnetic radiation. But the fact that the fourth-power law and also the displacement law were derived successfully proved the legitimacy of combining thermodynamics and electrodynamics. This was further confirmed by Einstein (1905) through his explanation of photon particles.

Wien continued his theoretical struggle further and tried to make a formula giving the spectral rate of energy of thermal radiation by applying the statistical distribution of molecular motions in relation to the heat capacity of a closed system containing a gas. The statistical theory used by Wien was originally developed by Maxwell and Boltzmann. During almost the same period, Rayleigh and Jeans also made a trial of making a formula assuming the state of electromagnetic standing waves in a vacuum closed system. Wien’s and Rayleigh-Jeans’ formulae fitted the opposite portions of the spectral rate of thermal radiation, respectively, but they never fitted the whole of measured spectral-distribution curve.

Planck (1900) made an intensive effort to forge a bridge between these two formulae and finally succeeded in establishing so-called Planck’s equation by assuming, rather unwillingly himself, that energy carried by thermal radiation does not have continuous nature, but discontinuous instead, that is quanta. Planck’s equation (See Eq. (4.49)) fitted very well the whole of spectral distribution of thermal radiation.

The integration of Planck’s equation for all wavelengths from zero to infinity gives theoretically the exact equation for the proportional constant appeared in fourth-power law described above. It was expressed by two integers: 2 and 15, π number, Boltzmann constant, the speed of light, and Planck constant. Precise measurement of thermal radiation in terms of the whole amount of energy and also of spectral distribution can determine experimentally the values of the proportional constant appeared in the fourth-power law and also the constant appeared in Wien’s displacement law.

Having these two values determined experimentally, the equations of the fourth-power proportional constant and the displacement law become a set of simultaneous equations for two unknown variables: Boltzmann constant and Planck constant. The results using a set of measured data available by contemporary accurate measurement are Boltzmann constant to be 1.381 × 10⁻²³ J/K, and Planck constant to be 6.626 × 10⁻³⁴ Js. Using these two values, the corresponding proportional constant of the fourth-power law turns out to be 5.676 × 10⁻⁸ W/(m²K⁴), which appears in most textbooks dealing with heat transfer.

Later, Einstein investigated theoretically the relationship between an increase in radiant entropy and volumetric expansion of a closed system and found that the energy carried by photon particles is expressed as the product of Planck constant and the frequency of electromagnetic radiation. With this as the re-starting point, Einstein revealed that it is possible to derive the momentum of photon particles and thereby derive their pressure exerted on the interior surface of the closed system. The result was exactly consistent with the electromagnetic pressure derived by combining electrodynamics and thermodynamics. This again proves the legitimacy of applying the thermodynamics to electromagnetic phenomena, while at the same time, proves the presence of photon particles.

Boltzmann constant is given as the ratio of so-called gas constant, 8.314 J/(mol K) to Avogadro number, which is the number of atomic particles existing in 22.4 liter of a low-pressure gas at 0 °C, so that if the value of Boltzmann constant is given from Planck’s equation together with Wien’s displacement law as mentioned above, then Avogadro number can be calculated. The result is 6.02 × 10²³. This value agrees very well with the other value obtained from the theoretical and experimental investigation of Brownian movement of fine particles, which are much greater than the size of single atomic or simple molecular particles.

Whether atomic particles really exist in nature or they are virtual tiny objects for making easier a variety of physical and chemical thoughts had been one of the very fundamental problems to be answered since the ancient Greek civilization, but it was finally proven in the beginning of twentieth century that atomic particles do exist in the course mentioned above.

Now that we have come to know how the concepts of radiant energy and entropy were developed by the early years of twentieth century, we no longer need to be suspicious or uncomfortable in using the relationship between the spectral energy, entropy, and absolute temperature of thermal radiation.

With this relationship in mind, we can calculate the values of spectral entropy from the spectral energy value and its corresponding radiant temperature to be derived from Planck’s equation. Energy, radiant temperature, and entropy values of arbitrary light sources including fluorescent or light-emitting-diode lamps were obtained in this manner for exergy calculation, whose results were discussed in Chap. 3.

For arbitrary long-wavelength radiation, the same procedure as for short-wavelength radiation can be applied to exergy calculation, but most materials such as building materials are quite close to black- or gray-body, whose emission of thermal radiation is uniformly proportional in any wavelength to black-body radiation given by Planck’s equation. For most building materials, the procedure to be taken is what was described in Sect. 4.4.2-a).