2.1.1 The Concept of Energy

We may never be actually able to observe the quantity named “energy,” but its effects are certainly and easily observable. Perhaps this also adds to the mystery of energy. Lindsay and Margenau presented a timeless review of the history of physical concepts in their book, Foundations of Physics (1936). Though dated, this text presents a very comprehensive treatment of basic concepts both in classical and quantum mechanics. Since we have seen little to compare with this and we don’t wish to compete with their treatment of mechanics, we quote their pointed and well-said statements:

All the problems of classical mechanics can be solved without reference to it (energy). The question at once arises: why then should it have been introduced at all? This is what we wish to discuss. We must first remark, however, that the idea if energy is historically much older than the name. Without doubt it goes back at least to Galileo and his observation with respect to machines that “what is gained in power is lost in speed,” referring to the fact that the force required to lift a weight (by means of a pulley system) multiplied by the distance through which the force has to be applied remains constant, though either factor in itself may vary. The concept of work is involved here. Its importance was, however, overshadowed by Galileo’s epoch-making discoveries of the laws of motion, and it was not until the time of Huyghens that it again became prominent in the concept of “vis viva” or “living force”, i.e., a quantity varying as the mass multiplied by the square of the velocity. The attribution of the term energy to the concept of “vis viva” did not come until the 19th century. (1936)

The origin of the word energy is Greek, and it means active, or the capacity to do work. In more recent times, the idea has taken hold that energy is some ethereal form of substance. In quantum and relativistic mechanics, energy is interchangeable with mass (matter) as indicated, for example, in the celebrated equation from Einstein’s Relativity Theory, E = mc².

Kinetic energy remains kinetic energy when, for example, a moving mass slows down by frictional losses. The collective kinetic energy of the mass is disbursed as kinetic energy (motion) of many more, smaller particles (molecules) and measured as a temperature rise. However, kinetic energy can be converted or swapped for potential energy when, for example, a mass loses velocity while slowing down as it moves against an attractive field of force such as gravity.

In more recent times, the idea has taken hold that energy is a substance. For those who wish to acquire a detailed knowledge of the history of the concept of energy and force, including the evolution of these concepts over the centuries, the books Concepts of Mass and Concepts of Force by Max Jammer would be helpful.

As far as we know, consistent with our observations, energy cannot be created or destroyed. We will ignore, in this book, the processes wherein energy and matter are equivalent, i.e., that matter can be transformed into energy and vice versa because all the phenomena associated with our lives do not occur at the atomic nuclear level. The availability of energy to do work can and does change. We can only transform energy from one of its two forms into the other, when possible, to suit our practical purposes, such as from kinetic energy to potential energy and back to kinetic. Ultimately, all these transformations resolve in an increase of entropy (disorder) in the universe and, therefore, an increase in world temperature. When all things are finally at the same temperature, there is no more useful energy.

Another penetrating and early analysis of the subjects of energy (kinetic and potential), mass, and force was put forth by James Clerk Maxwell, author of the electromagnetic theory of radiation, among other profound and basic contributions to physics. In his preface to Matter and Motion, he states:

Physical science, which up to the end of the eighteenth century had been fully occupied in forming a conception of natural phenomena as the result of forces acting between one body and another, has now fairly entered on the next stage of progress – that in which the energy of a material system is conceived as determined by the configuration and motion of that system, and in which the ideas of configuration, motion, and force are generalized to the utmost extent warranted by their physical definitions. (1877)

What is energy? It is not observable in the classical sense. Theoretical physicists have interminably debated its ability to be observed and have variously defined the concept over many years. First, it is necessary to identify or define what we mean by an “observable quantity.” Interestingly, there are few directly observable. The rest of these concepts, such as energy, momentum, temperature, etc., are inferred by various experiments and indirect observations. What we call momentum is calculated and seen as the force necessary to change the momentum of a body.

We are using the word “observable” in a rather sophisticated sense. It is necessary to divorce our ideas of what we directly experience from those of theories intended to explain the process or sensation. Force, a push or pull, is a directly observable magnitude, as are the velocity of a body in motion, pressure, volume, length, temperature, and even quantity of heat. In reality, the subject of what is legitimately observable is highly debatable. Not only are the semantics of the subject in question, but we must also identify the level of primitiveness or experience of the individual making the observation. However, such quantities as entropy and enthalpy in thermodynamics are not directly observable.

Implicitly, we think of energy as being some sort of substance or quality that enables one to overcome an opposing force and move an object to a different position spatially in opposing this force.

Examples of certain non-observables are electric and magnetic fields. We can observe their effects, but we are not able to see or observe the field directly. In the case of a magnetic field, the attraction force of a magnet for iron particles can be seen or experienced directly, but not the field itself. In fact, the magnetic field is an invention by physicists to structure a logical and working explanation of that observation. Similarly, we cannot experience an electric field directly, but surely the consequence of such a field is seen or felt when a statically electrified insulator, such as a rod of glass or plastic, gathers bits of fur or paper.

The dilemma we tend to fall into is the result of familiarity with the concepts and objects that we think we understand but unfortunately do not. The idea of energy in any form is very complex and sophisticated. Sometimes energy is described as having many different forms – chemical and mechanical energy and perhaps thermal energy as temperature. In actuality these are superficial manifestations of the same thing. We need to distinguish basic ideas and concepts from those involving superficial property assessments.

In deference to undue involvement in philosophic issues, let us take a deeper look into the matter of observables, dimensions, and measurable quantities in the physical sciences. As always, along with the benefits reaped from greater numbers of tools and concept evolution in physics, the more likely we become separated from basic contact with the outside, physical world. The tendency is to become too involved with word pictures and symbols, and then it becomes increasingly difficult to separate the reality that we “directly” perceive from the physical models generated in our minds and placed there by myriad conversations and teachings in physical science.

Returning to the subject of energy, for example, when we see a large mass such as an automobile moving at speed of, say, 60 mph, we might remark upon the large amount of kinetic energy the automobile has, especially if it collides with a rigid object and suffers severe damage. In this case, as in all others, we do not really make a direct observation of its “energy.” In fact, it is not possible to see or experience energy as such, only the consequences of its transference.

In order to make sense out of our observations, we simply invent concepts and weave a fabric of theories, hypotheses, and other explanatory structures. In this manner, we are attempting to not only acquire a better “understanding” of the world around us but also to be able to predict the results of our actions, and perhaps even control their outcome by making use of these concepts.

Now, let’s take a critical (analytical) look at the main subject of this book – energy. We cannot make direct measurements of this rather elusive substance. Only the effects or results of energy can be observed and measured. We cannot hold energy in our hands as we can grasp material substances such as a stone or stick of wood, nor can we see it as directly as we see wood, that same stone, or a source of colored light. Energy makes no sound that we can perceive. In fact, energy is more of a concept in classical physics than a “substance,” as in a material object. The idea of energy existing is mostly to explain motions and activities in life. If someone is very active and has great strength, we tend to say that the individual has a lot of energy, implying that he or she is in possession of a larger quantity of some substance that enables him to perform tasks greater or faster than normal. When an explosion takes place we think that a great amount of energy is released. Exactly what do we mean? Large forces are manifest when an explosion takes place, and we see solid objects torn apart and many particles of matter thrown away from the center of the explosion with great velocities. Knowing that large forces operating over very short periods of time are required to do this, the conclusion is that this large amount of stored (potential) energy is released in the form of much action and motion.

2.1.2 Kinetic Energy

The quantity we call kinetic energy implies motion (kinetic from the Greek signifying movement). It is the energy associated with physical bodies in motion. Potential energy is the latent ability to do work. Work and energy are essentially synonymous as employed in mechanics. However, doing work implies action.

Let us get a bit more involved in the details, units, and ways of measuring energy, always keeping in mind that we quantify on the basis of the results of transformations of what we refer to as energy.

Pushing on a body to move it against frictional resistance forces and the force of gravity are too familiar to all of us in our everyday experiences. The force we must exert through our leg muscles to climb a flight of stairs is frequently experienced and readily comprehensible.

The term “work” may have originated in times past because of the idea that effort had to be exerted to do the common tasks necessary to life. The amount of energy involved in lifting weights to certain heights is also defined as work. One can appreciate how we probably acquired the terminology and the idea that energy, or work, is defined as force times distance. In other words, the amount of energy required or produced is the product of the distance through which a force is acting times the force. This is simply expressed as

(2.1)

The above equation assumes that the force is constant throughout the distance it is acting. Before getting into the more quantitative aspects of energy, we should examine the basic units of measurement that are employed. That subject is usually referred to as dimension theory or dimensional analysis. Despite the seemingly abundant, and perhaps unlimited, number of phenomena and effects in nature ranging from electric, mechanical, thermal, magnetic to hydraulic, gravitational, and more, there is only a limited number of basic units that we must use in analyzing, measuring, and calculating these different processes.

Most processes can be described mathematically in terms of four basic parameters. Along with the most commonly used units, they include the following:

Dimension – Units

1. Mass (M) – grams, slugs
2. Length (L) – meters, feet
3. Time (t) – seconds
4. Electric charge (q) – coulombs

Let’s quickly explore the meaning of what has just been stated. Later in this book, we will return to the subject of dimensions when we pursue the subjects of energy conversion and use. All physical (mechanical) processes and properties can be described in terms of mass, length, and time. If we include the electric field, magnetic field, properties of conductance, permeability, electric current, etc., in our catalog of phenomena, then we need to add the fourth dimension, electric charge, to the list.

Units of measurement are necessary to quantify these parameters. There are many forms of the units in which energy is measured. Depending on whether the English or metric system is employed, the most common units are as follows.

In Equation (2.1) above we defined energy or work as the product of force multiplied by distance. The units of force (F) are found, from its definition, as the product of mass times acceleration, or

(2.2)

where M is mass, and a is acceleration, as illustrated in Figure 2.1.

Acceleration is the rate of change in velocity of a body (mass). Since velocity, v, is expressed as the rate of change in distance versus time, its units are v = L/t. The rate of change in velocity with time, similarly, is v/t = L/t².

We experience numerous types of forces. Among them are the forces due to gravitational attraction, attraction and repulsion of electric charges in an electric field, and magnetic attraction and repulsion forces. Somewhere in the development of these observations and concepts, the decision was made that all these forces are dimensionally the same. In order to remain consistent in the logic of mathematical representation, all types of forces have the dimensions of ML/t². How did this occur?

Mechanical force as mass times acceleration was probably first proposed by Newton in his Principia Mathematica as part of his Laws of Motion. Maintaining that a body in motion will remain in motion and that a body at rest will remain at rest unless acted upon by an external force seemed a reasonable and almost intuitive proposition after its statement. Unfortunately, the statement itself provides no method for measuring this force, and its utility is limited. In order for any postulate regarding physical interactions to be useful, there must be some way to quantify the results and to accurately predict future results given enough facts or data.

It seems that Newton may have felt that the idea of force was intuitively known and did not need further explanation. Even though most of us proceed through life without being bothered by much thought of the concept of force, that sentiment does not pervade all of theoretical physics. The ideas of force and energy are still subjects of great conjecture and debate.

The history of the development of physical concepts is not the prime concern here, but some knowledge of their evolution does serve to bring more closely to our attention and scrutiny a better appreciation of terms that we employ daily. Sometimes it is necessary to begin understanding or developing a body of knowledge in order to make certain basic assumptions on an entirely intuitive basis. As scientifically unsatisfying as that may be, it is unavoidable at times. One could draw a weak comparison to plane geometry (Euclid) with regard to its various axioms and the declaration that parallel lines never meet. Even the concept of straight lines is rather intuitive in nature.

Perhaps the best definition is that a force is required to change the motion of a body. Many problems arise in finding acceptable definitions for the basic parameters of physical science, namely, the abstract concepts of mass, time, force, and energy. However, we must learn to be satisfied with definitions that leave something to be desired in order to move on toward generating a working body of mechanics that enables us to design and build practical devices that serve our purposes.

An interesting definition of energy comes from the Grolier Encyclopedia, which states:

Energy can be measured in terms of mechanical work, but because not all forms of energy can be converted into useful work, it is more precise to say that the energy of a system changes by an amount equal to the net work done on the system … In classical physics, energy, like work, is considered a scalar quantity; the units of energy are the same as those of work. These units may be ergs, joules, watt-hours, foot-pounds, or foot-poundals, depending on the system of units being used. In modern science, energy and the three components of linear momentum are thought of as different aspects of a single four-dimensional vector quantity, much as time is considered to be one aspect of the four-dimensional space-time continuum … Energy exists in many different forms. The form that bodies in motion possess is called kinetic energy. Energy may be stored in the form of potential energy, as it is in a compressed spring. Chemical systems possess internal energy, which can be converted by various devices into useful work; for example, a fuel such as gasoline can be burned in an engine to propel a vehicle. Heat energy may be absorbed or released when the internal energy of a system changes while work is done on or by the system. (1993)

The force of gravity on ponderable bodies that have the quality or property of mass is given as

(2.3)

From experience we learned that the larger (more massive) the body the greater the force needed to accelerate that body to the same velocity on a smooth, low friction surface. The rationale of defining the mass and force is rather circuitous because we employ the same phenomena to assess each parameter. In determining the mass of an object, its weight is used. Hence, the more a body weighs the proportionately greater is its mass. The property of a body’s mass to resist being moved or accelerated is known as inertia. Thus, a mass that is acted upon by a gravitational force has a weight directly proportional to its mass, as illustrated in Figure 2.2. Since all bodies fall in a constant gravitational field with the same acceleration (Galileo), the gravitational and inertial masses are declared to be one and the same. This principle of equivalence explains why all bodies, when acted upon by gravity and permitted to fall freely with no opposing forces, will experience the same acceleration. Hence, the velocities they attain over the same vertical distance will be equal.

The force acting upon a mass with a gravitational mass, M_g is

(2.4)

The inertial mass, M_i, which is accelerated by some force, F, described in Equation 2.2 is expressed as

(2.5)

More properly, force is defined as directly proportional to the rate of change in momentum of a body, or

(2.6)

With this assumption of equivalence, or M_i = M_g, we arrive at a means of measuring force in a quantitative manner. A mass of 1 kg is thought to have a force of 1 kg times 9.80 m/sec², or 1 Newton. In the “c-g-s” system (centimeter-gram-sec), 1 gram of mass has a force of 1 dyne exerted upon it by gravity at the Earth’s surface. The acceleration constant of gravity in the metric system of measurements is 980 cm/sec².

In the English system of units, the acceleration, g, is 32 ft/sec², and in the metric system g = 980 cm/sec². By inspection we see that the unit of force, as defined above, is

(2.7)

Now, is that true for all types of forces, including electrical and magnetic? The answer apparently is yes. As the need arises in the development of our discussion of energy, we will explore each of these types of forces.

Returning again to the main topic, we will see how the idea of energy is explored and how a useable and quantitative definition is generated. The formulation of relationships in physics involves a lot of mathematical trickery. Some of these manipulations might even appear at times to employ the practice of self-deception in order to arrive at the desired answers to posed questions.

The derivation of the very familiar expression for kinetic energy, 1/2 mv², is interesting. How is it that the energy of a moving body with mass, m, is the product of its momentum, mv? Plus, where does the factor ½ come from?

Look again at how we have defined the idea of energy, or work, in Equation (2.1) as the scalar product of force times the distance. Without questioning further at this point how we have justified this leap of confidence, the next step is to quantify the idea.

If force can be defined as the product of the mass of a body times the acceleration produced by that force acting upon the body, as in equation (2.2), then an increment, dE, in energy, E, in moving a small distance, dx, can be represented as

(2.8)

where acceleration, a, is represented as the second derivative of distance with respect to time.

So far so good, if we accept the force times distance premise. It is now necessary to integrate distance in order to obtain an expression for a finite amount of energy produced by a force, constant or otherwise, moving through a distance, x. Thus,

(2.9)

Since acceleration is the first derivative of velocity, v, we substitute dv/dt for acceleration in equation (2.8) and obtain

(2.10)

If we make the dt term the denominator under dx, we obtain another velocity term under the integral sign and have the following upon integration:

(2.11)

which is the well-accepted formula for the instantaneous kinetic energy of a moving body.

An example of a device that stores kinetic energy is the flywheel (Figure 2.3). We can think of a flywheel as a kinetic battery. Flywheels offer advantages of high reliability and excellent cycle life, but are not suitable for small portable applications. The modern flywheel rotors are often made of composite materials for optimum strength and density properties, and the bearings are superconducting and electromagnetic to minimize friction.

Flywheels store rotational kinetic energy (U_rot) expressed as

(2.12)

where I is the moment of inertia (kg m²), and ω is the rotational velocity (rad/s). For a hollow cylinder or disc, the moment of inertia is equal to the mass times the radius r squared (I=mr²). For a solid cylinder, the moment of inertia is I=(1/2) mr². The speed of a modern high-speed fly-wheel with electromagnetic bearing can reach as high as 100,000 rpm in vacuum.

2.1.3 Gravitational Potential Energy

The problem of defining energy has been attacked in a rather pragmatic fashion. Perhaps, we can approach this concept from a different perspective. For many thousands of years, we have known that lifting weights requires doing what we generally now call “work”, and that weight lifting can be converted into other useful forms of work, such as turning a paddle-wheel, grinding grain, and powering woodworking tools. The energy stored after lifting a weight is referred to as gravitational potential energy.

Many thousands of years ago, early humans harnessed this energy by placing heavy stones on high altitudes to be later released for survival and defense. The modern examples of the potential gravitational energy usage include the clock pendulum, water behind a dam, swings, and rollercoasters.

To quantify the gravitational potential energy, we must identify relevant parameters including height and mass of the object. We calculate the gravitational potential energy (U) stored in an object, located at a certain position above the ground, by multiplying the mass of the object (m) by gravity (g) times the height (h) expressed as

(2.13)

where U is in Joules, m is in Kg, g is in m/s², and h is in meters.

2.1.4 Elastic Potential Energy

Humans have been using uncoiled mechanical springs in bows and arrows for thousands of years, for hunting and survival. When using bows and arrows, we are in fact harnessing the elastic potential energy that was stored in the unoiled springs. As time passed, upon the emergence of coiled springs, the usage of mechanical springs expanded to many common applications including watches.

Imagine, as you twist the knob on a watch, you are causing the spring to coil, and thus, you are powering the watch with elastic potential energy. The watch would then uncoil, gradually in time, during the operation and thus, release the stored energy. Figure 2.4 shows the schematics of the coiled springs in an Elgin watch that was commonly used from 1864 to 1968.

To quantify the potential energy stored in elastic coiled structures such as a spring, we express the elastic energy U as

(2.14)

where k is the spring constant, and x is the displacement of the spring.

For deformable bodies that can undergo deformation d or strain ε (the derivative of displacement with respect to position), the linear elastic strain energy (U) and strain energy density (u) (i.e. energy per volume), can be calculated as

(2.15)

(2.16)

where, F is the force on the body, k is the material stiffness (or spring constant), σ is the stress in the body, and E is the elastic modulus.

The strain energy density in one dimension can be converted to 2D or 3D based on the actual state of stresses and strains in the body. For example, if a body is subjected to plane stress (i.e., the normal and shear stresses in a plane are non-zero), the strain energy density (u) is expressed as

(2.17)

where x and y refer to the directions of the normal stresses and strains, and τ_xy is the γ_xy are the shear stress and strain, respectively, in the xy plane.

2.3.1 Nucleosynthesis and the Origin of Elements

When the first atoms were formed during the Big Bang, energy was stored in the chemical elements that are now prevalent in our modern stars, planets and cosmic bodies. Thus, chemical energy storage is as old as our universe. The age of the universe is estimated to be about 13.8 billion years based on the mapping of the universe conducted by Planck European Space Agency mission. The age of the universe and the major events in the universe are illustrated in Figure 2.6. Our universe is believed to have started (t=0 s) with the Big Bang event that was followed immediately by a rapid expansion referred to as the inflation (t=10^-32 s). Next, the protons began to form (t=10^-6 s) followed by nuclear fusion (t=10^-2 s to t=180 s).

The first atoms that formed during the Big Bang nucleosynthesis are known as the “light elements” consisting of Deuterium (²H), Tritium (³H), Helium-3 (³He), Helium-4 (⁴He), Lithium-7 (⁷Li) and Baryon-7 (⁷Be). Some of the primary nucleosynthesis reactions during the Big Bang are the following:

(2.28)

where p is for proton, n is for neutron, and γ is the photon gamma rays. The number on the top left of the element represents the number of neutrons in the isotope. For example, ²H is Deuterium or a hydrogen isotope with 2 neutrons, and ³H is Tritium or a hydrogen isotope with 3 neutrons. The formation of Helium-4 and Lithium-7 are also depicted in Figure 2.7.

The explosion of a star, known as Supernova, has led to the creation of several other relatively heavier elements. These elements include carbon, nitrogen, oxygen, neon, and sulfur, which are heavier than the “light elements” such as helium and lithium. Supernova is the last stage of a star’s life. The layered structure of a red giant star that is close to supernova event is shown in Figure 2.8. Each layer in the exploding star, contains specific elements, and the outer layers have relatively lighter elements compared to the inner layers of the star.

The chemical elements found in our bodies and on our planet can be all traced back to the stars and the Big Bang. It may be interesting to think of humans as star-based beings, composed of elements that are billions of years old and came from the stars. With this new perspective, let us take a look at the specific sources for different elements.

Jennifer Johnson, an observation astronomer at Ohio State University, has provided updated sources of chemical elements for our solar system. Figure 2.9 summarizes these sources based on Johnson’s work, and the previous investigation by researchers at Meteorite Laboratory at North Arizona University. The chemical elements in our solar system can be traced back specifically to the Big Bang fusion, cosmic rays fission, merging neutron stars, dying small (low mass) stars, exploding large (high mass) stars, and exploding white dwarfs. The human-made elements are also included. As astronomers collect more data and expand our understanding of the stellar events, more specific sources may be identified.

The investigation of the rate of expansion of the universe has led to some very interesting findings. To put it in simple terms, it appears that gravity alone cannot fully explain our observations of the universe. Therefore, scientists have proposed the presence of a matter and energy, known as the “Dark Matter” and “Dark Energy” that may have contributed to the rather puzzling behavior of the galaxy and the universe.

The dark matter is believed to be not directly observable, and essentially, it cannot emit light or interact with the electromagnetic radiation, hence, the reference to the term “dark”. Although not observable, the dark matter and dark energy are believed to cover most of the modern universe, more than 95%, as illustrated in Figure 2.10 (a). The composition in thepie chart is based on the results obtained by the Wilkinson Microwave Anisotropy Probe (WMAP) at NASA. “Dark energy” can be considered as the source of anti-gravity, responsible for accelerating the expansion of the universe.

2.3.2 Breaking and Forming the Chemical Bonds

During a chemical reaction, the chemical bonds can be either broken or formed. Depending on the nature of the chemical reaction, and the specific reactants and products involved, energy may be either released or absorbed in the reaction. If the energy is released during a reaction, it is known as an exothermic (heat is released) reaction, and if the reaction absorbs energy, it is an endothermic (heat is absorbed) reaction. The energy to break or form a chemical bond can be estimated as shown in Table 2.1 for selected types of bonds.

To determine whether a reaction absorbs or releases energy (endothermic or exothermic), we can use the formula of enthalpy of reaction as

(2.29)

A positive enthalpy indicates that the energy is absorbed (endothermic). A negative enthalpy indicates released energy (exothermic).

Another aspect of chemical reaction that can be determined is whether it is spontaneous or not. Through the formula for Gibbs free energy (GE), we can ascertain spontaneity of the chemical reaction. If GE is negative (∆G<0), then it is spontaneous, and the reaction will take place without the need for energy input. If GE is positive (∆G>0), then the reaction will not take place spontaneously, and it will require energy. If ∆G = 0, the reaction will neither go forward or reverse, and it is said to be in a state of equilibrium.

Imagine two chemical compounds A and B forming a by-product AB. The driving energy for this reaction (∆G_r^o) is the difference in the values of the standard Gibbs free energy (G energy) of the products (in this case AB) and the standard Gibbs free energy of the reactants, A and B:

(2.30)

If A and B are simple elements, this is called a formation reaction, and since the G energy of formation of elements is zero then:

2.3.3 Chemical vs. Electrochemical Reactions

One way to visualize the differences between a chemical and electrochemical reaction is by considering the transport pathways for the ions and electrons as illustrated in Figure 2.11. In a typical chemical reaction between two chemical compounds or elements A and B, a by-product AB can be formed directly at the site of A and B. The exchange of electrons and ions are made at the reaction site, and the reaction may continue until all A and B are consumed or converted to the product AB.

In an electrochemical reaction, such as in a battery, the elements or compounds A and B (i.e., electrodes) are not in contact, and are separated by the solid electrolyte or separator. Therefore, the pathways for the transport of ions and electrons are effectively separated. In other words, in an electrochemical cell, ions move through the electrolyte where no electron transport is allowed. At the same time, the electrons are sent to the external circuit, to power a device (current is supplied in the opposite direction).

2.3.4 Hydrogen

Hydrogen is another material commonly used to store chemical energy, to be later converted to other forms of energy like electrical energy. For example, in proton exchange membrane fuel cells (PEMFCs), hydrogen is converted into protons (hydrogen ion or H+) and electrons (e-). The protons transport through the fuel cell membrane, and they combine with the oxygen at the other electrode to form water (H₂O). The electrons generated from the hydrogen ionization (oxidation), exit the electrode and enter the external circuit to power the load.

Hydrogen can be stored using two main approaches: physical storage or chemical storage (Figure 2.12). The physical storage of hydrogenincludes liquid hydrogen, compressed (gaseous) hydrogen, and cryo-compressed hydrogen. A carefully designed, reliable and robust container is generally required to safely store and transport the hydrogen in its various physical forms, to prevent unwanted chemical reactions and catastrophic explosions.

The chemical storage of hydrogen involves chemical compounds that contain the hydrogen element and can be retrieved at a later time through a chemical process. These compounds include hydrides, carbohydrates, hydrocarbons, and ammonia and they are used as a medium to store the hydrogen.

A potential advantage of chemical storage is that some of the chemical compounds may be in the form of solid pellets that are relatively stable and easy to transport. On the other hand, releasing hydrogen from the chemical compounds may require special processing conditions including higher temperatures. For example, to release hydrogen from metal hydrides, a relatively high temperature around 120 °C (248 °F) to 200 °C (392 °F), may be needed, and that is mainly due to the very strong bonding of hydrogen to these compounds. One way to assess various hydrogen storage types is by comparison of their gravimetric and volumetric energy densities. For example, the metal hydrides may exhibit a relatively good volumetric energy density; however, their gravimetric energy density can be lower than that of hydrocarbon fuels.

2.4.1 Temperature

Temperature is another term that we use almost on a daily basis, like, “the temperature is hot or cold”, but its concept can be somewhat vague. What does a temperature really represent? Before we answer this question, let us consider the relevance to energy. Can the temperature play a role in how we identify whether thermal energy has transferred into or out of an object? The answer is yes. If we observe a temperature change in the object, then a transfer of thermal energy to or from an object has occurred. But, now we return to the question: “What is temperature?”

The temperature of an object can represent how fast the molecules in the object are vibrating. In other words, temperature is related to the kinetic energy of the molecules. Higher the temperature, faster the molecules vibrate, and more kinetic energy they possess. To find the exact relation between temperature and kinetic energy, various derivations and approaches including the kinetic theory model, and statistical thermodynamics approach have been employed, and are readily available in the literature. Here, we will not delve into any of those derivations, but we’ll discuss the main conceptual associations. Let us start with the famous equation for an ideal gas:

(2.32)

where P is pressure, V is volume, n is number of moles, R is the gas constant, and T is temperature. We can relate pressure P to the force exerted by the gas atoms inside a container. These forces can be eventually related (omitting the derivations) to the kinetic energy of the molecules as

(2.33)

where N is the number of molecules, and KE_avg is the average kinetic energy of the molecules (or particles). As discussed in previous sections, kinetic energy is also related to mass and velocity, and therefore, we can obtain an expression between temperature and the mean square average velocity of the molecules as

(2.34)

where m is the mass of the molecule (particle), v is the velocity of the molecules, k is the Boltzmann constant, and T is the absolute temperature. Rearranging the terms, we obtain the expression for temperature as

(2.35)

Here, we see that temperature is directly related to the mean square average of the velocity of the molecules or particles, and faster this is, higher is the temperature.

2.4.2 Thermal Energy Storage Types

Let us take a look at how thermal energy is stored. In general, thermal energy storage is classified into three main categories: (i) sensible heat storage, (ii) latent heat storage, and (iii) thermochemical energy storage, as depicted in Figure 2.13.

The concepts of sensible and latent heats are illustrated in Figure 2.14. As the material is heated, the temperature also begins to increase. This type of heat is referred to as the sensible heat, where both heat input and temperature are altered and the phase of the material remains the same (i.e., liquid or solid). In the case of a phase change material, as thetemperature reaches a certain temperature referred to as the phase change temperature T_pc, the material undergoes a phase-change, (e.g., solid to liquid) and thermal energy is stored in the material. This stored thermal energy at a relatively constant temperature, is referred to as latent heat. Similarly, when cooling from high temperatures, as the temperature decreases, the heat is released. When the temperature reaches T_pc, the material returns to its original phase (e.g., liquid to solid), and releases the stored thermal energy.

2.4.3 Phase Change Materials

Phase change materials can be classified into three categories: Organic, inorganic, and eutectic as shown in Figure 2.15. The most commonly known organic phase change materials are paraffin compounds, for example, used as candles. Organic PHMs can also be non-paraffin based. Examples of inorganic PHMs are salt hydrates and metallic materials.

A eutectic is a material composed of two or more components that has a minimum melting composition, where each individual component melts and freezes simultaneously. This leads to a mixture of the two components crystals. Therefore, a eutectic generally freezes or melts without undergoing any segregation. Eutectics can be composed of organic-organic components, inorganic-inorganic components or inorganic-organic components. More information on thermal storage can be obtained from several books and journal review articles, including a comprehensive review by Sharma, A., et al., 2009.