Thermodynamic Elements of Engines

Thermodynamic Elements of Engines

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The Little Heat Engine:
Heat Transfer in Solids, Liquids and Gases

The question now is wherein the mistake consists and how it can be removed.

Max Planck, Philosophy of Physics, 1936.

While it is true that the field of thermodynamics can be complex,1-8 the basic ideas behind the study of heat (or energy) transfer remain simple. Let us begin this study with an ideal solid, S1, in an empty universe. S1 contains atoms arranged in a regular array called a "lattice" (see Figure 1). Bonding electrons may be present. The nuclei of each atom act as weights and the bonding electrons as springs in an oscillator model. Non-bonding electrons may also be present, however in an ideal solid these electrons are not involved in carrying current. By extension, S1 contains no electronic conduction bands. The non-bonding electrons may be involved in Van der Waals (or contact) interactions between atoms. Given these restraint, it is clear that S1 is a non-metal.

Ideal solids do not exist. However, graphite provides a close approximation of such an object. Graphite is a black, carbon-containing, solid material. Each carbon atom within graphite is bonded to 3 neighbors. Graphite is black because it very efficiently absorbs light which is incident upon its surface. In the 1800's, scientists studied objects made from graphite plates. Since the graphite plates were black, these objects became known as "blackbodies". By extension, we will therefore assume that S1, being an ideal solid, is also a perfect blackbody. That is to say, S1 can perfectly absorb any light incident on its surface.

Let us place our ideal solid, S1, in an imaginary box. The walls of this box have the property of not permitting any heat to be transferred from inside the box to the outside world and vice versa. When an imaginary partition has the property of not permitting the transfer of heat, mass, and light, we say that the partition is adiabatic. Since, S1 is alone inside the adiabatic box, no light can strike its surface (sources of light do not exist). Let us assume that S1 is in the lowest possible energy state. This is the rest energy, Erest. For our ideal solid, the rest energy is the sum of the relativistic energy, Erel, and the energy contained in the bonds of the solid, Ebond.

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The relativistic energy is given by Einstein's equation, E = mc2. Other than relativistic and bonding energy, S1 contains no other energy (or heat). Simplistically speaking, it is near 0 Kelvin, or absolute zero.

That absolute zero exists is expressed in the form of the 3rd law of thermodynamics, the last major law of heat transfer to be formulated. This law is the most appropriate starting point for our discussion. Thus, an ideal solid containing no heat energy is close to absolute zero as defined by the 3rd law of thermodynamics. In such a setting, the atoms that make up the solid are perfectly still. Our universe now has a total energy (Etotal) equal to the rest mass of the solid (ETotal = Esolid = Erest= Erel + Ebond.

Now, let us imagine that there is a hypothetical little heat engine inside S1. We chose an engine rather than a source to reflect the fact that work is being done as we ponder this problem. However, to be strictly correct, a source of heat could have been invoked. For now, we assume that our little heat engine is producing hypothetical work and it is also operating at a single temperature. It is therefore said to be isothermal. As it works, the little heat engine releases heat into its environment.

It is thus possible to turn on this hypothetical little heat engine and to start releasing heat inside our solid. However, where will this heat go? We must introduce some kind of "receptacle" to accept the heat. This receptacle will be referred to as a "degree of freedom." The first degrees of freedom that we shall introduce are found in the vibration of the atoms about their absolute location, such that there is no net displacement of the atoms over time. The heat produced by our little heat engine will therefore begin to fill the vibrational degrees of freedom and the atoms in its vicinity will start vibrating. When this happens, the bonds of the solid begin to act as little springs. Let us turn on the heat engine for just a little while and then turn it off again. Now we have introduced a certain quantity of heat (or energy) inside the solid. This heat is in the immediate vicinity of the little heat engine (see Figure 2). As a result, the atoms closest to the heat engine begin to vibrate reflecting the fact that they have been heated. The total amount of energy contained in the vibrational degrees of freedom will be equal to Evib.

Since the little heat engine has been turned off, the heat produced will now start to equilibrate within the solid (see Figure 3). Thus, the area nearest the little heat engine becomes colder (the atoms nearest the heat engine slow down their vibration) and the areas away from our little engine heat up (they increase their vibration). As this happens, S1 is moving towards thermal equilibrium. That is, it is becoming isothermal - moving to a single uniform temperature. In this state, all the atoms in S1 share equally in the energy stored in the vibrational degrees of freedom. The driving force for reaching this thermal equilibrium is contained in the 2nd law of thermodynamics. This law states that heat must always move from hotter to colder regions in an irreversible manner.

That heat flows in an irreversible manner is the central theme of the 2nd law of thermodynamics. Indeed, no matter what mechanism will be invoked to transfer heat in nature, it will always be true that the macroscopic transfer of heat occurs in an irreversible manner.

So far, S1 is seeking to reach a uniform temperature or thermal equilibrium. For our ideal solid, thermal equilibrium can only be achieved through thermal conduction which in turn is supported by energy contained in the vibrational degrees of freedom. Thermal conduction is the process whereby heat energy is transferred within an object without the absolute displacement of atoms or molecules. If the little heat engine was kept on, then thermal conduction would constantly be trying to bring our solid to thermal equilibrium. If there were no other processes other than thermal conduction, and the engine was turned off, eventually one would think that the entire solid would come to a single new temperature and thermal equilibrium would be achieved. At this stage, our universe would have a total energy equal to that contained in the rest energy (Erel + Ebond) and in the vibrational degrees of freedom (ETotal= Esolid = Erest + Evib).

However, even though our little heat engine has been turned off, thermal equilibrium cannot be reached in this scenario. This is because there is another means of dissipating heat available to the solid. Thus, as the solid is heated, it dissipates some of the energy contained in its vibrational degrees of freedom into our universe in an effort to cool down. This is accomplished by converting some of the energy contained in the vibrational degrees of freedom into light!

The light that objects emit in an attempt to cool down is called thermal radiation. The word thermal comes in because we are dealing with heat. The word radiation comes from the fact that it is light (or radiation) which is being emitted.

This light is emitted at many different frequencies (see Figure 4). We represent the total amount of energy in this emission as Eem. Emission of light provides another means of dealing with heat. Thus, the emission of light joins vibration in providing for our stationary non-metallic solid the only degrees of freedom to which it can ever have access. However, the energy of emission becomes a characteristic of our universe and not of the solid. Thus, the universe now has a total energy given by ETotal= Esolid + Eem. As for the solid, it still has an energy equal only to that stored as rest energy and that contained in the vibrational degrees of freedom, Esolid= Erest + Evib. However, note that since all the heat energy of the solid was initially contained in its vibrational degrees of freedom, the energy of emission (Eem) must be related to the energy contained in Evib at the time of emission.

As stated above, light has the property that it cannot cross an adiabatic partition. Consequently, the light produced by heating the solid becomes trapped in our virtual box. If we kept our adiabatic walls close to the solid, eventually thermal equilibrium would be achieved between the solid and the radiation. In this scenario, the solid would be constantly emitting and absorbing radiation. Under a steady state regimen, all of the atoms in the solid would be sharing equally in the energy contained in the vibrational degrees of freedom. However, let us make the box infinitely large for now, so that it will take the light many years to reach the walls of the box and be reflected back towards the solid. For all purposes then, the light that the solid emits cannot return and hit the surface of the solid.

Up to this point, by turning on our little heat engine, we have been able to discuss two important processes. The first is thermal conduction. Thermal conduction is that process which tries to bring the internal structure of the solid to thermal equilibrium. In our ideal solid, the vibrations of the atoms are the underlying support for this process. The second process is thermal radiation (also called radiative emission). Through radiative emission, the solid is trying to come to thermal equilibrium with the outside world. There are only two means for an ideal solid to deal with heat. It can strive to achieve internal thermal equilibrium through thermal conduction supported by the vibrations of its atoms and it can dissipate some of the energy contained in its vibrational degrees of freedom to the outside world through thermal radiation.
, the light emitted in an attempt to reach or maintain thermal equilibrium will contain a continuous range of frequencies (see Figure 4). The intensity of the light at any given frequency will be given by the well known Planckian relation.9
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