A New Approach to Understanding Engineering Thermodynamics

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ABSTRACT. Engineering Thermodynamics is that engineering science in which students learn to analyze dynamic systems involving energy transformations ...
Proceedings of the ASME 2012 International Mechanical Engineering Congress & Exposition IMECE2012 November 9-15, 2012, Houston, Texas, USA

IMECE2012-85208

A NEW APPROACH TO UNDERSTANDING ENGINEERING THERMODYNAMICS FROM ITS MOLECULAR BASIS W. John Dartnall Faculty of Engineering, University of Technology, Sydney, No. 15 Broadway, Sydney, NSW, Australia, 2007. Email: [email protected] ABSTRACT Engineering Thermodynamics is that engineering science in which students learn to analyze dynamic systems involving energy transformations, particularly where some of the energy is in the form of heat. It is well known that people have difficulty in understanding many of the concepts of thermodynamics; in particular, entropy and its consequences. However, even more widely known concepts such as energy and temperature are not simply defined or explained. Why is this lack of understanding and clarity of definition prevalent in this subject? Older engineering thermodynamics textbooks (often containing the words “heat engines” in the title) had a strong emphasis in their early chapters on the general physical details of thermodynamic equipment such as internal and external combustion engines, gas compressors and refrigeration systems. The working fluid in these systems might expand or contract while heat, work and mass might cross the system boundary. The molecular workings within the thermodynamic fluid are not of prime concern to the engineer even though they are to a physicist or chemist. Modern engineering thermodynamics textbooks place great emphasis on mathematical systems designed to analyze the behavior and performance of thermodynamic devices and systems, yet they rarely show, at least early in their presentation, graphical images of the equipment; moreover, they tend to give only passing reference to the molecular behavior of the thermodynamic fluid. This paper presents some teaching strategies for placing a greater emphasis on the physical realities of the equipment in conjunction with the molecular structure of the working fluid in order to facilitate a deeper understanding of thermodynamic performance limitations of equipment. NOMENCLATURE = Average energy throughout the “atom’s” 𝜀𝑐 oscillatory cycle. = Equi-partition x-direction kinetic energy. 𝜀x

John A. Reizes Faculty of Engineering, University of Technology, Sydney, No. 15 Broadway, Sydney, NSW, Australia, 2007. Email: [email protected] 𝜀y 𝜀 𝜀𝐵 F

= = = =

Freq

=

KE KEM kB M N, n P S STP T v

= = = = = = =

xc/e x XM X

= = = = = = =

Y δɛ δɛs

= = =

δq δv δw

= = =

WM2D

Equi-partition y-direction kinetic energy. Atom’s average energy at a particular state Boltzmann equi-partition energy. Average inwardly directed force containing particle(s) in oscillation. The number of time-steps in a particular cell of a statistical distribution. Kinetic energy. Kinetic energy of mass M Boltzmann constant (1.381x10-23 J/K). Piston. Mass of piston. Number of particles. Pressure. Source of energy (heat). Standard temperature and pressure. Absolute temperature. Volume occupied by gas atom. Working Model 2 D software package. Minimum/maximum free path of oscillating atom. Free path in x direction of oscillating atom. X position of mass M. Horizontal dimension. Also: Gas atom’s oscillatory X space. Vertical dimension. Small increment in kinetic energy of gas atom. Small increment in the amount of heat received, on average, by the representative atom of the gas. Incremental (heat) energy input to gas system. Small increment in volume due to expanding gas. Small increment in work done, on average, by the representative atom of the expanding gas.

BACKGROUND It is acknowledged that students have difficulties in learning Engineering Thermodynamics. Christiansen and Rump [8], investigated students’ difficulties in applying basic science concepts in applied specialities such as Engineering

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Thermodynamics and made reference to two different styles for thermodynamics textbook writing: Euclidian (deductive in approach and presenting thermodynamics as a set of interconnected axioms) and Babylonian (emphasising useful models and systems with less focus on axioms). Modern Engineering Thermodynamics textbooks deal with a wider range of thermodynamic concepts than the Physics or Physical Chemistry textbooks of chemical engineers. Components that are Babylonian (analysis of processes related to equipment) and Euclidian (classical structure including the Laws of thermodynamics) occur in the Engineering Thermodynamics textbooks. In addition, Engineering Thermodynamics textbooks place a fundamental emphasis on systems, both closed and open, and on the concepts of Control Mass as well as Control Volume used in problem formulation and analysis. The Physics and Physical Chemistry textbooks concentrate more on closed systems employing Control Mass formulations and analyses. In summary, the Mechanical Engineering specialization currently endeavours to present thermodynamics from a number of different angles, leading Christiansen and Rump to conclude that “educators should take into account how subject matter is conceived in related scientific specialties when designing courses” [8]. They also point out that “students’ difficulties in applying basic science concepts in the applied specialties may be attributed to differences in teachers’ paradigmatic conception of subject-matter.” The authors of this paper have observed a historical evolution in the way that engineering thermodynamics textbooks have presented their complex subject. Until the mid-1900s, Engineering Thermodynamics textbooks, including textbooks on Heat Engines [9, 10, 11], often contained detailed, realistic diagrammatic representations of equipment and systems, performance graphs from actual experimental equipment, and a chapter on the Kinetic Theory of Gases. However, it was not common for these textbooks to specifically define (although they may imply) the concepts of closed and open system or the concepts of Control Mass as well as Control Volume used in problem formulation and analysis. In contrast, current textbooks [12, 13, 14], do not place emphasis on detailed diagrammatic representations of equipment and systems, or performance graphs from actual experimental equipment; nor do they have chapters devoted to the Kinetic Theory of Gases and Statistical Mechanics. They do have well-structured chapters containing phrases such as closed system, open system, control mass, control volume, and they include chapters entitled, for example, Second Law of Thermodynamics, Entropy and Exergy. Having established the laws of thermodynamics and a general basis for analyzing thermodynamic systems, however, they treat the various engineering systems such as heat engines, gas compression systems, refrigeration systems and vapor power cycle systems in a manner that is not greatly different from that of the older textbooks. Both the older textbooks and their modern counterparts have early chapters that deal with such topics/concepts as

dimensions and units; work, energy and heat; state and equilibrium; the First Law and elementary heat transfer. In comparing the two approaches, a view may be formed that the older books were more visual about the equipment and the molecular behaviour of the thermodynamic material, whereas the modern textbooks show more concern about the structure of classical thermodynamics and general techniques for analysis of thermodynamic systems. It appears that one assumption upon which a modern textbook rests is that a student will be provided with a visual laboratory experience in order to become familiar with the application of the broad concepts and general analytical techniques presented in the textbook. Unfortunately, this valuable laboratory experience does not always eventuate for students. This paper presents another visual and desirable student experience: computer-based experimentation with simple molecular dynamic simulations that demonstrate some of the thermodynamic concepts of recognised difficulty. INTRODUCTION Previous papers by the authors [1, 2, 3, 4, 5, 6 and 7] investigated simple, one-dimensional and two-dimensional molecular dynamic (MD) models illuminating the ideal gas law, ideal gas processes, the laws of thermodynamics and heat transfer. Molecular dynamic models are finding their way into the teaching of thermodynamics in science courses, but do not appear in engineering thermodynamics textbooks as they did earlier in their analytic form. The main purpose of this paper is to present some simple molecular dynamic models that are easily constructed by students and will be useful to them in visualising and comprehending such concepts as temperature, heat transfer and entropy. The complexities involved in the presentation of Engineering Thermodynamics were discussed in the previous section, and the need for students to have visual and “hands on” experience was commended. In the next section, a case is made for providing students with this practical experience by way of including MD simulation in engineering thermodynamics courses. The remainder of the paper presents examples and strategies developed by the authors in which MD simulations address thermodynamic concepts associated with known student-learning difficulty. COMPUTERS IN TEACHING AND MODELING USING COMPUTER SIMULATION AS PART OF STUDENTS’ HANDS-ON EXPERIENCE It is interesting to note that reserchers such as Anderson, Sharma and Taraban [15, 16] employed and tested the efficacy of computer-based active learning materials in an introductory engineering thermodynamics course which normally lacks laboratory experience. One component of their multi-faceted approach is to present images of thermodynamic equipment with text, voice-over and cursor-over pop-ups that enable additional graphics and information about the equipment. An example given by Anderson et al. is a sectioned image of a turbine building on which a pop-up shows an interior view of

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the turbine floor, with additional information accessible to the students by clicking on various features. This example addresses the lack of detailed, realistic diagrammatic representations of equipment and systems as well as the lack of student laboratory experience. Sokoloff et. al. [17] and van der Meij and de Jong [18] point out that modern, computer-based learning environments have improved real-time collection, display and analysis of data and the availability of many forms of dynamic representations. These authors are among many, including those of [19, 20 and 21], who predict and demonstrate numerous pedagogic and pragmatic benefits for students’ learning by means of active computer-based educational materials. Simulation of processes may enable students to experiment with large, expensive process equipment otherwise inaccessible to them; to perform a range of “what if” analyses, thereby efficiently gaining a comprehensive view of a complex technical situation; or, as shown by this paper, to view molecular situations that are too microscopic and rapid to observe by any other means than by computer simulation. It is the view of the authors that the inclusion of the molecular basis of thermodynamics, although desired by the engineering profession, has been waylaid because of the view that excessive teaching time would be required to address it by the traditional approach of Statistical Mechanics. CONCEPTUAL ONE-DIMENSIONAL MODEL OF A HEAT ENGINE It is possible for students to visualize otherwise abstract thermodynamic behavior by constructing simple onedimensional molecular dynamic models and observing the activities of both the mechanical and the thermodynamic elements. Figure 1 presents a model, an earlier version of which was described in [1, 2 and 7]. This model represents a heat engine. Suppose that an ideal gas is represented by a single, perfectly elastic “atom”, having mass, m, and for each cycle an average velocity, v, and average kinetic energy, 𝜀, while oscillating between a heat source/sink and a piston, having relatively large mass, M. The position of this perfectly elastic, frictionless piston constrains the one-dimensional space available to the oscillating atom. Any outward movement of the piston will result from the combined effect of the impulses from the atom, the piston’s own inertial resistance to its acceleration, FI, and a load force, FL, resisting its motion. The combination of FI and FL may be represented by a force F that resists the expansion of the one-dimensional ideal gas. S may alternatively act as a “thermal” source or sink. Both S and the atom are the thermodynamic elements of the system. When acting as a source, small increments of kinetic energy are imparted to the atom each time that it rebounds from this artifact, which may be thought of as a vibrating mass of “atoms”. When S is acting as a sink, the oscillating atom imparts small amounts of energy to S each time that it rebounds from it. In summary, heat is imparted to, or abstracted from, the oscillating atom, representing the gas, by its interaction with S. The gas does work on the piston by moving it relatively slowly

outwards and the piston may do work on the gas by slowly compressing the space in which the atom oscillates. atom

m, v,

piston, M

F

xc

S (source or sink)

x xe

FI

FL

Figure 1: A model of a one-dimensional heat engine with an atom that oscillates between an artifact, S, and a piston, M, receiving heat from S and doing work on M.

In [1], the gas was shown to be governed by a onedimensional ideal gas law, when F, x and 𝜀 are invariant: 𝐹𝐹 = 2ɛ. (1) This is equivalent to the ideal gas law in one dimension. SECOND LAW OF THERMODYNAMICS The Kelvin-Plank statement of the Second Law of Thermodynamics is: It is impossible for any device that operates on a cycle to receive heat from a single reservoir and produce a net amount of work. A remarkable consequence of the model of Figure 1 is that it may be used to derive this Second Law of Thermodynamics – admittedly within the limitations of the assumptions on which the model is based. Consider the isothermal expansion of an ideal gas. This is the only ideal process in which all the heat supplied may be converted perfectly to work. The model of Figure 1 requires that 𝜀𝑐 be constant throughout an expansion while the atom’s flights from S to the outward moving piston is at slightly higher speed than its return flights from the piston to S during each oscillation. This process was discussed in [1] and a molecular dynamic model suitable for student construction was discussed in [7]. In this process the source, S, needs to be capable of supplying sufficient energy (heat) to the atom to just balance the work done for each of the atom’s cycles, so that 𝜀𝑐 is constant throughout the process. One such cycle of the atom’s movement is illustrated in Figure 2. In the imagined isothermal expansion process, x will increase as the atom oscillates between S and M, the upper and lower energy levels of Figure 2 recurring at each cycle of the atom. The frequency of oscillation is inversely proportional to x. It was demonstrated in [5] that a required condition for a vibrating atom representing a source, such as S, to consistently impart energy to a returning oscillatory atom is that the source atom’s average energy level be at least equivalent to the energy of the imparting oscillatory atom, requiring, (2) 𝜀𝑆𝑆𝑆𝑆𝑆𝑆 ≥ 𝜀𝑐 + 𝛿𝛿�2 . 3

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Energy scale

atom travels away from S with energy

Source, S supplies to atom

atom travels from M with energy

average energy of atom throughout its cycle is

Piston, M , moving outwards receives from the atom

space, x , in which atom oscillates in its current cycle

Figure 2: Typical idealized cycle executed by the atom in which it receives 𝜹𝜹 units of energy from S in the form of heat and delivers this to the piston in the form of work.

Having executed a process involving perfect conversion of heat supplied to work, the student should attempt to cause the engine to repeat the process. First it will be necessary to compress the gas. One way to do this is to reverse the expansion process. A spring could reverse the momentum of the piston. On impact with the outward flying atom, the piston would add approximately 𝛿𝛿 to this atom because of its now inward motion. This would result in the atom having a return flight energy of approximately 𝜀𝑐 + 3𝛿𝛿�2 . On interacting with S the atom would impart approximately 𝛿𝛿 to S. If, in accordance with equation (2), the average energy of this source had been set at 𝜀𝑐 + 𝛿𝛿�2 it would now be forced to act as a sink. If an isothermal compression is performed at these new energy levels it would have a new mean energy, 𝜀𝑐1 , which will be 𝛿𝛿 above that of the expansion energy, 𝜀𝑐 . Therefore, a reversible process will not have been achieved. Indeed, work would need to be supplied from the surroundings. A cycle energy average of 𝜀𝑐 could only be achieved by replacing S, with a new S having a lower average energy level of, say 𝜀𝑐 − 𝛿𝛿�2. At this point the student would realize that a cycle has been constructed having a source and a sink; with the energy level of the sink lower than that of the source, this cycle has produced zero net-work. Once again, a perfect, reversible process has not been achieved because the source and the sink are not identical. With careful thought, the student will arrive at the realization that the source and sink energy levels could be made to approach each other by causing 𝛿𝛿 to be infinitesimally small. However, in the limit, no isothermal expansion will occur via this process. The student will have demonstrated that a reversible process is an ideal that can be only approximated in practice. If the energy level of S is reduced by a sizeable amount on the compression stroke, cycles having considerable net-work can be conceived [1], the most efficient of these having the efficiency of the Carnot cycle; however, such cycles do not have a single reservoir as required by the Kelvin-Plank statement of The Second Law. Processes other than the isothermal process lead to the same conclusion. Suppose that a heat source imparts energy 𝜀𝑐

to the working atom and then is replaced by an adiabatic wall. The atom does work on the moving piston. Were the piston to be reversed, the atom would follow the same history in reverse. This means that the energy of the atom would once again be 𝜀𝑐 , implying that no net-work would have been done. By introducing a sink, the energy of the atom could be lowered at the end of the expansion stroke, so that the energy of the atom at the end of compression would be less than 𝜀𝑐 . The cycle could begin again with the atom leaving the source with energy 𝜀𝑐 . Once again, a source and a sink are required to produce network. The student could ask, “Why not completely remove S during the piston’s return stroke?” This will lead the student to ponder what artifacts might replace it and create a cycle. If S is completely removed, or has zero kinetic energy, what will return the atom? In each case the Second Law of Thermodynamics will be endorsed. RESTRICTIONS OF THE MODEL’S IDEALISATIONS The supposition that, in the model of Figure 1, the atom representing the gas oscillates at close to constant kinetic energy level is far from realistic. When an attempt is made to construct a molecular dynamic model to represent an isothermal expansion, it will be found that construction of the oscillating atom and the piston, that is the mechanical end of the engine, using software such as [24], is conceivable. Provided the piston, M, has relatively large inertia compared to that of the gas atom, a student will find it easy to organize the software to compute the work done by the atom on the piston and plot this against the atom’s oscillatory distance, x as described in [7]. However, the artifact S, as described above, requires small packets, 𝛿𝛿, of energy to be imparted to or absorbed from the returning atom during an expansion or compression process, with perfect precision and timing. Although suitable software will allow such behavior to be programmed, as in [7], this is not realistic as a real heat source would return the atom with a wide range of kinetic energies in accordance with the Boltzmann Law. Statistical Mechanics, confirmed by numerous MD simulations described in papers [3, 4, 5 and 6], reveals that if at any time the working atom is to be in equilibrium with a realistic source, S, this atom’s kinetic energy will display a Boltzmann energydistribution having a time-averaged kinetic energy, 𝜀𝐵 , that is in equi-partition relationship with all the molecules of S. In other words, to be realistic, S must represent an assemblage of vibrating molecules of some kind. The time-averaged kinetic energy of each of the molecules of S would need to be a little above 𝜀𝐵 of the oscillating atom for S to be a source and a little below 𝜀𝐵 for S to be a sink [6]. At every point during the isothermal expansion, 𝜀𝐵 could only be established after many cycles with cycle-to-cycle variations of the working atom’s kinetic energy. Tackling this difficulty can lead students to distinguish between heat and work. Now, as discussed in [1] and as illustrated in Figure 3, it is possible to imagine an ideal gas represented by a single atom that is bouncing about in a two-dimensional space or in a three-

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dimensional cylinder. In these cases the time-averaged kinetic energy derived from the atom’s components of velocity in the x, y and, in the three-dimensional case, the z directions are equal – i.e. these energies would equi-partition. These models support the accepted ideal gas law, 𝑃𝑃 = (2/3)𝜀.

(3)

For the case where there are N atoms in the cylinder, 2

𝑃𝑃 = � � 𝑁𝑁.

(4)

𝑃𝑃 = 𝑁𝑘𝐵 𝑇.

(5)

3

The gas atom provided a further two degrees of freedom, making a total of 53 degrees of freedom in near equi-partition relationship. The simulation equi-partition energy was approximately 256*10 -25 J for the two DOFs of the source atoms. This was found by analyzing the energy records for a representative five of the source atoms, numbered 2 to 6 in Figure 4. These records are shown graphically in Figure 5 along with the average energy level of the gas atom (number 7), which was a little lower at about 254*10 -25 J and therefore near to energy equi-partition with the source atoms. 1 nm

Source atom (typical)

piston, M

X, (nm)

3

7 Atom representing the gas

4 5

F S (source or sink)

x xe

FI

FL

6

Spring (typical)

Scale, nm

Figure 3: A model of a three-dimensional heat engine having a single atom representing the ideal gas.

CONSTRUCTION OF A MOLECULAR DYNAMIC MODEL FOR TEACHING A two-dimensional MD model that students can construct and experiment with was trialled and is illustrated in Figure 4. One atom representing, say, Argon gas bounces about in twodimensional space, representing an ideal gas expansion by allowing the relatively massive piston of mass, M, to move slowly outwards, being driven by repeated impacts from the gas-atom. The boundary in which the centre of mass of the gas atom moves in its chaotic manner is indicated by a rectangle, of dimensions, X by Y (2.4 nm). The two-dimensional enclosure, serving the function of the cylinder of a three-dimensional equivalent, is the source. This enclosure is constructed by building a wall of anchored, perfectly elastic atoms whose collisions are frictionless. These source atoms vibrate in twodimensional motion, imparting energy to the gas-atom each time it collides with them. These vibrating wall atoms would also rebound from each other in a chaotic manner, redistributing energy between themselves and their associated springs. The relatively large number of energetic wall atoms, with associated springs, was sufficient to approximate an infinite energy source capable of slowly releasing energy to the gas-atom, which would do work in driving the piston outwards. In the trial simulation there were 17 wall atoms providing two degrees of freedom for each atom and a further degree of freedom for the atom’s spring, totaling 51 degrees of freedom.

piston, mass, M

Con-rod and crank– added for appearance only

Figure 4: A two-dimensional molecular dynamic model of a heat engine executing an expansion stroke. 300 250 Simulation average energy, (J*1025)

atom, m, v,

2

Y = 2.4 nm

Employing the new definition of the kelvin [23],

200 150 100 50

Source atoms 2 to 6 Gas atom, 7

0 2

3

4

5

6

7

Figure 5: Graphical summary of the equi-partition energy levels of five representative source atoms (numbered 2 to 6) and the simulation average energy for the gas atom (number 7).

Figure 6 is a graphical summary of the kinetic energy records from 24,000 time-steps, each of 0.000012 𝜇𝜇, for a sample of five of the source atoms, numbered 2 to 6, and the atom representing the gas, numbered 7. The energy averages over 1,000 time-steps of these averages for the representative source atoms are presented as graph (1) in the Figure. Energy averages of 1,000 time-steps, for the gas atom, are presented as graph (2).

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Several effects may be observed from this graph. Firstly, the energy levels of the 1,000 time-step averages varied considerably over simulation time. Secondly, the general energy-level trend, graph (3), of data of graph (1), was downward as a result of energy transferring slowly from the source atoms via the gas atom to cause a steady increase in kinetic energy of the piston. 450 (1) + Energy averages in 1,000 time-step cells of five source atoms (nos. 2 to 6) and the gas atom (no. 7).

Energy per particle averaged in periods of approximately 0.012 micro s, (J*1025)

400 350 300 250 200 150 100

(3) - - - Trend line for the energy averages of graph (1) of this chart.

(2) ● Energy averages in 1,000 time-step cells of the gas atom (no. 7).

50 Simulation time (μ s) 0 0.00 0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20 0.22 0.24 0.26 0.28

Frequency at energy level, Freq

Figure 6: Energy averages in 1,000 time-step cells indicating the considerable variability throughout the simulation. A general downward trend in the total energy of source and gas atoms is also indicated. 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0

Freq = 11,170 e -0.00424 ɛ R² = 99.45

0

500 1,000 1,500 2,000 Gas atom's energy level, ɛ , (J*1025)

Figure 7: Boltzmann energy distribution for the gas atom.

Force on piston, F (N*1016)

600

F = 285 X -1.00 R² = 0.93

500 400 300 200 100 0 0.0

0.5 1.0 1.5 2.0 2.5 3.0 Atom's oscillatory space, X (nm)

3.5

Figure 8: Load on piston, F, plotted against atom’s oscillatory space, X. This is analogous to a PV diagram for an isothermal process.

Figure 7 is a Boltzmann energy distribution constructed from the time-step energy records of the gas atom. The graph

shows the tallied number of time-steps for each 100*10-25 J wide energy cell. The coherent fit of the data to the classical Boltzmann negative exponential form concurs with the realnature heat transfer to the isothermally expanding gas mentioned earlier in this paper in the section entitled “Restrictions of the model’s idealizations”. The well-known isothermal process, in which pressure is inversely proportional to volume, was corroborated by the simulation results which are shown in Figure 8. Although the WM2D [24] software did not allow accurate direct recording of the atom’s impact forces against the piston, it was possible to derive time-averaged force from recorded simulation data for position, XM and kinetic energy KEM of the piston, M, since 𝐹 ∗ 𝛿𝛿𝑀 = 𝛿 𝐾𝐾𝑀 enabling F to be determined from the records of KEM and XM using the relationship,

𝐹=

𝛿𝐾𝐾𝑀 𝛿𝑋𝑀

.

(6)

The data was averaged over 1,000 time-steps prior to treatment with equation (6) due to the fact that the many impulses the piston received from the gas atom varied widely in magnitude in accordance with the Boltzmann Law. Finally, an interesting point to draw to the attention of students is the tiny magnitudes of scale present when working at the atomic level. The diameter of the gas atom was 0.384 nm, representing Argon gas, and the total time of the simulation was only 0.28 micro-seconds. Despite the fact that there were few degrees of freedom in the system and only one atom representing the gas, the wall-clock time for the simulation using a relatively powerful laptop computer was 1.25 hours. Students could ponder the improbability of being able to reverse this process, let alone one in which the gas contains approximately 1023 atoms or molecules and the source, sink and piston surface are comprised of similar numbers of atoms or molecules. TEACHING THERMODYNAMICS BY DISCUSSING THE BEHAVIOUR OF A REPRESENTATIVE ATOM A potentially efficient teaching approach is to introduce properties and behaviours that relate to one single atom, the average or representative atom of an ideal gas. Such a gas might be Helium, Neon or Argon at near room temperature and pressure. The behaviour of this atom has been shown, in references of Table 1, to represent the behaviour of all atoms of the gas under consideration. On averaging over sufficient time, this atom will occupy one nth of the total volume occupied by the gas. It will have a kinetic energy that is one nth of the total kinetic energy of the gas, and the pressure containing it will be the pressure containing the gas. With simple, time-efficient 1- and 2dimensional simulations such as described in the references of Table 1, students can demonstrate that the gas thermalizes; that is, equilibrium evolves. They can do this by taking averages over sufficient time, employing systems having relatively small numbers of particles. Each atom can be sown to exhibit a common Boltzmann time based energy distribution.

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Item

Table 1: Summary of some well-known general principles demonstrated by molecular-dynamic simulations. Simulation outcome observed over sufficient simulation time

1 An ideal gas law is established.

General principle

Comments relating to simulation outcomes. Note that the simulations contained ideal Reffrictionless mechanical elements, comprising masses and springs. Gravity was absent erence and the word atom is synonymous with particle.

A one dimensional ideal gas law was established in [1] and demonstrated in student constructed ideal gas-law simulations in which the relationship between Avogadro's number and the Boltzmann constant [1, 7] Ideal Gas Law may be confirmed experimentally by students was shown in [7]. One of the benefits of this approach is that it is computationally efficient, which is valuable in time-constrained student exercises.

The energy becomes equipartitioned between the energy Equi-partition 2 possessing elements (particles and law springs) of the systems.

Equi-partition of energy between atoms of one dimensional systems was demonstrated by introducing spring anchored walls or molecules comprising spring connected atoms in [3]. Two dimensional systems [3 to 6] of atoms demonstrated the equi-partition law with increasing precision as the number of atoms increased beyond three atoms, [6].

Each particle behaves as though it Pascal's 3 is contained by a common principle pressure.

The averaged inter-atomic forces in one-dimensional simulations, were found to demonstrate, with precision, the Pascal principle of equal pressure throughout the system. Employing atoms having a wide [3] variety of masses continued to confirm Pascal's principle with precision. The 1-D simulations enabled convenient evaluation of inter-particle forces.

Each particle behaves as though it is contained within a space that is Avogadro's 4 of the same size as that for all law atoms of the simulation.

A valuable aspect of one-dimensional simulations is that they conveniently, constrain the atoms, and spring connected atoms representing one-dimensional molecules, in such a way that it is possible to determine their average position over time. Even when a simulation contained atoms of extremely differing mass the mean positions of these atoms were found to equi-partition the space precisely.

Boltzmann energy distributions are Boltzmann established for each element, energy 5 resulting from simulations run over distribution sufficiently long time .

All simulations of [3 to 6] employed an unusually small number of particles. Each particle of a system was found to exhibit a Boltzmann energy distribution, derived from time steps of the simulation. All particles of a simulation produced similar Boltzmann time-based-distributions. It is more conventional to [3 to 6] construct Botzmann distributions from the energies, viewed at a typical point in time, of the spatial ensemble of a very large number of molecules.

[3, 5]

By ordering the time-averaged particle energies of a simulation and conveniently employing, a small The average spatial distribution of number of particles, it was possible to compute an average distribution of energy across the small Ergodic theory [6] 6 energy between the particles ensemble of particles. This distribution had a similar form to the time-based-distribution of item 4. Thus conforms to the Boltzmann Law. the Ergodic theory was demonstrated. Heat represented by vibrational mechanical energy is shown to 7 flow from 'hot' to 'cold', not the reverse.

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Heat transfer concept

Entropy for the simulated systems Entropy and their components is established

By constructing systems that would tend to equilibrium, for example, strings of atoms constrained at each end by spring anchored atoms that act as adiabatic walls, equilibrium could be demonstrated. It [5] was possible to demonstrate heat transfer by replacing the adiabatic walls by a source at one end and a sink at the other. Energy would flow steadily, with even gradient, from source to sink. Entropy of system components such as particles and springs, defined as δ(energy supplied)/(component kinetic or strain energy) was shown in [3] to behave in a way that was analoguous to entropy's classical [3] behaviour. Entropy of the system approached a maximum as equipartition of energy was established.

In a process that is analogous to classical textbook expositions of thermodynamics, the following treatment of a single atom representing an ideal monatomic, gas may be undertaken. Consider the single atom representing a gas in the engine of Figure 4 and employ the accepted sign conventions of the First Law of Thermodynamics to this situation. Let 𝛿𝜀𝑠 represent a small amount of energy imparted from the source, over a given time, to this atom. This energy will equate to the sum of two energies: δɛ - representing an increase in stored energy in the representative atom, and δw - representing the work done during the given time, by this atom, on the piston. In the more general case where some other load (e.g. surrounding gas) replaces the piston, work done by the representative atom

would be done on this load. Equation 7 summarizes the situation. (7) 𝛿𝜀𝑠 = 𝛿𝛿 + 𝛿𝛿 (J), If 𝛿𝜀𝑠 is interpreted as heat, the First Law of Thermodynamics can be said to apply to the single-atom gas, 𝛿𝛿 = 𝛿𝛿 + 𝑃𝑃𝑃 (J/atom).

(8)

If this single-atom gas is slowly heated at constant volume, it may be seen from equation (8) to have a specific heat at constant volume of, 𝑐𝑣 = 7

𝛿𝛿 𝛿𝛿

=1

(9)

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For a constant pressure process equation (3) may be differentiated yielding, 2

𝑃𝑃𝑃 = � � 𝛿𝛿 3

(J/atom). (10)

will redefine the kelvin in terms of fundamental physical constants. Instead of the kelvin being based on the triple point of water, as it is now, it will probably be defined as discussed by Mills et. al [23]:

(J/atom). (11)

“The kelvin, unit of thermodynamic temperature, is such that the Boltzmann constant is exactly 1.3806505×10−23 joule per kelvin.”

Substituting for Pδv in equation (8) yields, 2

5

𝛿𝛿 = 𝛿𝛿 + � � 𝛿𝛿 = � � 𝛿𝛿 3

3

So if the single-atom gas is slowly heated at constant pressure, it may be seen from equation (11) to have a specific heat at constant pressure of, 𝛿𝛿

𝑐𝑝 =

5

=( )

𝛿𝛿

(12)

3

A dimensionless entropy (on a per-atom basis) may be defined for this gas; see references [3, 22], 𝛿𝛿 = 𝛿𝛿 =

𝛿𝛿 𝜀

𝛿𝛿 𝜀

=

𝛿𝛿

𝛿𝛿

(dimensionless), (13)

2 𝛿𝛿

(dimensionless). (14)

𝜀

+𝑃

𝜀

3

𝜀

=

𝛿𝛿

𝜀2

+ � � 𝑙𝑙

+( )

Integrating equation (14) yields, 𝑠2 − 𝑠1 = 𝑙𝑙

𝜀1

2 3

𝑣

𝑣2 𝑣1

(dimensionless). (15)

Consider a constant entropy process, 0 = 𝑙𝑙

𝜀2 𝜀1

From which, On substituting for

𝜀1 𝜀2

2

+ � � 𝑙𝑙

𝜀1 𝜀2

3

𝑣

= � 2� 𝑣1

𝑣2

(dimensionless), (16)

𝑣1

2� 3

.

(17)

using the gas law equation, (3), 𝑃1 𝑃2

𝑣

= � 2� 𝑣1

5� 3

,

(18)

From which the familiar relationship (19) arises, 5� � 3

𝑃𝑣 �

= 𝑃𝑣 𝛾 = 𝐶.

This change will cause the definition of the kelvin to depend on the definitions of the second, the metre, and the kilogram. Throughout this paper, this definition is implied and temperature is represented by various symbols containing ɛ, as having energy units. In other words, the Boltzmann constant is merely a conversion factor between an equilibrium energy level of a system of molecules and the temperature of the substance – that is, temperature is really an energy level as implied in the previous sections. A remarkable outcome of this point, exposed in equations (13) to (16), is that entropy becomes dimensionless. The idea of dimensionless entropy is discussed by the authors in [3] and Leff in [22]. The authors of this paper now regard this to be how entropy should always have been defined. If entropy were defined in this dimensionless way, as an outcome of temperature being an “energy level”, it could then be likened to other dimensionless numbers such as the Reynolds or Nusselt numbers. It could be seen to define a trait of the system under investigation; it is certainly a property of the system, as is well-known. Like these other dimensionless numbers, it cannot be measured directly, but can be derived from two other directly measurable properties such as indicated in equation (15), where temperatures and volumes enable the change in entropy to be derived. Students learning about entropy could draw some comfort from its similarities, due to its dimensionless nature, to the other dimensionless numbers with which they are familiar. The idea that entropy is a dimensionless number is commensurate with the Boltzmann concept of entropy,

(19)

The above reasoning may be extended by replacing the atom of the single-atom gas by the representative atom (having the afore-mentioned time-averaged properties) of a homogeneous system of many atoms that is at equilibrium or that executes a quasi-equilibrium process. THE PROPOSED DEFINITION OF THE KELVIN AND DIMENSIONLESS ENTROPY A change in the definition of the kelvin is proposed by many metrologists from various standards laboratories, which is likely to be implemented by an international committee. This

𝑠 = 𝑘 log 𝑊.

(20)

W relates to the number of micro-states to which the thermodynamic substance has access at a particular macrostate. Here the Boltzmann constant acts as a conversion factor as it does with temperature. It gives the dimensions of J/K to entropy per unit quantity from log 𝑊, which, except for its dependence on the quantity of thermodynamic substance, in molecules, moles, kilograms or other units, is a dimensionless number. FURTHER WORK TO BE DONE An area where students are known to confuse concepts is between heat, temperature and internal energy. As stated below, these concepts are made physically and visually more evident with the models of this paper, particularly the model of Figure.4. However, the distinction between temperature and

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internal energy will be better served by having more than one atom – instead, say three atoms – in the gas enclosure. This could lead to discussions with students about the internal energy being represented by the total kinetic energy of all the atoms in this gas, whereas the temperature is the Boltzmann average energy of just a typical atom. It would be beneficial for students to trial these concepts, encountering ideas such as: treatment of the typical or representative atom; leading to discussion of the proposed change in definition of the kelvin and the concept of dimensionless entropy. In the authors’ opinion, this approach to presenting molecular and associated aspects of Engineering Thermodynamics is novel, and has not been trialed or implemented in the classroom. It is our intention to conduct trials, firstly with a small group of post-graduate research students who are keen to improve their fundamental understanding of thermodynamics. The authors welcome ideas and feedback from the engineering and thermodynamics teaching communities in order to further assess the worth of the ideas presented. CONCLUSIONS In engineering textbooks, expressions about heat and work crossing the boundary are customary. These expressions are abstractions that are not necessarily well grasped by students in a learning environment. The models discussed in this paper have required the physical modeling of matter in the form of atoms, interacting and exchanging energy, thus giving physical meaning to the words heat, work and “crossing the boundary”. Students’ confusion between heat, temperature and internal energy is well known. The differences between these entities could be pointed out in a visual way during discussion with students, when presenting the models, giving them a more hands-on understanding of the concepts. In conjunction with some of the molecular dynamic models of references listed in Table 1, concepts and the ideas presented with regard to treating the typical or representative atom, leading to the proposed change in definition of the kelvin and the concept of dimensionless entropy, provide a new approach to viewing the difficult concepts of thermodynamics. In this approach, when an ideal gas is at equilibrium, all molecules/atoms behave in a similar way, on average taken over sufficient time. Thus any one of them can represent the behavior of the whole gas. This average kinetic energy of the representative atom reveals the temperature of the gas. To obtain the internal energy of the gas, the energies of all the molecules/atoms must be summed. Heat transfer can be seen to be the cumulative effect of energy transfer due the difference in temperature of two molecular systems. Finally, entropy may be seen as a dimensionless number derived from other measurable properties of the molecular system in the familiar way of other dimensionless numbers such as the Reynolds or the Nusselt numbers. The approach recommended by this paper requires students of mechanical engineering to skillfully employ their most important and valued engineering science: Engineering Mechanics, thereby enhancing this skill in the process.

REFERENCES [1] Dartnall, W. J. and Reizes, J.A. 2005; “A Novel approach to the Teaching of Thermodynamic Cycles and the Laws of Thermodynamics”. 2005 ASME International Mechanical Engineering Congress and Exposition November 5-11, 2005, Orlando, Florida USA. [2] Dartnall, W. J. and Reizes, J.A. 2006; “A Novel Approach to the Teaching of Entropy Based on a recent Single Particle Heat Engine Model”. 2006 ASME International Mechanical Engineering Congress and Exposition November 5-10, 2006, Chicago, Illinois USA. [3] Dartnall, W. J. and Reizes, J.A. 2007; “Thermodynamics from a few Dynamic Particles raises Questions as to How Temperature and Entropy should be Perceived and Defined”. 2007 ASME International Mechanical Engineering Congress and Exposition November 11-15, 2007, Seattle, Washington, USA. [4] Dartnall, W. J., Reizes, J.A. and Anstis G. 2008; “Demystifying Thermodynamics by Connecting it with Mechanics”. 2008 ASME International Mechanical Engineering Congress and Exposition October 31 – November 6, 2008, Boston, Massachusetts, USA. [5] Dartnall, W. J., Reizes, J.A. and Anstis G. 2009; “Should Engineering Thermodynamics Include a Simplified Treatment of its Underlying Molecular Basis?” 2009 ASME International Mechanical Engineering Congress and Exposition November 13-19, 2009, Orlando, Florida USA. [6] Dartnall, W. J. and Reizes, J.A. 2010; “Developing Innovative Teaching Materials that use Molecular Simulations in Engineering thermodynamics” 2010 ASME International Mechanical Engineering Congress and Exposition November 12-18, 2010, Vancouver, British Columbia. [7] Dartnall, W. J. and Reizes, J.A. 2011; “Molecular Dynamic Simulation Models for Teaching Thermodynamic Principles” 2011 ASME International Mechanical Engineering Congress and Exposition November 11-17, 2011, Denver, Colorado, USA. [8] Christiansen, F. V. and Rump, Camilla. 2008; “Three Conceptions of Thermodynamics: Technical Matrices in Science and Engineering”. Res Sci Educ (2008) 38:545– 564. [9] Lewitt, E. H., 1946; “Thermodynamics Applied to Heat Engines. (3rd ed.)”. Pitman, London. [10] Wrangham, D. A., 1948; “Heat Engines. (2nd ed.)”. University Press, Cambridge. [11] Robinson, William and Dickson, J. M., 1954 “Applied Thermodynamics. (3rd ed.)”. Pitman, London. [12] Cengel, Y. A., and Boles, M. A., 2002; “Thermodynamics: an Engineering Approach”. 4th ed. WCB McGraw–Hill, Boston.

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[13] Van Wylen, G. J., Sonntag, R. E. and Borgnakke, C. 1998; “Fundamentals of Classical Thermodynamics (5th ed.)”, New York: Wiley. [14] Moran, M. J. and Shapiro, N. H. 1988; Fundamentals of Engineering Thermodynamics”. New York: Wiley. [15] Anderson, Edward E., Taraban, Roman and Sharma, M. P., 2005 “Implementing and Assessing Computer - based Active Learning Materials in Introductory Thermodynamics.” Int. J. Engng Ed. Vol. 21, No. 6, pp. 1168 – 1176. [16] Anderson, Edward E., Taraban, Roman and Sharma, M. P., 2002 “Implementing and Assessing Computer-Based Active Learning Materials in Introductory Thermodynamics.” American Society for Engineering Education. [17] Sokoloff, David R., Laws, Priscilla W. and Thornton, Ronald K. 2007; “Real Time Physics: active learning labs transforming the introductory laboratory.” Eur. J. Phys. 28 S83–S94. [18] van der Meij, Jan and de Jong, Ton, 2006; “Supporting students’ learning with multiple representations in a dynamic simulation-based learning environment.” Learning and Instruction 16, 199 – 212.

[19] Babich, Alexander and Konstantinos, Th. M., 2009; “Teaching of Complex Technological Processes Using Simulations.” Int. J. Engng Ed. Vol. 25, No. 2, pp. 209 – 220. [20] Tuttle, Kenneth L. and Wu, Chih 2000; “Intelligent Computer Assisted Instruction in Thermodynamics at the U.S. Naval Academy.” Department of Mechanical Engineering, 590 Holloway Rd, U.S. Naval Academy Annapolis, MD 21402. [21] Huang, Meirong and Gramoll, Kurt 2004; “Online Interactive Multimedia for Engineering Thermodynamics.” ASEE Conf., June 20 - 23, 2004, Salt Lake City, UT. [22] Leff , H. S., 1999; “What if entropy were dimensionless?” Am. J. Phys., Vol. 67, No. 12, December 1999. [23] Mills Ian M, et. al. 2006 “Redefinition of the kilogram, ampere, kelvin and mole: a proposed approach to implementing CIPM recommendation 1 (CI-2005)”. Metrologia 43 (2006) 227–246 [24] Working Model 2D version 5 motion simulation package by Design simulation Technologies, Inc, Canton, MI, USA. (Formerly by MSC Software Corporation. San Mateo, CA, USA.)

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