Associate Professor Yu. N. SOKOLOV.
Department of Thermodynamics, Tomsk Industrial Institute.

METHODS OF TEACHING TECHNICAL THERMODYNAMICS

“From living contemplation to abstract thought, and from it to practice—such is the dialectical path of the cognition of truth, the cognition of objective reality.”

Lenin.

Technical thermodynamics is one of the theoretical disciplines of higher technical education that is most difficult to master. The difficulties of mastering it are explained, on the one hand, by the abstractness of a number of concepts with which one has to operate here (quasi-static process, absolute temperature scale, entropy, etc.), and, on the other, by the circumstance that a student, when beginning the study of thermodynamics, in most cases does not yet have an idea of those heat-engineering units, the key to a proper understanding of the processes occurring in which thermodynamics is meant to provide. This circumstance places a special responsibility on the teacher conducting this subject, from whom a sensitive understanding of the audience is required.

Abstract thinking, with which one has to deal in studying the basic questions of thermodynamics, causes the greatest difficulties for students. Thermodynamics itself as a whole is a stage of abstract thought in the series of subjects of the heat-power cycle, and from this point of view it should perhaps have been preceded by a brief descriptive course on the fundamentals of heat engineering. This question, however, concerns the curriculum itself and is not subject to discussion here. Here we consider it necessary only to emphasize the importance of clearly and correctly distinguishing the three stages of the cognition of truth: living contemplation, abstract thought, and the application of its results to concrete reality, both in presenting the course of thermodynamics as a whole and in presenting its separate fundamental questions.

It should not be overlooked that a number of questions which seem elementary and obvious to the heat-engineering teacher may remain completely unclear to the student. This creates a gap in the basic features of his thinking and substantially impedes understanding of the main point to which special attention is being given in the given case.

Touching upon the significance of the course in technical thermodynamics, it is necessary to note that if for non-specialists (non-heat engineers) it is, to a certain degree, a general-education discipline, then for the heat engineer (in the curriculum of the heat-power specialty it is not without reason called “Theoretical Foundations of Heat Engineering”) the course in thermodynamics must provide the methodological foundations for all his further work and must ensure preparation for solving any heat-engineering questions connected with the analysis and reconstruction of existing thermal processes and with the formulation of problems of a scientific-research nature.

I. Sequence of Presentation of the Material

In passing to the concrete tasks of the methodology of teaching thermodynamics, we first of all encounter the question of the sequence of presenta-

of the material. This sequence in presenting thermodynamics acquires especially important significance both because of the aforementioned difficulties in mastering a number of questions, and for the reason that the very history of the development of this science did not always proceed in sufficient sequence. (Let us recall at least that Carnot’s work on the study of the so-called thermal cycle was carried out long before a clear understanding and proclamation of the principle of conservation of energy.)

A correctly chosen and well-thought-out system of exposition will ensure, on the one hand, good assimilation of the course, leaving students with logically coherent and integral material, and, on the other hand, will allow the lecturer himself to cope with the often difficult task of giving, within the rigidly allotted time, the extensive program material with the broadest and theoretically highest possible illumination.

In most higher educational institutions and in almost all existing textbooks of thermodynamics, the following scheme of presenting the course has been adopted, having become, so to speak, classical:

1. Basic gas laws.
2. The first law.
3. Processes of change of state of gases.
4. Cycles of gas machines (including the Carnot cycle).
5. The absolute temperature scale and the second law.
6. Entropy diagrams.
7. Basic properties of vapors.
8. Processes of change of state of vapor.
9. Cycles of steam-power plants.
10. Outflow, throttling, etc.

Such a scheme has evidently become established by virtue of the specific features of the development of the science itself and of some of its basic questions (especially the second law). Through the works of Clausius, Thomson, and a number of subsequent authors, the concept of entropy is introduced only through consideration of the Carnot cycle, and this makes it necessary to premise all the gas laws and processes of change of state of gases before the exposition of the second law. Meanwhile the scheme given above, from our point of view, has a number of substantial logical and methodological shortcomings, the principal ones being the following:

a) The whole system of exposition does not possess logical harmony; repeated transitions from general questions (basic laws) to particular ones (gas processes) and back again are unavoidable.

b) The applied part (gas processes and cycles) cannot be presented using the entire modern apparatus for studying thermodynamic questions (the \(T\)-\(S\) diagram is still unknown).

c) The aforementioned transitions from particular questions to general ones and back make it difficult to understand clearly the fundamental difference in the methods of thermodynamic analysis of processes of perfect (gases) and imperfect (vapors) thermodynamic agents¹.

d) The concept of “absolute temperature”² has to be introduced before the exposition of the second law, and this makes the assimilation of this abstract concept still more difficult.

e) The principal method of technical thermodynamics—the heat (entropy) diagram—is inevitably relegated to the very end of the section “gases”

and, quite naturally, this method, from the point of view of one studying thermodynamics for the first time, acquires the significance of a somewhat artificial appendage.

We advocate the following scheme for presenting the course of thermodynamics, tested by many years of experience at the Tomsk Industrial Institute:

I. Fundamental laws.

1. The thermal state of bodies. Heat capacity.
2. The first law of thermodynamics (including the concept of quasistatic and nonstatic processes, the introduction of the quantity “enthalpy,” the relation between it and internal energy, etc.).
3. The second law of thermodynamics (including general information on the \(T-S\) diagram, analysis of cyclic processes with illustrations in the heat diagram, the Carnot cycle¹, and the concept of the absolute temperature scale).

II. Gases.

1. Basic properties of ideal gases.
2. Processes of change of state of gases (with parallel illustration in the \(P-V\) and \(T-S\) coordinate systems).
3. Theoretical cycles of gas machines.

III. Vapors.

1. Basic properties of vapors (the inapplicability of a number of “gas” relations is emphasized).
2. Processes of change of state of vapor (wet and superheated, in parallel).
3. Theoretical cycles of steam-power installations.

IV. General part.

1. Moist air.
2. Efflux of gases and vapors.
3. Throttling of gases and vapors.
4. Mixing of gases and vapors.

We see the advantage of such a scheme above all in its logical coherence and in the clear separation of general and particular questions, which ensures a correct understanding of the limits of applicability of various relations.

Moreover, the concept of “entropy,” introduced in the exposition of the second law, is here immediately clothed in its practical applicability (the \(T-S\) diagram), which prevents the conclusions, common upon first acquaintance with thermodynamics, about the “artificiality” (and consequently the “worthlessness”) of this quantity.

Finally, one must not lose sight of the circumstance that the study of gas processes, with simultaneous and parallel illustration of them by the \(T-S\) diagram, introduces a number of advantages from the standpoint of visual clarity, simplification of the exposition², and reduction of the overall expenditure of time. (A repeated presentation of gas processes with the independent variables \(T\) and \(S\) is inevitably connected with returns to the preceding material, and therefore with a loss of time.)

It is quite understandable that ensuring the implementation of the proposed scheme is not as easy as it may seem at first glance; this, evidently, explains why most authors have taken another, less difficult, but, from our point of view, far from the most rational path. The difficulties encountered here consist above all in the method of presenting the second law and in introducing the concept of “absolute temperature” outside its connection with the laws of ideal gases. Below we shall consider this question in greater detail; here we shall merely note that the possibility is not excluded of simply postulating the expression $dQ = T . dS$ on the basis of analogies with other forms of energy ¹) (which we also allow for courses with a small number of hours allotted by the curriculum), and that, on the basis of the works of Schiller, Carathéodory, Afanas’eva-Ehrenfest, and others, a strictly mathematical derivation of this expression can be given quite independently of the properties of ideal gases.

It is impossible, of course, to pass over in silence the circumstance that the application of the proposed system of presenting the course can give a positive result only if the internal connection of the separate questions of the course has been deeply thought through and if they have been methodically prepared in the appropriate way.

2. Methodology for Presenting the Basic Questions.

The scope of the present work does not permit us to illuminate the methodological features of all the questions provided for by the normal program of a course in thermodynamics. Here we shall dwell only on certain individual questions that cause the greatest difficulties in their study or that are of special interest from the point of view of their applied significance.

The first law of thermodynamics, at least in the narrow understanding of this question, usually does not present serious difficulties for the student; however, under the general heading “The first law” one often has to touch upon a number of related questions, a clear understanding of which is very important from the point of view of preparation for mastering the subsequent questions of the course and, first of all, the second law of thermodynamics.

The presentation of the first law is usually begun with the principle of equivalence and the establishment of the relation between heat and work ($dQ = A . dL$). In itself this question cannot give rise to difficulties of a methodological nature. However, it should be borne in mind that, proceeding from the dependence $dQ = A dL$, the student first becoming acquainted with thermodynamics usually acquires a tendency to apply it to all cases of practice, completely disregarding the possibility of other (besides work) items of energy expenditure. This circumstance compels one to pay special attention to the device used by A. Lande in formulating the first law:

“The question arises of formulating the first fundamental law (the law of conservation of energy) without using the concept of quantity of heat: in most cases the latter should be defined conversely, proceeding from the first law, by introducing the concept of quantity of energy ²).”

We consider such an approach, from the methodological point of view, the most successful. It is only necessary to pay special attention not only to the formulation of the first law itself, but also to a clear representation of all the concepts connected with this law, among which the definition of the internal energy of a body may cause the greatest difficulties.

¹) As, for example, Prof. V. Schüle does (V. Schüle, Technische Thermodynamik, vol. I, book 1).

²) A. Lande, “Axiomatische Begründung der Thermodynamik durch Caratheodory,” Handbuch d. Physik, 9, 1926.

The traditional approach here should be considered to be the presentation of the internal energy of a body from the point of view of the molecular-kinetic hypothesis. We believe that technical thermodynamics, as a purely experimental science, can be entirely free of any hypotheses, and that kinetic theory should be used in it only for the purpose of explaining (and confirming) propositions established experimentally.

Thus, it proves highly desirable to give an idea of the internal energy of a body proceeding, above all, from thermodynamic processes known from everyday experience. This can quite well be done by considering adiabatic and isochoric processes in some very simple examples; moreover, the introduction of the terminology itself (adiabat, isochore) is not at all necessary here. In considering an adiabatic process, we state that the external work performed on a body is expended in increasing its internal energy and, conversely, that a body is able to perform external work only at the expense of the store of its internal energy. Consideration of an isochoric process \((dL = 0)\), on the contrary, makes it possible to characterize the change in the internal energy of a body at the expense of external heat exchange. We consider this method of explaining the concept of “internal energy” to be the one most consistent with its thermodynamic meaning, taking a minimum of time, at the same time providing the clearest elucidation of the meaning of this quantity and, finally, making it easy to proceed to the establishment of units of measurement for internal energy.

Obtaining the fundamental expression of the first law, \(dQ = dU + A dL\), by the usual compilation of a balance of all possible kinds of energy participating in the thermodynamic process of the body under consideration (up to and including the kinetic and potential energy of external bodies, as is done, for example, by Prof. Brandt), and the subsequent discarding of all “superfluous” terms of the resulting equation, we consider methodologically inexpedient, since such a device naturally creates an impression of artificiality, while the “unnecessary” terms of the equation find no application in the further exposition of the course.

A much more convenient approach to obtaining the analytical expressions of the first law is through the classification of possible thermodynamic processes. By dividing processes according to energetic features, it is easy to proceed from particular cases \((dQ = 0;\ dV = 0;\ dL = 0)\) to the general case \((dQ \ne 0;\ dU \ne 0;\ dL \ne 0)\) and to give for it the general analytical expression of the first law for any (quasistatic or nonstatic) process, as \(dQ = dU + A \cdot dL\). Dividing processes according to the rate of their occurrence leads to the concept of quasistaticity (irreversibility), to the definition of work as \(dL = p.dv\), and to the equation of the first law for quasistatic processes, as \(dQ = dU + Ap.dv\).

Finally, by considering cyclic processes (cycles) and applying the principle of conservation of energy to them, it is easy to obtain the usual expression for the energy balance of an ideal cycle, as \(Q_1 = AL + Q_2\), which may also be regarded as an analytical expression of the first law for cyclic processes. Only these three expressions of the first law have practical significance in technical thermodynamics, and special attention must be paid to their correct understanding.

In deriving the expression \(dL = p.dv\), one should particularly emphasize the difference between the work of overcoming external forces and the work of internal forces (in nonstatic processes) and, on the basis of comparing the work of external forces in nonstatic expansion and compression, show the irreversibility of nonstatic processes.

The correct understanding of the role of quasistatic processes in thermodynamic analysis, as a necessary abstraction, should also be...

special attention must be paid, for combining an indication of the non-static character of real processes with the predominant study in thermodynamics of only quasi-static processes, in most cases, gives rise to quite understandable perplexity.

In introducing the concept of a cyclic process, it is essential to note the circumstance that, in a quasi-static process, for any \((x)\) characteristic of state, \(\int dx = 0\), whereas for the characteristics of the process \(Q\) and \(L\), in general, the equality \(\int dr = 0\) is by no means obligatory.

We do not consider it necessary to develop the theory of differential equations in a course of technical thermodynamics; however, wherever they facilitate the exposition of questions that have substantial practical application, differential expressions should not be avoided. Thus, immediately after presenting the basic questions of the first law, we consider it advisable to give the expressions \(C_v = \left(\dfrac{\partial v}{\partial t}\right)_v\) and \(C_p = \left(\dfrac{\partial i}{\partial t}\right)_p\), since their use later on (Joule’s experiment, etc.) brings significant simplifications; and in presenting the question of the construction of heat diagrams we also employ other necessary differential relations, deriving them as needed.

If the derivation of the equation of the second law, \(dQ = TdS\), in accordance with the scheme of exposition of the course we have adopted, is to be given by the method of finding an integrating factor of heat energy (see below), then already in presenting the first law one should dwell especially on the possibility of expressing all thermodynamic quantities in the form:

\[ dZ = X \cdot dx + Y \cdot dy, \]

where \(X = f(x,y)\) and \(Y = \varphi(x,y)\) (Pfaff’s equations), emphasizing the circumstance that \(dZ\) may or may not be a complete differential.

The introduction of enthalpy (heat content) as a new characteristic of state, purely mathematically \((i = U + Apv)\), presents no difficulties; however, such an approach to this question naturally raises doubts about the necessity and applied significance of this quantity. The practical expediency of replacing \(U\) by \(i\) could be shown in the very process of deriving the expression \(dQ = X \cdot dt + Y \cdot dp\). The cumbersomeness of this method, however, can hardly be justified by the desire to show the significance of the characteristic \(i\); therefore we consider it methodically more correct to take \(i = U + Apv\) simply as a definition, at the same time giving a graphical representation of the quantities \(\int_1^2 p \cdot dv\) and \(\int_1^2 v \cdot dp\), and, a priori, pointing out the significance of the quantity \(vdp\) in thermodynamics (compressors, outflow). This will serve as the necessary methodological justification for introducing the new characteristic of state \(i\).

The second law of thermodynamics is indisputably the most difficult section of the course from the methodological point of view. The system we have adopted for presenting the program as a whole, leading to the rejection of “classical” methods of presenting the second law, makes the methodological preparation for the exposition of this question still more responsible.

The works of Carathéodory, Afanasyeva-Ehrenfest, and others, in the form in which they are given by the authors, of course cannot be presented to a student audience within the usual amount of time. Nevertheless, the Department of Thermodynamics of the Tomsk Industrial Institute (on the initiative and under the direction of its head, Acting Prof. G. I. Fuks) has for several years already practiced the derivation of the basic equation of the second law by proving the existence of an integrating factor of heat energy.

By a number of methodological measures it is possible here to obtain a not overly cumbersome theory, which can be presented in 4–5 lecture hours (the entire second law takes 8–12 lecture hours).

The essence of the method employed and the logical connection of the individual conclusions are given in the author’s “Logical Scheme of the Theoretical Justifications of the Second Law” (Table 1), upon careful consideration of which—

LOGICAL SCHEME
of the theoretical justifications of the second law of thermodynamics.

Objective: starting from the condition, to prove the existence of the factors \(T, S\) of heat energy and to investigate their properties.

Table 1.

it is not difficult to notice that we postulate not Carathéodory’s principle of adiabatic inaccessibility, but Schiller’s postulate, which we formulate as follows:

“If, during the course of a quasistatic process in an adiabatic system of \(n\) bodies in thermal equilibrium, \(n\) characteristics of state have returned to their initial values, then the \((n+1)\)-st characteristic also assumes its initial value.”

The main methodological question here is the task of showing the obviousness of this proposition. The author considers it most advisable to apply the following reasoning here:

1) A system of bodies is considered (for simplicity, only two) in an adiabatic shell, separated by an immovable diathermic wall.

2) The change in the characteristics of state \((p, v, t)\) and in the energetic characteristics \((Q, AL, \Delta U)\) of each body under a mechanical action on one of them is investigated; from this a conclusion is drawn concerning the possibility of the course of a quasistatic process in an adiabatic system of bodies in thermal equilibrium.

3) By investigating the change in the characteristics of state of the system under consideration when the process is reversed, it is shown that, if the bodies

occupy their initial volume, each of them acquires its own initial temperature, after which Schiller’s postulate becomes obvious.

The most difficult point in mastering the proposed scheme is the replacement of variables in order to obtain a new integrating factor \(T=\varphi(t)\) instead of \(\tau\). However, with the appropriate mathematical preparation of the audience, this difficulty is quite surmountable.

The principle of Carathéodory’s adiabatic inaccessibility, under the method of presenting the question considered above, proves to be established for quasistatic processes; however, for methodological reasons, we do not consider it necessary to proclaim it until it becomes necessary to investigate the question of the change in the entropy of an adiabatic system of bodies. The proof of its invariance in quasistatic processes presents no difficulties if one proceeds from the expressions \(dQ=T \cdot dS\) and \(dS=\sum dS_i\); consideration of the question of the change in the entropy of a system in nonstatic processes requires extending the principle of adiabatic inaccessibility to nonstatic processes, which can easily be accepted as a postulate. Under this condition, the possibility of only a one-sided change of entropy in nonstatic processes (not necessarily only an increase, but either only an increase or only a decrease) in an adiabatic system is easily proved, and by the simplest example of heat transfer from a hot body to a cold one it is shown that, under the conditions of the ordinary circle of phenomena, the entropy of the system of the bodies considered must increase.

With such a formulation of the question, the errors of idealistic interpretations of the second law become obvious, and the materialistic understanding of the essence of the second law can be easily explained; moreover, the entire interpretation of the second law becomes fully consistent with Engels’s brilliant statement¹ about the existence of phenomena in which the so-called (in thermodynamics) positive and negative processes exchange roles. It remains only to note the necessity of giving a natural-philosophical interpretation of the essence of the second law on the basis of a critique of idealistic conclusions, which, with the formulation of the question indicated above, proves to be easily feasible.

In accordance with what has been set forth above, we propose the following expanded plan for presenting the first and second laws, which can be carried out without particular difficulty in 16–20 lecture hours:

The First Law of Thermodynamics.

1. General concepts.
   a) The principle of conservation of energy.
   b) Heat as one of the forms of energy.
   c) Units of measurement of heat and work.
   d) The thermal equivalent of work.
   e) The internal energy of a body.

2. Thermodynamic processes and the analytical expression of the first law.
   a) Classification of processes according to energetic characteristics and the general form of the analytical expression of the first law.
   b) Classification of processes according to the rate of their course; work in nonstatic and quasistatic processes and the expression of the first law for the latter.

¹ Engels. Old Introduction to Dialectics of Nature.

c) Cyclic processes (cycles) and the expression of the first law for the ideal cycle of a heat engine.

1. Enthalpy—a new characteristic of state.
   a) Mathematical definition.
   b) The second form of the analytical expression of the first law.

2. The simplest differential equations following from the first law.
   a) Derivation of

\[ \left(\frac{\partial U}{\partial t}\right)_v = C_v \quad \text{and} \quad \left(\frac{\partial i}{\partial t}\right)_p = C_p . \]

b) On the meaning of expressions of the form \(dZ = Xdx + Ydy\) in thermodynamics.

The Second Law of Thermodynamics

1. General concepts.
   a) The first and second laws—a quantitative and qualitative evaluation of the various forms of energy.
   b) Positive and negative processes; the inequality of individual forms of energy.

2. Theoretical foundations of the second law for quasistatic processes.
   a) A system of \(n\) bodies in thermal equilibrium.
   b) Equations determining the course of quasistatic processes in an adiabatic system.
   c) Schiller’s postulate.
   d) The integrating divisor of thermal energy.
   e) Change of variables and transition to a new value of the integrating divisor.
   f) Investigation of the properties of the factors \(T\) and \(S\) of thermal energy.
   g) Analogies explaining the physical meaning of the factors \(T\) and \(S\).

3. Thermal diagrams.
   a) Construction and properties of the thermal diagram.
   b) Conditions for the continuous conversion of heat into work in heat engines and the efficiency of the ideal cycle.
   c) The Carnot cycle in the \(T\)-\(S\) diagram.
   d) The concept of the absolute temperature scale.

4. Change of entropy of a system of bodies.
   a) Constancy of the entropy of an adiabatic system of bodies in quasistatic processes.
   b) Carathéodory’s principle of adiabatic inaccessibility.
   c) One-sided change of entropy in nonstatic processes in an adiabatic system.
   d) Irreversibility of nonstatic processes.
   e) Conclusions of a natural-philosophical order and technical consequences following from the second law.

Processes of change of state of perfect gases cannot be classed among the methodologically difficult sections of thermodynamics. Here, however, questions of the greatest applied importance are considered; therefore a clear mastery of this section proves absolutely necessary. From this point of view, special attention in this section of the course must be devoted to creating such a (methodologically most correct) arrangement of the material as will ensure the greatest useful effect of the lecture hour.

The fundamental gas laws, the determination of the physical characteristics of gas mixtures, the internal energy and enthalpy of gases, and the first law for ga-

gases ($dQ=C_v\,dt+A p\,dv$ and $dQ=C_p\,dt-A v\,dp$), naturally, must be presented in advance. We believe that the relations $S=f(v,t)$, $S=\varphi(p,t)$, and $S=\psi(p,v)$ should also be established in advance, which (together with general familiarity with the $T$–$S$ diagram) will make it possible to present individual gas processes, illustrating them by both the $p$–$v$ and the $T$–$S$ diagrams.¹

In presenting processes of change of state, one is especially struck by the abundance of formulas (not difficult and easily obtained, but different for each of the characteristic processes), which are usually the chief “stumbling block” for those studying thermodynamics for the first time. Recognizing that the only way to master all these formulas can be systematic practice in applying them, we nevertheless come to the conclusion that their proper systematization can be of substantial help, and consequently special attention must be devoted to this systematization.

We see such systematization, for example, in a number of tables given in the explanations to the section “Gases” in Kvasnikov’s problem book.² However, even such a solution of the question does not satisfy us, since here no parallel is drawn between the methods of determining quantities of the same type for each of the principal characteristic processes. A complete solution of the question of systematization can evidently be achieved only by compiling a summary table of the principal formulas for all characteristic processes, as the author has done in Table II. The author has used such a summary table in teaching work at a higher technical educational institution since 1930, and practice has confirmed its methodological expediency.

The systematization of formulas must, naturally, also be reflected in the very (lecture) process of presenting this section. To this end we consider it necessary to present the individual questions:

a) The equation $f(p,v)=0$.

b) Determination of the final characteristics of the state from the initial ones and from the given character of the process.

c) Energy characteristics ($Q$, $\Delta U$, $\Delta i$, $A l$) of the process;

d) The equation $\varphi(S,T)=0$ and all questions connected with it.

These should be arranged in a definite and identical sequence in the presentation of each characteristic process, comparing, as far as possible, the results obtained with analogous ones for other processes.

The processes $V=\mathrm{Const}$ and $p=\mathrm{Const}$ prove most convenient for such comparison. They can easily (and expediently from the standpoint of saving time) be presented in parallel, in the full sense of the word, which the author accomplishes by dividing the “classroom blackboard” into two parts.

Processes of change of state of gases are usually concluded with the polytropic process as a generalizing one.³ The presentation of the polytropic process should begin from this generalization (i.e., from establishing the value of the “polytropic exponent” for the isochore, isobar, isotherm, and adiabat, and subsequently also the value of the heat capacity). Here, however, particular caution is needed with regard to the very definition of the concept “polytropic process.” Its definition as a process in which some definite fraction of heat is converted into work we regard as an archaism remaining from those times when the variability of heat capacity and technical thermodynamics in general were not taken into account. If this definition is extended also to cases—

…in the case of variable heat capacity, we naturally do not obtain the usual form of the polytropic equation \(pv^n=\mathrm{Const}\), while it is precisely this equation that is used in the practical solution of heat-engineering problems. We

Basic Thermodynamic Processes for Perfect Gases

Conventional designations:

\(L\) — work of the process, \(\mathrm{kg\,m/kg}\)
\(Q=AL\) — work of the process, \(\mathrm{kcal/kg}\)
\(Q_v=\Delta U\) — heat communicated, equal to the change of internal energy
\(Q_p=\Delta I\) — heat communicated, equal to the change of heat content
Time of process: \(\theta\), \(\mathrm{min}\)

Note: Equations marked with an asterisk are valid only when \(C_v=\mathrm{const}\).

Table II.

we consider it most expedient to define the polytropic process as one obeying the equation \(pv^n = Const\) \((n = Const)\), and to be consistent in this definition, retaining it also for the case of variable heat capacity.

Taking variable heat capacity into account in the analytical solution of the problem, even for the simplest dependences \(C_v = f(T)\), generally leads to very cumbersome formulas, and consequently the analytical method in this case \((C_v \ne Const)\) should not in general be recommended. We consider it entirely inexpedient, even in the “most special” course of technical thermodynamics, to engage in deriving complicated formulas (of the order of the adiabatic equation

\[ pv^k \cdot e^{\frac{bpv}{a}} = Const), \]

obtained when the dependence of heat capacity on temperature is taken into account.

THEORETICAL CYCLE OF A STEAM-POWER PLANT.

Diagram of a steam-power plant.

Designations:
\(K\) — boiler
\(P\) — superheater
\(T\) — turbine (machine)
\(K_d\) — condenser
\(E\) — economizer (feed-water heater)

Cycles in \(p\)-\(v\), \(T\)-\(s\), and \(i\)-\(s\) coordinates.

Energy balance.

\[ Q_1 = i_1 - i_2' = \text{area } 0 - 2' - 1' - 1 - b - 0 \]

\[ Q_2 = i_2 - i_2' = \text{area } 0 - 2' - 2 - b - 0 \]

\[ AL_t = i_1 - i_2 = \text{area } 2' - 1' - 1 - 2 - 2' \]

Cycle efficiency.

\[ \eta_t = \frac{AL_t}{Q_1} = \frac{i_1 - i_2}{i_1 - i_2'} = \frac{i_1' - i_2}{i_1' - i_2'} \]

Tab. III.

However, it is necessary to pay special attention to ensuring that the “limits of applicability” (whether only for \(C_v = Const\) or also for \(C_v \ne Const\)) of individual formulas are clearly mastered. For this purpose we mark formulas applicable when \(C_v = Const\) with a special sign (an asterisk in Table II).

Here we consider the theoretical cycle of a steam-power plant as an example of the methodological analysis of a question having direct applied significance. The absence of an element of “living contemplation” in a student who has come to a higher educational institution “from the school bench” is felt especially acutely in such questions, and it is absolutely necessary to fill this gap. From this point of view, we consider it expedient to begin the exposition of the question with an analysis of the theoretical scheme of a steam-power plant (see Table III), in which it is necessary to give (at least the most general) conception of its principal elements and of the processes taking place in them. Here it should also be noted wherein the “theoretical” character of the scheme consists, i.e., which losses we do not take into account and from which phenomena occurring in an actual installation we abstract.

Since the processes of heating a liquid, its evaporation, superheating, adiabatic expansion, and condensation have already been set forth earlier, the explanation of the individual elements of the cycle cannot present difficulties. The most advantageous illustration here is the \(T\)-\(S\) diagram, on which the individual elements of the cycle are plotted (in their actual sequence). At the same time, it is very important, from the standpoint of facilitating comprehension, to have before the eyes of the entire audience the above-mentioned schematic diagram and to coordinate the notation (see Table III).

The closure of the cycle already makes it possible to consider the energy balance of the theoretical installation. Here it is important to establish clearly the concepts of the “upper” and “lower” heat sources and to connect this abstraction with practice by pointing to the principal “source of losses” in a real steam-power installation. The \(i\)-\(S\) diagram here may be considered only from the point of view of the “working” method of determining the efficiency of the Rankine cycle, but the practical significance of such a determination must be emphasized by means of this example.

We consider it advisable to apply the Sankey diagram in thermodynamics as widely as possible; here, too, it lends particular visual clarity to the energy balance and, consequently, must be presented in parallel with the consideration of this question.

3. Lecture Methodology

The principal methodological questions confronting the lecturer, in our opinion, are: what to give? and how to give it? Our point of view on the first question has already been partly reflected above. Here it remains only to draw attention to the circumstance that the voluminous material of the syllabus must always be strictly correlated with the amount of time allotted for presenting the course. We regard as a common mistake of the beginning lecturer the desire to give, at all costs, the maximum amount of material, without taking into account the fact that this often entails excessive haste, irritates the audience, and, as a result, lowers assimilability to such an extent that the ultimate effect of expanding the volume of the material presented becomes negative.

The necessity of a high theoretical level in a higher-education course in thermodynamics is indisputable, but purely applied questions must in no way be sacrificed to it.

The solution of the problem of giving more and better material should be sought chiefly along the line of answering the question how to give it? Here, first of all, one must renounce all unnecessary (“beautiful”) digressions and, through methodological forethought and systematization of the material intended for presentation, make the lecture hour as compact as possible. This task is not an easy one. Its solution depends in large measure on the skill and experience of the lecturer, but it is precisely here that we see one of the possibilities for a concrete rationalization of the methods of work in the educational-production process.

Graphic illustrations are of the greatest importance in thermodynamics, and the question of the technique of using them in the lecture cannot be passed over in silence here. It is indisputable that to squander lecture time on drawing complex diagrams on the blackboard and writing out long tables is inadmissible. However, a thermodynamic scheme is usually not so complicated that, when skillfully used, it takes almost no additional time from the lecturer. The demonstration of ready-made diagrams, without drawing them in parallel on the blackboard, should be treated with a certain caution, since in this case the valuable methodological possibility is lost of concentrating the audience’s attention on the line or point of the diagram that has just been drawn; the drawings, so to speak, lose their dynamic—

ness. In addition, as a rule, it should be assumed that graphic material which has not been drawn on the board does not manage to be reproduced in students’ notes; and if the possibility of the student’s subsequent study of the drawing demonstrated in the lecture is not ensured, this circumstance must already be regarded as a negative one.

We consider optimal such an arrangement in which the lecturer has a demonstration drawing at his disposal, but accompanies its demonstration by drawing the simplest diagram on the board¹; the listeners, meanwhile, have previously reproduced copies of the demonstration drawing, or else are given the opportunity to reproduce these copies outside lecture time.

The most recent reform of the methods of work in higher education, shifting the center of gravity to the student’s independent work, does not relieve the teacher of the obligation to ensure the maximum effectiveness of this work. Hence the content of lectures must not be limited merely to bare theory, but must indicate the paths and methods of applying this theory to purely practical questions. From this point of view, we also consider the significance of an example in a lecture, which, obviously, need not at all be numerical, but must undoubtedly provide a general method for resolving the given question or an entire group of them.

Important from the methodological point of view, we also consider the clear presentation to listeners of that logical plan in accordance with which the lecture is delivered. Without speaking of the necessity of composing such a plan, with a clear subdivision of all the questions considered and an indication of their connection and subordination, when preparing for a lecture, we consider it necessary, in one way or another, to reflect this plan in the very process of its exposition.

The elementary task of singling out the headings of the individual questions considered by the lecturer acquires special significance from this point of view. These headings must be announced before the direct exposition of the question being touched upon, since only in this case will the listener have a clear idea of the principal thought of the material being presented.

We consider optimal such an arrangement in which each listener, before the beginning of the lecture, receives a detailed plan of its content; if this is impracticable for technical reasons, then the plan must be posted in advance for the information of those interested.

4. Methodology of Practical Classes.

The main task of practical classes in thermodynamics, naturally, must be mastery of the methodology of applying the fundamental thermodynamic propositions to the solution of concrete questions of heat engineering. Mastery of the theoretical questions of the course, apart from their application to practical questions, should be regarded as purposeless and practically unrealizable. Practical classes must thus take the form of exercises.

A methodologically correct organization of exercises must ensure, on the one hand, the final elimination of survivals of the brigade-laboratory method (cramming), and, on the other—the maximum effectiveness of the use of every hour of classes with the instructor, enabling the latter to transmit his experience and knowledge to students in the process of their independent—

and active work. This is possible provided that the role of the supervisor is not reduced merely to the duties of a consultant—in the final analysis, a passive one. The activity of the supervisor must be expressed in the ability to take proper account of the situation, to notice characteristic errors and difficulties in the independent work of individual students and, by generalizing them, to give a clear critique of the methods used for solving typical problems, making it possible to establish the most convenient and correct methodology for resolving all the basic questions of the given section. The methodological organization of exercises must ensure the fulfillment of this difficult task; for this reason we consider the following scheme for using the hours allotted by the curriculum for this type of class to be the most correct:

1) The entire group of students served by the given supervisor solves problems of the same type¹, assigned by the supervisor in accordance with a previously developed concrete calendar plan of classes.

2) The supervisor goes around to the students, gives individual instructions, and notes characteristic errors and difficulties.

3) At the end of the class the supervisor gives general instructions, in the nature of the critique, mentioned above, of the methods used in independent work. We consider this last stage necessary; however, it should not occupy more than 10–15% of the total time budget for the exercises.

5. The Student’s Independent Work.

In accordance with the reform of higher education being carried out pursuant to the June resolution of the Council of People’s Commissars and the Central Committee of the Party, the main emphasis must be placed on the student’s independent work. Meanwhile, the practice of the first semester of the 1936/37 academic year at the Tomsk Industrial Institute showed that the majority of students do not know how to organize their independent work properly and do not conduct it sufficiently systematically.

As a rule, the hours of noncompulsory classes in the first half of the semester were used at least insufficiently, and only in the last (pre-examination) period did the activity of the student body increase significantly. In this respect we see the main shortcoming in the implementation of the decree of the Party and the government both on the part of the student body and on the part of the institute’s professorial and teaching staff.

Systematic work on the course in thermodynamics is especially necessary because of the usual difficulties in mastering this discipline, and the basic task of the pedagogue (both as lecturer and as supervisor of practical classes) here should be recognized as the development among the student masses of an awareness of the need for systematic and everyday work on the course, as well as assistance in finding the correct and most effective methods of independent work.

Certain assistance must also be provided in the planning of the student’s independent work throughout the entire semester, and, from this point of view, the practice of assigning homework with an indication of the deadline for its completion should merit special attention.

We consider it possible to achieve an increase in activity in the use of noncompulsory hours by introducing an element of planning into this part of the work as well. Alongside the usual consultations, at which in most cases the initiative belongs to the student who has come for consultation, we consider it necessary (especially at the beginning of the se—

...semester) a portion of the optional hours should be devoted to exercises on a definite topic or section of the course, with interested students being notified of the specific content of these exercises, or even to lectures on a topic, for example, on methods of solving problems in the given section. Such an organization of the work will undoubtedly increase attendance at optional classes and will serve as an incentive to systematic work on the course from the very beginning of its study.

The tasks of the department should also include the creation of material conditions that ensure the highest productivity of the student’s independent work. From this point of view, we consider it necessary to create a subject room for thermodynamics, which should serve as the student’s workplace when independently working through the course. Here the student should have the opportunity to use the principal and any auxiliary literature, reference books, nomograms, calculating and measuring instruments, etc.; the main thing, however, should be considered the creation in such a room of a set of visual aids that would make it possible to study the program content of the course in the most productive and systematic way.

Not being able, within the framework of the present work, to analyze in detail the question of visual aids, we shall merely note that the practice of compiling diagram-tables similar to those considered above (in the methodological analysis of individual questions of the course) and posting them in the subject room, in order to ensure their use in students’ independent work, has yielded a positive result; and we shall give a list of diagrams and tables of the “first order,” which provide the minimum set of visual aids for the course in technical thermodynamics:

1. Heat capacities (by mass, by volume, molar, mean and true, $C_p$ and $C_v$).
2. Circular processes (cycles).
3. Logical scheme of the theoretical substantiation of the second law (Table I).
4. Processes of change of state of perfect gases (Table II).
5. Basic cases of the polytropic process ($n<1$, $1<n<k$, $n>k$).
6. Working process of a compressor.
7. Theoretical cycles of internal-combustion engines.
8. Principal properties of vapors.
9. Theoretical cycle of a steam-power plant (Table III).
10. Flow of gases and vapors.

It is quite clear that, as the equipment of the room is expanded, this list should be supplemented with material corresponding to the specific features of the programs of the thermodynamics courses served by the room.

1. The uniformity of the problems solved in the given classes is necessary, since only in this case will the supervisor have the opportunity to give the general instructions noted below. ↩↩↩↩↩

2. A. V. Kvasnikov, Problems and Exercises in Technical Thermodynamics. ONTI, 1934. ↩↩↩

3. It is hardly expedient to begin with the polytropic process and consider all the rest as its particular cases. ↩