© Copyright by Wydawnictwo Poligraf, 2025

© Copyright by Andrzej Tomczyński, 2025

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This book would likely has not been written if not for my Mom, Genowefa, because I still remember what she said to me when I was in the grammar school, and for what I never managed to thank her, because I wasn’t aware how strongly it shaped by thinking and my life. Thank you, Mom.

So, my Mom said to me, that if I understand what I am taught in the science classes, that it is good and I should feel good about it, but if I don’t understand something, then I need to try to remember something of these, or maybe even learn it by rote and then patiently wait until it is eventually clarified sometime during my education or life in general.

So whenever I had an opportunity, I have been talking, with my Friends, now with Women, and also with my daughter Izabela since she was born, about various theoretical and practical issues, including the topics I covered in this book, because such conversations can be inspirations, and also can solve some conundrums, exactly like my mother told me.

INTRODUCTION

From the Prof. W.R. Gundlach’s book Z Wysp Brytyjskich przez Turbinową Dolinę Szwajcarską w dwudziesty pierwszy wiek: szkic rozwoju cieplnych maszyn energetycznych od czasów Herona z Aleksandrii [From the British Isles through the Swiss Turbine Valley into the twenty-first century: a sketch of the development of thermal energy machines since the time of Heron of Alexandria], [1] I learned that when the Stirling engine has been patented (in 1816), it was impossible to perform calculations of the thermodynamic process happening in this endige, and that in 21st century many afficionados will still be working on the scientific and technical analysis of the achievements and construction intricacies of this peculiar engine.

Thus, as one of the aforementioned afficionados, I decided to perform calculation of the Stirling’s thermodynamic circuit, implementing them in a specific calculation model that consisted in the imitation of Stirling’s engine. I have quickly determined that these need to be iterative calculations what requires entropic functions of a gaseous medium ‘filling’ this computational model.

Looking for necessary literature, I haven’t found any that would consider entropic functions of some gaseous medium, being able to find only books showing, e.g. theoretical basics of the s-i plot (an entropy-enthalpy chart) for the dry air, with an attached printout of said chart on a A1 format sheet. Namely, it is a book Wykresy entropowe dla powietrza i spalin [Entropic diagrams for air and exhaust gases] by B. Kaczan, W.R. Gundlach and S. Czarnecki [2].

So I performed entropic functions for the dry air, consulted them with the aforementioned chart, and then made thermodynamic calculations for Stirling circuit. The results of this work, collected in the book Obliczenia pól pracy i pól ciepła silnika Stirlinga z  napędem rombowym [Calculations of work fields and heat fields of a Stirling engine with a rhombic drive], was published by the Łódź Technical University Publishing House in September 2022 [https://wydawnictwo.p.lodz.pl/ (in the window please enter ‘Tomczyński’ to select this book)].

I am aware that it would be better if I ‘filled’ the computational model of this engine by helium. Sure, I could have also made entropic functions for helium, but with the assumption that for helium I will use the real gas state equation formulated by Berthelot as ‘B2’. This is because in the book [2], all the information necessary to perform specific entropic and enthalpic functions are available only for the Berthelot’s ‘B2’ real gas state equation.

I would add that not only could I not know about some of this information at the time, but also explain why it is related to this equation.

Thus, I have presented the reasons why I wrote this book. I am a Stirling engine enthusiast, as I have already written about, but also an egocentric, what in this specific case means that I have been spending a lot of time to answer the following questions for myself, in order to later pass on the findings to others:

Why are there so many authors of the real gas state equations, and due to what specific merit have they became the ‘classics’ of thermodynamic books?

Why is it that no one used the real gas state equations by these authors to create thermal engineering tools, which could then be made available for enthusiasts who want to perform various thermal calculations?

I was to answer the first of the above questions by a well-known saying, or more precisely, its version immortalized by Honoré de Balzac 160 years ago, that the synonyms of beauty or ideals of perfect beauty are: a beautiful dancing woman, a thoroughbred horse at a gallop, and a frigate under full sail. To these, I would add also ‘graphs of equations’, including the graphs of the real gas state equations established by the authors cited above.

Of course, I will want to show this in this book, and I also think that I have already answered the second question above with this statement. I will also point out that the subtitle of this book (Functions calculating...) also relates to the answer to this second question.

Naturally, making such assumptions is associated with the need to do much more work than just that of analysing the real gas state equations and describing this process. Additionally, in a widely available application in MS® Excel, thermal functions of the real gas state equations will be created, which functions, as modern thermal engineering tools, will be applied to perform engineering calculations. The results of these engineering calculations will then be demonstrated as graphs throughout this book, and the buyers of this book will also be able to use them for other thermal calculations of dry air or as reference for performing thermal functions of other gases. It is worth noting that paper printouts, e.g. of thermal functions of gas equations of state, are similar thermal engineering tools, just more at home in the previous century.

With the above in mind, I’d like to inform you that in this book I will be using the ‘philosophical’ terms possibly concerning ‘theory’, namely, perhaps a bit oddly, ‘theory as such’ (usually omitting ‘as such’) and ‘theory concerning practice’. What is most significant about what I have written above is that I have chosen to identify specific thermal concrete issues by describing the successive equations of the real gas state and show them in diagrams, that is, to write and show a thermic ‘crime story’. I have chosen to use such a comparison because all the mysteries will be explained only at the end of the book.

I am aware that not everyone likes crime stories and such an approach to scientific theories. Hence, I apologize in advance to scientists and those readers who do not like detective literature – they can, if they wish, use only the Functions calculating..., which I made not only for the purpose of this book. This said, I wish remaining readers good intellectual and engineering ‘fun’.

At the same time, I must add that, just as the best real gas state equations were not established immediately, so certainly this book should also be corrected, supplemented and modified, and it would be good for this to be influenced by its readers, including users of thermal functions of real gas state equations. In other words, by enthusiast-theorists like me. This is because what I will be describing, is effectively a historical reconstruction of the thinking of the creators of the real gas state equations from the 19th and 20th centuries, and that in this reconstruction I am using my own and hopefully already 21st century thinking.

At this point, I will admit that I was also inspired to write this book J.C. Lennox’s God’s Undertaker: Has Science Buried God? [3], which among other things, includes two questions starting with ‘Why...?’ – a crucial question and a common one. This key question is: ‘Why does the Universe exist and why did Life, including mine, arise in it?’, and the common one is ‘Why did Aunt Matilda bake a cake?’. The author states that these are questions of intention, and for this reason science only plays a supporting role here; in the case of the first question, such a role is also played by philosophy. So it can be said that people have or support one of the two world-views and answer the first question like this: ‘Because that’s the way God wanted it, and he told us a little about it by revealing it to us’ or that ‘It was caused by Eternal Matter, because it has a nature of randomness and necessity, and this led to the creation of Life, including humans’.

Since I mentioned the philosophy that advises us to think, I will quote below a statement by the philosopher from Königsberg, Immanuel Kant (from the end of the Critique of Pure Reason), because it appeals to me and because I live in Elbląg, located not far from Königsberg, today called Królewiec. The quote is: ‘Two things fill my mind with ever-new and intensifying admiration and reverence the more often and permanently I reflect on them: the starry sky above me and the moral law within me’.

Returning to the question about Aunt Matilda, it can be said that science is only able to determine the ingredients of this cake or its caloric content, in order to assume that it is an occasional cake, i.e. one that is better than a cake baked every day. We will know the truth if Aunt Matilda tells us why she baked this cake, and this truth may also include a condition that we do not include in our assumptions, because, for example, Aunt Matilda could tell us ‘I baked the cake for my fourth grandson’s birthday, but my birthday is a day earlier, and he will get this cake when he visits to wish me a happy birthday’. With this in mind, I am aware that I have chosen to write a difficult and problematic book – to guess the intentions of the authors of the real gas state equations. In order to reduce the threat of criticism, I derive the formulas, reconstruct the 19th and 20th century establishment of physical quantities (taking into account the possibilities regarding these periods, and perhaps the ‘scientific fashions’ promoted at the time), make and describe graphs, and make available the thermal functions of dry air, so that all this can be quickly verified.¹

The error I found in the literature shows and describes its effects – for one can come to a false conclusion without knowing that they reference to the error. Anyway, I have to admit that one of the diagrams I made showed a beautiful justification of why the author of the real gas state equations, which is the reference to this diagram, deserved to become a ‘classic’ of thermodynamic books, but I had to completely change this justification because, double-checking my work, I discovered that I had made a mistake in deriving the equation of a certain function.

After this brief introduction, we can move on to identifying specific thermic specifics, as I promised before.

The following is a table from the book [2], as it compiles the real gas state equations, the theory and practical application of which I will present in this lecture.

Fig. I.1. Basic real gas state equations
Source: [2].

Some information necessary to recognize the theory of the real gas state equations present in the table in Fig. I.1 are present in the specific references that is also shown in this book.

Fig. I.2. References
Source: [2].

However, after determining this literature, I came to conclusion that it would be very difficult to reach its specific books, so I decided to refer to the Internet. After checking available sources, I found that the books and papers made available here and dealing with the subject of the real gas state equations provide information that is missing from the book [2] or that is the book that contains data absent in these online publications. In addition, graphs are published without showing the scale on the axes or information about the gas involved, and containing inaccurate technical descriptions. Thus for me these are more thermal artistic works, not thermal (thermodynamic) engineering works. This is because the latter need to reference specific engineering calculations, in the performance of which specific mathematical methods are applied and specific engineering tools are used.

Therefore, for the time being, I am attaching the ‘artistic’ diagrams, just to determine what phase of the gaseous dry air state I will address in this book. I also attach diagrams that were already so corrected by me to show this.

Fig. I.3. Family of van der Waals’ isotherms on the p-V-T diagram
Source: [4].

Fig. I.4. Family of van der Waals’ isotherms on the p-V-T diagram
Source: [4] with own correction.

Technically refining Fig. I.3 into Fig. I.4 with my own correction, I extracted the so-called ‘vapour’ phase from the vapour phase of some agent (substance) and called it ‘vapour (gas)’. In the case of water, its state in the vapour phase is called ‘dry (superheated) steam’. Under the thermal conditions (p, T) of the atmosphere that surrounds us, various substances are in solid, liquid or gas phase and can, mainly under the influence of temperature, change this phase state. Hence, for water, the nomenclature given above has been adopted.

For a substance such as air, we are surrounded by its ‘gaseous’ phase state, and it cannot be liquefied using any sort of pressure, unless its temperature is lowered below the critical temperature – for air it is crT_air = 132.5 [K] (crt_air = –140.65 [deg. C]), i.e. it is not brought to the ‘vapour (gas)’ gaseous state. Because of this, I have singled out this ‘vapour (gas)’ gas state and shown it in Fig. I.4, so that it is already clear at this stage that in the book I will not deal with this gas state of gases, i.e. with ‘vapour (gas)’, including dry air.

Thus, the word ‘gas’ used in the book means the vapour phase of a substance (dry air) temperature of which is equal to or greater than its critical temperature. Consequently, I also corrected Fig. I.5.

Fig. I.6. Van der Waals’ isotherms.
C – critical point, GP – phase transformation liquid-gas at a constant pressure
Source: [4] with own correction.

Fig. I.5. Van der Waals’ isotherms.
Cr – critical point, GP – phase transformation liquid-steam (of gas) at a constant pressure
Source: [4].

I also need to point out that in this book, the descriptions will not replace diagrams, as is practised in, e.g. Wikipedia [5] – see Fig. I.7 and Fig. 3.1.

Fig. I.7. Table row related to van der Waals’ equation
Source: [5].

And also for the fact that in Fig. I.7 the real gas state equation is presented in molar form, while in engineering practice such equations are used in mass form. Because of this and in order to have everything at hand, I begin this book with the perfect gas state equation – the ‘BC’ equation of Benoît Clapeyron (1834) [6].

¹ In addition to the sentence followed by this reference, it should be stated that in this book formulas and numbers have been copied from MS® Excel specific calculation sheets or macro sheets. In these worksheets, requirements for signs used in writing negative and decimal numbers had to be applied, and this applies to forms of ordinary numbers and forms of scientific numbers, and that requirements for the use of spaces in the formulas before or after the signs of mathematical operations used had to be applied.

In contrast, in the book, editorial formatting of the text is used, and this sometimes made it look strange when it was used without intrusive copying of numbers, formulas, and names of physical quantities in which the signs of mathematical operations are used.

Hence, in some cases, the comma was replaced with a dot in numbers, in formulas and in the titers of physical quantities the space was eliminated, etc.

Such interferences were made as little as possible, because in order to make the text in the book look pleasant, it was also decided that it could also be used, almost unchanged, in the above-mentioned MS® Excel sheets, in the case of verification of the author’s calculations or to perform analogous calculations relating to a medium other than air.

Therefore, it is noted that differences in notation, especially in decimal numbers, formulas, are not errors or lack of consistency, but that they are indications that in the same spreadsheets in MS® Excel different characters are used in the notation of numbers, and in formulas spaces are used or not used.

INDEX OF IMPORTANT INDICATIONS

Used in Chapter 0

1

m_sg; m_a

mass of the specific gas; air [kg]

2

M_sg; M_air

mole mass of the specific gas; air [kg/mol]

3

n = m_sg / M_sg;

n = m_air / M_air

number of moles of the specific gas; air [–]

4

R

universal gas constant [J/(mol*deg.)]

5

R_sg = R / M_sg;

R_air = R / M_air

gas constant of specific gas; air [J/(kg*K)]

6

V, V_sg; V_air

gas volume, specific gas; air [m ^3]

7

v_sg = V_sg / m_sg;

v_air = V_air / m_air

gas specific volume, specific gas; air [m^3/kg]

8

v, v_sg;

v_air

specific volume, specific volume of the specific gas; air [m^3/kg] – applied technical parameter

9

T, T_sg, t_sg;

T_air, t_air

gas temperature, temperature of the specific gas; air [K], (t)[deg. C] – applied technical parameter

10

p, p_sg;

p_air

gas pressure, pressure of the specific gas; air [N/m^2], [bar] – applied technical parameter

11

V_sgexp, V_sgexp,
m_sgexp; v_airexp,

V_airexp, m_airexp

respectively: specific volume, volume, mass of the specific gas; air determined in an experiment (cylinder with piston)

12

v_sgBC, m_sgBC;
v_ airBC,

m_airBC

respectively: specific volume, mass of the specific gas, volume of air calculated as not provided thermal equivalent, i.e. Benoît Clapeyron ‘BC’ state equation is satisfied

13

p_sgBC, T_sgBC;

p_ airBC, T_airBC

respectively: pressure, temperature of the specific gas; air calculated as unspecified thermal equivalent satisfying Benoît Clapeyron’s ‘BC’ state equation

14

crT_sg, crt_sg;

crT_air, crt_air

critical temperature of the specific gas; air [K],

[deg. C]

15

crp_sg; crp_air

critical pressure of the specific gas; air [N/m ^2], [bar]

16

t_N.C.

gas temperature in normal conditions [deg. C]

17

p_N.C.

gas pressure in normal conditions [bar]

18

VM_air

volume [m ^3] [litre] 1 mole of air [m ^3/kmol], (= [l/ mol]), regardless of equation it has been calculated with

19

compf_cpBC

compression factor in the critical point of the ‘BC’ Benoît Clapeyron’s equation – RN (rational number), quotient with ‘x = 1’ in the dividend and natural number ‘y = 1’ in the divisor

20

crv_sgBC;

crv_airBC

critical volume of the specific gas; air in the critical point of the ‘BC’ Benoît Clapeyron’s equation

Used in Chapter 1

21

a_sgVDW;

a_airVDW

coefficient in the correction increasing pressure of the specific gas; air in the ‘VDW’ van der Waals’ equation

22

Dep_sgVDW;

Dep_ airVDW

correction increasing pressure of the specific gas; air in the ‘VDW’ van der Waals’ equation

23

b_sgVDW;

b_airVDW

coefficient reducing the volume of the specific gas; air in the ‘VDW’ van der Waals’ equation

24

Dev_sgVDW;

Dev_ airVDW

reduced specific volume of the specific gas; air in the ‘VDW’ van der Waals’ equation, by the coefficient above (23)

25

p_sgVDW;

p_airVDW

pressure of the specific gas; air calculated as unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ equation

26

crv_sgVDW;

crv_airVDW

specific volume for the specific gas; air in the critical point of ‘VDW’ van der Waals’ equation

27

d(cp_sgWDW) /

d(crT_ sg, crv_sg)

derivate of the function cp_sgVDW =

= f (crT_sg, crv_sg) by crv_sg (25)

28

compf_VDW

compression factor in the critical point of the ‘VDW’ van der Waals’ equation – RN (rational number), quotient with ‘x = 3’ in the dividend and natural number ‘y = 8’ in the divisor

29

T_sgVDW;

T_airVDW

temperature of the specific gas; air calculated as unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ equation

30

v_sgVDW;

v_airVDW

specific volume of the specific gas; air, calculated as unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ equation (final result of the iterative calculations)

31

Cv_sgVDW2;

Cv_ airVDW2;

Cv_sgVDW1;

Cv_ airVDW1;

Cv_sgVDW0;

Cv_ airVDW0

coefficients of the specific volume of the specific gas; air; for the specific volume in second and third power and free expression, in the third degree polynomial equation used to calculate the unspecified thermal equivalent satisfying the ‘VDW’ van der Waals’ equation

32

LSEv_VDWit

left side of the polynomial equation used to calculate specific volume of the specific gas, air as a unspecified thermal equivalent satisfying the ‘VDW’ van der Waals’ equation

33

v_sgVDWit0;

v_airVDWit0

specific volume of the specific gas; air, assumed in the iteration 0 in the iterative calculations, where this value is calculated as a unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ equation

34

RSEv_VDWit

right side of the polynomial equation used to calculate specific volume of the specific gas, air as a unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ equation

35

Ul_ v_sgVDWit;

Ul_ v_airVDWit

upper limit of the iterated specific volume of the specific gas; air, calculated as a unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ state equation

36

Ll_ v_sgVDWit;

Ll_ v_airVDWit

lower limit of the iterated specific volume of the specific gas; air, calculated as a unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ state equation

37

v_sgVDWit1;

v_airVDWit1

specific volume of the specific gas; air, calculated in the iteration 1 (and subsequent ones) to determine whether it is an upper or lower limit of the calculated unspecified thermal equivalent fulfilling the ‘VDW’ van der Waals’ state equation, when upper and lower limit of this iterated value is calculated (determined) (35) and (36)

38

Mov_sgVDWit1_0|

module of difference of the specific volumes calculated in the subsequent iterations as a not provided thermal equivalents fulfilling the ‘VDW’ van der Waals’ state equation; written this way it applies to iteration 1 and iteration 0

Used in Chapter 2

39

a_sgD;

a_airD

coefficient in the exponent of the factor-correction, Euler’s number ‘e’, of the specific gas; air in the Dieterici’s ‘D’ equation

40

b_sgD;

b_airD

coefficient decreasing specific volume of the specific gas; air in the Dieterici’s ‘D’ equation

41

crv_sgD;

crv_airD

specific volume of the specific gas; air in the critical point of Dieterici’s ‘D’ equation

42

compf_cpD

compression coefficient in the critical point of the ‘D’ Dieterici’s equation – IN (irrational number), quotient with ‘x = 2’ in the dividend and number

‘y = e^2’ in the divisor

43

p_sgD;

p_airD

pressure of the specific gas; air, calculated as a not provided thermal equivalents fulfilling the ‘D’ Dieterici’s state equation

44

T_sgD; t_sgD;

T_airD; t_airD

temperature of the specific gas; air [K], [deg. C], calculated as a unspecified thermal equivalent fulfilling the ‘D’ Dieterici’s state equation (final result of the iterative calculations)

45

v_sgD;

v_airD

specific volume of the specific gas; air, calculated as a unspecified thermal equivalent fulfilling the ‘D’ Dieterici’s state equation (final result of the iterative calculations)

46

T_sgLitLlmin;

T_airLitLlmin

lower limit – minimal temperature assumed for the iterative calculations of the temperature of the specific gas; air as a unspecified thermal equivalent fulfilling the ‘D’ Dieterici’s state equation (final result of the iterative calculations)

47

T_sgUitULst;

T_airUitULst

temperature assumed to determine at the beginning of the iterative calculations whether it is an upper or lower limit of the iterated temperature of the specific gas; air, calculated as a unspecified thermal equivalent fulfilling the ‘D’ Dieterici’s state equation (final result of the iterative calculations)

48

v_sgLitUlmin;

v_ airLitUlmin

lower limit – minimal specific volume assumed for the iterative calculations of the specific volume of the specific gas; air, as a unspecified thermal equivalent fulfilling the ‘D’ Dieterici’s state equation (final result of the iterative calculations)

49

v_sgLitULst;

v_ airLitULst

specific volume assumed to determine at the beginning of the iterative calculations whether it is an upper or lower limit of the iterated specific volume of the specific gas; air, calculated as a unspecified thermal equivalent fulfilling the ‘D’ Dieterici’s state equation (final result of the iterative calculations)

50

Ul_TStit;

Ll_TStit

upper; lower start limit of the iterated temperature, refers to the specific gas, including air, calculated as a unspecified thermal equivalent fulfilling a specific equation when they are used for that purpose (e.g. they can relate to Dieterici’s ‘D’ state equation)

51

Ul_vStit;

Ll_vStit

upper; lower start limit of the iterated specific volume, refers to the specific gas, including air, calculated as a unspecified thermal equivalent fulfilling a specific equation when they are used for that purpose (e.g. they can relate to Dieterici’s ‘D’ state equation)

52

DeVarUl_Tit

minimal difference of temperatures assumed to establish either upper or lower limit described in (49)

53

DeVarUl_vit

minimal difference of specific volume assumed to establish either upper or lower limit described in (50)

54

MultDeVar

multiplier of minimal difference of temperatures of specific volume assumed to establish either upper or lower starting limit described in (49) or (50)

55

Ul / Ll_TStit

formula for establishing either upper or lower starting limit after using one of the limits (49) and (51), (53)

56

Ul / Ll_vStit

formula for establishing either upper or lower starting limit after using one of the limits (50) and (52), (53)

57

Ul_Tit / Ll_it

temperature of the specific gas; air calculated in the iteration 1 (and subsequent ones) to check whether it is a new upper or lower limit of the calculated unspecified thermal equivalent satisfying state equation, e.g. Dieterici’s ‘D’ equation, when upper and lower limit of this iterated value is already determined (calculated), e.g. in (49), (54)

58

Ul_vit / Ll_vit

specific volume of the specific gas; air calculated in the iteration 1 (and subsequent ones) to check whether it is a new upper or lower limit of the calculated unspecified thermal equivalent satisfying state equation, e.g. Dieterici’s ‘D’ equation, when upper and lower limit of this iterated value is already determined (calculated), e.g. in (50), (55)

59

Avl_Titb;

Avl_Tite

arithmetic average of the upper and lower limit of the iterated temperature, calculated as unspecified thermal equivalent satisfying state equation, e.g. Dieterici’s ‘D’ equation, at the beginning and end of every iteration step

60

Avl_vitb;

Avl_vite

arithmetic average of the upper and lower limit of the iterated specific volume, calculated as unspecified thermal equivalent satisfying state equation, e.g. Dieterici’s ‘D’ equation, at the beginning and end of every iteration step

Used in Chapter 3

61

a_sgB1;

a_airB1

coefficient in the correction increasing the pressure of the specific gas; air in the ‘B1’ Berthelot’s equation

62

b_sgB1;

b_airB1

coefficient decreasing the specific volume of the specific gas; air in the ‘B1’ Berthelot’s equation

63

crv_sgB1;

crv_airB1

specific volume of the specific gas; air in the critical point of the ‘B1’ Berthelot’s equation

64

crpcalc_airB1a

auxiliary critical point on the critical crt_sg (crt_air) – hyperbole line that is not a hyperbole, i.e. a line with monotonic descent bisected by line of calculated critical pressure, e.g. crprcalc_airB1 = 13.95 [bar])

65

compf_crpB1

compression coefficient in the critical point of the ‘B1’ Berthelot’s equation – RN (rational number), quotient with natural number ‘x = 4’ in the dividend and natural number ‘y = 9’ in the divisor

66

p_sgB1;

p_airB1

pressure of the specific gas; air, calculated as unspecified thermal equivalent satisfying the ‘B1’ Berthelot’s state equation

67

WT_sg1B1;

WT_ pow1B1;

WT_sg0B1;

WT_ pow0B1

temperature coefficients of the specific gas; air at the temperature to the first power and an absolute term in the polynomial quadratic equation from which the unspecified thermal equivalent satisfying ‘B1’ Berthelot’s state equation is calculated

68

Ll_T_sgB1;

Ll_t_sgB1;

Ll_T_airB1;

Ll_t_ powB1

lower limit (minimum) of the temperature of the specific gas; air assumed to exclude one of the roots of the polynomial quadratic equation from which the unspecified thermal equivalent satisfying ‘B1’ Berthelot’s state equation is calculated

69

Ul_T_sgB1;

Ul_t_sgB1;

Ul_T_airB1;

Ul_t_ powB1

upper limit (minimum) of the temperature of the specific gas; air assumed to exclude one of the roots of the polynomial quadratic equation from which the unspecified thermal equivalent satisfying ‘B1’ Berthelot’s state equation is calculated

70

T_sgB1; t_sgB1;

T_airB1; t_airB1

temperature of the specific gas; air [K], [deg. C], calculated as the unspecified thermal equivalent satisfying ‘B1’ Berthelot’s state equation – the result of calculations is one of the roots of the polynomial quadratic equation taking into account the temperatures provided in (68), (69)

71

crTcalc_sgB1,

crTcalc_ airB1;

crtcalc_sgB1,

crtcalc_airB1

critical calculational isotherm – [K], [deg. C], parameters calculated as the unspecified thermal equivalent satisfying ‘B1’ Berthelot’s state equation and playing the same role as the critical isotherm in the van der

Waals’ ‘VDW’ state equation

72

v_sgB1;

v_airB1

specific volume of the specific gas; air, calculated as the unspecified thermal equivalent satisfying ‘B1’ Berthelot’s state equation (final result of the iterative calculations as in (31)–(38))

73

Wv_sg2B1;

Wv_air2B1;

Wv_sg1B1;

Wv_air1B1;

Wv_sg0B1;

Wv_air0B1

specific volume coefficients of the specific gas; air at the volume to the second and third power and an absolute term in the third-level polynomial equation from which the unspecified thermal equivalent satisfying ‘B1’ Berthelot’s state equation is iteratively calculated

Used in Chapter 4

74

a_sgB2;

a_airB2

coefficient in the amendment increasing the pressure of the specific gas; air in the ‘B2’ Berthelot’s equation

75

b_sgB2;

b_airB2

coefficient decreasing the specific volume of the specific gas; air in the ‘B2’ Berthelot’s equation

76

crv_sgB2;

crv_airB2

specific volume of the specific gas; air in the critical point of the ‘B2’ Berthelot’s equation

77

crpcalc_airB2a

auxiliary critical point on the critical isotherm crt_sg (crt_air) – a hyperbole line that is not a hyperbole, i.e. a line with monotonic descent (bisected in point crpB2a by the non-natural (negative) line of the critical calculational pressure crprcalc_airB2 = –22.04 [bar])

78

compf_crpB2

compressibility factor in the critical point of the ‘B2’ Berthelot’s equation – RN (rational number), quotient with natural number ‘x = 9’ in the dividend and natural number ‘y = 32’ in the divisor

79

p_sgB2;

p_airB2

pressure of the specific gas; air, calculated as unspecified thermal equivalent satisfying the ‘B2’ Berthelot’s state equation

80

WT_sg1B2;

WT_ pow1B2;

WT_sg0B2;

WT_air0B2

temperature coefficients of the specific gas; air at the temperature to the first power and an absolute term in the polynomial quadratic equation from which the unspecified thermal equivalent satisfying ‘B2’ Berthelot’s state equation is calculated

81

T_sgB2;

t_sgB2;

T_airB2;

t_airB2

temperature of the specific gas; air [K], [deg. C], calculated as the unspecified thermal equivalent satisfying ‘B2’ Berthelot’s state equation – the result of calculations is one of the roots of the polynomial quadratic equation taking into account the temperatures provided in (68), (69)

82

v_sgB2;

v_airB2

specific volume of the specific gas; air calculated as unspecified thermal equivalent satisfying ‘B2’ Berthelot’s state equation (final results of the iterative calculations as in (31)–(38))

83

Wv_sg2B2;

Wv_air2B2;

Wv_sg1B2;

Wv_air1B2;

Wv_sg0B2;

Wv_air0B2

specific volume coefficients of the specific gas; air at the specific volume to the second and first power and an absolute term in the third-level polynomial equation from which the unspecified thermal equivalent satisfying ‘B2’ Berthelot’s state equation is iteratively calculated

84

v_sgB2min;

v_airB2min

quadratic function of the specific gas; air related to the temperature, v_airB2min = f(a2 * t^2 + a1 * t + a0), setting the limit of the specific volume from which the ‘B2’ Berthelot’s state equation can be applied

85

B_sgWB2;

B_airWB2

specific gas factor that is a function of temperature for ‘p’ in the virial equation that is an expansion of the ‘B2’ Berthelot’s equation into an exponential order

86

C_sgWB2;

C_airWB2

specific gas factor that is a function of temperature for ‘p^2’ in the virial equation that is an expansion of the ‘B2’ Berthelot’s equation into an exponentialorder