Thermal Engineering

Citation preview

I. SHVETS M. KONDAK N. KIRAKOVSKY I. NEDUZHY D. SHEVTSOV I. SHELUDKO

THERMAL ENGINEERING

PEACE PUBLISHERS MOSCOW

H. T. LUaeu,, M. A. Kondaic, H. . KupaKoacKua H. A. H edym uu, JX. C. lUeayoe, M. M. UJeAydbKO

OBLUAJI TEnJIOTEXHMKA

rOCyHAPCTBEHHOE HAVJHO-TEXHH'IECKOE H3AATEJlbCTBO MAIllHHOCTPOHTEJIbHOtt JIHTEPATyPM M oc

k

ea

I. SHVETS, M. KONDAK, N. KIRAKOVSKY, ]. NEDUZHY, D. SHEVTSOV, I. SHELUDKO

THERMAL ENGINEERING Translated fro m the Russian by S. SEMYONOV and R. FESENKO

Translation Editor Q. L E I E

PE ACE

P U B L I S H E R S . M O S C O W

UDC 621.036 + 662.61.9(075.8) = 20

The book has been written by a group of scientists working for many years at Higher Educational Institutions of the Soviet Union and is a textbook for students both of Higher and of Secondary Schools. The book contains the theoretical fundamentals of thermal engineering (engineering thermodynamics and heat transfer), con­ tains characteristics of fuels, describes combustion processes, boiler units and heat engines, such as steam engines, internal-combustion engines, steam and gas turbines and steam power plants. Besides being a textbook for students, the book will also be of interest to specialists in the field of thermal engineering.

CONTENTS

Foreword

................................................................................................................. 11 SECTION ONE. ENGINEERING THERMODYNAMICS

Chapter I. Basic Concepts of Therm odynam ics.....................................................13 1. Parameters of Gases and Relation between Them. The Equation of State for G a s e s ......................................................................... 13 2. Mixtures of G a s e s ......................................................................... 19 3. Thermodynamic P r o c e s s e s ..........................................................23 4. Work and Heat of a Process. Heat C a p a c ity ........................ 24 5. Internal Energy of a G a s ..........................................................33 6. Enthalpy of a G a s .........................................................................35 7. E n t r o p y ...........................................................................................£6 Chapter II. The First Law of Thermodynamics and Investigation of Thermo­ dynamic P ro c e s s e s .........................................................................37 1. The Law of Conservation and Conversion ofEnergy . . . . 37 2. The First Law of Therm odynam ics........................................... 38 3. Investigation of the Basic Thermodynamic Processes for Ideal G a s e s ................................................... .4 0 Chapter III. 1. 2. 3. 4. 5. 6.

Water Vapour and S t e a m ..........................................................£5 Process of Vaporization in p-v and T-s D ia g r a m s ............. 55 Determining the Parameters of Steam of Different States . . 58 The i s (Mollier) Diagram for S t e a m ............................................61 Thermodynamic Processes of Water Vapour (Steam) . . . . 62 Humid Air ..........................................................................................65 Flow and Throttling of Gases and V a p o u r s ........................ 68

Chapter IV. 1. 2. 3.

The Second Law of Therm odynam ics....................................... 76 Cyclic Processes and the Carnot C y c le ................................... 76 The Second Law of Therm odynam ics.......................................82 Mathematical Expression of the Second Law of Thermody­ namics and Change in the Entropy of an Isolated System 83

Chapter V. Ideal Cycles of Heat E n g in e s ........................................................... 89 1. Cycles of Internal-combustionE n g in e s ............................................89 2. Gas Turbine C y c l e s ....................................................................... 93

6

CONTENTS

3. 4. 5. 6. 7. 8. 9.

Compressor Processes ................................ 95 98 Basic Cycle of a Steam Power P l a n t ..................... Methods of Improving the Efficiency of the Basic Cycle . . 101 Heating and Power S y s te m s .......................................................... 103 Regenerative Cycle ................................................ 105 Steam-gas Cycle ............................. 107 Refrigeration C y c l e s .........................................................................108 SECTION TWO. HEAT TRANSFER

Chapter I. Kinds of Heat T r a n s f e r ....................................................................I l l 1. Conduction ....................................................................................... 112 2. Convection ........................................................................................117 3. Heat Transfer upon Change in Aggregate S t a t e ........................ 126 4. Radiation .......................... 130 Chapter

II. Heat-exchange E q u ip m en t................................................................140 1. Combined Heat Transfer .................................................................. 140 2. Calculation of Heat-exchange E q u ip m e n t.............................. 145 SECTION THREE. FUEL

Chapter /. Propertifs of F u e l ................................................................................ IEO 1. General In fo rm a tio n .........................................................................150 2. Composition of t Fuel ...................................................................151 3. Basic Specifications ofF u e l ...........................................................152 Chapter II. Kinds of Fuels and Processing T h e re o f....................................... 150 1. Solid F u e l s ........................................................................................150 2. Liquid Fuels ....................................................................................164 3. Gaseous Fuels ................................................................................ 167 4. Processing of SolidF u e l s ................................................................ 173 SECTION FOUR. BOILER INSTALLATIONS Chapter I. General Information on Boiler In sta lla tio n s.................................... 170 1. Classification of Boiler Installations ....................................... 170 2. General Information on Boiler U n i t s ..................................... 180 Chapter II. Combustion P ro c e sse s.........................................................................182 1. Combustion of Fuels and Ignition Temperature . . . . . . . 182 2. Air Required for Combustion ......................................................184 3. Excess-air Coefficient ..................................................................... 184 4. Volumes of Combustion Products Calculated from the Elemen­ tary Composition of F u e l ..............................................................184

CONTENTS

7

5. Volumes of Dry Combustion Products Determined by Flue Gas Analysis .................................................................................... 188 6. Volumes of Combustion Products of Gaseous F u e l s ................. 189 7. Enthalpy of Fuel Combustion P r o d u c t s ....................................190 Chapter III. Heat Balance of a Boiler U n i t .......................................................192 1. General E q u a tio n ..........................< ............................................. 192 2. Available H e a t ................................................................................ 192 3. Heat Utilized in BoilerU n i t ........................................................ 192 4. Heat Losses with FlueG a s e s ......................................................... 193 5. Heat Lossesdue to Chemically Incomplete Combustion . . 194 6. Heat Lossesdue to Mechanically Incomplete Combustion . . 195 7. Heat Lossesdue to External Cooling of Boiler Unit . . . . 195 8. Heat Lossesdue to Physical Heat of Slags and for Cooling of Beams and Panels not Included into Boiler Circulation Systems ............................................................................................196 9. Boiler Unit Efficiency ................................................................. 196 10. Fuel Consumption ......................................................................... 196 11. Evaporative Capacity of F u e l .......................................................197 Chapter IV. Temperatures and Heat Transfer in F u r n a c e ............................198 1. Temperatures in F u r n a c e ...............................................................198 2. Heat Transfer in F u r n a c e ...............................................................199 Chapter V. Furnaces 201 1. Classification of F u r n a c e s .............................................................. 201 2. Thermal Characteristics of F u r n a c e s ........................................... 205 3. Waterwalls ....................................................................................206 4. Gas-fired Furnaces ........................................................................206 5. Furnaces for Fuel O i l ..................................................................... 211 6. Pulverized-coal Furnaces ..............................................................216 7. Grate-fired Furnaces for Solid F u e l ...............................................225 Chapter VI. Boiler U n i t s ....................................................................................... 241 1. General Information and P a ra m e te rs ........................................... 241 2. Development of Natural Circulation B o i l e r s ............................ 252 3. Forced Circulation B o ile r s .............................................................. 260 Chapter VII. Superheaters, Water Economizers, Air H e a te r s .........................263 1. S u p erh eaters........................................................................................263 2. Water Economizers .........................................................................266 3. Air H e a t e r s ........................................................................................268 Chapter VIII. Heat Transfer in Convective Passes ofBoiler Units . . . . 270 Chapter IX. 1. 2. 3. 4.

Auxiliary Equipment, Settings and F r a m e .................................. 272 Draught Production Equipment .................................................. 272 Equipment for Flue Gas P u rific a tio n ....................................... 275 Boiler Unit Settings and F r a m e s ...................................................279 Feedwater Pumps and P i p i n g .......................................................280

8

CONTENTS

Chapter X. Water Conditions in Boiler Units and Feedwater Treatment . . 281 1. Boiler Water C o n d itio n s ................................................................. 281 2. Characteristics of Initial W a t e r .................................................. 282 3. Methods of Feedwater Treatment and EquipmentUsed . . . 282 SECTION FIVE. RECIPROCATING ENGINES Chapter I. Steam Engines .................................................................................. 286 1. Classification of Steam Engines ..............................................288 2. Working Cycle ............................. 289 3. Determination of Engine P o w e r .................................................. 290 4. Steam Engine Efficiencies .........................................................294 5. Steam C o n su m p tio n ........................................................................ 297 6. Valve Gear ....................................................................................... 297 7. Methods of Governing Engine P o w e r ...........................................303 8. Types of E n g i n e s .............................................................................305 Chapter

II. Reciprocating C om p resso rs.............................................................309

Chapter

III. Internal-combustion E n g in e s ........................................................ 313 1. Operating Principles and Classification of Engines . . . .314 2. Working Cycles of E n g in e s ..........................................................316 3. Utilization of Heat in E n g in e s ...................................................322 4. Indicated Efficiency .................................................................... 323 5. Mechanical Efficiency ................................................................ 329 6. Brake Thermal Efficiency and Parameters of Working Cycle 330 7. Comparison of; Theoretical and Actual IndicatorDiagrams . 335 8. Determination of Principal Dimensionsof E n g in e s ...................336 9. Heat Balance of E n g i n e ..............................................................338 10. Mixture F o rm a tio n .........................................................................340 11. Fuel Supply E q u ip m en t................................................................. 343 12. Flywheel and Unbalanced Forces of I n e r tia ............................ 349 13. Governing Systems and G o v e rn o rs ...........................................350 14. Timing G e a r ..................................................... ............................ 354 15. Fuel S y s t e m .................................................................................... 356 16. L u b ric a tio n ....................................................................................... 359 17. Cooling S y s te m s ............................................................................. 361 18. Electric I g n i t i o n ............................................................................. 364 19. Starting D e v ic e s ............................................................................. 367 20. Types of E n g in e s ............................................................................. 368 21. Further Development of Internal-combustion Engines . . . 374 SECTION SIX. TURBINES

Chapter

I. Steam T u rb in e s .................................................................................. 376 1. General In fo rm a tio n .........................................................................376 2. Processes in Turbine Nozzles and Moving B la d e s .....................378

CONTENTS

9

3. Heat Losses and Stage E fficien cy .................................................388 4. Turbine Heat C a lc u la tio n s............................................................092 5. Turbine Governing ......................................................................... 402 6. Steam Turbine D e s i g n ..................................................................... 405 7. Condenser Units ............................................................................. 413 Chapter II. Gas T u rb in e s ........................................................................................419 1. Development of Gas T u rb in e s .......................................................419 2. Fundamentals of Theory of Heat Processes in Gas Turbines 423 3. Heat Cycles .................................................................................... 428 4. Efficiencies of Gas Turbine In s ta lla tio n s ................................435 5. Constant-volume Combustion Gas T u rb in e s ............................ 436 6. Closed-cycle Gas Turbine In s ta lla tio n s ....................................438 7. Cycle Diagram Calculations ....................................................... 441 8. Compressors .................................................................................... 446 9. Gas Turbine Design ..................................................................... 448 10. Materials for and Cooling of Turbine Blades and Disks . . 452 11. Regenerators ....................................................................................454 12. Fuel and Combustion C h a m b e rs...................................................455 SECTION SEVEN. HEAT ELECTRIC GENERATING PLANTS Chapter I. Heat Electric GeneratingP l a n t s ........................................................458 1. Types of P l a n t s .................................................................................458 2. Layout and Equipment of Steam Turbine P l a n t s .....................463 3. Fuel F a c ilitie s.................................................................................... 469 4. Power Plant E co n o m y ..................................................................... 476 Chapter II. Automation in Power P l a n t s .........................................................480

FOREWORD

In training engineers of any speciality, particular attention should be paid to mastering the fundamentals of power engineer­ ing and to the theory and design of heat power plant equipment. This book on thermal engineering is intended for students not specializing in heat engineering. The book briefly treats various aspects of engineering thermodynamics, combustion processes and fuel gasification, also the fundamentals of theory and design of boiler units, steam and gas turbines, steam and internal-combus­ tion engines. The authors will greatly appreciate all comments and sug­ gestions from readers concerning the contents of this book.

Section

One

ENGINEERING THERMODYNAMICS

Chapter I BASIC CONCEPTS OF THERMODYNAMICS

I. Parameters of Gas and Relation Between Them. The Equation of State for Gases Engineering thermodynamics studies problems related to condi­ tions of the most advantageous conversion of thermal to mechani­ cal energy and vice versa in heat engines, heat-power and refri­ gerator units. Engineering thermodynamics is therefore the theo­ retical basis of heat engineering. Thermodynamics is based on two main experimental principles called the first and second laws of thermodynamics. Heat energy is converted into mechanical energy in heat engines through the agency of a working body capable of changing its volume. As a rule, gases and vapours are employed as the work­ ing bodies. That is why engineering thermodynamics studies the properties of gaseous substances. In gases the influence of molecular interaction forces and the size of the molecules is different in different states of the gases. One and the same gas may be assumed to be an ideal *, or per­ fect, gas under certain conditions (for instance, at a low pressure or high temperature), but under other conditions (a high pressure or low temperature) the gas should be treated as a real gas. Owing to molecular interaction the properties of real gases differ from those of ideal gases. * By an ideal gas is meant a gas in which the forces of molecular interac tion and the volume of the molecules can be neglected.

14

ENGINEERING THERMODYNAMICS

The physical quantities that determine the state of a gas are called parameters of state. The simplest parameters, which can be measured directly, are pressure, temperature and specific volume. The absolute pressure of a gas is the result of a large number of the molecules forming the gas striking the walls of the vessel. The force per unit of area is called the specific pressure or just the pressure p. In thermodynamics the m.k.s.a. system of units is used. Accordingly the unit of pressure will be kg/m2. As a pressure of 1 kg/m2 is very small, in engineering a pressure unit 104 times larger and equal to 1 kg/cm2 is used. This unit is called the technical atmosphere (at). 1 at = 1 kg/cm2= 10,000 kg/m2= 104 kg/m2. Besides the technical atmosphere, pressure is measured by standard atmospheres (atm), one standard atmosphere being equal to 1.0333 kg/cm2. Pressure may be measured by the height of a column of liquid (water, mercury, etc.): p — ~{h kg/m2, where y = specific gravity of liquid, kg/m3. Hence, the following relationships are true: 1 at = 735.6 mm Hg = 1 X 104 mm water column. It may be easily seem that for a standard atmosphere 1 atm = 760 mm H g = 1.033 X 104 mm water column. A pressure of 1 atm is employed to determine the so-called standard conditions (the state of a gas under a pressure of 760 mm Hg and a temperature of 0 deg C). There are distinguished absolute pressure, excess pressure and rarefaction (or vacuum). The liquid in the U-tube (Fig. 1) is subjected, on one hand, to the absolute pressure exerted by the gas filling the vessel — p a bs and, on the other, to the pressure of the surrounding medium (at­ mospheric air) — pbarThe difference between the absolute pressure p a bs and the pres­ sure of the surrounding medium pbar is called the excess, or gauge, pressure pg. Hence Pabs

'Pg~

I- P b a r •

0"1)

The pressure of the surrounding medium is measured with a barometer and is called barometric pressure.

BASIC CONCEPTS OF THERMODYNAMICS

15

Excess, or gauge, pressure is measured with a manometer or pressure gauge and is often called manometric pressure (pman or h). In some cases the pressure inside a vessel may be below at­ mospheric pressure (Fig. 2). The difference between barometric and absolute pressure is called rarefaction or vacuum (pT or pVac)-

Rarefaction is measured with a vacuum meter (pvac or h). Ob­ viously Pvac Pbar Pabs or Pabs Pbar Pvac' Absolute pressure measured in technical atmospheres is desig­ nated pata (absolute atmospheres), excess pressure—pg (gauge pressure). All thermodynamic formulas are based on absolute pressure measured in kg/m2, for only this pressure is a parameter of state. Temperature is a measure of the heat of a body, and it deter­ mines the direction of spontaneous heat flow. According to the kinetic theory of gases, the temperature is proportional to the mean kinetic energy of the translational velocity of molecules:

where T = absolute temperature. From the formula it follows that the absolute temperature T = 0 if the velocity of the molecules w = 0, which is unattain-

15

ENGINEERING THERMODYNAMICS

able. The Russian scientist M. V. Lomonosov was the first to in­ dicate that absolute zero is unattainable. In practice the measurement of temperature is based on the property of bodies to change their state with a varying degree of heating. The temperature measured in this way is called em­ pirical and is designated by t. The most rational temperature scale, independent of differing properties of bodies, is the so-called absolute, or thermodynamic, temperature scale proposed by Kelvin in 1848. The temperatures of this scale are designated by T deg K. A practical variant of the absolute scale is the international Centigrade scale in which the melting point of ice and the boiling point of •chemically pure water under standard conditions are respectively designated 0 deg and 100 deg. Temperature measured by means of this scale is designated t deg C. * The following relationship exists between the absolute temper­ ature and the temperature read on the Centigrade scale: r j K = /‘,C 4 - 273.16 or appr. /°C + 273, since absolute zero corresponds to a temperature o f—273.16 deg C. The specific volume is the volume of a substance of unit weight. u = -g- m3/kg. The reciprocal of the specific volume 1 a , , , v = i r = T kg/m is called the specific gravity. Apart from the parameters p, v and T, mentioned above, there are other parameters: entropy s, internal energy u and enthalpy, or heat content, i. These parameters will be considered below. Example 1. The pressure gauge of a steam boiler indicates an excess pres­ sure of 32 atm gauge, and the vacuum gauge shows a vacuum in the conden­ ser of 708.2 mm Hg. Determine the absolute pressure in the boiler and the condenser, if the barometer indicates a pressure pbar = 745 mm Hg, and express these pressures in kg/m2. The absolute pressure in the boiler 745 Pabs — Pg~h Pbar = 32 -fg = 33.01 ata. The absolute pressure in the condenser Pabs

= Pbar — Pvac = 745 — 708.2 = 36.8 mm Hg =

36 8

or Pabs

= 0.05 X 10< = 500 kg/m2.

C is the first letter of the Latin word centum (hundred).

- 0.05 ata

BASIC CONCEPTS OF THERMODYNAMICS

17

The Equation of State of an Ideal Gas. Experience and theory show that the parameters p, v and T of a homogeneous body are not independent. In a state of equilibrium the parameters have a property which may be expressed in the following form: depends on the nature of the body and varies for different bodies. The simplest equation of state, known Trom elementary physics, is the Clapeyron-Mendeleyev equation for an ideal gas based on the kinetic gas theory: pv — R T = 0 or pv = RT, (1-2) where p = absolute pressure, kg/m2; v = specific volume, m3/kg; T — absolute temperature, deg K; R = gas constant related to 1 kg of gas, and depending on the nature of the substance (specific gas con­ stant). From equation (1-2) we obtain the unit of the specific gas con­ stant R, namely, kg-m/kg-deg. Multiplying both sides of equation (1-2) by G, and, taking into account that Gv = E, we obtain the equation of state for an arbitrary amount of gas G: PV = G R T . (1-3) The equation of state for one mole of gas, i.e., for p kg can be derived from equation (1-3), if G = p kg, pVp = pRT (1-4) or (1-4a) 1* = ^ . where Ep = the volume of a mole of gas, m3/mole. According to Avogadro’s law, the volume of a mole is the same for all ideal gases under equal pressure and temperature. Thus, under standard conditions, the volume of one mole of an ideal gas Ep = 22.4 m3/mole. Since according to the combined Boyle and Gay-Lussac law p-j = const, it is also obvious that pR = const. The quantity pR is called the universal gas constant. 2 -8 1 4

18

ENGINEERING THERMODYNAMICS

The numerical value of the universal gas constant p/? can be obtained by substituting in equation (l-4a) the numerical values of the parameters under standard conditions (pst = 10,333 kg/m2^ V\i = 22.4 m3/mole, Tst = 273 deg K). PstVP Tst

10,333X22.4 273

= 848 kg-m/mole-deg.

The numerical value of the specific gas constant for any gas can be determined by the formula ' kg-m/kg-deg.

(1-5)

The Equation of State of Real Gases. In a real gas the forces of molecular interaction and the volume occupied by the mole­ cules are appreciable. Therefore, the relation between the parame­ ters and the function

0, i.e., for sufficiently rarefied gases the Van der Waals equa­ tion becomes pv = RT. There is not sufficient coincidence between data calculated by formula (1-6) and experimental results. A number of other equa­ tions of state have therefore been suggested, mainly based on empirical data and approximating more closely experimental data. The Soviet scientists, Professor of the Moscow Power Institute M. P. Vukalovich and Prof. I. I. Novikov, proposed in 1939 an equation of state of real gases based on the general theory of real gases, and taking into account the association of molecules into double and triple molecular groups. The very complicated Vukalovich-Novikov equation can be simplified to pv = R

T

[ l +

£ S p - ...],

where BX(T), B2(T) = known functions of temperature.

BASIC CONCEPTS OF THERMODYNAMICS

19

Example 2. Find the weight occupied by 6 cu m of carbon dioxide (C02) -at a pressure p = 4 atm gauge and a temperature t = 27 deg C. From formula (1-3) G =

pv kg. RT

■where p — 4 + I = 5 ata = 5 X 104 kg/in2; T = t + 273 = 300 deg K; A -J « Fc02

848 kg-m/kg-deg. 44

Substituting these values, we get

„

5X104X6

52 kg.

848X 300 44

Example 3. The specific gravity of air at standard conditions (pst = *=760 mm Hg and i,t= 0 deg C) is yJ(= 1.293 kg/m3. Determine the specific gravity of air at a pressure pt = 6 5 0 mm Hg and a temperature t\ — 17 deg C. Having written equation (I-‘2) for the two different states P stv st —

Piv, = RTi and upon dividing one equation by the other by members and rearranging, we get

PstVst _ P\V\ Tst

T,

‘

This expression may be rewritten as follows: P st __ P I TstT st _ Tfi^i

or Ti = Tif

P Jst P s tT i

1.293- ^ - X

273 290

1.08 kg/m3.

2. Mixtures of Gases In thermal engineering the working body is usually a mixture of several gases (for instance, air, products of fuel combustion) rather than a homogeneous gas. Let us consider some properties of a mixture of gases. The behaviour of each gas of a mixture is such as if it alone occupied the entire volume containing the mixture. The pressure that the component would have if it occupied the entire volume by itself at the temperature and volume of the mixture is called the partial pressure. 2*

20

ENGINEERING THERMODYNAMICS

According to Dalton’s law, the pressure of a mixture is equal to the sum of the partial pressures of its components: n P — Pi

+ / ,2+ /?3 + • • ■- \ ~ P n =

2 lP i


Gj

Upon dividing the numerator and denominator of the right part of equation (1-11) by G, we obtain ( 1- 12)'

The gas constant of the mixture can be calculated from the expression Rmix = ~ r kg-m/kg-deg. Km/jr

32

ENGINEERING THERMODYNAMICS

The molecular weight of a mixture can be expressed through the volume fractions by the following expression: I W = 2 IVV

(1-13)

The partial pressure can be calculated from the equation p .V = p \/l or whence (1-14)

Pi - P r i-

Substituting the values from equations (1-10) and (1-12) in equation (1-14), we obtain Pi = ~ ^ P -

d-15)

Example 4. Determine the apparent molecular weight, gas constant, specific ' gravity and partial pressure under standard conditions for a gas mixture con­ sisting of 6 kg of CO2, 3 kg of N2 and 1 kg of O2. The weight composition of the given mixture of gases is gC0' = G co2+ G n 0 + G02 = 6 + 3 + 1 = °'6' t By analogy, g Na = 0.3 and g 0a = 0.1. The molecular weights of the mixture components will be: H'COj = 44= H-Nj = 28; H*o2 = 32. The apparent molecular weight of the mixture of gases is calculated by for­ m ula (1-12) P m ix = —n

r ' 06

V

IL L i p-i 1

|

44 +

Q3

Q.i

28 +

32

= 36A

The gas constant can be calculated from formula (1-5):

R4R

RmU = 3 g^- = 23.1 kg-m/kg-deg. The specific gravity at 0 deg C and 760 mm Hg v . y-mix _ 36.4 'mlx ~ 22.4 22.4

1.62 kg/st m3.

BASIC CONCEPTS OF THERMODYNAMICS

23'.

Now we proceed to calculate the partial pressures of the gas mixture com­ ponents by formula (1-15): PcOj = S c o 2

P = a&^ T 760 = 376 mm Hg;

PN2= ^N2' ^ £- /, = a3-^ f-760 = 297 mm Hs: Po, = go —

p = 0-1^

760 = 87 mm Hg-

3. Thermodynamic Processes Reversible and Irreversible Processes. The equation of the form: 9 (p, v, T) considered above is the equation of the state of equi­ librium, i.e., of a state the parameters of which (pressure, specific volume and temperature) are the same over the entire mass of the gas and are equal to the respective parameters of the surrounding medium. The equality of pressure leads to mechanical equilibrium, while equality of temperature—to thermal equilibrium. The state of equilibrium cannot change without mechanical or thermal inter­ ference from the outside. The sequence in which the states of the working body change constitutes a thermodynamic process. A process going on so slowly that the state of equilibrium sets in in the working body at every moment of time is called an equilibrium process, other­ wise, it is referred to as a nonequilibrium process. Mechanical and thermal equilibrium, as well as the absence of friction, are the conditions necessary for an equilibrium process. This means that in cases of a change jn the state of bodies the change in volume should take an infinitely long time, while the pressure at any moment of time should be equal to the outside pressure (conditions of mechanical equilibrium). From the condi­ tion of thermal equilibrium it follows that the temperature of a body should always be equal to that of the surrounding medium, or to the temperature c)f the heat source from which heat is im­ parted to the body. It is very important to understand that in the case of a working body of varying temperature an equilibrium process is possible only if the outside source of heat changes its temperature in the same manner as the working body, i.e., when there is an infinite number of heat sources. Processes which can proceed both in the straight and the reverse directions through one and the same equilibrium states of the working body are called reversible processes. The direct and:

24

ENGINEERING THERMODYNAMICS

reverse processes should not result in any residual changes in the working body or in the surrounding medium. Processes that do not comply with this condition are said to be irreversible. It should be mentioned that there are no reversible processes in nature. The conception of process reversibility is an example of scientific abstract thinking that facilitates the solution of many problems of engineering thermodynamics. Each state of the work­ ing body in the process of an equilibrium change in state can be represented as a point with the coordinates p, v, and t. In the general case all three parameters may vary. As it is practically inconvenient to graphically illustrate processes in a three-dimen­ sional coordinate system, usually two-dimensional systems are used, in which the variation of any two parameters can be shown. The magnitude of the third parameter can be determined from the equation of state. The p-v system of coordinates has found extensive use in ther­ modynamics, but only processes of gases in the state of equilib­ rium can be represented in this system. It is impossible to show graphically the processes of gases not in equilibrium. A process in the course of which the working body, having ' passed through a number of successive states, returns to its ini­ tial state is called a cyclic process or a cycle. Conceptions of the equilibrium state, reversible and cyclic processes are of the greatest importance in engineering thermodynamics. 4. Work and Heat of a! Process. Heat Capacity Any thermodynamic process is the result of the mechanical or thermal action of the surrounding medium upon the working body (for instance, compression or expansion of the working body, heat transfer). In the first case the volume of the working body changes and mechanical work is performed. In the second case the quantity of energy imparted to the body during heat transfer is called the quantity of heat or the heat of the process. Thus, the concepts of “work” and “heat” are two forms of energy transmission; they relate to the process of the change in the state of a system and are the basic characteristics of thermo­ dynamic processes. The m.k.s.a. unit of work is the kilogram-metre (kg-m). There is a larger unit of work, namely, the horsepower hour (hp-hr). The horsepower (hp) is the industrial unit of power (i.e., work per second) equal to 75 kg-m/sec. Then, 1 hp-hr = 75 X 3,600 = 270,000 kg-m.

BASIC CONCEPTS OF THERMODYNAMICS

2S

The industrial unit of electrical work is the kilowatt-hour (kw-hr)—the work of one kilowatt for one hour. The kilowatt (kw) is the industrial unit of electrical power. 1 kw = 102 kg-m/sec = 1.36 hp. Hence 1 kw-hr = 102 X 3,600 = 367,200 kg-m = 1.36 hp-hr. The m.k.s.a. unit of heat is the kilogram-calorie (kcal). The standard kilogram-calorie is the amount of heat required to raise, the temperature of 1 kg of chemically pure water from 19.5 to 20.5 deg C at normal atmospheric pressure (760 mm Hg) in lati­ tude 45 deg at sea level. Besides the standard kilogram-calorie, there is an international (electrical) kilogram-calorie equal to — kw-hr. According to the law of conservation and conversion of energy, heat and work under certain conditions can be converted one into the other in the following equivalent proportions: Qo = AL0, where Q0 = quantity of heat converted into work, kcal; L0 = work obtained at the expense of the heat, kg-m; A = heat equivalent of mechanical work, kcal/kg-m. It has been established experimentally that 1 kg-m corresponds to kcal, i.e., A—

kcal/kg-m.

The heat equivalents of the larger units of work are: 1 hp-hr = 270,000 -^ - « 632 kcal/hp-hr; 1 kw-hr = 1.36 X 632 « 860 kcal/kw-hr. Calculation of Work. Let us assume that a working body con­ fined in a cylinder inside which a piston moves without friction performs external work overcoming a pressure p * (Fig. 3). A uniformly distributed absolute pressure p will force the piston, which has an area /, to move through an elementary distance dh. When the piston moves through the distance dh the gas will perform an infinitely small quantity of work dl = p f dh = p dv,

(1-16)

* In processes in which the volume increases the working body performs work against external forces. Upon compression, on the contrary, the external forces overcome the resilience of the body and perform work on it.

:26

ENGINEERING THERMODYNAMICS

where p f = force; dh. = distance; fd h — dv = corresponding change in the volume of the working body. In Fig. 3 this elementary work is represented by the shaded area. The work of 1 kg of the working body for a finite change in the volume from iq to v2 is determined by the expression va

I — J p d v kg-m/kg.

(1-17)

If G kg of the gas having a volume V participates in the process of expan­ sion, the work performed will be correspondingly va

va

L = Gl — Q J p d v = J p d V kg-m.(l-18) V,

«/,

In the general case the specific pres­ sure p is a variable and depends on v. For calculation of the integral the relationship p = f(v) for the given process must be known. This relationship is graphically shown in a p-v diagram by curve 1-2 (Fig. 3). It is obvious'that the work performed by the gas will depend on the nature of the curve representing the process and will be depicted by the area bound by the curve of the process, the two ordinates and the abscissa (area 1-2-3-4-1). The work per­ formed by a gas upon expansion is assumed to be positive, and upon compression— negative. Calculation of the Quantity of Heat. The quantity of heat in­ volved in a thermodynamic process can be calculated in three different ways. One of them employs the conception of entropy. Similar to elementary work being represented by the expression dl = pdv, the elementary quantity of heat can be represented by the expression dq = T d s kcal/kg, (1-19)

Fig. 3. Process of gas expan­ sion In a cylinder

where s = specific entropy of the body. The analogy existing between the expressions for dl and dq is not confined only to their mathematical form, but it also reflects the similarity in the nature of the two forms of energy exchange between the body and the surrounding medium. Indeed, work is a macrophysical form of transmitting energy directly observed in

2T

BASIC CONCEPTS OF THERMODYNAMICS

the surrounding macrocosm. Heat is the microphysical form of the manifestation of changes in energy, due to the uniformly random motion of the microscopic particles of a body. From expression (1-19) it follows that an infinitely small value of the entropy is equal to the ratio of an infinitely small quan­ tity of heat in the process dq to the temperature T at which the heat enters the system (for an infinitely small section of the process T may be assumed constant), i.e.,. ds = -Y~ kcal/kg-deg. In studying the properties of entropy (see Chapter IV, Par. 3), it will be shown that entropy is a parameter of state of great practical importance, because it makes possible the plotting of the so-called heat diagram in which entropy is plotted along the x-axis and absolute temperature— along the y-axis, as shown in Fig. 4. Just as in the p-v diagram the shaded area bound by the curve of the process and the x-axis represents work, it can be readily shown that in the T-s diagram the area bound by the curve of the process, the x-axis and two ordinates represents the amount of heat participating in the process. Indeed, integration of expression (1-19) gives q = j" T ds,

(1-20}

Si

where T = f(s). From Fig. 4 it follows that area t-2-2'-l'-l is equal to

J T dsi

Since the absolute temperature T is always positive, the sign of the change in entropy also determines the sign of dq, i.e., it allows us to judge whether heat is imparted to or taken away from the working body in the process under consideration. Just as the amount of work, the amount of heat depends on the nature of the process, i.e., on the shape of curve 1-2 of the process (Fig. 4). Another method of calculating the amount of heat follows from the concept of heat capacity. In the general case the supplying of heat to or removal thereof from a body in a thermodynamic process results in a change in the temperature of the body.

■28

ENGINEERING THERMODYNAMICS

The ratio of the amount of heat participating in the process Aq to the corresponding change in temperature At is called the heat capacity of the body in the given process: c

A? At '

( 1- 21)

Different amounts of heat are required to change the tempera­ ture of a gas by I deg C in different temperature ranges. The heat capacity of gases is therefore variable for a given process and depends mainly on temperature. Expression (1-21) shows the average amount of heat spent in a process upon raising the temperature of a gas by 1 deg C in the temperature range from t\ to t2. The heat capacity calculated in this manner is called the mean heat capacity cm:

The heat capacity corresponding to an infinitesimal change in temperature dt is called the true heat capacity. dq dt '

(1-23)

Heat capacity is usually related to a unit of substance (specific heat). Depending on the unit of substance selected, there are distinguished weight tfeat capacity, related to 1 kg of the working body (c kcal/kg-deg), volume heat capacity, related to 1 st m3 (c' kcal/st m3-deg) and molar heat capacity, related to 1 mole of the substance (pc kcal/mole-deg). These heat capacities are relat­ ed by the following equations: c= f

= c'vst;

(1-24)

c' cy v = — ^22.4 = — ‘"I where 22.4 = volume of a mole of gas under standard conditions. Since the amount of heat depends on the nature of the process, heat capacity depends on the conditions under which the process of heat exchange between the gas and the surrounding medium goes on. Among the different heat capacities possible for each working body of particular interest are the heat capacities characteristic of heat transfer in processes with a constant volume cv and a con­ stant pressure cp.

29

BASIC CONCEPTS OF THERMODYNAMICS

A certain relationship exists between these two heat capacities. For ideal gases this relationship is established by Mayer’s equa­ tion known from physics, cP— cv = AR

(1-25)

V-cp- v .c v = AvR = ^ 8 4 8 * 2 kcal,

(1-26)

or

i.e., cp > c„. This is due to the fact that in a constant-volume process the heat communicated to the gas serves entirely to change the internal energy, depending only on the velocity of the mole­ cules, which increases with a rise in temperature, whereas in a constant-pressure process the heat is spent not only for increasing the internal energy, but also to perform work against external forces. According to the Van der Waals equation, for real gases cp— cv > A R . This relationship is based on the circumstance that upon the expansion of real gases (at p = const) not only external work, but also internal work overcoming molecular cohesion is performed, which explains the great amount of heat spent in the process. The heat capacity of real gases therefore depends both on the tempera­ ture t and the pressure p. £

The ratio of these two heat capacities £ = —cv finds a wide application in heat engineering. The kinetic theory of heat capacity, which does not take into account the dependence of the heat capacity of an ideal gas on the temperature, establishes the following magnitudes of cv, cp and k depending on the valency of gases: k For monoatomlc gases . For diatomic gases . . . For polyatomic gases . .

3 5 7

5 7 9

1.6 1.4 1.3

The general expression for calculating the amount of heat parti­ cipating in a process is obtained from formula (1-23): dq = cdt

30

ENGINEERING THERMODYNAMICS

or

t, q — Jcdt.

(1-27)

ti

On the other hand, from formula (1-22) it follows that =

(1-28)

In order to integrate expression (1-27) we must know the re­ lation of the true heat capacity to temperature. The quantum theory of heat capacity which takes into account the energy of intramolecular oscillations of the atoms establishes the relation of heat capacity to B temperature in the form of the fol­ lowing exponential polynomial: c = a -\~ b 't-\-et2-\~ . . . .

Fig. 5. Relation of heat capacity to temperature i

Engineering calculations are often performed with the aid of binomial formulas, or calculations are based on a linear relation between heat capacity and tem­ perature: c = a -\-b t, (1-29)

where a, 6 = numerical factors depending on the nature of the gas involved and the character of the process. The relation between heat capacity and temperature is il­ lustrated in Fig. 5. Substituting expression (1-29) in (1-27), we get ti

/2

q = f c d t = f (a + bt)dt = (a + b *' + ‘x

(t2— 7,).

(1-30)

‘x

Upon comparing formulas (1-28) and (1-30), it can be readily seen that c j; ,= a + b ± ± ± . (1-31) Thus, the mean heat capacity depends on both the extreme tem­ peratures t\ and t2 which may vary within the widest limits. It would be practically impossible to compile tables for such a mul­ titude of intervals.

31

BASIC CONCEPTS OF THERMODYNAMICS

The amount of heat can be determined, however, without having to calculate the mean heat capacities in the temperature range from ti to t2. Indeed, expression (1-27) can be presented in the following form: ti

ti

t\