(For B.E. Mechanical Engineering Students) AIR WALK PUBLICATIONS (Near All India Radio) 80, Karneeshwarar Koil Street Mylapore, Chennai - 600 004. Ph.: 2466 1909, 94440 81904 Email: [email protected], [email protected]www.airwalkpublications.com Basic Mechanical Engineering As per New Syllabus of Biju Patnaik University of Technology [BPUT] ODISHA Dr. S.Ramachandran, M.E., Ph.D., Dr. A. Anderson, M.E., Ph.D., Sathyabama University Jeppiaar Nagar, Chennai - 600 119
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1.1 Introduction ..................................................................... 1.11.2 Role of Thermodynamics in Engineering and Science 1.21.3 Applications of Thermodynamics .................................. 1.21.4 Basic Concepts ................................................................ 1.31.5 Classical and Statistical Thermodynamics.................. 1.71.6 Thermodynamic Systems ............................................... 1.91.7 Closed System................................................................. 1.91.8 Open System ................................................................... 1.101.9 Isolated System............................................................... 1.101.10 Homogeneous and Heterogeneous Systems .............. 1.111.11 Property.......................................................................... 1.111.12 State, Path and Process .............................................. 1.121.13 Cycle ............................................................................... 1.131.14 Thermodynamic Equilibrium....................................... 1.141.15 Quasi-static (or) Quasi–equilibrium Process............. 1.141.16 Path Function and Point Function............................ 1.161.17 Zeroth Law of Thermodynamics................................. 1.171.18 Measurement of Temperature..................................... 1.17
1.18.1 Thermometry ...................................................... 1.181.18.2 Applications of thermometry............................ 1.181.18.3 Triple point of water ........................................ 1.181.18.4 Comparison of Thermometer ........................... 1.18
1.19 Equation of State ......................................................... 1.191.20 Celsius Temperature Scale.......................................... 1.191.21 Work ............................................................................... 1.21
1.21.1 Work Transfer - a Path Function .................. 1.221.22 Heat ................................................................................ 1.23
1.22.1 Heat Transfer – A Path Function .................. 1.24
Contents C.1
1.23 Heat Capacity ............................................................... 1.251.24 Reversible and Irreversible Process........................... 1.261.25 First Law of Thermodynamics ................................... 1.261.26 Joules Experiment ........................................................ 1.271.27 Perpetual Motion of Machine of First Kind ............ 1.291.28 First Law For A Closed System Undergoing Change of State............................................................. 1.29
1.28.1 Internal Energy ................................................. 1.291.28.2 Energy - A property of the system ................. 1.30
1.29 Specific Heat Capacities .............................................. 1.341.29.1 Specific heat....................................................... 1.341.29.2 Latent heat......................................................... 1.341.29.3 Specific heat at constant volume .................... 1.341.29.4 Enthalpy ............................................................. 1.351.29.5 Specific heat at constant pressure .................. 1.36
1.30 Thermodynamic Processes ........................................... 1.371.31 Non-flow Process (or) Closed System ........................ 1.37
1.31.1 Constant Volume Process ................................. 1.371.31.2 Constant Pressure Process (or) isobaric Process 1.401.31.3 Constant Temperature Process (or) isothermal
Process PV constant; T constant .............. 1.431.31.4 Reversible Adiabatic Process (or) isentropic Process ................................................................ 1.47
1.31.5 Polytropic process: PVn constant n n .... 1.511.32 Steady Flow Process (Open System) ......................... 1.57
1.33 Limitations of First Law of Thermodynamics ......... 1.671.34 The Second Law of Thermodynamics........................ 1.68
1.34.1 Thermal Energy Reservoirs.............................. 1.691.35 Heat Engines................................................................. 1.701.36 The Second Law of Thermodynamics: Kelvin-PlanckStatement ............................................................................... 1.721.37 The Second Law of Thermodynamics: Clausius Statement...................................................................... 1.741.38 Refrigerators and Heat Pumps................................... 1.751.39 Coefficient of Performance - COP.............................. 1.761.40 Equivalence of The Two Statements ......................... 1.77
1.41.1 The Carnot Principles (or) Carnot Theorem . 1.831.41.2 Corollaries of Carnot’s Theorem ..................... 1.86
1.42 Clausius Inequality....................................................... 1.861.43 Entropy........................................................................... 1.901.44 Third Law of Thermodynamics .................................. 1.991.45 Properties of Ideal Gas................................................ 1.1001.46 Properties of Pure Substances.................................... 1.1081.47 Thermodynamic Properties of Steam......................... 1.111
1.47.1 Property Tables.................................................. 1.1111.47.2 Saturated Liquid and Saturated Vapour States .................................................................. 1.1121.47.3 Saturated Liquid-Vapour Mixture .................. 1.1121.47.4 Superheated Vapour.......................................... 1.113
Module 2: Application of Thermodynamics and Introduction to Fluid Mechanics and Heat transfer
2.1 Air Compressors ............................................................. 2.1
Contents C.3
2.1.1 Applications of Air Compresser......................... 2.12.1.2 Reciprocating Air Compressors.......................... 2.12.1.3 Working Principle of Reciprocating Air Compressors 2.32.1.4 Rotary Compressors ............................................ 2.62.1.5 Roots Blower........................................................ 2.62.1.6 Vane Type Blower Compressor.......................... 2.72.1.7 Centrifugal compressor ....................................... 2.82.1.8 Axial Flow Compressor ...................................... 2.9
2.2 Steam Power Plant ........................................................ 2.102.2.1 Layout of Steam Power Plant........................... 2.102.2.2 Working of Steam Power Plant ........................ 2.132.2.3 Major Thermal Power Plants in India............ 2.13
2.3 Refrigerators and heat pump ....................................... 2.142.3.1 Parts of the household refrigerator .................. 2.152.3.2 Internal parts of the refrigerator ...................... 2.162.3.4 External parts of the refrigerator ..................... 2.17
2.4 IC Engines....................................................................... 2.202.4.1 IC Engine Components and Functions ............ 2.202.4.2 Basic Terms Connected With I.C. Engines ..... 2.252.4.3 Working Principle of Four Stroke Spark IgnitionEngine (4 Stroke Petrol Engine)................................. 2.262.4.4 Four Stroke Diesel Engine (4 Stroke CompressionIgnition Engine) ............................................................ 2.292.4.5 Two Stroke Engine ............................................. 2.312.4.6 Two Stroke Petrol Engine (si Engine) ............. 2.322.4.7 Two Stroke (Compression Ignition) Diesel Engine in Diesel Engine .................................... 2.34
2.5 Fluid Properties .............................................................. 2.402.5.1 Density (or) mass Density.................................. 2.402.5.2 Specific weight (or) Weight density .................. 2.41
2.5.3 Specific Volume v ............................................. 2.41
2.5.4 Specific gravity (or) Relative density s.......... 2.412.5.5 Temperature ......................................................... 2.422.5.6 Viscosity................................................................ 2.42
2.5.6.1 Kinematic Viscosity () .................................... 2.442.5.6.2 Variation of Viscosity with temperature ....... 2.44
2.5.7 Compressibility 1K
...........................................
2.44
2.5.7.1 Relationship between Bulk Modulus K and
pressure P of a Gas for Isothermal and
Isentropic Process................................................ 2.452.5.8 Vapour Pressure .................................................. 2.462.5.9 Cavitation ............................................................. 2.462.5.10 Gas and Gas laws ............................................ 2.462.5.11 Surface Tension................................................. 2.472.5.11.1 Surface Tension on Droplet .......................... 2.492.5.11.2 Surface Tension on a Hollow Bubble ......... 2.502.5.12 Capillarity .......................................................... 2.502.5.12.1 Expression for Capillary Rise ...................... 2.512.5.13 Thermodynamic Properties............................... 2.512.5.14 Newton’s law of viscosity ................................. 2.522.6 Types of Fluid ........................................................ 2.52
2.8 Buoyancy.......................................................................... 2.592.9 Principle of Conservation of Energy............................ 2.602.10 Bernoulli’s Theorem ..................................................... 2.60
2.10.1 Assumptions for derivation of Bernoulli’s theorem............................................................... 2.61
Contents C.5
2.10.2 Bernoulli’s theorem for Real Fluid................. 2.622.10.3 Bernoulli’s Theorem: Applications ................. 2.632.10.3.1 Venturi meter.................................................. 2.642.10.3.2 Orifice meter ................................................... 2.64
2.11 Pipe Flow....................................................................... 2.652.12 Reynolds Number ......................................................... 2.662.13 Loss of Head For A Given Length of Pipe (Hagen Poiseuille Formula) ....................................... 2.692.14 Law of Fluid Friction .................................................. 2.692.15 Frictional Loss in Pipe Flow ...................................... 2.712.16 Hydraulic Turbines....................................................... 2.72
2.16.1 Classification of Hydraulic Turbines ............. 2.722.16.2 Pelton Turbine (or) Pelton Wheel ................... 2.722.16.2.1 Working of A Pelton Wheel .......................... 2.732.16.3 Kaplan Turbine ................................................. 2.732.16.3.1 Working of Kaplan Turbine ......................... 2.742.16.4 Francis Turbine................................................. 2.752.16.4.1 Working of A Francis Turbine..................... 2.752.16.5 Difference Between Kaplan and Francis Turbine 2.752.16.6 Difference Between Impulse and Reaction HydraulicTurbines ......................................................................... 2.77
2.22 Direct Contact (or) Open Heat Exchangers ............. 2.922.23 Indirect Contact Type .................................................. 2.932.24 Classification of Heat Exchangers according............ 2.94
3.3 Casting ............................................................................. 3.53.3.1 Parts Made by Casting ...................................... 3.53.3.2 Mould in Casting................................................ 3.5
Contents C.7
3.3.3 Sand Casting ....................................................... 3.53.3.4 Sand Moulds ....................................................... 3.73.3.5 Steps/ Procedure for making sand mould for a two piece pattern ...................................... 3.83.3.6 Pattern .................................................................. 3.113.3.7 Special Casting Processes .................................. 3.113.3.8 Investment casting (Lost wax casting) ............. 3.133.3.9 Centrifugal Casting............................................. 3.14
3.4. Welding ........................................................................... 3.163.4.1 Types of Welding Processes ............................... 3.163.4.2 Welding Terminology .......................................... 3.163.4.3 Welding Processes................................................ 3.173.4.3.1 Oxy – Acetylene Welding (Gas Welding)....... 3.173.4.3.2 Electric Arc Welding Proceses ........................ 3.183.4.3.3 Arc Welding (or) Metal Arc Welding ............ 3.193.4.3.4 Gas Tungusten Arc Welding (GTAW) or Tungusten Inert Gas Welding (TIG) .............. 3.203.4.3.5 Gas Metal Arc Welding (GMAW) or Metal Inert Gas Welding (MIG) ..................... 3.213.4.4 Comparision of Gas Welding and Arc Welding 3.223.4.5 Electrodes ............................................................. 3.223.4.6 Resistance Welding.............................................. 3.223.4.6.1 Spot Welding .................................................... 3.22
3.8 Forging ........................................................................... 3.293.8.1 Classification of Forging .................................... 3.293.8.2 Smith die forging (or) Open die forging (or) Flat-die forging.................................................... 3.303.8.3 Power Hammers .................................................. 3.333.8.4 Power press (or) Forging press ......................... 3.343.8.5 Impression die forging (or) Closed die forging 3.353.8.6 Comparison between press forging and drop forging (Hammer Forging)................................. 3.35
3.9 Rolling of Metals ............................................................ 3.353.10 Engineering Materials.................................................. 3.383.11 Metals............................................................................. 3.39
3.12 Mechanical Properties of Materials ........................... 3.413.13 Steel................................................................................ 3.443.14 Effect of Alloying Additions on Steel ........................ 3.44
3.14.1 Effect of Manganese (Mn) in Steels ............... 3.443.14.2 Effect of Silicon (Si) in Steels ........................ 3.453.14.3 Effect of Chromium (Cr) in Steels ................. 3.453.14.4 Effect of Molybdenum (Mo) in Steels............. 3.453.14.5 Effect of Vanadium (V) in Steels ................... 3.453.14.6 Effect of Titanium (Ti) in Steels .................... 3.463.14.7 Effect of Tungsten (W) in Steels..................... 3.46
3.23 Plastics ........................................................................... 3.573.24 Properties and Applications of Some Common Thermoplastics ............................................................... 3.593.25 Properties and Applications of Some Common Thermosetting Polymers ............................................. 3.60
Module 4: Fasteners and Power Transmission Devices & Mechanical Measurements
Thermodynamics: Systems, Properties, Process, State, Cycle, Internalenergy, Enthalpy, Zeroth Law, First law and Second Law of Thermodynamics,Basic Concept of Entropy, Properpties of ideal gas., Propereties of puresubstances, Steam, formation, Types of Steam, Enthalpy, Specific volume,Internal energy and dryness fraction of steam, use of Steam tables, Relatednumerical.
1.1 INTRODUCTION
Thermodynamics can be defined as the science which deals with thetransformation of energy and its effects on the properties of the substance. Theword ‘Thermodynamics’ originates from two Greek words, ‘therme’ (heat) and‘dynamis’ (motion or power). Thermodynamics can also be called as the scienceof 3E’s i.e., Equilibrium, Energy and Entropy.
Thermodynamics is a branch of Science that deals with the relationshipamong heat, work and properties of system which are in equilibrium withone another.
One of the most fundamental laws of nature is the conservation ofenergy principle. It states that during an interaction, energy can change fromone form to another but the total amount of energy remains constant. i.e.,Energy can neither be created nor be destroyed, but it can be transformedfrom one form to another. For example,
1. A ball falling from a tower picks speed as a result of its potentialenergy being converted to kinetic energy.
2. A person eating more and working less i.e., a person who has agreater energy input (food) than energy output (exercise) will gain weight (storeenergy in the form of fat)
The first law of thermodynamics is based on the conservation ofenergy principle. The second law of thermodynamics reveals that energy hasquality as well as quantity and actual processes occur in the direction ofdecreasing quality of energy. For example, a cup of hot coffee in a roomeventually cools, but a cup of cool coffee never gets hot by itself.
The first and second laws of thermodynamics emerged simultaneously inthe 1850s, by the works of James Prescott Joule, William Rankine, RudolphClausius and Lord Kelvin.
1.2 ROLE OF THERMODYNAMICS IN ENGINEERING AND SCIENCE
Thermodynamics plays a vital role in the field of Engineering andScience. They are explained as follows.
Thermodynamics is very important in the design, analysis anddevelopment of Engineering systems such as the power-producing systemslike engines, turbines etc. and the power-consuming systems like compressor,pumps, etc.
The working principle of many scientific devices like thermometer,thermistor, thermostat etc... involves thermodynamics.
Good understanding and careful applications of the fundamentals ofthermodynamics leads to improved design, increased efficiency, optimumoperating conditions and decreased levels of environmental pollution etc.
1.3 APPLICATIONS OF THERMODYNAMICSThermodynamic principles are applied in various fields of Engineering
and Technology. Some of the fields are
(i) Internal combustion engine (ii) Gas turbines
(iii) Air conditioner & Refrigerator (iv) Compressor
(v) Steam and Nuclear power plants (vi) Rockets
(vii) Jet engines etc.
Applications of thermodynamics are found in various aspects of life
eg: A commonly encountered thermodynamics application is, ‘thehuman heart, which is constantly pumping blood to all parts of the body.The movement of blood and various energy conversions generates heat, withinthe body. (normally called as body heat). The generated heat is constantlyrejected to the environment and the process is called as metabolic heatrejection. However, this heat rejection can be controlled by adjusting ourclothing to the environment conditions.
Other application of thermodynamics involves increasing the heattransfer rate like exhaust fan in a house, cooling fans in the computer CPUand fins provided on the engine cylinders etc.
Some of the basic concepts of thermodynamics are discussed here. They are.
1.4.1 Units and DimensionsNowadays, the SI (System International) system of units has been used.
Any physical quantity can be characterised by dimensions. The arbitrarymagnitudes assigned to the dimensions are called Units. Some basicdimensions such as mass m, length L, time t and temperature T are selectedas primary (or) fundamental dimensions, while others such as velocity V,energy E, and volume V are expressed in terms of primary dimensions andare called secondary dimensions (or) derived dimensions.
A number of unit systems have been developed over the years. Butnowadays, the SI (System International)
system of units are widely used. The SI is simple and logical system. Somefundamental dimensions and their units in SI are given in the following table.
Dimensions Unit Symbol
Length metre m
Mass kilogram kg
Time second s
Temperature Kelvin K
Electric current Ampere A
Amount of light Candela cd
Amount of matter mole mol
1.4.2 Mass mIt is the quantity of matter contained in a body. Its unit is ‘kg’.
1.4.3 ForceIn SI, the unit for force is derived from Newton’s second law.
i.e., Force Mass Acceleration
F m a
In SI, the force unit is the Newton N and it is defined as the force required
to accelerate a mass of 1 kg at a rate of 1 m/s2. i.e., 1 N 1 kg m/s2
Thermodynamics 1.3
The weight of a body W is the force with which the body is attracted
towards the centre of the earth. It is the product of its mass m and the local
gravitational acceleration g, i.e., W mg, where g 9.80665 9.81 m/s2 atsea level. The mass m of a body will remain same, regardless of its locationin the universe. Its weight, however, will change with a change in gravitationalacceleration. A body will weigh less on top of a mountain since g decreaseswith altitude. At sea level, a mass of 1 kg will weigh 9.807 N.
1.4.4 PressurePressure is the normal force exerted
by a system against unit area of thebounding surface.
Pressure, P ForceArea
FA
The unit for pressure in the SI systemis the Pascal Pa, which is the force of 1
N acting on an area of 1 m2.
1 Pascal 1 Pa 1 N/m2
The unit of pascal is very small. Hence, very often, kilo-pascal (kPa)(or) mega-pascal (MPa) is used. Sometimes bar and standard atmosphereare also used which are not within the SI system of units.
1 bar 105 Pa 100 kPa 0.1 M Pa and
Standard atmosphere 1 atm 101.325 kPa 1.01325 bar
Note: The pressure P at a point in a fluid in equilibrium is the same in alldirections.
Most of the instruments indicate pressure relative to the atmosphericpressure where as the pressure of a system is its pressure above zero (or)relative to a perfect vacuum.
The pressure relative to the atmosphere is called gauge pressure.
The pressure relative to a perfect vacuum is called absolute pressure.
Absolutepressure
Gauge pressure Atmospheric pressure
The SI unit prefixes are used in all branches of engineering. Some standardprefixes in SI units for Mechanical Engineering branch are given in the table.
1.4.5 TemperatureIt is defined as the degree of intensity of heat or measure of hotness
or coldness of a body.
Temperature - Intensive property - C or K (Centigrade or Kelvin).
Eg: 30C 30 273 303 K
Normally, t is for C and T is for K.
1.4.6 Volume, Specific Volume and Density
Volume V is the space occupied by a substance and is measured in m3.
Volume Total volume V in m3
Specific volume: v
The specific volume v of a substance is defined as the volume per unit
mass and is measured in m3/kg.
Specific volume Volume
mass
Vm
v
Its unit is m3/kg
Density : It is the mass of the substance per unit volume. Its unit is
kg/m3.
Density is the mass per unit volume of a substance and is given
in kg/m3. mV
Thermodynamics 1.5
1.4.7 EnergyEnergy is the ability of the physical system to perform work. Energy
exists in several form such as heat energy, kinetic energy, potential energy,light energy, electrical energy, thermal energy and so on. However, theseenergy transformations are constrained by a fundamental principle known asthe “Conservation of Energy”, Principle. One way to state this principle is
“Energy can neither be created nor destroyed, but may transform fromone form into another form”.
Another approach is that,
“the total energy of the isolated system remains constant”.
In S.I units, energy is represented as Joule (J)
1 Joule 1 Nm kg m
s2 m 1 kg m2
s2
Different forms of energySome forms of energy are explained as follows.
3. Pressure EnergyIt is the energy possessed by any system (or body) by virtue of its
pressure.
Pressure energy Pv N
m2 m3
kg
Nmkg
J kg
Sp. pressure energy Pv in J/kg
4. Internal Energy: (U) in JIt is the energy possessed by any system (or body) due to its molecular
arrangement and motion of molecules. ‘u’ is the specific internal energy inJ/kg.
1.4.8 PowerThe rate of energy transfer (or) storage is called power. The unit of
power is Watts (W), kiloWatts kW (or) MegaWatts (MW).
Power Workdonetime taken
Nm
s
1 W 1 J/s 1 Nm/s; 1kW 1000 W
1.4.9 Dimensional HomogeneityIn engineering, all equations must be dimensionally homogeneous. i.e.,
every term in an equation must have the same unit.
Analyse the following equation, E 35 kJ 8 kJ/kg
where E is the total energy in kJ. Since the two terms on the right hand sidedo not have the same units, they can not be added to obtain the total energy.Multiplying the last term 8 kJ/kg m kg by mass will eliminate thekilograms in denominator, and the whole equation will become dimensionallyhomogeneous i.e., every term in the equation will have the same unit.
i.e., E 35 kJ 8 m kJ
1.5 CLASSICAL AND STATISTICAL THERMODYNAMICS
Normally, a substance consists of a large number of particles calledmolecules and the properties of substance naturally depend on the behaviourof these particles. There are two points of view from which the behaviour ofparticles can be studied. They are
Thermodynamics 1.7
1.5.1 Macroscopic viewpoint (Classical thermodynamics)In this point of view, a certain amount of matter is considered,
neglecting the events occurring at the molecular level. It does not require aknowledge on the behaviour of individual particles. It provides a direct andeasy way to the solution of engineering.
In general, macroscopic point of view is just like, “inventing a deviceor using a instrument, to measure the quantities directly, instead of findingit out through calculation, derivation and formulas”.
eg: ‘Pressure gauge’ is used to measure pressure directly withoutcalculating force on a given unit area.
Likewise, temperature can also be measured using thermometer.
However, macroscopic point of view is limited to those effects thatcan be perceived by human senses.
1.5.2 Microscopic viewpoint (Statistical thermodynamics)In this point of view, behaviour of each molecule has to be studied,
to find out the behaviour of the entire substance. Each molecule at a giveninstant has certain position, velocity and energy, and for each molecule theseparameters changes frequently as a result of collisions. And also, these effectscannot be perceived by human senses.
In general, microscopic point of view requires large number of dataand equations to find out the quantities through calculation and derivation.
eg: Impulse, Enthalpy, Entropy, Gibbs function, Helmhotz function etc.
3. Molecular effects can beperceived by human senses.
Molecular effects cannot be perceivedby human senses.
4. Classical thermodynamics,however can be derived fromstatistical thermodynamics.
Statistical thermodynamics cannot bedetermined using classicalthermodynamics.
5. eg: Pressure, Temperature eg: Enthalpy, Entropy
1.6 THERMODYNAMIC SYSTEMSThermodynamic system is known as a space
or constrained area upon which our attention isconcentrated on. It is the region to be studied. Thesystem has boundary (or) boundary line. Anythingoutside the boundary is called surroundings. In otherwords, the system and surroundings are separated byboundary. (Boundary may be real or imaginary).
1.6.1 Types of systemThe systems are classified into:
1. Closed system, 2. Open system, 3. Isolated system.
System may be considered to be closed (or) open, depending onwhether a fixed mass (or) a fixed volume in space is chosen for study.
1.7 CLOSED SYSTEMA closed system, is also known as a
control mass, in which only the energy istransferred from the surroundings into the systemand from the system to the surroundings. But thereis no mass transfer involved in closed system.
It consists of a fixed amount of mass andno mass can cross its boundary.
i.e., no mass can enter (or) leave a closedsystem as shown in Fig. 1.3 But energy, in theform of heat (or) work, can cross the boundary.And the volume of a closed system does not have to be fixed.
Eg. Piston and cylinder without valves Fig. 1.4.
SurroundingsBoundary
Surroundings
System
Fig. 1.2
C LOS EDSYS TEM
m = constan t
ene rg y Y ES
M ass N O
M ass can not c ross the bound aries of a closed sys tem , b ut ene rg y can.
Fig. 1.3
Thermodynamics 1.9
1.8 OPEN SYSTEMAn open system (control volume) is
a properly selected region in space.
i.e., Both energy and mass aretransferred from the system to surroundingsand from the surroundings to system.
The boundaries of a control volumeare called as control surface as shown inFig. 1.6 and they can be real (or) imaginary.
A system, which contains only a single phase, is called asHomogeneous system.
Eg. Mixture of air and water vapour.
A system, which consists of more than one phase, is called asHeterogeneous system.
Eg. Mixture of water and steam.
1.10.1 Pure substanceA Substance with homogeneous and invariable chemical composition
throughout its mass, even though change of phase takes place is called aspure substance. It is a one component system and may exist in one or morephases. Eg. Water.
1.10.2 State postulateThe number of independent intensive properties are required to fix the
state of the system is given by state postulate.
For a pure substance, this is called as two property rule. i.e., twointensive properties are required to fix the state of the pure substance.
Number of independent intensive properties can be found analyticallyusing Gibb’s rule
P F C 2
P - No. of phase, C - No. of components
F - No. of independent intensive properties
eg: For single liquid (or) single gas
P 1, C 1
F C P 2 1 1 2 2
1.11 PROPERTY
It is a characteristic of the system. The system is identified by somequantities like temperature, pressure, volume etc.
1. Intensive Property: The properties which do not depend on the massof the system, are called intensive property.
Intensive property - Independent of the mass of the system: Example- Temperature, pressure, density, surface tension, specific gravity, etc.
Thermodynamics 1.11
2. Extensive Property: Those properties which depend on the mass of thesystem are called extensive property. Example: Volume (If mass is more,then volume will be more) Force, eg: electric charge, weight, energy (heatand work) magnetization. (Fig 1.8).
The extensive property when divided bymass will become intensive property.
(e.g)
Volume m3Extensive
/ mass Specific volume m3/kgIntensive
Enthalpy kJExtensive
/mass Specific enthalpy kJ/kgIntensive
Generally, upper case letters are used todenote extensive properties (with mass ‘m’being a major exception), and lowercase lettersare used for intensive properties (with pressureP and temperature T being the obviousexceptions).
1.12 STATE, PATH AND PROCESSConsider a system does not
undergoing any change. At this point, allthe properties can be measured (or)calculated throughout the system, whichgives us a set of properties that completelydescribes the condition, or the state, of thesystem. At a given state, all the propertiesof a system have fixed values. If the valueof even one property changes, the statewill change to a different one. A systemis shown at different states in the Fig. 1.9.
Thermodynamics deals only withequilibrium states. The word equilibriumimplies a state of balance. In an equilibrium state, there are no unbalancedpotentials (or driving forces) within the system.
Exte ns ivep rop erties
In tens ivep rop erties
C riteria to diffe rentiate intensiveand extens ive properties .
There are many types ofequilibrium, and a system is not inthermodynamic equilibrium unless theconditions of all the relevant types ofequilibrium are satisfied.
For example, a system is inthermal equilibrium if the temperatureis the same throughout the system, asshown in Fig. 1.10.
i.e., the system experiences notemperature difference, which is thedriving force for heat flow.
Any change that a systemundergoes from one equilibrium state toanother is called a process, and the seriesof states through which a system passesduring a process is called the path of theprocess. To describe a process completely,one should specify the initial and finalstates of the process, as well as the pathit follows.
Refer the Fig. 1.11
At state (1), the pressure is P1 and
the volume is V1. Similarly at state (2),
the pressure is P2 and the volume is V2.
In between 1 and 2, there are so many statesA, B, C, , G.
1.13 CYCLEThe series of processes whose end
states are identical, is called as cycle. ReferFig 1.12.
12 The process A21 The process B121 Cycle
( ) Befo rea ( ) A fterbA c losed sys tem reaching therm a l
equ ilibrium
Fig. 1.10
XX
X
X
XX
X
P 2
P 1
V 2 V 1
V
1
2G
FE
D
CB A
S ta te2 S ta te 1
1- S ta teA - S ta teG - S tate2- S ta te
1 - 2 = P roce ss
Fig. 1.11
2
B
A
1
V
P
Fig. 1.12
Thermodynamics 1.13
The process A and the process B together form a cycle (or)thermodynamic cycle, because, the system has returned to its initial state atthe end of the process. i.e., for a cycle, the initial and final states are identical.
1.14 THERMODYNAMIC EQUILIBRIUM
When mechanical equilibrium, chemical equilibrium and thermalequilibrium are satisfied, then the system is said to be in thermodynamicequilibrium.
When any property remains same with respect to time, it is called asequilibrium or steady state.
Eg. Temperature and pressure at all points of a system are same onlyif the system exists in thermodynamic equilibrium.
1. Mechanical EquilibriumIf the system’s pressure remains same with respect to time, then the
system is said to be in mechanical equilibrium.
2. Chemical EquilibriumIf the system’s chemical composition remains same with respect
to time, then the system is said to be in chemical equilibrium.
3. Thermal EquilibriumIf the system’s temperature remains same with respect to time, then
the system is said to be in thermal equilibrium.
1.15 QUASI-STATIC (OR) QUASI–EQUILIBRIUM PROCESS
When a process proceeds in such a manner that the system remainsinfinitesimally close to an equilibrium state at all times, it is called aquasi-static, (or) quasi-equilibrium process. A quasi-equilibrium process canbe viewed as a sufficiently slow process that allows the system to adjustitself internally so that properties in one part of the system do not changeany faster than those at other parts.
A quasi-static process is an idealized process and is not a truerepresentation of an actual process. But many actual processes closelyapproximate it and they can be modeled as quasi-equilibrium with negligibleerror. Engineers are interested in quasi-equilibrium processes for two reasonsas follows.
1. They are easy to analyze.
2. Work-producing devices (like engine) deliver maximum work whenthey operate on quasi-static processes.
Therefore, quasi-static processes serve as standards to which actualprocesses can be compared.
Consider a system of gas contained in a cylinder Fig 1.13(a). Thesystem initially is in equilibrium state P1, V1 and t1. The weight on the piston
is balanced by the pressure of the gas. If the weight is removed, there willbe an unbalanced force between the system and surroundings, and under thegas pressure, the piston will move up till it hits the stops. The system comesto an equilibrium state - P2, V2 and t2. But the inter-mediate states are
non-equilibrium states and we cannot define its properties. So these two states(1 and 2) are joined by dotted line.
Now instead of single weight, we can place many number of smallweights on the piston Fig 1.13(b). If we remove weights one by one very
Stops
C ylinder
Fina l sta te
W eigh t
In itials tate
Systemboundary
P 1
P 2
1
2
P(P ressure )
V 1 V 2
V(Volum e)
Stops
C ylinder
W eigh ts
Systemboundary
P I S T O N
P V T1 1 1
P(Pressure)
Equilibrium sta tes
Q uas i-sta ticprocess
V (Vo lume)
Fig. 1.13 Quasi-Static process (or) Quasi-Equilibrium process
Thermodynamics 1.15
slowly from the piston, then the piston will move upward very slowly. Thechange of state of system from equilibrium state is infinitesimally small. Soevery state traced by the system is an equilibrium state. The process, joinedby all the equilibrium states passed by the system is called quasi-staticprocess. [‘Quasi’ means ‘almost’].
Quasi-static process is a reversible process.
A process which can be reversed in direction and the system retracesthe same equilibrium states is known as reversible process.
Infinite slowness is the characteristic feature of a quasi-static process.
1.16 PATH FUNCTION AND POINT FUNCTIONRefer the Fig 1.14. In this fig, there
are many quasi-static paths such asA, B or C from state 1 to state 2. Sincearea under each curve represents the workfor each process, the amount of workinvolved in path A, B and C are not equal.The work is not a function of the endstates of the process but it depends on thepath the system follows from state 1 tostate 2. Hence, work is called a path
function and d– W is inexact (or)imperfect differential.
Thermodynamic properties like pressure, volume and temperature arepoint functions, because, for a given state, there is a definite value for eachproperty. The change of property of a system is independent of the path, thesystem follows during the change of state and it depends only on the initialand final states of the system. Hence dV (or) dP is an exact (or) perfectdifferentials.
For example, the integration of property is
V1
V2
dV V2 V1
i.e., Change in volume depends only on the end states of the systemirrespective of the path of the system.
But, workdone in a quasi-static process between two given statedepends on the path followed. And the integration of work transfer is
1
2
dW W2 W1
Rather, 1
2
dW W1 2
1.17 ZEROTH LAW OF THERMODYNAMICS
If the systems A and B are inthermal equilibrium separately and A andC are in thermal equilibrium separately,then B and C should be in thermalequilibrium.
This law is the basis for temperaturemeasurement.
1.17.1 Equality of temperatureWhen two bodies at different temperatures are brought into contact,
after some time, they attains a common temperature and are then said to existin thermal equilibrium.
The zeroth law was first formulated by R.H Fowler. This law explainsthe fundamental principles of thermodynamics, but was recognised after theformulation of first and second laws of thermodynamics. So it was named as‘zeroth law’ since it should have preceded the first and second laws ofthermodynamics.
1.18 MEASUREMENT OF TEMPERATURE
The temperature of a body can be measured, by using a device knownas thermometer. A common thermometer consists of a small amount ofmercury in an evacuated glass tube. There are five different kinds ofthermometer.
1. Constant volume gas thermometer2. Constant pressure gas thermometer3. Electrical resistance thermometer4. Thermocouple 5. Mercury-in-glass thermometer
Therm
al e
quilib
rium
Thermal e quilibrium
B C
A
Fig. 1.15
Thermodynamics 1.17
1.18.1 ThermometryThermometry is the science and practice of temperature measurement.
Any measurable change in the thermometric probe (eg: extension of mercuryin the tube of mercury-in-glass thermometer) can be used to mark temperaturelevels, that should later be calibrated as internationally agreed unit (likecelsius, Fahrenheit, Kelvin, etc...)
Thermometry is broadly divided into two subfields.
(i) Contact thermometry (ii) Non-contact thermometry
1.18.2 Applications of thermometryTemperature is one of the most measured physical parameters in
science and technology, but thermometry is not only applied to measuringtemperature, but also an indirect measure of many material properties likethermal capacities, relative humidity (bymeans of wet bulb (or) dew pointtemperatures) and enthalpy changes (bythermal analysis).
1.18.3 Triple point of waterIt is a point at which three phases
(solid, liquid and gas) i.e., ice, liquidwater and water vapour exist inequilibrium with one another.
The temperature at which, thesestates exist in equilibrium are 273.16 Kor 273.16 degrees Kelvin.
1.18.4 Comparison of Thermometer
Thermometer Thermometric property
1. Constant volume gas thermometer Pressure P2. Constant pressure gas thermometer Volume V
3. Electric resistance thermometer Resistance R4. Thermocouple Thermal e.m.f E5. Liquid-in-glass thermometer Length L
1.19 EQUATION OF STATEThe equation which relates the properties P, v and T is known as an
Equation of state.
ie f P, v, T 0
The simplest form of equation of state for the ideal gas is given below.
Equation of state: Pv R
T [ R
universal gas constant in kJ kg mol.K
and v molar volume in m3 kg mol ]
Also: Pv RT [ R characteristic gas constant in kJ kg K]
v specific volume in m3 kgThe above equation - Equation of state is also called Characteristic gas equation.
1.20 CELSIUS TEMPERATURE SCALECelsius temperature scale is related to the kelvin scale by
t T 273.15
where t - celsius temperature, T - kelvin temperature
eg: At steam point, T 373.15 K i.e., kelvin temperature of steam at 1 atmis 373.15 K
t 373.15 273.15
t 100 C
Solved ProblemsProblem 1.1: The temperature t on a thermometric scale is defined in termsof a property k by the relation t a ln k b where a and b are constants.
The value of k are found to be 1.83 and 6.78 at the point 0C and steam
point 100C. Determine the temperature corresponding to a reading of kequal to 2.42 on thermometer.
Given:
k 1.83 when t 0; k 6.78 when t 100 k 2.42 when t ? t a ln k b
(i) 0 a ln 1.83 b
0.6043 a b 0 ...(1)
(ii) 100 a ln 6.78 b
1.914 a b 100 ...(2)
Thermodynamics 1.19
on solving, a 76.35 b 46.14
(iii) t 76.35 ln 2.42 46.14
76.35 0.8838 46.14 t
t 21.335 C
Problem 1.2: A New scale N of temperature is divided in such a way thatthe freezing point of ice is 100N and the boiling point 400N what is thetemperature reading on the new scale when the temperature in the celsiusscale is 150C. At what temp both the celsius and New temperature scalewould be same.
Given:
New temperature scale Celsius scale
Ice point 100N 0C
Boiling point 400N 100C
temperature = ? when 150C
Let us take celsius scale on the x - axis and new temperature scale onthe y - axis
Slope, m y
x N
C
400 100
100 0
300100
3
Slope-intercept form equation, is
y mx b (y - intercept)when x 0
N mc b (N 100 when C 0)
N 3C 100
Temperature on the new scale, when temp or celsius scale is 150C N 3 150 100 550 N
Also, the temp at which both scale readings are same.
At particular temperature, N C
From the relation N 3C 100
C 3C 100
C 3C 100 2C 100
C 50 C or N 50 N
So, at 50 both the scale would read the same value.
1.21 WORKIn mechanics, work is said to be
done by a force, as it acts upon a body,which moves the body in the direction ofthe forceWork Force distance moved F x
In thermodynamics, work is said tobe done by a system, if sole effect onthing’s external to system can be reducedto the raising of weight.
Refer Fig.1.17
Let A Cross sectional area of the
piston in m2
P = Pressure of gas at any time.
dl Distance moved by the pistonunder gas pressure.
Work done Force distance travelled
PA dl PAdl Pdv
Consider unit mass.
Then W Pdv [where v specific volume]
When the reversible process (Quasi-static process) takes place betweenstates 1 and 2, then
P1
P2
1
2
V 1 V 2dv
Quasi s ta ticp rocess
Fig. 1.17
Thermodynamics 1.21
Work done 1
2
Pdv
Therefore the work done by thegas during any reversible process is givenby the area under the Pv diagram.(Shaded area).
When the work is done by thesystem, it is positive work Fig 1.18(a).
When the work is done on thesystem, it is negative work Fig 1.18(b).
1.21.1 Work Transfer - a Path Function
Let us consider a system shown inFig 1.19.
The system moves from aequilibrium state 1 to a final equilibriumstate 2 through two different paths
1 A 2 and 1 B 2. The processes arequasi-static.
Since area under each curverepresents the work for each process.1 A 2 gives one value of work and
1 B 2 gives another value of work. Sothe work does not depend on the end stateof the process. But it depends on the pathof the system. So the work is called path
function. Hence d W is an inexact orimperfect differential.
Change in pressure P2 P1 ; Change in temperature t2 t1
So the properties of the system depend only on end states andindependent of the path the system follows. So the differentiation of pointfunctions are exact or perfect differentials.
Note:1. Thermodynamic properties are point functions, since they depend
only on the end states.
2. For a cyclic process, the initial and final states of the system arethe same, and hence, the change in any property is zero.
O dV 0, O dP 0, O dT 0
where the sign O denotes the cyclic integral for the closed path. Therefore,
cyclic integral of a property is always zero.
1.22 HEATHeat is defined as the form of
energy that is transferred between twosystems (or a system and its surroundings)by virtue of a temperature difference
The transfer of heat between twobodies in contact is called conduction.The transfer of heat between a wall anda fluid system in motion is calledconvection. The transfer of heat betweentwo bodies separated by empty space (or)gases through electromagnetic waves iscalled radiation.
The temperature difference is the driving force for heat transfer. Thelarger the temperature difference, the higher is the rate of heat transfer ReferFig 1.20.
Normally, the heat is transferred from high temperature system to thelow temperature system. The transfer of heat into a system is frequently
No hea ttransfer
8 J /sHeat
1 6 J /sHeat
Room air 25 Co
25 Co
Water
15 Co 5 Co
Water Water
Fig. 1.20
Thermodynamics 1.23
referred to as heat addition and is taken as positive. The transfer of heatout of a system is called heat rejection and is taken as negative.
Refer Fig. 1.21. A process during which there is no heat transfer iscalled an adiabatic process. An adiabatic process should not be confusedwith an isothermal process. Even though there is no heat transfer during anadiabatic process, the energy content and thus the temperature of a system canstill be changed by other means such as work.
As a form of energy, heat has energy units, kJ. The amount of heattransferred during the process between two states (states 1 and 2) is denoted byQ1 2. Heat transfer per unit mass of a system is denoted by q.
ie q Qm
kJ/kg
The rate of heat transfer i.e., the amount of heat transferred per unittime is denoted by Q
, where the overdot stands for the unit derivative (or)
“per unit time”. The heat transfer rate Q has the unit kJ/s, which is equivalent
to kW.
1.22.1 HEAT TRANSFER – A PATH FUNCTIONHeat transfer is also a path function. It depends on the path the system
follows. It does not depend on the end states. So dQ is inexact differential.
Refer Fig. 1.23. Area under TS diagram gives the heat transfer. Inthe processes A and B, the end states are same, but the path are different. Sothe heat transfer in two processes A and B are different.
Q1 2 1
2
d Q 1
2
Tds
Note
W1 2 W1 W2; W1 2 Work transfer from 1 to 2.
Q1 2 Q1 Q2; Q1 2 Heat transfer from 1 to 2.
s entropy (which will be explained later).
ds change in entropy.
Path functions have Inexact differentials designated by the symbol
(or) d–. Therefore, a differential amount of heat (or) work is represented by
Q or d– Q, W or d– W, respectively, instead of dQ (or) dW.
1.23 HEAT CAPACITY
Heat capacity (also called as the thermal capacity) is the ratio of theheat added to (or) heat removed from an object to the resulting temperaturechange.
It can also be defined as a product of mass and specific heat.
C mc kg
JkgK
JK
Fig. 1.23
Thermodynamics 1.25
It is denoted by capital letter C, Cp, Cv
The S.I unit of heat capacity is Joule/Kelvin.
1.24 REVERSIBLE AND IRREVERSIBLE PROCESS
Reversible processA process in which both system and its surroundings can be
simultaneously returned to their initial states after the process has beencompleted.
Irreversible processA process in which both system and its surroundings cannot be
simultaneously returned to their initial states after the process has beencompleted.
Reversible process Irreversible process
(i) Reversible process is an idealprocess
Irreversible process is a naturalprocess. (i.e.) all the processesoccurring in nature are irreversible.
(ii) Reversible process attainsequilibrium state at all the stages ofthe operation.
Irreversible process is in equilibriumonly at the initial and final stages ofoperation.
(iii) It is a extremely slow process. It occurs at measurable speed.
(iv) It takes infinite time for theprocess to occur.
It takes place in finite time.
(v) Workdone by a reversibleprocess is greater than thecorresponding irreversible process.
Workdone by an irreversible processis lesser than the correspondingreversible process.
(vi) It does not increase the entropyof the universe.
It increases the entropy of theuniverse (system & surroundings).
1.25 FIRST LAW OF THERMODYNAMICS
The first law of thermodynamics, also known as the conservation ofenergy principle, which provides a basis for studying the relationships amongthe various forms of energy and energy interactions. The first law ofthermodynamics states that energy can be neither created nor destroyed; itcan only change forms.
A rock at some elevation possesses some potential energy, and part ofthis potential energy is converted into kinetic energy as the rock falls.Experimental data show that the decrease in potential energy mg Z exactly
equals to the increase in kinetic energy m V2
2 V12
2 when the air resistance
is negligible;
The increase in the energy of a potato in an oven is equal to theamount of heat transferred to it.
According to First Law of Thermodynamics‘Heat can be converted into work and work can be converted into heat.’
For a closed system, Net heat transfer = Net work transfer.
O d– Q O d– W First law of Thermodynamics for closed system.
O is the cyclic integral for closed path. (i.e for cycle).
1.26 JOULES EXPERIMENT
Consider a closed system as shown in Fig 1.25. It consists of a knownmass of water, insulated container having thermometer and a paddle wheel.
Now a certain amount of work is done on the system by means of apaddle wheel. The work can be measured as
W1 2 Weight distance travelled
Fig. 1.24
Thermodynamics 1.27
The system was initially attemperature t1 and after work transfer, the
system temperature rises to t2. The
pressure is one atmospheric pressure. Theprocess 1-2 is shown in Fig 1.26.
Now the insulation is removed. Theheat transfer takes place between systemand surroundings till the system returns toits original temperature t1. The heat can be
measured as Q2 1 mCpt
The system executes a cycle 1 2 1.
Now we can see W1 2 is equal to Q2 1. It has been found that this
work W1 2 is always proportional to the heat Q2 1 and the constant of
proportionality is called the Joule’s equivalent (or) the mechanical equivalentof heat.
In this example, only two energy transfer occurs. If the cycle has somany heat and work transfers, we can get the same result as
develops work without any heat input. Therefore, thePMM 1 violates the I law of thermodynamics, sinceI law states that Q W in a cycle.
Hence this machine is impossible.
1.28 FIRST LAW FOR A CLOSED SYSTEM UNDERGOING CHANGE OF STATE
1.28.1 Internal Energy
Wcycle Qcycle applies only to systems undergoing cycles and
the algebraic sum of all energy transfer across system boundaries is zero. Butif a system undergoes a change of state during which both heat transfer andwork transfer are involved, the net energy transfer will be stored (or)accumulated within the system.
If Q heat transfer and
W work transfer, then the net energy transfer Q W will be storedin the system. Energy in storage is neither heat nor work and is given thename Internal Energy (or) simply energy of the system.
Therefore Q W E
where E is the increase in the energy of the system.
(or) Q E W
The change in energy of the system during a process is the sum of the changesin the internal, kinetic and potential energies and can be expressed as
No Hea t Input
Wo
rk o
utp
ut
ENGINE
Fig. 1.27
Thermodynamics 1.29
E U KE PE
where U m u2 u1; K.E 12
m V22 V1
2; PE mg Z2 Z1
When the initial and final states are specified, the values of the specificinternal energies u1 and u2 can be determined directly from the property tables
(or) thermodynamic property relations.
According to the first law of thermodynamics, when the systemundergoes a process, both heat transfer and work transfer takes place. Thenet energy transfer is stored within the system and is known as stored energy(or) total energy of the system.
Q W E First law of thermodynamics for process.
E P.E K.E UWhen P.E 0; K.E 0, then E U
Then Q W U. This is I law for process when P.E = 0 & K.E = 0. In otherwords, Q W U
1.28.2 Energy - A property of the systemConsider a system which changes its state
from state 1 to state 2 by following the path A,and returns from state 2 to state 1 by followingthe path B. (Refer the following Fig 1.28). Sothe system undergoes a cycle. writing the firstlaw for path A
QA EA WA ... (i)
and for the path B
QB EB WB ... (ii)
The processes A and B together constitute a cycle, for which
Similarly, if the system returns from state 2 to state 1 by followingthe path C instead of B,
then EA EC ... (iv)
From equations (iv) and (v)
EB EC ...(v)
Therefore, it is proved that change in energy between two states of asystem is the same, whatever path the system may follow in undergoing thatchange of state. Therefore, energy has a definite value for every state of thesystem. Hence, it is a point function and a property of the system.
SOLVED PROBLEMS
Problem 1.3: During a cycle, the sum of all heat transfers is 200 kJ. Thesystem completes 2 cycles per sec. Complete the following table, showingmethod for each item and compute the net rate of work output in kW.
Process Q in kW W in kW E in kW
a b 0 36 –
b c 350 0 –
c d 35 – 610d a – – –
Solution
Process a b, Q W E (I law of thermodynamics for process)
i.e. Q E W
0 E 36
So, Eab 36 kW
Process b c, Q E W
350 E 0 Eb c 350 kW
Process c d, Q E W
35 610 W Wc d 575 kW
Thermodynamics 1.31
Process d a, Qcycle 200 kJ
The system completes 2 cycles/sec, so
Q 200 2 kJ
1sec
400 kW
Q 400 kW Qa b Qb c Qc d Qd a
400 0 350 35 Qd a
So, Qd a 715 kW
Also O dE 0, since cyclic integral of any property is zero.
i.e. Ecycle 0
Ea b Eb c Ec d Ed a 0
36 350 610 Ed a 0
Ed a 296 kW
In Process d a, Q E W
715 296 W
Wd a 1011 kW
Process Q W E
a b 0 36 36b c 350 0 350
c d 35 575 610d a 715 1011 296Total Q 400 W 400 E 0
Problem 1.4: When a system is taken from state a to state b along pathacb, 84 kJ of heat flow into the system and the system does 32 kJ of work.
1. How much will the heat that flows into the system along the pathadb be, if the work done is 10.5 kJ?
2. When the system is returned from b to a along the curved path,the work done on the system is 21 kJ. Does system absorb orliberate heat? and how much of the heat is absorbed or liberated?
Problem 1.5: Air occupies a space of 0.3 m3 at a pressure of 2 bar and atemperature of 80C. It is heated at constant volume, until the pressure is 8bar. Determine (1) temperature at the end of the process (2) mass of air (3)change in internal energy (4) change in enthalpy (5) change in entropy duringthe process.
Constant volume processTo Find T2 Temp. at the end of the process
P1V1
T1
P2V2
T2
P1
T1
P2
T2 [... V1 V2]
T2 P2 T1
P1 800
353200
1412 K
To Find m - mass of air
P1V1 mRT1
m P1V1
RT1
200 0.3
0.287 353 0.59224 kg
[ . . . R 0.287 kJ/kg K for air]
To Find U change in internal energy
U mCvT2 T1 0.59224 0.7181412 353 450.32 kJ
[ . . . Cv 0.718 kJ/kg K for air]
To Find H change in enthalpy
H mCpT2 T1 0.59224 1.0051412 353 630.32 kJ
[ . . . Cp 1.005 kJ/kg K for air]
Thermodynamics 1.39
To Find S change in entropy
S mCv ln T2
T1 0.59224 0.718 ln
1412353
0.5895 kJ / kg K
Problem 1.6: A rigid tank contains air at 600 kPa and 150C. As a resultof heat transfer to the surroundings, the temperature and pressure inside thetank are 70C and 400 kPa, respectively. Determine the boundary work doneduring this process.
Solution:
A sketch of the system and the P-Vdiagram of the process are shown in Fig.
The boundary work can bedetermined as
W 1
2
PdV 0
because dV 0 ie a rigid tank has aconstant volume and dV 0 in the aboveW equation. Therefore, there is noboundary work done during this process.That is, the boundary work done during aconstant - volume process is always zero.This is also evident from the PV diagram of the process (ie the area underthe process curve is zero).
1.31.2 Constant Pressure Process (or) isobaric Process [P C; n 0]
During this process, (Heating (or) Cooling process) the pressureremains constant.
Heat transfer = Change in enthalpy in constant pressure process.
Also change in entropy S mCp ln T2T1
(or) S mCp ln V2
V1
For any process PVn constant
In constant pressure process n 0
So, PV0 C P C
Fig. 1.30 Constant Pressure Process (or) Isobaric Process
Thermodynamics 1.41
Problem 1.7: A frictionless piston - cylinder device contains 5 kg of water
vapour at 0.40 MPa and specific volume 0.5342 m3/kg. Heat is now
transferred to the system until the volume reaches 0.5951 m3/kg at constantpressure. Determine the workdone by the steam during this process.
Solution: A sketch of the system and P-V diagram of the process are shownhere
Assumption: The expansionprocess is quasi-equilibrium
W 1
2
Pdv P v2 v1
W mP v2 v1
5 0.4 103 0.5951 0.5342 121.8 kJ
The positive sign indicates that the work is done by the system. Thatis, steam uses 121.8 kJ of its energy to do this work. The magnitude of thiswork could also be determined by calculating area under the process curve(ie rectangle) on the P-V diagram, which is simply PoV for this case.
Problem 1.8: 1 kg of gas with initial volume 2.5 m3 is heated at constantpressure from 25C to 220C. Estimate the heat added, ideal work done,change in internal energy, change in enthalpy, change in entropy. Alsocalculate final pressure and final volume. Take Cp 0.984 kJ/kg K;
1.31.3 Constant Temperature Process (or) isothermal Process PV constant ; T constant
When a piston expands from high pressure to low pressure, thetemperature will fall. But in this process, heat is added continuously tomaintain the temperature constant. Similarly, during compression process, thepressure is raised. To avoid the temperature rise, the heat is liberatedcontinuously. So the temperature will remain constant.
When the system temperature is maintained constant during a process,the process is called isothermal process or constant temperature process.
Problem 1.9: A piston - cylinder device initially contains 0.5 m3 of air at
100 kPa and 80C. The air is now compressed to 0.1m3 in such a way thatthe temperature inside the cylinder remains constant. Determine the workdoneduring this process.
Solution: A sketch of the system and the PV diagram of the process areshown in Fig.
Assumptions: The compression process is quasi-equilibrium. Air can beconsidered as an ideal gas.
For an ideal gas at constant temp.
PV mRT C [constant]
(or) P CV
W 1
2
PdV 1
2
CV
dV
C 1
2
1V
dV C lnV 1
2 C ln
V2
V1
W1 2 P1V1 ln V2
V1
In the above equation, P1V1 can be replaced
by P2V2 (or) mRT. Also, V2V1
can be replaced by
P1P2
because P1V1 P2V2 ie V2V1
P1
P2
Thermodynamics 1.45
Substituting numerical values in the above equation.
W1 2 P1V1 ln V2
V1 100 0.5
ln
0.10.5
W1 2 80.472 kJ
The negative sign indicates that this work is done on the system (awork input), which is always the case for compression process.
Problem 1.10: A quantity of air has a volume of 0.5 m3 at a pressure of5 bar and 80C. It is expanded in a cylinder at a constant temperature toa pressure of 1 bar. Determine the amount of work done and heat transferduring expansion.
Change in Enthalpy H mCpT2 T1Change in Entropy S 0
For any process PVn C
For this process PV C
So, n
Problem 1.12: Air at 1.02 bar, 22C initially occupies a cylinder volume
of 0.015 m3 is compressed isentropically by a piston to a pressure of 6.8 bar.Determine, (i) the final temperature (2) the final volume (3) the work done.
1.31.5 Polytropic process: PVn constant n nGenerally all the processes are polytropic process. When the index of
expansion (or) compression ‘n’ is changed, the process name is changed.
When index n , it is called constant volume processwhen n 0, it is constant pressure process
when n 1, it is isothermal processwhen n , it is isentropic process.
Fig 1.33 (a)
Thermodynamics 1.51
When n n, then it is polytropic process.The n varies from 0 to .
During actual expansion and compression processes of gases, pressure
and volume are often related by PVn C, where n and C are constants. Aprocess of this kind is called a polytropic process. Let us derive a generalexpression for the workdone during a polytropic process.
We know, PVn C for polytropic process. p C
Vn CV n
Substitute the relation in W 1
2
PdV equation
We get W 1
2
PdV 1
2
CV n dv
C V n 1
1 n 1
2
PVn V2
1 n V11 n
1 n
P2V2
nV21 n P1V1
nV11 n
1 n
W P2V2 P1V1
1 n [. . . PVn P2V2
n P1V1n]
Note here that C PVn P1V1n P2V2
n
For an ideal gas, PV mRT,
P
P 1
P 2
V 1 V 2 V
2
1 P V P V1 1 2 2 = n n
P n = const.V
Schem atic and d iagram for a polytropic p rocess.P -V
The workdone equation can be written as W mR T2 T1
1 n and n 1
This PVn can be used for any process by substituting appropriate value for n. Polytropic Workdone can be calculated by replacing in the isentropicprocess workdone by n.
1. W P2V2 P1V1
1 n
mRT2 T11 n
(or) W P1V1 P2V2
n 1
mRT1 T2n 1
Q 0
Q U W mCvT2 T1 mRT2 T1
1 n
Note V2
V1 P1
P2
1n
; T2
T1 P2
P1
n 1n
2. U mCvT2 T1 mRT2 T1
1
Q U W . . . Cv
R 1
mRT2 T1
1
mRT1 T2n 1
mRT2 T1
1
mRT2 T1n 1
n 1 1 1 1
1 n 1 [ mRT2 T1 ]
n
1 n 1 [ mRT2 T1 ]
n 1
mRT1 T2
n 1 n 1
W . . . W
mRT1 T2n 1
Insert table from Pg 2.30, 2.31 (Kerala book Thermodynamics)
Thermodynamics 1.53
For
all
pro
cess
es,
P1V
1
T1
P
2V2
T2
an
d P
V
mR
T
ie P
1V1
mR
T1;
P2V
2
mR
T2
....
Pro
cess
pVn
CW
UQ
H S
Rem
arks
Wor
k do
ne
W
p
dv
Ch
ange
in
Inte
rnal
en
ergy
Hea
t tr
ansf
erC
hang
e in
En
thal
pyC
han
ge i
nE
ntr
opy
1.C
onst
ant
volu
me
proc
ess
n
V
C
0 (z
ero)
mC
vT
2
T1
mC
vT
2
T1
mC
pT
2
T1
mC
vln
T2
T1 (
or)
mC
vln
P2
P1
Pd
V
0
sinc
e
dV
0
Q
U
V1
V2
2.C
onst
ant
pres
sure
proc
ess
n
0
P
CW
P V
2
V1
(or)
W
mR T
2
T1
mC
vT
2
T1
mC
pT
2
T1
mC
pT
2
T1
mC
pln
T2
T1
(or)
mC
pln
V2
V1
Q
H
P1
P2
3.C
onst
ant
tem
pera
ture
proc
ess
(or)
Isot
herm
alpr
oces
s
n
1 P
V
CT
C
PV
ln V
2
V1 or
PV
ln P
1
P2
or
mR
T ln
V2
V1
0 (z
ero)
(s
ince
T1
T2)
Q
W
P
V ln
V2
V1
P
V ln
P1
P2
0 (z
ero)
(sin
ce
T1
T2)
.m
R ln
V2
V1 (
or)
mR
ln P
1
P2
Q
W
P1V
1
P2V
2
T1
T2
4.Is
entr
opic
proc
ess
(or)
Rev
ersi
ble
Adi
abat
icpr
oces
s
n
PV
C
T2
T1
P2
P1
1
T2
T1
V1
V2
1
P1V
1
P2V
2
1
(or)
mR T
1
T2
1
mC
V T
2
T1
or
U
W
Zero
(No
Hea
tT
ran
sfer
)
mC
p T
2
T1
0 (z
ero)
si
nce
s 1
s2
P1V
1 P
2V2 ;
T2
T1
P2
P1
1
;
V2
V1
P1
P2 1 ;
T2
T1
V1
V2
1
5.P
olyt
ropi
cpr
oces
sn
n; p
Vn
C
T2
T1
p 2 p 1 n
1
n
V2
V1
p 1 p 2 1 n
P1V
1
P2V
2
n
1
(or)
mR T
1
T2
n
1
mC
vT
2
T1
n
1
W
(or)
mC
nT
2
T1
mC
pT
2
T1
mC
nln T
2
T1
(or)
n
1
mR
ln V
2
V1
Pol
ytro
pic
sp.
heat
Cn
Cn
Cv
n
1
n ;
P1V
1n P
2V2n
T2
T1
V1
V2 n
1
The
rmod
ynam
ics
1.55
1.54
Bas
ic M
echa
nica
l E
ngin
eerin
g -
ww
w.a
irwal
kpul
icat
ions
.com
3. So in polytropic process Q n
1 W (or)
Q mCn T2 T1 where Cn CV n1 n
4. Change in enthalpy H mCpT2 T1
5. Change in entropy S S n 1
mR ln V2
V1 (or) S mCn ln
T2
T1
By studying all the non-flow processes we can draw a tabular column toknow W, Q, U, H, S for ready made use.
By referring the tabular column in the nearby pages, we can solve anytype of problem in non-flow process.
Problem 1.13: Air at 300C and 10 bar expands to 3 bar reversibly
following the law PV1.35 C. Determine the work done per kg of air, heattransfer and change in entropy.
When there is a mass transfer across the boundary, the system is calledas open system. Most of the engineering devices are open system, involvingthe flow of mass through them.
We can mark an imaginary box (broken line around the open system)and this is called control surface. The volume inside the control surface isknown as control volume. The mass of fluid is flowing through this controlvolume.
Steady flow means that the rate of flow of mass and energy areconstant. The properties will have fixed value at particular location and willnot alter with time.
m
1 m
2 m mass rate of flow in kg/sec
m 1 A1 C1 2 A2 C2
m
A1 C1
v1
A2 C2
v2
where density of fluid, C Velocity of fluid, v specific volume
A Area of cross section through which mass flows.
According to Ist Law, Q E W Q E2 E1 W
E1 Q E2 W
E1 energy at position 1 for one Kg of mass
E1 PE1 KE1 Flow work u1
P.E. = Potential energy
K.E. = Kientic energy; Flow work P1 v1
u Internal energy
Thermodynamics 1.57
E1 Z1 g C1
2
2 P1 v1 u1; E2 Z2 g
C22
2 P2 v2 u2
The steady flow energy equation on mass basis is given below.
Z1 g C1
2
2 P1 v1 u1 Q Z2 g
C22
2 P2 v2 u2 W
Here all energy values are in J/kg.
The steady flow energy equation for m
kg/sec of fluid is given asfollows. (Time basis)
m [Z1 g
C12
2 h1] Q
m
[Z2 g
C22
2 h2] W
where Q in J/s i.e., Watts [ h1 u1 P1 v1]
W
in J/s i.e., Watts h2 u2 P2 v2]
Z1 ht above datum at 1 in m
C1 Velocity of fluid at 1 in m/sec
h1 enthalpy of fluid at 1 in J/kg
A1 Area at inlet in m2
Similarly Z2, C2, h2 and A2
The steady flow energy equation
h1 C1
2
2 Z1 g Q h2
C22
2 Z2 g W in J/kg (or)
m h1
C12
2 Z1 g Q
m
h2
C22
2 Z2 g W
in Watts which can be
applied for
1. Nozzles and Diffuser 2. Throttling Device
3. Turbine and Compressor
4. Heat exchangers (Boiler, Condensor and evaporator)
Formulae to Remember1. The Steady Flow Energy Equation (SFEE) is given by per unit mass
h1 C1
2
2 Z1 g Q h2
C22
2 Z2 g W in J/kg (or)
m h1
C12
2 Z1 g Q
m
h2
C22
2 Z2 g W
in Watts
2. For boiler SFEE is Q h2 h1
3. For condenser Q h2 h1
4. For nozzle SFEE is C2
2 C12
2 h1 h2 [where h1, h2 are in J/kg ]
Final velocity C2 2 h1 h2 C12 m/s
If initial velocity is neglected
C2 2 h1 h2
5. For turbine SFEE is W h1 h2 J/kg
6. For rotary Compressor W h2 h1 J/kg
7. For Reciprocating Compressor W Q h2 h1 J/kg
Problem 1.14 A mass of 0.8333 kg/sec enters the control volume of a steadyflow system at 2 bar and 100C and an elevation of 100 m above the datum.The same mass leaves the control volume at 150 m elevation with a pressureof 10 bar and temperature of 300C. The entrance velocity is 40 m/sec andthe exit velocity is 20 m/sec. During the process 13.8889 kW of heat aretransferred to the control volume and the rise in enthalpy is 8 kJ/kg. Calculatethe power developed.
Solution
Given: m 0.8333 kg/sec; P1 2 Bar; T1 100 273 372 K ; Z1 100 m;
Z2 150 m; C1 40 m/sec; C2 20 m/sec; P2 10 bar;
T1 300 273 573 K ; Q 13.8889 kW 13.8889 103 W;
h2 h1 8000 J/kg; h1 h2 8000 J/kg
Steady flow Energy equation gives
Thermodynamics 1.59
m h1
C12
2 Z1 g Q
m
h2
C22
2 Z2g
W
Q is in Watts and W
is in Watts
W
Power m h1 h2
C1
2 C22
2 Z1 g Z2 g
Q
0.8333 8000
402 202
2 [100 150 9.81]
13.8889 103
7313.75 Watts
1.32.1 Nozzle: Nozzle is a device which is used to increase the velocity of fluid at
the expense of its pressure drop.
1.32.2 Diffusor: Diffusor is a device which is used to increase the pressure of fluid at
the expense of its kinetic energy (by velocity drop)
Problem 1.15: The velocity and enthalpy of fluid at the inlet of a certainnozzle are 50 m/sec and 2800 kJ/kg respectively. The enthalpy at the exit ofnozzle is 2600 kJ/kg. The nozzle is horizontal and insulated so that no heattransfer takes place from it. Find 1. Velocity of fluid at exit of the nozzle,
2. mass flow rate, if the area at inlet of nozzle is 0.09 m2 and specific
volume is 0.185 m3/kg 3. Exit area of the nozzle, if the specific volume at
Problem 1.16: A turbine operates under steady flow condition receivingsteam with the following data. Pressure P1 1.2 MPa; Temperature
t1 188C; enthalpy h1 2785 kJkg
; Velocity C1 33.3 m/sec and elevation
Z1 3 m. The steam leaves the turbine at 100 m/sec velocity, zero elevation,
pressure = 20 kPa, 2512 kJ/kg of enthalpy. Heat is lost to the surroundingat the rate of 0.27 kJ/sec. Determine the power output of the turbine, if themass flow rate of steam is 0.42 kg/sec.
Solution
P1 1.2 MPa 1.2 106 Pa P2 20 kPa 20 103 Pa
T1 188 273 461 K T2 ?
h1 2785 kJ/kg
2785 103 J/kg
h2 2512 103 J/kg
C1 33.3 m/sec C2 100 m/sec
Z1 3 m Z2 0
Q 0.27 103 J/sec 270 J/sec m
0.42 kg/sec
Applying S.F.E.E
m h1
C12
2 Z1 g
Q
m
h2
C22
2 Z2 g
W
Thermodynamics 1.63
W
m h1 h2
C12 C2
22
Z1 g Z2 g Q
0.42 2785 103 2512 103
33.32 1002
2 3 9.81 0
270
112535 J/sec 112.535 kW
Problem 1.17: In a Gas turbine installation, the gases enter the turbine atthe rate of 5 kg/sec with a velocity of 50 m/sec and enthalpy of 900 kJ/kgand leave the turbine with 150 m/sec. and enthalpy of 400 kJ/kg. The lossof heat from the gases to the surroundings is 25 kJ/kg. AssumeR 0.285 kJ/kgK Cp 1.004 kJ/kg K and inlet conditions to be at 100 kPa
and 27C. Determine the diameter of the inlet pipe.
Given m 5 kg/s, C1 50 m/s, h1 900 kJ/kg
C2 150 m/s, h2 400 kJ /kg
R 0.285 kJ /kg K, Cp 1.004 kJ /kg K, Q 25 kJ /kg
P1 100 kPa; t1 27 CThe steady flow equation is given by
1.32.5 Compressor: Compressor is a device used to increase the pressure of the fluid by
taking work input.
Problem 1.18: Air enters a compressorwith a velocity of 60 m/sec, pressure100kPa, temp 40C and leaves thecompressor with a velocity of 90 m/sec,500kPa and 120C. Consider the system isadiabatic. Find the power of motor for themass flow rate of 40kg/min. Write theassumptions made.
Solution
Given
C1 60 m/sec C2 90 m/sec
P1 100 103 Pa P2 500 103 Pa
T1 40 273 313 K T2 120 273 393 K
Q 0 adiabatic
m 40 kg/min
4060
0.6667 kg/sec
S.F.E.E
m h1
C12
2 Z1 g Q
Air Compressor
(1)
W
(2)
Q =0(a
diab
atic
)
Thermodynamics 1.65
m h2
C22
2 Z2 g W
Assume Z1 g Z2 g
(Since nothing is given about P.E.)
W
m h1 h2
C12 C2
2
2
h1 h2 Cp T1 T2 1.005 313 393
80.4 kJ/kg 80.4 103 J/kg
W
0.6667 80.4 103
602 902
2
55102.76 J/sec 55.103 KW
[– sign indicates work is done on the system]
1.32.6 Heat Exchanger: It is a device used to heat the fluid (or) cool the fluid. No work
transfer, change in kinetic energy is negligible. Eg. Boiler cooling plant.
(i) Boiler:
It is a device in which water (or) fluid is heated.
Problem 1.19: In a cooling plant water enters with enthalpy of 240 kJ/kgand leaves with enthalpy of 192 kJ/kg. Pump work is negligible.The exit pipe is 20 m above the inlet piple. Change in K.E. is negligible.Find the heat transfer from the water.
[– sign indicates heat is flowing from the water to surroundings]
Problem 1.20: The boiler generates 800 kg/hr of steam. The enthalpy of feedwater is 170 kJ/kg and the enthalpy of steam is 2800 kJ/kg. Neglectingpotential and kinetic energies, find the rate of heat transfer.
Solution
m
8003600
0.222 kg sec
S.F.E.E on time basis
m h1
C12
2 Z1 g Q
m
h2
C22
2 Z2 g W
Change in K.E. = 0; Change in P.E. = 0
W
0
The S.F.E.E equation becomes m h1 Q
m
h2
Q m
h2 h1 0.222 2800 103 170 103
584444.4 J/sec 584.44 kW
58.444 kW of heat is added to boiler water.
1.33 LIMITATIONS OF FIRST LAW OF THERMODYNAMICS
Limitations Description Examples
(i) No restriction in thedirection of heat flow.
First law does notspecify the direction ofthe process. It onlyestablishes a relationshipbetween heat and work,but failed to mention,whether the heat flowsfrom cold body to hotbody or not.
Consider heating of aroom by means of anelectric room heater.Room gets heated bypassing current through aresistor. But, transferringthe heat from room willnot generate electricalenergy.
Thermodynamics 1.67
Limitations Description Examples
(ii) Feasibility of theprocess is not specified.
It does not give anyinformation, whether theprocess is possible or not.
Consider two bodies Aand B. A is at highertemperature than B.When brought together,either the body A losesheat to body B or thebody A gains heat frombody B. First law failsto explain the processthat is possible.
(iii) Amount of energyconversion is notspecified.
Work can be convertedinto 100% equivalentheat but heat cannot beconverted into 100%equivalent work.
An electric heater canconvert 1 kJ of work to1 kJ heat but an ICengine cannot convert 1kJ heat to 1kJ work.
So heat and work arenot completelyinterchangeable forms ofenergy.
This is because someamount of heat getsdissipated duringcombustion process.
1.34 THE SECOND LAW OF THERMODYNAMICSA process must satisfy the first law to occur.
However, satisfying the first law alone does notensure that the process will actually take place. Acup of hot coffee left in a cooler room eventuallycools off (Fig. 1.35). This process satisfies the firstlaw of thermodynamics since the amount of energylost by the coffee is equal to the amount gained bythe surrounding air. Now let us consider the reverseprocess–the hot coffee getting even hotter in a coolerroom as a result of heat transfer from the room air.We all know that this process never takes place. Yet,doing so would not violate the first law as long as the amount of energy lostby the air is equal to the amount gained by the coffee.
Consider a paddle-wheel mechanism that is operated by the fall of amass (Fig. 1.36). The paddle wheel rotates as the mass falls and stirs a fluid
within an insulated container. As a result, thepotential energy of the mass decreases, and theinternal energy of the fluid increases inaccordance with the conservation of energyprinciple. However, the reverse process, raisingthe mass by transferring heat from the fluid tothe paddle wheel, does not occur in nature,although doing so would not violate the firstlaw of thermodynamics.
The first law places no restriction on thedirection of a process, but satisfying the firstlaw does not ensure that the process will actually occur. This inadequacy ofthe first law to identify whether a process can take place is remedied byintroducing another general principle, the second law of thermodynamics. Aprocess will not occur unless it satisfies both the first and the second lawsof thermodynamics. Before studying second law of thermodynamics, weshould know the following terms.
1.34.1 Thermal Energy ReservoirsA hypothetical body with a relatively
large thermal energy capacity (mass specific heat) that can supply or absorb finiteamounts of heat without undergoing anychange in temperature is called a thermalenergy reservoir, or just a reservoir. Inpractice, large bodies of water such as oceans,lakes, and rivers as well as the atmosphericair can be considered as thermal energyreservoirs because of their large thermalenergy storage capabilities or thermal masses(Fig. 1.37). The atmosphere, for example,does not warm up as a result of heat losses from residential buildings inwinter.
A reservoir that supplies energy in the form of heat is called a source,eg: Furnace. A reservoir that absorbs energy in the form of heat is called asink eg: atmosphere (Fig. 1.38). Thermal energy reservoirs are often referredto as heat reservoirs since they supply or absorb energy in the form of heat.
Fig. 1.37 Bodies with relativelylarge thermal masses can beconsidered as thermal energy
Fig. 1.36 Transferring heat toa paddle wheel will not
cause it to rotate
Thermodynamics 1.69
1.35 HEAT ENGINESWork can easily be converted to other forms
of energy, but converting other forms of energy towork is not that easy. The mechanical work done bythe shaft shown in Fig. 1.38 is first converted to theinternal energy of the water. This energy may thenleave the water as heat. Any attempt to reverse thisprocess will fail. That is, transferring heat to the waterwill not cause the shaft to rotate. From thisobservations, we conclude that work can be convertedto heat directly and completely, but converting heatto work requires the use of some special devices.These devices are called heat engines.
3. They reject the remaining waste heat to a low-temperature sink(the atmosphere, rivers, etc.).
4. They operate on a cycle.
1.35.1 Heat Engine CycleA heat engine cycle is a thermodynamic cycle in which there is a net
heat transfer to the system and a net work transfer from the system. Herethe heat engine is considered as a system.
The work-producing device that best fits into the definition of a heatengine is the steam power plant, which is an external-combustion engine.That is, combustion takes place outside the engine, and the thermal energyreleased during this process, is transferred to the steam as heat. The schematicof a basic steam power plant is shown in Fig 1.41
Q1 Qin Amount of heat supplied to water in boiler from a
high-temperature source (furnace)
Q2 Qout Amount of heat rejected from steam in condenser to a
low-temperature sink (the atmosphere, a river, etc.)
Wout Amount of work delivered by steam as it expands in turbine
Win Amount of work required to compress water to boiler pressure
Fig. 1.41 Schematic of a steam power plant
Thermodynamics 1.71
The net work output of this power plant is simply the differencebetween the total work output of the plant and the total work input.
Wnet Wout Win (kJ) WT WP
The efficiency of a heat engine or a heat engine cycle is defined as
Network output of the cycleTotal heat input to the cycle
Wnet
Q1
WT WP
Q1
By the first law of thermodynamics Q W
For a heat engine, Q1 Q2 WT WP
Q1 Q2
Q1
1 Q2
Q1
This is also known as the thermal efficiency of a heat engine.
1.36 THE SECOND LAW OF THERMODYNAMICS: KELVIN-PLANCK STATEMENT
Even under ideal conditions, a heat engine must reject some heat to alow-temperature reservoir in order to complete the cycle. That is, no heatengine can convert all the heat it receives to useful work. This limitation onthe thermal efficiency of heat engines forms the basis for the Kelvin-Planckstatement of the second law of thermodynamics, which is expressed asfollows:
It is impossible for any device that operates on a cycle to receive heatfrom a single reservoir and produce a net amount of work.
Efficiency Wnet
Q1
is always less than unity because heat transfer to the system cannot
be completely converted into work transfer. So, 100%. There is alwaysheat rejection.
Kelvin-Planck Statement in simple wordsIt is impossible to construct a
heat engine which will convert all heatenergy into equal amount of work in acyclic process. i.e., All heat cannot beconverted into work. Some heat will berejected to the surroundings (Sink).
That is, a heat engine mustexchange heat with a low-temperaturesink as well as a high-temperaturesource to keep operating. TheKelvin-Planck statement can also beexpressed as no heat engine can havea thermal efficiency of 100 percent(Fig. 1.42), or as for a power plant tooperate, the working fluid must exchange heat with the environment aswell as the furnace.
Perpetual motion of machine kind II - PMM2A heat engine which will produce work in a complete cycle by
exchanging heat with only one reservoir, is called PMM2. This engine is notpossible since it violates second law - Kelvin Planck Statement.
Fig. 1.42 A heat engine that violatesthe Kelvin-Planck statement of the
second law
Fig. 1.43 PPM2 and Possible Engine
Thermodynamics 1.73
1.37 THE SECOND LAW OF THERMODYNAMICS: CLAUSIUS STATEMENT
It is impossible to construct a device that operates in a cycle andproduces no effect other than the transfer of heat from a lower-temperaturebody to a higher-temperature body.
In otherwords, it is impossible toconstruct a device which will transfer theheat from cold body to hot body withoutexternal work.
[i.e., Heat cannot flow from cold body to hotbody without any external work]
It is a common knowledge that heatdoes not flow from a cold medium to awarmer one. The Clausius statement simplystates that a refrigerator, will not operateunless its compressor is driven by a externalpower source, such as an electric motor (Fig.1.44). This way, the net effect on thesurroundings involves the consumption ofsome energy in the form of work, in addition
Fig. 1.44 (a)&(b)
Fig. 1.44(c) A refrigerator thatviolates the Clausius statement
to the transfer of heat from a colder body to a warmer one. Therefore, ahousehold refrigerator is in complete compliance with the Clausius statementof the second law.
Both the Kelvin- Planck and the Clausius statements of the second laware negative statements. Until now, no experiment has been conducted thatcontradicts the second law, and this should be taken as sufficient evidenceof its validity.
1.38 REFRIGERATORS AND HEAT PUMPS
1.38.1 RefrigeratorWe all know that heat
flows in the direction ofdecreasing temperature, that is,from high-temperature medium tolow-temperature one. This heattransfer process occurs in naturewithout requiring any devices. Thereverse process, however, cannotoccur by itself. The transfer ofheat from a low-temperaturemedium to a high-temperature onerequires special devices calledrefrigerators.
Refrigeration is the process of maintainingthe temperature of a body below that of itssurroundings.
The working fluid used for this purpose iscalled Refrigerant, the equipment used is calledRefrigerator.
The refrigerant is compressed to higher pressureand
high temperature in the compressor and it getscooled in the condenser at constant pressure.
The high pressure refrigerant vapour isexpanded in the expander and the required quantityof refrigerant evaporates in the evaporator byabsorbing heat from the space to be cooled and thecycle is repeated.
Condenser
Evaporator
Expander
Com pressor
Fig 1.45 (a)
Fig. 1.45 (b)
Thermodynamics 1.75
Refrigerators, like heat engines, are cyclicdevices. The working fluid used in the refrigerationcycle is called a refrigerant.
A refrigerator is shown schematically in Fig.1.45 (a). In Fig. 1.45 (b), Q2 is the magnitude of the
heat removed from the refrigerated space attemperature T2. Q1 is the magnitude of the heat
rejected to the warm environment at temperature T1
and Win is the net work input to the refrigerator.
1.38.2 Heat PumpHeat pump is a device which maintains
temperature of a body greater than that of itssurroundings.
Another device that transfers heat from alow-temperature medium to a high-temperature oneis the heat pump, shown schematically in Fig. 1.46Refrigerators and heat pumps operate on the samecycle but differ in their objectives. The objective of a refrigerator is tomaintain the refrigerated space at low temperature by removing heat from it.
The objective of a heat pump, is to maintain a heated space at hightemperature. This is accomplished by absorbing heat from a low- temperaturesource, such as well water or cold outside air in water, and supplying thisheat to the high-temperature medium such as a house.
1.39 COEFFICIENT OF PERFORMANCE - COPThe efficiency of a refrigerator is expressed in terms of the Coefficient
of Performance (COP). It is a performance parameter. The objective of arefrigerator is to remove heat Q2 from the refrigerated space. To accomplish
this objective, it requires a work input of Win. Then the COP of a refrigerator
can be expressed as
COPR Desired outputRequired input
Q2
Win
This relation can also be expressed in rate form by replacingQ2 by Q
The conservation of energy principle for a cyclic device requires that,Win Q1 Q2
Then the COP relation can also be expressed as
COPR Q2
Q1 Q2
1Q1/Q2 1 ... (i)
Notice that the value of COPR can be greater than unity. That is the
amount of heat removed from the refrigerated space can be greater than thatof the amount of work input.
The measure of performance of a heat pump is also expressed in termsof the Coefficient Of Performance COPHP, defined as
COPHP Desired outputRequired input
Q1
Win
which can also be expressed as
COPHP Q1
Q1 Q2
11 Q2/Q1 ...(ii)
A comparison of Equations (i) and (ii) reveals that
COPHP COPR 1
for fixed values of Q2 and Q1
1.40 EQUIVALENCE OF THE TWO STATEMENTS
The Kelvin-Planck and the Clausius statements are equivalent in theirconsequences, and either statement can be used as the expression of thesecond law of thermodynamics. Any device that violates the Kelvin-Planckstatement also violates the Clausius statement, and vice versa. This can bedemonstrated as follows.
Consider a heat enginerefrigerator combination shown in Fig. 1.47 (a)operating between the same two reservoirs. The heat engine is assumed toviolate the Kelvin-Planck statement and have a thermal efficiency of 100percent, and therefore it converts all the heat Q1 to work W. This work is
now supplied to a refrigerator that removes heat in the amount of Q2 from
the low-temperature reservoir and rejects heat in the amount of Q2 Q1 to
the high-temperature reservoir. During this process, the high-temperaturereservoir receives a net amount of heat Q2 (the difference between
Thermodynamics 1.77
Q2 Q1 and Q1). Thus, the combination of these two devices can be viewed
as a refrigerator, as shown in Fig. 1.47 (b), that transfers heat in an amountof Q2 from a cooler body to a warmer one without requiring any input from
outside. This is clearly a violation of the Clausius statement. Therefore, aviolation of the Kelvin-Planck statement results in the violation of the Clausiusstatement.
It can also be shown in a similar manner that a violation of theClausius statement leads to the violation of theKelvin-Planck statement. Therefore, theClausius and the Kelvin-Planck statements aretwo equivalent expressions of the second lawof thermodynamics.
1.40.1 Perpetual-Motion MachinesA process cannot take place unless it
satisfies both the first and second laws ofthermodynamics. Any device that violateseither law is called a Perpetual-MotionMachine.
A device that violates the first law ofthermodynamics (by creating energy) is calleda Perpetual-Motion Machine of the first
Fig. 1.47 Proof that the violation of the Kelvin-Planck statement leads to theviolation of the Clausius statement
PUM PTURBINE
Sys tem bou ndary
W ne
t
.
Q��
.
A perpetual-motion machinethat v io lates the second law of therm odynamics (PM M2).
kind (PMM1), and a device that violates the second law of thermodynamicsis called a Perpetual- Motion machine of the second kind (PMM2).
1.41 CARNOT CYCLEThe best known reversible cycle is the Carnot cycle, first proposed in
1824 by French engineer Sadi Carnot. The theoretical heat engine that operateson the Carnot cycle is called the Carnot heat engine. The Carnot cycle iscomposed of four reversible processes-two isothermal and two adiabatic–and itcan be executed either in a closed or a steady-flow system.
However, Carnot cycle is an imaginary cycle. It gives maximumefficiency. We are studying carnot cycle to compare the efficiencies of othercycles (Air Standard cycles).
The efficiency of Carnot Cycle can be calculated as follows:The Carnot Cycle consists of following processes.
Process 1-2: Reversible Isothermal expansion process
PV C ; P1V1 P2V2
Process 2-3: Isentropic expansion process. P2V2 P3V3
; PV C
Carnot Cycle
Carnot Cycle
1
2
3
4
Q 1
Q 2
P-V diagram
Isotherm al Exp ansion
Isen trop icExp ansion
Isen trop iccom pressio n
Isotherm alcom pressio n
T 2
T 1
(P)
W c
W E
S ource T1
Q 1
Q 2
S ink T 2
S ystem A diaba tic
W E
W C
T 2
T 1
T
1 2
34
T-S diagram
s
Therm al C onductor(D iatherm ic C over)
Therm al Insulator(Adiabatic Cover)
V
T = const1
T = const2
Fig. 1.49 (a)
Thermodynamics 1.79
Process 3-4: Reversible Isothermal compression process.
P3V3 P4V4 ; PV C
Process 4-1: Isentropic Compression process. P4V4 P1V1
The area within the PV diagram represents the networkWnet WE WC or Q1 Q2 of this cycle. The net effects of the cycle are
(i) Heat is added to gas from the high temperature reservoir (source)at T1.
(ii) Heat is rejected from the gas to the low temperature reservoir(sink) at T2.
(iii) Work is produced.
First law states that
Wnet Heat added Heat rejected, Wnet Q1 Q2
Heat added to cylinder at T1, Q1 mRT1 ln V2
V1 (Isothermal process)
Heat rejected from the cylinder at T2, Q2 mRT2 ln V2
V1 (Isothermal process)
Efficiency of carnot cycle Work done
Heat supplied
Q1 Q2
Q1
mRT1 ln V2
V1 mRT2 ln
V2
V1
mRT1 ln V2
V1
T1 T2
T1
So Carnot cycle efficiency carnot Q1 Q2
Q1
T1 T2
T1
WnetQ1
Note: Carnot cycle requires the piston to travel very slowly duringisothermal expansion process to maintain the temperature constant at T1.
During remaining process (adiabatic expansion) of this cycle, it requires thepiston to travel quickly (very fast). This is not practically possible. So carnotcycle is an ideal cycle which gives maximum efficiency.
Problem 1.21: A Carnot engine working between 400C and 40C produces130 kJ of work. Calculate (i) the cycle efficiency; (ii) the heat added; (iii) theentropy changes during heat rejection process.
Solution:
Source temperature T1 400 273 673 K T2
Sink temperature T3 T4 40 273 313 K
Work produced Wnet 130 kJ
To Find Carnot Cycle Efficiency
carnot T1 T2
T1
673 313673
0.535 53.5%
To Find Heat Added
carnot Wnet
Q1
(where Q1 heat added,
Q2 heat rejected and
Wnet Q1 Q2
Q1 Wnet
carnot
1300.535
243 kJ
To Find Entropy Change during Heat Rejection Process s3 s4
Q T ds
Heat rejected Q2 T3 s3 s4
s3 s4 Q2
T3
But Wnet Q1 Q2
Q2 Q1 Wnet 243 130 113 kJ
Now s3 s4 Q2
T3
113313
0.361 kJ K
Carnot Cycle
1
2
3
4
Q 1
Q 2
P-V diagram
Isothe rm al Expansion
Isen trop icExpansion
Isen trop iccom press ion
Isothe rm alcom press ion
T 2
T1
(P)
W c
W E
T2
T1
T
1 2
34
T-S diagrams
V
Carnot Cycle
Thermodynamics 1.81
Problem 1.22: In a carnot cycle, the maximum temperature and pressureare given as 410C and 18 bar respectively. The ratio of isentropiccompression is 6 and isothermal expansion is 1.5. Assuming the volume of
air at the beginning of isothermal expansion as 0.18 m3, determine (i) thetemperature and pressure at all cordinal points (ii) change in entropy duringisentropic expansion.
Solution: Maximum pressure P1 18 bar ;
Max. temperature T1 T2 410 273 683 K;
Isentropic compression ratio V4V1
6
Isothermal expansion ratio V2V1
1.5
Volume of air at the beginning of isothermal expansion V1 0.18 m3
To Find T and P at all Cordinal PointsProcess 4-1 Isentropic Compression Process
Problem 1.23: A carnot (reversible) engine works between 300C and
30C. The heat supplied to the engine is 20 kJ. Determine 1. efficiency;2. Work output; 3. Heat rejection
Solution
T1 300 273 573 K
Q1 20kJ
T2 30 273 303 K
carnot T1 T2
T1
573 303573
0.4712 47.12%
carnot 0.4712 WQ1
Work output carnot Q1 0.4712 20 9.4241 kJ
Heat rejection Q2,W Q1 Q2
Q2 Q1 W 20 9.4241 10.5792 kJ
Problem 1.24: A refrigerator plant operates on reversed carnot heat enginecycle. It is maintained at a temperature of 5C and the heat is rejected at
the rate of 5 kW. The atmospheric temperature is 25C. Calculate the powerrequired to drive the plant.
Solution T2 5 273 268 K
T1 25 273 298 K;
Q2 5kW W Q1 Q2
COP T2
T1 T2
Q2
Q1 Q2
268
298 268 8.9333
Power Work in kW
Carnot COP Q2
W
H.E
T 2
Q 2
Q 1
T1 = 573 K
= 20 kJ
= 303 K
W
Q 2Q 1Ref
T2
Q 2
Q 1
T1 = 298
= 268 K
W = -
=5kW
Thermodynamics 1.85
Power
W Q2
Carnot COP
58.9333
0.5597 kW
1.41.2 Corollaries of Carnot’s Theorem
Corollary 1All reversible heat engines operating between the same thermal
reservoirs must have the same efficiency.
i.e., the efficiency of reversible heat engine is independent of the typeof engine, amount and nature of the working fluid used and also other factors.
Corollary 2The efficiency of a reversible heat engine is a function of their
respective temperature of the two thermal reservoirs. It can be evaluated byreplacing the ratio of heat transfers Q2 and Q1 by the ratio of temperature
T2 and T1 of their respective heat reservoirs.
Therefore, the efficiency of reversible heat engine
rev WQ1
Q1 Q2
Q1 1
Q2Q1
1 T2
T1
Q1
Q2 Rev
T1
T2
1.42 CLAUSIUS INEQUALITY
An important inequality that has major consequences inthermodynamics is the Clausius inequality. It was first stated by the Germanphysicist R. J.E. Clausius (1822-1888), one of the founders ofthermodynamics, and is expressed as
O QT
0
Here the cyclic integral of Q/T is always less than or equal to zero.This inequality is valid for all cycles, reversible or irreversible. The symbol
O (integral symbol with a circle in the middle) is used to indicate that the
integration is to be performed over the entire cycle.
The inequality of Clausius is used to find whether the cycle isreversible (or) irreversible.
0, the cycle is impossible, since it violates the second law.
Problem 1.25: A heat engine is supplied with 19 kW of heat from 565 Kreservoir and rejects heat to 282.5 K reservoir. Which of the following engineis irreversible engine, reversible engine and impossible engine:Case (a): If 14.0833 kW of heat is rejected.Case (b): If 4.75 kW of heat is rejected.Case (c): If 9.5 kW of heat is rejected.Use Carnot method and Clausius inequality method.
Solution
By Carnot Method
carnot T1 T2
T1
565 282.5
565
0.49558 0.5
Case (a): Q2 14.0833 kW
actual Q1 Q2
Q1
19 14.083319
0.258774
Since actual 0.25 carnot 0.5, it is irreversible engine.
Case (b): If Q2 4.75 kW
actual 19 4.75
19 0.75
Since actual0.75 carnot0.5, it is impossible engine.
Q 2Q 1H E
T 2
Q 2
Q 1
T 1 =5 65 K
= 282 .5 K
W = -
=19kW
Thermodynamics 1.87
Case (c): If Q2 9.5 kW
actual 19 9.5
19 0.5
Since actual carnot, it is reversible engine.
By Clausius Inequality Method
If QT
‘’, then irreversible engine.
If QT
0, then reversible engine.
If QT
‘’, then impossible engine.
Case (a): Q2 14.0833 kW
QT
Q1
T1
Q2
T2
19565
14.0833
282.5
0.0162, so it is irreversible engine.
Case (b): Q2 4.75 kW
QT
19
565
4.75282.5
0.0168 so it is impossible engine
Case (c): Q2 9.5 kW
QT
19565
9.5
282.5
0, so it is reversible engine.
Problem 1.26: An inventor claims to have developed an engine which givesan efficiency of 55% when working under a source of 1000 K and sink of500 K. Check whether his claim is correct or not.
Problem 1.27: A refrigerator removes heat from a refrigerated space at2C at a rate of 300 kJ/min and rejects heat to kitchen air at 26C at arate of 345 kJ/min. Verify whether it violates II law by (i) Clausius inequality(ii) Carnot theorem. (FAQ)
Solution
T2 2 273 275 K; Q2 30060
5 kW
T1 26 273 299 K; Q1 34560
5.75 kW
Clausius Inequality
QT
Q1
T1
Q2
T2
5.75
299
5275
[Q1 is‘’ because heat is removed
from the system]
1.048 10 3
Since it is ‘’ ve, it is irreversible refrigerator.
Carnot Principle
COPcarnot T2
T1 T2
275299 275
11.4583
COPactual Q2
Q1 Q2
55.75 5
6.6667
HE
T2
Q 2
Q 1
T1 =1000 K
= 500 K
W
R ef
T 2
Q 2
Q 1
T 1 = 299 K
= 2 75
W
= 5 .7 5kW
= 5kW
Thermodynamics 1.89
Since C OPactual6 . 6 6 6 7
C OPcarnot11.4583
, it is possible irreversible refrigerator.
Problem 1.28: An inventor claims that his engine develops 3 kW of workfor a heat addition of 240 kJ/min. The highest and lowest temperature of thecycle are 1527C and 327C. Would you agree his claim. [Use clausiusinequality method]. (FAQ)
Solution
W 3 kW Q1 24060
4 kW
T1 1527 273 1800 K;
T2 327 273 600 K
Clausius Inequality Method
QT
Q1
T1
Q2
T2
4
1800
1600
5.55 10 4
It violates the II law.
So it is impossible.
Extra: Carnot Principle
carnot T1 T2
T1
1800 6001800
0.6667
actual Q1 Q2
Q1
4 14
0.75
Since actual carnot, it is impossible. It violates the Carnot theorem
1.43 ENTROPY
Entropy can be viewed as a measure of molecular disorder, ormolecular randomness. As a system becomes more and more disordered, theposition of the molecules become less predictable and the entropy of thesystem increases. Thus, it is not surprising that the entropy of a substance islowest in the solid phase and highest in the gas phase (Fig. 1.50). In thesolid phase, the molecules of a substance continually oscillate about their
equilibrium positions, but they cannotmove relative to each other, and theirposition at any instant can be predictedwith good certainty. In the gas phase,however, the molecules move about atrandom, collide with each other, andchange direction, making it extremelydifficult to predict accurately themicroscopic state of a system at anyinstant. Associated with this molecularchaos is a high value of entropy.
Therefore, from a microscopic pointof view, the entropy of a system increaseswhenever the molecular randomness oruncertainty (i.e., molecular probability) of asystem increases. Thus, entropy is ameasure of molecular disorder, and themolecular disorder of an isolated systemincreases anytime if it undergoes a process.
The gas molecules will not rotate apaddle wheel inserted into the container andproduce work. This is because the gasmolecules, and the energy they possess,are disorganized. Probably the number ofmolecules trying to rotate the wheel in onedirection at any instant is equal to thenumber of molecules that are trying torotate it in the opposite direction, causingthe wheel to remain motionless. Therefore,we cannot extract any useful work directlyfrom disorganized energy (Fig. 1.51)
Now consider a rotating shaft shownin Fig. 1.52. This time the energy of themolecules is completely organized sincethe molecules of the shaft are rotating inthe same direction together. This organizedenergy can readily used to perform usefultasks such as raising a weight or generatingelectricity. Being an organized form of
SO LID
L IQ U ID
G A S
T h e le v e l o f m o le c u la r d iso rd er(e n tro p y) o f a su bs tan ce in c re a s e s
a s i t m elts o r e vap o ra tes .
E n trop y,kJ /kg K.
Fig 1.50
LOA D
D isorganized energy does not create m uch useful effect, no m atter how large it is.
Fig. 1.51 Disorganized energydoes not create much useful
effect, no matter how large it is
W E IGH T
W sh
Fig. 1.52 In the absence offriction, raising a weight by arotating shaft does not create
any disorder (entropy) and thusenergy is not degraded during
this process
Thermodynamics 1.91
energy, work is free of disorder or randomness and thus free of entropy. Thereis no entropy transfer associated with energy transfer as work. Therefore, inthe absence of any friction, the process of raising a weight by a rotating shaft(or a fly wheel) will not produce any entropy. Any process that does notproduce a net entropy is reversible, and thus the process described above canbe reversed by lowering the weight. Therefore, energy is not degraded duringthis process, and no potential to do work is lost.
1.43.1 Concept of EntropyEntropy is an extensive property of a system and sometimes it is also
referred as total entropy. Entropy per unit mass, designated as s, is anintensive property and has the unit kJ/kg K. The term entropy is generallyused refer to both total entropy and entropy per unit mass since the contextusually clarifies which one is meant.
The entropy change of a system during a process can be determinedby integrating Eq. (i) between the initial and the final states:
S S2 S1 1
2
QT
int rev
kJ/K
Note that the entropy is a property, andlike all other properties, it has fixed values atfixed states. Therefore, the entropy changeS between two specified state is the sameno matter what path, (reversible orirreversible) is followed during a process(Fig. 1.53).
There is a close link between entropyand molecular disorder. When work isdissipated into internal energy, the moleculardisorder increases. So entropy increases.When two gases are mixed together, there isa molecular disorder. So entropy increases. Soentropy can be defined as the measure of thedegree of molecular disorder existing in thesystem.
When heat is added to the system,molecular disorder increases so entropy increases.
The entropy of the isolated system either increases or remains constant
1
2
0.3 0.7
Revers ibleprocess
S ,kJ /K
T
Irrevers ib leprocess
S S S= - =0.4 kJ/k2 1
Fig. 1.53 The entropy changebetween two specified state isthe same whether the process
The combination of the system and the surroundings is known as the universe.
So dSuniverse dSsystem dSsurroundings 0
1.43.2 Characteristics of entropy1. Entropy of the system increases when heat is added to system.
2. Entropy of the system decreases, when heat is removed from thesystem.
3. Entropy is constant in isentropic process (Reversible adiabaticprocess).
Change in entropy S2 S1 1
R
2
dQT
where R represents reversible path.
1 Initial State, 2 Final State.Let a system is changed from initial state1 to final state 2 through a reversible path,R1. Then the system is again changed
back from 2 to 1 by another reversiblepath R2. Refer the Fig. 1.54. Now the two
reversible paths R1 and R2 form a cycle.
By using Clausius theorem
OR1 R2
dQT
0
The above cyclic integral can be replaced by the summation of twointegrals as follows.
1
R1
2
dQT
2
R2
1
dQT
0
P
R 1
R 2
2
V
1
Two revers ib le paths R and Rbetween tw o equ ilibrium
states 1 and 2
1 2
Fig. 1.54
Thermodynamics 1.93
(or) 1
R1
2
dQT
2
R2
1
dQT
Since R2 is a reversible path, we can write as follows.
1
R1
2
dQT
1
R2
2
dQT
Since R1 and R2 are any two reversible paths, the value 1
R
2
dQT
is
independent of the reversible path connecting 1 and 2. Therefore 1
R
2
dQT
represents Change in one property of the system and that property is knownas entropy.
S1 entropy at initial state;
S2 entropy at final state
1
R
2
dQT
change in entropy S2 S1
If two end states are very near, then we can write,
dQR
T dS
The unit of entropy S is J/K
Specific entropy is s and the unit is J/kgK
. . . s
Sm
J/kgK
Refer Fig. 1.55. When the system is brought from an initial state 1 toa final state 2 by an irreversible path, the irreversible path can be replacedby reversible path to integrate for evaluation of change in entropy in the
irreversible process. Because, the entropyis a point function. It is a property. It isindependent of the path followed. So wecan replace irreversible path by reversiblepath and calculate the entropy.
Heat which is supplied to a hightemperature substance is converted moreinto work than the heat supplied to a lowtemperature substance.
Entropy shows the possibility ofconversion of heat into work. The changein entropy is more when heat is added ata lower temperature and the change in entropy is less when heat is added ata higher temperature. i.e., for a maximum entropy change, there is aminimum availability for conversion into work and for a minimum entropychange, there is a maximum availability for conversion into work.
Solved problemsProblem 1.29: When water is flowing through turbine, its temperature israised from 35C to 40C due to friction. No heat transfer occurs duringthis process. Assume it is a constant volume process. Find the change inentropy of water while it is passing through the turbines.
Solution
Since friction causes the temperatureraise, this process is taken as irreversibleprocess. Refer the T S diagram. Theirreversible process is shown by dotted line.We can replace the irreversible process byreversible process, since the entropy changedepends on only the two end states. It isindependent of the path the system follows.
T2 40 273 313 K T1 35 273 308 K
We know dQrev TdS
dS dQrev
T
T
T
R
2
1
ds
ss1 s 2
Rev.pathw hich replaces
the irreversible path
Integration can be done onlyon a reversible path
Fig. 1.55
T
irreve rs ible path 2
dss
dQ rev
R evers ib le pa thconnecting the in itia l a nd fina l equ ilibr iumstates
1
Thermodynamics 1.95
dS m cv dT
T
Integrating on both sides, we get
S2 S1 mcv ln T2
T1
S2 S1 1 4.187 ln T2
T1
... mass is taken as 1 kg . Cv for water 4.187 kJ/KgK
1 4.187 ln
313308
0.0674
Change in entropy of water S2 S1 0.0674 kJ / K
Problem 1.30: In a closed system air is at a pressure of 1 bar, temperature
of 300 K and volume of 0.025 m3. The system executes the following processesduring the completion of thermodynamic cycle : 1-2; constant volume heataddition till pressure reaches 3.8 bar, 2-3; constant pressure cooling of air,3-1; isothermal heating to initial state. Determine the change in entropy ineach process. Take Cv 0.718 kJ/kg K, R 287 J/kg K. (FAQ)
Negative sign indicates there is a decrease in entropy.
Thermodynamics 1.97
(iii) Process 3 1 constant temperature process
S1 S3 S2 S3 S2 S1 0.0389 0.02779
0.0111 kJ/K
Problem 1.31: An heat exchanger circulates 5000 kg/hr of water to cooloil from 150C to 50C. The rate of flow of oil is 2500 kg/hr. The averagespecific heat of oil is 2.5 kJ/kgK. The water enters the heat exchanger at21C. Determine the net change in the entropy due to heat exchange process,and the amount of work obtained if cooling of oil is done by using the heatto run a Carnot engine with sink temperature of 21C. (FAQ)
SolutionThe Heat transfer by oil Qoil moil Cpoil Toil
As we discussed earlier that, for a carnot engine to have a maximumefficiency, the sink temperature T2 must be lowered. To achieve progressively
lower temperatures, the process becomes more and more difficult. This meansto attain absolute zero, infinite number of operations have to be performedwithout violating the other laws of thermodynamics. This observation iscontained in the third law of thermodynamics.
“It is impossible to attain the absolute zero by a finite number ofoperations”.
Just like second law, the third law has an equivalent number ofstatements.
(i) FowlerGuggenheim statement of the third law:
It states that “It is impossible by any procedure, no matter howidealized, to reduce any system to the absolute zero of temperature in a finitenumber of operations”.
(ii) Nernst-Simon statement of the third law: It states that “The entropies of all systems and the entropy changes in
all reversible isothermal process tend to zero as the temperature approacheszero”.
Mathematically,
lim T 0
S 0; lim T 0
S 0
Source
150 Co
S ink
21 Co
RevH.E
W
Q =173.611kJ1
Thermodynamics 1.99
This was experimentally discovered by Nernst. He observed that thenet change in entropy of a system is very small, when there is a change fromone lower temperature equilibrium state to another.
For example,
Take separate samples of sodium and chlorine at 4C and convert itinto sodium chloride at the same temperature.
This means that, near the absolute zero, all the systems are highlyordered and the entropy of all states is the same and the values of entropyare extremely small or zero.
In simple words, the principles of first law and second law can beused to calculate the entropy difference only, but to find the absolute valueof entropy, the third law of thermodynamics is required.
Note: Efficient people lead low-entropy (highly organized) lives. They havea place for everything (minimum uncertainty), and it takes minimum energyfor them to locate something. Inefficient people, on the other hand, aredis-organized and lead high-entropy lives. It takes them long time (if nothours) to find something they need, and they are likely to create a biggerdisorder as they are searching since they will probably conduct the searchin a disorganized manner. People leading high-entropy lifestyles are alwayson the run, and never seem to catch up.
1.45 PROPERTIES OF IDEAL GAS
A hypothetical gas which obeys the law Pv R
T (equation of state) iscalled an ideal gas.
An ideal gas is a theoritical gas consists of four gas variables i.e.,Pressure P, Volume V, Temperature T and number of mole of a gas
n. Lastly, the constant in the equation is R
, which is the universal gasconstant R
8.314 kJ/kg. mol. k
PV
R
T ...(1)
V
molar specific volume m3
kg. mole
Divide the above equation by molecular weight , then the equationbecomes,
Equation 1,2,3 is called the ideal gas equation of state.
At very low pressure or density, all gases and vapour approaches idealgas behaviour.
Practically, no ideal gas (perfect gas) exists in nature. However,hydrogen, oxygen, nitrogen and air behave as an ideal gas under normalcondition.
Any equation that relates the pressure, temperature and specific volumeof a substance is called an equation of state which is given here as
Pv RT
where the constant of proportionality R is called the gas constant.A gas that obeys this relation is called an ideal gas.
The ideal gas equation is very simple and thus very convenient to use.But, gases deviate from ideal-gas behaviour significantly. This deviation fromideal gas behaviour at a given temperature and pressure can accurately beaccounted by the introduction of a correction factor called the compressibilityfactor Z defined as
Z PVRT
or PV ZRT
Thermodynamics 1.101
It can also be expressed as
Z Vactual
Videal where Videal
RTP
Obviously, Z 1 for ideal gases.
For real gases, Z can be greater than (or) less than 1 when the gasdeviates from ideal - gas behaviour
But real gases do not conform to this equation of state with completeaccuracy. Ideal gas do not have intermolecular attractive forces. These idealgases obeys the equation of state at all ranges of pressure and temperature.
Practically, no ideal gas (perfect gas) exists in nature. However, hydrogen,oxygen, nitrogen and air behave as an ideal gas under normal condition.
1.45.1 Equation of stateThe equation which relates the properties P, v and T is known as an
Equation of state.
ie f P, v, T 0
The simplest form of equation of state for the ideal gas is given below.
Equation of state: Pv R
T [ R
universal gas constant in kJ kg mol.K
and v molar volume in m3 kg mol ]
Also: Pv RT [ R characteristic gas constant in kJ kg K ]
v specific volume in m3 kgThe above equation - Equation of state is also called Characteristic gas equation.
1.45.2 Relationship Among PropertiesA mole of a substance has a mass numerically equal to the molecular
weight of the substance.
1 gm mol of oxygen has a mass of 32 gm, 1 kg mol of oxygen hasa mass of 32 kg and so on.
For a certain gas, if m is its mass in kg, and is its molecular weight,then the number of kg moles of the gas, n, is given by,
n m
kg.moles
The functional relationship among the properties, pressure P, molar orspecific volume v and temperature, T, known as the equation of state and itis expressed as
Therefore equation of state of an ideal gas can be written as
PV mRT Pv RT
Pv R
T [ v molar volume v ]
PV n R
T [ n number of kg moles ]
PV N KT [ where K R
A and
A Avogadro number 6.023 1026 molecules/kg mol
N Total number of molecules nA ]
1.45.8 Comparison between Ideal and Real Gases
Ideal gas Real gas
(i) An Ideal gas is an hypothetical gas As the name implies, these gasesare real and exist in nature.
(ii) It has no definite volume Real gas has definite volume
(iii) It has no mass It has mass
(iv) Ideal gas follows the equationPV
R
TReal gas follows the equation P
a
v2 v b RT
(v) Ideal gas follows gas laws at alltemperatures pressures
Real gas do not follow gas lawsat all temperatures & pressures
(vi) No force of attraction or repulsionbetween molecules
Force of attraction or repulsionbetween molecules is significant
(vii) Ideal gases don’t exist in natureand they don’t behave like real gasat any temperatures & pressures
Real gases behave like an idealgas at high temperatures and lowpressures
Thermodynamics 1.107
1.46 PROPERTIES OF PURE SUBSTANCESA substance with a fixed chemical composition throughout is called
pure substance. For example water, nitrogen, helium and carbondioxide areall pure substance.
A pure substance does not need to be of a singlechemical element or compound. However, a mixture ofvarious chemical elements or compounds also qualifies asa pure substance as long as the mixture is homogeneous.Since air is a mixture of several gases, it is consideredas a pure substance because it has a uniform chemicalcomposition (Fig. 1.56). However a mixture of oil andwater is not a pure substance. Since oil is not soluble inwater, it will come up on top of the water, forming twochemically dissimilar regions.
A mixture of two or more phases of a puresubstance is still a pure substance as long as the chemicalcomposition of all phases is the same (Fig. 1.57) Amixture of ice and liquid water, for example,is a pure substance because both phaseshave the same chemical composition. Amixture of liquid air and gaseous air,however, is not a pure substance since thecomposition of liquid air is different fromthe composition of gaseous air, and thus themixture is no longer chemicallyhomogeneous. This is due to differentcomponents in air condensing at differenttemperatures at a specified pressure.
1.46.1 THERMODYNAMIC PROPERTIES OF PURE SUBSTANCES IN SOLID, LIQUID AND VAPOUR PHASES
Consider a piston-cylinder device containing liquid water at 20C and1 atm pressure (state 1. Fig. 1.58). Under these conditions, water exists inthe liquid phase, and it is called a compressed liquid, or a subcooled liquid,meaning that is not about to vaporize. Heat is now transferred to the wateruntil its temperature rises to, say, 40C. As the temperature rises, the liquidwater expands slightly, and so its specific volume increases. To accommodatethis expansion, the piston will move up slightly. The pressure in the cylinderremains constant at 1 atm during this process since it depends on the outside
N 2
A IR
Fig 1.56 Nitrogenand gaseous air
are pure
VAP O R
LIQUID
(a) H O2
VAP O R
LIQUID
(b) A IR
Fig 1.57 A mixture of liquid andgaseous water is a pure
substance, but a mixture ofliquid and gaseous air is not
barometric pressure and the weight of thepiston, both of which are constant. Wateris still a compressed liquid at this statesince it has not started to vaporize.
As more heat is transferred, thetemperature will keep rising until itreaches 100C (state 2, Fig. 1.59). At thispoint water is still a liquid, but any heataddition will cause some of the liquid tovaporize. That is, a phase-change processfrom liquid to vapour is about to takeplace. A liquid that is about to vaporizeis called a saturated liquid. Therefore,state 2 is saturated liquid state.
1.46.2 Saturated Vapour and Superheated VapourOnce boiling starts, the temperature will stop rising until the liquid
is completely vaporized. That is, the temperature will remain constant duringthe entire phase-change process if the pressure is held constant. During aboiling process, the only change we will observe is a large increase in thevolume and a steady decline in the liquid levelas result of more liquid turning to vapour.
Midway about the vaporization line (state3, Fig. 1.60), the cylinder contains equalamounts of liquid and vapour. As we continuetransferring heat, the vaporization process willcontinue until the last drop of liquid is vaporized(state 4, Fig. 1.61). At this point, the entirecylinder is filled with vapour. Any heat lossfrom this vapour will cause some of the vapourto condense (phase change from vapour toliquid). A vapour that is about to condense iscalled a saturated vapour. Therefore, state 4 isa saturated vapour state. A substance at statesbetween 2 and 4 is often referred to as asaturated liquid-vapour mixture since the
liquid and vapour phases coexist in equilibrium at thesestates.
Further transfer of heat will result in an increase inboth the temperature and the specific volume (Fig. 1.62) Atstate 5, the temperature of the vapour is, let us say,300C; and if we transfer some heat from the vapour, thetemperature may drop somewhat but no condensation willtake place as long as the temperature remains above100C (for P 1 atm. A vapour that is not about tocondense (i.e., not a saturated vapour) is called asuperheated vapour. Therefore, water at state 5 is asuperheated vapour. This constant-pressure phase-changeprocess is illustrated on a T v diagram in Fig 1.63.
If the entire process described here is reversed bycooling the water while maintaining the pressure at the samevalue, the water will go back to state 1, retracing the samepath, and in doing so, the amount of heat released willexactly match the amount of heat added during the heatingprocess.
STATE 4
P = 1atmVapo ur
T = 100 CO
Heat
Fir 1.61 At 1atm pressure,
the temperatureremains
constant at100C until the
last drop ofliquid is
vaporized(satured vapour)
P = 1atm
Super heatedVapour
T = 300 CO
H eat
Fig 1.62A s m ore hea t is trans fe rred, thetem perature o f the vapor s ta rts
1.46.3 Saturation Temperature and Saturation Pressure
Water started to boil at 100C. Strictly speaking, the statement “waterboils at 100C” is incorrect. The correct statement is “water boils at 100C”
and 1 atm pressure 1.01325 bar. If the pressure inside the cylinder wereraised to 500 kPa by adding weights on top of piston, the water would startboiling at 151.9C. That is, the temperature at which water starts boilingdepends on the pressure; therefore, if the pressure is fixed, so is the boilingtemperature.
At a given pressure, the temperature at which a pure substance changesphase is called the saturation temperature Tsat. Likewise, at a given
temperature, the pressure at which a pure substance changes phase is calledthe saturation pressure Psat. At a pressure of 101.325 kPa, Tsat is 100C.
Conversely, at a temperature of 100C, Psat is 101.325 kPa.
Saturation tables that list the saturation pressure against the temperature(or the saturation temperature against the pressure) are available for practicallyall substances. For water, steam table is available.
The amount of energy absorbed or released during a phase-changeprocess is called the latent heat. More specifically, the amount of energyabsorbed during melting is called the latent heat of fusion and is equivalentto the amount of energy released during freezing. Similarly, the amount ofenergy absorbed during vaporization is called the latent heat of vaporizationand is equivalent to the energy released during condensation. The magnitudesof the latent heats depend on the temperature or pressure at which the phasechange occurs. At 1 atm pressure, the latent heat of fusion of water is 333.7kJ/kg and the latent heat of vaporization is 2257.1 kJ/kg.
1.47 THERMODYNAMIC PROPERTIES OF STEAM
1.47.1 Property TablesFor most substances, the relationship among thermodynamic properties
are too complex to be expressed by simple equations. Therefore, propertiesare frequently presented in the form of tables. Some thermodynamic propertiescan be measured easily, but others cannot and are calculated by using therelations between them and measurable properties. The results of thesemeasurements and calculations are presented in tables in a convenient format.In the following discussion, the steam tables will be used to demonstrate theuse of thermodynamic property tables.
Thermodynamics 1.111
1.47.2 Saturated Liquid and Saturated Vapour StatesThe subscript f is used to denote properties of a saturated liquid, and
the subscript g to denote the properties of saturated vapour. Another subscriptcommonly used is fg, which denotes the difference between the saturatedvapour and saturated liquid values of the same property. For example,
vf specific volume of saturated liquid
vg specific volume of saturated vapour
vfg difference between vg and vf (that is, vfg vg vf)
The quantity hfg is called the enthalpy of vaporization (or latent heat
of vaporization). It represents the amount of energy needed to vaporize a unitmass of saturated liquid at a given temperature or pressure. It decreases asthe temperature or pressure increases, and becomes zero at the critical point.
1.47.3 Saturated Liquid-Vapour MixtureDuring a vaporization process, a substance exists as part liquid and
part vapour. That is, it is a mixture of saturated liquid and saturated vapour.To analyze this mixture properly, we need to know the proportions of theliquid and vapour phases in the mixture. This is done by defining a newproperty called the quality x as the ratio of the mass of vapour to the totalmass of the mixture:
Quality has significance for saturated mixtures only. It has no meaningin the compressed liquid or superheated vapour regions. Its value is between0 and 1. The quality of a system that consists of saturated liquid is 0 (or 0percent), and the quality of a system consisting of saturated vapour is 1 (or100 percent)
Let us consider a mixture of saturatedliquid water and water vapour in equilibriumat pressure P and temperataure T. Thecomposition of the mixture by mass will begiven by its quality x, and its state will bewithin the vapour dome (Fig. 1.65) Theproperties of mixture are as follows.
v vf x vfg; u uf x ufg
h hf x hfg; s sf x sfg
where vf, vfg, hf, hfg, sf and sfg are the
saturation properties at the given pressureand temperature.
1.47.4 Superheated VapourThe region to the right of the saturated vapour line and at temperature
above the critical point temperature, a substance exist as superheated vapour.Since the superheated region is a single-phase region (vapour phase only),temperature and pressure are no longer dependent properties and they canconveniently be used as the two independent properties in the tables.
Superheated vapour is characterized by
Lower pressures (P Psat at a given T)
Higher temperatures (T Tsat at a given P)
Higher specific volumes (v vg at a given P or T)
Higher internal energies (u ug at a given P or T)
Higher enthalpies (h hg at a given P or T)
T
sP=C
constant pressure line
P=C ,T=C
P=C
x
sf s sg
Fig 1.65 Property in two phaseregion
Thermodynamics 1.113
Higher entropies (s sg at a given P or T )
1.47.5 Thermodynamic Properties of SteamImportant Notes:
At critical point, the water suddenly flashes into vapour.
Critical pressure 221.2 bar
Critical temperature 374.15C; Critical volume 0.00317 m3/kg
Critical point is a point in which saturation liquid and saturation vapourcoexists.
Dryness Fraction: xIt is defined as the mass of steam present in 1 kg of mixture
(mixture = water + steam)
x mass of steam
mass of water mass of steam
Wetness Fraction: 1 xIt is defined as the mass of water present in 1 kg of mixture.
1 x mass of water
mass of mixture
If x 0.1 means 10% of steam present in the 100% mixture.
x 0.9 means 90% of steam present in the 100% mixture.
f stands for liquid (or) water.
hf enthalpy of liquid in kJ/kg.
g stands for gas, vapour (or) steam.
hg enthalpy of dry steam (simply steam) in kJ/kg.
Similarly, vf, vg, sf, sg, uf, ug, etc.
All the values vf, vg, hf, hg, sf and sg for different saturated pressure
and saturated temperature are given in steam table.
v vf xvfg v vg v for P and t in super heated steamtable.
h hf xhfg h hg h for P and t in super heated steamtable.
s sf xsfg s sg s for P and t in super heated steamtable.Degree of superheat t tsat
hfg means hg hf = Latent heat of vapourization.
vfg means vg vf; sfg means sg sf
NoteIf v vg, it is dry saturated steam.
If v vg, it is wet steam. If v vg, it is superheated steam.
Similarly, If s sg, it is dry saturated steam.
If s sg, it is wet steam; If s sg, it is superheated steam.
Similarly, If h hg, it is dry saturated steam.
If h hg, it is wet steam.
If h hg, it is superheated steam.
Similarly,
If t tsat, it is dry saturated steam (or) Wet stream.
If t tsat, it is superheated steam.
Thermodynamics 1.115
Properties Wet steam Dry saturated steam Superheated steamProperties can betaken from steamtable (pressuretable (or)temperature table)once pressure (or)temperature isgiven
Properties can betaken from steamtable (pressuretable (or)temperature table)once pressure (or)temperature isgiven
Once pressure andtemperature is given, wecan take all propertiesfrom superheated steamtable.
SpecificEnthalpy inkJ/kg
h hf x hfg h hgh can be taken for givenP and t from superheatedsteam table.(or)
h hg Cp steam t tsat
where Cp steam 2.1
to 2.3 kJ/kg KSpecificVolume in
m3/kg
v vf x vfg
or v x vg
[ . .
. vf ~ 0 ]
v vgv can be taken fromsuper heated steam table
(or) v vg
TTsat
Specificentropy in
kJkg K
s sf x sfg s sg(From super heated
steam table) (or) s
sg Cp steam ln
TTsat
Problem 1.35: Find the saturation temperature, changes in specific volumeand entropy during evaporation and the latent heat of vaporization of steamat 10 bar.
Change in entropy during evaporation, sfg 4.445 kJ/kg K
Latent heat of vaporization hfg 2013.6 kJ/kg
Problem 1.36: Saturated steam has an entropy of 6.66 kJ/kg K. What are itspressure, temperature, specific volume and enthalpy?
From steam table, for sg 6.66 kJ/kg K; P 8 bar;
t2 170.4C; vg 0.24026 m3/kg ; hg 2767.4 kJ/kg.
Problem 1.37: Find the specific volume of steam at 12 bar absolute, whenthe condition of steam is (a) wet steam with dryness fraction 0.9, (b) drysaturated steam and (c) super heated to the temperature of 300C.
Problem 1.41: 2 kg of water at 200C are contained in a 20 m3 vessel.Determine the pressure, enthalpy, mass and volume of vapour within thevessel. (FAQ)
Solution
Given: mf 2 kg, tsat 200 C, V 20 m3
From steam tables corresponding to tsat 200 C,
The saturation pressure Psat 15.549 bar
and vg 0.12716 m3/kg
But we know that, mg Vvg
20
0.12716 157.28 kg
The dryness fraction of given steam,
x mg
mf mg
157.282 157.28
0.987
The enthalpy of the vapour is given by h hf xhfg
From steam tables, corresponding to temperature 200C