Prepared by - Higher Technological Institute

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Prepared by:

MAIN CONCEPTS 1. Thermodynamics .................................................................................................. 1

1.1 Laws of Thermodynamics ................................................................................ 1

1.1.1 First law of thermodynamics ...................................................................... 1

1.1.2 Second law of thermodynamics .................................................................. 1

1.1.3 Third law of thermodynamics ..................................................................... 1

1.1.4 Zeroth law of thermodynamics ................................................................... 2

1.2 Application Areas of Thermodynamics ............................................................ 2

2. Dimensions and Units .......................................................................................... 2

2.1 Standard Prefixes in SI Units ............................................................................ 3

2.2 Conversion Between SI and English Units ....................................................... 4

2.2.1 Length ......................................................................................................... 4

2.2.2 Mass ............................................................................................................ 4

2.2.3 Force ........................................................................................................... 4

2.2.4 Work ........................................................................................................... 4

2.2.5 Power .......................................................................................................... 4

3. Temperature Measurement ................................................................................ 5

3.1 Temperature Scales ........................................................................................... 5

3.2 Conversion between Temperature Scales ......................................................... 5

4. Pressure Measurement ........................................................................................ 6

4.1 Absolute and Gauge Pressure ........................................................................... 6

4.2 Pressure Measurement Devices ........................................................................ 7

4.2.1 Mechanical pressure gage ........................................................................... 7

4.2.1.1 The Barometer ...................................................................................... 7

4.2.1.2 The Manometer .................................................................................... 8

C O N T E N T S

C O N T E N T S

4.2.1.3 Bourdon pressure gauge ....................................................................... 9

4.2.1.4 Deadweight tester ................................................................................. 9

4.2.2 Pressure transducers..................................................................................10

4.2.2.1 Strain gauge pressure transducer ........................................................10

4.2.2.2 Piezoelectric pressure transducer .......................................................11

Basic Information and Definitions

5. System, Boundary and Surroundings ..............................................................12

5.1 Classification of System .................................................................................12

5.2 Properties of a System ....................................................................................13

6. State and Equilibrium .......................................................................................14

7. Processes .............................................................................................................14

7.1 Iso- Process .....................................................................................................15

7.1 Steady and Uniform Process ...........................................................................15

8. Energy .................................................................................................................16

8.1 Forms of Energy .............................................................................................16

9. Phase of a Pure Substance .................................................................................16

9.1 Phases ..............................................................................................................16

9.1.1 Principal phases ........................................................................................16

9.1.2 Phase-change processes of pure substances .............................................17

10. Property Tables ................................................................................................20

10.1 Enthalpy ........................................................................................................20

10.2 Vapor quality ................................................................................................21

IDEAL-GAS AND BOUNDARY WORK

11. The ideal-gas Equation of State ......................................................................23

11.1 Gas Laws .......................................................................................................23

11.1.1 Boyle's law ..............................................................................................23

11.1.2 Charles's law ...........................................................................................24

11.1.3 Avogadro's law .......................................................................................24

11.1.4 Gay-Lussac's law ....................................................................................25

C O N T E N T S

11.2 Ideal-gas equation of state ............................................................................25

11.3 Water Vapor as an ideal gas .........................................................................26

12. Boundary Work................................................................................................26

FIRST LAW OF THERMODYNAMICS - CLOSED SYSTEM

13. Energy Balance and First law of thermodynamics .......................................29

13.1 Energy balance for closed systems ...............................................................29

14. Specific Heats ....................................................................................................32

14.1 Specific heat relations of ideal gases ............................................................33

14.2 Specific heat relations of incompressible substance.....................................34

MAIN CONCEPTS

Thermodynamics (A) – Lecture Notes Main Concepts

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1. Thermodynamics

The name thermodynamics stems from the Greek words “therme” (heat) and “dynamis”

(power) which is most descriptive of the early efforts to convert heat into power.

Today the same name is broadly interpreted to include all aspects of energy and energy

transformations, and relationships among the properties of matter, so Thermodynamics can be

defined as the science of energy.

It is difficult to give a precise definition for Energy and it can be viewed as the ability to

cause changes.

1.1 Laws of Thermodynamics

1.1.1 First law of thermodynamics

The first law of thermodynamics is an expression of the conservation of energy principle

Conservation of energy principle states that during an interaction, energy can change from

one form to another but the total amount of energy remains constant. That is, energy cannot be

created or destroyed.

1.1.2 Second law of thermodynamics

The second law of thermodynamics asserts that energy has quality as well as quantity, and

actual processes occur in the direction of decreasing quality of energy.

For example, a cup of hot coffee left on a table eventually cools, but a cup of cool coffee in

the same room never gets hot by itself. The high-temperature energy of the coffee is degraded

(transformed into a less useful form at a lower temperature) once it is transferred to the surrounding

air.

1.1.3 Third law of thermodynamics

The third law of thermodynamics is concerned when the temperature of the system absolute

zero.

The third law of thermodynamics states that it is impossible for any procedure to lead

absolute zero (isotherm T = 0) in a finite number of steps (establishing a temperature absolute zero

is unattainable in somewhat the same way as the speed of light).

Note: No matter how cold the system is, it can always be made colder, but it can never reach

absolute zero.


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1.1.4 Zeroth law of thermodynamics

When a body is brought into contact with another body that is at a different temperature, heat

is transferred from the body at a higher temperature to the one at a lower temperature until both

bodies attain the same temperature. At that point, the heat transfer stops, and the two bodies are

said to have reached thermal equilibrium.

Note: The equality of temperature is the only requirement for thermal equilibrium.

The zeroth law of thermodynamics states that if two bodies are in thermal equilibrium with

a third body, they are also in thermal equilibrium with each other.

1.2 Application Areas of Thermodynamics

All activities in nature involve some interaction between energy and matter; thus, it is hard

to imagine an area that does not relate to thermodynamics in some manner. Therefore, developing

a good understanding of the basic principles of thermodynamics has long been an essential part

of engineering education.

Thermodynamics is commonly encountered in many engineering systems and other aspects

of life, and it does not need to go very far to see some application areas of it.

The heart is constantly pumping blood to all parts of the human body, various energy

conversions occur in trillions of body cells.

Many ordinary household appliances are designed, in whole or in part, by using the

principles of thermodynamics. Some examples include the heating and air-conditioning systems,

the refrigerator, the humidifier, the pressure cooker and the water heater.

On a larger scale, thermodynamics plays a major part in the design of vehicles from ordinary

cars to airplanes and conventional or nuclear power plants.

2. Dimensions and Units

Any physical quantity can be characterized by dimensions. The magnitudes assigned to the

dimensions are called units.

Dimensions are classified into

• Primary or Fundamental dimensions such as mass “m”, length “L”, time “t”, and temperature

(T), and

• Secondary dimensions, or Derived dimensions that are expressed in terms of the primary

dimensions, such as velocity, energy, and volume.


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Despite strong efforts in the scientific and engineering community to unify the world with a

single unit system, two sets of units are still in common use today:

• English system, which is also known as the United States Customary System (USCS), and

• Metric SI (Le Système International d’ Unités), which is also known as the International

System.

The SI is a simple and logical system based on a decimal relationship between the various

units, and it is being used for scientific and engineering work in most of the industrialized nations,

including England.

The English system, however, has no apparent systematic numerical base, and various units

in this system are related to each other rather arbitrarily (12 in = 1 ft, 1 mile = 5280 ft, etc.), which

makes it confusing and difficult to learn. The United States is the only industrialized country that

has not yet fully converted to the metric system.

Table 2.1

The seven primary (or fundamental) dimensions and their units in SI.

Dimension Unit

Length meter (m)

Mass kilogram (kg)

Time second (s)

Temperature kelvin (K)

Electric current ampere (A)

Amount of light candela (cd)

Amount of matter mole (mol)

Note: All unit names are to be written without capitalization even if they were derived from

proper names [For example, the SI unit of force, which is named after Sir Isaac Newton, Standard

prefixes in SI units is newton (not Newton), and it is abbreviated as N]. Also, the full name of a

unit may be pluralized, but its abbreviation cannot [For example, the length of an object can be 5

m or 5 meters, not 5 ms or 5 meter]. Finally, no period is to be used in unit abbreviations unless

they appear at the end of a sentence [For example, the proper abbreviation of meter is m (not

m.)].

2.1 Standard Prefixes in SI Units

The SI is based on a decimal relationship between units. The prefixes used to express the

multiples of the various units. They are standard for all units and should memorize them because

of their widespread use.


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Table 2.2

Standard prefixes in SI units.

Prefix Multiple Prefix Multiple

yotta, Y 1024 deci, d 10-1

zetta, Z 1021 centi, c 10-2

exa, E 1018 milli, m 10-3

peta, P 1015 micro, μ 10-6

tera, T 1012 nano, n 10-9

giga, G 109 pico, p 10-12

mega, M 106 femto, f 10-15

kilo, k 103 atto, a 10-18

hecto, h 102 zepto, z 10-21

deka, da 101 yocto, y 10-24

2.2 Conversion Between SI and English Units

2.2.1 Length

English unit SI units 1 inch (in) = 25.4 mm

1 feet (ft) = 30.48 cm

2.2.2 Mass

English unit SI unit 1 pound-mass

(lbm) = 0.45359 kg

1 ton = 1000 kg

2.2.3 Force

English unit SI unit 1 pound-force

(lbf = 𝐥𝐛𝐦∙𝐟𝐭

𝐢𝐧𝟐)

= 4.448 newtons

(N)

2.2.4 Work

Work is an energy, which is viewed that force causing movement (or displacement).

English unit SI unit 1 Btu (British

thermal unit) =

1.0551 kJ

(J = N∙m)

2.2.5 Power

English unit SI units 1 horsepower

(hp) =

745.6 Watts

(W = J/s)


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3. Temperature Measurement Technically, Temperature can be defined as an indication of intensity of molecular activity.

3.1 Temperature Scales

The temperature scales used in the SI and in the English system today are

• Celsius scale (formerly called the centigrade scale; in 1948 it was renamed after the Swedish

astronomer “A. Celsius”) The temperature unit on this scale is the celsius, which is designated

by °C and

• Fahrenheit scale (named after the German instrument maker “G. Fahrenheit”). The

temperature unit on this scale is the fahrenheit which is designated by °F, respectively.

These are often referred to as two-point scales since temperature values are assigned at two

different points. On the Celsius scale, the ice and steam points were originally assigned the values

of 0 and 100°C, respectively. The corresponding values on the Fahrenheit scale are 32 and 212°F.

In thermodynamics, it is very desirable to have a temperature scale that is independent of

the properties of any substance or substances. Such a temperature scale is called a thermodynamic

temperature scale.

The thermodynamic temperature scale used in the SI and in the English system today, respectively

are

• Kelvin scale (named after the Irish-Scottish mathematical physicist and engineer “Lord

Kelvin”). The temperature unit on this scale is the kelvin, which is designated by K (not °K).

The lowest temperature on the Kelvin scale is absolute zero, or 0 K (corresponds to -273.15°C).

• Rankine scale (named after the Scottish mechanical engineer “William Rankin”). The

temperature unit on this scale is the rankine, which is designated by R.

Note: Using nonconventional refrigeration techniques, scientists have approached absolute

zero kelvin (they achieved 0.000000002 K).

3.2 Conversion between Temperature Scales

The Kelvin scale is related to the Celsius scale by

T(K) = T(°C) + 273.15 (3-1)

The Rankine scale is related to the Fahrenheit scale by

T(R) = T(°F) + 459.67 (3-2)

The temperature scales in the two-unit systems are related by

T(R) = 1.8T(K) (3-3)

T(°F) = 1.8T(°C) + 32 (3-4)


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Figure 3.1. Comparison of temperature scales.

4. Pressure Measurement

Pressure is defined as a normal force exerted by a fluid per unit area. Normally, the pressure

is used when dealt with a gas or a liquid. The counterpart of pressure in solids is normal stress.

Since pressure is defined as force per unit area, it has the SI unit of newtons per square meter

(N/m2), which is called a pascal (Pa). That is, 1 Pa = 1 N/m2.

Three other pressure units commonly used in practice, especially in Europe, are bar,

standard atmosphere (atm), and kilogram-force per square centimetre (kgf/cm2):

1 bar = 105 Pa

1 atm = 101,325 Pa

1 kgf/cm2 = 9.807 × 104 Pa

4.1 Absolute and Gauge Pressure

The actual pressure at a given position is called the Absolute pressure )Pabs), and it is

measured relative to absolute vacuum (i.e., absolute zero pressure). While the pressure measured

in relation to ambient atmospheric pressure “Patm” is called Gauge pressure “Pgage”. The gage

pressure. can be positive (above atmospheric pressure) or negative (below atmospheric pressure),

which sometimes called vacuum pressures “Pvac”. Absolute, gauge and vacuum pressures are

related to each other by

Pabs = Patm + Pgauge (4-1)

Pabs = Patm – Pvac (4-2)


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Figure 4.1. Absolute, gauge, and vacuum pressures.

4.2 Pressure Measurement Devices

Most pressure-measuring devices, however, are calibrated to read zero in the atmosphere,

and so they indicate the difference between the absolute pressure and the local atmospheric

pressure (i.e., gauge pressure)

4.2.1 Mechanical pressure gage

4.2.1.1 The Barometer

Atmospheric pressure is measured by a device called a barometer; thus, the atmospheric

pressure is often referred to as the barometric pressure.

Figure 4.2. Schematic diagram of the basic Barometer.

The Barometer (invented by the Italian physicist and mathematician “Evangelista

Torricelli”) is constructed by inverting a mercury-filled tube into a mercury container that is open

to the atmosphere, as shown in Figure 4.2. The pressure at point B is equal to the atmospheric

pressure, and the pressure at point C can be taken to be zero since there is only mercury vapor

above point C and the pressure is very low relative to “Patm” and can be neglected to an excellent

approximation.


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Writing a force balance in the vertical direction gives

Patm = ρgh (4-3)

where “ρ” is the density of mercury, g is the local gravitational acceleration, and “h” is the height

of the mercury column above the free surface.

A frequently used pressure unit is the standard atmosphere, which is defined as the pressure

produced by a column of mercury 760 mm in height at 0°C (ρHg = 13,595 kg/m3) under standard

gravitational acceleration (g = 9.807 m/s2). The unit “mmHg” is also called the torr in honour of

“Torricelli”. Therefore, 1 atm = 760 torr mmHg =760 torr.

Note: If the water is used instead of mercury to measure the standard atmospheric pressure,

a water column of about 10.3 m would be needed.

4.2.1.2 The Manometer

Manometer (invented by the Italian physicist and mathematician “Evangelista Torricelli”)

is commonly used to measure small and moderate pressure differences. A shown in Figure 4.3 the

Manometer consists of a glass or plastic U-tube containing one or more fluids such as mercury,

water, alcohol, or oil

Figure 4.3. The basic Barometer.

To keep the size of the manometer to a manageable level, heavy fluids such as mercury are

used if large pressure differences are anticipated.

The differential fluid column of height h is in static equilibrium, and it is open to the

atmosphere. Then the pressure at point 2 is determined directly from

P = ρgh + Patm (4-4)

where “ρ” is the density of the manometer fluid in the tube.

h


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4.2.1.3 Bourdon pressure gauge

Bourdon tube (named after the French engineer and inventor “Eugene Bourdon”) and as

shown in Figure 4.4 consists of a bent, coiled, or twisted hollow thin-wall, closed-end metal tube.

The closed-end of this tube is left free to move, while the other end is left open to allow fluid to

enter. A simple mechanical linkage transmits the movement of the free end of the tube to a pointer

moving around a dial.

When the tube is open to the atmosphere, the tube is undeflected, and the needle on the dial

at this state is calibrated to read zero (gauge pressure). When the fluid inside the tube is pressurized

or evacuated, the tube stretches or contracts, respectively and moves the needle in proportion to

the applied pressure.

Figure 4.4 Bourdon pressure gauge.

4.2.1.4 Deadweight tester

Another type of mechanical pressure gage called a deadweight tester is used primarily for

calibration (Calibration is the comparison of a measured dimension with a standard one in order

to determine the accuracy of the measuring instrument).

As shown in Figure 4.5, it is constructed with an internal chamber filled with a fluid (usually

oil), along with a tight-fitting piston, cylinder, and plunger. Weights are applied to the top of the

piston, which exerts a force on the oil in the chamber.

A deadweight tester measures pressure directly through the application of a weight that

provides a force per unit area. Since the piston cross-sectional area “Ac” is known, the reference

pressure is calculated as F/Ac. The reference pressure port is connected to a pressure sensor that is

to be calibrated.


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Figure 4.5. Deadweight tester.

4.2.2 Pressure transducers

Electronics have made their way into every aspect of life, including pressure measurement

devices. Modern pressure sensors, called pressure transducers, use various techniques to convert

the pressure effect to an electrical effect such as a change in voltage, resistance, or capacitance.

Pressure transducers are smaller and faster, and they can be more sensitive, reliable, and

precise than their mechanical counterparts.

4.2.2.1 Strain gauge pressure transducer

Strain gauge pressure transducer (discovered by the American mechanical

engineer “Arthur Claude Ruge”) and as shown in Figure 4.6 consists of a diaphragm which

supports a metallic foil pattern. The pressure change causes a resistance change due to the

distortion of the foil. The value of the pressure can be found by measuring the change in resistance

of the foil.

When the metallic foil is stretched it will become narrower and longer, changes that increase

its electrical resistance end-to-end. Conversely, when the foil is compressed, it will broaden and

shorten, leading to a decrease its electrical resistance end-to-end.

Figure 4.6. Strain gauge pressure transducer.


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4.2.2.2 Piezoelectric pressure transducer

Piezoelectric pressure transducer (also called solid-state pressure transducers) is a device

that uses the piezoelectric or press-electric effect (discovered by French physicist and professor

“Jacques Curie” along with his younger brother, “Pierre Curie”) to measure changes

in pressure, by converting them to an electrical charge (voltage).

Piezoelectric pressure transducer and as shown in Figure 4.7 consists of a thin membrane that

generates a voltage when deformed where the voltage generated is directly proportional to the

applied force.

Piezoelectric pressure transducer has a much faster frequency response compared to

diaphragm units and is very suitable for high-pressure applications but it is generally not as

sensitive as diaphragm-type transducers, especially at low pressures.

Figure 4.7. Piezoelectric pressure transducer.

Basic Information and

Definitions

Thermodynamics (A) – Lecture Notes Basic Information and Definitions

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5. System, Boundary and Surroundings

A system is defined as a quantity of matter or a region in space chosen for study. while the

mass or region outside the system is referred to as the surroundings. The contact surface shared

by both the system and the surroundings is called the boundary. The boundary is a zero-thickness

surface that neither contains any mass nor occupy any volume in space and separates the system

from its surroundings.

Figure 5.1. System, Boundary and Surroundings.

As illustrated in Figure 5.2, the boundary of a system can be real or imaginary and can be

fixed or movable.

Figure 5.2. System, Boundary and Surroundings.

5.1 Classification of System

Systems may be considered to be closed or open

• Closed system (also known as a control mass),

• Open system (also known as a control volume).

A closed system (or a control mass) and as shown in Figure 5.3, consists of a fixed amount

of mass, and no mass can cross its boundary, only energy can (i.e., no mass can enter or leave a

closed system But energy, in the form of heat or work, can cross the boundary).

Note: a special case, even energy is not allowed to cross the boundary, that system is called

an isolated system.

System System


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Figure 5.3. Closed system (control mass).

An open system (or a control volume) and as shown in Figure 5.4, is a properly selected

region in space, that both mass and energy can cross its boundary of a control

A large number of engineering problems involve mass flow in and out of a system and,

therefore, are modeled as control volumes, such as water heater, a car radiator, a turbine, and a

compressor.

Note: The boundaries of a control volume are called a control surface, and they can be real

or imaginary.

(a) A control volume (CV) with real and

imaginary boundaries (b) A control volume (CV) with fixed and

moving boundaries as w

Figure 5.4. Open system (Control volume).

5.2 Properties of a System

Any characteristic of a system is called a property. Some familiar properties are pressure

P, temperature T, volume V, and mass m. The list can be extended to include fewer familiar ones

such as viscosity, velocity and elevation.

Properties are considered to be either

• Intensive properties are those that are independent of the mass of a system, such as temperature,

pressure, and density.

• Extensive properties are those whose values depend on the size-or extent-of the system. Total

mass, total volume, and total momentum are some examples of extensive properties.

- Extensive properties per unit mass are called specific properties. Some examples of specific

properties are specific volume (v = V/m)


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Note: An easy way to determine whether a property is intensive or extensive is to divide the

system into two equal parts with an imaginary partition. Each part will have the same value of

intensive properties as the original system, but half the value of the extensive properties.

Generally, uppercase letters are used to denote extensive properties (with mass “m” being

a major exception), and lowercase letters are used for intensive properties (with pressure “P” and

temperature “T” being the obvious exceptions).

6. State and Equilibrium

The state is the condition a system not undergoing any change, at which all the properties

can be measured or calculated throughout the entire system.

Equilibrium implies a state of balance where there are no unbalanced potentials (or driving

forces) within the system. (A system in equilibrium state experiences no changes when it is isolated

from its surroundings). Thermodynamics deals with equilibrium states

There are many types of equilibrium,

• Thermal equilibrium: an equilibrium state at which the temperature is the same throughout the

entire system.

• Mechanical equilibrium: an equilibrium state at which pressure at any point of the system

remains unchanged with time.

• Phase equilibrium: an equilibrium state at which the mass of each phase in the system reaches

an unchanging level.

• Chemical equilibrium: an equilibrium state at which chemical composition of the system does

not change with time (i.e., no chemical reactions occur).

Thermodynamic equilibrium an equilibrium state at which the conditions of all the relevant

types of equilibrium (Thermal, mechanical, phase and chemical) are satisfied.

7. Processes

A process is any change that a system undergoes from one equilibrium state to another. The

path is the series of states through which a system passes during a process.

To describe a process completely and as illustrated in Figure 7.1, one should specify

• initial and final states of the process,

• the path the process follows, and

• interactions with the surroundings.


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Figure 7.1. A process between initial and final states and the process path.

A system is said to have undergone a cycle if it returns to its initial state at the end of the

process. That is, for a cycle the initial and final states are identical.

7.1 Iso- Process

The prefix “iso-” is used to designate a process for which a particular property remains

constant such as

• Isothermal process is a process during which the temperature remains constant.

• Isobaric process is a process during which the pressure remains constant.

• Isochoric (or isometric) process is a process during which the specific volume remains

constant.

7.1 Steady and Uniform Process

.The term Steady implies no change with time at a specific location.

Note: The opposite of steady is unsteady, or transient.

The term uniform, however, implies no change with location over a specified region at a

certain time.

A large number of engineering devices operate for long periods of time under the same

conditions, and they are classified as steady-flow devices. the steady-flow process, which can be

defined as a process during which a fluid flows through a control volume steadily.

Note: During the steady-flow process; the fluid properties can change from point to point

within the control volume, but at any fixed point they remain the same during the entire process.


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8. Energy

Energy can be viewed as the ability to do work

8.1 Forms of Energy

The total energy of a system (E or e = E/m) is made up of variable forms of energy, where

different forms of energy can be classified into:

• Macroscopic energy,

• Microscopic energy

Macroscopic energies are forms of energy that a system possesses as a whole with respect

to some external reference frames such as kinetic energy and Potential energy.

• Kinetic energy (KE or ke = KE/m) is the energy that a system possesses as a result of its motion

relative to an external reference.

• Potential energy (PE or pe = PE/m) ) is the energy that a system possesses as a result of its

elevation in a gravitational field.

Microscopic energies are forms of energy that are related to the molecular structure and the

degree of molecular activity and they are independent of outside reference frames.

• Internal energy (U or u = U/m) is the sum of microscopic energies (i.e., the sum of the kinetic

and potential energies of the molecules).

Note: a system in the gas phase is at a higher internal energy level than it is in the solid or

the liquid phase, because of the added energy to the molecules of a solid or liquid, to overcome

the molecular forces (i.e., intermolecular forces) and break away, turning the substance into a

gas.

9. Phase of a Pure Substance

A pure substance is a substance that has a fixed chemical composition such as water,

Nitrogen, Helium, etc.

9.1 Phases

Phase is a distinct molecular arrangement that is homogenous throughout and separated

from others by easily identifiable bounding surfaces.

9.1.1 Principal phases

The principal phases of a pure substance are:

• Solid

• Liquid

• Vapor


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Note: A substance may have several phases within a principal phase, each with a different

molecular structure, Carbon, for example, may exist as graphite or diamond in the solid phase.

9.1.2 Phase-change processes of pure substances

As a familiar substance, water is used to demonstrate the basic principles involved.

Note: all pure substances exhibit the same general behavior.

• Consider a piston–cylinder device containing liquid water at state 1 with temperature “T1” and

1 atm pressure (i.e., barometric pressure).

• Heat is now transferred to the water until its temperature rises up to “T2”. As the temperature

rises, the liquid water expands slightly, and so its specific volume increases. To accommodate

this expansion, the piston moves up slightly. The pressure in the cylinder remains constant at 1

atm during this process (Isobaric process) since it depends on the outside barometric pressure.

Water is still at liquid phase at this state since it has not started to vaporize.

• As more heat is transferred, the temperature keeps rising until it reaches state 2. At this point,

water is still a liquid, but any heat addition will cause some of the liquid to vaporize. That is, a

phase-change process from liquid to vapor is about to take place.

Note: A liquid that is about to vaporize is called a saturated liquid. Therefore, state 2 is a

saturated liquid state.

• The temperature at which a pure substance changes phase is called the saturation temperature

Tsat. Likewise, at a given temperature, the pressure at which a pure substance changes phase is

called the saturation pressure “Psat” (At barometric pressure of 101.325 kPa, “Tsat” of water is

99.97°C).

• Once phase-change (boiling) starts, the temperature stops rising (remains at “Tsat”) until the

liquid is completely vaporized. That is, the temperature will remain constant during the entire

phase-change process if the pressure is held constant.

• At state 3, the entire cylinder is filled with vapor that is on the borderline of the liquid phase.

Any heat loss from this vapor will cause some of the vapor to condense (phase change from

vapor to liquid).

Note: A vapor that is about to condense is called a saturated vapor. Therefore, state 3 is a

saturated vapor state.


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• Once the phase-change process is completed, up to state 4 a single-phase region is obtained

again (this time vapor), and further transfer of heat results in an increase in both the temperature

and the specific volume

Note: A vapor that is not about to condense is called a superheated vapor. Therefore, water

at state 4 is a superheated vapor.

This constant-pressure phase-change process (isobaric heat addition process) mentioned is

illustrated on a T-v diagram in Figure 9.1.

Figure 9.1. T-v diagram for the isobaric heating process of a pure substance.

Noting that if the entire process described here is reversed by cooling the water while

maintaining the pressure at the same value, the water will go back to state 1, retracing the same

path, and in so doing, the amount of heat released will exactly match the amount of heat added

during the heating process.

• Latent heat is the amount of heat absorbed or released during a phase-change process in a

thermodynamic system

- Latent heat of fusion is the amount of energy absorbed during melting and is equivalent to

the amount of energy released during freezing.

- Latent heat of vaporization is the amount of energy absorbed during vaporization and is

equivalent to the energy released during condensation.

• Sensible heat is the amount of heat absorbed or released causing a change in the temperature

of a thermodynamic system.

Note: At 1 atm pressure, the latent heat of fusion of water is 333.7 kJ/kg and the latent heat

of vaporization is 2256.5 kJ/kg.

3 T2 = T3 = Tsat ---------


19 | P a g e

A liquid-vapor saturation curve is a plot of “Tsat” versus “Psat” and it is shown in Figure

9.2. From this figure, it is clear that “Tsat” increases with “Psat”. Thus, a substance at a specified

pressure boils at the saturation temperature corresponding to that pressure (i.e., a substance at

higher pressures boils at higher temperatures).

Figure 9.2. The liquid-vapor saturation curve of a pure substance.

When the phase-change process of water at 1 atm pressure shown in Figure 9.1 is repeated

at different pressures the saturation vapor curve (T-v diagram) for water is developed and is

illustrated in Figure 9.3.

Note: the saturation vapor curve for any pure substance can be developed in the same

manner).

Figure 9.3. T-v diagram of constant-pressure phase-change processes of water at various pressures.

The saturated liquid states in Figure 9.3 can be connected by a line called the saturated

liquid line, and saturated vapor states in the same figure can be connected by another line, called

the saturated vapor line. These two lines meet at the point called the critical point, forming a

dome as shown in Figure 2.3. The critical point is the point at which the saturated liquid and

saturated vapor states are identical.


20 | P a g e

All the compressed liquid states are located in the region to the left of the saturated liquid

line, called the compressed liquid region.

All the superheated vapor states are located to the right of the saturated vapor line, called the

superheated vapor region.

Note: In these two regions, the substance exists in a single phase, a liquid or a vapor and all

the states that involve both phases in equilibrium are located under the dome, which is called the

saturated liquid-vapor mixture region, or the wet region.

10. Property Tables

For most substances, the relationships among thermodynamic properties are too complex to

be expressed by simple equations. Therefore, properties are frequently presented in the form of

tables.

Some thermodynamic properties can be measured easily, but others cannot and are

calculated by using the relations between them and measurable properties.

For each substance, the thermodynamic properties are listed in more than one table. In fact,

a separate table is prepared for each region of interest such as the superheated vapor, compressed

liquid, and saturated (mixture) regions.

The subscript “f” is used to denote properties of a saturated liquid, and the subscript “g” to

denote the properties of saturated vapor.. Another subscript commonly used is “fg”, which

denotes the difference between the saturated vapor and saturated liquid values of the same

property, such that

vfg = vf – vg (10-1)

, where

vf is the specific volume of saturated liquid

vg is the specific volume of saturated vapor

vfg is the difference between vg and vf

10.1 Enthalpy

Enthalpy (H or h =H/m) is a combination property associated with control volumes (open

systems) which is the sum of the internal energy of a system and the product of its volume

multiplied by the pressure and is given by:

H = U + PV , kJ (10-2)


21 | P a g e

or by unit mass,

h = u + Pv , kJ/kg (10-3)

The pressure unit pascal (Pa) is equivalent to Pa = N/m2 = N·m/m3 = J/m3, which is energy per

unit volume, and the product “Pv” or it's equivalent “P/ρ” has the unit J/kg, which is energy per

unit mass. Note that pressure itself is not a form of energy but a pressure force acting on a fluid

through a distance produces work, called flow energy.

The quantity “hfg” is called the enthalpy of vaporization (or latent heat of vaporization). It

represents the amount of energy needed to vaporize a unit mass of saturated liquid at a given

temperature or pressure. It decreases as the temperature or pressure increases and becomes zero at

the critical point.

10.2 Vapor quality

During a vaporization process, a substance exists as a part liquid and part vapor. That is, it

is a mixture of saturated liquid and saturated vapor. To analyze this mixture properly, the

proportions of the liquid and vapor phases in the mixture is defined by the quality “x”

The vapor quality “x” is defined as the ratio of the mass of vapor to the total mass of the

mixture:

x = 𝑚𝑣𝑎𝑝𝑜𝑟

𝑚𝑡𝑜𝑡𝑎𝑙 (10-4)

, where

mtotal = mliquid + mvapor = mf + mg (10-5)

Consider a tank that contains a saturated liquid–vapor mixture. The volume occupied by

saturated liquid is “Vf”, and the volume occupied by saturated vapor is “Vg”. The total volume

“V” is the sum of the two:

V = Vf + Vg (10-6)

V = mtv (10-7)

mtv = mf vf + mgvg (10-8)

mf = mt – mg (10-9)

mtv = (mt – mg)vf + mgvg (10-10)

Dividing by total mass “mt” yields,

v = (1 – x) vf + xvg (10-11)


22 | P a g e

Since x = mg/mf, this relation also be expressed as

v = vf + xvfg , m3/kg (10-12)

where vfg = vg - vf, solving for “x”, we obtain

𝑥 = 𝑣 − 𝑣𝑓

𝑣𝑔 (10-13)

The analysis given above can be repeated for internal energy and enthalpy with the

following results:

u = uf + xufg , m3/kg (10-14)

h = hf + xhfg , kJ/kg (10-15)

Ideal gas and Boundary

work

Thermodynamics (A) – Lecture Notes Ideal-gas and Boundary work

23 | P a g e

11. The ideal-gas Equation of State

Gas is one of the four fundamental states of matter consisting of particles that have neither

independent shape nor volume but tends to expand indefinitely and freely through space (this

means that the kinetic energy of the particles is greater than the potential energy of the

intermolecular forces).

A pure gas may be made up of individual atoms (e.g., noble

gas like neon), elemental molecules made from one type of atom (e.g., oxygen),

or compound molecules made from a variety of atoms (e.g., carbon dioxide). A gas mixture, such

as air, contains a variety of pure gases.

Gas and vapor are often used as synonymous words. The vapor phase of a substance is

customarily called a gas when it is above the critical temperature. Vapor usually implies a gas

that is not far from a state of condensation.

Ideal-gas is the gas that obeys three conditions

1- The particles can not exhibit any intermolecular forces (No intermolecular forces)

2- The particles occupy no intermolecular volume (pointless particles)

3- All collisions are perfectly elastic.

Note: All gases can be considered Ideal-gas in most usual conditions except at high pressures and

low temperatures relative to its critical temperature and pressure.

Equation of state is an equation that relates the pressure, temperature, and specific volume

of a substance.

Note: Property relations that involve other properties of a substance at equilibrium states are

also referred to as equations of state.

11.1 Gas Laws

The fundamental gas laws were experimentally developed at the end of the 18th century.

These laws relate the pressure, volume, and temperature of a gas. These laws are:

11.1.1 Boyle's law

Boyle's law states that for a given amount of a gas at a constant temperature, the volume of

the gas will vary inversely with pressure.


24 | P a g e

This law can be represented with these formulae:

V α 1

𝑃 (11-1)

or

P α 1

𝑉 (11-2)

or

PV = constant (11-3)

⸫ PiVi = PfVf (11-4)

, where “P” is the pressure (Pa), “V” is the volume of a gas (m3) and the subscripts “i” and “f”

represent the initial and final states, respectively.

11.1.2 Charles's law

Charles’s law states that the volume of a fixed amount of a gas is directly proportional to

its temperature if the pressure is kept constant.


V α 𝑇 (11-5)

or V

𝑇 = constant (11-6)

⸫ V𝑖

𝑇𝑖 =

V𝑓

𝑇𝑓 (11-7)

, where “T” is the temperature (K), “V” is the volume of a gas (m3) and the subscripts “i” and

“f” represent the initial and final states, respectively.

11.1.3 Avogadro's law

Avogadro’s law states that the pressure of the volume occupied by a gas is directly

proportional to the number of moles if the pressure and temperature are kept constant.


V α 𝑛 (11-8)

or V

𝑛 = constant (11-9)

⸫ V𝑖

𝑛𝑖 =

V𝑓

𝑛𝑓 (11-10)

, where “n” is the number of moles (mole number) of the gas, “V” is the volume of a gas (m3)

and the subscripts “i” and “f” represent the initial and final states, respectively.


25 | P a g e

11.1.4 Gay-Lussac's law

Gay-Lussac’s law states that the pressure of a gas is directly proportional to the temperature

if the volume and amount of gas are kept constant.


P α 𝑇 (11-11)

or 𝑃

𝑇 = constant (11-12)

⸫ 𝑃𝑖

𝑇𝑖 =

𝑃𝑓

𝑇𝑓 (11-13)

, where “P” is the pressure of the gas (Pa), “T” is the temperature of the gas (K) and the subscripts

“i” and “f” represent the initial and final states, respectively.

11.2 Ideal-gas equation of state

Ideal-gas equation of state, or simply the ideal-gas relation is a combination of the

empirical Boyle’s, Charles’s, Avogadro’s and Gay-Lussac’s laws, which relates the pressure,

temperature, volume and amount of ideal-gas through universal gas constant (Ru).


𝑃V

𝑛𝑇 = constant= Ru (11-14)

⸫ PV = nRuT (11-15)

, where “P” is the pressure (Pa), “T” is the temperature (K), “V” is the volume of a gas (m3),

“n” is the number of moles of the gas and “Ru” is the universal gas constant (= 8.31447 J/mol·K).

This equation can be written in different forms as follow:

⸪ Ru = M × R (11-16)

, and

m = n × M (11-17)

⸫ PV = nMRT (11-18)

, and

⸫ PV = mRT (11-19)

For the same mass (m), this equation can be rewritten as

𝑃𝑖V𝑖

𝑇𝑖 =

𝑃𝑓V𝑓

𝑇𝑓 (11-20)


26 | P a g e

or by dividing both sides of the same equation by mass (m)

Pv = RT (11-21)

11.3 Water Vapor as an ideal gas

At pressures below 10 kPa, water vapor can be treated as an ideal gas, regardless of its

temperature, with negligible error (less than 0.1%). At higher pressures, however, the ideal gas

assumption yields unacceptable errors, particularly in the vicinity of the critical point and the

saturated vapor line (over 100%).

In air-conditioning applications, the water vapor in the air can be treated as an ideal gas with

essentially no error since the pressure of the water vapor is very low. In steam power plant

applications, however, the pressures involved are usually very high; therefore, ideal-gas relations

should not be used.

12. Boundary Work

Moving boundary work (or Boundary work) is one form of mechanical work frequently

encountered in practice, which is associated with a moving boundary. For example, and as shown

in Figure 12.1 the expansion or compression of a gas in a piston-cylinder device. During this

process, part of the boundary (the inner face of the piston) moves back and forth.

Quasi-equilibrium process (or Quasi-Static) is a process can be viewed as a very slow

process that allows the system to adjust itself internally, so the properties in one part of the system

do not change any faster than that of the other.

The analyze of the moving boundary work is considered for a quasi-equilibrium process thus

the entire gas in the cylinder is at the same pressure at any given time.

Figure 12.1. Expansion or compression of a gas in a piston-cylinder device.


27 | P a g e

Consider the gas enclosed in the piston–cylinder device shown in Figure 12.2. A gas does a

differential amount of work “dw” as it forces the piston to move by a differential amount “ds”.

The initial pressure of the gas is “P”, the total volume is “V”, and the cross-sectional area of the

piston is “A”. If the piston is allowed to move a distance “ds”, the differential work done during

this process is

dW = F·ds = PA·ds (12-1)

⸪ F =PA (12-2)

⸫ dW = PA·ds (12-3)

⸪ AdS = dV (12-4)

⸫ dW = PdV (12-5)

Figure 12.2. Differential amount of work done by gas enclosed in the piston-cylinder device.

Note: According to this expression, the moving boundary work is sometimes called PdV

work.

The total boundary work done “W” during the entire process as the piston moves is obtained

by adding all the differential works from the initial state to the final state:

W = ∫ 𝑃𝑑V2

1 (12-6)

For ideal gas,

PV = nRuT (12-7)

For constant temperature,

⸫ P = 𝑛𝑅𝑢𝑇

𝑉 (12-8)

W = 𝑛𝑅𝑢𝑇 ∫𝑑𝑉

𝑉

2

1 (12-9)

W = 𝑛𝑅𝑢𝑇(𝑙𝑛V2 − 𝐿𝑛V1) (12-10)

W = 𝑛𝑅𝑢𝑇 ln (V2

𝑉1) (12-11)


28 | P a g e

The quasi-equilibrium expansion process described is shown on a P-V diagram in Figure

12.3. On this diagram, the differential area under the process curve “dA” is equal to “PdV”, which

is the differential work.

Figure 12.3. Differential amount of work done by gas enclosed in the piston-cylinder device.

The total area “A” under the process curve 1–2 is obtained by adding these differential areas:

Area = A = ∫ 𝑑𝐴2

1 = ∫ 𝑃𝑑V

2

1 (12-12)

A comparison of this equation with the last previous equation reveals that the area under the

process curve on a P-V diagram is equal, in magnitude, to the work done during a quasi-

equilibrium expansion or compression process of a closed system (On the P-v diagram, it

represents the boundary work done per unit mass).

The net work done during a cycle produced by cyclic devices (e.g., car engines) is the

difference between the work done by the system and the work done on the system.

As illustrated in Figure 12.4, net work output is produced from a cyclic device because the

work done by the system during the process 2-1 (area under path A) is greater than the work done

on the system during the process 1-2 (area under path B), and the difference between these two is

the net work done during the cycle (the colored grey area).

Substantially, boundary work is a path function (i.e., it depends on the path followed as well

as the end states)

Figure 12.4. The work produced by these devices during one part of the cycle.

Note: The use of the boundary work relation is not limited to the quasi-equilibrium processes

of gases only. It can also be used for solids and liquids.

Energy Balance and First law of

Thermodynamics - Closed System

Thermodynamics (A) – Lecture Notes First law of thermodynamics - Closed System

29 | P a g e

13. Energy Balance and First law of thermodynamics

Energy balance for any system undergoing any kind of process is defined by “First law of

thermodynamics”, where the first law of thermodynamics is a generalization of the conservation

of energy principle, and it states that “Energy can change from one form to another but the total

amount of energy remains constant (i.e., energy can be neither created nor destroyed during a

process; it can only change forms)”. Energy balance for any system undergoing any kind of

process can be expressed as

(Total energy entering the system) – (Total energy leaving the systems)

=

“Change in the total energy of the system”

(i.e., Change in internal, kinetic, potential, etc., energies)

𝐸in − 𝐸out = ∆𝐸𝑠𝑦𝑠𝑡𝑒𝑚 (kJ) (13-1)

The energy balance can be expressed on a per unit mass basis as

𝑒in − 𝑒out = ∆𝑒𝑠𝑦𝑠𝑡𝑒𝑚 (kJ/kg) (13-2)

or, in the rate form, as

�̇�in − �̇�out = 𝑑𝐸𝑠𝑦𝑠𝑡𝑒𝑚

𝑑𝑡 (kW) (13-3)

13.1 Energy balance for closed systems

A closed system (control mass) is a system that does not involve any mass flow across its

boundaries, The only two forms of energy interactions associated with a closed system are heat

transfer and work (i.e., Energy can cross the boundary of a closed system in two distinct forms:

heat and work).

An energy interaction is heat transfer if its driving force is a temperature difference [i.e.,

Heat is defined as the form of energy that is transferred between two systems (or a system and its

surroundings) by virtue of a temperature difference]. Otherwise, it is work. So, the energy balance

for a cycle can be expressed in terms of heat and work interactions.

In thermodynamics, the term heat simply means heat transfer, as Heat is energy in

transition. It is recognized only as it crosses the boundary of a system. Adiabatic process is a

process during which there is no heat transfer. There are two ways a process can be adiabatic:

Either the system is well insulated so that only a negligible amount of heat can pass through the

boundary, or both the system and the surroundings are at the same temperature and therefore

there is no driving force (temperature difference) for heat transfer.

https://energyeducation.ca/encyclopedia/Thermodynamics

https://energyeducation.ca/encyclopedia/Law_of_conservation_of_energy

https://energyeducation.ca/encyclopedia/Law_of_conservation_of_energy


30 | P a g e

Note: Even though there is no heat transfer during an adiabatic process, the energy content

and thus the temperature of a system can still be changed by other means such as work.

Work is the energy transfer associated with a force acting through a distance. A rising

piston, a rotating shaft, and an electric wire crossing the system boundaries are all associated with

work interactions (work done per unit time is called power).

Heat and work are directional quantities (i.e., path functions), and thus the complete

description of a heat or work interaction requires the specification of both the magnitude and

direction. One way of doing that is to adopt a sign convention. The generally formal sign

convention for heat and work interactions is as follows: heat transfer to a system and work done

by a system are positive; heat transfer from a system and work done on a system are negative.

If directions of heat and work transfers are unknown, in such cases, it is common practice to

use the classical thermodynamics sign convention and to assume heat to be transferred into the

system (heat input) in the amount of “Q” and work to be done by the system (work output) in the

amount of “W”, and then to solve the problem using the subscript “in” or “out” and solve for it.

A positive result indicates the assumed direction is right; a negative result, on the other

hand, indicates that the direction of the interaction is the opposite of the assumed direction (Note:

this intuitive approach is used as it eliminates the need to adopt a formal sign convention and the

need to carefully assign negative values to some interactions).

The energy balance relation in that case for a closed system becomes

𝑄 − 𝑊 = ∆𝐸 (13-4)

where net heat input is expressed as

𝑄 = 𝑄in − 𝑄out (13-5)

and the net work output is expressed as

𝑊 = 𝑊out − 𝑊in (13-6)

Note: Obtaining a negative quantity for “Q” or “W” simply means that the assumed

direction for that quantity is wrong and should be reversed.

Consider a piston-cylinder device that undergoes an isobaric expansion process. The

direction of heat transfer “Q” is taken to be to the system and the work “W” is expressed as the

sum of boundary “Wb” and other forms of work (such as electrical and shaft).


31 | P a g e

Then, the energy balance can be expressed as

𝐸in − 𝐸out = ∆𝐸𝑠𝑦𝑠𝑡𝑒𝑚 (13-7)

⸫ 𝑄in −𝑊b − 𝑊in = ∆𝑈 + ∆𝐾𝐸 + ∆𝑃𝐸 (13-8)

where “KE” is the kinetic energy which is defined as; the macroscopic energy that a system

possesses as a result of its motion relative to some reference frame. It is expressed as

KE = m 𝑉𝑒2

2 (kJ) (13-9)

or, on a unit mass basis,

ke = 𝐾𝐸

𝑚 =

𝑉𝑒2

2 (kJ/kg) (13-10)

, where “Ve” denotes the velocity of the system relative to some fixed reference frame.

Note: The kinetic energy of a rotating solid body is given by 0.5·I·ω2 where “I” is the

moment of inertia of the body and “ω” is the angular velocity and “PE” is the potential energy

which is defined as; the macroscopic energy that a system possesses as a result of its elevation in

a gravitational field. It is expressed as

PE = mgz (kJ) (13-11)

or, on a unit mass basis,

pe = 𝑃𝐸

𝑚 = gz (kJ/kg) (13-12)

, where “g” is the gravitational acceleration and “z” is the elevation of the center of gravity of a

system relative to some arbitrarily selected reference level.

Note: Most closed systems remain stationary during a process and thus experience no change

in their kinetic and potential energies (that is, ΔKE = ΔPE = 0). Closed systems whose velocity and

elevation of the center of gravity remain constant during a process are frequently referred to as

stationary systems

The tank is stationary and thus the kinetic energy change “∆𝐾𝐸” equal zero and the change

in the potential energy “∆𝑃𝐸” is very small and can be ignored. Therefore, internal energy “∆𝑈”

is the only form of energy of the system that may change during this process.

⸫ 𝑄 −𝑊b − 𝑊 = ∆𝑈 (13-13)

and

𝑄 − 𝑊 = ∆𝑈 + 𝑊b (13-14)


32 | P a g e

For a constant-pressure process, the boundary work “𝑊b” is given as

𝑊b = 𝑃∆𝑉 (13-15)

⸫ 𝑄 − 𝑊 = ∆𝑈 + 𝑃∆𝑉 (13-16)

⸪ ∆𝐻 = ∆𝑈 + 𝑃∆𝑉 (13-17)

⸫ 𝑄in − 𝑊 = ∆𝐻 (kJ) (13-18)

This equation is very convenient to use in the analysis of closed systems undergoing a

constant-pressure (iso-baric) quasi-equilibrium process since the boundary work is automatically

taken care of by the enthalpy terms, and one no longer needs to determine it separately.

14. Specific Heats

Specific heat is the energy required to raise the temperature of a unit mass of a substance by

one degree.

Specific heat is a property that enables the comparison of the energy storage capabilities of

various substances.

In thermodynamics, kinds of specific heats and as shown in Figure 14.1 are:

• Specific heat at constant volume “cv”,

• Specific heat at constant pressure “cp”,

Figure 14.1. Illustrations for the difference between constant-volume and constant-pressure

specific heats (values given are for helium gas).

Note: The specific heat at constant pressure “cp“ is always greater than “cv“ because at

constant pressure the system is allowed to expand and the energy for this expansion work must

also be supplied to the system.


33 | P a g e

An expression for the specific heat at constant volume “cv“ can be obtained by considering

a constant-volume process (and thus no expansion or compression work is involved). It yields

cvdT = du … at constant volume (14-1)

An expression for the specific heat at constant pressure “cp“ can be obtained by considering

a constant-pressure expansion or compression process. It yields

cpdT = dh … at constant pressure (14-2)

A common unit for specific heats is kJ/kg·°C or kJ/kg·K. Notice that these two units are

identical since ΔT(°C) and ΔT(K), and 1°C change in temperature is equivalent to a change of 1

K. and according to the previously mentioned equation Specific heat at constant volume “cv” and

Specific heat at constant pressure “cp” can be defined as

• Specific heat at constant volume “cv”, which is the change in the internal energy of a substance

per unit change in temperature as the volume is maintained constant.

• Specific heat at constant pressure “cp”, which is the change in the enthalpy of a substance per

unit change in temperature as the pressure is maintained constant.

In other words, “cv”, is a measure of the variation of internal energy of a substance with

temperature, and “cp” is a measure of the variation of enthalpy of a substance with temperature.

14.1 Specific heat relations of ideal gases

There are three ways to determine the internal energy “Δu” and enthalpy changes “Δh” of

ideal gases:

1- By using the tabulated “u” and “h” data. This is the easiest and most accurate way when

tables are readily available,

2- By using average specific heats. This is very simple and certainly very convenient when

property tables are not available (Note: The results obtained are reasonably accurate if the

temperature interval is not very large). This can be done by using the following relations

Δu = u2 – u1= cv,avg(T2 – T1) (14-3)

Δh = h2 – h1= cp,avg(T2 – T1) (14-4)

The average specific heats “cv,avg” and “cp,avg” are evaluated from this table at the average

temperature (T1 + T2)/2.


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Another way of determining the average specific heats is to evaluate them at “T1” and “T2”

and then take their average. Usually, both methods give reasonably good results, and one is not

necessarily better than the other.

Note: If the final temperature “T2” is not known, the specific heats may be evaluated at “T1”

or at the anticipated average temperature. Then “T2” can be determined by using these specific

heat values. The value of “T2” can be refined, if necessary, by evaluating the specific heats at the

new average temperature.

A special relationship between “cv” and “cp” for ideal gases can be obtained by as follows

Pv = RT (14-5)

⸪ h = u + Pv (14-6)

⸫ Pv = h – u (14-7)

h – u = RT (14-8)

h = u + RT (14-9)

by differentiating

dh = du + R dT (14-10)

⸪ dh = cpdT and du = cvdT (14-11)

⸫ cpdT = cvdT + R dT (14-12)

cp = cv + R (14-13)

This is an important relationship for ideal gases since it enables us to determine “cv” from

a knowledge of “cp” and the gas constant “R”.

Another ideal-gas property called the specific heat ratio “k”, defined as

𝑘 = 𝑐𝑝

𝑐v (14-14)

14.2 Specific heat relations of incompressible substance

Incompressible substance is a substance whose specific volume (or density) is constant. The

specific volumes of solids and liquids essentially remain constant during a process. Therefore,

liquids and solids are considered as incompressible substances.

The constant-volume and constant-pressure specific heats are identical for incompressible

substances. Therefore, for solids and liquids, the subscripts on “cp” and “cv” can be dropped, and

both specific heats can be represented by a single symbol “c”.


35 | P a g e

That is,

cp = cv = c (14-15)

Like those of ideal gases, the differential form of the change in internal energy yields

du = cvdT = cdT (14-16)

The change in internal energy between states 1 and 2 is then obtained by integration:

Δu = u2 – u1 ≅ cavg·ΔT ≅ cavg(T2 – T1) (14-17)

The differential form of the enthalpy change of incompressible substances can be determined

by differentiation to be

⸪ h = u + Pv (14-18)

⸫ dh =du + (Pdv + vdP) (14-19)

⸪v =constant , ⸫dv = 0

⸫ dh =du + vdP (14-20)

The enthalpy change between states 1 and 2 is then obtained by integration:

Δh = h2 – h1 = Δu + v·Δp (14-21)

⸫ Δh ≅ cavg ΔT+ v Δp (14-22)

For solids, the term “v·Δp” is insignificant and thus

Δh = Δu ≅ cavg ΔT (14-23)

For liquids, two special cases are commonly encountered:

1. Constant-pressure processes, (ΔP = 0): Δh = Δu ≅ cavg ΔT

2. Constant-temperature processes, (ΔT = 0): Δh = v Δp, then for a process between states 1

and 2, the enthalpy change can be expressed as

Δh = h2 – h1 = v(P2 – P1) (14-24)

By taking state 2 to be the compressed liquid state at a given “T” and “P” and state 1 to be

the saturated liquid state at the same temperature, the enthalpy of the compressed liquid can be

expressed as

h2 = hf @ T + vf @T (P2 – Psat @ T ) (14-25)

Note: At high temperature and pressures, this equation may overcorrect the enthalpy and

result in a larger error than the approximation h = hf @ T.

Prepared by - Higher Technological Institute

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