Thermodynamics

PREPARED BY:PRADEEP KUMAR GUPTA

ASSISTANT PROFESSOR

DEPARTMENT OF MECHANICAL ENGINEERING

Table of Content• Introduction

• Microscopic and Macroscopic Approach

• Microscopic Approach

• Macroscopic Approach

• Difference between Microscopic and Macroscopic Approach

• Definitions of thermodynamics terminology

• Surrounding

• Boundary

• Universe

• Thermodynamic systems

• State of system

• Classification of thermodynamic systems

• Concept of Continuum

• Equilibrium

• Thermodynamic Properties of a system

• Classification of properties of a thermodynamic system

• Path

• Process

• cycle

• Quasi-Static Process

• Energy and its forms

• Work

• Power

• Heat

• Types of Heat

• Laws of thermodynamics

• Zeroth law of thermodynamics

• Thermometry

• Definition of temperature

• Temperature Scales

• Conversion from Celsius to Fahrenheit or Fahrenheit to Celsius

• Conversion from Celsius to Kelvin

• Relation between Celsius, Fahrenheit and Kelvin

• First law of thermodynamics

• First law of thermodynamics for a cyclic process

• Internal energy and Enthalpy

• Thermodynamic Processes and calculation of work

• Joule’s experiment

• First law for non flow process

Table of Content continued……………• Constant volume process (Isochoric

process)• Constant pressure process (isobaric

process)• Constant Temperature process

(isothermal process)• Adiabatic process (isentropic

process)• Polytropic process• Steady Flow Energy Equation

(S.F.E.E.)• Applications of Steady flow energy

equation• Throttling Process

Definition of ThermodynamicsThermodynamics is a branch of science which deals with energy,phenomena of energy and related properties of matter, especially of lawsof transformation of heat into other forms of energy and vice versa.Microscopic and Macroscopic ApproachMicroscopic Approach This approach considers that the system is made up of a very large

numbers of the discrete particles known as molecules. These molecules have different velocities and energies.

The behaviour of system is found by using statistical method as the number of molecules is very large.

The properties like velocity, momentum, impulse, kinetic energy etc, which describes the molecule cannot be easily measured by instruments.

Large number of variables is needed to describe such a system. So approach is complicated.

Macroscopic Approach:

In this approach, we do not follow the behavior of individualmolecules but study the properties of particular mass of thesubstances.

The analysis of macroscopic system requires simple mathematicalformulae.

The values of the properties of system are their average values.

Only few properties are needed to describe such a system.

Difference between Microscopic and Macroscopic ApproachSr. No. Microscopic Approach Macroscopic Approach

1

This approach considers that the system is

made up of a very large numbers of the discrete

particles known as molecules. These molecules

have different velocities and energies.

In this approach, the behaviour of individual

molecules is not considered but studies the

properties of particular mass of the

substances.

2

The behaviour of system is found by using

statistical method as the number of molecules is

very large.

The analysis of macroscopic system requires

simple mathematical formulae.

3

The properties like velocity, momentum,

impulse, kinetic energy etc, which describes the

molecule cannot be easily measured by

instruments.

The properties like temperature and

pressure are required to describe the system

can be easily measured.

4

Large number of variables is needed to describe

such a system. So approach is complicated.

Only few properties are needed to describe

such a system.

Definitions of thermodynamics terminology

Thermodynamic systems: A thermodynamic system may be defined as the quantity of matteror definite region in space upon which some thermodynamic process is taking place.Thermodynamic systems are defined by using a real or imaginary boundary. Anything beyondreal or imaginary boundary is known as surroundings.

Surrounding: The space outside the thermodynamic system is known as surrounding.

Boundary: The line separating the system and surrounding is known as boundary.

Universe: The combination of system , surrounding and boundary is known as universe.

State of system: A state is a macroscopic condition of a thermodynamic system as described byits particular thermodynamic parameters. Some thermodynamic parameters are pressure,temperature, density, composition etc.

Classification of thermodynamic systems

Thermodynamic systems may be broadly classified in three categories:

1. Open system

2. Closed system

3. Isolated system

1.Open system: Open system is one in which matter (mass of working substance) as well as energy (heat and work)crosses the boundary of the system. As shown by figure (a) energy as well as water vapour is coming out from thesystem.

2.Closed system: Closed system is one in which only energy (heat and work) crosses the boundary of the systemwithout adding or losing of matter (mass of working substance). As shown by fig (b).

3.Isolated system: In an isolated system neither matter (mass of working substance) nor energy (heat and work)crosses the boundary of the system. As shown by fig (c).

Equilibrium

Equilibrium indicates the state of balance. In an equilibrium state there are no unbalanced potentialswithin the system. Equilibrium may be classified as:

(i) Chemical Equilibrium

(ii) Mechanical Equilibrium

(iii) Thermal Equilibrium

(i)Chemical Equilibrium: If there is no chemical reaction or diffusion of matter from one part of thesystem to another, the system is said to be in chemical equilibrium.

(ii)Mechanical Equilibrium: If there are no unbalanced forces in the system, the system is said to be inmechanical equilibrium.

(iii)Thermal Equilibrium: When a system is prevailing in chemical and mechanical equilibrium isseparated from its surroundings by a diathermic wall and if no spontaneous change in any property of thesystem, the system is said to be in state of thermal equilibrium.

Thermodynamic Properties of a system

Properties are those characteristics of the system which can be used for defining the system. Such asvolume, pressure, temperature, viscosity etc.

Classification of properties of a thermodynamic system

The thermodynamic properties may be classified into two categories:

1. Intensive property

2. Extensive property

1.Intensive property: Intensive properties are those properties which have samevalue for any part of the system or these are those properties that are independentof the mass of the system. Such as temperature, pressure and density.

2.Extensive property: Extensive properties are those properties which dependupon the mass of the system and do not maintain the same value for any path ofthe system. Such as mass, enthalpy, volume and energy etc.

Note: The ratio of extensive property of the system to the mass of the system givesthe intensive property. Such as the ratio of total volume (V) of the system to itstotal mass (m) is known as specific volume.

= V/m …………it is an intensive property.

Path: If all the changes of states of the system are plotted, then line joining thechange of states of the system is known as path.

Process: A process is a complete description of change of state of athermodynamic system through a specified path.

cycle: A thermodynamic cycle is defined as the series of state of changes such thatthe intial state is identical with the final state.

sv

Quasi-Static ProcessConsider a system which contained gas in a cylinder in fig. Initially it is in an equilibrium state,represents the properties P1, v1, T1. The weight on the piston just balance the force exerted bythe gas. When weight is removed from the piston the system become unbalanced. Theunbalanced force is between the system and the surrounding, and gas pressure will moves thepiston in upward direction till it hits the stop.

The system again comes to an equilibrium states, being described by the properties P2, v2, T2.But the immediate states passed through by the system are non-equilibrium states whichcannot be described by thermodynamic coordinates. Figure shows the points 1 and 2 as theinitial and final equilibrium states joined by dotted line.

Now if the single weight on the piston is made up of many very small pieces of weights andthese weights are removed one by one very slowly, at any instant of the upward travel of thepiston, if the gas is isolated, the departure of the state of the system from thermodynamicequilibrium state will be infinitesimally small. So every state passed through by the system willbe an equilibrium state.

Energy and its formsEnergy is defined as the capacity to do work or energy can also be defined as the capacity toexert a force in a given direction through a distance.

The unit of energy in SI (System international) system is Nm or Joule (J).

Forms of Energy1. Work

2. Heat

WorkWork is one of the basic modes of energy transfer.

In mechanics the action of a force on a moving body is identified as work. The work isdone by a force as it acts upon a body moving in the direction of force.

In thermodynamics, work transfer is considered as occurring between the system andthe surroundings. Work is said to be done by a system if the sole effect on the thingsexternal to the system can be reduced to the raising of a weight.

The work is done by a system, it is taken to be positive, and when work is done on asystem, it is taken to be negative.

(a)Work is Positive (+ve) (b) Work is negative (-ve)

Power: The rate of energy transfer is known as power or the rate of work transfer is known as power. The unit of power is J/s or Watt.

Heat

Heat is defined as the form of energy that is transferred across a boundary by virtue of atemperature difference. The temperature difference is the potential or force and heat transfer isthe flux.

Heat flow into a system is taken to be positive, and heat flow out of a system is taken asnegative.

A process in which no transfer of heat through boundary is known as adiabatic process.

(a)Heat transfer is Positive (+ve) (b) Heat transfer is negative (-ve)

The symbol used for heat transfer is Q. The unit of heat transfer in SI (System international) system is Nm or Joule (J). The rate of heat transfer is given by W or kW.

Types of Heat

1. Specific Heat: Specific heat is defined as the amount of heat required to raise the temperature of a unit mass (1kg) ofthe substance by unit degree (1oC or 1K) change in temperature.

The quantity of heat absorbed or rejected by a system during heating or cooling is measured by the formula as givenbelow:

Q=m×c×(T2-T1)

Where,

Q= heat gain or loose by the system in kJ,

m= mass of the substance in kilograms (kg),

c= specific heat in kJ/kgK

(T2-T1)= Temperature rise or drop in degree Celsius or Kelvin.

Types of specific heat: Basically there are two types of specific heats as given below:

(i)Specific heat at constant pressure (cp)

(ii)Specific heat at constant volume (cv)

(i)Specific heat at constant pressure (cp): It is defined as the amount of heatrequired to raise the temperature of a unit mass (1kg) of the substance by unit degree(1oC or 1K) change in temperature when the pressure is constant. It is represented bycp. Its unit is kJ/kgK.

(ii)Specific heat at constant volume (cv): It is defined as the amount of heatrequired to raise the temperature of a unit mass (1kg) of the substance by unit degree(1oC or 1K) change in temperature when the volume is constant. It is represented bycv. Its unit is kJ/kgK.

Specific heat of water: c=4.186 kJ/kgK

Specific heats of air: cp=1.005 kJ/kgK

cv=1.005 kJ/kgK

2.Latent heat of vaporization: It is defined as the amount of heat required toevaporated one kilogram of water at its saturation temperature (boiling point) withoutchange of temperature. It is represented by hfg. Its unit is kJ/kg. The latent heat ofvaporization of water or latent heat of steam is 2257 kJ/kg.

Laws of thermodynamics

There are three laws of thermodynamics given as under:

1.Zeroth law of thermodynamics

2.First law of thermodynamics

3.Second law of thermodynamics

Zeroth law of thermodynamics:

Zeroth law states that if two systems are at same time in thermal equilibrium with a thirdsystem, they are in equilibrium with each other.

If the system A and B are in thermal equilibrium with a third system C separately then the twosystems A and B will also be in thermal equilibrium with each other.

Thermometry: Thermometry is defined as that branch of science, in which the temperature is measuredwith accuracy and precision.

Definition of temperature: Temperature is defined as the measure of hotness and coldness of asubstance with reference to a standard value.

Temperature Scales

There are three types of temperature scales for the measurement of temperature.

(a) Celsius,

(b) Fahrenheit and

(c) Kelvin.

Celsius: Swedish astronomer Anders Celsius in 1742. It is also called as centigrade temperature scale, inthis scale freezing point of water is represented by 0 degree and boiling point is represented by 100degree. It has 100-degree intervals between the defined points so that sometimes it is called thecentigrade scale.

Fahrenheit: German physicist Daniel Gabriel Fahrenheit in 18th century. In this scale freezing point ofwater is 32 and boiling point of water is 212. The interval between the two (32-212) being divided into180 parts.

Kelvin: British physicist William Thomson, Baron Kelvin. It is defined as 1/ 273.16 of the triple point(equilibrium among the solid, liquid, and gaseous phases) of pure water. The Kelvin is written by symbol Kwith using degree (o). This scale has as its zero point absolute zero, the theoretical temperature at whichthe molecules of a substance have the lowest energy. The difference between the freezing and boilingpoints of water is 100 degrees in each, so that the Kelvin has the same magnitude as the degree Celsius.

Relation between Celsius, Fahrenheit and Kelvin𝐶

100=(𝐹 − 32)

180=(𝐾 − 273)

100

𝐶

5=(𝐹 − 32)

9=(𝐾 − 273)

5

First law of thermodynamics

First law of thermodynamics also states that, “the energy can neither be created nor be destroyed it canonly be transformed from one form to another.” According to this law, when a system undergoes athermodynamic process, both heat and work transfer takes place. The net energy is stored within thesystem and is termed as stored energy or total energy of the system. Mathematically it is written as:

δQ-δW=dE

First law of thermodynamics for a cyclic process

A process is cyclic if the initial and final states of thesystem are identical. A system represented by state 1undergoes a process 1-r-2 and returns to the initial statefollowing the path 2-s-1. All the properties of the systemare restored, when the initial and final state is reached.During the completion of these processes:

(a) Area 2-3-4-1-s-2 denotes the work done W1 by thesystem during expansion process 2-s-1.

(b) Area 4-3-1-s-4 denotes the work done W2 supplied tothe system during the compression process 4-s-1.

(c) Area 1-r-2-s-1 denotes the net work done (W1-W2)delivered by the system.

According to first law of thermodynamics, “when a closed system undergoes a thermodynamic cycle,the net heat transfer is equal to net work done.”

Or

“The cyclic integral of heat transfer is equal to cyclic integral of work done.” Mathematically it iswritten as:

𝛿𝑄 = 𝛿𝑊

On integrating the above equation for a thermodynamic state 1 to 2, we get,

1

2

𝛿𝑄 − 1

2

𝛿𝑊 = 1

2

𝑑𝐸

𝑄1−2 −𝑊1−2 = 𝐸2 − 𝐸1

Where,

Q1-2 = heat transferred to the system during the process 1 to 2.

W1-2= Work transfer by the system during the process 1 to 2.

E1 = Total energy of the system at state 1

E2 = Total energy of the system at state 2

Note: The total energy is the sum of potential energy, kinetic energy and internal energy of the system. It is mathematically written as:

𝐸 = 𝑃. 𝐸. +𝐾. 𝐸.+𝑈

𝐸 = 𝑚𝑔𝑧 + 𝑚𝑣2

2+ 𝑈

Where, P.E. = Potential energy,

K.E. = Kinetic energy,

U = Internal Energy.

Internal Energy: Internal energy of steam is define as the energy stored in the steam, above 0oC (freezing point) of water. It may be obtained by subtracting the work done during evaporation to the enthalpy of steam. It is represented by U. Mathematically it is written as,

Internal energy of steam=Enthalpy of steam-Workdone during evaporation

Enthalpy: It is defined as the amount of heat absorbed by water from 0oC (freezing point) to saturation point (sensible heat) plus heat absorbed during evaporation (latent heat). It is represented by hg.

So that,

Enthalpy=sensible heat + latent heat

Joule’s experiment

First law for non flow process

In thermodynamics there are number of processes where in one or another state parameter remains constant. The basic thermodynamic processes used to analyze,

(i) Relationship between various parameters such as temperature, pressure and volume,

(ii) For obtaining the work and heat in the process, and

(iii) For obtaining the alteration in internal energy.

Constant volume process (Isochoric process)

• An Isochoric process is a process during which the specific volume v remains constant. Some facts about constant volume process

Pressure, volume and temperature relationshipFor the initial state 1:

𝑃1𝑣1 = 𝑚𝑅𝑇1For the final state 2:

𝑃2𝑣2 = 𝑚𝑅𝑇2

We know that from general gas equation,𝑃1𝑣1𝑇1

=𝑃2𝑣2𝑇2

Since during the process specific volume is constant (v1=v2), so that

𝑃1

𝑇1=

𝑃2

𝑇2𝑜𝑟

𝑃

𝑇= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Since there is no expansion of gas (dV=0), no work is done on the system or by the system. From non flow energy equation𝛿𝑄 = 𝛿𝑊 + 𝛿𝑈

Since 𝛿𝑊 = 0 all the heat is used to change the internal energy of the system.

Therefore,𝛿𝑄 = 𝛿𝑈

Heat added during a constant volume process is given by 𝛿𝑄 = 𝑚𝑐𝑣(𝑇2 − 𝑇1)

Or it may be written as 𝛿𝑄 = 𝛿𝑈 = 𝑚𝑐𝑣(𝑇2 − 𝑇1)

Where cv= specific heat at constant volume

For unit mass (i.e. m=1)𝑑𝑈 = 𝑐𝑣𝑑𝑇

Or

𝑐𝑣 =𝑑𝑈

𝑑𝑇

Constant pressure process (isobaric process)An Isobaric process is one during which the pressure P remains constant.

Pressure, volume and temperature relationship

For the initial state 1: 𝑃1𝑣1 = 𝑚𝑅𝑇1

For the final state 2: 𝑃2𝑣2 = 𝑚𝑅𝑇2


=𝑃2𝑣2𝑇2

Since during the process pressure is constant (P1=P2), so that 𝑣1𝑇1

=𝑣2𝑇2

𝑜𝑟𝑣

𝑇= 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Mechanical work,

𝑊1−2 = 1

2

𝑃𝑑𝑣 = 𝑃(𝑣2 − 𝑣1)

For a non flow process,𝛿𝑄 = 𝛿𝑊 + 𝛿𝑈

Q1−2 = 𝑃 𝑣2 − 𝑣1 + 𝑈2 − 𝑈1Q1−2 = 𝑈2 + 𝑃2𝑣2 − 𝑈1 + 𝑃1𝑣1

Q1−2 = (ℎ2 − ℎ1)Since,

ℎ = 𝑈 + 𝑃𝑣ℎ2 − ℎ1 = Q1−2 = 𝑚𝑐𝑝(𝑇2 − 𝑇1)

Where cp= specific heat at constant pressureFor unit mass (i.e. m=1)

𝑑ℎ = 𝑐𝑝𝑑𝑇

Or

𝑐𝑝 =𝑑ℎ

𝑑𝑇Significance of gas constant R:During constant pressure process,Work done,

𝑊1−2 = 1

2

𝑃𝑑𝑣 = 𝑃(𝑣2 − 𝑣1)

Since P1=P2=P;𝑃1𝑣1 = 𝑅𝑇1𝑃2𝑣2 = 𝑅𝑇2

𝑊1−2 = 𝑃(𝑣2 − 𝑣1)𝑊1−2 = 𝑅(𝑇2 − 𝑇1)

𝑅 =𝑊1−2

(𝑇2 − 𝑇1)Thus the gas constant is equal to the work of 1 kg of gas in an isobaric process when the temperature changes by 1 degree.

Relationship between specific heats (cp and cv) and gas constant R:Let an unit mass of an ideal gas undergo constant volume and constant pressure processes separately through a temperature range from T1 to T2.During isochoric process:

𝑞1−2 =Q1−2𝑚

= 𝑐𝑣(𝑇2 − 𝑇1)

And 𝑊1−2 = 0

And 𝑐ℎ𝑎𝑛𝑔𝑒 𝑖𝑛 𝑖𝑛𝑡𝑒𝑟𝑛𝑎𝑙 𝑒𝑛𝑒𝑟𝑔𝑦𝑈2 − 𝑈1 = 𝑐𝑣(𝑇2 − 𝑇1)

During isobaric process:

𝑞1−2 =Q1−2𝑚

= 𝑐𝑝(𝑇2 − 𝑇1)

𝑊1−2 = 𝑃(𝑣2 − 𝑣1)Change in internal energy,

Q1−2 = 𝑃 𝑣2 − 𝑣1 + 𝑈2 − 𝑈1𝑈2 − 𝑈1 = Q1−2 − 𝑃 𝑣2 − 𝑣1

𝑈2 − 𝑈1 = 𝑐𝑝(𝑇2 − 𝑇1) − 𝑃 𝑣2 − 𝑣1For an ideal gas internal energy is a function of temperature {U=f(T)}. So that equating the two internal energy equations;

𝑐𝑝 𝑇2 − 𝑇1 − 𝑃 𝑣2 − 𝑣1 = 𝑐𝑣(𝑇2 − 𝑇1)

𝑐𝑝 𝑇2 − 𝑇1 − 𝑅(𝑇2 − 𝑇1) = 𝑐𝑣(𝑇2 − 𝑇1)

{Since, 𝑃1 = 𝑃2 = 𝑃; 𝑃1𝑣1 = 𝑅𝑇1; 𝑃2𝑣2 = 𝑅𝑇2}𝑐𝑝 − 𝑅 = 𝑐𝑣

Or𝒄𝒑 − 𝒄𝒗 = 𝑹

The ratio cp/cv is known as isentropic index and is expressed by 𝛾.

𝜸 =𝒄𝒑

𝒄𝒗We can write from the above to relations,

𝒄𝒑 = 𝑹𝜸

𝜸−𝟏, 𝒄𝒗 =

𝑹

𝜸−𝟏

Constant Temperature process (isothermal process)An isothermal process is one during which temperature T remains constant.

Pressure, volume and temperature relationship

For the initial state 1: 𝑃1𝑣1 = 𝑚𝑅𝑇1

For the final state 2: 𝑃2𝑣2 = 𝑚𝑅𝑇2


=𝑃2𝑣2𝑇2

Since during the process temperature is constant (T1=T2), so that 𝑃1𝑣1 = 𝑃2𝑣2 𝑜𝑟 𝑃𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡

Work done,

𝑊1−2 = 1

2

𝑃𝑑𝑣

Since, 𝑃𝑣 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐾)

𝑃 =𝐾

𝑣

Substituting the value of P in the equation of work done,

𝑊1−2 = 1

2𝐾

𝑣𝑑𝑣 = 𝐾

1

2 1

𝑣𝑑𝑣

𝑊1−2 = 𝐾 ln(𝑣) 12

𝑊1−2 = 𝐾{𝑙𝑛 𝑣2 − 𝑙𝑛 𝑣1 }

𝑾𝟏−𝟐 = 𝑷𝟏𝒗𝟏𝒍𝒏𝒗𝟐𝒗𝟏

= 𝑷𝟐𝒗𝟐𝒍𝒏𝒗𝟐𝒗𝟏

Or

𝑾𝟏−𝟐 = 𝑷𝟏𝒗𝟏𝒍𝒏𝑷𝟏

𝑷𝟐= 𝑷𝟐𝒗𝟐𝒍𝒏

𝑷𝟏

𝑷𝟐

{Since, 𝑃1𝑣1 = 𝑃2𝑣2}

Adiabatic process (isentropic process)

An adiabatic process is one in which no heat is gained or lost by the system during itsexpansion or compression. This will happen when the working substance remainsthermally insulated, so that no heat enters or leaves it during the process. It may benoted that adiabatic process may be reversible or irreversible. The reversible adiabaticprocess (frictionless adiabatic process) is known as isentropic process or constantentropy process. But if the friction is involved in the process, then the adiabaticprocess is irreversible, in this case entropy does not remain constant.

Some facts about isentropic process:

i. No heat enters or leaves the working substance.

ii. The temperature of the gas changes.

iii. The change in internal energy is equal to the work done.

iv. It is expressed by the relation Pvγ= constant.

Where γ is the isentropic index and its valve is 1.4.

Also, γ= cp/cv and R= cp-cv

Where cp= specific heat at constant pressure and,

cv= specific heat at constant volume.

R= gas constant.

Work done,

𝑊1−2 = 1

2

𝑃𝑑𝑣

Since, 𝑃1𝑣1

𝛾= 𝑃2𝑣2

𝛾= 𝑃𝑣𝛾 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐾)

𝑃 =𝐾

𝑣𝛾


𝑊1−2 = 1

2 𝐾

𝑣𝛾𝑑𝑣

𝑊1−2 = 𝐾 1

2 1

𝑣𝛾𝑑𝑣 =

𝐾

1 − 𝛾(𝑣2

1−𝛾− 𝑣1

1−𝛾)

As 𝐾 = 𝑃1𝑣1

𝛾= 𝑃2𝑣2

𝛾

By multiplying the first term inside the bracket by 𝑃2𝑣2𝛾

and the second term by 𝑃1𝑣1

𝛾, we get,

𝑾𝟏−𝟐 =𝑷𝟐𝒗𝟐 − 𝑷𝟏𝒗𝟏

(𝟏 − 𝜸)=𝑷𝟏𝒗𝟏 − 𝑷𝟐𝒗𝟐

(𝜸 − 𝟏)=𝑹(𝑻𝟏 − 𝑻𝟐)

(𝜸 − 𝟏)

From non-flow process,𝛿𝑄 = 𝛿𝑊+ 𝛿𝑈

{Since, 𝛿𝑄 = 0}𝛿𝑊+ 𝛿𝑈 = 0𝛿𝑊 = −𝛿𝑈

𝛿𝑊 = −(𝑈2 − 𝑈1)

Or

𝑼𝟏 − 𝑼𝟐 =𝑷𝟏𝒗𝟏 − 𝑷𝟐𝒗𝟐

(𝜸 − 𝟏)


=𝑃2𝑣2𝑇2

And as per the isentropic law,𝑃1𝑣1

𝛾= 𝑃2𝑣2

𝛾

The following relations can be set up,

𝑻𝟐𝑻𝟏

=𝑷𝟐

𝑷𝟏

𝜸−𝟏𝜸

=𝒗𝟏𝒗𝟐

𝜸−𝟏

Polytropic process

The polytropic process is also known as general law for the expansion and compression of gases, and it is expressed by the relation:

Pvn= constant

Where n is a polytropic index.

Work done

𝑊1−2 = 1

2

𝑃𝑑𝑣

Since, 𝑃1𝑣1

𝑛 = 𝑃2𝑣2𝑛 = 𝑃𝑣𝑛 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡 (𝐾)

𝑃 =𝐾

𝑣𝑛


𝑊1−2 = 1

2 𝐾

𝑣𝑛𝑑𝑣

𝑊1−2 = 𝐾 1

2 1

𝑣𝑛𝑑𝑣 =

𝐾

1 − 𝑛(𝑣2

1−𝑛 − 𝑣11−𝑛)

As 𝐾 = 𝑃1𝑣1

𝑛 = 𝑃2𝑣2𝑛

By multiplying the first term inside the bracket by 𝑃2𝑣2𝑛 and the second term by 𝑃1𝑣1

𝑛, we get,

𝑾𝟏−𝟐 =𝑷𝟐𝒗𝟐 − 𝑷𝟏𝒗𝟏

(𝟏 − 𝒏)=𝑷𝟏𝒗𝟏 − 𝑷𝟐𝒗𝟐

(𝒏 − 𝟏)

𝒘𝟏−𝟐 =𝑾𝟏−𝟐

𝒎=𝑷𝟏𝒗𝟏 − 𝑷𝟐𝒗𝟐(𝒏 − 𝟏)

=𝑹(𝑻𝟏 − 𝑻𝟐)

(𝒏 − 𝟏)

Change in internal energy𝑼𝟐 −𝑼𝟏 = 𝒎𝒄𝒗(𝑻𝟐 − 𝑻𝟏)𝒖𝟐 − 𝒖𝟏 = 𝒄𝒗(𝑻𝟐 − 𝑻𝟏)

Heat interaction𝑞1−2 = 𝑤1−2 + 𝑢2 − 𝑢1

𝑞1−2 =𝑃1𝑣1 − 𝑃2𝑣2(𝑛 − 1)

+ 𝑐𝑣(𝑇2 − 𝑇1)

𝑞1−2 =𝑃1𝑣1 − 𝑃2𝑣2(𝑛 − 1)

+𝑅(𝑇1 − 𝑇2)

(𝛾 − 1)

(Since, 𝑐𝑣 =𝑅

𝛾−1)

𝑞1−2 =𝑃1𝑣1 − 𝑃2𝑣2(𝑛 − 1)

+𝑃2𝑣2 − 𝑃1𝑣1(𝛾 − 1)

(Since, 𝑃1𝑣1 − 𝑃2𝑣2 = 𝑅(𝑇1 − 𝑇2))

𝑞1−2 =(𝛾 − 𝑛)

(𝛾 − 1)×𝑃1𝑣1 − 𝑃2𝑣2(𝑛 − 1)

𝒒𝟏−𝟐 =(𝜸 − 𝒏)

(𝜸 − 𝟏)× 𝑷𝒐𝒍𝒚𝒕𝒓𝒐𝒑𝒊𝒄 𝒘𝒐𝒓𝒌

The following relations can be set up,


=𝑃2𝑣2𝑇2

And as per the isentropic law,𝑃1𝑣1

𝑛 = 𝑃2𝑣2𝑛

𝑻𝟐𝑻𝟏

=𝑷𝟐

𝑷𝟏

𝒏−𝟏𝒏

=𝒗𝟏𝒗𝟐

𝒏−𝟏

Steady and unsteady flow process:

When fluid parameters at any point of the control volume remainconstant with respect to time, the flow process is called steady flowprocess. Let velocity, pressure, temperature etc. Are functions onlyof location and do not vary with time. If pressure is represented by

P then mathematically a steady flow is defined as𝜕𝑃

𝜕𝑡= 0, i.e., the

rate of change of pressure at a position is zero.

Whereas when the fluid parameters vary with respect to time, theflow process is known as unsteady flow process. If pressure isrepresented by P then mathematically a unsteady flow is defined as𝜕𝑃

𝜕𝑡≠ 0, i.e., the rate of change of pressure at a position is not equal to

zero.

Steady Flow Energy Equation (S.F.E.E.)

Assume the flow through a system as shown in figure. During a small timeinterval dt there occurs a flow of mass and energy into a fixed control volume;entry is at point 1 and exit at point 2.

The fluid enters the control volume at point 1 with a average velocity V1, pressureP1, specific volume v1 and internal energy U1.The fluid exit the control volume atpoint 2 and the corresponding values are V2, P2, v2, U2. During the fluid flow fromthe two sections, heat Q and mechanical work W may also cross the controlsurface.The following points are taken into consideration for energy balance

equation:

(i) Internal energy

(ii) Kinetic and potential energies.

(iii) Flow work

(iv) Heat and mechanical work which cross the

control volume.

From the law of conservation of energy, energy neither be created nor be destroyed we can write,

Total energy flow rate into the control volume = Total energy flow rate out of control volume

m(energy carried into the system)+m(flow work)+ rate of heat flow= m(energy carried out of the system)+m(flow work)+ rate of work transfer

m(I.E.+P.E.+K.E.)1 +m(flow work)1 + 𝑄= m(I.E.+P.E.+K.E.)2 +m(flow work)2 + 𝑊

Where, 𝑄 =𝑑𝑄

𝑑𝑡𝑎𝑛𝑑 𝑊 =

𝑑𝑊

𝑑𝑡

𝑚 𝑈1 + 𝑔𝑧1 +𝑉12

2+𝑚 𝑃1𝑣1 + 𝑄 = 𝑚 𝑈2 + 𝑔𝑧2 +

𝑉22

2+𝑚 𝑃2𝑣2 + 𝑊

Arranging the equation,

𝑚 𝑈1 + 𝑃1𝑣1 + 𝑔𝑧1 +𝑉12

2+ 𝑄 = 𝑚 𝑈2 + 𝑃2𝑣2 + 𝑔𝑧2 +

𝑉22

2+ 𝑊

𝑚 (𝑈1 + 𝑃1𝑣1) + 𝑔𝑧1 +𝑉12

2+ 𝑄 = 𝑚 (𝑈2 + 𝑃2𝑣2) + 𝑔𝑧2 +

𝑉22

2+ 𝑊

Since ℎ = 𝑈 + 𝑃𝑣 , 𝑠𝑜 𝑡ℎ𝑎𝑡 ℎ1 = 𝑈1 + 𝑃1𝑣1 𝑎𝑛𝑑 ℎ2 = (𝑈2 + 𝑃2𝑣2)

𝒎 𝒉𝟏 + 𝒈𝒛𝟏 +𝑽𝟏𝟐

𝟐+ 𝑸 = 𝒎 𝒉𝟐 + 𝒈𝒛𝟐 +

𝑽𝟐𝟐

𝟐+ 𝑾

This equation is known as steady flow energy equation (SFEE).

If the mass of fluid is taken as unity then steady flow energy equation is reduces to,

ℎ1 + 𝑔𝑧1 +𝑉12

2+ 𝑞

1−2= ℎ2 + 𝑔𝑧2 +

𝑉22

2+ 𝑤

All the terms represent energy flow per unit mass of fluid (J/kg).

Applications of Steady flow energy equation

Steady flow energy equation is commonly used in flow processes in many engineering plants. Some commonly used engineering systems which works on steady flow energy equation (SFEE) are as follows:

(i) Compressor

(ii) Condenser

(iii) Boiler

(iv) Turbine

(v) Nozzle and

(vi) Pump

(i) Compressor: Compressor is a device which is used tocompress the fluid (may be air) and deliver it at a high pressure andlarge flow rate. There are two types of compressors as follows:

(a) Rotary compressor

(b) Reciprocating compressor

(a) Rotary compressor: Rotary compressors are the deviceswhich are used to develop high pressure and have a rotor as theirprimary element. The characteristic features of flow through arotary compressor are:

Work is done on the system so that W is negative.

Negligible change in Potential energy.

Heat is lost from the system so that Q is negative

Steady flow energy equation may be written as follows:

𝑚 ℎ1 +𝑉12

2− 𝑄 = 𝑚 ℎ2 +

𝑉22

2−𝑊

Or

𝑊 = 𝑚 ℎ2 +𝑉22

2−𝑚 ℎ1 +

𝑉12

2+ 𝑄

If the change in velocity is negligible and the flow process is assumed as adiabatic (i.e. Q=0) due to very high flow rates, then

𝑊 = 𝑚(ℎ2 − ℎ1)

Reciprocating compressor: Reciprocating compressors are the devices which areused to develop high pressure and have a piston cylinder arrangement as theirprimary element. The characteristic features of flow through a rotary compressor are:

Work is done on the system so that W is negative.

Negligible change in Potential energy.

Heat is lost from the system so that Q is negative

Steady flow energy equation may be written as follows:

𝑚 ℎ1 +𝑉12

2− 𝑄 = 𝑚 ℎ2 +

𝑉22

2−𝑊

Or

𝑊 = 𝑚 ℎ2 +𝑉22

2−𝑚 ℎ1 +

𝑉12

2+ 𝑄

If the change in velocity is negligible, then𝑊 = 𝑚(ℎ2 − ℎ1) + 𝑄

ii. Condenser: Condenser is a type of heat exchanger. It is used to transfer heatfrom one fluid to another. The characteristic features of a condenser are asfollows:

No mechanical work (i.e., W=0).

No change in kinetic and potential energies.

No external heat interaction (Since it is perfectly insulated).

Heat is absorbed by the one fluid (Steam) to the another fluid (coolant), sothat heat is taken negative.

Thus steady flow energy equation reduces to;

𝑚 ℎ1 + 𝑔𝑧1 +𝑉12

2+ 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 +

𝑉22

2+𝑊

ℎ1 − 𝑄 = ℎ2𝑄 = ℎ1 − ℎ2

(iii) Boiler: Boiler is an equipment used for generation of steam. Thermal energy released bycombustion of fuel is transferred to water which vaporizes and gets converted into steam.

The characteristic features of a boiler are as follows:

No mechanical work (i.e., W=0).

No change in kinetic and potential energies

Height change between inlet and exit point is negligible.


𝑚 ℎ1 + 𝑔𝑧1 +𝑉12

2+ 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 +

𝑉22

2+𝑊

ℎ1+𝑄 = ℎ2𝑄 = ℎ2 − ℎ1

(iv) Turbine: Turbine is a device which converts thermal energy into useful work. Inturbine fluids expand from high pressure to a low pressure. The work output from theturbine may be used to drive a generator to produce electricity. The characteristicfeatures of a turbine are as follows:

Negligible change in velocity so that negligible change in kinetic energy.

Negligible change in potential energy.

Isentropic expansion takes place since the walls of turbine are thermally insulated.


𝑚 ℎ1 + 𝑔𝑧1 +𝑉12

2+ 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 +

𝑉22

2+𝑊

𝑊 = 𝑚(ℎ2− ℎ1)

(v) Nozzle: Nozzle is a device of varying cross-section used for increasing the velocityof a flowing stream at the expense of its pressure drop. In nozzle pressure energy ofthe fluid is converted into kinetic energy. It is used in turbines, fuel pumps and jet

engines etc.

The characteristic features of a nozzle are as follows:

No mechanical work (i.e. W=0)

Flow is isentropic (i.e. Q=0)

Change in height between entry and exit is negligible. (i.e. z1=z2)


ℎ1 +𝑉12

2= ℎ2 +

𝑉22

2

Let V1 is known then,

𝑉2 = 2 ℎ1 − ℎ2 + 𝑉12

(vi) Pump: A pump is a device which takes the fluid from a low level and delivers it to a high level. The characteristic features of a pump are as follows:

Flow is assumed to be adiabatic (i.e. Q=0)

No change in internal energy.

Work is done on the system, so that work is taken negative.


𝑚 ℎ1 + 𝑔𝑧1 +𝑉12

2+ 𝑄 = 𝑚 ℎ2 + 𝑔𝑧2 +

𝑉22

2+𝑊

𝑚 𝑔𝑧1 +𝑉12

2= 𝑚 𝑔𝑧2 +

𝑉22

2−𝑊

Throttling Process:

Throttling is an irreversible expansion process. In this process theexpansion of fluid takes place from high pressure to low pressure. Thisprocess occurs when the fluid is flowing across a restriction (partiallyclosed valve or a small orifice) placed in the flow passage.

This process occurs in a flow through a porous plug as shown infigure. In this process a steady stream of gas at a given pressure (P1) andtemperature (T1) flows through a porous plug contained in a thermallyinsulated horizontal tube. The Fluid exits at a reduced pressure (P2).Throttling process is used for obtaining the dryness fraction of wetsteam.

The characteristic features of a pump are as follows:

Change in kinetic and potential energies are negligible.

No mechanical work (i.e. W=0)

No heat loss as the tube is thermally insulated (i.e. Q=0)

Thus steady flow energy equation for unit mass reduces to;

ℎ1 + 𝑔𝑧1 +𝑉12

2+ 𝑄 = ℎ2 + 𝑔𝑧2 +

𝑉22

2+𝑊

ℎ1 = ℎ2

As we know that ℎ = 𝑐𝑝. 𝑇

Where cp= specific heat at constant pressure

So that we can write, 𝑐𝑝. 𝑇1 = 𝑐𝑝. 𝑇2

Or𝑇1 = 𝑇2

Thermodynamics

Engineering