Thermo Theory

THERMODYNAMICS

THERMODYNAMICS (THEORY)

CONTENTS

Unit Chapters Page No.

1 Basic Concepts (Introduction) 2 - 13

2 Work & Heat 14 - 19

3 First Law of Thermodynamics 20 - 23

4 Second Law of Thermodynamics 24 - 31

5 Entropy 32 - 39

6 Available Energy, Availability & Irreversibility 40 - 45

7 Properties of Pure Substances 46 - 48

8 Air Cycles 49 - 59

9 Rankine Cycles 60 - 69

10 Refrigeration 70 - 75

11 Psychrometry 76- 83

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Published at : Ascent Gate Academy

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Rajnandgaon (Chhattisgarh) Mob : 09993336391

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MECHANICAL ENGINEERING

1 INTRODUCTION

THERMODYNAMICS : -

Thermodynamics is the science of energy transfer & its effect on physical properties of substance.
Some medium is required to convert heat to work or work to heat.

Eg.:- In I.C. engine. we give heat as input to the air due to which its temperature increases & then
piston dose work.

In refrigerator, freon gas is used as medium.

↓ Medium

Gas ↓
→ Ideal
→ Real gas Pure Substance
→ Water
→ Refrigerant

MICROSCOPIC & MACROSCOPIC APPROACH :-

* In microscopic thermodynamics, the behaviour of the gas is described by summing up behaviour of each
molecule (Statistical thermodynamics)
* In macroscopic thermodynamics, the behaviour of gas is described by the net effect of action of all the
molecules which can be perceived by human senses. (Classical thermodynamics)

Thermodynamic System, Surrounding and Boundary,

* A thermodynamic system is defined as quantity of matter or region in space upon which attention is
concentrated in the analysis of problem.
* Everything external to system is called surrounding or environment.
* The plane which separates system & environment is called Boundary.
* System & surrorunding together comprise a Universe.

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THERMODYNAMICS

Types of System :-
i) Closed System : It is a system of fixed mass i.e. no mass transfer. There may be energy transfer into or out
of the system.
Eg. Gas confined in piston cylinder arrangement

work transfer
No mass trasfer

Gas surrounding
(System)

Fig. : Closed System

ii) Open System :- Both mass & energy transfer across boundary .

Eg. : Turbine m

<>

<w

Fig. : Turbine (open system) ↓ m

iii) Isolated System:- There is no interaction of mass & energy across boundary

Eg. :- Thermos flask. 111111111111111111222222222222222222..3333....4444....5555..6666......7777..8888....9999..0000....1111.....2222. .3333.....4444..5555....6666.....111111111111111117777. 22222222222222222888899

CONTROL VOLUME & CONTROL SURFACE

Air out
↑

Motor Air
Compresse
Control Volume
↑ → Control Surface.
Air in

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MECHANICAL ENGINEERING

For thromodynamic analysis of open System, attention is focused on a certain volume in space to
which continously mass goes in & from which continously mass comes out, is known as control volume
bounded by a surface called control surface.
Property :- A thermodynamic property refers to the characterstics which can be used to describe the
condition or state of the system
Eg. :- Temperature, Pressure, chemical composition, colour, volume, energy etc.

Features :- 1) Measurable characterstics which describes the system.
ii) Definite unique value at particular state.
iii) Doesnot depend on path followed by system.
iv) Differential is exact.

Two types :-
a) Intensive properties :- Independent of mass.
Eg. : Pressuessure, temperature, density, composition, Viscosity, Thermal conductivity, electrical potential
etc.
b) Extensive property : - dependent on mass
Eg. Energy, Enthalpy, entropy, volume etc.
.

Note :- Specific extensive properties are intensive properties.
Eg. : - Sp.energy, Sp.Volume, Sp. Enthalpy, etc.

THERMODYNAMIC STATE, PROCESSES & CYCLES

* When all the properties of a system has a definite value, the system is said to exist in a definite state

* Any operation in which one or more properties of system changes is called change of state

* The succession of states passed through during change of state is called path

* When the path is completely specified, the change of state is called Process.

* A thermodynamic cycle is defined as a series of state changes such that the final state is

identical with initial satate.

a
<
<

<
↑

P1

<

b

2

<

a - b : - Process
1 - 2- 1 : - Cycle

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THERMODYNAMICS

Homogenous & Heterogenous systems :-

* A quantity of matter which is homogenous throughout in chemical composition & physical
structure is called phase.
* A system consisting of single phase is called Homogenous system
* A system consisting more than one phase is heterogenous system.

Thermodynamic Equillibrium:-

When no change in macroscopic property is registered if the system is isolated from its surrounding then
system is said to be in thermodynamic equillibrium.

If 3 equilibrium conditons are satisfied, system is in thermodynamic equilibrium
a) Mechanical equilibrium :- No unbalanced forces
b) Chemical equilibrium :- No chemical reaction
c) Thermal equilibrium :- when the system existing in mechanical & chemical equilibrium is separated from
surrounding by a diathermic wall, & if there is no spontaneous change in its properties then the system is
said to exist in state of thermal equllibrium.
** When condition of any one of three type of equilibrium. does not exists, system is said to be in non
equilibrium state.

Quasi - Static Process.

1 Equillibriu
>
↑ > x State
x

P1 V1 T1 x
Px

x Quasi static
x Process
x
x
xx2

V→>

A process carried out in such a way that each point of the path is in equillibrium is known as Quasi static or
reversible process. Infinite slowness is the characterstic of Quasi static process.

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MECHANICAL ENGINEERING

POINT FUNCTION & PATH FUNCTION :-

Those functions which depend only on intial & final states and doesnot depend on path followed is

called point function.

Eg. : -All those function thermodynamic properties (pressure, temperature, volmue etc.)

which depends on path is called path function

Eg : - Heat, work etc.

Check for property : -

dp = Mdx + Mdy

is a thermodynamic property if its defferential is exact.

i.e. ∂M = ∂N y
∂y x ∂x

A differential is said to be exact if its integration between two end states can be carried out without

requiring any information variation of variables.

Eg. : - i) pdv + vdp.

= d (pv)

Integrating
∫d(pv) = pv
∴ an exact diffeiential & hence property of system

Eg. ii) pdv
∫d(pv) - requires relationship between P & V, so inexact differential

P & V, so inexact defferential

Eg. iii) V dp + P dV
T T

If differential is exact.

∂M = ∂N y
i.e. ∂y x ∂x

Then this is property

Compare with dp = Mdx + N dy

M= VR N = P = V
T = P, T R

∂M = ∂v ∂R
∂y ∂v T p = ∂v P p = 0
x

∂N ∂ P p= ∂R =0
∂x y = ∂p T v
∂p V

∂M x = ∂N
∂y ∂x y

∴ given function is exact differential & hence property of systum.

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THERMODYNAMICS

Zeroth law of thermodymanics.

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adiabatic wall

When two separate systems are individually in thermal equilibrium with the third system, then the two
systems are in thermal equlliberium with each other.
Significance :- It gives us the concept of temperature & defines the isotherm.

↑

x (X3, y3)
x (X2, y2)
x (X , y )

11

↑

* if the series of states can be found on a system which are in thermal equilibrium with one state of another
system then locus of all these state is called as isotherm

Thermometry :-Act of measuring temperature with accuracy & precision The sensing element or device
has certain physical characterstics which changes with change in value as a function of temperature is called
thermometric property and corresponding substance is thermetric substence.

Types of thermometers :-

i) Liquid in glass thermometer :

a) Mercury - high temperature measurement (upto 3000C)

b) Alcohol - Low temperature measurement (upto - 1110C)

ii) Gas thermometer : Expansion & contraction type

a) Constant pressure type : b) Constant volume type :

VαT ; V1 = V2 PαT ; P = P2
T T2 1 T2

1 T1

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MECHANICAL ENGINEERING

iii) Resistance thermomoter : Function on the principle of change in resistance with temperature

Rt = R0 ( 1 + At + Bt2)
R0 = Resistance at 00C
A, B = Constant

t = Temperature

Resistance thermometer function on wheat stone bridge principle. S
P

Q Rt

Here, P, Q & S are known resistance & R is unknown resistance.
PS
Q= R

S.Q.
Or Rt = P.

Subsitituting value of Rt, t can be determined
(Resistance R is kept at a place whose temperature is to be measured).

iv) Thermocouple :Athermocouple is manufactured by joining two dissimilar metals belonging to thermo-
electric series. One is a higher order element & other is lower order element & two junctions are formed.
One junction is kept at ice point & other is kept at the point whose temperature is to be measured.

Due to difference in temperature, an emf is created & this emf is related to temperature as
e = at + bt2
(It works on seeback effect prinicple).

emf

↑

cold Hot
V) Pyrometer : Works on principle of radiation. Used to measure temperature of distant objects.

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THERMODYNAMICS

Ideal Gas and their behaviour .

* An ideal gas has no intermolecular forces of attraction or repulsion, doesnot change its phase during

thermodynamic process & obeys a set of common rules geverning change of properties.

Relationship between P, V & T given by.

f(P.V.T.) = 0 or P = f (V,T)

Ideal gas obeys the law

Pv = RT

Note : As P → 0, & T → 0 , the real gas approaches ideal gas behaviour.
∴ At reduced pressure & elevated temperataure the gas molecules lie far apart & molecular force of

attraction between them tends to be small.

* Gas deviates from ideal behaviour at high press, & low temp. i.e. at high densities.

Note :- For real gas, sp. heats are found to vary with temperature.

Perfect gas is a gas which is ideal & has const value of Cp & Cv.

>

Basic Gas laws := > isothermal
I) Boyles law :- At const temp
T1 T2 T3
Vα 1
P >

or PV = Const. T1 < T2 < T3.

* Valid only at very low tempertaure pressure or at moderately high temp V→

II) Charles law :-
V α T at const pressure

or V = C at P = C
T

OR
P α T (at const. volume)

or P = C
T at V = C

NOTE :- Gay lussac & Regnault found that at const. pressure change in volume of any perfect gas
corresponding to unit degree temp change is given by 1/2 73 of its volume at O0C,
Let Vo = volume of gas at O0 C,

V = Volume of gas at t0 C.
t

V =V t
t0 1+ 273

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MECHANICAL ENGINEERING

Ideal gas equation
PV = n R T.
R = Universal gas Const = 8.314 kj/mok
R = R.M.
R = Characterstic gas const (KJ/kg k).
M = Molecular mass
PV = n M R T

or PV = m R T [ ∴n M = m = mass of gas]

Cp - Cv = R

Cp =γ
Cv
8.314
Eg : - For O2’ Ro2 = 32 = 0.262 KJ/qk

8.314
For air R = 28.96= 0.287 KJ/qk

Vander Waals equation of gas

The concept of perfect gas Pv = RT based on following assumptions :-

1) There is little or no attraction between molecules of gas.

II) Volume occupied by molecules themselves is neglisibly small compared to volume of gas.

When pressure is very small & temperature very high, this assumption is valid and real

gas obeys ideal gas equation But as pressure increases & also volume of molecles become appreciable

compared to total gas volume. Vander waal introduced correction factor.

a (V - b) = RT
P + V2

* The coefficient a introduced to account for the existance of mutual attraction between molecules. The

term a/V2is called force of cohesion.

* The coefficient b was introduced to account for volumes of molecules & is known as co-volume.
Note :- Real gases conform more closely with vander waals equation of state. Not obeyed by real gas in
all ranges of press & temperature.

Compressibility Factor (Z)

Actual volume of the gas
Z = Volume predicted by ideal gas eqtion (PV = RT)

or

V = PV
Z = RT/P RT

Note :- i) Factor Z is dimensionless & becomes unity for ideal gas at all press & temp

ii) Magnitude of Z for a given gas gives an indication of extent of deviation of gas from ideal gas

behaviour.

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THERMODYNAMICS

Compressibility Chat :-

>

T1 < T2 < T3

T1
Z T2

T3

P >
I) As p → o , Z → 1

This shows gas behave more or less like perfect gas as pressure is reduced
II) If Z < 1 → actual density is greater than predicted by ideal gas equation

If Z > 1 → actual density is less than predicted by ideal gas equation

Specific heat at const. press (Cp) & Const. Volume CCv)
* Specific heat at const vol. (Cv) is defined as ratio of change in internal energy to change in temp at
const. volume

du
Cv = dt v = c
or du = Cvdt.
or dU = mCvdt.

* Cp is defined as ratio of change in enthalpy with respect to temperature at const. Pressure

dh
Cp = dt p = c
or dh = Cpdt.
or dH = mCpdt.

* Enthalpy of a gas is defined as

h = u + Pv [PV = flow work]

or H = U + pV or Cp = γ Cv.

Cp - Cv = R, Cp =γ
Cv

∴ R
Cv = γ - 1

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MECHANICAL ENGINEERING

Cp = γ Cv

Cp = γR
γ-1

γ = 1.67 - For manoatomic gaser. (Eg. He, Ar, Kr, Xe etc.)
γ = 1.4
γ = 1.33 - For diatomic gaser. (Eg. H2, O2, N2 ect.)
- For polyatemic gases or triatemic gases (Eg. SO2, NO2, ect.)

R
If R = M ,

R

Cv = MCv = γ - 1,

Cp = MCp = Cp) mas = γR
γ-1

Cv & Cp are molar sp. heats at const. Vol. & Const. Pressure resp.

(KJ/mo.k)

Note:- Value of γ depends only on molecular structure of gas i.e. whether gas is monoatomic, diaatomic

or polyatomic.

* Value of cp & Cv dependes only on γ & R i.e. number of atoms in molecule & molecular weight of gas.

They one independent of temp or press of gas.

Note : For real gas increases spand Cv with temperature but the ratio Cp - Cv = R is constant .
* γ = Cp / Cv decreases with temperature for real gas.

< }CP Real gas

CV

CP, CV, γ }CP idal gas

CV

γ − ideal gas
γ − Real gas

T→ <

Degrees of Freedom.
The number of independent quantities, which must be known for describing comptelely state of motion of
body is called “Degree of Freedom”

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THERMODYNAMICS

1) Monoatomic gas : - (3 DOF)

2) Diaatomic gas :- (5 DOF)

3) Triatomic gas :- (6 DOF)

Note :- Cp = γ1 + 2 (Here n = DOF)
Cv n

************

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not what someone else wants for you”

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MECHANICAL ENGINEERING

2 WORK AND HEAT TRANSFER

Definition : Work is said to be done if sole effect external to the system can be reduced to raising of
weight.
* Work done by the system is positive and when the work is done on a system, it is taken to be negative.
* Work is a path function and inexact or imperfect differential.
* For firctionless quasistatic (reversible) process work done is calculated by

W = ∫pdV >
>
P W = ∫pdV
>>
>

V
* Area of PV curve, when projected on volume axis gives us closed system or non-flow work.

NOTE :-

* For point function, total change in property in the cycle is zero.

∫ dV = 0, ∫ dp = 0, ∫ dT = 0 ∫ dU = 0, ∫ dh = 0,

* For path fuction, net change in the cycle may be non-zero.
w = ∫ pdV ≠ 0, ∫ dQ ≠ 0

Work Done in Various Reversible Processes

n =γ n=∞

P n =1

>Compression n =0
n =0
> Expansion

n =1

PVn = C n=γ
n=∞
>
V

(i) Constant pressure (isobaric) w 1- 2 = p(v2 - v1) = nR (T2 - T1)
(ii) Constant volume (isochoric) w 1-2 = 0

(iii) Constant Temperature (isothermal) w = p V ln p1 = nRT ln p1 = nRT ln v2 = p1 V1 ln V2
1-2 1 1 p2 p2 v1 V1

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THERMODYNAMICS

(iv) Abiabatic (isentropic) (PVγ = C) w 1-2 = (p1v1 - p2v2) = nR (T1 - T2)
( γ - 1)
n = No. of moles γ−1
γ = Adiabatic constant
= p1v1 p γ−1/γ
( γ - 1) 1- 2

p
1

PV -P V
11 2 2

* forPolytropic process work done =
n -1

where process is given as Pvn = Constant
n = polytropic index

Slope of Isothermal curve on PV Plane

Slope = dp = -P
dv V

Slope of adiabadic Curve on PV Plane

dp = - γP
dv V

Slope of adiabatic Curve on T-V Plane
TV γ − 1 = C

dT = (1 − γ ) T
dV V

Slope of adiabatic curve on T - P Plane

dT = 1−γ T
dV 1 V

Slope of isothermal curve on ln P Vs
Slope = m = -1

Slope of adiabatic curve on ln P - ln plane
m=−γ

Slope of adiabatic curve on ln T - ln V plane
m=1−γ

Slope of adiabatic curve on ln T VS ln P plane

γ−1
m= 1

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Indicator diagram

Mean Effective Pressure (Pm)

Pm = ad × k
ld

ld → length of the p-v diagram along γ-axis
ad → area of the p-v diagram
k → Spring constant (N/cm2 x cm travel)
W.D. in one engine cycle = Pm.A.L.
W.D./min = Pm.LAN (for 2 stroke)

W.D./min = Pm.LAN (for 4 stroke)
2

* The power developed inside the cylinder of engine is called Indicated power.

Indicated Power (for two stroke engine)
IP = pm (LAN)k [ Here K = No. of cylinders ]
60

Indicated Power (for four stroke engine)
IP = pm (LAN)k
2 × 60

Brake power : The power available at crank shaft is called Brake power (B.P.) or shaft power which is less
than Indicated power.

(BP) = 2πNT
60

Mechanical efficiency
(η mech) = BP
IP

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THERMODYNAMICS

L → stroke of the piston

A → cross - sectional area of the cylinder π D2
N → r.p.m. of the crank shaft 4

K → number of cylinders

T → Torque on the crank shaft

Note : If working fluid is made to work on both sides of piston, it is said to be Double acting engine Work

developed is twice that of single acting

ηIndicated Thermal

η Indicated power in watt
ith = mf x C.V.

mf = Mass of fuel supplied

C.V. = Lower calorific value in kJ/kg

ηBrakeThermal bth

η B.P. (watt)
bth = mf x c.v.
ηbth
∴ ηith = B.P. ηme [ ηme = Mechanical Efficiency]
IP

∴ ηbth = ηm x ηith

Specific fuel consumption : Mass of fuel require to be supplied to an engine to develop unit kw power/hr.

Sfc = mf kg/kwh
BP

Ideal gas equation for various processes

(a) Constant volume process = P1 = T1
P2 T2

(b) Constant pressure process = v1 = T1
P1vv12 = TP22 v2
(c) Isothermal process =

(d) Adiabatic process = T2 = P2 γ−1 v γ−1
γ 1

T1 P1 v2

(e) Polytropic process = T2 = P2 n− 1 v n−1
n 1

T1 P1 v2

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MECHANICAL ENGINEERING

* Flow work : The flow work in an open system represents the energy transferred across the system
boundary as a result of the energy imparted to the fluid by a pump, blower or compressor to make the fluid
flow across the control volume. It is analogous to displacement work.
Flow work per unit mass

Wflow = pvdm

Flow work per unit mass = p.v. ( p → pressure , v → specific volume )

Free Expansion :
* Expansion of a gas against vacuum is called free expansion.
* There is no work transfer involved in free expansion.

HEAT TRANSFER

* Heat is a form of energy that is transferred across a boundary by virtue of temperature difference.
3 means :- conduction, convection and Radiation.
* Direction of heat taken from high temperature to low temperature.
* Heat flow into the system is taken as possitive & heat flow out of system is taken as negative.
* System does not has heat, it has energy when it flows it is converted to heat or work.
* Heat does not always cause temperature rise (Latent heat). When temperature rise of system occurs it
may not be due to heat transfer. Temperature rise may be caused due to work transfer also.
* Heat & work are boundary phenomenon. They come into existance only when they cross system’s boundary.
* They are transient in nature.( lasting only for short time).

* Heat transfer in polytropic process = γ-n x W.D.
γ-1

* Heat like work is also a path function so is inexact or imperfect differential.

* At constant volume Q = mCvdt

Cv = Specific heat at constant volume
* At constant pressure

Q = mCpdt
Cp = Specific heat at constant pressure

Cp = γ
C

v

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THERMODYNAMICS

Latent Heat :-
* The latent heat of fusion is the amount of heat transferred to melt unit mass of solid into liquid or to
freeze unit mass of liquid to solid at a constant pressure and temperature.

QF = ml
F

lF = Latent heat of fusion

* The latent heat of vapourization is the amount of heat transferred to vaporize unit mass of liquid into

vapour or condense unit mass of vapour into liquid at a constant pressure and temperature.

QV = m l V
lV = Latent heat of vapourization
* The latent heat of sublimation is the amount of heat transferred to convert unit mass of solid to vapour

directly or vice-versa.

* The latent heat of fusion is not much affected by pressure whereas latent heat of vapourization is highly

sensitive to pressure.

* Heat required (Q) for transforming a solid at temperature T1 to gas at temperature T2 where freezing
temperature is Tf and boiling temperature is Tv.
Q = Q1 + Q2 + Q3 + Q4 + Q5
Q1 = Heat required to raise the temprature of solid from T1 to Tf

= mc1 (Tf - T1)
Q2 = Heat required for phase change from solid to liquid = ml f
Q3 = Heat required to raise the temperature of liquid from Tf to Tv

= mc2 (Tv - Tf)
Q4 = Heat required for phase change from liquid to vapour

= m.l v
Q5 = Heat required to raise the temperature of vapour from Tv to T2

= mc3 (T2 - TV)
c1, c2, c3 are specific heat in solid, liquid and gasesous phase respectively.

lF lv are latent heat of fusion and vapouization respectively.

*******************

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MECHANICAL ENGINEERING

3 FIRST LAW OF THERMODYNAMICS

THEORY
* First law states the principle of conservation of energy.
* In cyclic process

∫ ∫dw = j dQ
j → joules equivalent
j = 1 when, Q and w are in joule.
j = 1/4.18 cal/joule
* System undergoging change of state

Q = ∆E + W
Q = Heat transfer
W = Work transfer
∆E = Change in Total energy

* Energy is a point fuction and a property of the system. Energy is an extensive property while specific

energy is an intensive property.

* Total energy of system is given as

E = Ek + Ep + U
Where,

Ek = Kinetic energy of system
Ep = Potential energy of system
U = Internal energy of system

Note : The internal energy depends only on temperature. The enthalpy also depends only on temperature.

* Heat transfer at constant volume

QV = (∆u)V = ∫T 2

CvdT

T1

Cv = specific heat at constant volume

* Heat transfer at constant volume increases the internal energy of the system.

* Enthalpy is given as

H = U + PV

or specific enthalpy

h = u + pv

* Heat transfer at constant pressure

Qp = (∆h) = ∫TT2 CpdT
1

Cp = Specific heat at constant pressure

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THERMODYNAMICS

* Heat transfer at constant pressure increases the enthalpy of the system.
* For an isolated system
dQ = 0, dw = 0 & dE = 0
* Energy of an isolated system is always constant.
* PMM1 - There can be no machine which would continuously supply mechanical work without some
other form of energy appearing simultaneously. Such a fictitious machine is called perpetual motion machine
of first kind (PMM-1)
* A PMM-1 is thus impossible according to I law.

STEADY FLOW PROCESS
* In a flow if thermodynamic properties do not change with time at different locations, the process is
steady flow process.
* In steady flow process there is no accumulation of mass or energy in control volume i.e. conservation
of mass and energy occurs.

1 dQ 2 Flow out Let ,
dτ A1, A2 → cross-section of stream, m2
flow in w2 w1, w2 → mass flow rate, kg/s
w1 Steady flow device 2 Shaft p1, p2 → pressure, (absolute), kN/m2
v1, v2, → specific volume, m3/kg
1 Control volume dwx u1, u2 → specific internal energy, K/kg
dτ V1, V2, → velocity, m/s
Z1 Z1, Z2 → elevation above an arbitrary datum, m
C. S.

Z2

Datum

11223344556677889900112233445566778899001122334455667788990011221122334455667788990011

Steady flow process

dQ → net rate of heat transfer through the control sufrace, KJ/s
dτ

dW → net rate of shaft work through the control sufrace, KJ/s
dτ

* By conservation of mass

w1 = w2

A1. V1 = A2.V2
v1 v2

w1, w2 → mass flow rate entering and leaving the control volume

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MECHANICAL ENGINEERING

Conservation of energy (Steady flow energy equation SFEE)

* Per unit time basis

w1 h1 + v12 + Z1g dQ w2 h2 + v22 + Z2g dW
2000 1000 + dτ = 2000 1000 + dτ

* Per unit mass basis

h1 v12 + Z1g dQ = h2 v2 + Z2g dW
2 + dm 2 + dm

2

* For one fluid stream, SFEE (per unit time) is used.
* For more than one fluid stream, SFEE (per unit mass) is used.

Nozzle and Diffuser

dQ = 0, dwx = 0, z1 = z2, V1 = 0

V2 = 2000 (h1 - h2) m/s for nozzle

V1 = 2000 (h2 - h1) m/s for diffuser

where, h is in kJ.
* A nozzle is a device which increases the velocity or K.E. of fluid at the expense of its pressure drop.
* A diffuser is a device which increases the pressure of fluid at the expense of K.E.

Fig. : Steady Flow Process Involving Two Fluid Streams at the Inlet and Exit of the nozzle

Throttling Device

* When a fluid flow through a narrow passage like an orifice, partially opened valve, there is an
appreciable drop in pressure. The process is throtting process.

* Throtting is an isenthalpic process.

dQ
dm = 0

dw
x =0

dm

z1 = z2, V1, V2 are negligible

h1 = h2 Fig. : Flow Through a Valve

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THERMODYNAMICS

Turbine and compressor

dQ
dm = 0, z1 = z2, V1, V2 are negligible

dWx = (h1 - h2) for turbine
dm

dWx = (h2 - h1) for compressor Fig. : Flow Through a Turbine
dm

Heat Exchanger

dQ
=0

dt

dw
=0

dt

z1 = z2, V1, V2 are negligible Fig. : Steam Condenser
wCh1 + wsh2 = wch3 + wsh4
ws(h2 - h4) = wc(h3 - h1)

Note : Comparison between SFEE and Bernoulli’s eq. Bernoulli’s eq. is a special and limiting case of SFEE

while the Bernoulli’s eq. given as
PV

+ + Z = Constant
pg 2 g

applies only to frictionless, incompressible fluids SFEE can be applied to viscous and compressible fluid as
well.

*****************

“Someone’s sitting in the shade today because someone
planted a tree a long time ago”

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MECHANICAL ENGINEERING

4 SECOND LAW OF THERMODYNAMICS

THEORY

* Second law of thermodynamics is a directional law which tells the feasibility of any thermodynamic
process.
* Work is said to be high grade energy and heat is a low grade energy. The complete conversion of low
grade energy into high grade energy in a cycle is impossible while the complete conversion of high grade
energy into low grade energy is possible.
* A heat engine works on a thermodynamic cycle in which there in net heat transfer to the system and a
net work transfer from the system.

CHE = Cyclic heat engine
MER= Mechanical energy resrvoir
* Boiler, turbine, condenser and pump, all four together consitute a heat engine.

Thermal Efficiency

η = Wnet = Q1 - Q2 = 1 - Q2
th Q1 Q1 Q1

where

Q = heat supplied
1

Q2 = heat rejected
Wnet = Qnet (First law of thermodynamics)

* A thermal energy reservoir (TER) is defined as a large body of infinite heat capacity which is capable of
absorbing or rejecting an unlimited quantity of heat without any appreciable changes in its thermodynamic
properties.

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THERMODYNAMICS

Second Law of Thermodynamics
* Kelvin Palnck Statement : It is impossible for a heat engine to produce net work in a complete cycle if
it exchanges heat only with bodies at a single fixed temperature such fictitious machine is called PMM-2.
* PMM2 is impossible according to Kelvin Plank statement.
* Clausius Statement: It is impossible to construct a device which operating in a cycle will produce no
effect other than the tranfer fo heat from cooler body to a hotter body.
* Heat always flow from a body at higher temperature to a body at lower temperature but it can’t flow in
reverse direction spontaneously. Some work must be done to achieve this.

Refrigerator & Heat Pump

Coefficient of Performance (COP)
* COP = desired effect

work input

* (COP)ref = Q2 Q2
wnet Q1 - Q2
=

(COP)H.P. = Q1 = Q1
Wnet Q1 - Q2

(COP) H.P. = (COP)ref + 1

Note :
* The basic cycle in refrigerator and heat pump is same. But the main purpose of refrigerater is to

maintain temperature which is less than that of surrounding while in heat pump, the main purpose is to
maintain temperature which is more than that of surrounding.

* PMM-3: Continual motion of a movable device in complete absence of friction is known as
PMM-3,

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MECHANICAL ENGINEERING

Reversibility and lrreversibility : Any process, when reversed, if doesn’t leave any effect on system
and surrounding, is called reversible process.
All spontaneous processes are irreversible process.

Causes of Irreveribilities :

1) Irreversibility due to lack of Equillibrium :-

a) Heat transfer through a finite temperature difference

b) Lack of pressure equillibrium within the interior of the system or between the system and the sur

rounding

c) Free expansion is irreversible 111111111111111222222222222222333333333333333444444444444444555555555555555666666666666666111111111222222222777777777777777333333333888888888888888444444444A999999999999999555555555>666666666000000000000000777777777G111111111111111888888888999999999222222222222222a333333333333333s444444444444444555555555555555666666666666666777777777777777888888888888888B999999999999999000000000000000111111111111111>222222222222222V333333333333333a444444444444444c555555555555555c666666666666666u777777777777777m

> > insulatiot

diaphram

2) Irreversitbility due to dissipative effect :-
a) Friction
b) Paddle wheel work transfer
c) Transfer of electricity through resistor.

Note :
A process will be reversible when it is performed in such a way that the system is at all time infinitisimally near
state of thermodynamic equillibrium & in absence of dissipative effect of any form Reversible process are
purely ideal, limiting cases of actual process.

Carnot Cycle:

* Carnot cycle is reversible cycle.
* It consist of 4 processes

i) Reversible isothermal heat addition process (1-2)
ii) Reversible adiabatic expension process (2-3)
iii) Reversible isothermal heat rejection process (3-4)
iv) Reversible adiabatic compression process (4-1)

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THERMODYNAMICS

> 1 2
> >

T > >

>
43

>

S>

Fig. : T-s diagram for carnot cycle>

adiabatic cover111111111111111222222222222222333333333333333S444444444444444o555555555555555u666666666666666r777777777777777c111111111eQ222222222t11333333333444444444555555555111111111111111‘‘`’‘‘‘‘222222222222222’’’333333333333333``````>`444444444444444’’’’’’``’555555555555555d’’`’’i`666666666666666’’a````t777777777777777’‘‘h````e888888888888888`’’r’’’’999999999999999’e’’``’m’``000000000000000``‘‘’’i`111111111111111c``’’``’222222222222222c’’o````````333333333333333v444444444444444er555555555555555666666666666666777777777777777888888888888888999999999999999000000000000000111111111111111222222222222222333333333333333444444444444444>555555555555555666666666666666777777777777777>WWpE>

Q2

Sink t2

Efficiency of Carnot cycle

η = 1 - T2 or η= T1 - T2
rev. T1 T1

Note :

* In carnot cycle, the compressive or back work is very high. Net work is less, work ratio is less, mean

effective pressure is less but η is very high.
th

Work ratio = Wnet
Expansion work

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MECHANICAL ENGINEERING

* The η of carnot cycle is independent of working fluid & depends only upon temperature of source & sink.
* If T2 = 0, η = 100%. This means absence of heat sink, which is violation of kelvin planck statement of II
law
* η α (T1 - T2) Higher the temperature difference higher is the η.
* η increases as T1 increses or T2 decreases
* During isothermal heat exchange, head of engine should act as perfect conductor. During adiabatic expension
head of engine should act as perfect insulator or which is practically impossible. So engine can have η .
cartot η .

* Isothermal process is very slow process. Adiabatic process is very fast process. For carnot cycle to
happen, the engine should run slowly & fast alternatively & so achieveing carnot η is impossible.

T1 Wnet = Q1 - Q2
Q1
>> W= Q1 - Q2 η= Q1 - Q2 = 1 - Q2
E > Q1 Q1
Q2
Also η = 1 - T2
T2 T1

∴ Q2 = T2 for reversible engine
Q1 T1

Also η = W = W
Q1 Q2 + W

or ⇒ Q2 = 1−η
W η

Ques. : What should be done to improve the η of control cycle.

Sol : η = 1 - T2
T1

To improve η , either increase T1 or decrease T2 .

dη T2 (i)
d T1 T2 = c = T12

dη - 1 T1 ( ii )
T1 = c = = T12
d T2 T1

∴T1 > T2

∴ Slope dη T1 = c > dη
d T1 T2 = c
dT
2

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THERMODYNAMICS

So, changing lower temperature will have great improvement in η when compared to changing high
temperature.

Note : Lowering sink temperature has higher improvement in η when compared to raising source
temperature.

Carnot’s Theorem : It states that of all engines running between given constant temperature source & given
constant temperature sink, none has higher η than a reversible engine.

Corllary of Carnot theorem : The η of all reversible heat engines operating between same temperature
level is same.

COP of reversible (carnot) refrigerater & Heat pump :-

COP ) ref= Q T2
2= T1 - T2

Q1 - Q2

⇒ T1 = 1 + 1
T2 Cop.

COP )HP. = Q1 = T1
Q1 - Q2 T1 - T2

⇒ T2 = 1 - 1
T Cop.

1

Note : In refrigerator changing lower temperature has greater improvement in COP when compared to
changing higher temperature.

COUPLED ENGINES T
For equal work output 1

>>E1 Q1 W >= Q -Q
1

Q

⇒ T= T +T Arithemetic mean T
12
E2 Q W2>= Q1 - Q2
2
Q2
For equal Efficiency : T2>

⇒ T = T1 + T2 Geometric mean
Note :

(i) For equal work output condition the intermediate temperature is arithmetic mean of two temperature.

(ii) For equal efficiency intermediate temperature is geometric mean of two temperature.

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MECHANICAL ENGINEERING

Over all efficiency of coupled cycle :
⇒ (1 - η0) = (1 - η1) (1 - η2)

⇒ η = η + η - η η
0 1 2 1 2

Note :
(i) The overall efficiency of a coupled cylce is greater than individual efficiency of a cycle
(ii) I law is a quantitative law, II law is qualititive law. It imposes a limitation on the amount of heat that can be
converted to work. II law is directional law.
Note : If enough engines are placed in series to make the total work output = Q then by I law heat rejected
from last engine will be zero. By II law, however, the operation of a cyclic heat engine with zero heat
rejection cannot be achieved Temperature of heat rejection also approaches zero as a limit.

Third law of Thermodynamics :-
It is impossible by any procedure, no matter how idealized to reduce any system to the absolute zero

temperature in a finite number of operations.

Coupled Regrigerator

T T2 T1
C1 = T - T T - T2
, C2 = >> Q1 W= Q -Q
1 1
R1 >
Q

or T

T 1 T 1 Q W2 = Q1 - Q
1 C1 T2 C
1+ , 1+ R2 >
T 2 Q2
= = >

T2

For equal work input

T = T1 + T2 arithmetic mean
T2

For equal COP’s

T = T1 T2 Geometric mean

Overall COP of coupled cycle :

T1

R>W

T
2

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THERMODYNAMICS

⇒ 11 1
1+ = 1+ 1+
C0 C1
C2

⇒ 1 11 1
= + C2 + C1C2
C0 C1

Note : Overall COP of a coupled refrigerator is less than the individual COPs

For 3 refrigerators :

11 1 1
1+ C0 = 1+ 1+ 1+
C1
C2 C3

Heat Pump in Series :

T1

Q1>> W= Q -Q
1
HP1 >
Q

T

Q W2 = Q - Q2

HP2 >
Q2
>

T2

for equal work input
T = T1 - T2
2

for equal COP

T = T1 T2

Overall COP of coupled HP :

1 1 1 1 1 1 1
C0 + C2 C1C2 1- = 1- 1 - C2
= - C1
C1 C0

***************************

“When you’re out of quality, you’re out of business”

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MECHANICAL ENGINEERING

5 ENTROPY

Note :

* Two reversible adiabatic paths cannot intersect each other which violates the Kelvin-Plank’s

statement.

* Calusius Theorem: Any reversible path may be substituted by a reversible zigzag path between the

same end states, consisting of a reversible adiabatic followed by a reversible isotherm and then by reversible

adiabatic such that the heat transferred during the isothermal process is the same as that transferred during

the original process.

* Clausius in inequality ∫ dQ
T <0

∫ dQ - Reversible cycle
T =0

∫ dQ - Irreversible cycle
T <0

∫ dQ - Impossible cycle
T >0

Note : dQ
T
* For a reversible process quantity dosen’t depend on the path and hence

∫ f dQ = 0 is a point fuction f and is a property of the sytem.
iT

This property is known as ENTROPY.

ENTROPY
* It is the degree of molecular disorderness. Greater the entropy, greater is the disorderness and less is the
efficiency.

i.e. dQ for a reversible process.
T = dS

For irreversible process

dS > dQ
T

Thus dS > dQ
T

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THERMODYNAMICS

For isolated system,
dQ = 0

∴ ds )iso > 0

i.e dS) = 0 - for reversible process
iso
dS > 0 - for irrversible process
iso

** Thus Entropy of an isolated system can never decrease. It always increase or remain constant.
only for reversible process. This is known as Principle of increase of Entropy, or simply Entropy
Principle

Universe = System + Surrounding
Universe is isolated system.

∴ dS univ > 0
or dS sys + dS surrounding > 0

Note :-
* For a system only, the change in entropy can be positive , negative or zero. But for universe,

change in entropy is always positive
* Net effect of an irreversible process is an entropy increase of whole universe.
* The entropy increase of an isolated system is a measure of the extent of irreversibility of process

undergone by the system.
* The entropy of an isolated system always increase & becomes maximum at the state of

equillibrium. When the system is at equillibrium, any concievable change in entropy would be
zero.
* If the system is taken from i to f. by an irreversible path since entropy is a Point or state function
and
entropy change is independent of path followed, the irrversible path is to be replaced by
reversible path for evaluation of entropy change in irreversible process.

Sf - Si = ∫f dQrev. = ∆S) irreversible path
1 T

Temp - Entropy plot

>

dQ i > Qrev. = ∫ f TdS,
rev f
↑ >
dS = T Τ

dQ rev. = TdS.
Qrev. = ∫ f TdS.

i

S ↑
Note: - Area under T - S diagram gives heat transfer in the process.

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MECHANICAL ENGINEERING

For adiabatic dQrev = 0
∴ dS = 0

or s = Constant
Note : A reversible adiabatic process is an Isentrepic process

Reasons for Increase of Entropy :
The Entropy of any closed system can increase in two ways : -

a) By heat interaction in which there is Entropy transfer:- The heat inflow increase the molecular
disorder, there is flow or disorder along with heat. This is known as Entropy transfer:

b) By internal irreversibilities or dissipotive effects which is known as entropy production or entropy
generation.
dS = deS + diS
⇒ dS = dQ + diS
T

deS = Entropy increase due to external heat interaction
diS = Entropy increase due to irreversibility

∴ dS > dQ rev
T

or diS > 0

{ {If the entropy decrease of the system due to heat loss to surrounding −∫ dQ
T

is equal to entropy increase of the system due to internal irreversibility such as friction etc. { ∫ dis} In this
case entropy of system before & after the process will remain same { ∫ ds = 0}

Note : -
* An isentropic process need not be adiabatic or reversible. But if isentropic process is reversible, it must
be adiabatic. Also if isentropic process is adiabatic, it must be reversible. An adiabatic process need not
be isentropic, since entropy can also increase due to friction. But if process is adiabatic & reversible it
must be isentropic.

Note : -
* There is entropy change due to heat transfer but there is no entropy change associated with work
transfer.

If work is dissipated adiabatically into system energy of the system increase, there is an entropy
increase in the system but there is no such entropy transfer into it.

Note :-
* The II law states that, in general, any thernodynamic process is accompnied by entropy generation.
* The entropy generation also depends on path the system follows. Sgen is ∴ not a thermodynamic
property.
* The amount of entropy generation quantifies the intrinsic irreversibility of process.

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THERMODYNAMICS

By II law,

dQ rev = TdS.
By I law, for closed non flow system.

dQ = dU + PdV

TdS = dU + PdV

Now, H = U + pV
or dH = dU + PdV + VdP.
dH = TdS + VdP.

or TdS = dH - Vdp

Note :-
a) dQ = dE + dW - For any process reversible or irrversible & for any system
b) dQ = dU + dW - For any process undergone by closed stationary system
c) dQ = dU + pdV - For closed system when only pdV work is present

True only for reversible (Quasi static) process.
d) dQ = TdS - true only for reversible process
e) TdS = dU + pdV - for any process reversible or irrversible undergone by closed system

it is a relation among property which are independent of path
f) TdS = dH - VdP - This equation related only the properties of closed system. There is no path

function term in the equation. Hence the equation holds good for any process.

Note : -
*Anatural process which is inherently irreversible is indicated by a dotted line connecting initial & final
states.

The entropy change of a system between two identifiable equation states is same whether the
intervening process is reversible or change of state is irreversible.
*A process always occurs in such a direction so as to cause an increase in entropy of universe.

* An irreversible process always tends to take the system (isolated) to a state of greater disorder.
The entropy of a sytem is a measure of the degree of molecular disorder existing in the system

when heat is imparted to a system, the disorderly motion of molecules increase & so entropy of system
increase. The reverse occurs when heat is removed from the system.

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MECHANICAL ENGINEERING

* Entropy change of an ideal gas

∆S = Cv ln T2 + R ln V2
T1 V1

1) ForaclsoonSst2a-ntSv1o=luCmplenprTTo21ce+ssR ln P1
P2

∆S = Cv ln T2 = Cv ln P2
T1 P1

2) For constant press process

∆S = Cp ln T2 = Cp ln V
T1 2

V1

3) Isothermal process

∆S = R ln V2 = R ln P1
V1 P2

4) Reversible adiabatic process

dQ = 0

ds = 0

5) Polytropic process

dQ γ − n dw γ − n pdv
ds = T = γ − 1 T = γ − 1 T

ds = γ−n R. dv
γ−1 v

or ds = γ−n Rln v2
γ−1 v1

γ−n 1
γ−1
or ds = Rln T2 1-n
T
1

ds = (1- γ−n 1) Rln T2
n) (γ − T1

or ∴
P1V1n = P2V2n
1
1 γ−n P1 n γ−n
ds = γ − 1 P2 ds = γ − 1
or V2 = P1 n , Rln , x R ln P1
V1 P2 nP

2

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THERMODYNAMICS

* Entropy change of a reservoir (infinite body)

dS = dQ for sink (heat addition)
+
T

dQ
= - T For source (heat rejection)

* Entropy change for a finite body (a finite system)
dQ = mCdT

or TdS = mCdT

∫ dS = mC ∫ dT
T

S2 - S1 = mCln T2
T1

Entropy change during phase change

dQ mL
dS = T =T

L = Latent heat

T = Phase change temperature

* Reversible adiabatic work in steady flow process (open system work)

SFEE dQ = dh - VdV + gdZ + dWx
dQ = TdS = dh - VdP

Wx = -∫ 2 VdP neglecting KE and PE
1

Application of entropy principle
* Transfer of heat trough a finite Temperature difference

(∆S) suniv =- Q + Q =Q T1 - T2
T1 T2 T1 T2

∴ ∴ (∆S) suniv > 0
T1 > T2

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MECHANICAL ENGINEERING

* Mixing of two fluids

final temp (t ) = m1c1t1 + m2c2t2
f m1c1+m2c2

(∆S)univ = m1c1 ln Tf + m2c2 ln Tf
T1 T2

* Maximum work obtainable from two finite identical bodies at T1 and T2.

Final temperature of the two bodies (Tf)

____
Tf = √ T1T2

__ __
Wmax = Cp (√ T1 - √ T2)2

* T_h_e_final temperature of two bodies initially at T1 & T2, can range from (T1 + T2)/2 with no delivery of
work √ T1T2 with maximum delivery of work.

* Maximum work obtainable with finite body & TER

Wmax = Cp (T - To)- To ln T
To

T = Temp. of body

To = Temp of TER
* Adiabatic dissipation of work

(∆S) univ = Cp ln Tf , (∆S)surr = 0
T1

(∆S) univ = Cp In T
f

T1

III Law of Thermodynamics :
“The entropy of a pure crystaline substance at absolute zero temperature is zero”

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THERMODYNAMICS

Nernst symonds statement of III Law :-
“Entropy of any system vanishes at absolute zero temperature”

lt ∆ S = 0
∆t→0

Entropy change due to mixing process :-

∆ Smix = - nR [Σ xi ln xi]
Where

n = No. of moles = n1 + n2
R = Universal constant
Xi = Mole fraction of ith gas
Note : If same gases are mixed, there is no entropy change due to mixing process but if different gases ae
mixed, there is entropy change due to mixing.

Mole fractions Xi = n1 X2 = n2 n1 n2
n1 + n2 n1 + n2 p1 p2
v1 v2
∆ SI = n1 - Cv ln Tf + R ln Vf T1 T2
T1 V1
(n1 + n2)
∆S = n - Cv ln Tf + R ln Vf pf
II 2 T V2
vf Tf
2

∆ Smix = (n1 + n2) . R [ x1 ln xi + x2 ln x2 ]

Entropy change of system = ∆SI + ∆ S2 +∆ Smix

***************************

“Ones best success comes after their greatest disappointments”

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MECHANICAL ENGINEERING

6 AVAILABLE ENERGY, AVAILABILITY & IRREVERSIBILITY

__________________________________________________________

High grade energy Low grade energy
__________________________________________________________

Mechanical work, Heat derived from nuclear

Electrical Energy, fusion, nuclear fission, chemical

Water Power, Wind reaction, combustion.

Power, Tidal power

Kinetic Energy

___________________________________________________________

* The part of low grade energy which is available for conversion to high grade energy is referred to as
available energy, which is also known as exergy while the part of low grade energy which, according to
second law, must be rejected is called unavailable energy. Unavailable energy is known as anergy.
* The maximum work output that can be obtained from a certain heat input in a cyclic heat engine is
called the available energy (AE) and the minimum energy that has to be rejected to the sink in called as
unavailable energy (UE)

Q1 = AE + UE
or Q1 = Exergy + Anergy

Wmax = 1 - To Q1
T1

To = Minimum heat rejection temperature
T1 = Heat addition temperature

* In a cyclic Heat Engine (when heat is withddrawn at constant temperature)

η= T1 - T2 = W
T1 Q1

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THERMODYNAMICS

For Wmax T2 should be minimum (surrounding temperature T0)

W max = 1 - T0 Q1
T1

⇒ W max = Q1 - Q1 . T0
T1

⇒ A.E. = W max = Q1 - T0 ∆ S.

* Unavailable energy is h. eat rejected or the product of lowest temperature in the cycle and change in
entropy of the system during the process of heat supply.
* Unavalilable energy = heat rejected = To (S2 - S1)
* Decrease in available energy due to irreversible process when heat is transferred through a finite tem-
perature difference.

Decrease in A.E. = To (∆S’ - ∆S)
To : lowest feasible temperature in the process
∆S’ : entropy change due to irreversible heat transfer
∆S’ :entropy change due to reversible heat transfer

Note :

* Greater the temperature difference between the bodies, greater will be the percentage of unavailable

energy in total heat input.

* Available energy from a finite energy source:

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MECHANICAL ENGINEERING

For a hot gas, transferring heat to a working fluid working in a cycle

∫ ∫2 T0 mCpdT

AE = Q - To dS = mCpdT - To
1T T
2 2
W =max mv 1 mv 2

H1 - ToS1 + 2 + mgZ1 - H2 - ToS2 + 2 + mgZ2

or available energy = w max = Q1 - Q2

AE = mgCpg (T - To ) - To l n T
To

Note :

* Same amount of heat has more available energy when it is transferred from higher temperature body

than when it is transferred from lower temperature body.

* The degradation is more for energy loss at a higher temperature than that at a lower temperature.

* The first law states that the energy is always conserved quantity-wise while the second law emphasizes

that energy always degrades quality wise.

* Work done in all reversible processes between the same end states is the same for a closed system

exchanging energy form only one reservoir.

AVAILABILITY
TheAvailability of a given system is defined as the maximum useful which (total work - pdV which) that

is obtainable in a process in which the system comes to quit which its sorrounding.

Availability of Non flow process :-

- Q = (U0 - U1) + Wexp. Atm P.V.T.
∴ Wexp. = (U - U ) - Q.
>
10
Wexp.
Weng. = 1- T0 Q = Q - T0 (S1 - S0) >>
T1 Q

Wmax. = [ (U1 - U0) - Q ] + [ Q - T0 (S1 - S0)] > Weng

Wmax. = U1 - U0) - T0 (S1 - S0) T0 (S1 - S0)

Wnet = Wmax. - Wsurr

= (U1 - U0) - T0 (S1 - S0) - P0 (V0 - V1)
= (U1 + P0V1 - T0 S.) - (U0 + P0V0 - T0S0)
= Qi - Q2

Q = U + P0V - T0S is Non flowAvailability fn

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THERMODYNAMICS

Availability of Steady Flow System

> Outlet
Inlet
P1V1,T1S >

>> P0V0T0, S0

Q

> Weng

T0 (S1 - S0)

SFEE :-

V2 V02
U1 + P1V1 + 1 gz1 - Q = V0 + p0V0 + 2 + gZ0 + Ws.

2

⇒ Ws = ( H1 + mV12 + mgz1) - ( H0 + mV02 + mgz0) - Q
2 2

Weng = (1 - T0 ) Q.
T1

= Q - T0 (S1 - S0)

)Wu max = Ws + Weng.

= ( H1 - T0 S1 + mV12 + mgz1) - (H -T S + mV02 + mgz0)
2 0 00 2

B = H - T0S = Keenan function.

)Wmax u = (B + mV12 + mgz ) - ( B0 + mV02 + mgz0)
1 2 1 2

)Wmax u = ψ1 − ψ2

ψ = B1 + mV12 + mgz = steady flow availability fn.
2

When K.E. & P.E. neglected.

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MECHANICAL ENGINEERING

)Wu max = B1 - B0

)Wu max = (H1 - H0 ) - T0 (S1 - S0 )

HELMHOLTZ & GIBB’S FUNCION
for non flow reversible process
Wmax = (U1 - T0S1) - (U2 - T0S2)
Process is constant temperature

Replacing T0 by T1 & T2 (so the process is at constant temperature) we have
Wmax = (u1 - T1S1) - (u2 - T2S2)
= A1 - A2.
(U - TS) is called Hemlholtz function
The maximum work of the process is equal to decrease in Helmholtz function of the system. It’s useful-

ness is restricted to non flow process only.
For flow process :
Wmax = (U1 + P1V1 - T0S1) - (U2 + P2V2 - T0S2)
Replacing T by T & T

0 12

Wmax = (U1 + P1V1 - T1S1) - (U2 + P2V2 - T0S2)
= (H1 - T1S1) - (H2 - T2S2)
= G1 - G2

H - TS is Gibb’s function

Irreversibility and GOUY-STODOLA theorem
The actual work done by the system is always less than the idealized reversible work and the difference

between the two is called the irreversibility (I)
I = Wmax - W ≥ 0
= To (∆Ssys. + ∆Ssurr.) = T0∆Suniv. for both flow and non flow process

or I = T0 Sgen.

* The gouy-stodola theorem states that the rate of loss of available energy or exergy in a process in
proportional to the rate of entropy generation.
* Availability or exergy balance.

While energy is always conserved, exergy is the quality of energy which usually degrades.
Energy in -Energy out = 0
Exergy in - Exergy out = Exergy Destroyed.

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THERMODYNAMICS

Second Law Efficiency

ηΙΙ = Wact = Wact
Wmax A.E.

ηΙ = Wact = W x Wmax = ηΙ x η
Q1 Wmax Q carnot

1

ηΙ = ηΙΙ η
carnot

or Minimum exergy intake to perform the given task
ηΙΙ = Actual exergy intka to perfom the same task

ηΙΙ = Amin { A=Available Energy or Exergy }
A

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“The best way to predict the future, is to invent it”

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MECHANICAL ENGINEERING

7 PROPERTIES OF PURE SUBSTANCE

* A pure substance is a substance of constant chemical composition throughout its mass. It is one
component system. It may exist in one or more phases.

* A saturation state is a state from where a change of phase may occur without a change of pressure or
temperature.

* The line passing through all the saturated solid states is called the saturated solid line. The lines passing
through all the saturated liquid state with respect to solidification and vaporization respectively are
kown as saturated liquid lines and similarly saturated vapour lines are drawn.

* The saturated liquid line with respect to vapourization and saturated vapour lines meet at critical point.

* Critical Point :
(i) Critical Temperature:At critical temperature a liquid completely changes to vapour and vice-versa.

Also above critical temperature a vapour cannot be liquefied by any amount of pressure.
(ii) Critical Pressure:At critical temperature the minimun pressure required to transform a vapour to liquid

is called critical pressure.

Pc

>P

* Transformation of solid to vapour directly is called sublimation.

* Transformation of vapour to solid directly is called ablimation.

* Triple point is a line on P-V diagram and a point on P-T diagram where all the three phases liquid, solid

and gas exist. curve

Critical point

Vaporization curvve

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THERMODYNAMICS

* Following is the decreasing arrangement of substance triple point

(i) C2H2 > CH4 > C2H6
(ii) H2O > CO2 > NH3 > N2 > O2 > H2

* For water
* Critical pressure (PC) = 221.2 bar
* Critical temperature (tC) = 274.15 oC
* Critical volume (VC) = 0.00317 m3/kg.
* Triple point (P) = 4.58 mm Hg
* Triple point (T) 273.16 k

* The slope of vaporization and sublimation curve for all substance are positive. The slope of the fusion
curve for most substances is positive but for water, it is negative.

* Water expands upon freezing while volume of other substances decreases upon freezing.
* Mollier diagram (or h-s diagram)

* The slope of constant pressure lines is given as

∂h
∂s = T

* Dryness fraction : It indicates the mass fraction of vapour in a liquid vapour mixture.

Dryness fraction (x) = mv
mv + m l

* Dryness fration: It indicates the mass fraction of vapour in a liquid vapour mixture.

Mv = mass of vapour

Ml = mass of liquid
* For saturated liquid, x = 0
* For saturated vapour, x = 1
* Dryness quality lines originate from critical line in various diagrams (P-V, h-s diagram etc.)

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MECHANICAL ENGINEERING

* Various properties of pure substance based on dryness fraction.
v = (1 - x)vf + xvg = vf + xvfg
s = (1 - x)sf + xsg = sf + xsfg
h = (1 - x)hf + xhg = hf + xhfg

For given pressure and temperature

v, v v = specific volume of moist vapour, liquid, vapour
f, g

vfg = vg, - vf

h, hf, hg = apecific enthalpy of moist vapour, liquid, vapour

hfg = hg - hf

s, sf, sg = specific entropy of moist vapour, liquid vapour

sfg = sg, - sf
Note :-
* In between the liquid line wrt vapouization and saturated vapour line, there exist equilirium between

liquid and vapour during phase change. The energy or heat given to the system in this region in called
latent heat of vapourization.
* As the pressure decreases, this region is reduced and at critical point, latent heat of vapourization
becomes zero.
* At critical point and above, a liquid upon heating suddenly flashes into vapour and vapour while
cooling, condensed into liquid without any transition zone.
* A gas or a pure substance require two known properties (P,V,T,H etc.) to decribe it completely i.e. it
have two degrees of freedom.
* The difference between temperature of the superheated vapour and that of saturated vapour at the
same pressure is called degree of superheat.

Throttling Calorimeter
* It is used to measure dryness fraction of pure substance.
* Throtting is an irreversible and adiabatic but not isentropic.

∴ h1 = h2
∴ hfp1 + x1hfgp1 = h2

∴ x1 = h2 - h
fp1

Tfgp1

* It is very difficult to measure quality of two phase system (liquid + vapour) so it is throttled to bring it to
single phase that is vapour phase and then with the measurement of pressure and temperature other

properties of two phase system can be known by calculation.

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THERMODYNAMICS

8 AIR CYCLES

GAS POWER CYCLES :

The thermodynamic cycle working with air as the coking fluid is called anAir standard cycle and the
thermal efficienty of an air standard cycle is calledAir Standard Efficiency.

Thermal η = W.D. = Q1 - Q2
Heat i / p Q1

Heat Engine :-Adevice in which chemical energy of fuel is converted to heat energy which is further utilised
to produce mechanical work.
a) External Comburtion engines
b) Internal Comburtion engines

i) S.I. (Petrol Engines)
ii) C.I. (Diesel Engines)

Note : In both S.I. & C.I. engines working fluid is never restored to initial state. These engines execute only
mechanical cycle & not thermodynamic cycle. (In thermal cycle, working fluid undergoes changes in fluid
properties but eventually it returns to initial state)

The mechanical cycle executed by ICE can be convered to equivalent thermodynamics cycle if air is
taken as working fluid & addition & rejection of heat are due to process of heat transfer alone.

CARNOT CYCLE (REVERSIBLE CYCLE): Two reversible adiabatics and two reversible isotherms.

η = Wnet = 1 - Q2 = 1 - T2
Q1 Q T1
1

Note : Wc = W41 i.e. Large back work is big drawback of carnot gas cycle.

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MECHANICAL ENGINEERING

Stirling Cycle

η = 1 - T2
T1

Note : The efficiency of cycle is less than the efficiency of carnot cycle but regenerative stirling cycle has the
same efficiency as the carnot cycle.

Ericsson cycle

Two reversible isotherms & Two reversible isobars.

With ideal regeneraction η = 1 - T2
T1

Note : The efficiency of the Ericsson cycle is less than that of the carnot cycle but with regeneration Ericsson
and Stirling cycle efficiency is equal to carnot cycle.

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