Td Chapter 2

8/8/2019 Td Chapter 2

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As the time passes, heat is transferred from hot body to cold body. The hot body is cooled and the cold body is heated. After some time, thermal

equilibrium is attained. Both the bodies attain the same properties including electric resistance, length of alcohol or mercury column, e.m.f. generated,

intensity of radiation. When this happens, both the bodies or systems are said to have the equality of temperature or are at the same temperature. Bothbodies are also in thermal equilibrium.

2.1.3 Zeroth Law

Fig. 2.2 Three systems in thermal equilibrium

Fig. 2.2 Shows three system. System 1 and System 3 are brought in contact with each other. When thermal equilibrium is attained, both the systems have

the equity of temperature or are at the same temperatures. Now again systems 2 and 3 are brought in contact with each other, when thermal equilibrium is

attained, both the systems 2 and 3 have equity of temperatures or are at the same temperature. Now, finally if systems 1 and 2 are brought in contact with

each other, it will have also equality of temperature. This observation can be stated as zeroth law.

Zeroth law states that if two systems are in thermal equilibrium with a third system, then the two systems will also be in thermal equilibrium with

each other.


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Zeroth law of thermodynamics can be utilized for the measurement of temperature. If we consider thermometer a third system, and if the

thermometer is in thermal equilibrium with another system, the temperature of the another system will be equal to the temperature given by the

thermometer.

2.2 First Law of Thermodynamics, Internal Energy and Enthalpy

2.2.1 Introduction

Energy is a combination of two Greek words meaning capacity of doing work. First law of thermodynamics is nothing but law of conservation of energy.

Energy can neither be created nor destroyed. It can be converted from one form to another form. From first law, precise definition of energy is derived.

Energy is an abstract quantity. It cannot be perceived with eyes. In mechanics, potential energy, kinetic energy and work are considered. In

thermodynamics, kinetic energy, potential energy, work, heat, internal energy etc. are dealt with.

Potential energy is the energy possessed by a system by virtue of its elevation from a given datum.

It is given by:

P.E. = mgz, .(2.1)

where m is mass, g is acceleration due to gravity and z elevation of the system from the datum. The kinetic energy is the energy possessed by a system byvirtue of its velocity.

It is given by

K.E = mve2 ..(2.2)

where m is the mass and ve is the velocity of the system.

2.2.2 First law of Thermodynamics for a Closed System Executing a Cycle

The first law of thermodynamics for a closed system executing a cycle, states that the net heat supplied to the system is equal to net work done by

the system.

Mathematically,

= WQ ..(2.3)Units of Q and W as generally used in thermodynamics are KJ. Symbol denotes the cyclic integral.


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(a) Paddle rotating in water (b) P-V diagram for a cycle

Fig. 2.3 First law of thermodynamics for a cycle

Fig. 2.3 (a) represents a insulated container having water and a paddle inside it. The paddle can be rotated by a pulley and weight arrangement. Now,paddle is rotated and work is done on the water. Water temperature increases from t1 to t2 by the work, W12.

Insulation of container is removed and water is allowed to transfer heat to the surroundings. When temperature changes from t2 to t1, the original value,

heat transfer Q12 to the surroundings is calculated. It is observed that the work done on the system, W12 is equal to heat lost to surroundings, Q12, during the

cycle 1-2-1.

2.2.3 First Law of Thermodynamics for a closed system executing a process

Fig. 2.4 shows a system undergoing two cycles 1-a-2-b-1 and 1-a-2-c-1


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Fig. 2.4 P-V diagram for two cycles

Cycle 1-a-2-b-1 consists of two processes a and b. Applying first law to the cycle

1-a-2-b-1, we get:

WWQQ

b

b

a

a

b

b

a

a

+=+1

2

2

1

1

2

2

1

.. (2.4)

Again applying first law of thermodynamics to the cycle 1-a-2-c-1, we obtain

WWQQ

c

c

a

a

c

c

a

a

+=+1

2

2

1

1

2

2

1

.. (2.5)

Subtracting Eq. (1.10) from Eq.(1.9), we get

=1

2

1

2

1

2

1

2

b

b

c

c

b

b

c

c

WWQQ

or WQWQc

c

b

b

= 1

2

1

2

)(

It is observed that (Q W) is same whether we follow path b or c. If we take other path d or e, same conclusions will be drawn. So it can be concluded

that the quantity WQ 2

1

is independent of path followed. It only depends on two states, hence a property. Q - W is denoted by dE. E is called

stored energy.


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Q - W = dE

or Q = dE + W ..(2.6)

For process 1-2, on integration Eq. (2.6) becomes

Q12 = E2-E1 + W12 ...(2.7)

2.23 Internal Energy and Stored Energy

Following are the important points which should be considered for studying and analyzing internal and stored energy.

(1) E is called stored energy, total energy or only energy.

(2) Energy is a property derived from first law of thermodynamics when applied to a closed system executing a process.

(3) E represents total energy i.e. sum of microscopic energy and macroscopic energy, which a system possess at a given state. Macroscopic typesinclude Kinetic and potential energy. Microscopic energy includes Kinetic energy, Chemical energy, nuclear energy , vibrational energy and

electronic energy of molecules. This microscopic energy is called internal energy. So, total energy is sum of kinetic energy, potential energy, work

of all types, heat and internal energy.

or E = U + KE + PE + Work + Heat

E = U + mv2 + mgz + W + Q (2.8)

(4) Law of conservation of energy is well known. For the universe or for an isolated system, Q = 0 and W = 0. First law of thermodynamics for a

closed system undergoing a process is simplified to:

dE = 0 or E1 = E2 ..(1.14)

Therefore, it can be stated that energy of the universe or an isolated system remains constant. This is called the law of conservation of energy. The lawis valid for all the processes whether reversible or irreversible.

2.2.4 Perpetual Motion Machine of First Kind, PMMFKA system or machine which produces work continuously without involving any expenditure of energy is called perpetual motion machine of first

kind. Such machine is not feasible as it violates the first law of thermodynamics.

2.2.5 Enthalpy

(1) Enthalpy H is defined as

H = U + p V ..(2.29)


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Enthalpy may be expressed as the sum of internal energy and product of pressure p and volume V.

Specific enthalpy h is given by

h = u + pvwhere u is specific internal energy, p is the pressure and v is the specific volume.

(2) Enthalpy is the sum and product of properties so property itself. H is an extensive property and h is an intensive property.

(3) H = U + pV

dH = dU + pdV + Vdp

From first law of thermodynamics

Q = dU + pdV

dH = Q + V dp .. (2.10)

If p is constant

dH = Q (2.11)

In a constant pressure process from state 1 to state 2, change in enthalpy is equal to heat supplied.

(4) At 00 C, enthalpy is taken to be zero.

(5) In an open system

H = U + pVpV is called the flow work. Flow work is due to pressure and moves the fluid in or out of a control volume.

(6) In throttling process, enthalpy remains constant.

2.2.6 First Law of Thermodynamics Applied to an open system or to a Flow Process

In a flow process, mass enters and leaves a specified volume. Attention is concentrated on a specified region in space. Here, specified volume is an

open system. Steam turbine and petrol engine are the examples of the open system. In a steam turbine, steam enters at high pressure and leaves at low

pressure. In a petrol engine, air and petrol enter and flue gases leave the engine. In steady flow process, mass and energy flow rates and properties

within the specified volume do not change with time.


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Fig. 2.5 First Law for an Open System

Fig. 2.5 displays an open system. Dotted lines show the control volume. At the inlet, force due to pressure p1A1 is moving a distance of l1 and the exit,

force due to pressure p2A2 is moving a distance of l2. Therefore, flow work at the entrance is p1A1l1 or p1V1 and at the exit it is p2A2l2 or p2V2. Now applyinglaw of conservation of energy to the control volume, we obtain

E1 + p1V1 + Q = E2 + p2V2 + W + (E2-E1)C.V. .(2.12)where

E1 = Stored energy entering the control volume

= m1 (u1 + ve2/2 + gz1)

E2 = Stored energy leaving the control volume

= m (u2 + ve2/2 + gz2)

(E2 E1)C.V. = Change in stored energy of the control volume.

m1 = Mass entering the control volume

m2 = Mass leaving the control volume

Q = Heat transferred to the control volumeW = Work done by the control volume on the surroundings.

Eq.(2.12) can be rearranged as

Q = (E2-E1)CV + W + m2 (h2 + ve22/2 + gz2) m1(h1 + ve1

2 / 2 + gz1) ..(2.13)

We know that, h1 = u1 + m1p1v1 and h2 = u2 + m2 p2v2For steady flow process, ( E2 E1)CV = 0, m1 = m2 = m

So above equation is reduced to

Q = W + m { (h2 + ve22 / 2 + gz2 ) (h1 + ve1

2 / 2 + gz1) } (2.14)

In differential form, above equation is simplified to


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Q = W + m { (h2 + ve22 / 2 + gz2 ) (h1 + ve1

2 / 2 + gz1) }..(2.15)

2.2.7 Application of First Law of Thermodynamics applied to a control volume or Applications of Steady Flow Equation.

We will apply 1st law of Thermodynamics to certain important systems.

(1) Steam Turbine

Fig. 2.6 Energy balance

in a steam turbine

Fig. 2.6 shows a steam

turbine in which high pressure steam with

enthalpy mh1 is entering

and low pressure steam of

enthalpy mh2 is leavingthe turbine. The work

done by the turbine is W and heat loss to the surroundings is negligible. From law of conservation of energy to the control volume:

Energy entering = Energy leaving or

m h1 = m h2 + W

W = m(h1 h2) .. (2.16)

Same value of work done will be obtained by applying steady flow equation to the control volume.

(2) Air Compressor


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Fig.2.7 Energy balance in an air compressor

Fig. 2.7 shows an air compressor. Low pressure air enters the control volume. The work is done on the compressor. High pressure air leaves the control

volume. From 1st law of thermodynamics, energy entering is equal to energy leaving the control volume, or

m h1 + W = m h2

or W = m(h2 h1) ..(2.17)

Same value of work done on the compressor will be obtained if we apply steady state energy equation to the control volume.

(3) Pump

Fig. 2.7 Energy balance in a water pump


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Fig. 2.7 shows a water pump. Water enters the control volume at low pressure and leaves at high pressure. Work is done on the system. Like forcompressor, work done is given by:W = m (h2-h1) .(2.18)

(4) Steam Boiler

Fig.2.8 Energy balance in a boiler

Fig.2.8 depicts water entering the control volume at atmospheric pressure with enthalpy m h1 and leaving as steam at high pressure with enthalpy m h 2.

Heat is supplied to the system by the product of combustion. From first law of thermodynamics, energy entering is equal to energy leaving the control

volume or

m h1 + Q = m h2or Q = m (h2 h1) .. (2.29)

Similar analysis can be carried for I.C. engines, nozzles, domestic refrigerator and heat exchanger.

2.7 Second Law of Thermodynamics

2.7.1 Limitations of First Law of Thermodynamics

(1) It is observed that heat flows from a system of higher temperature to a system of lower temperature and never from lower temperature system to

higher temperature system. Heat is transferred from hot water in a container to the atmosphere but never from atmosphere to hot water by itself.

From first law of thermodynamics, both processes, heat transfer from hot water to atmosphere and from atmosphere to hot water, are possible.Heat lost and heat gain must be equal in both the processes. According to second law of thermodynamics heat will only be transferred from high


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temperature to lower temperature and not vice versa. First law of thermodynamics is a quantitative statement and second law of thermodynamics is

a qualitative statement.

(2) It is also observed in nature that all work can be converted into heat but all heat cannot be converted into work. In power plants, all heat generated

from combustion of coal is not converted into work, but a portion of input heat has to be rejected in the condenser. Let us consider another

example of a running car. If the brakes are applied and the car is stopped, kinetic energy of the car is converted into frictional heat. If the wheel

and the brakes are cooled to the original temperature, car does not move. In the first process kinetic energy in the form of work was completely

transformed into heat, but in the second process heat was not converted into kinetic energy or work. According to first law of thermodynamics

work can be converted into heat and heat into work. It will be stated by second law that all heat cannot be converted into work.

2.7.2 Heat Engine, Heat Pump and Refrigerator

The main function of a heat engine is to produce work. The main object of a heat pump is to heat a space. The main task of a refrigerator is to cool a

space. The performance of an engine is measured by its efficiency and the performance of heat pump and refrigerator is given by coefficient of

performance.

A heat source is a system of infinite energy from which heat is taken without changing its temperature. A heat sink is a system of infinite energy towhich heat is rejected without affecting its temperature.

A heat engine is a system which operates in a cycle and partially converts heat into work. A steam turbine, boiler, condenser or pump are not steam

engines because individually they execute a process. If we combine all the four units together as in steam power plant, they can be treated as steam engine.

Fig. 2.9 shows a heat engine.

Fig. 2.9 Heat Engine

The engine receives heat Q1 from the source and rejects heat Q2 to the sink. The work done by the engine is W. The efficiency is given by:


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1Q

W=

From 1st law soQQW ,21 =

1

21

Q

QQ =

)20.2..(....................11

2

Q

Q=

A heat pump is a system which operates in a cycle and transfers heat from low temperature system to high temperature system when work is done on it.

Fig. 2.10 represents a heat pump or a refrigerator.

Fig. 2.10 Heat pump or a Refrigerator

Heat pump or refrigerator extracts heat Q2 from a low temperature system and transfer heat Q1 to the higher temperature system. Work is also done on the

system. For heat pump, coefficient of performance, COP is defined as the ratio of heat transferred to the high temperature system to work done on it or

W

QPOC 1.. =

From first law of thermodynamics,

W = Q1 Q2 , so

)21.2....(............21

1

QQ

QPOC

=

For refrigerator, coefficient of performance is defined as the ratio of heat extraction from cold system or sink to work done on it.


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W

QPOC 2.. =

)22.2......(............21

2

QQ

QPOC

=

2.72 Statements of Second Law of ThermodynamicsLike first law of thermodynamics, second law is based on certain facts observed in nature. These observations have never been violated. The statement of

this law cannot be proved mathematically.

All heat can not be converted into work. This observation is included in Kelvin-Planck statement.

1. Kelvin-Planck Statement

It is impossible to construct a device which operates in a cycle and produces no effect than producing work and exchanges heat with a single

reservoir.

2. Claucius Statement

It was also discussed that heat is transferred from high temperature system to low temperature system and not vice versa. This observation is

incorporated in the Claucius statement.It states that it is impossible to construct a device which operates in a cycle and produces no other effect than transferring heat from a lower

temperature system to higher temperature system.

2.7.4 Perpetual Motion Machine of Second Kind, PMMSK

A system which operates in a cycle and converts all heat into work and exchanges heat with a single reservoir is called perpetual motion machine

of second kind, PMMSK. Such system can not exist as it is violation of Kelvin-Plancks statement.

2.7.5 Reversible Process and Carnot Engine Efficiency

Reversible process is a process which once having taken place, can be reversed and doing so leaves no change in either the system or the

surroundings. A process which is not reversible is called irreversible process. If a wire is pulled down by a force, its length increases. If the force is

removed it regains its original length and state also. This process is a reversible.

Friction, heat transfer and free expansion make a process irreversible. A reversible process is also quasistatic process, but a quasistatic process maynot be reversible.


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(a) Carnot Engine (b) p-V diagram of a Carnot Engine

Fig. 2.11 Carnot Engine

Fig.2.11 shows a Carnot engine which has the maximum efficiency. It consists of undermentioned four reversible processes.In a steam power plant, thesefour processes are given below:

1-2 It is a reversible isothermal process in which heat is transfer to water in the boiler.2-3 It is a reversible adiabatic process in which work is done by the steam.

3-4 It is a reversible isothermal process in which heat is lost from exhaust steam in the

Condenser

4-1 It is a reversible adiabatic process in which work is done on the pump.

Efficiency of the Carnot cycle is the ratio of work done to heat supplied.

1Q

W=

From 1st law of thermodynamics,

W = Q1 Q2 so

)23.2......(..........1

21

Q

QQ =

During isothermal process 1-2, heat supplied Q1 is given by:

Q1 = W 1 2 = p1V1 ln V2 / V1Heat lost in the condenser, Q2 , during the isothermal process is given by:


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Q2 = W 3 4 = p3V3 ln V3 / V4

Putting values of Q1 and Q2 in Eq. (1.29) and simplifying, efficiency is given by:

)30.1......(..........1

21

T

TT =

2.8 Clausius Inequality

Whenever a system executes a cycle, the cyclic integral ofTQ around the cycle is less than or equal to zero or

0TQ

The equality sign applies to a reversible process and inequality sign to a irreversible process. Clausius inequality is applicable to both an engine and a

refrigerator.

1. Consider a reversible heat engine operating between reservoirs at temperatures T1 and T2

as shown in the diagram.

For Carnot cycle

0

02

2

1

1

1

21

2

1

=

=

=

TQ

orT

Q

T

Qor

T

TT

Q

Q


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2.Let us now consider an irreversible engine executing a cycle between the same two same reservoirs at temperatures T1 and T2 as shown in the figure.

3. Similarly, consider a reversible refrigerator operating

between two reservoirs at temperatures T1 and T2 as shown in the figure.

Now for a refrigerator working on Carnot cycle,

0

02

2

1

1

2

1

2

1

21

2

21

22

=

=+

=

=

==

TQ

orT

Q

T

Qor

T

T

Q

Q

or

TT

T

QQ

Q

W

QCOP

Since, the efficiency of irreversible engine is less

than that of reversible engine,

22

2

2

1

1

222121

0

,

QQasT

Q

T

Q

T

Q

engineleirreversibforThus

QQorQQQQ

orWW RIR

=


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4. Let us now consider an irreversible refrigerator operating between the same two reservoirs and absorbing the same amount of heat Q2 as shown in the

figure. Since COP of

2.9 Entropy

2.9.1 Definition of Entropy

From last article for a reversible Carnot engine Considering heat entering a system positive and heat leaving a system negative above equation can be re-

written as:

Cyclic integral of Q / T is zero, therefore Q / T is a property and denoted by dS.

)25.2.........(..........0

0

02

2

1

1

=

=

=+

TQ

or

T

Qor

T

Q

T

Q

irreversible refrigerator is always less than that of the

reversible refrigerator, it requires more work to absorb

for the same heat absorbed Q2. Hence,

0

.sin

0

,.

11

2

2

1

1

11

2121

+=

T

Qorsrefrigerat

andenginesleirreversibandreversibleallforSo

QQcegreater

istermnegativeasT

Q

T

Q

T

Q

orregrigeratleirreversibaforHenceQQ

orQQQQorWW RIR


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where S is called the entropy.

)27.2(....................RT

QSd

=

+Here R means reversible process. Also for reversible process,

S2 S1 = )33.1.........(..........1

2

1

Q

Change in entropy of a system executing a reversible process is given by 2

1T

Q.

It can be shown that for an irreversible process < )28.2......(..........0TQ

Combining Eq. (1.31) and Eq. (1.34), we get

)35.1.....(..........0TQ

This statement is called Claucius Inequality. Equal sign stands for reversible process and inequality sign stands for irreversible process.

2.9.2 Entropy Change in a Irreversible Process and Principle of Entropy Increase

Fig. 2.12 A reversible and an Irreversible cycles

Fig. 1.29 shows two cycles. Cycle 1-a-2-b-1consists of two reversible processes a and b. Cycle 1-a-2-c-1 consists of a reversible process a and a

irreversible process c.

From Claucius inequality for cycle 1-a-2-b-1.


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=

+

=

2

1

1

2

)30.2...(..........00ba T

Q

T

Qor

T

Q

From Claucius inequality, for cycle 1-a-2-c-1,

)31.2....(..........00

2

1

1

2

1

2

1

2 cbT

Q

T

Q

But by definition, =

1

2

, sosdT

Q

b

>1

2

1

2 rITQsd

or )32.2(....................rIT

Qsd

>

In general,

)33.2........(..........T

Qsd

The equality sign is true for reversible process and inequality sign holds for irreversible process. For an isolated system or for the Universe, Q = 0,)34.2....(..........0unisd

Entropy of an isolated system or the universe increases for an irreversible process and remains constant for a reversible process. This statement is calledthe principle of entropy increase.

2.10 Third Law of Thermodynamics

It states that entropy of a pure substance is zero at the absolute zero temperature. It helps to calculate absolute value of entropy.


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