thermodynamics lecture · pdf filethermodynamics lecture notes june 14, 2015 aeronautical...

THERMODYNAMICS LECTURE NOTES June 14,

2015

AERONAUTICAL ENGINEERING

UNIT I

Introduction to Thermodynamics: Thermodynamics is the science that deals with heat and

work and those properties of substance that bear a relation to heat and work. Thermodynamics is

the study of the patterns of energy change. Most of this course will be concerned with

understanding the patterns of energy change. More specifically, thermodynamics deals with (a)

energy conversion and (b) the direction of change. Basis of thermodynamics is experimental

observation. In that sense it is an empirical science. The principles of thermodynamics are

summarized in the form of four laws known as zeroth, first, second, and the third laws of

thermodynamics. The zeroth law of thermodynamics deals with thermal equilibrium and

provides a means of measuring temperature. The first law of thermodynamics deals with the

conservation of energy and introduces the concept of internal energy. The second law of

thermodynamics dictates the limits on the conversion of heat into work and provides the yard

stick to measure the performance of various processes. It also tells whether a particular process is

feasible or not and specifies the direction in which a process will proceed. As a consequence it

also introduces the concept of entropy. The third law defines the absolute zero of entropy.

System, Surrounding, Boundaries, Universe, Types of Systems:

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

region outside the system is called the surroundings. The real or imaginary surface that

separates the system from its surroundings is called the boundary. The boundary of a system can

be fixed or movable. Note that the boundary is the contact surface shared by both the system and

the surroundings. Mathematically speaking, the boundary has zero thickness, and thus it can

neither contain any mass nor occupy any volume in space. System and surroundings when put

together result in universe. These terms are illustrated in figure 1.1.

Figure 1.1: System, surroundings and boundary

Two types of exchange can occur between system and surroundings: (1) energy exchange (heat,

work, friction, radiation, etc.) and (2) matter exchange (movement of molecules across the

boundary of the system and surroundings).

Based on the types of exchange which take place or don't take place, there are three types of

systems:

A closed system (also known as a control mass) consists of a fixed amount of mass, and no

mass can cross its boundary. That is, no mass can enter or leave a closed system, as shown in

figure 1.2. But energy, in the form of heat or work, can cross the boundary; and the volume of a


2015


closed system does not have to be fixed. Closed system is the system having only energy

interactions at its boundary, for example, boiling water in a closed pan etc.

Figure: 1.2: Closed System

An open system, or a control volume, as it is often called, is a properly selected region in space.

It usually encloses a device that involves mass flow such as a compressor, turbine, or nozzle.

Flow through these devices is best studied by selecting the region within the device as the control

volume. Both mass and energy can cross the boundary of a control volume as shown in figure

1.3. The open system is one in which the energy and mass interactions take place at the system

boundary, for example automobile engine etc.

Figure: 1.3: Open System

Isolated system refers to the system which has neither mass interaction nor energy interaction

across system boundary, for example Thermos Flask etc. Thus, the isolated system does not

interact with the surroundings/systems in any way.


2015


Control Volume:

Control volume is defined as a volume which encloses the matter and the device inside a

control surface.

Everything external to the control volume is the surroundings with the separation given

by the control surface.

The surface may be open or closed to mass flows and it may have flows from energy in

terms of heat transfer and work across it.

The boundaries may be moveable or stationary.

In the case of a control surface that is closed to the mass flow, so that no mass can enter

or escape the control volume, it is called a control mass containing same amount of

matter at all times.

Concept of Continuum:

Matter is made up of atoms that are widely spaced in the gas phase. Yet it is very convenient to

disregard the atomic nature of a substance and view it as a continuous, homogeneous matter with

no holes, that is, a continuum.

In Macroscopic approach of thermodynamics the substance is considered to be continuous

whereas every matter actually comprises of myriads of molecules with intermolecular spacing

amongst them. In engineering thermodynamics where focus lies upon the gross behaviour of the

system and substance in it, the statistical approach is to be kept aside and classical

thermodynamics approach be followed. In classical thermodynamics, for analysis the atomic

structure of substance is considered to be continuous.

For facilitating the analysis this concept of continuum is used in which the substance is treated

free from any kind of discontinuity. As this is an assumed state of continuum in substance so the

order of analysis or scale of analysis becomes very important. Thus, in case the scale of analysis

is large enough and the discontinuities are of the order of intermolecular spacing or mean free

path then due to relative order of discontinuity being negligible it may be treated continuous. In

the situations when scale of analysis is too small such that even the intermolecular spacing or

mean free path are not negligible i.e. the mean free path is of comparable size with smallest

significant dimension in analysis then it cannot be considered continuous and the microscopic

approach for analysis should be followed.

For example, whenever one deals with highly rarefied gases such as in rocket flight at very high

altitudes or electron tubes, the concept of continuum of classical thermodynamics should be

dropped and statistical thermodynamics using microscopic approach should be followed.

Thus, in general it can be said that the assumption of continuum is well suited for macroscopic

approach where discontinuity at molecular level can be easily ignored as the scale of analysis is

quite large. The concept of continuum is thus a convenient fiction which remains valid for most

of engineering problems where only macroscopic or phenomenological information are desired.

For example, let us see density at a point as a property of continuum. Let us take some mass of

fluid Δm in some volume ΔV enveloping a point „P‟ in the continuous fluid. Average mass

density of fluid within volume ΔV shall be the ratio (Δm/ΔV). Now let us shrink the volume ΔV

enveloping the point to volume ΔV′. It could be seen that upon reducing the volume, ΔV′ may be

so small as to contain relatively few molecules which may also keep on moving in and out of the


2015


considered very small volume, thus average density keeps on fluctuating with time. For such a

situation the definite value of density cannot be given. Therefore, we may consider some limiting

volume ΔVlimit such that the fluid around the point may be treated continuous and the average

density at the point may be given by the ratio (Δm/ ΔVlimit).

Thus, it shows how the concept of continuum although fictitious is used for defining density at a

point as given below,

Average density at the point = limΔV→ΔVlimit(Δm/ΔV)

Macroscopic and Microscopic Approaches:

Microscopic approach uses the statistical considerations and probability theory, where we deal

with “average” for all particles under consideration. This is the approach used in the disciplines

known as kinetic theory and statistical mechanics.

In the macroscopic point of view, of classical thermodynamics, one is concerned with the time-

averaged influence of many molecules that can be perceived by the senses and measured by the

instruments.

The pressure exerted by a gas is an example of this. It results from the change in momentum of

the molecules, as they collide with the wall. Here we are not concerned with the actions of

individual molecules but with the time-averaged force on a given area that can be measured by a

pressure gage.

From the macroscopic point of view, we are always concerned with volumes that are very large

compared to molecular dimensions, and therefore a system (to be defined next) contains many

molecules, and this is called continuum.

The concept of continuum loses validity when the mean free path of molecules approaches the

order of typical system dimensions.

The State Postulate

The state of the system is described by its properties. Once a sufficient number of properties are

specified, the rest of the properties assume some values automatically. The number of properties

required to fix a state of a system is given by the state postulate:

The state of a simple compressible system is completely specified by two independent, intensive

properties.

The system is called a simple compressible system in the absence of electrical, magnetic,

gravitational, motion, and surface tension effects.

The state postulate requires that the two properties specified be independent to fix the state. Two

properties are independent if one property can be varied while the other one is held constant.

Temperature and specific volume, for example, are always independent properties, and together

they can fix the state of a simple compressible system. Thus, temperature and pressure are not

sufficient to fix the state of a two-phase system. Otherwise an additional property needs to be

specified for each effect that is significant. An additional property needs to be specified for each

other effect that is significant.


2015


Equilibrium

When the property of a system is defined, it is understood that the system is in

equilibrium.

If a system is in thermal equilibrium, the temperature will be same throughout the system.

If a system is in mechanical equilibrium, there is no tendency for the pressure to change.

In a single phase system, if the concentration is uniform and there is no tendency for

mass transfer or diffusion, the system is said to be in chemical equilibrium.

A system which is simultaneously in thermal, mechanical, and chemical equilibrium is

said to be in thermal equilibrium.

Property

In thermodynamics a property is any characteristic of a system that is associated with the

energy and can be quantitatively evaluated.

The property of a system should have a definite value when the system is in a particular

state.

Thermodynamic property is a point function.

Properties like volume of a system that depend on the mass of a system are called

extensive properties.

Properties like pressure or temperature which do not depend on the system mass are

called intensive properties.

The ratio of extensive property to the mass of the system are called specific properties

and therefore become intensive properties.

Substance can be found in three states of physical aggregation namely, solid, liquid and

vapor which are called its phases.

If the system consists of mixture of different phases, the phases are separated from each

other by phase boundary.

The thermodynamic properties change abruptly at the phase boundary, even though the

intensive properties like temperature and pressure are identical.

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

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

as viscosity, thermal conductivity, modulus of elasticity, thermal expansion coefficient, electric

resistivity, and even velocity and elevation.

Properties are considered to be either intensive or extensive.

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

temperature, pressure, and density.

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

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

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

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

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

uppercase letters are used to denote extensive properties (with mass m being a major exception),


2015


and lowercase letters are used for intensive properties (with pressure P and temperature T being

the obvious exceptions). Extensive properties per unit mass are called specific properties. Some

examples of specific properties are specific volume (v= V/m) and specific total energy (e= E/m).

Process:

Any change that a system undergoes from one equilibrium state to another is called a process,

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

process as shown in figure 1.4. To describe a process completely, one should specify the initial

and final states of the process, as well as the path it follows, and the interactions with the

surroundings.

A process is path followed by a system in reaching a given final state of equilibrium state

starting from a specified initial state. An actual process occurs only when the equilibrium state

does not exist. An ideal process can be defined in which the deviation from thermodynamic

equilibrium is infinitesimal. All the states the system passes through during a quasi-equilibrium

process may be considered equilibrium states. For non-equilibrium processes, we are limited to a

description of the system before the process occurs and after the equilibrium is restored. Several

processes are described by the fact that one property remains constant. The prefix iso- is used to

describe such processes. A process is said to be reversible if both the system and its surroundings

can be restored to their respective initial states by reversing the direction of the process.

Reversible: if the process happens slow enough to be reversed.

Irreversible: if the process cannot be reversed (like most processes).

Isobaric: process done at constant pressure

Isochoric: process done at constant volume

Isothermal: process done at constant temperature

Adiabatic: process where q=0

Cyclic: process where initial state is equal to final state

Figure 1.4: Process

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

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

Quasi-static Process:

When a process proceeds in such a manner that the system remains infinitesimally close to an

equilibrium state at all times, it is called a quasi-static, or quasi-equilibrium, process. A quasi-

equilibrium process can be viewed as a sufficiently slow process that allows the system to adjust

itself internally so that properties in one part of the system do not change any faster than those at

other parts.


2015


This is illustrated in figure: 1.5. When a gas in a piston-cylinder device is compressed suddenly,

the molecules near the face of the piston will not have enough time to escape and they will have

to pile up in a small region in front of the piston, thus creating a high-pressure region there.

Because of this pressure difference, the system can no longer be said to be in equilibrium, and

this makes the entire process non quasi-equilibrium. However, if the piston is moved slowly, the

molecules will have sufficient time to redistribute and there will not be a molecule pileup in front

of the piston. As a result, the pressure inside the cylinder will always be nearly uniform and will

rise at the same rate at all locations. Since equilibrium is maintained at all times, this is a quasi-

equilibrium process.

Figure 1.5: quasi static process

It should be pointed out that a quasi-equilibrium process is an idealized process and is not a true

representation of an actual process. But many actual processes closely approximate it, and they

can be modeled as quasi equilibrium with negligible error. Engineers are interested in quasi

equilibrium processes for two reasons. First, they are easy to analyze; second, work-producing

devices deliver the most work when they operate on quasi equilibrium processes. Therefore,

quasi-equilibrium processes serve as standards to which actual processes can be compared.

Note that the process path indicates a series of equilibrium states through which the system

passes during a process and has significance for quasi equilibrium processes only. For non quasi-

equilibrium processes, we are not able to characterize the entire system by a single state, and thus

we cannot speak of a process path for a system as a whole. A non quasi-equilibrium process is

denoted by a dashed line between the initial and final states instead of a solid line.

Concept of Temperature:

Although we are familiar with temperature as a measure of “hotness” or “coldness,” it is not easy

to give an exact definition for it. Based on our physiological sensations, we express the level of

temperature qualitatively with words like freezing cold, cold, warm, hot, and red-hot. However,

we cannot assign numerical values to temperatures based on our sensations alone. Furthermore,

our senses may be misleading. A metal chair, for example, will feel much colder than a wooden

one even when both are at the same temperature.

Fortunately, several properties of materials change with temperature in a repeatable and

predictable way, and this forms the basis for accurate temperature measurement. The commonly

used mercury-in-glass thermometer, for example, is based on the expansion of mercury with

temperature. Temperature is also measured by using several other temperature-dependent

properties.


2015


It is a common experience that a cup of hot coffee left on the table eventually cools off and a

cold drink eventually warms up. That is, when a body is brought into contact with another body

that is at a different temperature, heat is transferred from the body at higher temperature to the

one at lower temperature until both bodies attain the same temperature as shown in figure 1.6. At

that point, the heat transfer stops, and the two bodies are said to have reached thermal

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

Figure 1.6: Thermal Equilibrium

Zeroth Law of Thermodynamics:

We cannot assign numerical values to temperatures based on our sensations alone. Furthermore,

our senses may be misleading. Several properties of material changes with temperature in a

repeatable and predictable way, and this forms the basis of accurate temperature measurement.

The commonly used mercury-in-glass thermometer for example, is based on the expansion of

mercury with temperature. Temperature is also measured by using several other temperature

dependant properties. Two bodies (eg. Two copper blocks) in contact attain thermal equilibrium

when the heat transfer between them stops. The equality of temperature is the only requirement

for thermal equilibrium.

The Zeroth Law of Thermodynamics: If two bodies are in thermal equilibrium with a third body,

they are also in thermal equilibrium with each other. This obvious fact cannot be concluded from

the other laws of thermodynamics, and it serves as a basis of temperature measurement. By

replacing the third body with a thermometer, the zeroth law can be restated two bodies are in

thermal equilibrium if both have the same temperature reading even if they are not in contact.

Temperature Scales:

All temperature scales are based on some easily reproducible states such as the freezing and

boiling point of water, which are also called the ice-point and the steam-point respectively.

A mixture of ice and water that is in equilibrium with air saturated with water vapour at 1atm

pressure is said to be at the ice-point, and a mixture of liquid water and water vapour (with no

air) in equilibrium at 1atm is said to be at the steam-point.

Celsius and Fahrenheit scales are based on these two points (although the value assigned to these

two values is different) and are referred as two-point scales.

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

properties of the substance or substances.

Such a temperature scale is called a thermodynamic temperature scale.(Kelvin in SI)


2015


Ideal gas temperature scale:

The temperatures on this scale are measured using a constant volume thermometer. Based on the

principle that at low pressure, the temperature of the gas is proportional to its pressure at constant

volume. The relationship between the temperature and pressure of the gas in the vessel can be

expressed as

T = a + b.P

Where the values of the constants a and b for a gas thermometer are determined experimentally.

Once a and b are known, the temperature of a medium can be calculated from the relation above

by immersing the rigid vessel of the gas thermometer into the medium and measuring the gas

pressure.

Ideal gas temperature scale can be developed by measuring the pressures of the gas in the vessel

at two reproducible points (such as the ice and steam points) and assigning suitable values to

temperatures those two points. Considering that only one straight line passes through two fixed

points on a plane, these two measurements are sufficient to determine the constants a and b in the

above equation. If the ice and the steam points are assigned the values 0 and 100 respectively,

then the gas temperature scale will be identical to the Celsius scale. In this case, the value of the

constant a (that corresponds to an absolute pressure of zero) is determined to be –273.150C when

extrapolated.

The equation reduces to T = bP, and thus we need to specify the temperature at only one point to

define an absolute gas temperature scale. Absolute gas temperature is identical to

thermodynamic temperature in the temperature range in which the gas thermometer can be used.

We can view that thermodynamic temperature scale at this point as an absolute gas temperature

scale that utilizes an ideal gas that always acts as a low-pressure gas regardless of the

temperature.

At the Tenth international conference on weights and measures in 1954, the Celsius scale has

been redefined in terms of a single fixed point and the absolute temperature scale. The triple

point occurs at a fixed temperature and pressure for a specified substance. The selected single

point is the triple point of water (the state in which all three phases of water coexist in

equilibrium), which is assigned the value 0.01 C. As before the boiling point of water at 1 atm.

Pressure is 100.0 C. Thus the new Celsius scale is essentially the same as the old one. On the

Kelvin scale, the size of Kelvin unit is defined as “the fraction of 1/273.16 of the thermodynamic

temperature of the triple point of water, which is assigned a value of 273.16K”. The ice point on

Celsius and Kelvin are respectively 0 and 273.15 K.

Constant volume gas thermometer:

The constant volume gas thermometer is the most accurate laboratory thermometer and is used

for the calibration of other thermometers. When we heat a gas keeping the volume constant, its

pressure increases and when we cool the gas its pressure decreases. The relationship between

pressure and temperature at constant volume is given by the law of pressure. According to this


2015


law, the pressure of a gas changes by (1/273) of its original pressure at 00C for each degree

centigrade (or Celsius) rise in temperature at constant volume.

If Po is the pressure of a given volume of a gas at 0oC and Pt is the pressure of the same volume

of the gas (i.e., at constant volume) at toC, then

--------------------eq(1)

Where is constant and is known as coefficient of increase of pressure.

It consists of a glass bulb B connected to a tube A, through a capillary glass tube „C‟. The tube A

is connected to a mercury reservoir R which is clamped on the board and can be lowered or

raised whenever required to keep the volume of the air constant. The capillary tube C is provided

with a three way stopper S and can be used to connect capillary and bulb as well as to disconnect

tube from bulb B. A pointer is provided such that the end P is projecting inside from the upper

part of A. A scale calibrated in 0oC is provided between A and R. It is as shown in figure 1.7.

Figure 1.7: Constant Volume Gas Thermometer

The whole apparatus is leveled by adjusting the leveling screws. By adjusting the stopper, the

bulb „B‟ is filled with air or some gas and the pointer is adjusted so that tip of the pointer just

touches the level of mercury in the tube A. After filling the bulb, it is kept in an ice bath for

some time till the air inside the bulb attains the temperature of ice at which the mercury level

becomes stationary. Now the reservoir R is adjusted so that the level of mercury in the tube A

just touches the tip of the pointer P. The difference between the mercury levels in the two tubes

is noted and let it be ho. If Po is the pressure exerted by the air in the bulb, then

--------------------eq(2)

Now ice bath is removed and the bulb B is surrounded with steam. Again when the mercury level

in both the tubes becomes stationary, tube R is adjusted so that the mercury column in this

position in tube A touches the tip of the pointer.

Let the difference between the mercury columns levels in A and R be h100. The pressure of the

enclosed gas in B will be given by,


2015


--------------------eq(3)

--------------------eq(4)

Finally the bulb is brought in contact of the body whose temperature is to be found out. After the

mercury level becomes stationary, the tube R is adjusted so that the mercury level touches the tip

of the pointer again. Let ht be the difference between the heights of mercury levels and if Pt is

the pressure of the air inside the bulb B, then we have,

--------------------eq(5)

--------------------eq(6)

Dividing equation (6) by equation (4), we get

--------------------eq(7)

Substituting Pt and P100 in equation (7), we get,

--------------------eq(8)

Joule’s Experiments:

Let us take a closed system which permits work interaction and heat interaction both, as in case

of stirring in a container, as shown in figure 1.8. As a result of stirring it is seen that the

temperature of water gets raised up. This rise in temperature can be accounted by quantifying the

amount of heat supplied for raising this temperature. Thus, it is obvious that for any closed

system undergoing a cycle

i.e., the net heat interaction is proportional to the work interaction. Also the constant is known as

“Joule‟s mechanical equivalent of heat”. Joule‟s constant is described as;

Thus, J is a numerical conversion factor which could be unity if the heat is also given in joules.

For any cyclic process in the closed system the relationship between heat and work shall be, (if

the consistent units are used)


2015


Figure 1.8: Joule‟s Experiment

Energy in State and in Transition, Types:

Energy can exist in numerous forms such as thermal, mechanical, kinetic, potential, electric,

magnetic, chemical, and nuclear, and their sum constitutes the total energy E of a system.

In thermodynamic analysis, it is often helpful to consider the various forms of energy that make

up the total energy of a system in two groups: macroscopic and microscopic.

The macroscopic forms of energy are those a system possesses as a whole with respect to some

outside reference frame, such as kinetic and potential energies. The microscopic forms of energy

are those related to the molecular structure of a system and the degree of the molecular activity,

and they are independent of outside reference frames. The sum of all the microscopic forms of

energy is called the internal energy of a system and is denoted by U.

The macroscopic energy of a system is related to motion and the influence of some external

effects such as gravity, magnetism, electricity, and surface tension. The energy that a system

possesses as a result of its motion relative to some reference frame is called kinetic energy (KE).

The energy that a system possesses as a result of its elevation in a gravitational field is called

potential energy (PE).

Internal Energy:

The molecule as a whole can move in x, y and z directions with respective components

of velocities and hence possesses kinetic energy.

There can be rotation of molecule about its center of mass and than the kinetic energy

associated with rotation is called rotational energy.

In addition the bond length undergoes change and the energy associated with it is called

vibrational energy.

The electron move around the nucleus and they possess a certain energy that is called

electron energy.

The microscopic modes of energy are due to the internal structure of the matter and

hence sum of all microscopic modes of energy is called the internal energy.

Bulk kinetic energy (KE) and potential energy (PE) are considered separately and the other

energy of control mass as a single property (U).

The total energy possessed by the body is given by:

E = KE + PE + U


2015


Work:

Whenever a system interacts with its surroundings, it can exchange energy in two ways- work

and heat. In mechanics, work is defined as the product of the force and the displacement in the

direction of the force. Work done when a spring is compressed or extended: According to

Hooke's law

Spring force = - k (x – x0)

Where k is the spring constant, x0 is the equilibrium position, and x is the final position. The

negative sign shows that the direction of the spring force is opposite the direction of the

displacement from x0. The external force is equal in magnitude but opposite in sign to the spring

force, so

External force (force of your hands) = k (x –x0).

Now, we want to calculate the work done when we stretch the spring from position 1 to position

2.

W = F dx = k (x – x0) d(x-x0) = 1/2 k [(x2-x0)2 - (x1-x0)

2]

Work done when a volume is increased or decreased

Consider a gas in a container with a movable piston on top. If the gas expands, the piston moves

out and work is done by the system on the surroundings. Alternatively, if the gas inside

contracts, the piston moves in and work is done by the surroundings on the system. Why would

the gas inside contract or expand? It would if the external pressure, Pex, and the internal

pressure, Pin, were different.

To calculate the work done in moving the piston, we know that the force = pressure times area

and then work equals pressure times area times distance or work equals pressure times the

change in volume.

So, W = the integral of (Pex) dV

The differential work done (dW) associated with a differential displacement (dl) is given by

dW = F dl

For a piston cylinder assembly,

dW = F dl = PA (dl) = P dV

If the gas is allowed to expand reversibly from the initial pressure P to final pressure P, then the

work done is given by

W = ∫ p dV

The integral represents the area under the curve on a pressure versus volume diagram.

Therefore the work depends on the path followed and work is a path function and hence

not a property of the system.

The above expression does not represent work in the case of an irreversible process.


2015


The thermodynamic definition of work is “ Work is said to be done by a system on the

surrounding if the sole effect external to the system could be reduced to the raising of a

mass through a distance”.

Heat:

Heat like work, is a form of energy. The energy transfer between a system and its surroundings is

called heat if it occurs by virtue of the temperature difference across the boundary. The two

modes of energy transfer work and heat depend on the choice of the system. Heat energy moves

from a hotter body to a colder body upon contact of the two bodies. If two bodies at different

temperatures are allowed to remain in contact, the system of two bodies will eventually reach a

thermal equilibrium (they will have the same temperature). A body never contains heat. Rather

heat is a transient phenomenon and can be identified as it crosses the boundary.

Note that a quantity that is transferred to or from a system during an interaction is not a property

since the amount of such a quantity depends on more than just the state of the system. Heat and

work are energy transfer mechanisms between a system and its surroundings, and there are many

similarities between them:

1. Both are recognized at the boundaries of a system as they cross the boundaries. That is,

both heat and work are boundary phenomena.

2. Systems possess energy, but not heat or work.

3. Both are associated with a process, not a state. Unlike properties, heat or work has no

meaning at a state.

4. Both are path functions (i.e., their magnitudes depend on the path followed during a

process as well as the end states).

Path functions have inexact differentials designated by the symbol d. Therefore, a differential

amount of heat or work is represented by δQ or δW, respectively, instead of dQ or dW.

Properties, however, are point functions (i.e., they depend on the state only, and not on how a

system reaches that state), and they have exact differentials designated by the symbol d.

A small change in volume, for example, is represented by dV, and the total volume change

during a process between states 1 and 2 is

That is, the volume change during process 1–2 is always the volume at state 2 minus the volume

at state 1, regardless of the path followed. The total work done during process 1–2, however, is

That is, the total work is obtained by following the process path and adding the differential

amounts of work (δW) done along the way. The integral of δW is not W2 _ W1 (i.e., the work at

state 2 minus work at state 1), which is meaningless since work is not a property and systems do

not possess work at a state.


2015


Reversibility and Irreversibility:

Thermodynamic processes may have the change of state occurring in two ways. One is the

change of state occurring so that if the system is to restore its original state, it can be had by

reversing the factors responsible for occurrence of the process. Other change of state may occur

such that the above restoration of original state is not possible. Thermodynamic system that is

capable of restoring its original state by reversing the factors responsible for occurrence of the

process is called reversible system and the thermodynamic process involved is called reversible

process. Thus, upon reversal of a process there shall be no trace of the process being occurred,

i.e. state changes during the forward direction of occurrence of a process are exactly similar to

the states passed through by the system during the reversed direction of the process. It is quite

obvious that the reversibility can be realized only if the system maintains its thermodynamic

equilibrium throughout the occurrence of process.

The irreversibility is the characteristics of the system which forbids system from retracing the

same path upon reversal of the factors causing the state change. Thus, irreversible systems are

those which do not maintain equilibrium during the occurrence of a process. Various factors

responsible for the nonattainment of equilibrium are generally the reasons responsible for

irreversibility. Presence of friction, dissipative effects etc. have been identified as a few of the

prominent reasons for irreversibility. The reversible and irreversible processes are shown on p-v

diagram in figure 1.9. by „1–2 and 2–1‟ and „3–4 and 4–3‟ respectively.

Figure 1.9: Reversible and Irreversible Process

Causes for irreversibility:

Generic types of irreversibilities are due to;

(i) Friction,

(ii) Electrical resistance,

(iii) Inelastic solid deformations,

(iv) Free expansion

(v) Heat transfer through a finite temperature difference,

(vi) Non equilibrium during the process, etc.

(i) Friction: Friction is invariably present in real systems. It causes irreversibility in the process

as work done does not show equivalent rise in kinetic or potential energy of the system. Fraction

of energy wasted due to frictional effects leads to deviation from reversible states.


2015


(ii) Electrical resistance: Electrical resistance in the system also leads to presence of dissipation

effects and thus irreversibilities. Due to electric resistance dissipation of electrical work into

internal energy or heat takes place. The reverse transformation from heat or internal energy to

electrical work is not possible, therefore leads to irreversibility.

(iii) Inelastic solid deformation: Deformation of solids, when of inelastic type is also

irreversible and thus causes irreversibility in the process. If deformation occurs within elastic

limits then it does not lead to irreversibility as it is of reversible type.

(iv) Free expansion: Free expansion as discussed earlier in chapter 3, refers to the expansion of

unresisted type such as expansion in vacuum. During this unresisted expansion the work

interaction is zero and without expense of any work it is not possible to restore initial states.

Thus, free expansion is irreversible.

(v) Heat transfer through a finite temperature difference: Heat transfer occurs only when there

exist temperature difference between bodies undergoing heat transfer. During heat transfer if

heat addition is carried out in finite number of steps then after every step the new state shall be a

non-equilibrium state. In order to have equilibrium states in between, the heat transfer process

may be carried out in infinite number of steps. Thus, infinitesimal heat transfer every time causes

infinitesimal temperature variation. These infinitesimal state changes shall require infinite time

and process shall be of quasi-static type, therefore reversible. Heat transfer through a finite

temperature difference which practically occurs is accompanied by irreversible state changes and

thus makes processes irreversible.

(vi)Non equilibrium during the process: Irreversibilities are introduced due to lack of

thermodynamic equilibrium during the process. Non equilibrium may be due to mechanical in

equilibrium, chemical in equilibrium, thermal in equilibrium, electrical in equilibrium etc. and

irreversibility are called mechanical irreversibility, chemical irreversibility, thermal

irreversibility, electrical irreversibility respectively. Factors discussed above are also causing non

equilibrium during the process and therefore make process irreversible.

First Law of Thermodynamics:

The first law of thermodynamics, also known as the conservation of energy principle, provides a

sound basis for studying the relationships among the various forms of energy and energy

interactions. Based on experimental observations, the first law of thermodynamics states that

energy can be neither created nor destroyed during a process; it can only change forms.

Therefore, every bit of energy should be accounted for during a process.

Thus first law of thermodynamics states that “in a closed system undergoing a cyclic process, the

net work done is proportional to the net heat taken from the surroundings” or “for any cycle of a

closed system the net heat transfer equals the net work”.

For non-cyclic process: Let us now take up a system undergoing a non-cyclic process where

transfer of heat and work take place and there is some change in the state of system i.e., initial

and final states are different.


2015


If we assume the system to have the heat interaction ΔQ and work interaction ΔW, then from the

basic principles it can be said that: Energy lost = Energy gained; as the energy can neither be

created nor destroyed.

Therefore, between states 1–2 one can write energy balance as,

Q1–2 – W1–2 = U1–2

Where Q1–2, W1–2 and U1–2 are the heat, work and stored energy values. This stored energy is

called as internal energy for a system having negligible electrical, magnetic, solid distortion and

surface tension effects.

For elemental interactions, dQ = dU + dW

Thus, the first law of thermodynamics for non-cyclic processes can be given by

∫ dQ = ∫ dU + ∫ dW

The first law of thermodynamics for cyclic process can be employed,

Corollaries:

First law of thermodynamics based on law of energy conservation has proved to be a powerful

tool for thermodynamic analysis. But over the period of time when it was applied to some real

systems, it was observed that theoretically first law stands valid for the processes which are not

realizable practically. It was then thought that there exist certain flaws in first law of

thermodynamics and it should be used with certain limitations. Say for example let us take a

bicycle wheel and paddle it to rotate. Now apply brake to it. As a result of braking wheel comes

to rest upon coming in contact with brake shoe. Stopping of wheel is accompanied by heating of

brake shoe.

Examining the situation from first law of thermodynamics point of view it is quite satisfying that

rotational energy in wheel has been transformed into heat energy with shoe, thus causing rise in

its temperature: Now, if we wish to introduce the same quantity of heat into brake shoe and wish

to restore wheel motion then it is not possible simply, whereas theoretically first law permits the

conversion from heat to work (rotation of wheel in this case) as well.

Therefore, it is obvious that first law of thermodynamics has certain limitations as given below:

(i) First law of thermodynamics does not differentiate between heat and work and assures full

convertibility of one into other whereas full conversion of work into heat is possible but the vice-

versa is not possible.

(ii) First law of thermodynamics does not explain the direction of a process. Such as theoretically

it shall permit even heat transfer from low temperature body to high temperature body which is

not practically feasible. Spontaneity of the process is not taken care of by the first law of

thermodynamics.

Perpetual motion machine of the first kind (PMM-I) is a hypothetical device conceived, based on

violation of First law of thermodynamics. Let us think of a system which can create energy as

shown below.


2015


Figure 1.10: PMM-I, based on violation of First law of thermodynamics

Here a device which is continuously producing work without any other form of energy supplied

to it has been shown in figure 1.10 (a), which is not feasible. Similarly a device which is

continuously emitting heat without any other form of energy supplied to it has been shown in (b),

which is again not feasible. Above two imaginary machines are called Perpetual Motion

Machines of 1st kind.

First law applied to a Process and Flow System:

Let us consider an open system as shown in figure 1.11having inlet at section 1–1 and outlet at

section 2–2. The cross-section area, pressure, specific volume, mass flow rate, energy at section

1–1 and 2–2 are

Section 1–1 = A1, p1, v1, m1, e1

Section 2–2 = A2, p2, v2, m2, e2

Open system is also having heat and work interactions Q, W as shown in figure above.

Applying the energy balance at the two sections, it can be given as,

Energy added to the system + Stored energy of the fluid at inlet = Stored energy of the fluid at

outlet

Figure 1.11: I Law Applied to Open System

Steady Flow Energy Equation:

Steady flow refers to the flow in which its properties at any point remain constant with respect to

time. Steady system is the system whose properties are independent of time, i.e. any property at a

point in system shall not change with time.

Let us take an open system having steady flow. Figure 1.12 shows steady flow system having

inlet at section 1–1, outlet at section 2–2, heat addition Q and work done by the system W.

Figure 1.11: Steady Flow System


2015



2015


UNIT II Limitations of the First Law The first law of thermodynamics is a law of conservation of energy. It does not specify the direction of the process. All spontaneous processes processed in one direction only. The first law of thermodynamics does not deny the feasibility of a process reversing itself. The first law of thermodynamics does not provide answers to the following questions. Is a particular process / reaction feasible? To what extent do the process / reaction proceed? Is complete conversion of internal energy into work possible? There exists a law which determines the direction in which a spontaneous process proceeds. The law, known as the second law of thermodynamics, is a principle of wide generality and provides answer to the above questions. Thermal reservoir It is a large body from which a finite quantity of energy can be extracted or to which a finite quantity of energy can be added as heat without changing its temperature. A source is a thermal reservoir at high temperature from which a heat engine receives the energy as heat. A sink is a low temperature thermal reservoir to which a heat engine rejects energy as heat. Heat Engine A heat engine is a device which converts the energy it receives at heat, into work. It is a cyclically operating device. It receives energy as heat form a high temperature body, converts part of it into work and rejects the rest to a low temperature body. A thermal power plant is an example of a heat engine.

Figure 2.1: Heat Engine

In the boiler, the working fluid receives a certain amount of heat (Q1) from the hot combustion products. The superheated steam enters a turbine where it undergoes expansion performing the shaft work (WT). The low pressure steam enters a condenser where it exchange energy as heat at constant pressure with the cooling water and emerges as the condensate. The condensate rejects a certain amount of heat (Q2) to the cooling water. The low pressure condensate from the condenser enters the pump. Work (WP) is done on the pump to elevate the condensate to the boiler pressure and return it to the boiler.


2015


Heat pump Heat Pump is cyclically operating device which absorbs energy form a low temperature reservoir and reject energy as heat to a high temperature reservoir when work is performed on the device. Its objective is to reject energy as heat to a high temperature body (space heating in winter). The atmosphere acts as the low temperature reservoir. Refrigerator A refrigerator is a cyclically operating device which absorbs energy as heat from a low temperature body and rejects energy as heat to a high temperature body when work is performed on the device. The objective of this device is to refrigerate a body at low temperature. Usually it uses atmosphere as the high temperature reservoir.

Figure 2.2 : Heat pump/Refrigerator


2015


Parameters of performance Let QH and QL represents the amount of energy absorbed as heat from the low temperature reservoir and the energy rejected as heat to the high temperature reservoir respectively, Let W be the work done on the device to accomplish the task. QH and QL =W

Second Law of Thermodynamics: Kelvin-Planck and Clausius Statements KELVIN PLANCK STATEMENT: It is impossible to construct a cyclically operating device such that it produces no other effect than the absorption of energy as heat from a single thermal reservoir and performs an equivalent amount of work.

Figure 2.3: Kelvin Planck Statement

CLAUSIUS STATEMENT: Heat always flows from a body at higher temperature to a body at a lower temperature. The reverse process never occurs spontaneously. Clausius statement of the second law gives: It is impossible to construct a device which, operating in a cycle, will produce no effect other than the transfer of heat from a low-temperature body to a high temperature body.

Figure 2.4: Clauisus Statement


2015


Equivalence / Corollaries Let us suppose that the Kelven Planck statement is incorrect.

Figure 2.5: Equivalence of Kelvin Planck and Clausius Statement

It is possible to construct a device I which, working cyclically, absorbs energy a heat from a source at temperature and performs an equivalent amount of work. Next consider a device II which absorbs amount of energy from a low temperature body at and delivers energy as heat to a high temperature reservoir at. To accomplish this, work is done on the device. The device II does not violate the Clausius statement. For device II, we can write. Now combine I and II. The work delivered by device I is used by device, II. This combined device (which is no more aided by any external agency) working cyclically, is not producing any effect other than the transfer of energy as heat from the low temperature reservoir to the high temperature reservoir. This is in violation of the Clausius statement. To prove that violation of the Clausius statement leads to violation of Kelvin Planck statement, let us assume that the Clausius statement is incorrect. That is, it is possible to constructs a device I such that it transfers energy as heat from a body at lower temperature to a body at higher temperature unaided by any external agency. Consider another device II which receives energy as heat from a body at higher temperature, delivers work and rejects energy as heat to the body at a low temperature. Device II does not violent Kelvin Planck statement.

Figure 2.6: Equivalence of Clausius Kelvin and Planck Statement

PMM of Second kind A perpetual motion machine of the second kind, or PMM2 is one which converts all the heat input into work while working in a cycle. Carnot’s principle: The most efficient heat engine cycle is the Carnot cycle, consisting of two isothermal processes and two adiabatic processes. The Carnot cycle can be thought of as the most efficient heat engine cycle allowed by physical laws. When the second law of thermodynamics states that not all the supplied heat in a heat engine can be used to do work, the Carnot efficiency sets the limiting value on the fraction of the heat which can be so used.


2015


In order to approach the Carnot efficiency, the processes involved in the heat engine cycle must be reversible and involve no change in entropy. This means that the Carnot cycle is an idealization, since no real engine processes are reversible and all real physical processes involve some increase in entropy. Carnot cycle and its specialties The Carnot cycle when acting as a heat engine consists of the following steps: 1. Reversible isothermal expansion of the gas at the "hot" temperature, T1 (isothermal heat addition or absorption). During this step the gas is allowed to expand and it does work on the surroundings. The temperature of the gas does not change during the process, and thus the expansion is isothermal. The gas expansion is propelled by absorption of heat energy Q1 and of entropy ΔS= Q1/T1 from the high temperature reservoir. 2. Isentropic (reversible adiabatic) expansion of the gas (isentropic work output). For this step the mechanisms of the engine are assumed to be thermally insulated, thus they neither gain nor lose heat. The gas continues to expand, doing work on the surroundings, and losing an equivalent amount of internal energy. The gas expansion causes it to cool to the "cold" temperature, T2. The entropy remains unchanged. 3. Reversible isothermal compression of the gas at the "cold" temperature, T2. (isothermal heat rejection). Now the surroundings do work on the gas, causing an amount of heat energy Q2 and of entropy ΔS= Q2/T2 to flow out of the gas to the low temperature reservoir. (This is the same amount of entropy absorbed in step 1, as can be seen from the Clausius inequality.) 4. Isentropic compression of the gas (isentropic work input). Once again the mechanisms of the engine are assumed to be thermally insulated. During this step, the surroundings do work on the gas, increasing its internal energy and compressing it, causing the temperature to rise to T1. The entropy remains unchanged. At this point the gas is in the same state as at the start of step 1.

Figure 2.7: Carnot Cycle PV diagram Figure 2.8: Carnot Cycle TS diagram Thermodynamic scale of Temperature A temperature scale, which does not depend on the thermodynamic property of the substance can be established by making use of the fact that the efficiency of a reversible heat engine does not depend on the nature of the nature of the working fluid but depends only upon the temperature of the reservoirs between which it operates. The establishment of thermodynamic temperature scale is also a consequence of the second law of thermodynamics. It can be understood by the following figure.


2015


Figure 2.9: Thermodynamic scale of Temperature

Clausius Inequality Consider a system undergoing a reversible cycle. The given cycle may be sub-divided by drawing a family of reversible adiabatic lines. Every two adjacent adiabatic lines may be joined by two reversible isotherms.

Figure 2.10: Clausius Inequality


2015


The equality holds good for a reversible cycle and the inequality holds good for an irreversible cycles. The complete expression is known as Clausius Inequality. Entropy Entropy is a measure of the "disorder" of a system. The disorder refers to is really the number of different microscopic states a system can be in, given that the system has a particular fixed composition, volume, energy, pressure, and temperature. Clausius inequality can be used to analyze the cyclic process in a quantitative manner. The second law became a law of wider applicability when Clausius introduced the property called entropy. By evaluating the entropy change, one can explain as to why spontaneous processes occur only in one direction.

Consider that, it is possible to connect the states 1 and 2 by several reversible paths and see that

has the same value irrespective of the path as long as the paths are reversible. Therefore is

an exact differential of some function which we identify as entropy. Hence it can be said that there

exists a function S, called entropy, the change in entropy is expressed as

The following facts should be kept in mind while calculating the change in entropy for a process

1. For a reversible process 2. Entropy is a state function. 3. The entropy change of a system is determined by its initial and final states only, irrespective of

how the system has changed its state. In analyzing irreversible processes, it is not necessary to make a direct analysis of the actual process. One can substitute the actual process by a reversible process connecting the final state to the initial state, and the entropy change for the imaginary reversible process can be evaluated.

Principle of Entropy Increase Now let us apply the above results to evaluate the entropy of the universe when a system interacts with


2015


the surroundings.

Figure 2.11: Principle of Entropy Increase

Finally entropy increase means

The energy of the universe in constant (first law).

The entropy of the universe tends towards a maximum (second law)


2015


Energy Equation, Availability and Irreversibility Available Energy: In order to determine that part of the energy which can be converted into work by an engine, we require consider a thermal reservoir at constant temperature T from which a quantity of energy Q is being absorbed as heat. Consider the Carnot cycle as shown in a T-S diagram below:

Figure 2.12: Carnot TS Diagram

The area 1234 represents the portion of energy delivered as work by the reversible engine and is called available energy. The area 43BA represents the portion of the energy which is rejected to the atmosphere. The portion of energy is not available for conversion into work is called unavailable energy. Therefore, the available energy represents the portion of the energy supplied as heat which can be converted into work by means of a reversible engine. It is also known as EXERGY.

The availability of a given system is defined as the maximum useful work that can be obtained in a process in which the system comes to equilibrium with the surroundings or attains a dead state. Clearly, the availability of a system depends on the condition of the system as well as those of the surroundings. Thermodynamic Potentials Eight properties of a system, namely pressure (p), volume (v), temperature (T), internal energy (u), enthalpy (h), entropy (s), Helmholtz function (f) and Gibbs function (g). h, f and g are sometimes referred to as thermodynamic potentials. g = h – T.s is known as Gibb’s function or free energy function f = u – Ts is known as Helmholtz function. This gives maximum possible output when the heat Q is transferred at constant temperature and is the case with a very large source.


2015


Gibbs and Helmholtz Functions The work done in a non-flow reversible system (per unit mass) is given by : W = Q – (u0 – u1) = T.ds – (u0 – u1) = T (s0 – s1) – (u0 – u1) i.e., W = (u1 – Ts1) – (u0 – Ts0) The term (u – Ts) is known as Helmholtz function. This gives maximum possible output when the heat Q is transferred at constant temperature and is the case with a very large source. If work against atmosphere is equal to p0 (v0 – v1), then the maximum work available, Wmax = W – work against atmosphere

= W – p0 (v0 – v1) = (u1 – Ts1) – (u0 – Ts0) – p0 (v0 – v1) = (u1 + p0v1 – Ts1) – (u0 + p0v0 – Ts0) = (h1 – Ts1) – (h0 – Ts0)

i.e., Wmax = g1 – g0 where g = h – T.s is known as Gibb’s function or free energy function. The maximum possible available work when system changes from 1 to 2 is given by Wmax = (g1 – g0) – (g2 – g0) = g1 – g2 Maxwell Relations If dx is called an exact differential then dx = Mdy + Ndz where M=(∂x/∂y)z and N=(∂x/∂z)y We know from the first law of thermodynamics, dQ = du + pdv Tds = du + pdv du = Tds – pdv ….eq(1) We know enthalpy as h=u+pv, dh = du + pdv + vdp dh = Tds + vdp ….eq(2) Helmholtz free energy function f=u-Ts, df = du – Tds – sdT df = – pdv – sdT ….eq(3) Gibb’s free energy function g= h-Ts, dg = dh – Tds – sdT dg = vdp – sdT ….eq(4) Therefore the Maxwell’s equations can be rewritten as follows when above equations are considered as exact differentials.


2015


Elementary Treatment of the Third Law of Thermodynamics ‘Third law of thermodynamics’, an independent principle uncovered by ‘Nernst’ and formulated by ‘Planck’, states that the “Entropy of a pure substance approaches zero at absolute zero temperature.” This fact can also be corroborated by the definition of entropy which says it is a measure of molecular disorderness. At absolute zero temperature substance molecules get frozen and do not have any activity, therefore it may be assigned zero entropy value at crystalline state. Although the attainment of absolute zero temperature is impossible practically, however theoretically it can be used for defining absolute entropy value with respect to zero entropy at absolute zero temperature. Second law of thermodynamics also shows that absolute zero temperature can’t be achieved. Third law of thermodynamics is of high theoretical significance for the sake of absolute property definitions and has found great utility in thermodynamics.


2015


UNIT – III Perfect Gas Laws An equation of state of a particular substance is an empirical result, and it cannot be deduced from the laws of thermodynamics. Nevertheless the general form of the equation may be predicted from hypotheses about the microscopic structure of matter. This type of prediction has been developed to a high degree of precision for gases, and to a lesser extent for liquids and solids. The simplest postulates about the molecular structure of gases lead to the concept of the perfect gas which has the equation of state pv = RT. Experiments have shown that the behaviour of real gases at low pressure with high temperature agrees well with this equation. Boyle’s law: It states that volume of a given mass of a perfect gas varies inversely as the absolute pressure when temperature is constant. Charle’s law: It states that if any gas is heated at constant pressure, its volume changes directly as its absolute temperature. Equation of State Let us imagine a series of experiments in which the volume of a substance is measured over a range of temperatures while the pressure is maintained constant, this being repeated for various pressures. The results might be represented graphically by a three-dimensional surface, or by a family of constant pressure lines on a v-T diagram. It is useful if an equation can be found to express the relation between p, v and T, and this can always be done over a limited range of states. No single equation will hold for all phases of a substance, and usually more than one equation is required even in one phase if the accuracy of the equation is to match that of the experimental results. Equations relating p, v and T are called equations of state or characteristic equations. Specific and Universal Gas constants Avogadro discovered that V0 is the same for all gases at the same pressure and temperature and therefore it may be seen that MR = a constant ; R0 and thus pV0 = R0T R0 is called the molar or universal gas constant and its value is 8.3143 kJ/kg mol K. Various Non-flow processes, properties, end states, Heat and Work Transfer A process occurs when the system undergoes a change in a state or an energy transfer at a steady state. A process may be non-flow in which a fixed mass within the defined boundary is undergoing a change of state. Example : A substance which is being heated in a closed cylinder undergoes a non-flow process. Closed systems undergo non-flow processes. A process may be a flow process in which mass is entering and leaving through the boundary of an open system. In a steady flow process mass is crossing the boundary from surroundings at entry, and an equal mass is crossing the boundary at the exit so that the total mass of the system remains constant. In an open system it is necessary to take account of the work delivered from the surroundings to the system at entry to cause the mass to enter, and also of the work delivered from the system at surroundings to cause the mass to leave, as well as any heat or work crossing the boundary of the system.


2015


Throttling and Free Expansion Processes The throttling process is as shown in figure below. The velocity increases at the restriction, with a consequent decrease of enthalpy, but this increase of kinetic energy is dissipated by friction, as the eddies die down after restriction. The steady-flow energy equation implies that the enthalpy of the fluid is restored to its initial value if the flow is adiabatic and if the velocity before restriction is equal to that downstream of it.

Figure 3.1: Throttling Process

Consider two vessels 1 and 2 interconnected by a short pipe with a valve A, and perfectly thermally insulated as shown in figure. Initially let the vessel 1 be filled with a fluid at a certain pressure, and let 2 be completely evacuated. When the valve A is opened the fluid in 1 will expand rapidly to fill both vessels 1 and 2. The pressure finally will be lower than the initial pressure in vessel 1. This is known as free or unresisted expansion. The process is highly irreversible; since the fluid is eddying continuously


2015


during the process.

Figure 3.2: Free Expansion Process

Flow processes A process may be a flow process in which mass is entering and leaving through the boundary of an open system. In a steady flow process mass is crossing the boundary from surroundings at entry, and an equal mass is crossing the boundary at the exit so that the total mass of the system remains constant. In an open system it is necessary to take account of the work delivered from the surroundings to the system at entry to cause the mass to enter, and also of the work delivered from the system at surroundings to cause the mass to leave, as well as any heat or work crossing the boundary of the system. Deviations from perfect Gas Model The behavior of real gases usually agrees with the predictions of the ideal gas equation to within 5% at normal temperatures and pressures. At low temperatures or high pressures, real gases deviate significantly from ideal gas behavior. In 1873, while searching for a way to link the behavior of liquids and gases, the Dutch physicist Johannes vander Waals developed an explanation for these deviations and an equation that was able to fit the behavior of real gases over a much wider range of pressures. Vander Waals realized that two of the assumptions of the kinetic molecular theory were questionable. The kinetic theory assumes that gas particles occupy a negligible fraction of the total volume of the gas. It also assumes that the force of attraction between gas molecules is zero. The first assumption works at pressures close to 1 atm. But something happens to the validity of this assumption as the gas is compressed. Imagine for the moment that the atoms or molecules in a gas were all clustered in one corner of a cylinder, as shown in the figure below. At normal pressures, the volume occupied by these particles is a negligibly small fraction of the total volume of the gas. But at high pressures, this is no longer true. As a result, real gases are not as compressible at high pressures as an ideal gas. The volume of a real gas is therefore larger than expected from the ideal gas equation at high pressures. The deviation from ideal behavior is large at high pressure. The deviation varies from gas to gas. At lower pressures (<10 atm) the deviation from ideal behavior is typically small, and the ideal gas law can be used to predict behavior with little error. Deviation from ideal behavior is also a function of temperature: As temperature increases the deviation from ideal behavior decreases; as temperature decreases the deviation increases, with a maximum deviation near the temperature at which the gas becomes a liquid. Two of the characteristics of ideal gases included: The gas molecules themselves occupy no appreciable volume; the gas molecules have no attraction or repulsion for each other.


2015


Vader Waals Equation of State Vander Waals’ equation (for a real gas) may be written as: (p+(a/v2))(v-b) = RT The constants a and b are specific constants and depend upon the type of the fluid considered, ‘v’ represents the volume per unit mass and R is the gas constant. where units of R = 8314 Nm/kg mol K, a = Nm4/(kg-mol)2 b =m3/kg.mol Compressibility charts Let us plot to show the variation of volume with pressure for various constant values of temperature. In this case, the gradient of a curve at any state is (∂v/∂p)T . When this gradient is divided by the volume at that state, we have a property known as the compressibility K of the substance. Since this gradient is always negative, i.e., the volume of a substance always decreases with increase of pressure when the temperature is constant, the compressibility is usually made a positive quantity by defining it as K= (-1/v)*( (∂v/∂p)T). K can be regarded as a constant for many purposes for solids and liquids. The following figure shows the compressibility.

Figure 3.3: Compressibility

The compressibility factor (Z) of any gas is a function of only two properties, usually temperature and pressure, so that Z = f(Tr, pr) except near the critical point. The value of Z for any real gas may be less or more than unity, depending on pressure and temperature conditions of the gas. The general compressibility chart is plotted with Z versus pr for various values of Tr. This is constructed by plotting the known data of one or more gases and can be used for any gas.

Figure 3.4: Compressibility charts


2015


Variable specific Heats They are usually expressed in a form: cp = a + KT + K1T2 + K2 T3 cv = b + KT + K1T2 + K2T3 where a, b, K, K1 and K2 are constants. Phase Transformations: Triple point at critical state properties during change of phase If the vapour pressure of a solid is measured at various temperatures until the triple point is reached and then that of the liquid is measured until the critical point is reached, the result when plotted on a p-T diagram appears as shown in figure below.

Figure 3.5: p-T diagram for a substance such as water

If the substance at the triple point is compressed until there is no vapour left and the pressure on the resulting mixture of liquid and solid is increased, the temperature will have to be changed for equilibrium to exist between the solid and the liquid. Measurements of these pressures and temperatures give rise to a third curve on the p-T diagram, starting at the triple point and continuing indefinitely. The points representing the coexistence of (i) solid and vapour lie on the sublimation curve, (ii) liquid and vapour lie on the vapourisation curve, (iii) liquid and solid lie on the fusion curve. In the particular case of water, the sublimation curve is called the frost line, the vapourisation curve is called the steam line, and the fusion curve is called the ice line. The slopes of sublimation and the vapourisation curves for all substances are positive. The slope of the fusion curve, however may be positive or negative. The fusion curve of most substances have a positive slope. Water is one of the important exceptions. Triple point: The triple point is merely the point of intersection of sublimation and vapourisation curves. It must be understood that only on p-T diagram is the triple point represented by a point. Dryness Fraction The term dryness fraction is related with wet steam. It is defined as the ratio of the mass of actual dry steam to the mass of steam containing it. It is usually expressed by the symbol ‘x’ or ‘q’. If ms = Mass of dry steam contained in steam considered, and


2015


mw = Weight of water particles in suspension in the steam considered, Then, x= ms/(ms+mw) Thus if in 1 kg of wet steam 0.9 kg is the dry steam and 0.1 kg water particles then x = 0.9. Clausius – Clapeyron Equation Clausius-Claperyon equation is a relationship between the saturation pressure, temperature, the enthalpy of evaporation, and the specific volume of the two phases involved. This equation provides a basis for calculations of properties in a two-phase region. It gives the slope of a curve separating the two phases in the p-T diagram. Let us consider the change of state from saturated liquid to saturated vapour of a pure substance which takes place at constant temperature. During the evaporation, the pressure and temperature are independent of volume.

This is known as Clausius-Claperyon or Claperyon equation for evaporation of liquids. Property tables/Gas Tables/Steam tables In engineering problem, for any fluid which is used as working fluid, the six basic thermodynamic properties required are : p (pressure), T (temperature), v (volume), u (internal energy), h (enthalpy) and s (entropy). These properties must be known at different pressure for analysing the thermodynamic cycles used for work producing devices. The values of these properties are determined theoretically or experimentally and are tabulated in the form of tables which are known as ‘Steam Tables’. The properties of wet steam are then computed from such tabulated data. Tabulated values are also available for superheated steam. It may be noted that steam has only one saturation temperature at each pressure. Following are the thermodynamic properties of steam which are tabulated in the form of table: p = Absolute pressure (bar or kPa) ; ts = Saturation temperature (°C) ; hf = Enthalpy of saturated liquid (kJ/kg) ; hfg = Enthalpy or latent heat of vapourisation (kJ/kg) ;


2015


hg = Enthalpy of saturated vapour (steam) (kJ/kg) ; sf = Entropy of saturated liquid (kJ/kg K) ; sfg = Entropy of vapourisation (kJ/kg K) ; sg = Entropy of saturated vapour (steam) (kJ/kg K) ; vf = Specific volume of saturated liquid (m3/kg) ; vg = Specific volume of saturated vapour (steam) (m3/kg). Also, hfg = hg – hf ...... Change of enthalpy during evaporation sfg = sg – sf ...... Change of entropy during evaporation vfg = vg – vf ...... Change of volume during evaporation. The above mentioned properties at different pressures are tabulated in the form of tables as under : The internal energy of steam (u = h – pv) is also tabulated in some steam tables.

Mollier charts Dr. Mollier, in 1904, conceived the idea of plotting total heat against entropy, and his diagram is more widely used than any other entropy diagram, since the work done on vapour cycles can be scaled from this diagram directly as a length; whereas on T-s diagram it is represented by an area.

Figure 3.6: Mollier Charts


2015


Various Thermodynamic processes and energy Transfer

Steam Calorimetry

Steam calorimeter, in which the mass of steam condensed on a body, is used to calculate the

amount of heat supplied.

The dryness fraction of steam can be measured by using the following calorimeters:

1. Throttling calorimeter

2. Separating and throttling calorimeter.

The dryness fraction of wet steam can be determined by using a throttling calorimeter which is

illustrated diagrammatically in figure below:

The steam to be sampled is taken from the pipe by means of suitable positioned and dimensioned

sampling tube. It passes into an insulated container and is throttled through an orifice to

atmospheric pressure. Here the temperature is taken and the steam ideally should have about 5.5

K of superheat.

The throttling process is shown on h-s diagram in shown below by the line 1-2. If steam initially

wet is throttled through a sufficiently large pressure drop, then the steam at state 2 will become

superheated. State 2 can then be defined by the measured pressure and temperature. The

enthalpy, h2 can then be found and hence


2015


Figure 3.7: Throttling calorimeter

Figure 3.8: Throttling process

Hence the dryness fraction is determined and state 1 is defined.

Separating and Throttling Calorimeter:

If the steam whose dryness fraction is to be determined is very wet then throttling to atmospheric

pressure may not be sufficient to ensure superheated steam at exit. In this case it is necessary to

dry the steam partially, before throttling. This is done by passing the steam sample from the main

through a separating calorimeter as shown in figure below. The steam is made to change

direction suddenly, and the water, being denser than the dry steam is separated out. The quantity

of water which is separated out (mw) is measured at the separator, the steam remaining, which

now has a higher dryness fraction, is passed through the throttling calorimeter. With the


2015


combined separating and throttling calorimeter it is necessary to condense the steam after

throttling and measure the amount of condensate (ms). If a throttling calorimeter only is

sufficient, there is no need to measure condensate, the pressure and temperature measurements at

exit being sufficient.

Figure 3.9: Separating and throttling calorimeter

thermodynamics lecture · pdf filethermodynamics lecture notes june 14, 2015 aeronautical...

Documents