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EAS109

THERMO-FLUID MECHANICS

STUDY GUIDE (5CU)

Course Development Team

Head of Programme : Mr. Koh Pak Keng

Course Developer : Mr. J. Selva Raj

Production : Educational Technology & Production Team

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Production, Singapore University of Social Sciences.

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Release V1.3

CONTENTS

COURSE GUIDE

1. Welcome ............................................................................................................. 1

2. Course Description and Aims ......................................................................... 1

3. Learning Outcomes .......................................................................................... 4

4. Learning Material ............................................................................................. 4

5. Assessment Overview ...................................................................................... 5

6. Course Schedule ................................................................................................ 6

7. Learning Mode .................................................................................................. 6

STUDY UNIT 1

FLUID MECHANICS PRINCIPLES

Learning Outcomes ......................................................................................... SU1-1

Overview ........................................................................................................... SU1-1

Chapter 1 Basic Concepts relating to Fluid Mechanics .............................. SU1-2

Summary ........................................................................................................ SU1-19

Solutions .......................................................................................................... SU1-19

References ...................................................................................................... SU1-19

STUDY UNIT 2

KINEMATICS AND DIMENSIONAL ANALYSIS


Overview ........................................................................................................... SU2-1

Chapter 2 Fluid Kinematics and Dimensional Analysis ............................ SU2-2

Summary ........................................................................................................ SU2-16

Solutions .......................................................................................................... SU2-16

Reference ........................................................................................................ SU2-16

STUDY UNIT 3

THERMODYNAMIC PRINCIPLES AND SYSTEMS


Overview ........................................................................................................... SU3-1

Chapter 3 First Law of Thermodynamics for Closed and Open Systems SU3-2

Summary ........................................................................................................ SU3-23

References ...................................................................................................... SU3-23

STUDY UNIT 4

THERMODYNAMIC CYCLES


Overview ........................................................................................................... SU4-1

Chapter 4 Second Law of Thermodynamics ................................................ SU4-2

Summary ........................................................................................................ SU4-35

References ....................................................................................................... SU4-35

STUDY UNIT 5

FLUID MECHANICS APPLICATIONS


Overview ........................................................................................................... SU5-1

Chapter 5 Energy and Momentum Equations ............................................. SU5-2

Summary ........................................................................................................ SU5-16

References ...................................................................................................... SU5-16

STUDY UNIT 6

VISCOUS INTERNAL AND EXTERNAL FLOW


Overview ........................................................................................................... SU6-1

Chapter 6 Internal and External Flow .......................................................... SU6-2

Summary ........................................................................................................ SU6-51

Solutions .......................................................................................................... SU6-51

Reference ........................................................................................................ SU6-51

COURSE GUIDE

EAS109 COURSE GUIDE

1

1. Welcome

(Access video via iStudyGuide)

Welcome to the course EAS109 Thermo-Fluid Mechanics, a 5 credit unit (CU) course.

This Study Guide will be your personal learning resource to take you through the

course learning journey. The guide is divided into two main sections – the Course

Guide and Study Units.

The Course Guide describes the structure for the entire course and provides you

with an overview of the Study Units. It serves as a roadmap of the different learning

components within the course. This Course Guide contains important information

regarding the course learning outcomes, learning materials and resources,

assessment breakdown and additional course information.

2. Course Description and Aims

This course provides the student with introductory knowledge and understanding

of fluid mechanics and engineering thermodynamics.

Examples of engineering applications relating to fluid flow and thermodynamic

principles will be used for case studies. It is an integral part of undergraduate

curriculum for students majoring in aeronautical engineering and shall deliver the

essential concepts through both the theoretical and practical know-how needed in

this field of engineering.

As a part of the blended learning process you should spent approximately 150 hours

for the whole course. The study activities that will take up the 150 hours would

include attending lectures, carrying out laboratory investigations, self-reading

preparations before scheduled lectures, preparations for quizzes, laboratory report

writing and the final examination. Additionally the total study hours could include

group study discussions with your fellow course mates and discussions with your

main Course Tutor.

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Course Structure

This course is a 5-credit unit course presented over 6 weeks.

There are six Study Units in this course. The following provides an overview of each

Study Unit. Each face-to-face seminar alternates with an online learning session

using Collaborate©.

Study Unit 1 – FLUID MECHANICS PRINCIPLES

This unit initiates a conceptual foundation for fluid mechanics. It defines a fluid,

identifies its key properties and introduces basic fluid calculations. In Topic 1, we lay

the foundation for the systematic study of fluid mechanics by key definitions. A

working definition of a fluid precedes its key properties – density, viscosity and

vapour pressure. Next we look at pressure and flow rate. These are the two

components which define a state of fluid. In Topic 2, we study hydrostatic pressure

by calculating the effect of effective pressure on an object in relation to the depth in a

fluid. It would well worth your time to view videos relating to the basic concepts we

will present in Chapter 1.

Study Unit 2 – KINEMATICS AND DIMENSIONAL ANALYSIS

In this unit, we study the Eulerian and Lagrangian descriptions of fluid flow. The

definition of the streakline, streamline, pathline and timeline precede the different

visualizing techniques to ‘see’ fluid flow. We then study the difference between

rotational and irrotational flow regions using the flow property vorticity.

We will also learn elements of Dimensional Analysis as a key topic in research and

design in fluid mechanics. Dimensional analysis, using Buckingham’s Theorem,

helps us derive coherent and dimensionally accurate fluid flow parameters that can

be used in modeling and design.

Study Unit 3 – THERMODYNAMIC PRINCIPLES AND SYSTEMS

In this unit, the First Law of Thermodynamics for closed systems will be studied. We

first identify the first law of thermodynamics as simply a statement of the

conservation of energy principle for closed (fixed mass) systems and develop the

general energy balance applied to closed systems.

Next, we apply the first law of thermodynamics as the statement of the conservation

of energy principle to control volumes (open systems). The theory used is then

EAS109 COURSE GUIDE

3

applied to solve energy balance problems for common steady-devices such as

nozzles, compressors, turbines, and throttling valves.

Study Unit 4 – THERMODYNAMIC CYCLES

In this unit, the Kelvin–Planck and Clausius statements of the second law of

thermodynamics are introduced. Definition of entropy enables us to quantify the

second-law effects. Hence we can apply the second law of thermodynamics to

processes and recall the increase of entropy principle. This will allow us to calculate the

entropy changes that take place during processes for pure substances,

incompressible substances, and ideal gases. We will also apply the second law of

thermodynamics to the Carnot, Otto, Diesel and Brayton cycles and perform

calculations to enable heat supplied, heat rejected and thermal efficiencies be

calculated for a variety of given operating conditions.

Study Unit 5 – FLUID MECHANICS APPLICATIONS

In this unit, you will apply the conservation of energy and momentum as used in fluid

mechanics. We will derive the Bernoulli Equation and identify the assumptions made in its

derivation. We will also ‘drill’ deeper to understand the use and limitations of the Bernoulli

equation, and apply it to solve a variety of fluid flow problems. Subsequently, the

momentum equation is explained. This is Newton’s 2nd Law applied to fluid mechanics.

First, we identify the various kinds of forces acting on a control volume and use control

volume analysis to determine the thrust devices and passive elements in fluid flow.

Study Unit 6 – VISCOUS INTERNAL AND EXTERNAL FLOW

In this unit, we look at laminar and turbulent flow in pipes and analyse fully developed pipe

flow. We will calculate losses associated with pipe flow in piping networks and determine

the pumping power requirements. Finally, we will look at the effects of flow regime on the

drag coefficients associated with flow over cylinders and spheres initially. We will also learn

the fundamentals of flow over airfoils and calculate the drag and lift forces acting on airfoils.

We will develop an understanding of the various physical phenomena associated with

external flow such as drag, friction and pressure drag, drag reduction, and lift.

EAS109 COURSE GUIDE

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3. Learning Outcomes

Knowledge & Understanding (Theory Component)

By the end of this course, you should be able to:

Define the fundamentals of Thermodynamics and Fluid Mechanics.

Recognize interrelationships underpinning the laws governing these two

disciplines.

Key Skills (Practical Component)

By the end of this course, you should be able to:

Apply learnt content to explain and calculate thermodynamic and fluid flows

and processes.

Identify basic thermodynamic and fluid equations equipment for

fundamental flow study.

Describe and conduct thermodynamic and fluid experiments.

Interpret computer simulations results and extract data to perform analyses.

4. Learning Material

The following is a list of the required learning materials to complete this course.

Required Textbook(s)

[1] Selva Raj J., Loh H. C., (2015) Thermofluids I (5th Ed.) Singapore: McGraw-Hill.

[2] Selva Raj J., Wan K.H., (2016) Thermofluids II (3rd Ed.) Singapore: McGraw-Hill.

[3] Cengel Y. A., Turner R. H., Cimbala, J. M. (2011) Fundamentals of Thermal-Fluid

Sciences (4th Ed.). and Chapter 11 (3rd Ed) Singapore: McGraw-Hill.

Special Requirement

Lab equipment is critical to the pedagogical requirements of the course. The facilities

of the Applied Thermodynamic Labs I & II located in Singapore Polytechnic is used.

Website(s):

There are no specific requirements. However you are will explore material available

online to view and engage in learning beyond the classroom.

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5. Assessment Overview

The overall assessment weight for this course is as follows:

Assessment Description Weight Allocation

Online Quiz This test consists of 40 MCQs 12%

Laboratory Reports 5 reports are submitted after each

laboratory class over 5 weeks 12%

Pre-Class Quiz An on-line MCQ preparation for

the forthcoming seminar 6%

Examination An open book examination

consisting of 4 questions 70%

TOTAL 100%

The following section provides important information regarding Assessments.

SUSS’s assessment strategy consists of two components, Overall Continuous

Assessment (OCAS) and Overall Examinable Component (OES) that make up the

overall course assessment score.

For SST courses: the component weights are 30% OCAS and 70% OES.

To be sure of a pass result, you need to achieve scores of at least 40% in each

component. Your overall rank score is a weighted average of both components.

Continuous Assessment:

The table above details the three components. An online quiz is followed by

laboratory reports and a laboratory quiz.

Examination:

The final (2-hour) written exam will constitute the other 70 percent of overall student

assessment It will test the ability to articulate answers based on content delivered in

all the six study units. To prepare for the exam, you are advised to review Specimen

or Past Year Exam Papers available on Learning Management System.

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Passing Mark:

To successfully pass the course, you must obtain a minimum passing mark of 40

percent for each of the two TMA components. That is, students must obtain at least a

mark of 40 percent for the combined assessments and also at least a mark of 40

percent for the final exam. For detailed information on the Course grading policy,

please refer to The Student Handbook (‘Award of Grades’ section under Assessment

and Examination Regulations). The Student Handbook is available from the Student

Portal.

Non-graded Learning Activities:

Activities for the purpose of self-learning are present in each study unit. These

learning activities are meant to enable you to assess your understanding and

achievement of the learning outcomes. The type of activities can be in the form of

Quiz, Review Questions, Application-Based Questions or similar. You are expected

to complete the suggested activities either independently and/or in groups.

6. Course Schedule

To help monitor your study progress, you should pay special attention to your

Course Schedule. It contains study unit related activities including Assignments,

Self-assessments, and Examinations. Please refer to the Course Timetable in the

Student Portal for the updated Course Schedule.

Note: You should always make it a point to check the Student Portal for any

announcements and latest updates.

7. Learning Mode

The learning process for this course is structured along the following lines of

learning:

(a) Self-study guided by the six Study Units. Independent study will require at

least 6 hours per week.

(b) Working on tutorial questions or laboratory reports, either individually or in

groups.

(c) Classroom Seminar sessions (3 hours each session, 6 sessions in total).

EAS109 COURSE GUIDE

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iStudyGuide

You may be viewing the iStudyGuide version, which is the mobile version of the

Study Guide. The iStudyGuide is developed to enhance your learning experience

with interactive learning activities and engaging multimedia. Depending on the

reader you are using to view the iStudyGuide, you will be able to personalize your

learning with digital bookmarks, note-taking and highlight sections of the guide.

Interaction with Instructor and Fellow Students

Although flexible learning – learning at your own pace, space and time – is a

hallmark at SUSS, you are encouraged to engage your instructor and fellow students

in online discussion forums. Sharing of ideas through meaningful debates will help

broaden your learning and crystallise your thinking.

Academic Integrity

As a student of SUSS, it is expected that you adhere to the academic standards

stipulated in The Student Handbook, which contains important information

regarding academic policies, academic integrity and course administration. It is

necessary that you read and understand the information stipulated in the Student

Handbook, prior to embarking on the course.

EAS109

Thermo-Fluid Mechanics

STUDY UNIT 1

FLUID MECHANICS PRINCIPLES

EAS109 STUDY UNIT 1

SU1-1

Learning Outcomes

At the end of this unit, you are expected to:

Explain the basic concepts relating to Fluid Mechanics.

Define a fluid.

Identify the important fluid properties – density and viscosity.

Define pressure.

Apply the various expressions of pressure in fluid calculations.

Define fluid flow.

Define hydrostatic pressure.

Apply hydrostatic equations to solve static fluid applications.

Overview

This unit initiates a conceptual foundation for fluid mechanics. It defines a fluid,

identifies its key properties and introduces basic fluid calculations.

In Topic 1, we lay the foundation for the systematic study of fluid mechanics by key

definitions. A working definition of a fluid precedes its key properties – density,

viscosity and vapour pressure. Next, we look at pressure and flow rate. These are the

two components which define a state of fluid.

In Topic 2, we study hydrostatic pressure by calculating the effect of effective pressure

on an object in relation to the depth in a fluid. It would well worth your time to view

videos relating to the basic concepts we will present in Chapter 1.

EAS109 STUDY UNIT 1

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Chapter 1: Basic Concepts relating to Fluid Mechanics

Symbols / Units

READ:

Chapter 1 of Thermofluids I

Chapter 10 of Fundamentals of Thermal-Fluid Sciences

Topic 1 Fluid Flow – Basic Concepts

1.1 Identify elements in Fluid Flow

1.1.1 Definition of a Fluid

Non-technical Definition

A fluid takes the shape of its container. Alternately, a fluid is a substance that flows.

Technical definition

Fluids are material which cannot resist a shear stress without moving.

1.1.2 Definition of Fluid Statics

The fluid which is at rest is called static fluid.

1.1.3 Definition of Fluid Kinematics

The study of fluid flow when considering its motion is called fluid kinematics.

We will fill in the important parameters as we proceed through the course.

1. Density

2. RD vs. SG

3. Viscosity – Dynamic & Kinematic Viscosity

4. Vapour Pressure/Saturation Pressure

5. Pressure/Head

6. Static, Dynamic & Total (Stagnation) Pressures/Head

7. Atmospheric, Gauge, Vacuum & Absolute Pressures/Head

8. Volume Flow Rate, Mass Flow Rate

9. Fluid Power

10. Efficiencies (pump, mechanical, electrical)

EAS109 STUDY UNIT 1

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1.1.4 Definition of Fluid Dynamics

When internal (shear stresses) and external forces (pressures) are considered on the

moving fluid, this study is called fluid dynamics.

When a fluid is compressible, its density can change. Gases are compressible. When a

fluid is incompressible, its density does not change. Liquids are practically

incompressible except at huge pressures (>20MPa).

1.2 Identify the Important Fluid Properties – Density and Viscosity

ACTIVITY:

Please view the following videos

(i) Fluid Properties_Density.mpg

(ii) Fluid Properties_Viscosity.mpg

1.2.1 Density

The mass per unit volume of material is called the density, symbol ρ. The density of a

gas changes according to the pressure, but that of a liquid may be considered constant

unless the relevant pressure changes are very high. The units of density are kgm-3 (SI).

The ratio of the density of a material ρ to the density of water (at 40C), is called the

relative density (RD). This is often called the specific gravity (SG), a term which is

sometimes confusing.

For liquids and solids, the specific gravity is generally identified as the ratio of the

materials density relative to water both at the same conditions, i.e. 40C and 1.01 bar.

For gases, the specific gravity is the ratio of the material’s density to the density of air,

both at the same conditions as identified above. The term ‘Specific Gravity’ predates

the SI units.

The density of gases and vapours are greatly affected by the pressure. For so called

perfect gases, the density can be calculated from the formula, pV = mRT. For a gas of 1

kg, the volume per unit mass, is called the specific volume, which is generally

expressed by the symbol υ

υ = 1/ρ

1.2.2 Viscosity

This measures the internal friction of the fluid and how it interacts with gravity in fluid

flow.

Consider, for example, the simple shear between two plates:

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Couette Flow

Consider a fluid in 2-D steady shear between two infinite plates, as shown in the

sketch. The bottom plate is fixed, while the upper plate is moving at a steady speed of

U.

Figure 1-1 Development of Laminar Flow (Selva Raj & LohHC , 2015, p. 4)

The velocity profile, u(y) is linear, i.e. u(y) = Uy/h. Also notice that the velocity of the

fluid matches that of the wall at both the top and bottom walls. This is known as the

no slip condition.

The top plate will experience a friction force to the left, since it is doing work trying to

drag the fluid along with it to the right. The fluid at the top of the channel will

experience an equal and opposite force (i.e. to the right). Similarly the bottom plate will

experience a friction force to the right, since the fluid is trying to pull the plate along

with it to the right. The fluid at the bottom of the channel will feel an equal and

opposite force, i.e. to the left. In fluid mechanics, shear stress, defined as a tangential

force per unit area, is used rather than force itself.

Define (Greek letter "tau") as the friction force per unit area acting on the fluid, as

illustrated below:

In a simple 2-D shear flow such as this, the shear stress is directly proportional to the

slope of the velocity profile. In fact, the constant of proportionality is the coefficient of

viscosity itself.

Mathematically,

The SI unit for viscosity is the Pas (Pascal second). Also, acceptable equivalent units

are kg/sm and Ns/m2. (Prove these two equivalent units for viscosity.)

EAS109 STUDY UNIT 1

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1.2.3 Kinematic Viscosity

Kinematic viscosity, . The SI unit for kinematic viscosity is [m2s-1].

Definition of kinematic viscosity ν =μ

ρ

Note the definition of specific volume υ = 1/ρ

[Unfortunately, in print the symbols for kinematic viscosity and specific volume look

very similar.]

1.2.4 Newtonian /Non-Newtonian Fluids

Liquids that obey Newton’s Viscosity Relationship are call Newtonian fluids. We will

discuss non-Newtonian fluids (liquids and gases) in the class.

Figure 1-2 Shear Stress vs. Rate of Strain for Non-Newtonian Fluids (Cengel, Turner &

Cimbala, 2012, p. 389)

EAS109 STUDY UNIT 1

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Typical Non-Newtonian Fluids

1.2.5 Pressure

Pressure is the force per unit area at a point. The absence of pressure occurs in a

complete vacuum. A complete vacuum is really a theoretical concept.

The normal pressure experienced on the surface of the earth is called the atmospheric

pressure. Pressures are measured relative to the local atmospheric pressure. These

pressures are usually called “positive pressures” or “gauge pressures”.

Figure 1-3, below shows the relationship between the gauge pressure and the absolute

pressure for two measurements: a pressure less than atmospheric and a pressure

greater than atmospheric.

Figure 1-3 Identifying Pressure Types (Selva Raj & Loh HC 2015, p. 10)

We will identify at least 8 different non-Newtonian fluids in the class.

1 cement

2 milk

3 blood and saliva

4 tomato ketchup

5 corn-starch

6 non-drip paint

7 ___________

8 ___________

As you can now see, in the natural world, most fluids are non-Newtonian.

EAS109 STUDY UNIT 1

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pabs = pgauge + patm

The SI unit of pressure is the Pascal (Pa) (N/m2). The dimensional formula for pressure

is ML-1T-2.

In considering fluid pressures it has been found convenient in hydrostatics and in fluid

dynamics to use fluid head as a method of measuring pressure. Consider Figure 1-4.

Figure 1-4 Manometric Head Visualised

A quantity of fluid is exposed to atmospheric pressure on its surface.. The liquid will

be forced up the tube until the gravity force resulting from the level of fluid in the tube

balances the force due to the atmospheric pressure at the surface of the open vessel. If

the area of the tube is A, the density of the fluid = ρ, and the pressure at the top of the

tube is “zero” or a vacuum, the pressure, hρg is atmospheric pressure. For mercury

this height is 76 cm and for water 10.34. Prove this.

1.2.6 Vapour Pressure

This is defined as the pressure at which a liquid will boil (vaporise). Vapour pressure

rises as temperature rises. For example, suppose you are camping on a high mountain

(3,000 m in altitude). The atmospheric pressure at this elevation is about 70kPa. At a

temperature of around 900C, the vapour pressure of water is also around 70kPa. From

this it can be stated that at 3,000m of elevation, water boils at around 900C, rather than

the common 1000C at standard sea level pressure.

Vapour pressure is important to fluid flows because, in general, pressure in a flow

decreases as velocity increases. This can lead to cavitation, which is generally

destructive and undesirable. In particular, at high speeds the local pressure of a liquid

sometimes drops below the vapour pressure of the liquid. In such a case, cavitation

occurs. In other words, a "cavity" or bubble of vapour appears because the liquid

vaporises or boils at the location where the pressure dips below the local vapour

pressure.

Cavitation is not desirable for several reasons. First, it causes noise (as the cavitation

bubbles collapse as they migrate into regions of higher pressure). Second, it can lead

to inefficiencies and reduction of heat transfer in pumps and turbines (turbo-

h

X X

EAS109 STUDY UNIT 1

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machines). Finally, the collapse of these cavitation bubbles causes pitting and corrosion

of blades and other surfaces nearby.

1.3 Expressions of Pressure in Fluid Calculations

1.3.1 Atmospheric Pressure (patm)

The pressure exerted by the weight of the mass of the air per square metre on the

earth’s surface is known as the atmospheric pressure. Atmospheric pressure at sea-

level is 101.3 kPa. Atmospheric pressure is measured with the barometer.

1.3.2 Gauge Pressure (pg)

Gauge pressure is that pressure that is measured above or below atmospheric pressure.

A pressure gauge measures this pressure usually in a pipeline or a container. If the

pressure shows a ‘0’ (zero) reading then the pressure is atmospheric pressure. If it is

positive then the pressure is above atmospheric pressure.

1.3.3 Vacuum (pvac)

If the pressure gauge reading is negative, then the pressure is below atmospheric

pressure. It is referred to as vacuum (pressure).

1.3.4 Absolute Pressure (pabs)

Absolute pressure = atmospheric pressure + gauge pressure

Or, if the pressure is lower than atmospheric pressure,

Absolute pressure = atmospheric pressure – vacuum pressure

1.3.5 Static Pressure (p)

The pressure acting at a point in a fluid is called the static pressure. This pressure can

be:

(i) created by the pump;

(ii) because this point is submerged in a fluid.

Static pressure difference between two points in a fluid causes the fluid to flow from

the point of higher static pressure to the point of lower static pressure.

NOTE: Whenever pressure is used without an adjective, static pressure is implied.

1.3.6 Dynamic Pressure (pd)

In a fluid moving at an average velocity, U, the pressure created by causing it to stop

is equal to its kinetic energy (KE). This pressure is called dynamic pressure.

EAS109 STUDY UNIT 1

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i.e. dynamic pressure, pd = 2

2

1U

where = density ][3m

kg,

U = velocity[s

m]

1.3.7 Stagnation Pressure (Po)

Stagnation Pressure (note the capital ‘P’) at a point in the flow field of a liquid is

defined as the sum of the static and dynamic pressure.

Stagnation Pressure = Static pressure + Dynamic pressure

1.3.8 Pressure Head (h) [Unit: metre of liquid]

Sometimes, it is convenient to express pressure at a point in terms of the pressure an

equivalent height a fluid column would exert at that point. The height of the column

of liquid, h, which would produce the pressure, p is given by the equation

g

ph

where h = the height of the column of liquid, [m of the liquid],

p = pressure [2m

N],

33 m

J

m

Nm

so g

ph

, can be considered as energy per unit weight. (Prove this!)

= density ][3m

kg and

g = acceleration due to gravity = 9.81 [2s

m]

Since, Stagnation Head = Static Head + Dynamic Head

gg

p

g

P

2

U20

Head is expressed in [m of the liquid], e.g. 10 mH2O. Please note that stating 10m alone

is not enough.

Po = p + 2U

2

1

EAS109 STUDY UNIT 1

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1. Expressions for pressure readings

2. Converting pressure gauge readings

3. Pressure and Velocity head calculations

(Access videos via iStudyGuide)

1.3.9 Pressure Measuring Devices

(a) Bourdon Gauge

This consists of a spiral tube of oval section (Bourdon Tube) which straightens as

pressure is applied to it internally. One end of the tube is fixed and the movement of

the other end is magnified using gears. A pointer which deflects over a calibrated scale

reads pressures above atmospheric pressure. The compound gauge shown below is

calibrated to read pressures above and below atmospheric pressures. Note the units

used. Pressures above atmospheric are in [bar]; pressure readings below atmospheric

are in [mmHg].

Figure 1-5 Pressure Measuring Devices (Selva Raj & Loh HC, 2015, pp. 15 &16)

(b) Piezometer

The simplest type of pressure measuring device is the piezometer which is a

transparent tube at the top and with the other end fitted at the point where pressure is

EAS109 STUDY UNIT 1

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to be measured.

(c) U-Tube Manometer

The U-Tube Manometer is an extension of the piezometer where a double limb tube or

‘U-Tube’ is used. By using mercury as the indicating fluid, it is possible to measure

much higher pressures than would be possible with a piezometer. A differential

manometer is one where each limb is connected to different points in a fluid flow. This

enables the height of the indicating liquid column to read pressure differential between

the two points. The U-tube manometer in Figure 1-5 has the left limb connected to the

fluid whilst the right limb is ‘connected’ to the atmosphere.

Note: Piezometers and manometers read static pressure. Why? .

Interpreting U-tube manometer readings


1.4 Definition of Fluid Flow

1.4.1 Flow Rate

When we refer to flow rate in Fluid Mechanics, especially when we deal with liquid

flow, we usually imply volumetric flow rate. This is because fluids are incompressible

in most engineering situations. As such, the density is constant. Also most flow rate

measuring devices measure the volume flow.

1.4.2 Steady Flow

If the fluid velocity remains constant at a boundary (i.e. does not change with time),

the flow is said to be ‘steady’ at that boundary. (Please note that this does not mean

that the fluid velocity across any boundary or section has the same value.)

Illustration – the difference between steady and unsteady flow

Water from a tap flows into a container fitted with an outlet hole such that the water

cannot flow out as quickly as it is flowing in. Under these conditions, the flow from

the container is unsteady even though the tap flow is steady. This is because the outlet

velocity from the container will change with time as the level in the container rises.

After some time, the water level is constant and the outlet velocity from the container

will also be constant. The flow has then become steady.

EAS109 STUDY UNIT 1

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Figure 1-6 Steady Flow (Selva Raj & Loh HC, 2014, p. 18)

1.4.3 Volumetric Flow Rate (Q) [Unit:s

m 3

]

The volumetric flow rate (usually only referred to as the flow rate) is the volume of

fluid per unit time crossing a given boundary in a flow field.

t

V

time

VolumeQ [

s

m 3

]

If the flow is steady, then Q is a constant. This is called steady flow.

The unit of volumetric flow rate iss

m 3

. Since this is a very large measure, other units

are used to represent most flows. The most commonly used areh

m 3

ormin

litres. These are

not SI units. Also, note that 1000 litres = 1 m3.

1.4.4 Average Velocity (U) [Unit:s

m]

Each fluid particle in a flowing fluid, e.g. pipe flow, will most likely not have the same

velocity at a given cross-section. The average velocity at any given cross section or

boundary is the average or mean velocity of all the particles of the fluid crossing the

section in the direction perpendicular to it. If the fluid is flowing in the x direction, it

will be U. Since there is no net flow in the y direction the average velocity in the y

direction is zero. In fluid mechanics, when we refer to velocity of flow we usually mean

the average velocity.

EAS109 STUDY UNIT 1

SU1-13

1.4.5 Relationship between Volumetric Flow Rate, Q [s

m 3

] and Average Velocity of

Flow, U [s

m]

Consider a cross-sectional area A [m2] through which fluid is flowing with average

velocity U [s

m]. The volume of fluid moved in time t [sec] is Volume moved = UtA

Note that the distance move by the fluid is Ut.

But UAt

UtAQ

t

moved Volume

Therefore UAQ

Figure 1-7 Relationship between Flow Rate and Flow Velocity (Selva Raj & Loh HC,

2015, p. 20)

1.4.6 Mass Flow Rate (

m ) [Unit: s

kg]

Mass flow rate is related to volumetric flow rate, Q by

m = Q

EAS109 STUDY UNIT 1

SU1-14

1.4.7 The Continuity Equation

Figure 1-8 The Flow Continuity Concept (Selva Raj & Loh HC, 2015, p. 21)

1.4.8 Law of Conservation of Mass

When applied to fluid flows, the law of conservation of mass states that in a steady

flow, the rate at which mass enters a control volume per unit time is the same at which

it leaves per unit time, provided the control volume does not increase in mass.

Consider a fluid entering a pipe at section 1 and leaving at section 2. Applying the Law

of Conservation of Mass, we have

1

m = 2

m

This gives rise to the continuity equation which can be written

1Q1 = 2Q2

If the fluid is incompressible then 1 = 2. Therefore Q1 = Q2.

We usually express it in the following way which is useful for solving fluid flow

problems,

U1A1 = U2A2

where U1 = average velocity of flow at section 1 in s

m

A1 = cross-sectional area at section 1 in m2

U2 = average velocity of flow at section 2 ins

m

A2 = cross-sectional area at section 2 in m2

The Continuity Equation


EAS109 STUDY UNIT 1

SU1-15

Topic 2 Definition of Hydrostatic Pressure

2.1 Introduction

Fluid statics: Deals with problems associated with fluids at rest. The fluid can be either

gaseous or liquid. Fluid statics is generally referred to as hydrostatics when the fluid

is a liquid and as aerostatics when the fluid is a gas.

In fluid statics, there is no relative motion between adjacent fluid layers, and thus there

are no shear (tangential) stresses in the fluid trying to deform it. The only stress we

deal with in fluid statics is the normal stress, which is the pressure, and the variation

of pressure is due only to the weight of the fluid. The topic of fluid statics has

significance only in gravity fields.

The design of many engineering systems such as water dams and liquid storage tanks

requires the determination of the forces acting on the surfaces using fluid statics.

2.2 Hydrostatic Forces on Submerged Plane Surfaces

A plate, such as a gate valve in a dam, the wall of a liquid storage tank, or the hull of a

ship at rest, is subjected to fluid pressure distributed over its surface when exposed to

a liquid.

On a plane surface, the hydrostatic forces form a system of parallel forces, and we often

need to determine the magnitude of the force and its point of application, which is

called the centre of pressure.

When analysing hydrostatic forces on submerged surfaces, the atmospheric pressure

can be subtracted for simplicity when it acts on both sides of the structure.

Figure 2-1 Hydrostatic Force (Cengel, Turner & Cimbala, 2012, p. 449)

EAS109 STUDY UNIT 1

SU1-16

Figure 2-2 Hydrostatic Force on an Inclined Surface (Cengel, Turner & Cimbala, 2012,

p. 449)

The illustrations above show the hydrostatic force on an inclined plane surface

completely submerged in a liquid.

Please refer to the 2 illustrations (Figure 2-3) below while reading the following text.

The pressure at the centroid of a surface is equivalent to the average pressure on the

surface. The resultant force acting on a plane surface is equal to the product of the

pressure at the centroid of the surface and the surface area, and its line of action passes

through the centre of pressure

Figure 2-3 Resultant Force and Centroid Pressure for a Plane Surface (Cengel, Turner &

Cimbala, 2012, p. 450)

EAS109 STUDY UNIT 1

SU1-17

Figure 2-4 Hydrostatic Force acting on the Right Surface of a Vertical Plate (Cengel, Turner

& Cimbala, 2012, p. 452)

Hydrostatic force acts on the right surface of a submerged vertical rectangular plate.

Figure 2-5 Hydrostatic Force acting on the Top Surface of a Horizontal Plate (Cengel,

Turner & Cimbala, 2012, p. 452)

Hydrostatic force acts on the top surface of a submerged horizontal rectangular plate.

EAS109 STUDY UNIT 1

SU1-18

2.3 Hydrostatic Forces on Submerged Curved Surfaces

Figure 2-6 Determination of the Hydrostatic Force acting on a Submerged Curved Surface

(Cengel, Turner & Cimbala, 2012, p. 454)

The equations we use to determine hydrostatic force acting on a submerged curved

surface are listed below:

When a curved surface is above the liquid, the weight of the liquid and the vertical

component of the hydrostatic force act in the opposite directions.

The hydrostatic force acting on a circular surface always passes through the centre of

the circle since the pressure forces are normal to the surface and they all pass through

the centre.

Figure 2-7 Hydrostatic Force on a Curved Surface


Figure 2-8 Hydrostatic Force on a Semi-circular

Surface (Cengel, Turner & Cimbala, 2012, p. 455)

EAS109 STUDY UNIT 1

SU1-19

Summary The basic concept of fluid mechanics was introduced in this study unit. Definition of

key fluid properties and basic fluid calculations were highlighted. The foundation for

the systematic study of fluid mechanics was laid with the key definitions and a

working definition of density, viscosity and vapour pressure provided. Two

components which define a state of fluid, pressure and flow rate were also highlighted.

The hydrostatic pressure was studied by calculating the result of effective pressure on

an object in relation to the depth in a fluid. Examples of hydrostatic forces acting on

submerged flat and curved surfaces were also studied.

TUTORIAL: Classification of Fluid Flows

Problem 10-1C to 10-3C

Vapour & Cavitation

Problem 10-7C & 10-10

Viscosity

Problem 10-13C

Problem 10-26

Hydrostatics

11-3C to 11-5C, 11-12 & 11-30

TUTORIAL:

Exercises Complete the exercises in “Review of Basic Fluid Mechanics.docx”.

Solutions

Refer to “Guides to Tutorial Questions Seminar 1.docx”.

References

[1] Selva Raj, J., Loh, H. C. (2015). Thermofluids I (5th ed.). Singapore: McGraw-Hill.

[2] Cengel, Y. A., Turner, R. H., Cimbala, J. M. (2012). Fundamentals of Thermal-Fluid

Sciences (4th ed.). Singapore: McGraw-Hill.

[3] Munson, B. R., Rothmaver, A. P., Okiishi, T. H., Huebsch, W. W. (2012).

Fundamentals of Fluid Mechanics (7th ed.). USA: Wiley. (Chapter 1 & 2, the reference has

links to many interesting video files).

EAS109


STUDY UNIT 2

KINEMATICS AND

DIMENSIONAL ANALYSIS

EAS109 STUDY UNIT 2

SU2-1

Learning Outcomes


Distinguish between the Eulerian and the Lagrangian descriptions of fluid

flow.

Define the streakline, streamline, pathline, timeline.

Identify the different visualising techniques based on viewing photograph

images.

Distinguish between rotational and irrotational flow regions using the flow

property vorticity.

Define a dimensional analysis equation.

Apply dimensional analysis to define accurate fluid flow parameters, in

particular the Mach Number, Reynolds Number.

Overview

In this unit, we study the Eulerian and Lagrangian descriptions of fluid flow. The

definitions of the streakline, streamline, pathline and timeline precede the different

visualising techniques to ‘see’ fluid flow. We then study the difference between

rotational and irrotational flow regions using the flow property vorticity.

We will also learn Dimensional Analysis as a key topic in research and design in fluid

mechanics. Dimensional analysis, using Buckingham’s Theorem, helps us derive

coherent and dimensionally accurate fluid flow parameters that can be used in

modelling and design.

EAS109 STUDY UNIT 2

SU2-2

Chapter 2: Fluid Kinematics and Dimensional Analysis

READ:

Chapter 11 of Fundamentals of Thermal-Fluid Sciences 3E

Topic 3: Basic Fluid Kinematics

3.1 Eulerian and Lagrangian Fluid Flow

Kinematics: This is the study of motion. In this unit we will study the motion of fluids

and descriptors of its motion.

Fluid kinematics: The study of how fluids flow and how to describe fluid motion.

There are two distinct ways to describe motion: Lagrangian and Eulerian.

Lagrangian description: To follow the path of individual objects. This method requires

us to track the position and velocity of each individual fluid parcel (fluid particle) and

take to be a parcel of fixed identity.

A more common method is Eulerian description of fluid motion.

In the Eulerian description of fluid flow, a finite volume called a flow domain or

control volume is defined, through which fluid flows in and out. Instead of tracking

individual fluid particles, we define field variables, functions of space and time, within

the control volume.

The field variable at a particular location at a particular time is the value of the variable

for whichever fluid particle happens to occupy that location at that time.

For example, the pressure field is a scalar field variable. We define the velocity field as

a vector field variable.

The velocity and acceleration field can be expanded in Cartesian coordinates as:

Collectively, these (and other) field variables define the flow field.

EAS109 STUDY UNIT 2

SU2-3

Eulerian and the Lagrangian - details

In the Eulerian description, we don’t really care what happens to individual fluid

particles; rather we are concerned with the pressure, velocity, acceleration, etc. of

whichever fluid particle happens to be at the location of interest at the time of interest.

While there are many occasions in which the Lagrangian description is useful, the

Eulerian description is often more convenient for fluid mechanics applications.

Experimental measurements are generally more suited to the Eulerian description.

In the Eulerian description, one defines field variables, such as the pressure field and

the velocity field, at any location and instant in time.

Example 3.1: A 2-Dimensional velocity field

Figure 3-1 2D Velocity Field (Cengel, Turner & Cimbala, 2011, p. 428)

Example 3.1 describes a flow field near the bell mouth inlet of a hydroelectric dam.

Velocity vectors are shown as arrows. The scale is shown by the top arrow, and the

solid black curves represent the approximate shapes of some streamlines, based on the

calculated velocity vectors.

The stagnation point is indicated by the circle. The shaded region represents a portion

of the flow field that can approximate flow into an inlet.

EAS109 STUDY UNIT 2

SU2-4

The Acceleration Field

Figure 3-2 Acceleration and Velocity Vector Orientation (Cengel, Turner & Cimbala, 2011,

p. 429)

Newton’s second law applied to a fluid particle; the acceleration vector is in the same

direction as the force vector (black arrow), but the velocity vector may act in a different

direction.

The Material Derivative

The total derivative operator d/dt in this equation is given a special name, the material

derivative; it is assigned a special notation, D/Dt, in order to emphasise that it is formed

by following a fluid particle as it moves through the flow field. Other names for the

material derivative include total, particle, Lagrangian, Eulerian, and substantial

derivative.

The material derivative D/Dt is defined by following a fluid particle as it moves

throughout the flow field. In this illustration, the fluid particle is accelerating to the

right as it moves up and to the right.

Figure 3-3 Composition of the Material Derivative (Cengel, Turner & Cimbala, 2011, p. 430)

Local acceleration Advective (convective)

acceleration

Total acceleration

EAS109 STUDY UNIT 2

SU2-5

The components of the acceleration vector in Cartesian coordinates are shown below:

Figure 3-4 Distinguishing the X and Y Components of the Velocity Vector (Cengel, Turner

& Cimbala, 2011, p. 429)

When following a fluid particle, the x-component of velocity, u, is defined as

dxparticle/dt. Similarly, v = dyparticle/dt and w = dzparticle/dt. Movement is shown here only

in two dimensions for simplicity.

Example 3.2: Flow of water through the nozzle of a garden hose (below) illustrates that

fluid particles may accelerate, even in a steady flow. In this example, the exit speed of

the water is much higher than the water speed in the hose, implying that fluid particles

have accelerated even though the flow is steady.

Figure 3-5 Acceleration is possible with Steady Flow (Cengel, Turner & Cimbala, 2011, p. 430)

EAS109 STUDY UNIT 2

SU2-6

Example 3.3: Material acceleration in a steady velocity field

Acceleration vectors for the velocity field of shown in Figure 3-6. The scale is shown

by the top arrow, and the solid black curves represent the approximate shapes of some

streamlines, based on the calculated velocity vectors. (See the figure below. The

stagnation point is indicated by the circle.)

Please refer to Examples 11-1 and 11-2 in p. 427 and p. 430 respectively for specifics of

creating this flow field.

Figure 3-6 A Typical Material Acceleration in a Steady Flow field (Cengel, Turner &

Cimbala, 2011, p. 431)

3.2 Flow Patterns and Flow Visualisation

Flow visualisation: The visual examination of flow field features. While quantitative

study of fluid dynamics requires advanced mathematics, much can be learned from

flow visualisation. Flow visualisation is useful not only in physical experiments but in

numerical solutions as well [computational fluid dynamics (CFD)].

In fact, the very first thing an engineer using CFD does after obtaining a numerical

solution is simulate some form of flow visualisation.

x

EAS109 STUDY UNIT 2

SU2-7

Figure 3-7 The Photograph shows a Spinning Baseball. Here the flow speed is about 23 m/s

and the ball is rotated at 630 rpm. (Cengel, Turner & Cimbala, 2011, p. 431)

3.2.1 Streamlines

Definition of a Streamline: A curve that is everywhere tangent to the instantaneous

local velocity vector.

Streamlines are useful as indicators of the instantaneous direction of fluid motion

throughout the flow field. For example, regions of recirculating flow and separation of

a fluid off of a solid wall are easily identified by the streamline pattern. Streamlines

cannot be directly observed experimentally except in steady flow fields.

Figure 3-8 Streamline Definition: Resolving the Velocity Vector into the X and Y Directions


The figure shows streamlines for a steady,

incompressible, two-dimensional velocity

field. Streamlines for the velocity field of

velocity vectors (short arrows) are

superimposed for comparison.

The velocity vectors point everywhere

tangent to the streamlines. Note that speed

cannot be determined directly from the

streamlines alone. Figure 3-9 Streamlines for a Steady, Incompressible, Two-dimensional Velocity Field.


EAS109 STUDY UNIT 2

SU2-8

3.2.2 Pathlines

Definition of a Pathline: The actual path travelled by an individual fluid particle over

some time period.

A pathline is a Lagrangian concept in that we simply follow the path of an individual

fluid particle as it moves around in the flow field. Thus, a pathline is the same as the

fluid particle’s material position vector (xparticle(t), yparticle(t), zparticle(t)) traced out over

some finite time interval.

In the photograph, pathlines produced by

white tracer particles suspended in water

and captured by time-exposure

photography; as waves pass horizontally,

each particle moves in an elliptical path

during one wave period.

Figure 3-10 Definition of a Pathline (Cengel, Turner & Cimbala, 2011, p. 433)

Figure 3-11 Pathline Visualisation Example (Cengel, Turner & Cimbala, 2011, p. 433)

3.2.3 Streaklines

Definition of a Streakline: The locus of fluid particles that have passed sequentially

through a prescribed point in the flow.

Streaklines are the most common flow pattern generated in a physical experiment. If

you insert a small tube into a flow and introduce a continuous stream of tracer fluid

(dye in a water flow or smoke in an air flow), the observed pattern is a streakline.

In the figure on the right, a streakline is formed by

continuous introduction of dye or smoke from a

point in the flow. Labelled tracer particles (1

through 8) were introduced sequentially.

Figure 3-12 Streakline Definition (Cengel, Turner & Cimbala, 2011, p. 434)

EAS109 STUDY UNIT 2

SU2-9

Figure 3-13 Streaklines Example: Streaklines produced by Fluid introduced Upstream


Summary – Streamlines, Pathlines and Streaklines

Streaklines, streamlines and pathlines are identical in steady flow but they can be quite

different in unsteady flow. The main difference is that a streamline represents an

instantaneous flow pattern at a given instant in time, while a streakline and a pathline

are flow patterns that have some age and thus a time history associated with them. A

streakline is an instantaneous snapshot of a time-integrated flow pattern. A pathline,

on the other hand, is the time-exposed flow path of an individual particle over some

time period.

3.2.4 Timelines

Definition of a timeline: A set of adjacent fluid particles that were marked at the same

(earlier) instant in time.

Timelines are particularly useful in situations where the uniformity of a flow (or lack

thereof) is to be examined.

In the figure on the right, timelines are formed

by marking a line of fluid particles, and then

watching that line move (and deform) through

the flow field; timelines are shown at t = 0, t1,

t2, and t3.

Figure 3-14 Timeline Definition (Cengel, Turner & Cimbala, 2011, p. 436)

EAS109 STUDY UNIT 2

SU2-10

Figure 3-15 Timeline Example (Cengel, Turner & Cimbala, 2011, p. 436)

In the photograph above, timelines produced by a hydrogen bubble wire are used to

visualise the boundary layer velocity profile shape. Flow is from left to right, and the

hydrogen bubble wire is located to the left of the field of view. Bubbles near the wall

reveal a flow instability that leads to turbulence.

3.2.5 Surface Flow Visualisation Techniques

The direction of fluid flow immediately above a solid surface can be visualised with

tufts—short, flexible strings glued to the surface at one end that point in the flow

direction.

Tufts are especially useful for locating regions of flow separation, where the flow

direction suddenly reverses.

A technique called surface oil visualisation can be used for the same purpose—oil

placed on the surface forms streaks called friction lines that indicate the direction of

flow.

If it rains lightly when your car is dirty (especially in the winter when salt is on the

roads), you may have noticed streaks along the hood and sides of the car, or even on

the windshield. This is similar to what is observed with surface oil visualisation.

Lastly, there are pressure-sensitive and temperature-sensitive paints that enable

researchers to observe the pressure or temperature distribution along solid surfaces.

3.3 Vorticity and Rotationality

A kinematic property of great importance to the analysis of fluid flows is the vorticity

vector, defined mathematically as the curl of the velocity vector. Vorticity is equal to

twice the angular velocity of a fluid particle.

EAS109 STUDY UNIT 2

SU2-11

The direction of a vector cross product is determined by the

right-hand rule.

The vorticity vector is equal to twice the angular velocity vector

of a rotating fluid particle. If the vorticity at a point in a flow

field is nonzero, the fluid particle that happens to occupy that

point in space is rotating. The flow in that region is called

rotational.

Likewise, if the vorticity in a region of the flow is zero (or

negligibly small), fluid particles there are not rotating; the flow

in that region is called irrotational.

Physically, fluid particles in a rotational region of flow rotate

end over end as they move along in the flow.

Figure 3-17 Vorticity Vector and Angular Velocity (Cengel, Turner & Cimbala, 2011, p. 437)

Figure 3 -18 Distinguishing between Rotational and Irrotational Flow (Cengel, Turner &

Cimbala, 2011, p. 437)

The illustration above demonstrates the difference between rotational and irrotational

flow: fluid elements in a rotational region of the flow rotate, but those in an irrotational

region of the flow do not.

Figure 3-16 Vector Cross Product Right Hand Rule (Cengel, Turner & Cimbala, 2011, p. 437)

EAS109 STUDY UNIT 2

SU2-12

3.3.1 Comparison of Two Circular Flows

The figures below illustrate streamlines and velocity profiles for (a) flow A, solid-body

rotation and (b) flow B, a line vortex. Flow A is rotational, but flow B is irrotational

everywhere except at the origin.

Figure 3-19 Flow A, Solid-body Rotation

(Cengel, Turner & Cimbala, 2011, p. 439) Figure 3-20 Flow B, a Line Vortex


EAS109 STUDY UNIT 2

SU2-13

Topic 4: Dimensional Analysis

4.1 Introduction

The dimensions of a physical quantity refer to the base quantities, length [L], mass [M]

and time [T] that constitute it. The dimensions of area are always length squared, [L^2].

The dimensions of velocity are always length/time, [L/T]. What are the dimensions of

pressure, flow and power? We will consider these three parameters relating to fluid

flow extensively in this course.

Next, dimensions can be used to help in working out relationships between fluid flow

properties.

This style of calculation is where mathematics helps us to be systemic is called

Dimensional Analysis. You will find this approach ideal when working with fluid flow

where we are not really sure of the relationships between variables.

4.2 Dimensions and Units

There are rules to use when working with dimensions:

1. Add or subtract quantities only if they have the same dimensions.

2. Dimensions can be divided and multiplied in algebraic manner.

3. We do not use ‘=’ (the equal sign) but the equivalence sign ‘ ‘.

Let us check the dimensions of volumetric flow rate.

Its formula/definition is Q = v*A. [Q] [v]*[A] [M/S]*[M^2] [M^3/S}

As an exercise, derive the dimensions of pressure, density and viscosity (both

kinematic and dynamic viscosity) listed in the table below.

EAS109 STUDY UNIT 2

SU2-14

The following table lists dimensions of some common physical quantities:

Quantity SI Unit Dimension

power

Watt W

N m/s

kg m2/s3

Nms-1

kg m2s-3

ML2T-3

pressure

Pascal P,

N/m2,

kg/m/s2

Nm-2

kg m-1s-2

ML-1T-2

density kg/m3 kg m-3 ML-3

relative density a ratio

no units

[1]

no dimension

Viscosity

(dynamic)

N s/m2

kg/m s

N sm-2

kg m-1s-1

M L-1T-1

4.3 Dimensional Homogeneity

An equation is said to be dimensionally homogenous if the dimensions of the terms

on its left hand side are the same as the dimensions of the terms on its right hand side.

If an expression or variable is dimensionless, all its variables cancel one another. We

say also that this variable has no units. A good example is the Reynolds Number,

Please prove that it is indeed dimensionless. Another good example is the Mach

number, where c = the sonic velocity and u = the flow velocity.

Dimensions of some of the physical quantities are given below for your use.

From the table above, we can see that in SI.

Please note that in SI all capital letters in units are derived and not the original base

quantities (length [L], mass [M] and time [T]).

(a) Pressure – [ML-1T-2] is N/m2

(b) Dynamic viscosity – [ML-1T-1] can be Pas or kg/ms or Ns/m2

EAS109 STUDY UNIT 2

SU2-15

4.4 Results of Dimensional Analysis

Dimensional analysis is used in fluid mechanics to prepare analysis of experiments.

These results from these experiments, especially on models, allow us to develop a

prototype from which we can scale up our results, and, predict reasonably well its

results and output from model we make to the final prototype.

Unfortunately, in fluid mechanics, the effects of viscosity are very difficult to replicate

accurately in model testing and the leap from model to prototype usually requires

extensive wind tunnel testing. This results in coefficients, the lift coefficient, the drag

coefficient and the friction coefficient which we will see as we progress through the

course and during the laboratory classes.

4.5 Buckingham's Theorem

In the Buckingham π theorem, a certain number n of physical variables and the original

equation under investigation can be rewritten in terms of a set of p=n−k dimensionless

parameters π1, π2 ,…….., πp constructed from the original variables, where k is the

number of physical dimensions involved.

A reference to look up before your Laboratory Class 5 on how Buckingham’s theorem

can be used to analyse the parameters in pipe flow is available at:

http://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=fl&chap_sec=06.1&page=theory

4.6 Common Groups

During dimensional analysis, several groups will appear again and again for different

problems. These often have special names after prominent engineering experimenters.

You will recognise the Reynolds Number, and the Mach Number.

We will understand and appreciate these powerful dimensionless groups when we

study velocity of flow in a pipe and the transition between two types of fluid flow –

laminar and turbulent flow.

http://ecourses.ou.edu/cgi-bin/eBook.cgi?doc=&topic=fl&chap_sec=06.1&page=theory

EAS109 STUDY UNIT 2

SU2-16

Summary

The Eulerian and Lagrangian descriptions of fluid flow were discussed. The different

flow visualisation techniques together with the definition of the streakline, streamline,

pathline and timeline were also identified and defined. In addition, the difference

between rotational and irrotational flow regions using the flow property vorticity was

also distinguished.

The application of Dimensional Analysis, using Buckingham’s Theorem, was also

carried out to help us derive coherent and dimensionally accurate fluid flow

parameters for the modelling and design.

TUTORIAL:

Lagrangian vs. Eulerian Descriptors

Example 11-8C & 11-9C

Visualisation

Example 11-27C & 11-29C

Vorticity & Rotationality

Example 11-36C

Solutions

Refer to “Guides to Tutorial Questions Seminar 1.docx”.

Reference



EAS109


STUDY UNIT 3

THERMODYNAMIC PRINCIPLES

AND SYSTEMS

EAS109 STUDY UNIT 3

SU3-1

Learning Outcomes


Identify the first law of thermodynamics as a statement of the conservation of energy

principle to closed (fixed mass) systems.

Apply the general energy balance to closed systems.

Identify the first law of thermodynamics as the statement of the conservation of

energy principle to control volumes (open systems).

Recall that the energy carried by a fluid stream crossing a control surface as the sum

of internal energy, flow work, kinetic energy, and potential energy of the fluid.

Recall the combination of the internal energy and the flow work to the property

enthalpy.

Define the conservation of mass principle for closed thermodynamic systems.

Apply the conservation of mass principle to steady control volumes (open systems).

Apply the first law of thermodynamics to steady systems.

Solve energy balance problems for common closed systems such as pistons operating

within cylinders, a fixed volume, pressure and temperature systems.

Solve energy balance problems for common steady-flow devices such as nozzles,

compressors, turbines and throttling valves.

Overview

In this unit, the First Law of Thermodynamics for closed systems will be studied. We first

identify the first law of thermodynamics as simply a statement of the conservation of energy

principle for closed (fixed mass) systems and develop the general energy balance applied to

closed systems.

Next, we apply the first law of thermodynamics as the statement of the conservation of energy

principle to control volumes (open systems). The theory used is then applied to solve energy

balance problems for common steady-flow devices such as nozzles, compressors, turbines,

and throttling valves.

Before you proceed to learn about thermodynamics processes, please view these introductory

calculations to enable you to understand and manipulate property values.

1. Thermodynamic Properties

2. Properties, States and Processes

3. Determining Property Changes during Processes


EAS109 STUDY UNIT 3

SU3-2

Chapter 3: First Law of Thermodynamics for Closed and Open

Systems

Topic 5: First Law of Thermodynamics for Closed Systems

5.1 Energy Analysis of Closed Systems

Fig. A Fig. B

Figure 5-1 Moving Boundary Work (Cengel, Turner & Cimbala, 2012, p. 146)

Moving Boundary Work

Moving boundary work = (PdV): This expansion and compression work in a piston-cylinder

device. The gas does a differential amount of work Wb as it forces the piston to move by a

differential amount ds as shown in the following equation.

Integrating from 12

Quasi-equilibrium process: A process during which the system remains nearly in

equilibrium at all times

Wb is positive is for expansion processes

Wb is negative for compression processes

In Fig. B, the work associated with a moving boundary is called boundary work.

READ: Chapter 3 of Thermofluids I

Chapter 3, 5 and 6 of Fundamentals of Thermal-Fluid Sciences

EAS109 STUDY UNIT 3

SU3-3

Figure 5-2 Area under the P-V Diagram (Cengel, Turner & Cimbala, 2012, p. 147)

The area under the process curve on a P-V diagram represents the boundary work

Figure 5-3 Work Done during the Process depends on the Path (Cengel, Turner & Cimbala,

2012, p. 147)

The boundary work done during a process depends on the path followed as well as the end

states.

The net work done during a cycle is the difference between the work done by the system and

the work done on the system.

EAS109 STUDY UNIT 3

SU3-4

5.2 Polytropic, Isothermal, Isobaric and Isochoric Processes

Figure 5-4 Polytropic Process (Cengel, Turner & Cimbala, 2012, p. 150)

This is a polytropic process where C and n (the polytropic exponent) are constants.

The work done for a polytropic process is given as follows:

For an ideal gas,

pV = mRT

When n=1 (an isothermal process) then

For a constant pressure process

What is the boundary work for a constant volume process?

5.3 Energy Balance for Closed Systems

The energy balance for any system undergoing any process can be expressed as:

EAS109 STUDY UNIT 3

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When expressed as a rate, it takes the following form:

Note the units kW instead of kJ.

The total quantities are related to the quantities per unit time is given by

Now, the energy balance per unit mass is

This energy in differential form is:

The energy balance per cycle is given by:

Follow the procedure of this video example to build your confidence when working with

processes.

Non-flow process calculations


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Energy balance when the sign convention is used (i.e. heat input and work output are positive

while heat output and work input are negative).

For a cycle E = 0 so Q = W

Figure 5-5 A Thermodynamic Cycle (Cengel, Turner & Cimbala, 2012, p. 152)

NOTE: The first law of thermodynamics cannot be proven mathematically, but no process in

nature is known to have violated the first law, and this should be taken to be sufficient proof.

5.4 Energy Balance

5.4.1 Expansion or Compression Process

In a general analysis for a closed system undergoing a quasi-equilibrium constant-pressure

process, Q is to the system and W is from the system.

But by definition

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An example of a constant pressure process

Figure 5-6 A Constant Pressure Process (Cengel, Turner & Cimbala, 2012, p. 133)

5.5 Specific Heats

5.5.1 Specific Heat at Constant Volume, cv

This is the energy required to raise the temperature of the unit mass of a substance by one

degree as the volume is maintained constant.

5.5.2 Specific Heat at Constant Pressure, cp

This is the energy required to raise the temperature of the unit mass of a substance by one

degree as the pressure is maintained constant.

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

one degree in a specified way. Is the following statement True or False? The cp is always

greater than cv for perfect gases.

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Figure 5-7 Specific Heats of a Gas (Cengel, Turner & Cimbala, 2012, p. 157)

As an illustration, in the diagram above, the constant-volume and constant-pressure specific

heats cv and cp values are for helium gas.

Also, the equations in the figure are valid for any substance undergoing any process. Note that

cv and cp are properties. cv is related to the changes in internal energy and cp to the changes in

enthalpy. A common unit for specific heats is kJ/kg · °C or kJ/kg · K. Are these units identical?

The specific heat of a substance changes with temperature.

Formal definitions of cv and c

p are partial derivatives shown below:

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5.6 Internal Energy, Enthalpy and Specific Heats of Ideal Gases

5.6.1 Joule’s Experiment

Joule showed using this experimental apparatus that u = u(T). Also it follows from the

definition of h and the specific heats of gas that the following relations hold (in the black box).

NOTE: For ideal gases, u, h, cv, and cp vary with temperature only

Figure 5-8 Relationships between the Various Thermodynamic Properties (Cengel, Turner &

Cimbala, 2011, pp. 174-175)

The proof of the relationships in the black box can be shown as follows:

Internal energy change and enthalpy change of an ideal gas are shown below:

At low pressures, all real gases approach ideal-gas behaviour, and therefore their specific

heats depend on temperature only. The specific heats of real gases at low pressures are called

ideal-gas specific heats, or zero-pressure specific heats, and are often denoted cp0 and cv0.u and h

data for a number of gases have been tabulated below for your reference. These tables are

obtained by choosing an arbitrary reference point and performing the integrations by treating

state 1 as the reference state.

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GRAPH

The Ideal-gas constant-pressure specific heats for some common gases

Figure 5-9 Specific Heats at Constant Pressure for Common Gases


TABLE

In the preparation of ideal-gas tables, 0K is chosen as the reference temperature. Refer to the

table for air

Figure 5-10 Table Values used for Easy Extraction of Data (Cengel, Turner & Cimbala, 2012, p. 159)

Internal energy and enthalpy change when specific heat is taken constant at an average value.

For small temperature intervals, the specific heats may be assumed to vary linearly with

temperature.

Figure 5-11 Internal Energy and Enthalpy Calculations (Cengel, Turner & Cimbala, 2012, p. 160)

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The relation u = cv T is valid for any kind of process, constant-volume or not.

There are three ways of calculating u and h

1. By using the tabulated u and h data. This is the easiest and

most accurate way when tables are readily available.

2. By using the cv or c

p relations as a function of temperature

and performing the integrations. This is very inconvenient

for hand calculations but quite desirable for computerised

calculations. The results obtained are very accurate.

3. By using average specific heats. This is very simple and

certainly very convenient when property tables are not

available. The results obtained are reasonably accurate if

the temperature interval is not very large.

5.6.2 Specific Heat Relations of Ideal Gases

The relationship between cp, cv and R

Specific heat ratio

and the following dh = cpdT and du = c

vdT

The cp of an ideal gas can be determined from knowledge of cv and R. The specific ratio varies

with temperature, but this variation is very mild. For monatomic gases (helium, argon, etc.),

its value is essentially constant at 1.667. Many diatomic gases, including air, have a specific

heat ratio of about 1.4 at room temperature.

5.7 Internal Energy, Enthalpy and Specific Heats of Solids and Liquids

Please watch a sample calculation first.

Joule’s Law


Incompressible substance: A substance whose specific volume (or density) is constant. Solids

and liquids are incompressible substances.

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The specific volumes of incompressible substances remain constant during a process. The cv and

cp values of incompressible substances are identical and are denoted by c.

Figure 5-12 Internal Energy, Enthalpy and Specific Heats of Solids and Liquids (Cengel, Turner &

Cimbala, 2012, p. 165)

5.8 Summary of Formulae for Calculations

Internal Energy Changes

Enthalpy Changes

Topic 6: Mass and Energy Analysis of Control Volumes

6.1 Conservation of Mass

Conservation of mass: Mass, like energy, is a conserved property, and it cannot be created

or destroyed during a process.

Closed systems: The mass of the system remains constant during a process.

Control volumes: Mass can cross the boundaries, and so we must keep track of the amount

of mass entering and leaving the control volume.

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Mass is conserved even during chemical reactions

Mass m and energy E can be converted to each other through 𝐸 = 𝑚𝑐2 where c is the speed

of light in a vacuum, which is c = 2.9979 108 m/s. The mass change due to energy change is

absolutely negligible.

6.1.1 Mass and Volume Flow Rates

Mass flow rate

The volume flow rate is the volume of fluid flowing through a cross section per unit time.

6.1.2 Definition of Average Velocity

The average velocity Vavg is defined as the average speed through a cross section.

Figure 6-1 Defining Average Velocity (Cengel, Turner & Cimbala, 2012, p. 187)

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6.1.3 Conservation of Mass Principle

The conservation of mass principle for a control volume: The net mass transfer to or from a

control volume during a time interval t is equal to the net change (increase or decrease) in

the total mass within the control volume during t.

Example 6.1: Conservation of mass principle for an ordinary bathtub

Consider a bathtub where water enters through a faucet. An overflow controls the mass

within the tub.

The mass quantities are illustrated in the picture.

General conservation of mass

General conservation of mass in rate form

or

Figure 6-2 Defining Average Velocity (Cengel, Turner & Cimbala, 2012, p. 188)

6.2 Mass Balance for Steady-flow Processes

During a steady-flow process, the total amount of mass contained within a control volume

does not change with time (mCV = constant).

Then the conservation of mass principle requires that the total amount of mass entering a

control volume equal the total amount of mass leaving it.

For steady-flow processes, we are interested in the amount of mass flowing per unit time,

that is, the mass flow rate.

For multiple inlets and exits

Many engineering devices such as nozzles, diffusers, turbines, compressors, and pumps

involve a single stream (only one inlet and one outlet), i.e.

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The figure illustrates conservation of mass principle for a two-inlet–one-outlet steady-flow

system.

Figure 6-3 Mass Balance for Steady-flow Processes (Cengel, Turner & Cimbala, 2012, p. 190)

Special Case: Incompressible Flow

The conservation of mass relations can be simplified even further when the fluid is

incompressible, which is usually the case for liquids.

There is no such thing as a “conservation of volume” principle.

However, for steady flow of liquids, the volume flow rates as well as the mass flow rates

remain constant since liquids are essentially incompressible substances.

During a steady-flow process, volume flow rates are not necessarily conserved although mass

flow rates are.

Steady, incompressible

Steady, incompressible flow (single stream)

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6.3 Flow Work and the Energy of a Flowing Fluid

Flow work, or flow energy: The work (or energy) required to push the mass into or out of the

control volume. This work is necessary for maintaining a continuous flow through a control

volume.

Since 𝐹 = 𝑃𝐴

In the absence of acceleration, the force applied on a fluid by a piston is equal to the force

applied on the piston by the fluid.

Figure 6-4 Applied Force on a Fluid (Cengel, Turner & Cimbala, 2011, p. 193)

The schematic for flow work is shown below.

Figure 6- 5 Flow Work done on a Fluid (Cengel, Turner & Cimbala, 2012, p. 193)

The formulae:

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6.3.1 Total Energy of a Flowing Fluid

The flow energy is automatically taken care of by enthalpy. In fact, this is the main reason for

defining the property enthalpy. The total energy consists of three parts for a non-flowing fluid

and four parts for a flowing fluid.

Figure 6-6 Total Energy in a Moving Fluid (Cengel, Turner & Cimbala, 2011, p. 194)

6.3.2 Energy Transport by Mass

Please refer to the figure below:

The product is the energy transported into control volume by mass per unit time.

When the kinetic and potential energies of a fluid stream are negligible

When the properties of the mass at each inlet or exit change with time as well as over the

cross section

Figure 6-7 Energy Transport by Mass (Cengel, Turner & Cimbala, 2012, p. 194)

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6.4 Energy Analysis of Steady-flow Systems

6.4.1 Introductory Remarks

Many engineering systems such as power plants operate under steady conditions. Under

steady-flow conditions, the mass and energy contents of a control volume remain constant.

Under steady-flow conditions, the fluid properties at an inlet or exit remain constant (do not

change with time).

Figure 6-8 Mass Balance for a Steady-flow Process (Cengel, Turner & Cimbala, 2012, p. 196)

Mass and Energy balances for a steady-flow process

The mass balance is written as follows:

The energy balance is written as follows:

Figure 6-9 Energy Balance for a Steady-flow Process (Cengel, Turner & Cimbala, 2012, p. 197)

Finally, the Energy Equation can be written as

Example

A water heater in

steady operation.

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6.4.2 Energy Balance Relations with Sign Conventions

(i.e., heat input and work output are positive)

When kinetic energy and potential energy changes are negligible

Under steady operation, shaft work and electrical work are the only forms of work a simple

compressible system may involve.

Figure 6-10 Work Forms – Shaft and Electrical Work in Simple Compressible Systems (Cengel,


Steady-Flow Engineering system using vapour.

Understanding the Steam Plant


Some energy unit

equivalents

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6.5 Some Steady-flow Engineering Devices

(1) NOZZLES and DIFFUSERS

(2) TURBINES and COMPRESSORS

(3) THROTTLING VALVES

Many engineering devices operate essentially under the same conditions for long periods of

time. The components of a steam power plant (turbines, compressors, heat exchangers, and

pumps), for example, operate nonstop for months before the system is shut down for

maintenance. Therefore, these devices can be conveniently analysed as steady-flow devices.

Figure 6-11 Gas Turbine (Cengel, Turner & Cimbala, 2012, p. 199)

A modern land-based gas turbine used for electric power production. This is a General Electric

LM5000 turbine. It has a length of 6.2 m, it weighs 12.5 tons, and produces 55.2 MW at 3600

rpm with steam injection.

6.5.1 Nozzles and Diffusers

Nozzles and diffusers are commonly utilised in jet engines, rockets, spacecraft, and even

garden hoses.

A nozzle is a device that increases the velocity of a fluid at the expense of pressure.

A diffuser is a device that increases the pressure of a fluid by slowing it down.

The cross-sectional area of a nozzle decreases in the flow direction for subsonic flows and

increases for supersonic flows. The reverse is true for diffusers.

Nozzles and diffusers are shaped so that they cause large changes in fluid velocities and thus

kinetic energies.

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Figure 6-12 Nozzles and Diffusers (Cengel, Turner & Cimbala, 2012, p. 200)

6.5.2 Turbines and Compressors

Turbine drives the electric generator in steam, gas, or hydroelectric power plants. As the fluid

passes through the turbine, work is done against the blades, which are attached to the shaft.

As a result, the shaft rotates, and the turbine produces work.

Compressors, as well as pumps and fans, are devices used to increase the pressure of a fluid.

Work is supplied to these devices from an external source through a rotating shaft.

A fan increases the pressure of a gas slightly and is mainly used to mobilise a gas.

A compressor is capable of compressing the gas to very high pressures. (In the figure, examine

the energy balance for the compressor.)

Pumps work very much like compressors except that they handle liquids instead of gases.

Figure 6-13 Nozzles and Diffusers (Cengel, Turner & Cimbala, 2012, p. 203)

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6.5.3 Throttling Valves

Throttling valves are any kind of flow-restricting devices that cause a significant pressure drop

in the fluid.

What is the difference between a turbine and a throttling valve?

The pressure drop in the fluid is often accompanied by a large drop in temperature, and for that

reason throttling devices are commonly used in refrigeration and air-conditioning

applications.

In a throttling device, . Since

Figure 6-14 Throttling Process (Cengel, Turner & Cimbala, 2012, p. 206)

Examples of throttling devices

Figure 6-15 Throttling Valves (Cengel, Turner & Cimbala, 2012, p. 206)

The temperature of an ideal gas does not

change during a throttling (h = constant)

process since h = h(T).

During a throttling process, the enthalpy of a

fluid remains constant. But internal and flow

energies may be converted to each other.

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Other processes to view:

1. Understanding the adiabatic process part 1

2. The Characteristic Gas Equation

3. Heat removal in refrigeration

4. Understanding the constant pressure process

5. Understanding the polytropic process

6. Understanding the adiabatic process part 2


Summary The First Law of Thermodynamics for closed systems was studied by first identifying it as

simply a statement of the conservation of energy principle for closed (fixed mass) systems and

then developing the general energy balance applied to closed systems. The first law of

thermodynamics was also applied as the statement of the conservation of energy principle to

control volumes (open systems). The theory used was then applied to solve energy balance

problems for common steady-devices such as nozzles, diffusers, compressors, turbines, and

throttling valves.

TUTORIAL: Chapter 3 – Q3-2C, Q3-3C, Q3-4, Q3-38, Q3-40

Chapter 5 – Examples 5-1 to 5-10, Q5-73, Q5-74

Chapter 6 – Examples 6-1 to 6-13, Q6-28, Q6-38, Q6-135

References

[1] Cengel, Y. A., Turner, R. H., Cimbala, J. M. (2012). Fundamentals of Thermal-Fluid Sciences

(4th ed.). Singapore: McGraw-Hill.

[2] Selva Raj, Loh H.C. (2015). Thermofluids 1 (5th ed.). Singapore: McGraw-Hill.

(Links, through QR Codes, for all the videos in this unit can be also found here.)

EAS109


STUDY UNIT 4

THERMODYNAMIC CYCLES

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Learning Outcomes


State the second law of thermodynamics.

Describe the Kelvin–Planck and Clausius statements of the second law of

thermodynamics.

Define entropy to quantify the second-law effects.

Apply the second law of thermodynamics to solve problems on processes.

Recall the increase of entropy principle.

Calculate the entropy changes that take place during processes for pure

substances, incompressible substances, and ideal gases.

Describe the second law of thermodynamics to cycles and cyclic devices.

Recall the Carnot, Otto, Diesel and Brayton cycles by plotting the p-v and T-s

diagrams for each of these cycles.

Perform calculations for the Carnot, Otto, Diesel and Brayton cycles to

determine heat supplied, heat rejected and thermal efficiencies given a set of

operating conditions.

Overview

In this unit, the Kelvin–Planck and Clausius statements of the second law of

thermodynamics are introduced. Definition of entropy enables us to quantify the

second-law effects. Hence we can apply the second law of thermodynamics to

processes and recall the increase of entropy principle. This will allow us to calculate the

entropy changes that take place during processes for pure substances, incompressible

substances, and ideal gases. We will also apply the second law of thermodynamics to

the Carnot, Otto, Diesel and Brayton cycles and perform calculations to enable heat

supplied, heat rejected and thermal efficiencies be calculated for a variety of given

operating conditions.

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Chapter 4: Second Law of Thermodynamics

READ:




Topic 7: Statements of the Second Law and Entropy

7.1 Introduction to the Second Law

1. A cup of hot coffee does not get hotter in a cooler room.

2. Transferring heat to a paddle wheel will not cause it to rotate.

3. Transferring heat to a wire will not generate electricity.

4. These processes cannot occur even though they are not in violation of the first

law.

5. Processes occur in a certain direction, and not in the reverse direction.

6. A process must satisfy both the first and second laws of thermodynamics to

proceed.

Figure 7-1(a)-(e) Processes occur in One Direction (Cengel, Turner & Cimbala 2012, p. 238)

(a)

(b)

(c)

(d)

(e)

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7.2 Major Uses of the Second Law

1. The second law may be used to identify the direction of processes.

2. The second law also asserts that energy has quality as well as quantity. The first

law is concerned with the quantity of energy and the transformations of energy

from one form to another with no regard to its quality. The second law provides

the necessary means to determine the quality as well as the degree of

degradation of energy during a process.

3. The second law of thermodynamics is also used in determining the theoretical

limits for the performance of commonly used engineering systems, such as heat

engines and refrigerators, as well as predicting the degree of completion of

chemical reactions.

7.3 Thermal Energies – Sources and Sinks

Bodies with relatively large thermal masses can be modelled as

thermal energy reservoirs. A hypothetical body with a relatively

large thermal energy capacity (mass x specific heat) that can supply

or absorb finite amounts of heat without undergoing any change

in temperature is called a thermal energy reservoir, or just a

reservoir.

In practice, large bodies of water such as oceans, lakes, and rivers

as well as the atmospheric air can be modelled accurately as

thermal energy reservoirs because of their large thermal energy

storage capabilities or thermal masses.

A source supplies energy in the form of heat and a sink absorbs it.

Figure 7-2 Sources and Sinks (Cengel, Turner & Cimbala, 2012, p. 239)

7.4 Heat Engines

Heat engines are devices that convert heat to work.

1. They receive heat from a high-temperature source (solar energy, oil furnace,

nuclear reactor, etc.).

2. They convert part of this heat to work (usually in the form of a rotating shaft).

3. They reject the remaining waste heat to a low-temperature sink (the atmosphere,

rivers, etc.).

4. They operate on a cycle.

Heat engines and other cyclic devices usually involve a fluid to and from which heat

is transferred while undergoing a cycle. This fluid is called the working fluid.

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Figure 7-3 Work to Heat Conversions (Cengel, Turner & Cimbala, 2012, p. 240)

Work can always be converted to heat directly and completely, but the reverse is not

true. Part of the heat received by a heat engine is converted to work, while the rest is

rejected to a sink.

7.4.1 Example: A steam power plant

Figure 7-4 Simple Steam Power Plant (Cengel, Turner & Cimbala, 2012, p. 241)

A portion of the work output of a heat engine is consumed internally to maintain

continuous operation.

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Figure 7-5 Concept of Net Heat (Cengel, Turner & Cimbala, 2012, p. 241)

7.4.2 Thermal efficiency

Some heat engines perform better than others (convert more of the heat they receive to

work). Even the most efficient heat engines reject almost one-half of the energy they

receive as waste heat.

Figure 7-6 Concept of a Heat Engine (Cengel, Turner & Cimbala, 2012, p. 241)

Important Formulae when calculating overall thermal efficiencies of heat engines

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7.4.3 Can We Save Qout

?

Figure 7-7 It Is Necessary to Complete the Heat Engine Cycle (Cengel, Turner & Cimbala,

2012, p. 243)

In a steam power plant, the condenser is the device where large quantities of waste

heat are rejected into rivers, lakes, or the atmosphere. Can we not just take the

condenser out of the plant and save all that waste energy?

The answer is, unfortunately, a firm no for the simple reason that without a heat

rejection process in a condenser, the cycle cannot be completed.

A heat-engine cycle cannot be completed without rejecting some heat to a low-

temperature sink. Every heat engine must reject some energy by transferring it to a

low-temperature reservoir in order to complete the cycle, even under idealised

conditions.

7.5 The Second Law of Thermodynamics: Kelvin-Planck Statement

It is impossible for any device that operates on a cycle to receive heat from a single

reservoir and produce a net amount of work. No heat engine can have a thermal efficiency

of 100 percent, or as for a power plant to operate, the working fluid must exchange heat with

the environment as well as the furnace.

The impossibility of having a 100% efficient heat engine is not due to friction or other

dissipative effects. It is a limitation that applies to both the idealised and the actual heat

engines.

The following statement is impossible - All heat is converted to work. This heat engine

violates the Kelvin–Planck statement of the second law.

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Figure 7-8 Kelvin Plank Statement Illustrated (Cengel, Turner & Cimbala, 2012,

p. 245)

7.6 The Second Law of Thermodynamics: Clausius Statement

It is impossible to construct a device that operates in a cycle and produces no effect

other than the transfer of heat from a lower-temperature body to a higher-temperature

body.

It states that a refrigerator cannot operate unless its compressor is driven by an external power

source, such as an electric motor.

This way, the net effect on the surroundings involves the consumption of some energy

in the form of work, in addition to the transfer of heat from a colder body to a warmer

one.

To date, no experiment has been conducted that contradicts the second law, and this

should be taken as sufficient proof of its validity.

Figure 7-9 Clausius Statement Illustrated (Cengel, Turner & Cimbala, 2012, p. 246)

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A refrigerator that makes it possible to remove heat from a low temperature to a high

temperature with work input into the system. This is an example of the practical use

of the Clausius statement of the second law.

7.7 Equivalence of the Kelvin-Plank and Clausius Statements

The Kelvin–Planck and the Clausius statements are equivalent in their consequences,

and either statement can be used as the expression of the second law of

thermodynamics. Any device that violates the Kelvin–Planck statement also violates

the Clausius statement, and vice versa.

7.8 Definitions of Reversible and Irreversible Processes

Reversible process: A process that can be reversed without leaving any trace on the

surroundings.

Irreversible process: A process that is not reversible.

All the processes occurring in nature are irreversible.

Why are we interested in reversible processes?

(1) They are easy to, and

(2) They serve as idealised models (theoretical limits) to which actual processes can be

compared. Some processes are more irreversible than others. We try to approximate

reversible processes. Why?

Figure 7-10 Reversible and Irreversible Processes (Cengel, Turner & Cimbala, 2012, p. 252)

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7.8.1 Irreversibilities

The factors that cause a process to be irreversible are called irreversibilities. They

include friction, unrestrained expansion, mixing of two fluids, heat transfer across a

finite temperature difference, electric resistance, inelastic deformation of solids, and

chemical reactions. The presence of any of these effects renders a process irreversible.

Figure 7-11 More Examples of Irreversible Processes (Cengel, Turner & Cimbala, 2012,

pp. 253-254)

7.8.2 Internally and Externally Reversible Processes

Internally reversible process: No irreversibilities occur within the boundaries of the

system during the process.

Externally reversible: No irreversibilities occur outside the system boundaries.

Totally reversible process: It involves no irreversibilities within the system or its

surroundings. A totally reversible process involves no heat transfer through a finite

temperature difference, no non-quasi-equilibrium changes, and no friction or other

dissipative effects.

Friction renders

a process irreversible.

(a) Heat transfer through a

temperature difference is

irreversible, and

(b) the reverse process is

impossible.

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Figure 7-12 Examples of Reversible Processes (Cengel, Turner & Cimbala, 2012, p. 255)

7.9 The Carnot Cycle – consists of 4 processes to make a cycle

Reversible Isothermal Expansion (process 1-2, TH = constant)

Reversible Adiabatic Expansion (process 2-3, temperature drops from TH to T

L)

Reversible Isothermal Compression (process 3-4, TL = constant)

Reversible Adiabatic Compression (process 4-1, temperature rises from TL to T

H)

A reversible process

involves no internal

and external

irreversibilities.

Totally and

internally reversible

heat transfer

processes

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Figure 7-13 The Carnot Cycle (Cengel, Turner & Cimbala, 2012, p. 256)

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Figure 7-14 The Combination of a Reversible and Irreversible Heat Engine (Cengel, Turner

& Cimbala, 2012, p. 258)

1. The efficiency of an irreversible heat engine is always less than the efficiency of

a reversible one operating between the same two reservoirs.

2. The efficiencies of all reversible heat engines operating between the same two

reservoirs are the same.

7.10 The Carnot Heat Engine

The Carnot heat engine is the most efficient of all heat engines operating between the

same high- and low-temperature reservoirs.

No heat engine can have a higher efficiency than a reversible heat engine operating

between the same high- and low-temperature reservoirs.

Only a fraction of heat can be converted to work. This fraction depends on the

temperature difference between the hot and the cold source.

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Figure 7-15 The Carnot Heat Engine Efficiencies (Cengel, Turner & Cimbala, 2012,

pp. 262-263)

Points to note:

The fraction of heat that can be converted to work is a function of source temperature.

The higher the temperature of the thermal energy corresponds to the higher its quality.

Questions

1. How do you increase the thermal efficiency of a Carnot heat engine?

2. How do you improve actual heat engine efficiency?

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Topic 8: Entropy

8.1 Formal Definition of Entropy

Figure 8-1 The Formal Definition of Entropy (Cengel, Turner & Cimbala, 2012, p. 280)

The equality in the Clausius inequality holds for totally or just internally reversible

cycles and the inequality for the irreversible ones.

Clausius inequality

This is the formal

definition of

entropy

The system

considered in the

development of

Clausius inequality

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8.1.1 Properties of Entropy

1. The entropy change between two specified states is

the same whether the process is reversible or

irreversible. (FIG.A)

2. Entropy is an extensive property.

3. Entropy is a quantity whose cyclic integral is zero

(i.e. a property like volume). (FIG. B)

4. A special case: An internally reversible isothermal

heat transfer process. This equation is particularly

useful for determining the entropy changes of

thermal energy reservoirs.

Figure 8-2 Properties of Entropy (Cengel, Turner & Cimbala, 2012, pp. 281-282)

FIG. A

FIG. B

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8.1.2 The Increase of Entropy Principle

Figure 8-3 The Increase of Entropy Principle (Cengel, Turner & Cimbala, 2012, p. 283)

1. A cycle is composed of a reversible and an irreversible process. (FIG. C)

2. The equality holds for an internally reversible process and the inequality for an

irreversible process (See FIG A also).

3. Some entropy is generated or created during an irreversible process, and this

generation is due entirely to the presence of irreversibilities.

4. The entropy generation Sgen is always a positive quantity or zero.

FIG. C

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8.1.3 Examples

FIG. A

1. The entropy change of an isolated system is the sum of the entropy changes of its

components, and is never less than zero.

Figure 8-4 Entropy of Isolated Systems - A (Cengel, Turner & Cimbala, 2012, p. 285)

FIG. B

1. A system and its surroundings form an isolated system.

Figure 8-5 Entropy of Isolated Systems - B (Cengel, Turner & Cimbala, 2012, p. 285)

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Figure 8-6 Entropy Change can be Negative (Cengel, Turner & Cimbala, 2012, p. 285)

1. Processes can occur in a certain direction only, not in any direction. A process must

proceed in the direction that complies with the increase of entropy principle, that is,

Sgen ≥ 0. A process that violates this principle is impossible.

2. Entropy is a non-conserved property, and there is no such thing as the conservation of

entropy principle. Entropy is conserved during the idealised reversible processes only

and increases during all actual processes.

3. The performance of engineering systems is degraded by the presence of

irreversibilities, and entropy generation is a measure of the magnitudes of the

irreversibilities during that process. It is also used to establish criteria for the

performance of engineering devices.

4. The entropy change of a system can be negative, but the entropy generation cannot.

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8.2 Isentropic Processes

A process during which the entropy remains constant is called an isentropic process.

Figure 8-7 Entropy is Constant for Reversible Adiabatic Processes (Cengel, Turner &

Cimbala, 2012, pp. 290 & 292)

During an internally reversible, adiabatic (isentropic) process, the entropy remains

constant. The isentropic process appears as a vertical line segment on a T-s diagram.

8.2.1 Property Diagrams involving Entropy

Figure 8-8 Extracting Properties from Thermodynamic Diagrams (Cengel, Turner &

Cimbala, 2012, pp. 292 & 293)

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1. On a T-S diagram, the area under the process curve represents the heat transfer

for internally reversible processes.

2. For adiabatic steady-flow devices, the vertical distance ∆h on an h-s diagram is a

measure of work, and the horizontal distance ∆s is a measure of irreversibilities.

8.3 What is Entropy?

1. The level of molecular disorder (entropy) of a substance

increases as it melts or evaporates.

2. A pure crystalline substance at absolute zero temperature is in

perfect order, and its entropy is zero (the third law of

thermodynamics).

3. Disorganised energy does not create much useful effect, no

matter how large it is.

4. In the absence of friction, raising a weight by a rotating shaft does

not create any disorder (entropy), and thus energy is not

degraded during this process.

5. The paddle-wheel work done on a gas increases the level of

disorder (entropy) of the gas, and thus energy is degraded

during this process.

6. During a heat transfer process, the net entropy increases. (The

increase in the entropy of the cold body more than offsets the

decrease in the entropy of the hot body.)

Figure 8-9 Concept of Entropy (Cengel, Turner & Cimbala, 2012, p. 294)

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8.3.1 The Tds relations

1. The T ds relations are valid for both reversible and irreversible

processes and for both closed and open systems.

Figure 8-10 Tds Relations (Cengel, Turner & Cimbala, 2012, p. 298)

2. Differential changes in entropy in terms of other properties

8.3.2 The Entropy Change of Ideal Gases

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8.3.3 Constant Specific Heats (Approximate Analysis)

Figure 8-11 Constant Specific Heats (Cengel, Turner & Cimbala, 2012, p. 303)

8.3.4 Isentropic Processes of Ideal Gases

(Constant Specific Heats – Approximate analysis)

Under the constant-specific

heat assumption, the specific

heat is assumed to be constant

at some average value.

We set this equation to zero to

get the following relations.

The isentropic relations of

ideal gases are valid for

the isentropic processes of

ideal gases only.

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8.3.5 Reversible Steady-flow Work

Figure 8-12 Reversible Work Relations for Steady Flow and Closed Systems (Cengel,


When kinetic and

potential energies are

negligible

The larger the specific

volume, the greater the

work produced (or

consumed) by a steady-

flow device.

Reversible work

relations for steady-

flow and closed

systems

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Topic 9: Application of Second Law to Thermodynamic Cycles

9.1 Basic Considerations in the Analysis of Power Cycles

Most power-producing devices operate on cycles.

Ideal cycle: A cycle that resembles the actual cycle closely but is made up totally of

internally reversible processes.

Reversible cycles such as Carnot cycle have the highest thermal efficiency of all heat

engines operating between the same temperature levels. Unlike ideal cycles, they are

totally reversible, and unsuitable as a realistic model. The analysis of many complex

processes can be reduced to a manageable level by utilising some idealisations.

Thermal efficiency of heat engines

The idealisations and simplifications in the analysis of power cycles:

1. The cycle does not involve any friction. Therefore, the working fluid does not

experience any pressure drop as it flows in pipes or devices such as heat

exchangers.

2. All expansion and compression processes take place in a quasi-equilibrium

manner.

3. The pipes connecting the various components of a system are well insulated, and

heat transfer through them is negligible.

Figure 9-1 Thermodynamic Cycle Graphs -Ts and pv (Cengel, Turner & Cimbala, 2012,

p. 489)

On a T-s diagram, the ratio of the area enclosed by the cyclic curve to the area under

the heat-addition process curve represents the thermal efficiency of the cycle. Any

modification that increases the ratio of these two areas will also increase the thermal

efficiency of the cycle. Care should be exercised in the interpretation of the results from

ideal cycles.

On both P-v and T-s

diagrams, the area

enclosed by the process

curve represents the

network of the cycle.

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9.2 The Carnot Cycle and its Value in Engineering

The Carnot cycle is composed of four totally reversible processes: isothermal heat

addition, isentropic expansion, isothermal heat rejection, and isentropic compression.

For both ideal and actual cycles: Thermal efficiency increases with an increase in the

average temperature at which heat is supplied to the system or with a decrease in the

average temperature at which heat is rejected from the system.

Figure 9-2 Steady Flow Carnot Engine (Cengel, Turner & Cimbala, 2012, p. 490)

Air-standard assumptions:

1. The working fluid is air, which continuously circulates in a closed loop and

always behaves as an ideal gas.

2. All the processes that make up the cycle are internally reversible.

3. The combustion process is replaced by a heat-addition process from an external

source.

4. The exhaust process is replaced by a heat-rejection process that restores the

working fluid to its initial state.

Cold-air-standard assumptions: When the working fluid is considered to be air with

constant specific heats at room temperature (25°C).

Air-standard cycle: A cycle for which the air-standard assumptions are applicable.

P-v and T-s diagrams

of a Carnot cycle. A steady-flow Carnot engine

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9.3 An Overview of Reciprocating Engines

There are 2 types we will look at:

Spark-ignition (SI) engines

Compression-ignition (CI) engines

The nomenclature for reciprocating engines is shown in the diagrams below:

r = compression ratio

MEP = Mean Effective Pressure

Figure 9-3 Cycle Properties and Terms Used (Cengel, Turner & Cimbala, 2012, pp. 493-494)

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9.4 Otto Cycle

The Ideal Cycle for Spark-ignition Engines

The actual and ideal cycles in spark-ignition engines and their P-v diagrams.

The illustrations below show various strokes of the 4-stroke Otto Cycle.

Figure 9-4 Otto Cycle (Cengel, Turner & Cimbala, 2012, p. 494)

Four-stroke cycle

1 cycle = 4 stroke = 2 revolution

Two-stroke cycle

1 cycle = 2 stroke = 1 revolution

The T-s diagram of the ideal Otto Cycle.

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The Two-stroke Otto Cycle Engine

The two-stroke engines are generally less efficient than their four-stroke counterparts

but they are relatively simple and inexpensive, and they have high power-to-weight

and power-to-volume ratios.

You will use these formulae to calculate thermal efficiencies of the Otto Cycle.

Figure 9-5 Otto Cycle pv Graph (Cengel, Turner & Cimbala, 2012, p. 494)

1. The thermal efficiency of the Otto cycle increases with the specific heat ratio k of

the working fluid.

2. Thermal efficiency of the ideal Otto cycle as a function of compression ratio

(k=1.4).

Figure 9-6 Compression Ratio and Thermal Efficiency (Cengel, Turner & Cimbala, 2012,

pp. 496-497)

NOTE: In SI engines, the compression ratio, r is limited by auto-ignition or engine

knock.

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9.5 Diesel Cycle The Ideal Cycle for Compression-ignition Engines

1-2 isentropic compression

2-3 constant-volume heat addition

3-4 isentropic expansion

4-1 constant-volume heat rejection

Figure 9-7 Diesel Cycle Graphs Ts and pv (Cengel, Turner & Cimbala, 2012, p. 500)

In diesel engines, only air is compressed during the compression stroke, eliminating

the possibility of auto-ignition (engine knock). Therefore, diesel engines can be

designed to operate at much higher compression ratios than SI engines, typically

between 12 and 24. The spark plug is replaced by a fuel injector, and only air is

compressed during the compression process.

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You will use these formulae to calculate thermal efficiencies of the Diesel Cycle.

Figure 9-8 Compression Ratio and Diesel Engine Performance (Cengel, Turner & Cimbala,

2012, p. 501)

Dual Cycle: A more realistic ideal cycle model for modern, high-speed compression

ignition engine. The P-v diagram of an ideal dual cycle shows input of heat at constant

volume and constant pressure.

Some questions to consider:

1. Diesel engines operate at higher air-fuel ratios than gasoline engines. Why?

2. Despite higher power to weight ratios, two-stroke engines are not used in

automobiles. Why?

3. The stationary diesel engines are among the most efficient power producing

devices (about 50%). Why?

4. What is a turbocharger? Why is it mostly used in diesel engines compared to

gasoline engines?

Thermal efficiency of the ideal

Diesel cycle as a function of

compression and cut-off ratios

(k=1.4).

Cut-off ratio,

rc

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Figure 9-9 Dual Cycle Graph for pv (Cengel, Turner & Cimbala, 2012, p. 501)

Example 3: The dual combustion (high speed diesel) cycle

https://www.youtube.com/watch?v=bHc94CSqoSI

https://www.youtube.com/watch?v=bHc94CSqoSI

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9.6 Brayton Cycle

The Ideal Cycle for Gas-turbine Engines

The combustion process is replaced by a constant-pressure heat-addition process from

an external source, and the exhaust process is replaced by a constant-pressure heat-

rejection process to the ambient air.

1-2 Isentropic compression (in a compressor)

2-3 Constant-pressure heat addition

3-4 Isentropic expansion (in a turbine)

4-1 Constant-pressure heat rejection

Figure 9-10 Brayton Cycle (Cengel, Turner & Cimbala, 2012, pp. 507- 508)

An open-cycle gas-turbine engine. A closed-cycle gas-turbine engine.

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T-s and P-v diagrams for the ideal Brayton cycle is shown below.

You will use these formulae to calculate thermal efficiencies of the Brayton Cycle.

Figure 9-11 Brayton Cycle Graphs and Efficiency (Cengel, Turner & Cimbala, 2012, p. 508)

The thermal efficiency of the ideal Brayton cycle is a function of the pressure ratio.

Figure 9-12 Brayton Cycle Graphs Efficiency vs. Pressure ratio (Cengel, Turner & Cimbala,

2012, p. 509)

The two major application areas of gas-turbine engines are aircraft propulsion and

electric power generation.

Pressure ratio, rp

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The highest temperature in the cycle is limited by the maximum temperature that the

turbine blades can withstand. This also limits the pressure ratios that can be used in

the cycle.

The air in gas turbines supplies the necessary oxidant for the combustion of the fuel,

and it serves as a coolant to keep the temperature of various components within safe

limits. An air–fuel ratio of 50 or above is not uncommon.

9.6.1 Development of Gas Turbines

The following improvements have resulted in very high performing and efficient gas

turbines:

1. Increasing the turbine inlet (or firing) temperatures

2. Increasing the efficiencies of turbo-machinery components (turbines,

compressors)

3. Adding modifications to the basic cycle (intercooling, regeneration or

recuperation, and reheating)

Deviation of Actual Gas-Turbine Cycles from Idealised Ones

Reasons: Irreversibilities in turbine and compressors, pressure drops, heat losses and

the resulting isentropic efficiencies of the compressor and turbine lead to a deviation

of an actual gas-turbine cycle from the ideal Brayton cycle.

Figure 9-13 Brayton Cycle Sources of Losses (Cengel, Turner & Cimbala, 2012, p. 509)

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Summary

In this unit, we were introduced to the Kelvin–Planck and Clausius statements of the

second law of thermodynamics. We were also introduced to the definition of entropy

which enables us to quantify the second-law effect and apply it to processes and recall

the increase of entropy principle. We also applied the second law of thermodynamics to

the Carnot, Otto, Diesel and Brayton cycles and performed calculations to enable heat

supplied, heat rejected and thermal efficiencies be evaluated for a variety of operating

conditions.

References



[2] Selva Raj, J., Wan, K. H. (2016). Thermofluids II (3rd ed.). Singapore: McGraw-Hill.

(There are similar concepts in Chapter 3, treated in an introductory manner.)

TUTORIAL:

Chapter 7 Examples 7-1 to 7-3, 7-5, Q7-15, Q7-20, Q7-66

Chapter 8 Examples 8-1, 8-9 to 8-11, Q8 -8C, Q8-14C, 8-23

Chapter 9 Examples 9-1 to 9-5, Q9-64, Q9-70, Q9-77

EAS109


STUDY UNIT 5

FLUID MECHANICS APPLICATIONS

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Learning Outcomes


Derive the Bernoulli Equation along a streamline and identify the

assumptions made in its derivation.

Explain the use and limitations of the Bernoulli equation.

Apply the Bernoulli Equation to solve a variety of fluid flow problems.

Adapt Newton’s Second Law for use in fluid mechanics.

Describe the concept of the control volume in steady, inviscid and

incompressible fluid flow.

Identify the various kinds of external forces acting on a control volume.

Apply the control volume analysis to determine the thrust devices and

passive elements in fluid flow

Overview

In this unit, you will apply the conservation of energy and momentum as used in fluid

mechanics. We will derive the Bernoulli Equation and identify the assumptions made

in its derivation. We will also ‘drill’ deeper to understand the use and limitations of

the Bernoulli equation, and apply it to solve a variety of fluid flow problems.

Subsequently, the momentum equation is explained. This is Newton’s 2nd Law

applied to fluid mechanics. First, we identify the various kinds of forces acting on a

control volume and use control volume analysis to determine the thrust devices and

passive elements in fluid flow.

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Chapter 5: Energy and Momentum Equations

READ:

Chapter 6 of Thermofluids II


Topic 10: Energy Equation and Bernoulli Equation

10.1 Mechanical Energy and Efficiency

Mechanical energy: The form of energy that can be converted to mechanical work

completely and directly by an ideal mechanical device such as an ideal turbine

Mechanical energy of a flowing fluid per unit mass:

Flow energy + kinetic energy + potential energy (See paragraph 3, p. 456)

Mechanical energy change

NOTES:

1. The mechanical energy of a fluid does not change during flow if its pressure,

density, velocity, and elevation remain constant.

2. In the absence of any irreversible losses, the mechanical energy change

represents the mechanical work supplied to the fluid (if emech > 0) or extracted

from the fluid (if emech < 0).

Figure 10-1 Methods for Calculating Power Output (Cengel, Turner & Cimbala, 2012,

p. 492)

Note where start and end points are taken.

Mechanical energy is illustrated by an ideal hydraulic turbine coupled with an ideal

generator. In the absence of irreversible losses, the maximum produced power is

proportional to (a) the change in water surface elevation from the upstream to the

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downstream reservoir or (b) (close-up view) the drop in water pressure from just

upstream to just downstream of the turbine.

10.2 The Bernoulli Equation

10.2.1 Bernoulli Equation – A Short Introduction

An approximate relation between pressure, velocity, and elevation, and is valid in

regions of steady, incompressible flow where net frictional forces are negligible.

Despite its simplicity, it has proven to be a very powerful tool in fluid mechanics.

The Bernoulli approximation is typically useful in flow regions outside of boundary

layers and wakes, where the fluid motion is governed by the combined effects of

pressure and gravity forces.

Figure 10-2 Application of the Bernoulli Equation to Calculate Power Requirements


The Bernoulli equation is an approximate equation that is valid only in inviscid regions of

flow where net viscous forces are negligibly small compared to inertial, gravitational,

or pressure forces. Such regions occur outside of boundary layers and wakes.

10.2.2 Derivation of the Bernoulli Equation

Consider the forces acting on a fluid particle along a streamline.

Figure 10-3 Derivation of the Bernoulli Equation along a Streamline (Cengel, Turner &

Cimbala, 2012, p. 473)

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The sum of the kinetic, potential, and flow energies of a fluid particle is constant along

a streamline during steady flow when compressibility and frictional effects are

negligible.

Bernoulli Equation

Steady, incompressible flow

The Bernoulli equation between any two points on the same streamline

10.2.3 Properties of the Bernoulli Equation

The Bernoulli equation states that the sum of the kinetic, potential, and flow energies

of a fluid particle is constant along a streamline during steady flow.

1. The Bernoulli equation can be viewed as the “conservation of mechanical energy

principle.”

This is equivalent to the general conservation of energy principle for systems that

do not involve any conversion of mechanical energy and thermal energy to each

other, and thus the mechanical energy and thermal energy are conserved

separately.

2. The Bernoulli equation states that during steady, incompressible flow with

negligible friction, the various forms of mechanical energy are converted to each

other, but their sum remains constant.

In other words, there is no dissipation of mechanical energy during such flows

since there is no friction that converts mechanical energy to sensible thermal

(internal) energy.

3. Despite the highly restrictive approximations used in its derivation, the Bernoulli

equation is commonly used in practice since a variety of practical fluid flow

problems can be analysed to reasonable accuracy with it.

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10.2.4 Force Balance across Streamlines

Force balance in the direction n normal to the streamline yields the following relation

applicable across the streamlines for steady, incompressible flow:

For flow along a straight line, R → and this equation reduces to P/ + gz = constant

or P = gz + constant, which is an expression for the variation of hydrostatic pressure

with vertical distance for a stationary fluid body.

The variation of pressure with elevation in steady, incompressible flow along a straight

line is the same as that in the stationary fluid (but this is not the case for a curved flow

section).

Figure 10-4 Variation of Pressure with Elevation (Cengel, Turner & Cimbala, 2012, p. 475)

10.2.5 Static, Dynamic, and Stagnation Pressures

The kinetic and potential energies of the fluid can be converted to flow energy (and

vice versa) during flow, causing the pressure to change.

Multiplying the Bernoulli equation by the density gives

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The sum of the static, dynamic, and hydrostatic pressures is called the total pressure.

Therefore, the Bernoulli equation states that the total pressure along a streamline is

constant.

10.2.6 Pitot-Tube

The sum of the static and dynamic pressures is called the stagnation pressure. It

represents the pressure at a point where the fluid is brought to a complete stop

isentropically.

Figure 10-5 The Pitot-Static Tube (Cengel, Turner & Cimbala, 2012, p. 476)

The photograph highlights the static, dynamic, and stagnation pressures. The photo

shows the close-up of a Pitot-static probe, showing the stagnation pressure hole and

two of the five static circumferential pressure holes.

Streaklines , in Figure 10-5, are produced by coloured fluid introduced upstream of an

airfoil. Since the flow is steady, the streaklines are the same as streamlines and

pathlines.

10.2.7 Limitations on the Use of the Bernoulli Equation

1. Steady flow: The Bernoulli equation is applicable to steady flow.

2. Frictionless flow: Every flow involves some friction, no matter how small, and

frictional effects may or may not be negligible.

3. No shaft work: The Bernoulli equation is used in a flow section that does not

include a pump, turbine, fan, or any other machine or impeller. Such devices

destroy the streamlines and carry out energy interactions with the fluid particles.

When these devices exist, the energy equation should be used instead.

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4. Incompressible flow: Density is taken constant in the derivation of the Bernoulli

equation. The flow is incompressible for liquids and also by gases at Mach

numbers less than about 0.3.

5. No heat transfer: The density of a gas is inversely proportional to temperature,

and thus the Bernoulli equation should not be used for flow sections that involve

significant temperature change such as heating or cooling sections.

6. Flow along a streamline: Strictly speaking, the Bernoulli equation is applicable

along a streamline. However, when a region of the flow is irrotational and there is

negligibly small vorticity in the flow field, the Bernoulli equation becomes

applicable across streamlines as well.

Figure 10-6 Application and Limitations of the Bernoulli Equation (Cengel, Turner &

Cimbala, 2012, p. 497)

Frictional effects and components that disturb the streamlined structure of flow in a

flow section make the Bernoulli equation invalid. It should not be used in any of the

flows shown here.

When the flow is irrotational, the Bernoulli equation becomes applicable between any

two points along the flow (not just on the same streamline).

10.3 Examples of the Use of Bernoulli’s Equation

10.3.1 Example 1: Spraying Water into the Air

Figure 10-7 Spraying Water into the Air (Cengel, Turner & Cimbala, 2012, p. 481)

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10.3.2 Example 2: Water Discharge from a Large Tank

Figure 10-8 Water Discharge from a Tank (Cengel, Turner & Cimbala, 2012, p. 481)

10.3.3 Example 3: Velocity Measurement by a Pitot Tube

Figure 10-9 Velocity Measure by a Pitot Tube (Cengel, Turner & Cimbala, 2012, p. 483)

A force F acting through a moment arm r generates a torque T

This force acts through a distance s

Shaft work

The power transmitted through the shaft is the shaft work done per unit time

Figure10-10 Energy Transmission through Rotating Shaft (Cengel, Turner & Cimbala, 2012,

pp. 74-75)

2 12( )

1

P P

gV

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Energy transmission through rotating shafts is commonly encountered in practice.

Shaft work is proportional to the torque applied and the number of revolutions of

the shaft

Energy equation in terms of heads

10.3.4 Example 4: Fan Selection for Air Cooling of a Computer

Energy equation between 3 and 4 (see diagram below)

Energy equation between 1 and 2 (see diagram below)

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Figure 10-11 Energy Calculations (Cengel, Turner & Cimbala, 2012, p. 482)

Topic 11: Momentum Equation

11.1 Newton’s Laws

Newton’s laws: Express relations between motions of bodies and the forces acting on

them.

Newton’s first law: A body at rest remains at rest, and a body in motion remains in

motion at the same velocity in a straight path when the net force acting on it is zero.

Therefore, a body tends to preserve its state of inertia.

Newton’s second law: The acceleration of a body is proportional to the net force acting

on it and is inversely proportional to its mass.

Newton’s third law: When a body exerts a force on a second body, the second body

exerts an equal and opposite force on the first.

Therefore, the direction of an exposed reaction force depends on the body taken as the

system.

Linear momentum or just the momentum of the body: The product of the mass and the

velocity of a body.

Newton’s second law is usually referred to as the linear momentum equation.

Conservation of momentum principle: The momentum of a system remains constant

only when the net force acting on it is zero.

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Figure11-1 The Principle of Conservation of Momentum (Cengel, Turner & Cimbala, 2012,

p. 506)

11.2 Conservation of Momentum

Conservation of momentum principle: The momentum of a system remains constant

only when the net force acting on it is zero.

Linear momentum is the product of mass and velocity, and its direction is the direction

of velocity.

Newton’s second law is also expressed as the rate of change of the momentum of a body is

equal to the net force acting on it.

11.3 Choosing a Control Volume

A control volume can be selected as any arbitrary region in space through which fluid

flows, and its bounding control surface can be fixed, moving, and even deforming

during flow.

Many flow systems involve stationary hardware firmly fixed to a stationary surface,

and such systems are best analysed using fixed control volumes.

When analysing flow systems that are moving or deforming, it is usually more

convenient to allow the control volume to move or deform.

In deforming control volume, part of the control surface moves relative to other parts.

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Figure 11-2 Examples of Control Volumes (Cengel, Turner & Cimbala, 2012, p. 508)

11.4 Forces acting on a Control Volume

The forces acting on a control volume consist of body forces that act throughout the

entire body of the control volume (such as gravity, electric, and magnetic forces) and

surface forces that act on the control surface (such as pressure and viscous forces and

reaction forces at points of contact).

Only external forces are considered in the analysis.

Figure 11-3 Total Momentum Force (Cengel, Turner & Cimbala, 2012, p. 509)

The total force acting on a control volume is composed of body forces and surface

forces; body force is shown on a differential volume element, and surface force is

shown on a differential surface element.

The most common body force is that of gravity, which exerts a downward force on

every differential element of the control volume.

Surface forces are not as simple to analyse since they consist of both normal and

tangential components.

Normal stresses are composed of pressure (which always acts inwardly normal) and

viscous stresses.

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Shear stresses are composed entirely of viscous stresses.

The gravitational force acting on a differential volume element of fluid is equal to its

weight; the axes have been rotated so that the gravity vector acts downward in the

negative z-direction.

A common simplification in the application of Newton’s laws of motion is to subtract

the atmospheric pressure and work with gauge pressures.

This is because atmospheric pressure acts in all directions, and its effect cancels out in

every direction.

This means we can also ignore the pressure forces at outlet sections where the fluid is

discharged to the atmosphere since the discharge pressure in such cases is very near

atmospheric pressure at subsonic velocities.

Figure 11-4 The Most Convenient Control Volume (Cengel, Turner & Cimbala, 2012, p. 510)

Atmospheric pressure acts in all directions, and thus it can be ignored when performing force balances since its effect cancels out in every direction. Cross section through a faucet assembly, illustrating the importance of choosing a control volume wisely; CV B is much easier to work with than CV A.

Steady Flow

The net force acting on the control volume during steady flow is equal to the difference

between the rates of outgoing and incoming momentum flows.

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Figure 11-5 Total Force acting on the Control Volume during Steady Flow (Cengel, Turner

& Cimbala, 2012, p. 500)

The net force acting on the control volume during steady flow is equal to the difference

between the outgoing and the incoming momentum fluxes.

Steady Flow with One Inlet and One Outlet

One inlet and one outlet

Along x-coordinate

A control volume with only one inlet and one outlet is shown below.

Figure 11-6 Reaction Force on a Curved Vane (Cengel, Turner & Cimbala, 2012, p. 520)

The reaction force on the support caused by a change of direction of water is equal and

opposite to the momentum force on the water.

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11.5 Flows with No External Forces

In the absence of external forces, the rate of change of the momentum of a control

volume is equal to the difference between the rates of incoming and outgoing

momentum flow rates.

Figure 11-7 Thrust generated from a Rocket (Cengel, Turner & Cimbala, 2012, p. 521)

The thrust needed to lift the space shuttle is generated by the rocket engines as a result

of momentum change of the fuel as it is accelerated from about zero to an exit speed

of about 2000 m/s after combustion.

Momentum Force Calculations

https://www.youtube.com/watch?v=cp6xpCh2Vk

https://www.youtube.com/watch?v=cp6xpCh2Vk

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Summary

The conservation of energy and momentum was applied in fluid mechanics. The

Bernoulli Equation was derived and the assumptions were made in its derivation. The

use and limitations of the Bernoulli equation were studied and the equation was

applied to solve a variety of fluid flow problems. Subsequently, the momentum

equation which is Newton’s 2nd Law was explained. We identified the various kinds

of forces acting on a control volume and used control volume analysis to determine

the thrust devices and passive elements in fluid flow.

TUTORIAL: Q12-4C, Q12-8C, Q12-10C, Q12-19, Q12-33, Q12-34, Q12-60

References

[1] Selva Raj, J., Wan, K. H. (2016). Thermofluids II (3rd ed.). Singapore: McGraw-Hill.



EAS109


STUDY UNIT 6

VISCOUS INTERNAL AND

EXTERNAL FLOW

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Learning Outcomes


Describe the two key flows - laminar and turbulent flow in pipes.

Calculate losses associated with pipe flow in piping networks.

Estimate the pumping power requirements for simple series and parallel pipe

systems.

Describe external flows over solid bodies.

Calculate the lift and drag force associated with flow over common geometries.

Describe the effects of flow regime on the drag coefficients associated with

flow over cylinders and spheres.

Recall the types of flow over airfoils.

Calculate the drag and lift forces acting on airfoils.

Demonstrate an understanding of the various physical phenomena associated

with external flow such as drag, friction and pressure drag, drag reduction,

and lift.

Overview

In this unit, we look at laminar and turbulent flow in pipes and analyse fully developed

pipe flow. We will calculate losses associated with pipe flow in piping networks and

determine the pumping power requirements. Finally, we will look at the effects of flow

regime on the drag coefficients associated with flow over cylinders and spheres

initially.

We will also learn the fundamentals of flow over airfoils and calculate the drag and lift

forces acting on airfoils. We will develop an understanding of the various physical

phenomena associated with external flow such as drag, friction and pressure drag,

drag reduction, and lift.

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CHAPTER 6: Internal and External Flow

READ:



Topic 12: Internal Flow

12.1 Introduction

Liquid or gas flow through pipes or ducts is commonly used in heating and cooling

applications and fluid distribution networks.

The fluid in such applications is usually forced to flow by a fan or pump through a

flow section.

We pay particular attention to friction, which is directly related to the pressure drop and

head loss during flow through pipes and ducts. The pressure drop is then used to

determine the pumping power requirement.

Figure 12-1 Circular Pipes vs. Rectangular Pipes (Cengel, Turner & Cimbala, 2012, p. 538)

Circular pipes can withstand large pressure differences between the inside and the

outside without undergoing any significant distortion, but noncircular pipes cannot.

Theoretical solutions are obtained only for a few simple cases such as fully developed

laminar flow in a circular pipe. Therefore, we must rely on experimental results and

empirical relations for most fluid flow problems rather than closed-form analytical

solutions.

Figure 12-2 Determining Average Velocity (Cengel, Turner & Cimbala, 2012, p. 538)

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12.2 Flow Rate and Flow Velocity

The value of the average velocity Vavg at some stream-wise cross-section is determined

from the requirement that the conservation of mass principle be satisfied.

The average velocity for incompressible flow in a circular pipe of radius R.

Average velocity Vavg is defined as the average speed through a cross section. For fully

developed laminar pipe flow, Vavg is half of the maximum velocity.

Figure 12-3 Laminar and Turbulent Flows (Cengel, Turner & Cimbala, 2012, p. 539)

12.3 Laminar and Turbulent Flow

Laminar flow is encountered when highly viscous fluids such as oils flow in small

pipes or narrow passages.

Laminar: Smooth streamlines and highly ordered motion.

Turbulent: Velocity fluctuations and highly disordered motion.

Transitions: The flow fluctuates between laminar and turbulent flows.

Most flows encountered in practice are turbulent.

The behaviour of coloured fluid injected into the flow in laminar and turbulent flows

in a pipe.

Laminar and turbulent flow regimes of candle smoke.

Figure 12-4 Reynolds Number (Cengel, Turner & Cimbala, 2012, p. 540)

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The transition from laminar to turbulent flow depends on the geometry, surface

roughness, flow velocity, surface temperature, and type of fluid.

The flow regime depends mainly on the ratio of inertial forces to viscous forces (Reynolds

number).

At large Reynolds numbers, the inertial forces, which are proportional to the fluid

density and the square of the fluid velocity, are large relative to the viscous forces, and

thus the viscous forces cannot prevent the random and rapid fluctuations of the fluid

(turbulent).

At small or moderate Reynolds numbers, the viscous forces are large enough to

suppress these fluctuations and to keep the fluid “in line” (laminar).

Critical Reynolds number, Recr: The Reynolds number at which the flow becomes

turbulent. The value of the critical Reynolds number is different for different

geometries and flow conditions.

The Reynolds number can be viewed as the ratio of inertial forces to viscous forces

acting on a fluid element.

For flow in a circular pipe

Figure 12-5 The 3 Flow Regimes in Pipe Flow (Cengel, Turner & Cimbala, 2012, p. 541)

In the transitional flow region of 2000 Re 4,000, the flow switches between laminar

and turbulent seemingly randomly.

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12.4 The Entrance Region of Internal Flow

Velocity boundary layer: The region of the flow in which the effects of the viscous

shearing forces caused by fluid viscosity are felt.

Boundary layer region: The viscous effects and the velocity changes are significant.

Irrotational (core) flow region: The frictional effects are negligible and the velocity

remains essentially constant in the radial direction.

Figure 12-6 Development of the Boundary Layer in Pipe Flow (Cengel, Turner & Cimbala,

2012, p. 541)

The illustration above shows development of the velocity boundary layer in a pipe.

The developed average velocity profile is parabolic in laminar flow, but somewhat

flatter or fuller in turbulent flow.

Hydrodynamic entrance region: The region from the pipe inlet to the point at which

the boundary layer merges at the centreline.

Hydrodynamic entry length Lh: The length of this region.

Hydrodynamically developing flow: Flow in the entrance region. This is the region

where the velocity profile develops.

Hydrodynamically fully developed region: Thıs is the region beyond the entrance

region in which the velocity profile is fully developed and remains unchanged

Fully developed: This is when both the velocity profile and the normalised

temperature profile remain unchanged.

Hydrodynamically fully developed

Figure 12-7 Constant Wall Shear Stress in Fully Developed Flow (Cengel, Turner &

Cimbala, 2012, p. 542)

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Head loss

12.5 Pressure Drop and Head Loss in Pipes

A pressure drop due to viscous effects represents an irreversible pressure loss, and it

is called pressure loss PL.

The equation above D’Arcy relates the pressure loss for all types of fully developed

internal flows.

12.5.1 Circular Pipe, Laminar Flow and Turbulent Flow

In laminar flow, the friction factor is a function of the Reynolds number only and is

independent of the roughness of the pipe surface. In turbulent flow, both the Reynolds

Number and the roughness play a significant role in determining the friction factor.

The head loss represents the additional height that the fluid needs to be raised by a

pump in order to overcome the frictional losses in the pipe.

Dynamic pressure Darcy friction factor

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12.6 Laminar Flow for Pipes

Poiseuille’s Law for Horizontal pipe

Figure 12-8 The Effect of Pipe Diameter in Pumping Power (Cengel, Turner & Cimbala,

2012, p. 546)

For a specified flow rate, the pressure drop and thus the required pumping power is

proportional to the length of the pipe and the viscosity of the fluid, but it is inversely

proportional to the fourth power of the diameter of the pipe.

The relation for pressure loss (and head loss) is one of the most general relations in

fluid mechanics, and it is valid for laminar or turbulent flows, circular or noncircular

pipes, and pipes with smooth or rough surfaces.

The pumping power requirement for a laminar flow piping system can be reduced by

a factor of 16 by doubling the pipe diameter.

Figure 12-9A Pumping Power Requirements


Figure 12-9B Velocity Profiles in Laminar and Turbulent

Flow (Cengel, Turner & Cimbala, 2012, p. 552)

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12.7 Turbulent Flow for Pipes

Most flows encountered in engineering practice are turbulent, and thus it is important

to understand how turbulence affects wall shear stress.

Turbulent flow is a complex mechanism dominated by fluctuations, and it is still not

fully understood. We must rely on experiments and the empirical or semi-empirical

correlations developed for various situations. Turbulent flow is characterised by

disorderly and a rapid fluctuation of swirling regions of fluid, called eddies,

throughout the flow. These fluctuations provide an additional mechanism for

momentum and energy transfer.

In turbulent flow, the swirling eddies transport mass, momentum, and energy to other

regions of flow much more rapidly than molecular diffusion, greatly enhancing mass,

momentum, and heat transfer. As a result, turbulent flow is associated with much

higher values of friction, heat transfer, and mass transfer coefficients.

Figure 12-10 Mixing of Fluid Particles in Turbulent Flow (Cengel, Turner & Cimbala, 2012,

p. 552)

The intense mixing in turbulent flow brings fluid particles at different momentums

into close contact and thus enhances momentum transfer.

12.7.1 Turbulent Velocity Profile

The velocity profile in fully developed pipe flow is parabolic in laminar flow, but much

fuller in turbulent flow. The very thin layer next to the wall where viscous effects are

dominant is the viscous (or laminar or linear or wall) sublayer.

The velocity profile in this layer is very nearly linear, and the flow is streamlined. Next

to the viscous sublayer is the buffer layer, in which turbulent effects are becoming

significant, but the flow is still dominated by viscous effects.

Above the buffer layer is the overlap (or transition) layer, also called the inertial

sublayer, in which the turbulent effects are much more significant, but still not

dominant.

Above that is the outer (or turbulent) layer in the remaining part of the flow in which

turbulent effects dominate over molecular diffusion (viscous) effects.

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12.7.2 The Moody Diagram

Figure 12-11 The Moody Chart (Cengel, Turner & Cimbala, 2012, p. 1029

Appendix Figure A-27)

Observations from the Moody Chart

For laminar flow, the friction factor decreases with increasing Reynolds number, and

it is independent of surface roughness.

The friction factor is a minimum for a smooth pipe and increases with roughness. The

Colebrook equation in this case ( = 0) reduces to the Prandtl equation.

The transition region from the laminar to turbulent regime is indicated by the shaded

area in the Moody chart. At small relative roughness, the friction factor increases in the

transition region and approaches the value for smooth pipes.

At very large Reynolds numbers (to the right of the dashed line on the Moody chart),

the friction factor curves corresponding to specified relative roughness curves are

nearly horizontal, and thus the friction factors are independent of the Reynolds

number. The flow in that region is called fully rough turbulent flow or just fully rough

flow because the thickness of the viscous sub-layer decreases with increasing Reynolds

number, and it becomes so thin that it is negligibly small compared to the surface

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roughness height. The Colebrook equation in the fully rough zone reduces to the von

Kármán equation.

12.8 Types of Fluid Flow Problems

Determine the pressure drop (or head loss) when the pipe length and diameter are

given for a specified flow rate (or velocity).

Determine the flow rate when the pipe length and diameter are given for a specified

pressure drop (or head loss).

Determine the pipe diameter when the pipe length and flow rate are given for a

specified pressure drop (or head loss).

The three types of problems encountered in pipe flow.

Figure 12-12 Three Types of Flow Problems (Cengel, Turner & Cimbala, 2012, p. 555)

To avoid tedious iterations in head loss, flow rate, and diameter calculations, these

explicit relations that are accurate to within 2 percent of the Moody chart may be used.

Figure 12-13 Swamee-Jain Expressions for Pipe Flow (Cengel, Turner & Cimbala, 2012,

p .556)

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12.9 Minor Losses

12.9.1 Loss Coefficients

The fluid in a typical piping system passes through various fittings, valves, bends,

elbows, tees, inlets, exits, enlargements, and contractions in addition to the pipes.

These components interrupt the smooth flow of the fluid and cause additional losses

because of the flow separation and mixing they induce.

In a typical system with long pipes, these losses are minor compared to the total head

loss in the pipes (the major losses) and are called minor losses.

Minor losses are usually expressed in terms of the loss coefficient KL.

Figure 12-14 Equivalent Length Method for Losses across a Valve (Cengel, Turner &

Cimbala, 2012, p. 560)

For a constant-diameter section of a pipe with a minor loss component, the loss

coefficient of the component (such as the gate valve shown) is determined by

measuring the additional pressure loss it causes and dividing it by the dynamic

pressure in the pipe.

Head loss due to component

When the inlet diameter equals outlet diameter, the loss coefficient of a component can

also be determined by measuring the pressure loss across the component and dividing

it by the dynamic pressure:

KL = PL /(V2/2).

When the loss coefficient for a component is available, the head loss for that component

is

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12.9.2 Equivalent Lengths

Minor losses are also expressed in terms of the equivalent length Lequiv.

The head loss caused by a component (such as the angle valve shown above) is

equivalent to the head loss caused by a section of the pipe whose length is the

equivalent length.

Figure 12-15 Head Loss across a Right-angled Valve (Cengel, Turner & Cimbala, 2012,

p. 561)

12.9.3 Summarising Losses

Total head loss (general)

Total head loss (D = constant)

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12.10 Other Techniques that Account for Minor Losses

12.10.1 The Effect of Rounding of a Pipe Inlet on the Loss Coefficient

The head loss at the inlet of a pipe is almost negligible for well-rounded inlets (KL =

0.03 for r/D > 0.2) but increases to about 0.50 for sharp-edged inlets.

Figure 12-16 Head Loss at Pipe Exit (Cengel, Turner & Cimbala, 2012, p. 562)

Figure 12-17 Head Loss at Pipe Entrances (Cengel, Turner & Cimbala, 2012, p. 563)

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Figure 12-18 Head Loss at Transition Elements (Cengel, Turner & Cimbala, 2012, p. 563)

Figure 12-19 Head Loss at Bends and Tees (Cengel, Turner & Cimbala, 2012, p. 564)

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12.10.2 Other Considerations

All the kinetic energy of the flow is “lost” (turned into thermal energy) through friction

as the jet decelerates and mixes with ambient fluid downstream of a submerged outlet.

The losses during changes of direction can be minimised by making the turn “easy” on

the fluid by using circular arcs instead of sharp turns.

The large head loss in a partially closed valve is due to irreversible deceleration, flow

separation, and mixing of high-velocity fluid coming from the narrow valve passage.

Sudden expansion

Figure 12-20 Head Loss Valves and Elbows (Cengel, Turner & Cimbala, 2012, pp. 565-566)

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12.11 Piping Networks and Pump Selection

The photograph below shows a piping network in an industrial facility.

For pipes in series, the flow rate is the same in each pipe, and the total head loss is the

sum of the head losses in individual pipes.

For pipes in parallel, the head loss is the same in each pipe, and the total flow rate is the

sum of the flow rates in individual pipes.

Figure 12-21 Typical Industrial Pipe Network (Cengel, Turner & Cimbala, 2012, p. 567)

12.11.1 Pipe Configurations

PIPES IN SERIES PIPES IN PARALLEL

Figure 12-22 Series and Parallel Pipe (Cengel, Turner & Cimbala, 2012, p. 563)

The relative flow rates in parallel pipes are established from the requirement that the

head loss in each pipe be the same. The flow rate in one of the parallel branches is

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proportional to its diameter to the power of 2 and is inversely proportional to the

square root of its length and friction factor.

The analysis of piping networks is based on two simple principles:

Conservation of mass throughout the system must be satisfied. This is done by requiring the

total flow into a junction to be equal to the total flow out of the junction for all junctions

in the system.

Pressure drop (and thus head loss) between two junctions must be the same for all paths between

the two junctions. This is because pressure is a point function and it cannot have two

values at a specified point. In practice, this rule is used by requiring that the algebraic

sum of head losses in a loop (for all loops) be equal to zero.

12.11.2 Piping Systems with Pumps and Turbines

Figure 12-23 Pumping Systems (Cengel, Turner & Cimbala, 2012, p. 569)

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When a pump moves a fluid from one reservoir to another, the useful pump head

requirement is equal to the elevation difference between the two reservoirs plus the

head loss.

The efficiency of the pump–motor combination is the product of the pump and the

motor efficiencies.

12.11.3 The Operating Point

Characteristic pump curves for centrifugal pumps, the system curve for a piping

system, and determination of the operating point are shown below.

Figure 12-24 Characteristic Curves for Pumps and Pipe Systems (Cengel, Turner &

Cimbala, 2012, p. 570)

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Topic 13: External Flow

13.1 Introduction and Definitions of Terms

Fluid flow over solid bodies frequently occurs in practice, and it is responsible for

numerous physical phenomena such as the drag force acting on automobiles, power

lines, trees, and underwater pipelines; the lift developed by airplane wings; upward

draft of rain, snow, hail, and dust particles in high winds; the transportation of red

blood cells by blood flow; the entrainment and disbursement of liquid droplets by

sprays; the vibration and noise generated by bodies moving in a fluid; and the power

generated by wind turbines.

The flow fields and geometries for most external flow problems are too complicated

and we have to rely on correlations based on experimental data.

1A Free-stream velocity: The velocity of the fluid approaching a body (V or u or U

)

1B Two-dimensional flow: When the body is very long and of constant cross section

and the flow is normal to the body.

1C Axisymmetric flow: When the body possesses rotational symmetry about an axis

in the flow direction. The flow in this case is also two-dimensional.

1D Three-dimensional flow: Flow over a body that cannot be modelled as two-

dimensional or axisymmetric such as flow over a car.

2A Incompressible flows: (e.g., flows over automobiles, submarines, and buildings).

2B Compressible flows: (e.g., flows over high-speed aircraft, rockets, and missiles).

Compressibility effects are negligible at low velocities (flows with Ma < 0.3).

3A Streamlined body: If a conscious effort is made to align its shape with the

anticipated streamlines in the flow. Streamlined bodies such as race cars and

airplanes appear to be contoured and sleek.

3B Bluff or blunt body: If a body (such as a building) tends to block the flow. Usually

it is much easier to force a streamlined body through a fluid.

A fluid moving over a stationary body (such as the wind blowing over a building), and

a body moving through a quiescent fluid (such as a car moving through air) are

referred to as flow over bodies or external flow.

Figure 13-1 External Flow Commonly Encountered (Cengel, Turner & Cimbala 3E, 2008,

p. 580)

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1B Two-dimensional, 1C axisymmetric, and 1D three-dimensional flows

It is much easier to force a streamlined body than a blunt body through a fluid.

Figure 13-2 Classifying External Flow (Cengel, Turner & Cimbala 3E, 2008, p. 581)

13.2 Drag and Lift

A body meets some resistance when it is forced to move through a fluid, especially a

liquid. A fluid may exert forces and moments on a body in and about various

directions.

Drag: The force a flowing fluid exerts on a body in the flow direction. The drag force

can be measured directly by simply attaching the body subjected to fluid flow to a

calibrated spring and measuring the displacement in the flow direction. Drag is usually

an undesirable effect, like friction, and we do our best to minimise it. But in some cases,

drag produces a very beneficial effect and we try to maximise it (e.g., automobile

brakes). High winds knock down trees, power lines, and even people as a result of the

drag force.

Figure 13-3 The Development of Lift and Drag (Cengel, Turner & Cimbala 3E, 2008, p. 582)

The resultant of the pressure and viscous forces acting on a two-dimensional body is

decomposed into lift and drag forces.

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Lift: The components of the pressure and wall shear forces in the direction normal to

the flow tend to move the body in that direction, and their sum is called lift.

The fluid forces may generate moments and cause the body to rotate.

Rolling moment: The moment about the flow direction.

Yawing moment: The moment about the lift direction.

Pitching moment: The moment about the side force direction.

(a) Drag force acting on a flat plate parallel to the flow

depends on wall shear only.

(b) Drag force acting on a flat plate normal to the flow depends

on the pressure only and is independent of the wall shear,

which acts normal to the free-stream flow.

During a free fall, a body reaches its terminal velocity when the

drag force equals the weight of the body minus the buoyant

force.

The drag and lift forces depend on the density of the fluid, the

upstream velocity, and the size, shape, and orientation of the

body.

It is more convenient to work with appropriate dimensionless

numbers that represent the drag and lift characteristics of the

body. Figure 13-4 Drag Forces – Parallel and Normal to Flow (Cengel, Turner & Cimbala, 2012,

p. 592)

These numbers are the drag coefficient CD, and the lift coefficient

CL. The drag and lift forces depend on the density of the fluid, the

upstream velocity, and the size, shape, and orientation of the

body.

Figure 13-5 Equilibrium Forces at Terminal Velocity (Cengel, Turner & Cimbala, 2012,

p. 593)

It is more convenient to work with appropriate dimensionless numbers that represent

the drag and lift characteristics of the body.

These numbers are the drag coefficient CD, and the lift coefficient CL.

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Figure 13-6 Lift and Drag Force calculated over the Aerofoil Area (Cengel, Turner &

Cimbala, 2012, pp. 593-594)

NOTE: The area A, projected on a plane normal to the direction of flow is called the

Frontal Area.

Airplane wings are shaped and positioned to generate sufficient lift during flight while

keeping drag at a minimum. Pressures above and below atmospheric pressure are

indicated by plus and minus signs, respectively.

13.3 Friction and Pressure Drag

The drag force is the net force exerted by a fluid on a body in the direction of flow due

to the combined effects of wall shear and pressure forces.

The part of drag that is due directly to wall shear stress is called the skin friction drag

(or just friction drag) since it is caused by frictional effects, and the part that is due

directly to pressure is called the pressure drag (also called the form drag because of its

strong dependence on the form or shape of the body).

The friction drag is the component of the wall shear force in the direction of flow, and

thus it depends on the orientation of the body as well as the magnitude of the wall

shear stress.

For parallel flow over a flat surface, the drag coefficient is equal to the friction drag

coefficient.

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Friction drag is a strong function of viscosity, and increases with increasing viscosity.

Figure 13-7 Friction Drag and Pressure (Form) Drag (Cengel, Turner & Cimbala, 2012,

p. 594)

Drag is due entirely to friction drag for a flat plate parallel to the flow; it is due entirely

to pressure drag for a flat plate normal to the flow; and it is due to both (but mostly

pressure drag) for a cylinder normal to the flow. The total drag coefficient CD is lowest

for a parallel flat plate, highest for a vertical flat plate, and in between (but close to that

of a vertical flat plate) for a cylinder.

Figure 13-8 Combining Friction Drag and Pressure (Form) Drag (Cengel, Turner &

Cimbala, 2012, p. 594)

13.3.1 Reducing Drag by Streamlining

Streamlining decreases pressure drag by delaying boundary layer separation and thus

reducing the pressure difference between the front and back of the body but increases

the friction drag by increasing the surface area. The end result depends on which effect

dominates.

Figure 13-9 Relationship between Friction, Pressure and Total Drag (Cengel, Turner &

Cimbala, 2012, p. 595)

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The variation of friction, pressure, and total drag coefficients of a streamlined strut

with thickness-to-chord length ratio for Re = 4×104. Note that CD for airfoils and other

thin bodies is based on planform area rather than frontal area.

Figure 13-10 Drag Changes with Aspect Ratio for an Elliptical Cylinder (Cengel, Turner &

Cimbala, 2012, p. 596)

The variation of the drag coefficient of a long elliptical cylinder with aspect ratio.

Here CD is based on the frontal area bD where b is the width of the body.

The drag coefficient decreases drastically as the ellipse becomes slimmer.

The reduction in the drag coefficient at high aspect ratios is primarily due to the boundary layer

staying attached to the surface longer and the resulting pressure recovery.

Streamlining has the added benefit of reducing vibration and noise.

Streamlining should be considered only for blunt bodies that are subjected to high-velocity fluid

flow (and thus high Reynolds numbers) for which flow separation is a real possibility.

Streamlining is not necessary for bodies that typically involve low Reynolds number

flows.

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13.3.2 Flow Separation

Flow separation in a waterfall. Flow separation over a backward-facing step along a wall.

Figure 13-11 Flow Separation (Cengel, Turner & Cimbala, 2012, p. 596)

Flow separation: At sufficiently high velocities, the fluid stream detaches itself from

the surface of the body.

The location of the separation point depends on several factors such as the Reynolds

number, the surface roughness, and the level of fluctuations in the free stream, and it

is usually difficult to predict exactly where separation will occur.

Separated region: When a fluid separates from a body, it forms a separated region

between the body and the fluid stream. This is a low-pressure region behind the body

where recirculating and backflows occur. The larger the separated region, the larger

the pressure drag. The effects of flow separation are felt far downstream in the form of

reduced velocity (relative to the upstream velocity).

Wake: The region of flow trailing the body where the effects of the body on velocity

are felt. Viscous and rotational effects are the most significant in the boundary layer,

the separated region, and the wake.

Figure 13-12 Flow Separation and the Wake Region during Flow over a Tennis Ball


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Figure 13-13 Flow Effect of Angle of Attack on Separation (Cengel, Turner & Cimbala,

2012, p. 597)

At large angles of attack (usually larger than 15°), flow may separate completely from

the top surface of an airfoil, reducing lift drastically and causing the airfoil to stall. An

important consequence of flow separation is the formation and shedding of circulating

fluid structures, called vortices, in the wake region. The periodic generation of these

vortices downstream is referred to as vortex shedding. The vibrations generated by

vortices near the body may cause the body to resonate to dangerous levels.

13.4 Drag Coefficients of Common Geometries

The drag behaviour of various natural and human-made bodies is characterised by

their drag coefficients measured under typical operating conditions. Usually, the total

(friction + pressure) drag coefficient is reported. The drag coefficient exhibits different

behaviour in the low (creeping), moderate (laminar), and high (turbulent) regions of

the Reynolds number. The inertia effects are negligible in low Reynolds number flows

(Re < 1), called creeping flows, and the fluid wraps around the body smoothly.

Drag Coefficient Creeping Flows

Stokes Law

Stokes law is often applicable to dust particles in the air and suspended solid particles

in water.

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The drag coefficient for many (but not all) geometries remains essentially constant at

Reynolds numbers above about 104.

DATA SHEET 1

Figure 13-14(B) Drag Coefficients at Low Reynolds Numbers (Cengel, Turner & Cimbala,

2012, p. 598)

Note that friction drag can be high at low

Reynolds Numbers because of the

influence of viscosity.

Figure 13-14(A) Drag Coefficients for Common

Geometries (Cengel, Turner & Cimbala, 2012, p. 598)

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DATA SHEET 2

Figure 13-14(C) Drag Coefficients for Common Geometries (Cengel, Turner & Cimbala,

2012, p. 599)

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DATA SHEET 3

Figure 13-14(C) Drag Coefficients for 3D Bodies (Cengel, Turner & Cimbala, 2012, p. 600)

DATA SHEET 4

Figure 13-14(D) Drag Coefficients for Common Geometries (Cengel, Turner & Cimbala,

2012, p. 600)

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DATA SHEET 5

Figure 13-14(E) Drag Coefficients for 3D Bodies - 2 (Cengel, Turner & Cimbala, 2012, p. 600)

13.4.1 Observations from the Drag Coefficient Tables

The orientation of the body relative to the direction of flow has a major influence on the

drag coefficient. For blunt bodies with sharp corners, such as flow over a rectangular

block or a flat plate normal to flow, separation occurs at the edges of the front and back

surfaces, with no significant change in the character of flow. Therefore, the drag

coefficient of such bodies is nearly independent of the Reynolds number. The drag

coefficient of a long rectangular rod can be reduced almost by half from 2.2 to 1.2 by

rounding the corners. The drag coefficient of a body may change drastically by

changing the body’s orientation (and thus shape) relative to the direction of flow.

Figure 13-15 Drag Coefficients depend on Body Orientation (Cengel, Turner & Cimbala,

2012, p. 601)

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13.5 Biological Systems and Drag

The concept of drag also has important consequences for biological systems.

The bodies of fish, especially the ones that swim fast for long distances (such as

dolphins), are highly streamlined to minimise drag (the drag coefficient of dolphins

based on the wetted skin area is about 0.0035, comparable to the value for a flat plate

in turbulent flow).

Airplanes, which look somewhat like big birds, retract their wheels after takeoff in

order to reduce drag and thus fuel consumption.

The flexible structure of plants enables them to reduce drag at high winds by changing

their shapes. Large flat leaves, for example, curl into a low-drag conical shape at high

wind speeds, while tree branches cluster to reduce drag. Flexible trunks bend under

the influence of the wind to reduce drag, and the bending moment is lowered by

reducing frontal area.

Horse and bicycle riders lean forward as much as they can to reduce drag.

Birds teach us a lesson on drag reduction by extending their beak forward and folding

their feet backward during flight.

Figure 13-16 Birds reduce Drag Coefficients during Flights (Cengel, Turner & Cimbala 3E,

2008, p. 593)

13.6 Drag Coefficients of Vehicles

The drag coefficients of vehicles range from about 1.0 for large semitrailers to 0.4 for

minivans, 0.3 for passenger cars, and 0.2 for race cars. The theoretical lower limit is

about 0.1. In general, the more blunt the vehicle, the higher the drag coefficient.

Installing a fairing reduces the drag coefficient of tractor-trailer rigs by about 20

percent by making the frontal surface more streamlined. As a rule of thumb, the

percentage of fuel savings due to reduced drag is about half the percentage of drag

reduction at highway speeds.

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Figure 13-17 Streamlining reduces Drag (Cengel, Turner & Cimbala, 2012, p. 602)

The sleek-looking Toyota Prius© above has a drag coefficient of 0.26—one of the lowest

for a passenger car. Streamlines around an aerodynamically designed modern car

closely resemble the streamlines around the car in the ideal potential flow (assumes

negligible friction), except near the rear end, resulting in a low drag coefficient. The

aerodynamic drag is negligible at low speeds, but becomes significant at speeds above

about 50 km/h (13.8 m/s).

At highway speeds, a driver can often save fuel in hot weather by running the air

conditioner instead of driving with the windows rolled down. The turbulence and

additional drag generated by open windows consume more fuel than does the air

conditioner.

13.6.1 Superposition

The shapes of many bodies encountered in practice are not simple. But such bodies can

be treated conveniently in drag force calculations by considering them to be composed

of two or more simple bodies.

A satellite dish mounted on a roof with a cylindrical bar, for example, can be

considered to be a combination of a hemispherical body and a cylinder. Then the drag

coefficient of the body can be determined approximately by using superposition.

13.6.2 Drafting

The drag coefficients of bodies following other moving bodies closely can be reduced

considerably due to drafting (i.e., falling into the low pressure region created by the

body in front). The drag coefficient of a racing bicyclist can be reduced from 0.9 to 0.5

by drafting.

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Figure 13-18 Drafting reduces Drag (Cengel, Turner & Cimbala, 2012, p. 603)

13.7 Parallel Flow over Flat Plates

13.7.1 Definitions

Velocity boundary layer: The region of the flow above the plate bounded by in

which the effects of the viscous shearing forces caused by fluid viscosity are felt. The

boundary layer thickness is typically defined as the distance y from the surface at which

u = 0.99V.

The hypothetical curve of u = 0.99V divides the flow into two regions:

Boundary layer region: The viscous effects and the velocity changes are significant.

Irrotational flow region: The frictional effects are negligible and the velocity remains

essentially constant.

Figure 13-19 The Development of the Boundary Layer for Flow over a Flat Plate, and the

Different Flow Regimes. Not to scale. (Cengel, Turner & Cimbala, 2012, p. 605)

The turbulent boundary layer can be considered to consist of four regions,

characterised by the distance from the wall:

• viscous sublayer

• buffer layer

• overlap layer

• turbulent layer

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Figure 13-20 Development of a Boundary Layer on a Surface is due to the No-slip

Condition and Friction. (Cengel, Turner & Cimbala, 2012, p. 606)

Friction coefficient on a flat plate

Friction force on a flat plate

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Figure 13-21 For Parallel Flow over a Flat Plate the Drag Force is equal to the Friction Force.


13.7.2 Laminar and Turbulent Flows

The transition from laminar to turbulent flow depends on the surface geometry, surface

roughness, upstream velocity, surface temperature, and the type of fluid, among other things,

and is best characterised by the Reynolds number.

The Reynolds number at a distance x from the leading edge of a flat plate is expressed

as:

V upstream velocity

x characteristic length of the geometry (for a flat plate, it is the length of the plate in

the flow direction)

For flow over a smooth flat plate, transition from laminar to turbulent begins at about

Re 1105, but does not become fully turbulent before the Reynolds number reaches

much higher values, typically around 3106. In engineering analysis, a generally

accepted value for the critical Reynolds number is

The actual value of the engineering critical Reynolds number for a flat plate may vary

somewhat from about 105 to 310

6 depending on the surface roughness, the turbulence

level, and the variation of pressure along the surface.

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13.7.3 Friction Coefficient

The friction coefficient for laminar flow over a flat plate can be determined

theoretically by solving the conservation of mass and momentum equations

numerically.

For turbulent flow, it must be determined experimentally and expressed by empirical

correlations.

The variation of the local friction coefficient for flow over a flat plate is shown in the

sketch below. The vertical scale of the boundary layer is greatly exaggerated.

Figure 13-22 Variation of Local Friction Factor over a Flat Plate (Cengel, Turner & Cimbala,

2012, p. 607)

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13.7.4 Determining the Average Friction Coefficient over the Entire Plate

NOTE: The calculations above are for laminar flow across the entire flat plate (Cengel,


When the laminar flow region is not disregarded plate (Cengel, Turner & Cimbala,

2012, p. 608), integration of the expression below

gives the expression

Hence we get an expression for the friction coefficient with laminar and turbulent

regions considered.

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For laminar flow, the friction coefficient depends only on the Reynolds number, and

the surface roughness has no effect.

For turbulent flow, surface roughness causes the friction coefficient to increase several

fold, to the point that in the fully rough turbulent regime the friction coefficient is a

function of surface roughness alone and is independent of the Reynolds number.

where

= the surface roughness

L = the length of the plate in the flow direction.

This relation can be used for turbulent flow on

rough surfaces for Re > 106, especially when /L > 10

4

For turbulent flow, surface roughness may

cause the friction coefficient to increase several

fold.

Figure 13-23 For Turbulent Flow, Surface Roughness may increase Friction Coefficient


The friction coefficient for parallel flow over smooth and rough flat plates for both

laminar and turbulent flows is shown below.

Figure 13-24 Friction Coefficients for Parallel Flow over Smooth and Rough Plates (Cengel,


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From the figure, note the following:

Cf increases several times with roughness in turbulent flow.

Cf is independent of the Reynolds number in the fully rough region.

This chart is the flat-plate analogue of the Moody chart for pipe flows.

13.8 Flow over Cylinders and Spheres

Flow over cylinders and spheres are frequently encountered in practice.

- The tubes in a shell-and-tube heat exchanger involve both internal flow

through the tubes and external flow over the tubes.

- Many sports such as soccer, tennis, and golf involve flow over spherical balls.

Figure 13-25 Laminar Boundary Layer Separation (Cengel, Turner & Cimbala, 2012, p. 610)

At very low velocities, the fluid completely wraps around the cylinder. Flow in the

wake region is characterised by periodic vortex formation and low pressures.

Laminar boundary layer separation with a turbulent wake; flow over a circular

cylinder at Re=2000

For flow over cylinder or sphere, both the friction drag and the pressure drag can be

significant.

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The high pressure in the vicinity of the stagnation point and the low pressure on the

opposite side in the wake produce a net force on the body in the direction of flow.

Figure 13-26 Average Drag Coefficient for Cross-flow over a Smooth Circular Cylinder and

a Smooth Sphere (Cengel, Turner & Cimbala, 2012, p. 611)

The drag force is primarily due to friction drag at low Reynolds numbers (Re<10) and

to pressure drag at high Reynolds numbers (Re>5000).

Both effects are significant at intermediate Reynolds numbers.

13.8.1 Observations from CD Curves

For Re<1, we have creeping flow, and the drag coefficient decreases with increasing

Reynolds number. For a sphere, it is CD=24/Re. There is no flow separation in this

regime.

At about Re=10, separation starts occurring on the rear of the body with vortex

shedding starting at about Re=90. The region of separation increases with increasing

Reynolds number up to about Re=103. At this point, the drag is mostly (about 95

percent) due to pressure drag. The drag coefficient continues to decrease with

increasing Reynolds number in this range of 10<Re<103.

In the moderate range of 103<Re<10

5, the drag coefficient remains relatively constant.

This behaviour is characteristic of bluff bodies. The flow in the boundary layer is

laminar in this range, but the flow in the separated region past the cylinder or sphere

is highly turbulent with a wide turbulent wake.

stagnation point low pressure

Separation starts Dramatic drop in CD

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There is a sudden drop in the drag coefficient somewhere in the range of 105<Re<10

6

(usually, at about 2105). This large reduction in CD is due to the flow in the boundary

layer becoming turbulent, which moves the separation point further on the rear of the

body, reducing the size of the wake and thus the magnitude of the pressure drag. This

is in contrast to streamlined bodies, which experience an increase in the drag coefficient

(mostly due to friction drag) when the boundary layer becomes turbulent.

There is a “transitional” regime for 2105<Re<210

6, in which CD dips to a minimum

value and then slowly rises to its final turbulent value.

Flow separation occurs at about = 80° (measured

from the front stagnation point of a cylinder) when

the boundary layer is laminar and at about = 140°

when it is turbulent.

The delay of separation in turbulent flow is caused

by the rapid fluctuations of the fluid in the

transverse direction, which enables the turbulent

boundary layer to travel farther along the surface

before separation occurs, resulting in a narrower

wake and a smaller pressure drag.

Flow visualisation of flow over

(a) a smooth sphere at Re = 15,000, and

(b) a sphere at Re = 30,000 with a trip wire.

The delay of boundary layer separation is clearly

seen by comparing the two photographs.

Figure 13-27 Delay in the Boundary Layer creates a Smaller Wake (Cengel, Turner &

Cimbala, 2012, p. 612)

Roughening the surface can be used to great advantage in reducing drag. Golf balls are

intentionally roughened to induce turbulence at a lower Reynolds number to take

advantage of the sharp drop in the drag coefficient at the onset of turbulence in the

boundary layer (the typical velocity range of golf balls is 15 to 150 m/s, and the

Reynolds number is less than 4105). The occurrence of turbulent flow at this Reynolds

number reduces the drag coefficient of a golf ball by about half. For a given hit, this

means a longer distance for the ball.

For a table tennis ball, however, the speeds are slower and the ball is smaller—it never

reaches speeds in the turbulent range. Therefore, the surfaces of table tennis balls are

made smooth.

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Surface roughness may increase or decrease the drag coefficient of a spherical object,

depending on the value of the Reynolds number.

Figure 13-28 Roughening the Surface can be used to great advantage in reducing Drag


13.9 Lift

Lift: The component of the net force (due to viscous and pressure forces) that is

perpendicular to the flow direction.

A = planform area: the area that would be seen by a person looking at the body from

above in a direction normal to the body

Lift coefficient

Frontal area for a cylinder and sphere

Drag Force Relationship

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Definition of various terms associated with an airfoil.

Figure 13-29 Definition of Various Terms associated with Flow over an Aerofoil (Cengel,


For an aircraft, the wingspan is the total distance between the tips of the two wings,

which includes the width of the fuselage between the wings.

The average lift per unit planform area FL/A is called the wing loading, which is simply

the ratio of the weight of the aircraft to the planform area of the wings (since lift equals

the weight during flying at constant altitude).

Airfoils are designed to generate lift while keeping the drag at a minimum.

Some devices such as the spoilers and inverted airfoils on racing cars are designed for

avoiding lift or generating negative lift to improve traction and control.

Lift in practice can be taken to be due entirely to the pressure distribution on the surfaces of the

body, and thus the shape of the body has the primary influence on lift.

Then the primary consideration in the design of aerofoils is minimising the average

pressure at the upper surface while maximising it at the lower surface. Pressure is low

at locations where the flow velocity is high, and pressure is high at locations where the

flow velocity is low. (Bernoulli’s Principle) Lift at moderate angles of attack is

practically independent of the surface roughness since roughness affects the wall

shear, not the pressure.

For airfoils, the contribution of viscous effects to lift is usually negligible since wall

shear is parallel to the surfaces and thus nearly normal to the direction of lift.

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Figure 13-30 Negligible Contribution of Wall Shear to Lift


Figure 13-31 Counterclockwise Starting Vortex+ Clockwise Resultant Circulation causes

Lift to be generated (Cengel, Turner & Cimbala, 2012, p. 615)

Flow starts with no lift, but the lower fluid stream separates at the trailing edge when

the velocity reaches a certain value. This forces the upper separated fluid stream to

close in at the trailing edge, initiating a clockwise circulation around the aerofoil. This

clockwise circulation increases the velocity of the upper stream while decreasing that

of the lower stream causing LIFT

Figure 13-32 Irrotational and Actual Flow Past Symmetrical and Non-symmetrical Two-

dimensional Airfoils (Cengel, Turner & Cimbala, 2012, p. 615)

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13.9.1 National Advisory Committee for Aeronautics

The graph below shows the variation of the lift-to-drag ratio with angle of attack for a

two-dimensional airfoil.

Figure 13-33 Variation of Lift-to-Drag Ratio with Angle of Attack (Cengel, Turner &

Cimbala, 2012, p. 616)

It is desirable for airfoils to generate the most lift while producing the least drag.

Therefore, a measure of performance for airfoils is the lift-to-drag ratio, which is

equivalent to the ratio of the lift-to-drag coefficients CL/CD.

The CL/CD ratio increases with the angle of attack until the airfoil stalls, and the value

of the lift-to-drag ratio can be of the order of 100 for a two-dimensional airfoil.

One way to change the lift and drag characteristics of an airfoil is to change the angle

of attack. On an airplane, the entire plane is pitched up to increase lift, since the wings

are fixed relative to the fuselage. Another approach is to change the shape of the airfoil

by the use of movable leading edge and trailing edge flaps. The flaps are used to alter the

shape of the wings during takeoff and landing to maximise lift at low speeds. Once at

cruising altitude, the flaps are retracted, and the wing is returned to its “normal” shape

with minimal drag coefficient and adequate lift coefficient to minimise fuel

consumption while cruising at a constant altitude.

Note that even a small lift coefficient can generate a large lift force during normal

operation because of the large cruising velocities of aircraft and the proportionality of

lift to the square of flow velocity.

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The lift and drag characteristics of an airfoil during takeoff and landing can be changed

by changing the shape of the airfoil by the use of movable flaps.

Figure 13-34 Flaps to assist Takeoff (Cengel, Turner & Cimbala, 2012, p. 616)

Figure 13-35 Effect of Flaps on Lift and Drag (Cengel, Turner & Cimbala, 2012, p. 617)

Lift and drag increase with angle of attack. Lift increases at a higher rate.

The maximum lift coefficient increases from about 1.5 for the airfoil with no flaps to

3.5 for the double-slotted flap case. The maximum drag coefficient increases from

about 0.06 for the airfoil with no flaps to about 0.3 for the double-slotted flap case.

The angle of attack of the flaps can be increased to maximise the lift coefficient.

The minimum flight velocity can be determined from the requirement that the total

weight W of the aircraft be equal to lift and CL = CL, max: For a given weight, the landing

Why are there flaps? Can they be removed?

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or takeoff speed can be minimised by maximising the product of the lift coefficient and

the wing area, CL, maxA. One way of doing that is to use flaps. Another way is to control

the boundary layer, which can be accomplished simply by leaving flow sections (slots)

between the flaps.

Slots are used to prevent the separation of the boundary layer from the upper surface

of the wings and the flaps. This is done by allowing air to move from the high-pressure

region under the wing into the low-pressure region at the top surface. A flapped airfoil

with a slot prevents the separation of the boundary layer from the upper surface and

to increase the lift coefficient.

Figure 13-36 Slots in Aerofoil to prevent Separation (Cengel, Turner & Cimbala, 2012,

p. 618)

The maximum lift coefficient increases from about 1.5 for the airfoil with no flaps to

3.5 for the double-slotted flap case.

The maximum drag coefficient increases from about 0.06 for the airfoil with no flaps

to about 0.3 for the double-slotted flap case.

The angle of attack of the flaps can be increased to maximise the lift coefficient.

The diagram below shows the variation of the lift coefficient with the angle of attack

for a symmetrical and a non-symmetrical airfoil.

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Figure 13-37 Variation of Lift Coefficient with Angle of Attack (Cengel, Turner & Cimbala,

2012, p. 618)

CL increases almost linearly with the angle of attack , reaches a maximum at about

=16°, and then starts to decrease sharply. This decrease of lift with further increase in

the angle of attack is called stall, and it is caused by flow separation and the formation

of a wide wake region over the top surface of the airfoil. Stall is highly undesirable

since it also increases drag.

At zero angle of attack ( = 0°), the lift coefficient is zero for symmetrical airfoils but

nonzero for non-symmetrical ones with greater curvature at the top surface. Therefore,

planes with symmetrical wing sections must fly with their wings at higher angles of

attack in order to produce the same lift. Observations show that the lift coefficient can

be increased several times by adjusting the angle of attack (from 0.25 at =0° for the

non-symmetrical airfoil to 1.25 at =10°).

Figure 13-38 Variation of Drag Coefficient with Angle of Attack (Cengel, Turner &

Cimbala, 2012, p. 618)

The graph above shows the variation of the drag coefficient of an airfoil with the angle

of attack.

The drag coefficient increases with the angle of attack, often exponentially.

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Therefore, large angles of attack should be used sparingly for short periods of time for

fuel efficiency.

Finite-Span Wings and Induced Drag

Tip vortices that interact with the free stream impose forces on the wing tips in all

directions, including the flow direction.

The component of the force in the flow direction adds to drag and is called induced drag.

The total drag of a wing is then the sum of the induced drag (3-D effects) and the drag

of the airfoil section (2-D effects).

Figure 13-39 Vortex Cores leaving the Trailing Edge of a Rectangular Wing (Cengel, Turner

& Cimbala, 2012, p. 619)

The photograph shows trailing vortices from a rectangular wing with vortex cores

leaving the trailing edge at the tips (top view).

Figure 13-40 Visualising of Tip Vortices (Cengel, Turner & Cimbala, 2012, p. 619)

The photograph shows a crop duster flies through smoky air to dramatically illustrate

the tip vortices produced at the tips of the wing.

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Aspect ratio: The ratio of the square of the average span of an airfoil to the planform

area. For an airfoil with a rectangular planform of chord c and span b,

The aspect ratio is a measure of how narrow an airfoil is in the flow direction. The lift

coefficient of wings, in general, increases while the drag coefficient decreases with

increasing aspect ratio. Bodies with large aspect ratios fly more efficiently, but they are

less manoeuvrable because of their larger moment of inertia (owing to the greater

distance from the centre).

Induced drag is reduced by (a) wing tip feathers on bird wings and (b) endplates or other

disruptions on airplane wings.

Figure 13-41 Induced Drag Reduction (Cengel, Turner & Cimbala, 2012, p. 620)

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Summary

Laminar and turbulent flow in pipes and fully developed pipe flow were studied.

Losses associated with pipe flow in piping networks and the pumping power

requirements were calculated. In addition, the effects of flow regime on the drag

coefficients associated with flow over cylinders and spheres were investigated. The

fundamentals of flow over aerofoils and the drag and lift forces acting on aerofoils

were evaluated. Various physical phenomena associated with external flow such as

drag, friction and pressure drag, drag reduction, and lift were also looked into.

TUTORIAL:

INTERNAL FLOW

Laminar & Turbulent Flow

Q14-2C, Q14-9C

Flow in Pipes

Q14-26

Piping Systems

Q14-75

EXTERNAL FLOW

Drag & Lift and Coefficients

Q15-9C, Q15-21

Lift, Drag & Stall

Q15-64C, Q15-76

Solutions

The solutions to questions posed in Study Unit 6 can be viewed in “Guides to tutorial

questions INTERNAL FLOW.docx” and ”Guides to tutorial questions EXTERNAL

FLOW.docx”.

Reference



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