CHAPTER ONE

INTRODUCTION

1.1. Background of the study

Fluid mechanics deals with the study of all fluids under static and

dynamic situations. Fluid mechanics is a branch of continuous

mechanics which deals with a relationship between forces motions

and statical conditions in a continuous material. This study area

deals with many and diversified problems such as surface tension

fluid statics flow in enclose bodies or flow round bodies (solid or

otherwise) flow stability etc. In fact almost any action a person is

doing involves some kind of a fluid mechanics problem. Researchers

distinguish between orderly flow and chaotic flow as the laminar

flow and the turbulent flow. The fluid mechanics can also be

distinguished between a single phase flow and multiphase flow (flow

made more than one phase or single distinguishable material).

Fluid flow in circular and noncircular pipes is commonly

encountered in practice. The hot and cold water that we use in our

homes is pumped through pipes. Water in a city is distributed by

extensive piping networks. Oil and natural gas are transported

hundreds of miles by large pipelines. Blood is carried throughout

our bodies by veins. The cooling water in an engine is transported

by hoses to the pipes in the radiator where it is cooled as it flows.

Thermal energy in a hydraulic space heating system is transferred

to the circulating water in the boiler and then it is transported to

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the desired locations in pipes. Fluid flow is classified as external

and internal depending on whether the fluid is forced to flow over a

surface or in a conduit. Internal and external flows exhibit very

different characteristics. In this chapter we consider internal flow

where the conduit is completely filled with the fluid and flow is

driven primarily by a pressure difference. This should not be

confused with open-channel flow where the conduit is partially filled

by the fluid and thus the flow is partially bounded by solid surfaces

as in an irrigation ditch and flow is driven by gravity alone. We

then discuss the characteristics of flow inside pipes and introduce

the pressure drop correlations associated with it for both laminar

and turbulent flows. Finally we present the minor losses and

determine the pressure drop and pumping power requirements for

piping systems. Pipes 611

14–5Liquid 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. A typical piping system

involves pipes of different diameters connected to each other by

various fittings or elbows to direct the fluid valves to control the

flow rate and pumps to pressurize the fluid. The terms pipe duct

and conduit are usually used interchangeably for flow sections. In

general flow sections of circular cross section are referred to as

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pipes (especially when the fluid is a liquid) and flow sections of

noncircular cross section as ducts (especially when the fluid is a

gas). Small-diameter pipes are usually referred to as tubes. Given

this uncertainty we will use more descriptive phrases (such as a

circular pipe or a rectangular duct) whenever necessary to avoid any

misunderstandings. You have probably noticed that most fluids

especially liquids are transported in circular pipes. This is because

pipes with a circular cross section can withstand large pressure

differences between the inside and the outside without undergoing

significant distortion. Noncircular pipes are usually used in

applications such as the heating and cooling systems of buildings

where the pressure difference is relatively small the manufacturing

and installation costs are lower and the available space is limited

for duct work. Although the theory of fluid flow is reasonably well

understood 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. Noting that the experimental results are

obtained under carefully controlled laboratory conditions and that

no two systems are exactly alike we must not be so naive as to view

the results obtained as ―exact.‖ The fluid velocity in a pipe changes

from zero at the surface because of the no-slip condition to a

maximum at the pipe center. In fluid flow it is convenient to work

with an average or mean velocity _m which remains constant in

incompressible flow when the cross-sectional area of the pipe is

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constant. The mean velocity in heating and cooling applications

may change somewhat because of changes in density with

temperature. But in practice we evaluate the fluid properties at

some average temperature and treat them as constants. The

convenience of working with constant properties usually more than

justifies the slight loss in accuracy.

Also the friction between the fluid layers in a pipe does cause a

slight rise in fluid temperature as a result of the mechanical energy

being converted to sensible thermal energy. But this temperature

rise due to fictional heating is usually too small to warrant any

consideration in calculations and thus is disregarded. For example

in the absence of any heat transfer no noticeable difference can

be detected between the inlet and exit temperatures of water flowing

in a pipe. The primary consequence of friction in fluid flow is

pressure drop and thus any significant temperature change in the

fluid is due to heat transfer.

1.2. Historical Developments

The continuous scientific development of fluid mechanics started

with Leonardo da Vinci (1452–1519). Through his ingenious work

methods were devised that were suitable for fluid mechanics

investigations of all kinds. Earlier efforts of Archimedes (287–212

B.C.) to understand fluid motions led to the understanding of the

hydro mechanical buoyancy and the stability of floating bodies. His

discoveries remained however without further impact on the

development of fluid mechanics in the following centuries.

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Something similar holds true for the work of Sextus Julius

Frontinus (40–103) who provided the basic understanding for the

methods that were applied in the Roman Empire for measuring the

volume flows in the Roman water supply system. The work of

Sextus Julius Frontinus also remained an individual achievement.

For more than a millennium no essential fluid mechanics insights

followed and there were no contributions to the understanding of

flow processes. Fluid mechanics as a field of science developed only

after the work of Leonardo da Vinci. His insight laid the basis for

the continuum principle for fluid mechanics considerations and he

contributed through many sketches of flow processes to the

development of the methodology to gain fluid mechanics insights

into flows by means of visualization. His ingenious engineering art

allowed him to devise the first installations that were driven fluid

mechanically and to provide sketches of technical problem solutions

on the basis of fluid flows. The work of Leonardo da Vinci was

followed by that of Galileo Galilei (1564–1642) and Evangelista

Torricelli (1608–1647). Whereas Galileo Galilei produced important

ideas for experimental hydraulics and revised the concept of

vacuum introduced by Aristoteles Evangelista Torricelli realized

the relationship between the weight of the atmosphere and the

barometric pressure. He developed the form of a horizontally ejected

fluid jet in connection with the laws of free fall. Torricelli’s work was

therefore an important contribution to the laws of fluids flowing out

of containers under the influence of gravity. Blaise Pascal (1623

1662) also dedicated himself to hydrostatics and was the first to

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formulate the theorem of universal pressure distribution. Isaac

Newton (1642–1727) laid the basis for the theoretical description of

fluid flows. He was the first to realize that molecule-dependent

momentum transport which he introduced as flow friction is

proportional to the velocity gradient and perpendicular to the flow

direction. He also made some additional contributions to the

detection and evaluation of the flow resistance. Concerning the jet

contraction arising with fluids flowing out of containers he engaged

in extensive deliberations although his ideas were not correct in

all respects. Henri de Pitot (1665–1771) made important

contributions to the understanding of stagnation pressure which

builds up in a flow at stagnation points. He was the first to

endeavor to make possible flow velocities by differential pressure

measurements following the construction of double-walled

measuring devices. Daniel Bernoulli (1700–1782) laid the

foundation of hydromechanics by establishing a connection

between pressure and velocity on the basis of simple energy

principles. He made essential contributions to pressure

measurements manometer technology and hydro mechanical

drives. Leonhard Euler (1707–1783) formulated the basics of the

flow equations of an ideal fluid. He derived from the conservation

equation of momentum the Bernoulli theorem that had however

already been derived by Johann Bernoulli (1667–1748) from energy

principles. He emphasized the significance of the pressure for the

entire field of fluid mechanics and explained among other things the

appearance of cavitations in installations. The basic principle of

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turbo engines was discovered and described by him. Euler’s work

on the formulation of the basic equations was supplemented by

Jean le Rond d’Alembert (1717–1783). He derived the continuity

equation in differential form and introduced the use of complex

numbers into the potential theory. In addition he derived the

acceleration component of a fluid element in field variables and

expressed the hypothesis named after him and proved before

by Euler that a body circulating in an ideal fluid has no flow

resistance. This fact known as d’Alembert’s paradox led to long

discussions concerning the validity of the equations of fluid

mechanics as the results derived from them did not agree with the

results of experimental investigations. The basic equations of fluid

mechanics were dealt with further by Joseph de Lagrange (1736–

1813) Louis Marie Henri Navier (1785–1836) and Barre de Saint

Venant (1797–1886). As solutions of the equations were not

successful for practical problems however practical hydraulics

developed parallel to the development of the theory of the basic

equations of fluid mechanics. Antoine Chezy (1718–1798)

formulated similarity parameters in order to transfer the results of

flow investigations in one flow channel to a second channel. Based

on similarity laws extensive experimental investigations were

carried out by Giovanni Battista Venturi (1746–1822) and also

experimental investigations were made on pressure loss

measurements in flows by Gotthilf Ludwig Hagen (1797–1884) and

on hydrodynamic resistances by Jean-Louis Poiseuille (1799–1869).

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This was followed by the work of Henri Philibert Gaspard Darcy

(1803–1858) on filtration i.e. for the determination of pressure

losses in pore bodies. In the field of civil engineering Julius

Weissbach (1806–1871) introduced the basis of hydraulics into

engineers’ considerations and determined by systematic

experiments dimensionless flow coefficients with which engineering

installations could be designed. The work of William Froude (1810–

1879) on the development of towing tank techniques led to model

investigations on ships and Robert Manning (1816–1897) worked

out many equations for resistance laws of bodies in open water

channels. Similar developments were introduced by Ernst Mach

(1838–1916) for compressible aerodynamics. He is seen as the

pioneer of supersonic aerodynamics providing essential insights

into the application of the knowledge on flows in which changes of

the density of a fluid are of importance. In addition to practical

hydromechanics analytical fluid mechanics developed in the

nineteenth century in order to solve analytically manageable

problems. George Gabriel Stokes (1816–1903) made analytical

contributions to the fluid mechanics of viscous media especially to

wave mechanics and to the viscous resistance of bodies and

formulated Stokes’ law for spheres falling in fluids. John William

Stratt Lord Rayleigh (1842–1919) carried out numerous

investigations on dynamic similarity and hydrodynamic instability.

Derivations of the basis for wave motions instabilities of bubbles

and drops and fluid jets etc. followed with clear indications as to

how linear instability considerations in fluid mechanics are to be

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carried out. Vincenz Strouhal (1850–1922) worked out the basics of

vibrations and oscillations in bodies through separating vortices.

Many other scientists who showed that applied mathematics can

make important contributions to the analytical solution of flow

problems could be named here. After the pioneering work of

Ludwig Prandtl (1875–1953) who introduced the boundary layer

concept into fluid mechanics analytical solutions to the basic

equations followed e.g. solutions of the boundary layer equations

by Paul Richard Heinrich Blasius (1883–1970). With Osborne

Reynolds (1832–1912) a new chapter in fluid mechanics was

opened. He carried out pioneering experiments in many areas of

fluid mechanics especially basic investigations on different

turbulent flows. He demonstrated that it is possible to formulate the

Navier–Stokes equations in a time-averaged form in order to

describe turbulent transport processes in this way. Essential work

in this area by Ludwig Prandtl (1875–1953) followed providing

fundamental insights into flows in the field of the boundary layer

theory. Theodor von Karman (1881–1993) made contributions to

many sub-domains of fluid mechanics and was followed by

numerous scientists who engaged in problem solutions in fluid

mechanics. One should mention here without claiming that the list

is complete Pei-Yuan Chou (1902–1993) and Andrei Nikolaevich

Kolmogorov (1903–1987) for their contributions to turbulence

theory and Herrmann Schlichting (1907–1982) for his work in the

field of laminar–turbulent transition and for uniting the fluid-

mechanical knowledge of his time and converting it into practical

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solutions of flow problems. The chronological sequence of the

contributions to the development of fluid mechanics outlined in the

above paragraphs can be rendered well in a diagram as shown in

Fig. 1.2.

Fig. 1.1 Diagram listing the epochs and scientists contributing to

the development of fluid mechanics.

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1.3. Significance of the study

Flows occur in all fields of our natural and technical environment

and anyone perceiving their surroundings with open eyes and

assessing their significance for themselves and their fellow beings

can convince themselves of the far reaching effects of fluid flows.

We somewhat arbitrarily classify these in two main categories: i)

physical and natural science and ii) technology. Clearly the second

thesis often of more interest to an engineering student but in the

modern era of emphasis on interdisciplinary studies the more

scientific and mathematical aspects of fluid phenomena are

becoming increasingly important.

Fluids in technology

It is easily recognized that a complete listing of fluid applications

would be nearly impossible simply because the presence of fluids in

technological devices is ubiquitous. The following provide some

particularly interesting and important examples from an

engineering standpoint.

1. Internal combustion engines—all types of transportation systems

2. Turbojet scramjet rocket engines—aerospace propulsion

systems

3. Waste disposal

(a) Chemical treatment

(b) Incineration

(c) Sewage transport and treatment

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4. Pollution dispersal—in the atmosphere (smog); in rivers and

oceans

5. Steam gas and wind turbines and hydroelectric facilities for

electric power generation

6. Pipelines

(a) Crude oil and natural gas transferral

(b) Irrigation facilities

(c) Office building and household plumbing

7. Fluid/structure interaction

(a) Design of tall buildings

(b) Continental shelf oil-drilling rigs

(c) Dams bridges etc.

(d) Aircraft and launch vehicle airframes and control systems

8. Heating ventilating and air-conditioning (HVAC) systems

9. Cooling systems for high-density electronic devices—digital

computers from PCs to supercomputers

10. Solar heat and geothermal heat utilization

11. Artificial hearts kidney dialysis machines insulin pumps

12. Manufacturing processes

(a) Spray painting automobiles trucks etc.

(b) Filling of containers e.g. cans of soup cartons of milk plastic

bottles of soda

(c) Operation of various hydraulic devices

(d) Chemical vapor deposition drawing of synthetic fibers wires

rods etc.

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Members of the Academia

We also want to draw the attention of the reader to the importance

of fluid mechanics in the field of chemical engineering where many

areas such as heat and mass transfer processes and chemical

reactions are influenced strongly or rendered possible only by flow

processes. In this field of engineering it becomes particularly clear

that much of the knowledge gained in the natural sciences can be

used technically only because it is possible to let processes run in a

steady and controlled way. In many areas of chemical engineering

fluid flows are being used to make steady-state processes possible

and to guarantee the controllability of plants i.e. flows are being

employed in many places in process engineering. Fluid flow

provides some examples of fluid phenomena often studied by

physicists astronomers biologists and others who do not

necessarily deal in the design and analysis of devices.

The study of fluid flow is significant to tackle other negative

effects on our natural environment that are the devastations that

hurricanes and cyclones can cause. When rivers lakes or seas leave

their natural beds and rims flow processes can arise whose

destructive forces are known to us from many inundation

catastrophes. This makes it clear that humans not only depend on

fluid flows in the positive sense but also have to learn to live with

the effects of such fluid flows that can destroy or damage the entire

environment.

We conclude from the various preceding examples that there is

essentially no part of our daily lives that is not influenced by fluids.

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As a consequence it is extremely important that engineers be

capable of predicting fluid motion. In particular the majority of

engineers who are not fluid dynamicists still will need to interact

on a technical basis with those who are quite frequently; and a

basic competence in fluid dynamics will make such interactions

more productive.

1.4. Problem statement

Fluid mechanics is a science that makes use of the basic laws of

mechanics and thermodynamics to describe the motion of fluids.

Here fluids are understood to be all the media that cannot be

assigned clearly to solids no matter whether their properties can be

described by simple or complicated material laws. Gases liquids

and many plastic materials are fluids whose movements are covered

by fluid mechanics. Fluids in a state of rest are dealt with as a

special cases of flowing media i.e. the laws for motionless fluids are

deduced in such a way that the velocity in the basic equations of

fluid mechanics is set equal to zero.

In fluid mechanics however one is not content with the

formulation of the laws by which fluid movements are described

but makes an effort beyond that to find solutions for flow problems

i.e. for given initial and boundary conditions. To this end there are

three major flow problems encountered in fluid mechanics:

(a) Analytical fluid mechanics problems:

Analytical methods of applied mathematics are used in this field to

solve the basic flow equations taking into account the boundary

conditions describing the actual flow problem.

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(b) Numerical fluid mechanics problems:

Numerical methods of applied mathematics are employed for fluid

flow simulations on computers to yield solutions of the basic

equations of fluid mechanics.

(c) Experimental fluid mechanics problems:

This sub-domain of fluid mechanics uses similarity laws for the

transferability of fluid mechanics knowledge from model flow

investigations. The knowledge gained in model flows by

measurements is transferred by means of the constancy of known

characteristic quantities of a flow field to the flow field of actual

interest.

The above-mentioned methods have until now in spite of

considerable developments in the last 50 years only partly reached

the state of development which is necessary to be able to describe

adequately or solve fluid mechanics problems especially for many

practical flow problems.

1.5. Objective of the study

The general objective of this study is to examine the head losses in

flow through horizontal and vertically mounted orifices with

statistical methods of data reliability. The goal of these experimental

remains to test the reliability of the result from the heat transfer

and fluid mechanics trainer. The results however can only attain

this objective through these:

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1. To convert volume flow rate in m/s

-1

to m

3

s

-1

and also h

1

and

h

2

in mm to m. also convert D

1

and D

2

in mm to m.

2. To compute P

1

P

2

V

1

V

2

A

1

A

2

and ∆H

L

for the set points of

90

0

75

0

60

0

45

0

30

0

and 15

0

using the analytical equations.

3. Plot H

L

versus V

2

/2g and discuss the plot.

4. To test the statistical hypotheses of the result

5. To provide suggestion for further improvement

1.6. Scope of the study

The study will make a great emphasis on the performance of head

losses in pipe flow using fluid mechanics and heat transfer trainer.

It tends to explain the statistical reliability of the experimental

results and the usefulness of such results.

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