Head Loses In Horizontal And Vertical Orificemeter A Comparative Evaluation And Analyses With Application Of Statistical Method Of Data Reliability

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
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
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
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.
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
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
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).
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
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).
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
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
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.

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
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.
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.
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.
(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:
1. To convert volume flow rate in m/s-1 to m3s-1 and also h1 and
h2 in mm to m. also convert D1 and D2 in mm to m.
2. To compute P1, P2, V1, V2, A1, A2, and ∆HL for the set points of
900, 750, 600, 450, 300, and 150 using the analytical equations.
3. Plot HL versus V2/2g 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.