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 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 22

  1. Pollution dispersal—in the atmosphere (smog); in rivers and oceans
  2. Steam, gas and wind turbines, and hydroelectric facilities for electric power generation
  3. Pipelines

(a) Crude oil and natural gas transferral

(b) Irrigation facilities

(c) Office building and household plumbing

  1. 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

  1. Heating, ventilating and air-conditioning (HVAC) systems
  2. Cooling systems for high-density electronic devices—digital computers from PCs to supercomputers
  3. Solar heat and geothermal heat utilization
  4. Artificial hearts, kidney dialysis machines, insulin pumps
  5. 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. 23

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 ΔHL 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.