Mechanical Principles B-2017-2018 (Part I & Part II)(Students Version) PDF

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SCHOOL OF ENGINEERING & BUILT ENVIRONMENT DEPARTMENT OF ENGIEERING

MECHANICAL PRINCIPLES B (M1H321923)  FLUID MECHANICS NOTES  THERMODYNAMICS NOTES 2017-2018

Prof M. El-Sharif Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

Prof M. El-Sharif

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MECHANICAL PRINCIPLES B (M1H321923) Part I: Fluid Mechanics

Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

Prof M. El-Sharif

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Chapter 1 1.1

Fluids Mechanics and Fluid Properties

Introduction

Matter exists in two states; the solid and the fluid, the fluid state being commonly divided into the liquid and gaseous states. Solids differ from liquids and liquids from gases in the spacing and latitude of motion of their molecules, these variables being large in a gas, smaller in a liquid, and extremely small in a solid. Thus it follows that intermolecular cohesive forces are large in a solid, smaller in a liquid, and extremely small in a gas. What is fluid mechanics? As its name suggests it is the branch of applied mechanics concerned with the statics and dynamics of fluids - both liquids and gases. The analysis of the behaviour of fluids is based on the fundamental laws of mechanics which relate continuity of mass and energy with force and momentum together with the familiar solid mechanics properties. 1.2

Objectives of this section 

Introduce the concept of dimension and units



Define the nature of a fluid.



Show where fluid mechanics concepts are common with those of solid mechanics and indicate some fundamental areas of difference.



Introduce viscosity and show what are Newtonian and non-Newtonian fluids



Define the appropriate physical properties and show how these allow differentiation between solids and fluids as well as between liquids and gases.

1.3 System of Units As any quantity can be expressed in whatever way you like it is sometimes easy to become confused as to what exactly or how much is being referred to. This is particularly true in the field of fluid mechanics. Over the years many different ways have been used to express the various quantities involved. Even today different countries use different terminology as well as different units for the same thing - they even use the same name for different things e.g. an American pint is 4/5 of a British pint ! To avoid any confusion on this course we will always use the SI (metric) system which you will already be familiar with. It is essential that all quantities are expressed in the same system or the wrong solutions will results. Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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Despite this warning you will still find that this is the most common mistake when you attempt example questions. 1.3.1 Dimension and Units Dimension = A dimension is the measure by which a physical variable is expressed quantitatively. Unit = A unit is a particular way of attaching a number to the quantitative dimension. Thus length is a dimension associated with such variables as distance, displacement, width, deflection, and height, while centimetres or meters are both numerical units for expressing length. 1.3.2 The SI System of units The SI system consists of six primary units, from which all quantities may be described. For convenience secondary units are used in general practice which are made from combinations of these primary units. 1.3.3 Primary Units The six primary units of the SI system are shown in the table below: Quantity Length Mass Time Temperature Current Luminosity

SI Unit metre, m kilogram, kg second, s Kelvin, K ampere, A candela

Dimension L M T θ I Cd

In fluid mechanics we are generally only interested in the top four units from this table. Notice how the term 'Dimension' of a unit has been introduced in this table. This is not a property of the individual units, rather it tells what the unit represents. For example a metre is a length which has a dimension L but also, an inch, a mile or a kilometre are all lengths so have dimension of L. (The above notation uses the MLT system of dimensions, there are other ways of writing dimensions)

1.3.4 Derived Units There are many derived units all obtained from combination of the above primary units. Those most used are shown in the table below: Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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Quantity velocity acceleration force energy (or work)

power

pressure ( or stress)

density specific weight relative density viscosity surface tension

SI Unit m/s m/s2 N kg m/s2 Joule J N m, kg m2/s2 Watt W N m/s kg m2/s3 Pascal Pa, N/m2, kg/m/s2 kg/m3 N/m3 kg/m2/s2 a ratio no units N s/m2 kg/m s N/m kg /s2

-1

ms ms-2 kg ms-2

Dimension LT-1 LT-2 M LT-2 ML2T-2

kg m2s-2 ML2T-3 -1

Nms kg m2s-3 -2

ML-1T-2

Nm kg m-1s-2 kg m-3 Nm-3 kg m-2s-2

N sm-2 kg m-1s-1 Nm-1 kg s-2

ML-3 ML-2T-2 1 no dimension M L-1T-1 MT-2

The above units should be used at all times. Values in other units should NOT be used without first converting them into the appropriate SI unit. If you do not know what a particular unit means - find out, else your guess will probably be wrong. 1.3.5 Dimensional Analysis. The fundamental quantities mainly relevant to fluid mechanics are Mass, Length and Time. All other quantities can be expressed in terms of them (e.g. velocity = length/time). The unit schemes give a way of measuring these quantities. All equations must be dimensionally consistent dim(X) = dim(Y ) The fundamental mass M, length L and time T constituents that make up X and Y must be the same. e.g.

Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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dim(Δs) = dim(a) dim(t2)

L

L 2 T L T2

Since the LHS and RHS have the same dimensions it is possible for the equation to be correct. It is only possible to add or subtract two quantities if they have the same dimension (e.g. you can’t add apples and oranges). The equation X=A+B can only be true if dim(X) = dim(A) + dim(B)

Recommendation: In working with problems with complex or mixed system units, at the start of the problem convert all parameters with units to the base units being used in the problem, e.g. for S.I. problems, convert all parameters to kg, m, & s. Then convert the final answer to the desired final units. 1.4

Fluids

There are two aspects of fluid mechanics which make it different to solid mechanics: 1.

The nature of a fluid is much different to that of a solid

2.

In fluids we usually deal with continuous streams of fluid without a beginning or end. In solids we only consider individual elements.

As previously stated that liquid and gas are called fluids and are in contrast to solids they lack the ability to resist deformation. To appreciate this fundamental difference between solids and fluids let us subject block of a solid and block of a fluid to a shearing forces and observe their behaviour. Behaviour of a solid when subjected to a tensile force

If a solid body is subjected to forces that tend to stretch, compress or shear the object, its shape changes, reaching a new status of equilibrium between external and Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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internal forces. If the object returns to its original shape when the external forces are removed, it is said to be elastic. Most objects are elastic for forces up a certain maximum, called the elastic limit. If the acting forces exceed the elastic limit the object remains permanently deformed or it is fractured. A solid can resist a deformation force while at rest, this force may cause some displacement but the solid does not continue to move indefinitely. Behaviour of a fluid when subjected to a shearing force Consider two flat plates of infinite length placed a distance h apart as shown in Figure below. The lower plate is fixed while the upper plate is allowed to move.

Shear Force F

Moving Plate A h

B

A

D

C

A’

B’

B

Fluid

C

D

Fixed Plate Shearing force, F, acting on a fluid element. The deformation is caused by shearing forces which act tangentially to a surface. Referring to the figure above, we see the force F acting tangentially on a rectangular element ABDC. This is a shearing force and produces the (dashed lined) rhombus element A’B’DC. Because a fluid cannot resist the deformation force, it moves, it flows under the action of the force. Its shape will change continuously as long as the force is applied. This is a fundamental difference between solids and fluids and hence is used as a base for the definition of a fluid: A Fluid is a substance which deforms continuously, or flows, when subjected to shearing forces however small it may be. and conversely this definition implies the very important point that: If a fluid is at rest there are no shearing forces acting. All forces must be perpendicular to the planes on which they are acting. 1.4.1 Fluid Properties Solids, liquids and gases are all composed of molecules in continuous motion. However, the arrangement of these molecules, and the spaces between them, differ, giving rise to the characteristics properties of the three states of matter. In solids, the molecules are densely and regularly packed and movement is slight, each molecule Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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F

being strained by its neighbours. In liquids, the structure is loser; individual molecules have greater freedom of movement and, although restrained to some degree by the surrounding molecules, can break away from the restraint, causing a change of structure. In gases, there is no formal structure, the spaces between molecules are large and the molecules can move freely.

In a solid the atoms are tightly bound by intermolecular forces.

In a liquid the intermolecular forces keep the atoms close together. But, lack of long range order with disruption of molecular forces makes it possible for groups of atoms to slide past each other.

In a gas the molecules can move independently of each other.

1.4.2 Continuum Hypothesis In this course, fluids will be assumed to be continuous substances, and, when the behaviour of a small element or particle of fluid is studied, it will be assumed that it contains so many molecules that it can be treated as part of this continuum. Quantities such as velocity and pressure can be considered to be constant at any point, and changes due to molecular motion may be ignored. Variations in such quantities can also be assumed to take place smoothly, from point to point. 1.4.3 Fluids Properties: 1.4.3.1 Density There are three ways of expressing density: a. Mass density (ρ): The fluid mass density (ρ) is one of the primary fluid properties. For a fluid, the mass density is the mass m per unit volume V, so Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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ρ= mass per unit volume



massof fluid volumeof fluid



(Units: kg/m3)

m V

b. Specific Volume (v): The specific volume is the volume per unit mass and is the reciprocal of the density,

v

1



(Units: m3/kg)

(specific volume is mainly used in thermodynamics) Temperature and pressure do not have much effect on the density of liquids as shown below for the density of water (they do on gases). The density of water is 1000 kgm−3 at 4oC and about 958 kgm−3 at 100oC.

c. Specific Weight ( ): The specific weight () of a fluid is designated as the weight force per unit volume.

weight of fluid unit volume of fluid mass of fluid gravitational accel  unit volume of fluid



Because the weight (a force), W, related to its mass, m, by Newton’s second law of motion in the form Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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W=m×g

In which g is the acceleration due to the local force of gravity.

m g   g V kg m N    g [ 3  2 ]  3 m m s



(Units: N/m3 or kg/m2/s2)

The specific weight of water is 9.80 kNm−3

d.

Relative Density (RD):

The relative density of a fluid is the ratio of the fluid mass density to the mass density of water at a specific temperature. For solids and liquids this standard mass density is the maximum mass density for water (which occurs at 4oC) at atmospheric pressure. The reference temperature is usually 4oC since ρ is largest here.

 RD =

massdensity of a substance massdensityof water@ 4o C

RD 

s

 w @ 4o C

The specific density is a ratio of densities so it is dimensionless.

Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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Example: Given that 5.6m3 of oil weighs 46,800 N, calculate its density (ρ) and relative density (RD). Solution:

Weight of Unit Volume   g  Density  

g g



46800 8360N / m3 5.6

8360  852 kg/m3 9.81

RelativeDensity (RD) 

 oil 852  0.852 =  water 1000

1.4.3.2 Pressure Convenient to work in terms of pressure, p, which is the force per unit area. What is pressure?

The pressure force arises due to the continual transfer of momentum from individual molecules during collisions with the walls. The SI unit for pressure is the Pascal Pa and as it is a stress 1 Pa = 1 N/1 m2. The pressure force is an example of a stress, i.e. a force that is applied over an area. The pressure force acts in a direction perpendicular to the surface. Hence it is convenient to work in terms of pressure, p, which is the force per unit area.

Pressure

Force Area over whichthe forceis applied P

F A

(Units: N/m2)

Units: Newtons per square metre, N/m2, kg/m s2 (kg m-1s-2). Also known as a Pascal, Pa, i.e. 1 Pa = 1 N/m2 Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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Also frequently used is the alternative SI unit the bar, where 1bar = 105 N/m2 Standard atmosphere = 101325 Pa = 101.325 kPa 1 bar = 100 kPa (kilopascals) 1 mbar = 0.001 bar = 0.1 kPa = 100 Pa Uniform Pressure: If the pressure is the same at all points on a surface uniform pressure There are two pressures, absolute pressure and gauge pressure. (note, gauge pressure is sometimes written as gage or gauge

Pressure definitions There are 3 different working definitions for pressure. These are absolute, gauge or differential pressure.

The absolute pressure is the force per unit area that the molecules inside a chamber exert on the chamber walls. Always positive.

The chamber is immersed in the atmosphere. The gauge pressure is the difference between the absolute pressure and the atmospheric pressure. Can be negative if pressure in chamber is less than atmosphere.

The differential pressure refers to the pressure between two different chambers.

Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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Mechanical Principles B:

Tutorials I

Fluid Properties

Problem 1: Given that the density of a liquid is 837 kg/m3, find its weight per unit volume and relative density. (Ans: 8210 N/m3, 0.837) Problem 2: A cylindrical tank (diameter=10m and depth=5m) contains water at 20oC and is brimful. If the water is heated to 50oC, how much water will spill over the edge of the tank. The density of water at 20oC is 998.2 kg/m3 and at 50oC is 988.14 kg/m3

(Ans: 4m3).

Problem 3: An elephant exerts a force of 5000N by pressing his foot on the ground. If the area of his foot is 0.02 m2, determine the pressure exerted by his foot. (Ans: 250 kN/m2).

Problem 4: A tank contains 1000 kg of water. If the base of the tank has an area of 20 m2, determine the pressure exerted by the water on the base. (Ans: 490.5 N/m2).

Problem 5: Determine the force on an area of 11.2 m2 caused by a uniform pressure of 120 N/m2. (Ans: 1344 kN). Problem 6: A pipeline of length 120m holds 3792 kg of kerosene of relative density 0.78, determine the diameter of the pipeline. (Ans: 0.228 m)

Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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1.4.3.3 Newton’s Law of Viscosity How can we make use of these observations? We can start by considering a 3d rectangular element of fluid, like that in the figure below.

Fluid element under a shear force The shearing force F acts on the area on the top of the element. This area is given by A  z  x . We can thus calculate the shear stress which is equal to force per unit area i.e.

Shear force Areain shear F  A

Shear Stress 

Now let us consider a 2-D view

The deformation which this shear stress causes is measured by the size of the angle Ф and is know as shear strain.

Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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It has been found experimentally that the rate of shear stress (shear stress per unit time, / time ) is directly proportional to the shear stress. If the particle at point E (in the above figure) moves under the shear stress to point E’ and it takes time t to get there, it has moved the distance x. For small deformations we can write

Shear Strain,  

x y Φ x x1   t ty t y u  y

rate of shear strain

Where

x u t

is the velocity of the particle at E.

Using the experimental result that shear stress is proportional to rate of shear strain then:

  constant The term

u y

u y

is the change in velocity with y, or the velocity gradient, and may be written in

the differential form

du dy

. The constant of proportionality is known as the dynamic viscosity,

, of the fluid, giving



Where



 

du dy

du dy

This is known as Newton’s law of viscosity.

is the shear stress in N/m2 is the absolute viscosity, or the dynamic viscosity or just the viscosity. N s/m2 or kg/m s is the velocity gradient s-1

There are two ways of expressing viscosity: Coefficient of Dynamic Viscosity: Mechanical Principles B: (M1H321923) Fluid Mechanics Notes:

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

 (Units: N s/m2 or Pa s or kg/m s)

 du   dy 

The unit Poise is also used where 10 P = 1 Pa s Water Mercury Olive oil Pitch Honey Ketchup

μ = 8.94 × 10 −4 N s/m2 μ = 1.526 × 10−3 N s/m2 μ = 0.081 N s/m2 μ = 2.3 × 108 N s/m2 μ = 2000 – 10000 N s/m2 μ = 50000 – 100000 N s/m2 (non-Newtonian)

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