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Ch1 Introduction

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INTRODUCTION
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Page 1: Ch1 Introduction

INTRODUCTION

Page 2: Ch1 Introduction

Characteristics of fluids

• A fluid may be liquid, vapour or gas. It has no permanent shape but takes up the shape of a containing vessel or channel or is shaped by external forces (e.g. the atmosphere).

• A fluid consists of atoms/molecules in random motion (translation) and in continual collision with the surroundings.

• Fluids are readily deformable, and flow.

• Solids have ‘frozen’ molecules that vibrate and do not translate. Solids resist change of shape.

• A fluid is defined as a substance that deforms continuously when acted on by a shearing stress of any magnitude.

Page 3: Ch1 Introduction

• For a solid, application of a shear stress causes a deformation which, if modest, is not permanent and solid regains original position.

Attached plates Solid

Characteristics of fluids

Page 4: Ch1 Introduction

• For a fluid, continuous deformation takes place with an infinite number of layers sliding over each other. Deformation continues until the force is removed.

• A fluid is a substance for which a shear stress tends to produce unlimited deformation.

Fluid

Characteristics of fluids

Page 5: Ch1 Introduction

Dimensions and Units

• Fluid characteristics are described qualitatively in terms of basic dimensions: length, L, time, T, and mass, M.

• All theoretically derived equations are dimensionally homogeneous.

• For a quantitative description units are required

• Two system of units will be used:

– International System (SI); m, s, kg, K

– British Gravitational (BG) System; ft, s, lb, ºF or ºR

Page 6: Ch1 Introduction

• From a microscopic point of view a fluid is not a continuous and homogeneous substance.

• We take the engineering macroscopic view such that we can examine a sufficiently large ‘particle’ of fluid to allow the concept of velocity and density ‘at a point’.

• Density is the mass per unit volume, and is a macroscopic concept. Density at a point is:

• Specific volume, specific weight, specific gravity

Properties and characteristics of fluids

Vm

v 0lim

Page 7: Ch1 Introduction

• Pressure = (normal force) / area. The pressure at a point is:

• In the absence of shear forces (fluid at rest or in uniform motion) pressure at a point is independent of direction

Properties and characteristics of fluids

AF

pA 0lim

Page 8: Ch1 Introduction

• Perfect gas law

– In this course all gases obey the perfect gas law

Properties and characteristics of fluids

RTpormRTpV

Page 9: Ch1 Introduction

Viscosity

• For elastic solids shearing strain is proportional to the shearing stress

• For fluids shearing stress is proportional to the rate of shearing strain

• For Newtonian fluids shearing stress is linearly proportional to the rate of shearing strain

• The study of non-Newtonian fluids is called rheology

• Viscosity is very sensitive to temperature

Page 10: Ch1 Introduction

Example 1.5: The velocity distribution for the flow of a Newtonian fluid between two wide, parallel plates is given by the equation

where V is the mean velocity. The fluid has a viscosity of 0.0r lb·s/ft2. When V = 2 ft/s and h = 0.2 in. determine: (a) the shearing stress acting on the bottom wall, and (b) the shearing stress acting on a plane parallel to the walls and passing through the centerline (midplane)

23

12

V yu

h

Page 11: Ch1 Introduction

Example 1.5: The velocity distribution for the flow of a Newtonian fluid between two wide, parallel plates is given by the equation

where V is the mean velocity. The fluid has a viscosity of 0.0r lb·s/ft2. When V = 2 ft/s and h = 0.2 in. determine: (a) the shearing stress acting on the bottom wall, and (b) the shearing stress acting on a plane parallel to the walls and passing through the centerline (midplane)

Solution.

Shearing stress

Velocity distribution

(a) Along the bottom wall, y = -h shearing stress

(b) Along the midplane, y = 0 shearing stress

23

12

V yu

h

du

dy

0du

dy

2

3du Vy

dy h

3du V

dy h

2bot wall 14.4 lb/ft

midplane 0

Page 12: Ch1 Introduction

Properties and characteristics of fluids

• Compressibility: all fluids are compressible, especially gases. Most liquids can be regarded as incompressible for most purposes.

• The bulk modulus of elasticity, Ev , is a property which is used to account for compressive effects:

• Speed of sound is the velocity at which small disturbances propagate in a fluid. For ideal gases speed of sound:

v

pE

c kRT

Page 13: Ch1 Introduction

• Vapor pressure is a pressure exerted by a vapour on the fluid when they are in equilibrium in a closed vessel

• Vapor pressure is a function of temperature

• A liquid boils when the pressure is reduced to vapor pressure

• When the liquid pressure is dropped below the vapor pressure due to flow phenomena, we call the process cavitation

• Cavitation is the formation and subsequent collapse of vapor bubbles in a flowing fluid

Vapor Pressure

Page 14: Ch1 Introduction
Page 15: Ch1 Introduction

• Liquid, being unable to expand freely, will form an interface with a second liquid or gas

• This surface phenomenon is due to unbalanced cohesive forces acting on the liquid molecule on the fluid surface

• The intensity of molecular attraction per unit length along any line in the surface is called the surface tension coefficient, (N/m)

• The value of surface tension decreases as temperature increases

• If the interface is curved, then there is a pressure difference across the interface, the pressure being higher on the concave side (drop of fluid, bubble)

• Capillary action in small tubes, which involves a liquid-gas solid interface, is also caused by surface tension

Surface Tension

Page 16: Ch1 Introduction

Effect of capillary action in small tubes. (a) Rise of column for a liquid that wets the tube. (b) Free-body diagram for calculating column height. (c) Depression of column for a nonwetting liquid.

The height h is governed by the value of the surface tension, , tube radius, R, specific weight of the liquid, , and the angle of contact, .

Page 17: Ch1 Introduction

Effect of capillary action in small tubes. (a) Rise of column for a liquid that wets the tube. (b) Free-body diagram for calculating column height. (c) Depression of column for a nonwetting liquid.

The height h is governed by the value of the surface tension, , tube radius, R, specific weight of the liquid, , and the angle of contact, .

2 2 cos

2 cos

R h R

hR

Page 18: Ch1 Introduction

Example 1.8: Pressures are sometime determined by measuring the height of a column of liquid in a vertical tube. What diameter of clean glass tubing is required so that the rise of water at 20ºC in a tube due to capillary action (as opposed to pressure in the tube) is less than 1.0 mm?

Page 19: Ch1 Introduction

Example 1.8: Pressures are sometime determined by measuring the height of a column of liquid in a vertical tube. What diameter of clean glass tubing is required so that the rise of water at 20ºC in a tube due to capillary action (as opposed to pressure in the tube) is less than 1.0 mm?

Solution

For water at 20ºC (from Table B.2), = 0.0728 N/m and = 9.789 kN/m3. Since 0º it follows that for h = 1.0 mm,

And the minimum required tube diameter, D, is

2 cosh

R

2 cosR

h

3 3 3

2 0.0728 N/m 10.0149 m

9.789 10 N/m 1.0 mm 10 m/mmR

2 0.0298 m 29.8 mmD R

Page 20: Ch1 Introduction

Problem 1.65: A 12-in.-diameter circular plate is placed over a fixed bottom plate with a 0.1-in. gap between the two plates filled with glycerin. Determine the torque required to rotate the circular plate slowly at 2 rpm. Assume that the velocity distribution in the gap is linear and that the shear stress on the edge of the rotating plate is negligible.

Page 21: Ch1 Introduction

Problem 1.65: A 12-in.-diameter circular plate is placed over a fixed bottom plate with a 0.1-in. gap between the two plates filled with glycerin. Determine the torque required to rotate the circular plate slowly at 2 rpm. Assume that the velocity distribution in the gap is linear and that the shear stress on the edge of the rotating plate is negligible.

Solution

Torque due to shearing stress on plate:

where Thus

and

dT r dA

2dA rdr

2dT r rdr

2

02

RT r dr

Page 22: Ch1 Introduction

Problem 1.65: A 12-in.-diameter circular plate is placed over a fixed bottom plate with a 0.1-in. gap between the two plates filled with glycerin. Determine the torque required to rotate the circular plate slowly at 2 rpm. Assume that the velocity distribution in the gap is linear and that the shear stress on the edge of the rotating plate is negligible.

Solution

Torque due to shearing stress on plate:

Velocity distribution

Searing stress

2

02

RT r dr

du V r

dy

du r

dy

Page 23: Ch1 Introduction

Problem 1.65: A 12-in.-diameter circular plate is placed over a fixed bottom plate with a 0.1-in. gap between the two plates filled with glycerin. Determine the torque required to rotate the circular plate slowly at 2 rpm. Assume that the velocity distribution in the gap is linear and that the shear stress on the edge of the rotating plate is negligible.

Solution

Torque due to shearing stress on plate:

Searing stress

Torque

2

02

RT r dr

du r

dy

43

0

2 2

4

R RT r dr

Page 24: Ch1 Introduction

Problem 1.65: A 12-in.-diameter circular plate is placed over a fixed bottom plate with a 0.1-in. gap between the two plates filled with glycerin. Determine the torque required to rotate the circular plate slowly at 2 rpm. Assume that the velocity distribution in the gap is linear and that the shear stress on the edge of the rotating plate is negligible.

Solution

Torque

4

4 2

lb s rev rad 1 min 62 0.0313 2 2

2 ft min rev 60 s 12 ft0.0772 ft lb

0.14 ft 4

12

RT

Page 25: Ch1 Introduction

END OF LECTURE

Page 26: Ch1 Introduction

Dimensions Associated with Common Physical Quantities

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Page 27: Ch1 Introduction

Density of water as a function of temperature

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Page 28: Ch1 Introduction

(a) Deformation of material placed between two parallel plates. (b) Forces acting on upper plate.

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Page 29: Ch1 Introduction

Behavior of a fluid placed between two parallel plates

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du

dy

Page 30: Ch1 Introduction

Linear variation of shearing stress with rate of shearing strain for common fluid

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Page 31: Ch1 Introduction

Variation of shearing stress with rate of shearing strain for several types of fluids, including common non-Newtonian fluids.

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Page 32: Ch1 Introduction

Dynamic (absolute) viscosity of some common fluids as a function of temperature

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Page 33: Ch1 Introduction

Forces acting on one-half of a liquid drop

22

2i e

R p R

p p pR

Pressure drop across the surface of the droplet

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Page 34: Ch1 Introduction

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