Chapter 4 fluid mechanics

FLUID MECHANICSCLD 10603

INTERNAL FLOW IN PIPES AND CONDUITS

CLD 10603 Chapter 4: Internal Flow in Pipe & Conduits

Steam pipe bridge in a geothermal power plant. Pipe flows are everywhere, often occurring in groups or networks. They are designed using the principles outlined in this chapter.

CLD 10603 3 Chapter 4: Internal Flow in Pipe & Conduits

Universiti Kuala Lumpur Malaysian Institute of Chemical and Bioengineering technology

Student should be able to: Identify the characteristics of laminar flow and

turbulent flow Employ Poiseulle and Darcy equation in basic

problems Recall Bernoulli’s equation Employ Bernoulli’s equation in flow measurement Identify major losses Identify minor losses

Objectives

The three regimes of viscous flow:

a) Laminar flow at low Re;

b) Transition at intermediate Re;

c) Turbulent flow at high Re



Types of Flow

Laminar Flow Flow structure is characterized by smooth

motion in lamina or layer. There is no macroscopic mixing of adjacent fluid layer.



Types of Flow

- E.g. : A thin filament of dye injected into a laminar flow appears as a single line. There is no dispersion of dye throughout the flow, except the slow dispersion due to molecular motion.

Laminar Flow

- Re < 2000

- 'low' velocity

- Dye does not mix with water

- Fluid particles move in straight lines

- Simple mathematical analysis possible

- Rare in practice in water systems.



Types of Flow

Turbulent Flow Flow structure is characterized by random,

three-dimensional motion of fluid particles superimposed on the mean motion.



Types of Flow

- E.g.: If a dye filament injected into a turbulent flow, it disperses quickly throughout the flow field; the line of dye breaks up into myriad entangled threads of dye.

Turbulent Flow

- Re > 5000

- 'high' velocity

- Dye mixes rapidly and completely

- Particle paths completely irregular

- Average motion is in the direction of the flow



Types of Flow

Turbulent Flow - Cannot be seen by the naked eye - Changes/fluctuations are very difficult to

detect. Must use laser. - Mathematical analysis very difficult - so

experimental measures are used - Most common type of flow.



Types of Flow

Laminar Turbulent

Re < 2000 Re > 4000

Inefficient mixing – low radial mixing

Radial mixing can be enhanced

Broad residence time distribution

Short residence time distribution

Low power High power



Types of Flow

The primary parameter correlating the viscous behaviour of all Newtonian fluids.

The ratio of inertial forces to viscous forces.

Where V = velocity (m/s), D = diameter (m),

is kinematic viscosity (m2/s).



Reynolds Number

uDVD

Re

Used to determine whether a flow is laminar or turbulent.

Reynolds Number is dimensionless. Re < 2000: Laminar Flow Re between 2000 to 4000 : Transitional

Flow Re > 4000 : Turbulent Flow



Reynolds Number

Question 1

Water flow through a pipe 25 mm in diameter at a velocity of 6 m/s. Determine whether the flow will be laminar or turbulent assuming that the viscosity of water is 1.30 x 10-3 kgm-1s-1 and its density 1000 kgm-3.



Reynolds Number

Question 2

If oil of S.G 0.9 and viscosity 9.6 x 10-2 kgm-1s-1 is pumped through the same pipe, what type of flow will occur?



Reynolds Number

The total quantity of fluid flowing in unit time past any particular cross section of a stream is called the discharge or flow at that section.

It can be measured either in terms of:

i. volume (volumetric flowrate = Q = Au)

Unit : m3/s

ii. mass (mass flowrate = = Qρ).

Unit : kg/s



Discharge

m

Question 3

Water flowing at the rate of 10 gal/min in pipe having an inside diameter of 0.0525 m. Calculate the Reynolds Number if = 8.007 x 10-4 Pa.s. Given H2O = 996 kg/m3.



Reynolds Number

The Bernoulli’s equation states that:

P + ½ u2 + gz = constant

Where:

P = pressure

= density

g = acceleration of gravity

z = height

u = velocity

Chapter 4: Internal Flow in Pipe & Conduits CLD 10603 34


Bernoulli's Equation

It may be written as:




2

222

1

211

22z

g

u

g

pz

g

u

g

p

Bernoulli's equation has some restrictions in its applicability, they are: Flow is steady; Density is constant (incompressible fluid) Friction losses are negligible. The equation relates the states at two

points along a single streamline




Flow through a small orifice



Bernoulli's Application

Orifice

A actual

Vena contractor

u1 = 0 (negligible)

P1 = 0 (P atm)

P2 = 0 (P atm)

Take z2 as the datum z1= h, z2= 0

2

222

1

211

22z

g

u

g

pz

g

u

g

p




g

uz

2

22

1

2

222

1

211

22z

g

u

g

pz

g

u

g

p

12 2gzu




Discharge through the orifice is given by:

Q = Au

Q actual = A actual u actual

= cd A orifice u theoretical

= cd A orifice 2gh



Flow Measurement - Orifice

Question 5

A 100 mm diameter standard orifice discharges water under a 6.1 m head. What is the volumetric flow rate? Given cd = 0.594




Question 6

The actual velocity in the contracted section of a jet of liquid flowing from a 50 mm diameter orifice is 8.53 ms-1 under a head of 4.57 m. If the measured discharge is 0.0114 m3s-1,what is the value of the coefficient of discharge (cd)?




A device to measure the local velocity along a streamline.

u2 = 0 (stagnation point)

z1=z2= 0



Flow Measurement – Pitot Tube

2

222

1

211

22z

g

u

g

pz

g

u

g

p




2

222

1

211

22z

g

u

g

Pz

g

u

g

P

)(2

)(2

2

121

1221

2112

PPu

PPu

g

u

g

PP

Where:P = h(man -)g




)(21 P

u

A device for measuring discharge in a pipe.

Consists of a rapidly converging section which increases the velocity of flow and hence reduces the pressure.

Then returns to the original dimensions of the pipe by a gently diverging 'diffuser' section.



Flow Measurement - Venturi




A Venturi Meter







Discharge can be expressed in terms of the manometer reading , giving:




22

21

21

12

AA

gh

AACQ

man

dactual

Question 7

A Venturi meter is being used to measure flow in a pipeline of diameter 250 mm which carries water. When the pressure difference between the throat and the entrance of the Venturi meter is 300 mm on a mercury manometer, determine the flow in the pipeline. The Venturi meter has a throat diameter of 80 mm and a coefficient of discharge of 0.97. The relative density of mercury is 13.6.




Question 8

A Venturi meter measures the flow speed of water through a pipe. The speed to be measured is v1, the speed in the pipe where the cross-sectional area is A1 = 4.10 × 10–4 m2. Pressure meters measure the pressure in this pipe (P1), and the pressure (P2) at a narrow section of pipe with cross-sectional area A2 = 3.24 × 10–4 m2. Assume water behaves as an ideal, non-viscous fluid. (Density of water is 1000 kg/m3.) If the pressure difference, P1 – P2 is measured to be 81.0 Pa, calculate the speed v1 of the water in the pipe.






Roughness Friction factor Darcy Weisbach Equation Hagen Poiseuille Law Energy Equation Major head loss Minor head loss

Internal Flow

Relative pipe roughness is the ratio of the pipe surface roughness () to its diameter (D).

Relative pipe roughness = /D



Roughness



Roughness

Types of Roughness

The loss of pressure or head due to the resistance to flow in the pipe and fittings. Friction loss is influenced by pipe size and fluid velocity, and is usually expressed in feet of head.



Friction Factor

The friction coefficient (f), depends on the flow (laminar, transient or turbulent ) and the roughness of the tube or duct.

For fully developed laminar flow the roughness of the duct or pipe can be neglected.

f = 64/Re



Friction Factor

For hydraulically smooth pipes (= 0) in turbulent flow:

f = 0.3164 / Re 0.25

For rough pipe in turbulent flow, use the Moody Diagram to obtain f.



Friction Factor



Friction Factor

Proposed by Julius Weisbach, a German professor who in 1850 published the first modern textbook on hydrodynamics.

An important and widely used equation in hydraulics.

It enables calculation of the head loss due to friction within a given run of pipe.

Valid for duct flows of any cross section and for laminar and turbulent flow.



CLD 10603 26


Darcy Weisbach Equation

Where:

hl = head loss

f = friction factor

L = pipe length

D = pipe diameter

V = average flow velocity

g = acceleration of gravity



Darcy Weisbach Equation

g

V

D

Lfhl 2

2

2

2

2gA

Q

D

Lf

Question 4

Oil of absolute viscosity 0.101 N.s/m2 and S.G 0.850 flows through 3000 m of 300 mm cast iron pipe at the rate of 0.0444 m3/s.

1) What is the head loss in the pipe?



hf = Friction @ Major head loss

hm = Minor head loss

mf hhzg

u

g

pz

g

u

g

p 2

222

1

211

22



Energy Equation

- Head loss due to the friction of fluid flowing through a length of pipe.

- For Laminar Flow

- For Turbulent Flow



Major Losses

g

V

D

Lhl 2

642

g

V

D

Lfhl 2

2

Energy losses that come from fittings, entrances and exits.

E.g.:

i. Valves and pipe bends

ii. Pipe entrances (for fluid flowing from a tank to a pipe) and pipe exits (fluid flowing from a pipe to a tank).



Minor Losses

g

VKh LL 2

2

Documents

Chapter 4 fluid mechanics