Chapter 8

Introduction to Fluid MechanicsProf. Miguel Canals

Chapter 8

Internal Incompressible Viscous Flow

Viscous Flow in Pipes

Sections 8.3-8.8

Viscous pipe flow: applications

• Fluid flow in pipes is important for:

• Animals and Plants circulation systems.

• In our homes.

• City water.

• Irrigation system.

• Sewer water system

Viscous pipe flow: simplifying assumptions

1. The pipe is assumed to be completely full of the flowing fluid.• If not, flow is channel flow and

not pipe flow due to presence of free surface, which allows surface waves and complicates the dynamics

2. Flows are usually assumed to be fully developed.

3. A flow may be laminar, transitional or turbulent:• Laminar: Re < 2100• Transitional: 2100 < Re < 4000• Turbulent: Re > 4000

General characteristics of viscous pipe flow

From http://www-mdp.eng.cam.ac.uk

Fully developed pipe flow: laminar flow

• The Hagen-Poiseuille equation for laminar flow can be obtained from direct analysis of the Navier Stokes equation.• It indicates that pressure drop is independent of surface roughness.

•This same result can be obtained from dimensional analysis.•Pressure drop is usually written as:

•Where f is a dimensionless quantity called the Darcy friction factor. For fully developed laminar flow:

Fully developed pipe flow: laminar flow

D

L

D

L

VDV

p

Re

6464

21 2

2

2

1V

D

Lfp

Re

64laminarf

Fully developed pipe flow: turbulent flow

www.cmmt.csiro.au

• Due to difficulty in understanding turbulence, most information about turbulent pipe flow is based on experimentshttp://www.youtube.com/watch?v=NplrDarMDF8

•Roughness effects now play an important role

•Flow separation occurs, leading to energy dissipation and the resulting head loss

•Importance of roughness measured in terms of an equivalent roughness ε

• 7 variables, 3 fundamental dimensions

• 7-3 = 4 dimensionless groups:

• The parameter ε/D is called the relative roughness

Dimensional analysis of turbulent pipe flow

DD

Lf

U

P,Re,

2

0),,,,,,( VDLPf

Values if equivalent roughness (from textbook)

Pipe Equivalent roughness, ε

Feet Millimetres

Rivited steel 0.003 – 0.03 0.9 - 9.0

Concrete 0.001 – 0.01 0.3 – 3.0

Wood stave 0.0006 –

0.003

0.18 – 0.9

Cast iron 0.00085 0.26

Galvanised iron 0.0005 0.15

Commercial steel

or wrought iron

0.00015 0.045

Drawn tubing 0.000005 0.0015

Plastic, glass 0.0 (smooth) 0.0 (smooth)

• The Darcy-Weisbach equation is a phenomenological equation which describes the head loss due to friction along a pipe, given the cross-sectional average velocity of the flow in the pipe:

• It can be obtained from dimensional analysis (previous slide).• Describes head loss due to friction in both laminar and turbulent flow•This can be used in the energy (per unit weight) equation:

The Darcy–Weisbach equation

g

V

D

LfH l

2

2

loutoutout

ininin Hz

g

Vpz

g

Vp

22

22

Head losses

minormajor lll HHH

• Head losses are divided into major and

minor losses

Major losses• Major losses are energy losses due to friction in the boundary layer

•The friction factor depends on Reynolds number and equivalent roughness

•The Darcy friction factor, f, is usually selected from the Moody diagram

•Moody diagram: a family of curves that relate the friction factor to the Reynolds number and the relative roughness of a pipe.

g

V

D

LfH

majorl2

2

Re,D

f

Moody diagram

Moody diagram

Minor Losses• Are mainly due to geometric properties

of pipes

• Flow separation and associated viscous effects will tend to decrease the flow energy and hence the losses

• The phenomenon is fairly complicated.

• All the physics we do not understand are bundled into the loss coefficient KL

Valves Bends T joints Expansions Contractions

g

VKh Ll

2

2

minorRe),geometry(fKL

Example

ExampleWater at 40oF flows through the coils of the heat exchanger as shown at a rate of 0.9 gal/min. Determine the pressure drop between the inlet and outlet of the horizontal device.

Multiple Pipe Systems• Governing mechanisms for the flow in multiple pipe systems

are the same as for the single pipe systems

Multiple Pipe Systems

• Pipe network problems can be solved using node and loop concepts.

• Complex pipe networks usually require numerical solutions

Pumps, fans and blowers• Power: Is the rate at which work is performed or at which

energy is converted

• Dimensions of Power

• Units of power– SI: N*m/s = J/s = Watt

– BG: 550 lbf*ft/s = 1 Horsepower, from wikipedia: “Watt determined that a

horse could turn a mill wheel 144 times in an hour (or 2.4 times a minute). The wheel was 12 feet in radius, therefore the horse travelled 2.4 × 2π × 12 feet in one minute. Watt judged that the horse could pull with a force of 180 pounds”

Wt

WP

31 MLTFLTW

Pump powerpump

Intake

(V1,P1)

Outlet

(V2,P2)

pump a sit' 0 if turbine,a sit' 0 if 12 ΔpΔpppΔp

pumppump

22

QΔW then and If

22

p z zVV

gzVp

gzVp

mW

inoutinout

ininin

outoutout

pump

www.fluent.com

http://www.youtube.com/watch?v=A0GWe6Bgps4

In terms of heads:

• This allows us to express energy balance in a pipe system with pumps:

pumppumppumppump HmgQγQΔW HP

min22

22

LLoutoutout

pumpininin HHz

g

VpHz

g

Vpmajor

The energy equation: Effects of shaft work

lossturbineoutoutout

pumpininin HHz

g

VpHz

g

Vp

22

22

Pump or turbine

in

out

Pump adds energy

Turbine subtracts energy

Friction subtracts energy

Pump efficiency• Compares the amount of work or power we get out of the

pump to the amount of power we are putting into the pump

• pump efficiencies range from 15% to over 90%

• Some factors affecting efficiency:– Turbulence

– Vibrations

– Etc…

http://www.youtube.com/watch?v=jUuoeZS34vs&feature=related

http://www.youtube.com/watch?v=VSGb4c0YMpA

Example

1 m3 = 267.2 gallons

• Same principle as wind turbines

• Available Power:

• Larger density of seawater allows power generation at low speeds (>1 m/s)

• Benefit: High predictability

Tidal turbines

www.windenergy.co.uk/framestidal.htm

2

3AuP

Modeled tidal currents (J. Gonzalez)

Wave energy

About 80% of the energy is contained within a quarter of a

wavelength from the surface

~25-40 kW per m of exposed coastline in Atlantic (average)

WEC: Oscillating water column

WEC: Floating devices: Pelamis

http://www.windprospect.com.au/FCKfiles/Image/Cutaway2.jpg