1

Chapter 6

• Losses due to Fluid Friction

2

Objectives

To measure the pressure drop in the straight section of smooth, rough, and packed pipes as a function of flow rate.

To correlate this in terms of the friction factor and Reynolds number.

To determine the influence of pipe fittings on pressure drop

3

Calculation of Head (Energy) Losses:

In General: When a fluid is flowing through a pipe, the fluid experiences some

resistance due to which some of energy (head) of fluid is lost.

Energy Losses

(Head losses)

Major Losses Minor losses

loss of head due to pipe

friction and to viscous

dissipation in flowing

water

Loss due to the change of

the velocity of the flowing

fluid in the magnitude or in

direction as it moves

through fitting like Valves,

Tees, Bends and Reducers.

4

Losses due to Friction Mechanical energy equation between locations 1 and 2 in the absence of

shaft work:

For flow in a horizontal pipe and no diameter change (V1=V2), then :

21 PP

F

Thus, the shear

stress at the wall

is responsible for

the losses due to

friction

Losshg

zzg

V

g

V

g

P

g

P

F)()

22()( 21

2

2

2

121

Hagen-Poiseuille Law-E5.10

4

oD

128x QF

Because of (5.4) :

D

xF w

4

(6.13) (6.14)

m

Qu

Fwhere

=Friction head/unit mass

Or F/g= hLoss= 128μQL/(πρgD4)

g

PP

21

LhF/g

OR F in J/kg

and

5

Total head loss , hL (=F/g), is regarded as the sum of

major losses and minor losses

hL major, due to frictional effects in fully developed flow in

constant area tubes,

hL minor, resulting from entrance, fitting, area changes,

and so on.

Losses due to Friction

6

Losses due to Friction/The Friction Factor In order to determine an expression for the losses due to friction we must

resort to experimentation.

D

V LF

2

By introducing the friction factor, f:

D.2

V f

2LF

where

)2/)(/( 2VDL

Ff

(6.15)

where L=length of the pipe,

D=diameter of the pipe,

V=velocity,

2/

)/(2V

LDPf

or

21 PP

F

f is called Darcy friction factor

7

Friction Factor

0)2

(2

Lwork PPV

gzP

The Darcy friction factor f is defined as

We know the wall shear stress

2/

)/(2V

LDPf

L

PDτw

4

2/2Vf w

(E6-21)

The friction factor is the ratio between wall shear stress

and flow inertial force.

orLmajorLL PPP min,,

now later

and

8

Major Loss and Friction Factor

0)2

(2

Lwork PPV

gzP

With the introduction of friction factor, we can calculate major loss by

(E6-22) 2

2

,

V

D

LfP MajorL

Friction factor Pipe geometry factor Dynamic pressure

Therefore, our job now is find the friction factor f for various flows.

orLmajorLL PPP min,,

9

Friction Factor –

Case1: Laminar Pipe Flow For a pipe with a length of L, the pressure gradient is constant, the pressure drop

based on Hagen-Poiseuille Law ,

Dividing both sides by the dynamic pressure V2/2 and L/D

We have

2

42

4

4

/32

/)4/(128

/)(128

,/128

DVLP

DLDVP

DLVAP

VAQDLQP

Re

64f

Re

6464

2/

/32

2/

)/(2

2

2

VDL

D

V

DVL

V

LDPf

10

11

Example 1

losses in

12

According to Darcy

13

Example 2

14

15

Case 2: Turbulent Flow When fluids flow at higher flow rates, the streamlines are not steady, not

straight and the flow is not laminar.

Generally, the flow field will vary in both space and time with fluctuations

that comprise "turbulence”

When the flow is turbulent the velocity and pressure fluctuate very

rapidly. The velocity components at a point in a turbulent flow field

fluctuate about a mean value.

n

R

r

V

u/1

max

1

Time-averaged velocity profile

can be expressed in terms of

the power law equation, n =7

is a good approximation.

16

Friction Factor-Turbulent Pipe Flow •For a laminar flow, the friction factor can be analytically derived.

•It is impossible to do so for a turbulent flow so that we can only

obtain the friction factor from empirical results.

•In addition, most pipes, except glass tubing, have rough surfaces.

•The pipe surface roughness is quantified by a dimensionless

number, relative pipe roughness (ε / D ), where ε is pipe

roughness and D is pipe diameter.

•For laminar pipe flow, the flow is dominated by viscous effects

hence surface roughness is not a consideration.

•However, for turbulent flow, the surface roughness may emerge

beyond the laminar sublayer and affect the flow to a certain

degree.

•Therefore, the friction factor f can be generally written as a

function of Reynolds number and pipe relative roughness

There are several theoretical models available for the

prediction of shear stresses in turbulent flow.

17

Surface Roughness

Additional dimensionless group /D

need to be characterized

Thus more than one curve on friction factor-

Reynolds number plot

Fanning diagram or Moody diagram

Depending on the laminar region.

If, at the lowest Reynolds numbers, the laminar portion

corresponds to f =16/Re Fanning Chart

(or f = 64/Re Moody chart)

18

Pipe Surface Roughness

19

Friction Factor of Turbulent Flow

If the surface protrusions are

within the viscous layer, the pipe

is hydraulically smooth;

If the surface protrusions extend

into the buffer layer, f is a

function of both Re and /D;

For large protrusions into the

turbulent core, f is only a

function of /D.

)(Re, Df

)( Df

1/ 4

0.316

Ref

20

Friction Factor for Smooth, Transition,

and Rough Turbulent flow

1

f 4.0 * log Re* f 0.4

Smooth pipe, Re>3000

1

f 4.0 * log

D

2.28

Rough pipe, [ (D/ε)/(Re√ƒ) <0.01]

1

f 4.0 * log

D

2.28 4.0 * log 4.67

D /

Re f1

Transition function

for both smooth and

rough pipe

f P

L

D

2U 2

f 0.079Re0.25Or

21

Fanning Diagram

f =16/Re

1

f 4.0 * log

D

2.28

1

f 4.0 * log

D

2.28 4.0 * log 4.67

D /

Re f1

22

Friction Factor – The Moody Chart

23

Friction Factor – The Moody Chart

24

Example: Comparison of Laminar or

Turbulent pressure Drop

• Air under standard conditions flows through a 4.0-mm-

diameter drawn tubing with an average velocity of V = 50 m/s.

For such conditions the flow would normally be turbulent.

However, if precautions are taken to eliminate disturbances to

the flow (the entrance to the tube is very smooth, the air is dust

free, the tube does not vibrate, etc.), it may be possible to

maintain laminar flow.

• (a) Determine the pressure drop in a 0.1-m section of the tube

if the flow is laminar.

• (b) Repeat the calculations if the flow is turbulent.

Straight and horizontal pipe and same diameters give same velocity:

Z1=Z2=0 V1=V2 and thus Losshgp /

25

Solution1/2

flowTurbulent700,13.../VDRe

Under standard temperature and pressure conditions

=1.23kg/m3, μ=1.7910-5Ns/m

The Reynolds number

kPaVD

fp 179.0...2

1 2

If the flow were laminar and using Darcy friction

f=64/Re=`…=0.00467

kPaVD

fp 179.0...2

14 2

f=16/Re=`…=0.001167

If the flow were laminar and using Fanning friction

26

Solution2/2

kPaVD

fp 076.1...2

1 2

If the flow were turbulent

From Moody chart f=Φ(Re, smooth pipe) =0.028

From Fanning chart f=Φ(Re, smooth pipe) =0.007

kPaVD

fp 076.1...2

14 2

27

Example

Straight and horizontal pipe and same diameters give

same velocity:

Z1=Z2=0 V1=V2 and thus

pressurepumpghp Loss

28

29

30

Example: Determine Head Loss

• Crude oil at 140°F with γ=53.7 lb/ft3 and μ= 810-5

lb·s/ft2 (about four times the viscosity of water) is

pumped across Alaska through the Alaska pipeline, a

799-mile-along, 4-ft-diameter steel pipe, at a

maximum rate of Q = 2.4 million barrel/day = 117ft3/s,

or V=Q/A=9.31 ft/s. Determine the horsepower

needed for the pumps that drive this large system.

31

Solution1/2

The energy equation between points (1) and (2)

Assume that z1=z2, p1=p2=V1=V2=0 (large, open tank)

ftg

V

Dfhh PL 17700...

2

2

L2

2

22P1

2

11 hzg2

Vphz

g2

Vp

hP is the head provided to the

oil by the pump.

Minor losses are negligible because of the large length-

to-diameter ratio of the relatively straight, uninterrupted

pipe.

f=0.0124 from chart ε/D=(0.00015ft)/(4ft), Re=…..

32

Solution2/2

The actual power supplied to the fluid

power=Q∆P =Qρgh

unit of power (SI): Watt =N.m/s=J/s

hpslbft

hpgQhP 202000

/550

1...Power