Pipe Flows – Head Losses

Trans-Alaska Pipeline: 800 miles (1290 km), 48 in. (1.2 m) diameter

Pipe wall thickness ~0.5 in. Takes oil 11.9 days to travel the length at an average speed of 3.7 mph (6.0 kph)

Pipe Flows – Head Losses

! "#$+ 𝛼 '()

*$+ 𝑧,

-./= ! "

#$+ 𝛼 '()

*$+ 𝑧,

12− 𝐻5 + 𝐻6

where

𝛼 = 72 𝑅𝑒; < 2300(laminar)1 𝑅𝑒; > 2300(turbulent)

𝐻6 =�̇�-/PQR,-2T'

�̇�𝑔

Steady, Laminar, Viscous, Newtonian Fluid Flow Through a Circular Pipe Assumptions: 1. steady flow 2. fully developed flow in the z direction 3. no body forces 4. axi-symmetric flow with no swirl component Continuity Equation and Navier-Stokes Equation in the z direction:

Boundary Conditions: 1. No slip at the pipe circumference 2. Flow velocity remains finite Velocity profile:

𝑢X =𝑅*

4𝜇[−

𝑑𝑝𝑑𝑧^_1 −

𝑟*

𝑅*a

𝑅𝑒; =𝑢b𝐷𝜈 < 2300

Average velocity:

𝑢b =1𝜋𝑅*

f 𝑢X(𝑟)(2𝜋𝑟𝑑𝑟)Rgh

Rgi=12𝑢jkl =

𝐷*

32𝜇[−

𝑑𝑝𝑑𝑧^

Wall shear stress:

𝜏n = 𝜇𝑑𝑢𝑑𝑟oRgh

=𝑅2[𝑑𝑝𝑑𝑧^ = −

4𝜇𝑢b𝑅

𝑓; ≡ r4𝜏ns)tu(

)r =

32𝜇𝜌𝑢b𝑅 =

64𝑅𝑒;

( )

2 2

2 2 2

1 1 0

1 1

r z

z z z z z z zr z z

ru u ur r r z

uu u u u u u upu u r gt r r z z r r r r z

q

q

q

r µ rq q

¶ ¶ ¶+ + =

¶ ¶ ¶é ù¶ ¶ ¶ ¶ ¶ ¶ ¶¶ ¶æ ö æ ö+ + + = - + + + +ê úç ÷ç ÷¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶ ¶è øè ø ë û

R r z

Pipe Flows – Head Losses

Revisiting the average velocity equation:

𝑢b =𝐷*

32𝜇o∆𝑝𝐿o ⟹

∆𝑝s)#.(

)=64𝜇𝜌𝐷𝑢b

𝐿𝐷 = 64

𝜇𝜌𝑢b𝐷

𝐿𝐷 =

64𝑅𝑒;

𝐿𝐷

𝑘 ≡∆𝑝s)#.(

)

𝑘|}~�� = 𝑓; [𝐿𝐷^

_𝑝𝜌𝑔 + 𝛼

𝑉b*

2𝑔 + 𝑧a-./

= _𝑝𝜌𝑔 + 𝛼

𝑉b*

2𝑔 + 𝑧a12− 𝐻5 + 𝐻6

Pipe Flows – Head Losses

ME 309 Comprehensive Exam Formula Sheet Last updated: 2019 Sep 17

Page 6 of 9

Colebrook Formula: Haaland Formula

1f≈ −2.0log10

ε D3.7 + 2.51

ReD f

⎛

⎝⎜⎜

⎞

⎠⎟⎟

1f≈ −1.8log10

6.9ReD

+ ε D3.7

⎛⎝⎜

⎞⎠⎟

1.11⎡

⎣⎢⎢

⎤

⎦⎥⎥

ME

309

Com

preh

ensi

ve E

xam

For

mul

a Sh

eet

Last

upd

ated

: 20

19 S

ep 1

7

Pa

ge 6

of 9

Col

ebro

ok F

orm

ula:

Haa

land

For

mul

a

1 f≈−2.0log

10εD 3.7+

2.51

ReD

f

⎛ ⎝⎜ ⎜

⎞ ⎠⎟ ⎟

1 f≈−1.8log

106.9 Re

D

+εD 3.7

⎛ ⎝⎜⎞ ⎠⎟1.1

1⎡ ⎣⎢ ⎢

⎤ ⎦⎥ ⎥

Pipe Flows – Head Losses

C. Wassgren Last Updated: 29 Nov 2016 Chapter 11: Pipe Flows

5. Other Losses The loss due to the viscous resistance caused by the pipe walls is referred to as a major loss. Pressure losses may occur due to viscous dissipation resulting from fluid interactions with other parts of a pipe system such as valves, bends, contractions/expansions, inlets, and connectors. These losses are known as minor losses. The names can be misleading since it’s not uncommon in pipe systems to have most of the pressure loss resulting from the minor losses (e.g., a pipe system with a large number of bends and valves, but short sections of straight pipe). What causes these minor losses? The pressure loss results primarily from viscous dissipation in regions with large velocity gradients, such as in a recirculation zone as shown in the figure below. A closely related phenomenon known as the vena contracta acts to effectively reduce the diameter at entrances and bends. The recirculation zone also results in a pressure loss. Although minor loss coefficients can be determined analytically for certain situations, most frequently the loss coefficient for a particular device is found experimentally. Essentially, one measures the pressure drop across the device, Δp, and forms the loss coefficient, k, using,

212

pkVρ

Δ= , (43)

where ρ is the fluid density and V is the average speed through the device. Many tables with experimentally determined loss coefficients have been generated. Notes: 1. When using a loss coefficient, it is important to know what velocity has been used to form the

coefficient. For example, the loss coefficient for a contraction is typically based on the speed downstream of the contraction, while the loss coefficient for an expansion is based on the speed upstream of the expansion.

2. Minor losses are sometimes given in terms of equivalent lengths of pipe. An equivalent minor loss of

10 pipe diameters worth of a particular type of pipe means that the major loss caused by a pipe of that type, 10 diameters in length will give the same pressure loss as the minor loss. Thus, a loss coefficient and equivalent pipe length, Le, can be related by,

eDLk fD

⎛ ⎞= ⎜ ⎟⎝ ⎠. (44)

3. For non-circular pipes or pipes that are not completely filled, the same method of determining the

friction factor and loss coefficients are used, except that a hydraulic diameter, Dh, is used in place of the diameter. The hydraulic diameter is defined as,

recirculation zone

recirculation zone

Reffective

1066

C. Wassgren Last Updated: 29 Nov 2016 Chapter 11: Pipe Flows

5. Other Losses The loss due to the viscous resistance caused by the pipe walls is referred to as a major loss. Pressure losses may occur due to viscous dissipation resulting from fluid interactions with other parts of a pipe system such as valves, bends, contractions/expansions, inlets, and connectors. These losses are known as minor losses. The names can be misleading since it’s not uncommon in pipe systems to have most of the pressure loss resulting from the minor losses (e.g., a pipe system with a large number of bends and valves, but short sections of straight pipe). What causes these minor losses? The pressure loss results primarily from viscous dissipation in regions with large velocity gradients, such as in a recirculation zone as shown in the figure below. A closely related phenomenon known as the vena contracta acts to effectively reduce the diameter at entrances and bends. The recirculation zone also results in a pressure loss. Although minor loss coefficients can be determined analytically for certain situations, most frequently the loss coefficient for a particular device is found experimentally. Essentially, one measures the pressure drop across the device, Δp, and forms the loss coefficient, k, using,

212

pkVρ

Δ= , (43)

where ρ is the fluid density and V is the average speed through the device. Many tables with experimentally determined loss coefficients have been generated. Notes: 1. When using a loss coefficient, it is important to know what velocity has been used to form the

coefficient. For example, the loss coefficient for a contraction is typically based on the speed downstream of the contraction, while the loss coefficient for an expansion is based on the speed upstream of the expansion.

2. Minor losses are sometimes given in terms of equivalent lengths of pipe. An equivalent minor loss of

10 pipe diameters worth of a particular type of pipe means that the major loss caused by a pipe of that type, 10 diameters in length will give the same pressure loss as the minor loss. Thus, a loss coefficient and equivalent pipe length, Le, can be related by,

eDLk fD

⎛ ⎞= ⎜ ⎟⎝ ⎠. (44)

3. For non-circular pipes or pipes that are not completely filled, the same method of determining the

friction factor and loss coefficients are used, except that a hydraulic diameter, Dh, is used in place of the diameter. The hydraulic diameter is defined as,

recirculation zone

recirculation zone

Reffective

1066

Pipe Flows – Head Losses

ME 309 Comprehensive Exam Formula Sheet Last updated: 2019 Sep 17

Page 7 of 9

Average Roughness of Commercial Pipes Material (new) ft mm Riveted 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 Galvanized iron 0.0005 0.15 Asphalted cast iron 0.0004 0.12 Commercial steel or wrought iron 0.00015 0.045 Drawn tubing 0.000005 0.0015 Plastic, glass 0.0 (smooth) 0.0 (smooth)

Table of Minor Loss Coefficients Component K Component K

a. Elbows Regular 90o, flanged 0.3 Regular 90o, threaded 1.5 Long radius 90o, flanged 0.2 Long radius 90o, threaded 0.7 Long radius 45o, flanged 0.2 Regular 45o, threaded 0.4

b. 180o return bends 180o return bends, flanged 0.2 180o return bends, threaded 1.5

c. Tees Line flow, flanged 0.2 Line flow, threaded 0.9 Branch flow, flanged 1.0 Branch flow, threaded 2.0

d. Union, threaded 0.06

h. Sudden Contraction/Expansion:

e. Valves

Globe, fully open 10 Angle, fully open 2 Gate, fully open 0.15 Gate, ¼ closed 0.26 Gate, ½ closed 2.1 Gate, ¾ closed 17 Swing check, forward flow 2 Swing check, backward flow ¥ Ball valve, fully open 0.05 Ball valve, 1/3 closed 5.5 Ball valve, 2/3 closed 210

f. Entrances Re-entrant 0.8 Sharp-edged 0.5 Slightly rounded 0.2 Well rounded 0.04 g. Exits

Re-entrant, sharp-edged, slightly rounded, well-rounded 1

Pipe Flows – Head Losses

http://mechanicstips.blogspot.com/2016/02/types-of-valves.html