Hydraulics of pipelines

K 141 HYAE Hydraulics of pipelines 1

Application of Bernoulli equation BEcontinuity equation CE

Zgv

gph

gv

gph +

α+

ρ+=

α+

ρ+

22

222

2

211

1

Z – loss head (losses):

• friction losses Zt (in distance L)

• local losses Zm

mt ZZZ +=

BE 1 - 2

CE 1 - 2

221121 SvSvQQ ⋅=⋅⇒=

K 141 HYAE Hydraulics of pipelines 2

FRICTION LOSSES

uniform flow ∅D = const, v = constiE = hydraulic slope (friction slope) = slope of EL

Zt = Zt (v, D, L, character of walls of pipeline)

K 141 HYAE Hydraulics of pipelines 3

Equilibrium of forces in elementary volume EV

( ) dsOSpSdpp 0 ⋅⋅τ=⋅−⋅+

SO

dsdp

0 ⋅τ=

for circle pipeline ∅D:D4

4DD

SO

4DS,DO 2

2

=⋅π⋅π

=⇒⋅π

=⋅π=

D4

dsdp 0 ⋅τ

=

K 141 HYAE Hydraulics of pipelines 4

Darcy-Weisbach equation

2v

D1

D4v

8D4

dsdpi

220 ⋅ρ⋅⋅λ=⋅⋅ρ⋅

λ=

⋅τ==

uniform flow: gZpLZ

LgpELPL t

t ⋅ρ⋅=Δ⇒=⋅⋅ρ

Δ⇒

g2v

DLZ

2

t ⋅⋅λ=

(for ds = L, dp = Δp)

g2v

D1

LZi

2t

E ⋅⋅λ==

Expression of τ0 under turbulent flow

2

00f

v21volumeunitinenergykinetic

sshearstresf⋅ρ⋅

τ=

τ=

202

0f v

8v8f4 ⋅ρ⋅

λ=τ⇒

⋅ρτ⋅

=⋅=λ

coefficient of friction loss λ= f (Re, Δ/D)ν

vDRe =

friction coefficient:

K 141 HYAE Hydraulics of pipelines 5

iE = a vbfriction slope

b = 1LF

1 < b < 2TF

transit z.

b = 2quadratic zone

iE

vk ↔ Rek = 2320

vmvk v

LP - laminar flow

TP - turbulent flow

K 141 HYAE Hydraulics of pipelines 6

Effect of dimensions of projections on flow in close vicinity topipeline wall

turbulent flow

δ δ

δδ

laminar flow

K 141 HYAE Hydraulics of pipelines 7

FRICTION FACTOR λMoody diagram 1 Laminar flow

Poiseuille

2 Region of transition3 Smooth pipes

4 Turbulent flow Colebrook - White- transitional zone

20,25

3,71 Dlogλ =

⎛ ⎞⎜ ⎟Δ⎝ ⎠

0,25680,11Re D

Δ⎛ ⎞λ = +⎜ ⎟⎝ ⎠

1 2,512log3,71DRe

⎛ ⎞Δ= − +⎜ ⎟λ λ⎝ ⎠

64Re

λ =

0,250,3164Re

λ =

3 ← 4 → 5Altshul

NikuradseΔD

λ200Re >

5 Rough turbulent quadraticzone

Blasius

K 141 HYAE Hydraulics of pipelines 8

Notes:

- tap water T = 12°C → ν = 1,24.10-6 m2s-1

- hydraulic roughness Δ = 0,01 mm ÷1 mm

- recommended velocity for water pipeline 0,5 ÷ 1,5 ms-1

tables

K 141 HYAE Hydraulics of pipelines 9

LOCAL LOSSES IN PIPELINES

loss coefficient ζi

INLET (common, curved, streamlined, suction basket,backflow valve, grid, …)

ζv = 0,5 0,05 ÷ 0,2 2 ÷ 20

TAB

Total losses = friction losses + local losses

g2vZ

2

imi ⋅ξ=

∑ ∑ ⎟⎟⎠

⎞⎜⎜⎝

⎛ξ+=+=

2gv

DL

λZ2j

ij

jjmt ZZ

K 141 HYAE Hydraulics of pipelines 10

CHANGE OF DIRECTION (bend, knee pipe, segment knee pipe)

BRANCHING,CONNECTIONCHANGE OF PROFILE (sudden x continuous,

enlargement x thinning)

MEASURING DEVICE (pipe flow meters)

OUTLET

6 -10°

ζu = 0large reservoirζu= 1

⎟⎠⎞

⎜⎝⎛=

DΔα,,

Drfζ s

s

⎟⎟⎠

⎞⎜⎜⎝

⎛= δ,

DDfζ

2

1z

⎟⎟⎠

⎞⎜⎜⎝

⎛= α,

QQ,

QQ,

DD,

DDfζ

1

3

1

2

3

2

3

1o

K 141 HYAE Hydraulics of pipelines 11

slide valve valve

CLOSURES (valves, backflow valves, taps, slide valves)

( )openningtype,fζu =

ball tap

K 141 HYAE Hydraulics of pipelines 12

HYDRAULIC CALCULATIONS OF PIPELINES

3 kinds of equations:

← geometry and roughness of pipe,discharge

- Bernoulli equation ← elevations and pressure relations,boundary conditions

calculation: Q, v, D, L, H, p, Z

- continuity equation

- equation of losses

m1

Slide 13

m1 maresovai; 24.3.2006

K 141 HYAE Hydraulics of pipelines 13

2 2A A B Bp v p vH Zg 2g g 2g

α α+ + = + + ∑

ρ ρ

∑ ∑ ∑ ∑ ∑ ⎟⎠⎞

⎜⎝⎛ +=+=

2gvζ

DLλZZZ

2

imt

BE

CE jj2211 SvSvSvQ ⋅=⋅=⋅=

equation of losses

K 141 HYAE Hydraulics of pipelines 14

⇒ EL ≡ PL

Note:

so called „hydraulically long pipeline“

02gαvZ

2gαv 2

t

2≈⇒<<

ttm ZZZ =⇒<< Z

K 141 HYAE Hydraulics of pipelines 15

• open and large reservoirs

• outlet from pipe to open space

BE: H = ∑Z... total head loss

BR:

exit loss

no exit loss

pB = 0

pA = pB = 0 (overpressures)vA = vB = 0

∑+= Z2gαvH

2

K 141 HYAE Hydraulics of pipelines 16

UNDERPRESSURES IN PIPELINES

always to check!

|pva|max ≈ (6 ÷ 8).104 Pa

≈ (6 ÷ 8) m w. col.va

max

pgρ

underpressures → cavitation

• places with high elevation siphon tube(top above upper reservoir level)

BE 0-1:

s

ΣZEL

PL ∑

∑

−−−=⇒

⇒===

+++=++

Z2gαvs

ρgp

ρgp

pp0,v0,H

Z2gαv

ρgps

2gαv

ρgpH

21a1

a00

211

200

vamax

max

pfor sρg

⇒

K 141 HYAE Hydraulics of pipelines 17

Description of system:lower reservoir → suction pipe SP → pump Č → delivery pipe VP →upper reservoir (event. outlet to open space)

Solved problems :- determination of Q, D- checking of underpressure in SP- determination of input power of Č

Types of losses:SP:Zs → friction, suction basket, backflow valve, knee pipe, bendsVP:Zv → friction, closures, knee pipe,

bends

PIPELINE – PUMP SYSTEM

K 141 HYAE Hydraulics of pipelines 18

g2v

gpp

gp 2

sČava α++=ρ−

=ρ

= Sgsva ZHH

p A

BE: lower reservoir - Č

s

2sČ

gSa Z

g2v

gp

Hg

p +α+ρ

+=ρ

for open large reservoir( ) 0v,0pp ApA ≈== aAp

practically Hva < (6 ÷ 8) m w. col.

determination of Q, D - basic equations (BE, CE)

checking of underpressure in SP

K 141 HYAE Hydraulics of pipelines 19

geodetic gradient:total head:

g2v

g2v

gp

gp 2

A2

BAB −+ρ

−ρ

+++= VSgd ZZHH

Z+=++= gVSgd HZZHH

gvgsg HHH +=

For open large reservoirs0vv,pp BA ≈=== aBAp

Input power [W]

efficacy: η … practically up to 0,9

dρgQHη1P =

determination of necessary input power

K 141 HYAE Hydraulics of pipelines 20

VA

Branched net

Closed loop

PiPELINE SYSTEMS

- branched- close loop- combined

V - distribution reservoir

K 141 HYAE Hydraulics of pipelines 21

WATER HAMMER

quick manipulation with closure, pump starting or failure⇓

rapid change of pressure (water hammer),in particular at long pipelines

protection against water hammer:• sufficiently slow manipulation with closures• air chambers, surge chambers (equilibrating towers)• ...