flow in pipes, pipe networks continuity equation – mass balance (g54) bernoulli equation –...

FLOW IN PIPES, PIPE NETWORKSFLOW IN PIPES, PIPE NETWORKS

Continuity equation – mass balance (G54)

2211 SuSu

Bernoulli equation – mechanical-energy balance (G71 – 74)

zeghpu

ghpu

22

222

211

212

1 22

z

z

ppppe

21

Turbulent flow: 1

Laminar flow: (neglectable) 02

22 u

mechanical-energy loss

Mechanical-energy loss for flow in Mechanical-energy loss for flow in pipepipe

Mechanical-energy loss due to skin friction for incompressible fluid (liquids) (G90 – 96)

2

2u

d

lez

Laminar flow: (pipe with circular cross-section A = 64) Re

A

Turbulent flow: (noncircular cross-section G105) kRef ,

du

Re d

kk av

chanelofperimeterwetted

chanelofareationalcross

O

Sde

sec4

4

Friction factor

Reynolds number relative roughness of pipe wall

Dependence of friction factor on Reynolds number and relative roughness of pipe k*

8,0log0,21

Re

klog214,11

29,0

727,0log2

Re

k

Re

64 Smooth pipes

Rough pipes

Values of absolute roughness kav of pipes from different materials

Type resp. material of pipe kav

[mm]

glass, brass, copper, drawn tubing seamless, steel drawn tubes, new steel welded tubes, new steel tubes, slightly corroded steel tubes, corroded steel tubes, galvanized cast iron, new cast iron, corroded cast iron asphalt dipped PVC concrete, smooth concrete, rough asbestos cement tubes

0,0015 0,0025 0,03 0,06 0,04 0,1 0,15 0,4 0,5 1,5 0,1 0,15 0,2 0,6 1 1,5

0,1 0,15 0,002

0,3 0,8 1 3

0,03 0,1

EXAMPLEEXAMPLE:: Friction loss for flow in pipeFriction loss for flow in pipe

56 l•s-1 of liquid with temperature 25°C flow in horizontal slightly corroded steel tubes with length 600 m with inside diameter d = 150 mm. Determine value of pressure drop and loss due to skin friction in pipe.Liquid:

a) waterb) 98 % aqueous solution of glycerol ( = 1255 kg•m-3, = 629 mPa•s)

EXAMPLEEXAMPLE:: Friction loss for flow in pipe with Friction loss for flow in pipe with noncircular cross-sectionnoncircular cross-section

Determine value of pressure drop in heat exchanger pipe in pipe with annulus cros-section. 98 % aqueous solution of glycerol with temperature 25°C ( = 1255 kg•m-3, = 629 mPa•s) has mass flow rate 40 kg•min-1. Outside diameter of inside tube is d1 = 32 mm and inside diameter of outside tube is d2 = 51 mm. Length of exchanger is L = 25 m.

2

2uez

Friction losses in expansion, contraction, pipe fittings and valves (G98-102)

2

2u

d

le ez dle

loss coefficient

1

22 15,0

S

SContraction

Expansion2

2

11 1

S

S

Gradual expansion (diffuser)

400 tr 1

Pipe entrance

Elbowto

2/1

21,0

d

ro

d

r

d

lt

2

Tee

2

2up

Valves

http://www.jmahod.cz

AA – Check valve, screwed

BB – Back straight-way valve

CC – Check valve, casted

DD – Back angle valve

EE – Check angle valve

FF – Check oblique valve

ŠŠ – Gate valves

EXAMPLE:EXAMPLE: Determination of pump head pressure Determination of pump head pressure

Determinate head pressure of pump which give flow rate 240 l·min-1 of water with temperature 15 °C. Water is pumping up to storage tank with pressure over liquid surface 0.2 MPa. Pipes are made from slightly corroded steel tubes with outside diameter 57 mm and thickness of wall 3 mm.

Basic cases for pipe designBasic cases for pipe design

Calculation of pipe diameter at given flow rate without demand of loss (the most frequently case G107)

u

VS

S

d4

Calculation of flow rate at given loss and pipe diameter

Given: mechanical-energy loss ez

dimensions of pipe (l, d, kav)

liquid density and viscosity

l

dedu

lu

deRe zz

2

32

2

222

22 22

lu

deu

d

le zz 2

2 2

2

,, kRefdu

Re

l

dedRe z

22

d

ReuReRe

1

kRef ,1

kRef ,1

384

2300

642300

krkrkr ReRe

Re

k 51,2

71,3log2

1

Calculation of pipe diameter at given loss and flow rate

Given: flow rate mechanical-energy loss ez

pipe length lliguid density and viscosity

V2

4

d

Vu

53

35 128

l

eVRe z

střk

V

k

Re 4 1551 Rek,λRefλ/

lV

de

lu

de zz2

52

2 8

2

d

VduRe

4

d

kkkRef stř ,,

u

RedReRe

5

5 1

1551 Rek,λRefλ/

2930,0

5

937,0

1

5 5,4369,0log2

ReRek

Re

11242300

64230055 krRe

EXAMPLE:EXAMPLE: Calculation of flow rate Calculation of flow rate

84 % aqueous solution of glycerol ( = 1220 kg•m-3, = 99.6 mPa•s) is in tank with height of liquid surface over bases 11 m. Glycerol gravity outflow to second tank with height of liquid surface over same bases 1 m. Pipe is made from steel with outside diameter 28 mm and thickness of wall 1.5 mm and its length is 112 m. Determine volumetric flow rate of glycerol. Losses of fittings and valves are neglectable.

EXAMPLE:EXAMPLE: Calculation of pipe diameter Calculation of pipe diameter

Solution of ETHANOL ( = 970 kg•m-3, = 2,18 mPa•s) gravity outflow from open tank with flow rate 20 m3•h-1 via pipe with length 300 m to second open tank. Liquid surface in upper tank is 2.4 m over liquid surface of second tank. Which pipe diameter is necessary for required flow rate. Pipe is made from steel with average roughness 0.2 mm. Losses of fittings and valves are express as 10 % from pipe length.

DesignDesign of pipe networks of pipe networks

Procedure of solving:

1) Bernoulli equation for all pipes

2) Continuity equation for all nodes

3) Solve system of equations

Compressible flow of gasesCompressible flow of gases

Isothermal compressible flow (G107-110)

pvp

c

2

Velocity of compression wave (velocity of sound in fluid)

Bernoulli equation

0d02

ddd

2

1

2

1 22

22 uu

d

lpuuu

0dd

d2

1

2

1 22

zepp

uup

u

02

d 2

dlu

d

pudu

Mass velocity (density of mass flow) .const uw

02

d 2

dlu

d

pudu

0d2

dd

2

2

3

2

ld

wpw

State equation for ideal gas pRT

MconstT

M

RTpdd.,

2

1

2

1 02

0d2

1d

dp

p

p

p

l

ld

ppwRT

M

p

p

0ln 22

212

2

2

1

d

lpp

wRT

M

p

p

Maximum flow for compressible flow of gas 0d/d 2 pw

0d

d1

2

22

21322

2

p

wpp

wRT

Mp

wRT

M

p

22krkr w

M

RTp

kr

krkrkr

kr

krkr

pvp

wu

01ln2

1

2

1

d

l

p

p

p

p

krkr

EXAMPLE:EXAMPLE: Pressure drop for flow of Methane Pressure drop for flow of Methane

Methane flow in long-distance (3 km) pipe from storage tank withhead pressure 0.6 MPa. Pipe is made from slightly corroded steel tubes with outside diameter 630 mm and thickness of wall 5 mm. Determine pressure drop for Methane mass flow rate 40 kg·s-1. Suppose isothermal flow with temperature 20 °C ( dynamic viscosity of Methane is 1,1·10-5 Pa·s).