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FLOW IN PIPES, PIPE NETWORKS
Continuity equation – mass balance (G54)
2211 SuSu
Bernoulli equation – mechanical-energy balance (G71 – 74)
zeghpu
ghpu
22
2
22
211
2
12
122
z
z
ppppe
21
Turbulent flow: 1
Laminar flow: (neglectable)02
2
2 u
mechanical-energy loss
Mechanical-energy loss for flow in
pipe
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 ,
duRe
d
kk av
chanelofperimeterwetted
chanelofareationalcross
O
Sde
sec4
4
Friction factor
Reynolds number relative roughness of pipe wall
Values of constant A for various shapes of cross-section
Shape of cross-section Charact. length
Hydraulic diameter
Equation for computation of parameter A A
Circle
d – 64
Annulus
= 10-2
= 10-1
= 0,5
d2 – d1
ln
11
164
22
2
A
80,11
89,37
95,25
Slit
2h – 96
Rectangle
h/b = 10-2
h/b = 10-1
h/b = 1 hb
bh
2
...5,3,155
2
2tgh
11921
1
1
96
n h
bn
nb
h
b
hA
94,71
84,68
56,91
Ellipse
b/a = 0,1
b/a = 0,25
b/a = 0,5
ba
ab
4
2
2
1
1128
a
b
a
b
A
106,84
87,04
71,11
Isosceles
triangle
= 60°
a = b
= 90° 2sin1
sin
a
12tg
1
2
54
2tg1
2tg2
22
tg148
22
2
2
Bkde
B
B
A 53,33
52,71
2
1
d
d
Dependence of friction factor on Reynolds number and relative
roughness of pipe k*
8,0log0,21
Re
klog214,11
29,0
727,0log2
Rek
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
EXAMPLE: Friction 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) water
b) 98 % aqueous solution of glycerol ( = 1255 kg•m-3, = 629 mPa•s)
EXAMPLE: Friction loss for flow in pipe with
noncircular 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 e
z dle
loss coefficient
1
22 15,0
S
S
Contraction
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
A – Check valve, screwed
B – Back straight-way valve
C – Check valve, casted
D – Back angle valve
E – Check angle valve
F – Check oblique valve
Š – Gate valves
EXAMPLE: 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 design
Calculation of pipe diameter at given flow rate without
demand of loss (the most frequently case G107)
u
VS
Sd
4
Geankopolis, C. J.: Transport Processes and Separation Process Principles. 4th edition. New Jersey: Publishing as Prentice Hall PTR,
2003.1026 p. ISBN 0-13-101367-X.
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
2
2 22
lu
deu
d
le zz 2
2 2
2
,, kRef
duRe
l
dedRe z
22
d
ReuReRe
1
kRef ,1
kRef ,1
3842300
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 l
liguid density and viscosity
V2
4
d
Vu
53
3
5 128
l
eVRe z
střk
V
k
Re 4
1551 Rek,λRefλ/
lV
de
lu
de zz
2
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: 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: 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.
Design 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 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
zepp
uup
u dd
d2
1
2
1 22
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 0
20d
2
1d
dp
p
p
p
l
ld
ppwRT
M
p
p
0ln 2
2
2
12
2
2
1
d
lpp
wRT
M
p
p
u w 2d du w
Maximum flow for compressible flow of gas 0d/d 2 pw
0d
d1
2
2
2
2
1322
2
p
wpp
wRT
Mp
wRT
M
p
22
krkr wM
RTp
kr
krkrkr
kr
krkr
pvp
wu
01ln
2
1
2
1
d
l
p
p
p
p
krkr
EXAMPLE: 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).
