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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).