Page 1
Daniel Bernoulli (Groningen, 8 February 1700
– Basel, 8 March 1782) was a Dutch-
Swiss mathematician and was one of the many
prominent mathematicians in the Bernoulli
family. He is particularly remembered for his
applications of mathematics to mechanics,
especially fluid mechanics, and for his
pioneering work in probability and statistics.
Bernoulli's work is still studied at length by
many schools of science throughout the world.
21
2
222
2
111
22 H
g
uzy
g
uzy
0
Page 2
Julius Ludwig Weisbach (born 10 August 1806 in
Mittelschmiedeberg (now Mildenau), Erzgebirge,
died 24 February 1871, Freiberg) was a German
mathematician and engineer.
Henry Philibert Gaspard Darcy (June 10,
1803 – January 3, 1858) was
a French engineer who made several
important contributions to hydraulics.
2g
u
4ΔΗ
2
f R
L
0
Page 3
Antoine de Chézy (September 1, 1718
– October 5, 1798) was a French
hydraulics engineer. He is known for
the Chézy formula, which concerned
the velocity of pipe flow.[1] He died
in 1798 after being director of the École
nationale des ponts et chaussées for less
than a year.[2] His son was the
orientalist Antoine-Léonard de Chézy.
𝑉 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑓
http://chezy.sdsu.edu/
0
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CU06997 Fluid Dynamics
Sewer calculation 12.1 Introduction (page 401)
12.2 Design of a simple pipe system (page 401-404)
12.3 Series, parallel and branched pipe systems (page 404-408)
Reader : Sewer systems module for HPE (link on VLD)
5.4 Hydraulics
1
Page 5
Energy loss [m]
2g
u
2g
u
4ΔΗ
22
f R
L
Total Head
Pressure Head
1
• Turbulent flow
• Friction loss (wrijvingsverlies)
Going to look at other formulas
for calculating friction loss
𝑉 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑓 𝑆𝑓 =ΔH
𝐿
Page 6
Combined sewer / gemengd rioolstelsel
50 m
Ø300 PVC
Ø500 concrete
Ø250 PVC
pump
P4 P3 P2
GL (ground level) +6.00 m
IL +4,00 m IL +3,90 m
IL +3,73 m
Rain water
Waste water
Rain
Waste
+5,5 m
IL (Invert level) +3,53 m
P1
1
Page 7
Sewer
Location
Sewer
Overflow
2
Page 8
Length
Chezy formula
Chezy formula describes the mean velocity of uniform, turbulent flow
ΔH
𝑉 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑓
𝑉 = Mean Fluid Velocity [m/s]
R = Hydraulic Radius [m]
𝑆𝑓 = Hydraulic gradient [1]
𝐶 =8𝑔
𝜆 Chezy coefficient [m1/2/s]
𝑆𝑓 =ΔH
𝐿
3
Total Head
Pressure Head
Page 9
Chezy coefficient 𝐶 = 18 ∙ 𝑙𝑜𝑔12𝑅
𝑘
C = Chezy coefficient [m1/2/s]
R = Hydraulic Radius [m]
kS = surface roughness [m]
3
Page 10
Surface roughness kS [m] Equivalent Sand Roughness,
Material (ft) (mm)
Copper, brass 1x10-4 - 3x10-3 3.05x10-2 - 0.9
Wrought iron,
steel 1.5x10-4 - 8x10-3 4.6x10-2 - 2.4
Asphalt-lined
cast iron 4x10-4 - 7x10-3 0.1 - 2.1
Galvanized iron 3.3x10-4 - 1.5x10-
2 0.102 - 4.6
Cast iron 8x10-4 - 1.8x10-2 0.2 - 5.5
Concrete 10-3 - 10-2 0.3 - 3.0
Uncoated Cast
Iron 7.4x10-4 0.226
Coated Cast Iron 3.3x10-4 0.102
Coated Spun
Iron 1.8x10-4 5.6x10-2
Cement 1.3x10-3 - 4x10-3 0.4 - 1.2s
Wrought Iron 1.7x10-4 5x10-2
Uncoated Steel 9.2x10-5 2.8x10-2
Coated Steel 1.8x10-4 5.8x10-2
Wood Stave 6x10-4 - 3x10-3 0.2 - 0.9
PVC 5x10-6 1.5x10-3
Compiled from Lamont (1981), Moody (1944), and
Mays (1999)
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Page 11
Head loss sewer pipe
𝑉 = 𝐶 ∙ 𝑅 ∙ 𝑆𝑓 𝑄 = 𝑉 ∙ 𝐴 𝑆𝑓 = 𝑖 =∆𝐻
𝐿 Combine
∆𝐻 = 𝐿𝑄2
𝐶2 ∙ 𝑅ℎ ∙ 𝐴𝑠2
∆𝐻 = Head Loss, energy loss [m]
Q = discharge pipe [m3/s]
L = length of the pipe [m]
C = Chezy coefficient [m1/2/s]
R = Hydraulic Radius [m]
A = Wetted Area, flow surface [m2]
Sf ,i = slope of hydraulic gradient [-] 3
Page 12
Overflow
50 m
Ø300 PVC
Ø500 concrete
Ø250 PVC
pump
P4 P3 P2
GL (ground level) +6.00 m
IL +4,00 m IL +3,90 m
IL +3,73 m
Rain water
Waste water
Rain
Waste
+5,5 m
IL (Invert level) +3,53 m
P1
4
Page 13
Overflow / Weir 𝑄 = 𝑚 ∙ 𝐵 ∙ 𝐻32
Q = discharge overflow [m3/s]
m = runoff coefficient (1,5 – 1,8) [m1/2/s]
B = Width crest overflow [m]
H = Head at overflow [m]
measured from top crest!!
In example m = 1,8 H
Energy line
4
Page 14
Calculating sewer systems
50 m
Ø300 PVC
Ø500 concrete
Ø250 PVC
pump
P4 P3 P2
GL (ground level) +6.00 m
IL +4,00 m IL +3,90 m
IL +3,73 m
Rain
Waste
Rain
Waste
+5,5 m
IL (Invert level) +3,53 m
P1
5
Page 15
Question 1
50 m
Ø300 PVC
Ø500 concrete
Ø250 PVC
Pump
P4 P3 P2
GL +6.00 m
IL +4,00 m IL +3,90 m
IL +3,73 m
Rain
Waste
Rain
Waste
+5,5 m
IL +3,53 m
P1
5
Page 16
Question 2
50 m
Ø300 PVC
Ø500 concrete
Ø250 PVC
Pump
P4 P3 P2
GL +6.00 m
IL +4,00 m IL +3,90 m
IL +3,73 m
Rain=0
Waste=10l/s
Rain=0
Waste=10l/s
+5,5 m
IL +3,53 m
Q=20 l/s
Q=10 l/s
P1
5
Page 17
Partially filled pipe
𝐼𝑛𝑝𝑢𝑡: 𝑄𝑝𝑎𝑟𝑡
𝑄𝑓𝑢𝑙𝑙= 0,17
𝑂𝑢𝑡𝑝𝑢𝑡: ℎ
𝐷= 0,27
𝑂𝑢𝑡𝑝𝑢𝑡: 𝑢𝑝𝑎𝑟𝑡
𝑢𝑓𝑢𝑙𝑙= 0,75
5
Page 19
Question 3
50 m
Ø300 PVC
Ø500 concrete
Ø250 PVC
Pump=20 l/s
P4 P3 P2
GL +6.00 m
IL +4,00 m IL +3,90 m
IL +3,73 m
Rain=66 l/s
Waste=10 l/s
Rain=225 l/s
Waste=10 l/s
+5,5 m
IL +3,53 m
P1
5
Page 20
Question 3b
50 m
Ø300 PVC
Ø500 beton
Ø250 PVC
Pump=20 l/s
P4 P3 P2
GL +6.00 m
IL +4,00 m IL +3,90 m
IL +3,73 m
Rain=66 l/s
Waste=10 l/s
Rain=225 l/s
Waste=10 l/s
+5,5 m
IL +3,53 m
P1
5
Page 21
Question 3c
50 m
Ø300 PVC
Ø500 beton
Ø250 PVC
Pump=20 l/s
P4 P3 P2
GL +6.00 m
Rain=66 l/s
Waste=10 l/s
Rain=225 l/s
Waste=10 l/s
+5,5 m
Q=66 l/s
Q=291 l/s
P1
In example m = 1,8 5
Page 22
Strategy [situation with overflow]
Preparation
Information available for each pipe
- Diameter, R, L, k, C
- Discharge and Velocity
Information Overflow / weir
- Width, m
- Discharge
- Level crest in m N.A.P.
𝐶 = 18 ∙ 𝑙𝑜𝑔12𝑅
𝑘
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Page 23
1. Calculate H at weir
2. Calculate ∆H each pipe
3. Water level at weir (P1) = level crest weir + H at weir
4. Water level at P2 = Water level at weir + ∆Hweir(p1) – p2
5. Water level at P3 = Water level at P2 + ∆H p2– p3
6. Water level at P4 = Water level at P3 + ∆H p3– p4
Strategy [situation with overflow]
Steps
All levels in m N.A.P.
𝑄 = 𝑚 ∙ 𝐵 ∙ 𝐻32 ∆𝐻 = 𝐿
𝑄2
𝐶2 ∙ 𝑅ℎ ∙ 𝐴𝑠2
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Page 24
• In manhole velocity is low so water level (y) ≈ total head (H)
Otherwise you have to take the velocity head (u2/2g) into
account
• Pipes are submerged
• Steady situation
• Turbulent flow
• Subcritical flow [stromend] (discussed later, most of the time
flow is subcritical
Strategy [situation with overflow]
Remarks
With subcritical flow [stromend] the downstream situation affects
the upstream situation. So that is why you start at the weir and
work back to P3.
5
Page 25
Question 3de
50 m
Ø300 PVC
Ø500 beton
Ø250 PVC
Pump=20 l/s
P4 P3 P2
GL +6.00 m
Rain=66 l/s
Waste=10 l/s
Rain=225 l/s
Waste=10 l/s
+5,5 m
Q=66 l/s
v=0,93 m/s
I=1:244
Q=291 l/s
v=1,48 m/s
I=1:166
Q=0 l/s
v=0 m/s
I=0 P1
In example m = 1,8 5