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.
2 2
u u
y1  z1   y 2  z2 
1
 H12
2
2g 2g
0

2. Henry Philibert Gaspard Darcy (June 10,
1803 – January 3, 1858) was
a French engineer who made several
important contributions to hydraulics.
L u2
ΔΗ f    
4 R 2g
Julius Ludwig Weisbach (born 10 August 1806 in
Mittelschmiedeberg (now Mildenau), Erzgebirge,
died 24 February 1871, Freiberg) was a German
mathematician and engineer.
0

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

4. 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

5. Energy loss [m] • Turbulent flow
• Friction loss (wrijvingsverlies)
Total Head
Pressure Head
2 2
L u u
ΔΗ f      
4 R 2g 2g
Going to look at other formulas
for calculating friction loss
ΔH
1 𝑉= 𝐶∙ 𝑅 ∙ 𝑆𝑓 𝑆𝑓 =
𝐿

6. Combined sewer / gemengd rioolstelsel
Rain water Rain
Waste water Waste
GL (ground level) +6.00 m
P4 P3 P2 P1 +5,5 m
Ø500 concrete
Ø300 PVC
Ø250 PVC IL +4,00 m
IL +3,90 m
IL +3,73 m
50 m
pump
1 IL (Invert level) +3,53 m

7. Sewer
Location
Sewer
Overflow
2

8. Chezy formula 𝑉= 𝐶∙ 𝑅 ∙ 𝑆𝑓
Chezy formula describes the mean velocity of uniform, turbulent flow
𝑉= Mean Fluid Velocity [m/s]
Total Head R= Hydraulic Radius [m]
Pressure Head 𝑆𝑓 = Hydraulic gradient [1]
8𝑔
𝐶= Chezy coefficient [m1/2/s]
𝜆
ΔH
𝑆𝑓 =
𝐿
ΔH
3 Length

9. 12𝑅
Chezy coefficient 𝐶 = 18 ∙ 𝑙𝑜𝑔
𝑘
C= Chezy coefficient [m1/2/s]
R= Hydraulic Radius [m]
kS = surface roughness [m]
3

10. Surface roughness kS [m]
Equivalent Sand Roughness,
Material (ft) (mm)
Copper, brass 1x10-4 - 3x10-3 3.05x10-2 - 0.9
Wrought iron,
1.5x10-4 - 8x10-3 4.6x10-2 - 2.4
steel
Asphalt-lined
4x10-4 - 7x10-3 0.1 - 2.1
cast iron
3.3x10-4 - 1.5x10-
Galvanized iron 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
7.4x10-4 0.226
Iron
Coated Cast Iron 3.3x10-4 0.102
Coated Spun
1.8x10-4 5.6x10-2
Iron
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)
3

11. Head loss sewer pipe
∆𝐻
Combine 𝑉= 𝐶∙ 𝑅 ∙ 𝑆𝑓 𝑄= 𝑉∙ 𝐴 𝑆𝑓 = 𝑖 =
𝐿
𝑄2
∆𝐻 = 𝐿 2
𝐶 ∙ 𝑅ℎ ∙ 𝐴2
𝑠
∆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]
3 Sf ,i = slope of hydraulic gradient [-]

12. Overflow
Rain water Rain
Waste water Waste
GL (ground level) +6.00 m
P4 P3 P2 P1 +5,5 m
Ø500 concrete
Ø300 PVC
Ø250 PVC IL +4,00 m
IL +3,90 m
IL +3,73 m
50 m
pump
4 IL (Invert level) +3,53 m

13. 3
Overflow / Weir 𝑄= 𝑚∙ 𝐵∙ 𝐻2
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!!
Energy line
In example m = 1,8 H
4

14. Calculating sewer systems
Rain Rain
Waste Waste
GL (ground level) +6.00 m
P4 P3 P2 P1 +5,5 m
Ø500 concrete
Ø300 PVC
Ø250 PVC IL +4,00 m
IL +3,90 m
IL +3,73 m
50 m
pump
5 IL (Invert level) +3,53 m

15. Question 1
Rain Rain
Waste Waste
GL +6.00 m
+5,5 m
P4 P3 P2 P1
Ø500 concrete
Ø300 PVC
Ø250 PVC IL +4,00 m
IL +3,90 m
IL +3,73 m
50 m
Pump
5 IL +3,53 m

16. Question 2
Rain=0 Rain=0
Waste=10l/s Waste=10l/s
GL +6.00 m
+5,5 m
P4 P3 P2 P1
Ø500 concrete
Q=10 l/s
Q=20 l/s Ø300 PVC
Ø250 PVC IL +4,00 m
IL +3,90 m
IL +3,73 m
50 m
Pump
5 IL +3,53 m

17. Partially filled pipe
𝑄 𝑝𝑎𝑟𝑡
𝐼𝑛𝑝𝑢𝑡: = 0,17
𝑄 𝑓𝑢𝑙𝑙
𝑢 𝑝𝑎𝑟𝑡
𝑂𝑢𝑡𝑝𝑢𝑡: = 0,75
𝑢 𝑓𝑢𝑙𝑙
ℎ
𝑂𝑢𝑡𝑝𝑢𝑡: = 0,27
𝐷
5

18. Table
5

19. Question 3
Rain=66 l/s Rain=225 l/s
Waste=10 l/s Waste=10 l/s
GL +6.00 m
P4 P3 P2 P1 +5,5 m
Ø500 concrete
Ø300 PVC
Ø250 PVC IL +4,00 m
IL +3,90 m
IL +3,73 m
50 m
Pump=20 l/s
5 IL +3,53 m

20. Question 3b
Rain=66 l/s Rain=225 l/s
Waste=10 l/s Waste=10 l/s
GL +6.00 m
P1
P4 P3 P2 +5,5 m
Ø500 beton
Ø300 PVC
Ø250 PVC IL +4,00 m
IL +3,90 m
IL +3,73 m
50 m
Pump=20 l/s
5 IL +3,53 m

21. Question 3c
Rain=66 l/s Rain=225 l/s
Waste=10 l/s Waste=10 l/s
GL +6.00 m +5,5 m
P4 P3 P2
Ø500 beton
Ø300 PVC P1
Ø250 PVC Q=291 l/s
Q=66 l/s
50 m
Pump=20 l/s
5 In example m = 1,8

22. Strategy [situation with overflow]
Preparation
Information available for each pipe 12𝑅
- Diameter, R, L, k, C 𝐶 = 18 ∙ 𝑙𝑜𝑔
- Discharge and Velocity 𝑘
Information Overflow / weir
- Width, m
- Discharge
- Level crest in m N.A.P.
5

23. Strategy [situation with overflow]
Steps
3
𝑄2
𝑄= 𝑚∙ 𝐵∙ 𝐻2 ∆𝐻 = 𝐿 2
𝐶 ∙ 𝑅ℎ ∙ 𝐴2
𝑠
All levels in m N.A.P.
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
5

24. Strategy [situation with overflow]
Remarks
• 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
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

25. Question 3de
Rain=66 l/s Rain=225 l/s
Waste=10 l/s Waste=10 l/s
GL +6.00 m +5,5 m
P4 P3 P2
Q=0 l/s Ø500 beton
v=0 m/s P1
Ø300 PVC
I=0
Ø250 PVC Q=291 l/s
Q=66 l/s v=1,48 m/s
v=0,93 m/s I=1:166
Pump=20 l/s I=1:244
5 50 m In example m = 1,8