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

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

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

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

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

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

Sewer

Location

Sewer

Overflow

2

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

Chezy coefficient 𝐶 = 18 ∙ 𝑙𝑜𝑔12𝑅

𝑘

C = Chezy coefficient [m1/2/s]

R = Hydraulic Radius [m]

kS = surface roughness [m]

3

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)

3

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

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

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

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

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

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

Partially filled pipe

𝐼𝑛𝑝𝑢𝑡: 𝑄𝑝𝑎𝑟𝑡

𝑄𝑓𝑢𝑙𝑙= 0,17

𝑂𝑢𝑡𝑝𝑢𝑡: ℎ

𝐷= 0,27

𝑂𝑢𝑡𝑝𝑢𝑡: 𝑢𝑝𝑎𝑟𝑡

𝑢𝑓𝑢𝑙𝑙= 0,75

5

Table

5

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

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

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

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

5

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

5

• 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

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