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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
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
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]          • 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
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
Sewer
    Location
    Sewer
    Overflow




2
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
12𝑅
Chezy coefficient                         𝐶 = 18 ∙ 𝑙𝑜𝑔
                                                        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,
                                         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
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     [-]
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
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
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
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
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
Partially filled pipe

                        𝑄 𝑝𝑎𝑟𝑡
                 𝐼𝑛𝑝𝑢𝑡:        = 0,17
                        𝑄 𝑓𝑢𝑙𝑙

                                                𝑢 𝑝𝑎𝑟𝑡
                                        𝑂𝑢𝑡𝑝𝑢𝑡:        = 0,75
                                                𝑢 𝑓𝑢𝑙𝑙
       ℎ
𝑂𝑢𝑡𝑝𝑢𝑡: = 0,27
        𝐷




5
Table




5
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
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
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
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
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
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
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

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Cu06997 lecture 8_sewers

  • 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
  • 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