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CU06997 Fluid Dynamics
    Flow in pipes and closed conduits

    4.1 Introduction (page 91)
    4.2 The historical context (page 91-93)
    4.3 Fundamental concepts of pipe flow (page 94-97)
    4.4 Laminar flow (page 97-100)
    4.5 Turbulent flow (page 100 – 111)



1
Pipe with head loss
     2         2
     u         u
h1  1
         h4  4
                   H14
     2g        2g         Q  u1  A1  u4  A4

                                              Head loss



                                      Total
                                      Head

                                      Pressure
                                      Head




1
Reynolds number:
p 93 (pipe), p 127 (open channel)
                                           𝜌∙ 𝑉∙ 𝐷   𝑉∙ 𝐷
                                      𝑅𝑒 =         =
𝜇=   Absolute viscosity     [m2/s]
                                              𝜇        𝜈
𝜐=   Kinematic viscosity    [kg/ms]
     water, 20°C= 1,00 ∙ 10−6                   𝑉. 4𝑅
𝜌 = Density of liquid       [kg/m3]        𝑅𝑒 =
𝑉 = Velocity                [m/s]                  𝜈
D = Hydraulic diameter      [m]
R=   Hydraulic Radius = D/4 [m]
𝑅𝑒 = Reynolds Number        [1]

𝑹𝒆 > 𝟒𝟎𝟎𝟎 Turbulent flow
𝑹𝒆 < 𝟐𝟎𝟎𝟎 Laminar flow

1
Laminar flow, frictional head loss
[Energieverlies tgv wrijving]

                                                  Total Head

         32 ∙ 𝜇 ∙ 𝐿 ∙ 𝑉                          Pressure Head

    ℎ𝑓 =
           𝜌 ∙ 𝑔 ∙ 𝐷2

    ℎ𝑓 =   frictional head loss ∆H     [m]
    𝜇=     Absolute viscosity          [kg/ms]
    𝐿=     Length between the Head Loss [m]
    𝑉=     mean velocity               [m/s]
    D=     Hydraulic Diameter          [m]
    𝜌=     Density of liquid           [kg/m3]
2   𝑔=     earths gravity              [m/s2]
Laminar flow, wall shear stress
[Schuifspanning]


          4∙ 𝜇∙ 𝑉
      𝜏0=
             𝑅

     τ0 =   shear stress at solid boundary [N/m2]
      𝜇=    Absolute viscosity             [kg/ms]
      𝑉=    mean velocity                  [m/s]
     R=     Hydraulic Radius               [m]

2
Head loss /Energy loss [m]
• Turbulent flow                                u   2

• Friction loss (wrijvingsverlies)   ΔΗ              [m]
                                                2g
• Local loss (lokaal verlies)

    • ΔH = Head loss or Energy loss [m]
    • u2/2g = Velocity head [m]
    • ξ (ksie) = Loss coëfficiënt [1]


3
2         2
Darcy-Weisbach                     L u        u
                       ΔΗ f           
                                  4 R 2g      2g
                         Total Head
                                                  L
                                         f  
                         Pressure Head



                                                 4R
             •   ΔH = Head loss by friction        [m]
             •   u2/2g = Velocity head             [m]
             •   L = Length                         [m]
             •   λ = (lamda) = Friction coëfficiënt[1]
             •   ξ (ksie) = Loss coëfficiënt       [1]
3            •   R = hydraulic radius              [m]
Remarks friction loss Darcy-Weisbach
    • λ (boundary roughness) depends on material and
      construction. λ often between 0,01 and 0,10
    • λ is not a constant, depends on “boundary layer”.
      “Smooth” or “Rough”, Most of the time “Smooth”


    How to calculate λ !!!
    • During exams Fluid Dynamics, the λ will be given



3
Colebrook-White transition formula
 1                   𝑘𝑠     2,51
       = −2 ∙ 𝑙𝑜𝑔         +
     𝜆            3,70 ∙ 𝐷 Re∙ 𝜆
 𝜆=    Friction coefficient  [1]
D=     Hydraulic Diameter 4R [m]
kS =   surface roughness     [m]
        (k-waarde)

Difficult to solve
Could use figure 4.5 page 105
Nowadays computers?

3
Moody diagram




3
Colebrook-White and Darcy Weisbach
                                       𝑘𝑠         2,51υ
𝑉 = −2 2𝑔 ∙ 𝐷 ∙ 𝑆 𝑓 ∙ 𝑙𝑜𝑔                   +
                                    3,70𝐷       D 2𝑔∙𝐷∙𝑆 𝑓
              ℎ𝑓
with 𝑆 𝑓 =
               𝐿


        𝑉=         Average velocity                   [m/s]
       D=          Hydraulic Diameter (4R)            [m]
       kS =        surface roughness                  [m]
        𝜐=         Kinematic viscosity                [kg/ms]
       Sf =        slope of hydraulic gradient        [-]
       hf =        frictional head loss (∆Hf)         [m]
        𝐿=         Length between the Head Loss       [m]
3
Turbulent flow ,
    Mean boundary shear stress
     𝜏0 = 𝜌 ∙ 𝑔 ∙ 𝑅 ∙ 𝑆0
    τ0 =    shear stress at solid boundary [N/m2]
    R=      Hydraulic Radius        [m]
     𝑆0 =   Slope of channel bed [1]


    In sewer minimum shear stress value
    (0.5 – 1.5 N/m2)




3
Local head losses




                            2
                        u
           ΔΗ l              [m]
                        2g
4
Head loss Sudden Pipe Enlargement




             V1  V2 
                      2
                          ∆𝐻 𝑙 = (1 −
                                        𝐴1 2 𝑉1
                                           ) ∙
                                                2
    ΔΗ l                               𝐴2     2𝑔
                 2g
4
Head loss Sudden Pipe Enlargement

∆𝐻 𝑙 = (1 −
              𝐴1 2 𝑉1
                 ) ∙
                      2
                           𝜉 𝑙 = (1 −
                                        𝐴1 2
                                           )          (𝑉1 − 𝑉2 )2
              𝐴2     2𝑔                 𝐴2     ∆𝐻 𝑙 =
                                                          2𝑔


    ∆𝐻 𝑙 =   Head Loss due to sudden pipe enlargement [m]
     𝜉𝑙 =    Loss coefficient due to sudden pipe enlargement   [1]
     𝐴=      Wetted Area [m2]
     𝑉=      Mean Fluid Velocity [m/s]
     𝑔=      earths gravity [m/s2]
    1=       Before enlargement
    2=       After enlargement


4
Head loss Sudden Pipe Contraction




4
                                                        2
        𝐴1        2
                 𝑉2                                    𝑉2
∆𝐻 𝑙 = ( − 1)2 ∙
        𝐴3       2𝑔
                    and 𝐴3 ≅ 0,6 ∙ 𝐴2    ∆𝐻 𝑙 = 0,44 ∙
                                                       2𝑔
∆𝐻 𝑙 = Head Loss due to sudden pipe contraction        [m]
𝑉2 = Mean Fluid Velocity after sudden pipe contraction [m/s]
𝑔 = earths gravity                                     [m/s2]
Local head loss coefficients
                 𝑢2   𝑘 𝑙 = 𝜉𝑙
    ∆𝐻 𝑙 = 𝑘 𝑙 ∙
                 2𝑔




4

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Cu06997 lecture 6_flow in pipes 1_2013

  • 1. CU06997 Fluid Dynamics Flow in pipes and closed conduits 4.1 Introduction (page 91) 4.2 The historical context (page 91-93) 4.3 Fundamental concepts of pipe flow (page 94-97) 4.4 Laminar flow (page 97-100) 4.5 Turbulent flow (page 100 – 111) 1
  • 2. Pipe with head loss 2 2 u u h1  1  h4  4  H14 2g 2g Q  u1  A1  u4  A4 Head loss Total Head Pressure Head 1
  • 3. Reynolds number: p 93 (pipe), p 127 (open channel) 𝜌∙ 𝑉∙ 𝐷 𝑉∙ 𝐷 𝑅𝑒 = = 𝜇= Absolute viscosity [m2/s] 𝜇 𝜈 𝜐= Kinematic viscosity [kg/ms] water, 20°C= 1,00 ∙ 10−6 𝑉. 4𝑅 𝜌 = Density of liquid [kg/m3] 𝑅𝑒 = 𝑉 = Velocity [m/s] 𝜈 D = Hydraulic diameter [m] R= Hydraulic Radius = D/4 [m] 𝑅𝑒 = Reynolds Number [1] 𝑹𝒆 > 𝟒𝟎𝟎𝟎 Turbulent flow 𝑹𝒆 < 𝟐𝟎𝟎𝟎 Laminar flow 1
  • 4. Laminar flow, frictional head loss [Energieverlies tgv wrijving] Total Head 32 ∙ 𝜇 ∙ 𝐿 ∙ 𝑉 Pressure Head ℎ𝑓 = 𝜌 ∙ 𝑔 ∙ 𝐷2 ℎ𝑓 = frictional head loss ∆H [m] 𝜇= Absolute viscosity [kg/ms] 𝐿= Length between the Head Loss [m] 𝑉= mean velocity [m/s] D= Hydraulic Diameter [m] 𝜌= Density of liquid [kg/m3] 2 𝑔= earths gravity [m/s2]
  • 5. Laminar flow, wall shear stress [Schuifspanning] 4∙ 𝜇∙ 𝑉 𝜏0= 𝑅 τ0 = shear stress at solid boundary [N/m2] 𝜇= Absolute viscosity [kg/ms] 𝑉= mean velocity [m/s] R= Hydraulic Radius [m] 2
  • 6. Head loss /Energy loss [m] • Turbulent flow u 2 • Friction loss (wrijvingsverlies) ΔΗ    [m] 2g • Local loss (lokaal verlies) • ΔH = Head loss or Energy loss [m] • u2/2g = Velocity head [m] • ξ (ksie) = Loss coëfficiënt [1] 3
  • 7. 2 2 Darcy-Weisbach L u u ΔΗ f       4 R 2g 2g Total Head L f   Pressure Head 4R • ΔH = Head loss by friction [m] • u2/2g = Velocity head [m] • L = Length [m] • λ = (lamda) = Friction coëfficiënt[1] • ξ (ksie) = Loss coëfficiënt [1] 3 • R = hydraulic radius [m]
  • 8. Remarks friction loss Darcy-Weisbach • λ (boundary roughness) depends on material and construction. λ often between 0,01 and 0,10 • λ is not a constant, depends on “boundary layer”. “Smooth” or “Rough”, Most of the time “Smooth” How to calculate λ !!! • During exams Fluid Dynamics, the λ will be given 3
  • 9. Colebrook-White transition formula 1 𝑘𝑠 2,51 = −2 ∙ 𝑙𝑜𝑔 + 𝜆 3,70 ∙ 𝐷 Re∙ 𝜆 𝜆= Friction coefficient [1] D= Hydraulic Diameter 4R [m] kS = surface roughness [m] (k-waarde) Difficult to solve Could use figure 4.5 page 105 Nowadays computers? 3
  • 11. Colebrook-White and Darcy Weisbach 𝑘𝑠 2,51υ 𝑉 = −2 2𝑔 ∙ 𝐷 ∙ 𝑆 𝑓 ∙ 𝑙𝑜𝑔 + 3,70𝐷 D 2𝑔∙𝐷∙𝑆 𝑓 ℎ𝑓 with 𝑆 𝑓 = 𝐿 𝑉= Average velocity [m/s] D= Hydraulic Diameter (4R) [m] kS = surface roughness [m] 𝜐= Kinematic viscosity [kg/ms] Sf = slope of hydraulic gradient [-] hf = frictional head loss (∆Hf) [m] 𝐿= Length between the Head Loss [m] 3
  • 12. Turbulent flow , Mean boundary shear stress 𝜏0 = 𝜌 ∙ 𝑔 ∙ 𝑅 ∙ 𝑆0 τ0 = shear stress at solid boundary [N/m2] R= Hydraulic Radius [m] 𝑆0 = Slope of channel bed [1] In sewer minimum shear stress value (0.5 – 1.5 N/m2) 3
  • 13. Local head losses 2 u ΔΗ l    [m] 2g 4
  • 14. Head loss Sudden Pipe Enlargement V1  V2  2 ∆𝐻 𝑙 = (1 − 𝐴1 2 𝑉1 ) ∙ 2 ΔΗ l  𝐴2 2𝑔 2g 4
  • 15. Head loss Sudden Pipe Enlargement ∆𝐻 𝑙 = (1 − 𝐴1 2 𝑉1 ) ∙ 2 𝜉 𝑙 = (1 − 𝐴1 2 ) (𝑉1 − 𝑉2 )2 𝐴2 2𝑔 𝐴2 ∆𝐻 𝑙 = 2𝑔 ∆𝐻 𝑙 = Head Loss due to sudden pipe enlargement [m] 𝜉𝑙 = Loss coefficient due to sudden pipe enlargement [1] 𝐴= Wetted Area [m2] 𝑉= Mean Fluid Velocity [m/s] 𝑔= earths gravity [m/s2] 1= Before enlargement 2= After enlargement 4
  • 16. Head loss Sudden Pipe Contraction 4 2 𝐴1 2 𝑉2 𝑉2 ∆𝐻 𝑙 = ( − 1)2 ∙ 𝐴3 2𝑔 and 𝐴3 ≅ 0,6 ∙ 𝐴2 ∆𝐻 𝑙 = 0,44 ∙ 2𝑔 ∆𝐻 𝑙 = Head Loss due to sudden pipe contraction [m] 𝑉2 = Mean Fluid Velocity after sudden pipe contraction [m/s] 𝑔 = earths gravity [m/s2]
  • 17. Local head loss coefficients 𝑢2 𝑘 𝑙 = 𝜉𝑙 ∆𝐻 𝑙 = 𝑘 𝑙 ∙ 2𝑔 4