This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Frictional head loss (laminar flow) h f=32 ∙ μ ∙L ∙Vρ ∙g ∙ D2
h f = frictional head loss [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] g=¿ earths gravity [m/s2]
Wall shear stress (laminar flow) τ0=¿ 4 ∙ μ ∙V
R¿
τ0 = shear stress at solid boundary [N/m2]μ=¿ Absolute viscosity [kg/ms] V=¿ mean velocity [m/s]R=¿ Hydraulic Radius [m]
Turbulent flow in pipes and closed conduits
Head loss / Energy loss ∆H=ξ ∙ u2
2g
∆ H=¿ Head Loss [m]
u2
2g=¿ Velocity Head [m]
ξ=¿ Loss coefficient [1]g=¿ earths gravity [m/s2]
Frictional head losses
Darcy-Weisbach ∆H f= λ∙ LD∙ u
2
2 g=ξ f ∙
u2
2 g
∆ H f=¿ Head Loss due to friction [m] λ=¿ Friction coefficient [1]
u2
2g=¿ Velocity Head [m]
D = Hydraulic Diameter 4R [m]L=¿ Length between the Head Loss [m]g=¿ earths gravity [m/s2]
Sf = slope of hydraulic gradient [-]hf = frictional head loss [m]L=¿ Length between the Head Loss [m]g=¿ earths gravity [m/s2]
9
Local head losses
Sudden Pipe Enlargement ∆H l=(V 1−V 2)
2
2g
∆ H l=(1−A1A2
)2
∙V 12
2g ξ l=(1−A1A2
)2
∆H l=¿ Head Loss due to sudden pipe enlargement [m] ξ l=¿ 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 enlargement2= After enlargement
Sudden Pipe Contraction ∆H l=(A1A2
−1)2
∙V 22
2gA1≅ 0,6 ∙ A2
∆H l=0,44 ∙V 22
2 g
∆H l=¿ Head Loss due to sudden pipe contraction [m] V❑=¿ Mean Fluid Velocity after sudden pipe contraction
[m/s] g=¿ earths gravity [m/s2]
Tapered Pipe Enlargement ξ t=n ∙(A1A2
−1)2
∆H t=¿ Head Loss due to tapered pipe enlargement [m] ξ s=¿ Loss coefficient due to tapered pipe enlargement [1]A=¿ Wetted Area [m2]1= Before enlargement2= After enlargement
10
n=¿ factor which depends on the widening angel α
11
Submerged Pipe Outlet ∆H o=1∙v12
2g
∆ H o=¿ Head Loss due to submerged pipe outlet [m] v1=¿ Mean Fluid Velocity before pipe outlet [m/s] ξo=1 Loss coefficient due to submerged pipe outlet [1]g=¿ earths gravity [m/s2]
Pipe Bends ∆H b=ξb ∙v2
2g
∆H b=¿ Head Loss due to pipe bend [m] v❑=¿Mean Fluid Velocity [m/s] ξb=¿ Loss coefficient due to pipe bend [1]g=¿ earths gravity [m/s2]
Tabel 4.5 only applies for α = 90o and a smooth pipe.
With α = 90o and a rough pipe, increase ξ with 100%
∆H tot=¿ Total Head Loss Culvert [m] ξ tot=¿ Sum of Loss coefficients [1]vc=¿ Mean Fluid Velocity Culvert [m/s] ξ i=¿ Loss coefficient due to contraction [1]ξw=¿ Loss coefficient due to friction [1]ξo=¿ Loss coefficient due to outlet [1]μ=¿ Contraction coefficient [1]g=¿ earths gravity [m/s2]λ=¿ Friction coefficient [1]R=¿ Hydraulic Radius [m]l=¿ Length between the Head Loss [m]
13
Culvert submerged 2 q=m⋅Ac⋅√2g⋅ΔH tot
m= 1√ξ tot
q=¿ Flow rate Culvert [m3/s]m=¿ Discharge coefficient [m]A=¿ Wetted Area Culvert [m2]∆H tot=¿ Total Head Loss Culvert [m] ξ tot=¿ Sum of Loss coefficients [1]g=¿ earths gravity [m/s2]
Culvert partly submerged
Free flow broad crested weir
(Volkomen lange overlaat) h3≤23 H
qv=cv⋅b⋅H32
q=¿ Discharge Culvert [m3/s]b=¿ Width weir [m]cv=discharge coefficient free flow broad crested weir [m1/2/s]H=¿ Head Loss upstream [m] h3=¿ Water level downstream [m]Submerged flow broad crested weir
(Onvolkomen lange overlaat) h3>23 H
qv=col⋅b⋅h3⋅√2 g⋅(H−h3 )
col≈1
√ξ totq=¿ Discharge Culvert [m3/s]b=¿ Width weir [m]col=discharge coefficient submerged flow broad crested weir [1]H=¿ Head Loss upstream [m] h3=¿ Water level downstream [m]Total head (H) and water level (h) measured from crest weir (bed culvert)
14
Open channel flow
V average=QA
=V 1 A1+V 2 A2+V 3 A3
A1+A2+A3
Frictional head losses, turbulent flowMean boundary shear stress τ 0=ρ ∙ g ∙R ∙ S0
τ0 = shear stress at solid boundary [N/m2]R=¿ Hydraulic Radius [m]S0=¿ Slope of channel bed [1]
Chezy V=C ∙√R ∙S f
V=¿ Mean Fluid Velocity [m/s]R=¿ Hydraulic Radius [m]
15
Sf=¿ Slope energy / total head [1]
C=√ 8gλ Chezy coefficient [m1/2/s]
Manning V=R23 ∙ S f
12
n
Q=1n ∙ A
53
P23
∙ S f
12
V=¿ Mean Fluid Velocity [m/s]R=¿ Hydraulic Radius [m]Sf=¿ Slope Total Head [1]A=¿ Wetted Area [m2]P=¿ Wetter Perimeter [m]n=¿ Mannings roughness coefficient [s/m1/3]
C=R16
n
Specific energy H s= y+V 2
2 g
V=¿ Mean Fluid Velocity [m/s]
y= pρ ∙g
=¿ Pressure Head / water depth [m]
Equilibrium / normal depth [m] yn=3√ q2
b2 ∙C2∙ S0 S0=S f
yn = normal depth, equilibrium depth [m]q = discharge [m3/s]b = width [m] S0=¿ bed slope [1]Sf=¿ Hydraulic gradient caused by friction [1]
C=√ 8gλ Chezy coefficient [m1/2/s]
Backwater, direct step method ∆ x=∆ y ∙( 1−Fr2
S0−Sf)
Δx= horizontal distance from point [m]Δy= waterdepth [m]Fr = Froude number [-]S0=¿ bed slope [1]Sf=¿ Hydraulic gradient caused by friction [1]