Frictional head loss (laminar flow)

Transcription

Frictional head loss (laminar flow)
CU06997 Fluid Dynamics / CU03287 Waterstroming: formulas
Henk Massink , 31-1-2014
Contents
CU06997 Fluid Dynamics / CU03287 Waterstroming: formulas ............................................................. 1
Principal symbols / units ......................................................................................................................... 2
Fluid statics .............................................................................................................................................. 3
Visualisation flow, streamlines, streaklines, streamtube........................................................................ 4
Total Head or Bernoulli’s Equation.......................................................................................................... 4
Turbulent and Laminar flow, Reynolds Number ..................................................................................... 6
Laminar flow in pipes and closed conduits ............................................................................................. 7
Turbulent flow in pipes and closed conduits .......................................................................................... 7
Frictional head losses ......................................................................................................................... 7
Local head losses ................................................................................................................................. 9
Partially full pipes .............................................................................................................................. 11
Culverts.............................................................................................................................................. 11
Open channel flow ................................................................................................................................ 13
Frictional head losses, turbulent flow .............................................................................................. 13
Subcritical and Supercritical flow ...................................................................................................... 15
Hydraulic structures .......................................................................................................................... 16
Sewers ................................................................................................................................................... 17
1
Principal symbols / units
A=
Wetted Area
Cross-sectional area of flow
b=
width
C=
Chézy coefficient
Cv =
velocity coefficient
Cc =
contraction coefficient
d, D =
diameter
Dm =
Hydraulic mean depth
E=
Energy
Es =
specific energy
F=
Force
Fr =
Froude Number
g=
gravitational acceleration
H=
head
hf , ∆H = frictional head loss
hL =
local head loss
kL =
local loss coefficient
kS =
surface roughness
L=
length
m=
mass
n=
Manning’s roughness coefficient
p* =
piezometric pressure
p=
pressure
P=
wetted perimeter
Ps =
crest height
Q=
discharge, flow rate
q=
discharge per unit channel width
R, r =
radius
R=
Hydraulic radius
Re =
Reynolds Number
Sc =
slope of channel bed to give critical flow
Sf ,I =
slope of hydraulic gradient
S0 =
slope of channel bed
Ss =
slope of water surface
u,v =
velocity
V=
mean velocity
V=
volume
ū=
average velocity
y=
water depth
yc =
critical depth
yn =
normal depth
z=
height above datum, potential head
δ=
boundary layer thickness
λ=
friction factor
µ=
absolute viscosity
ν=
kinematic viscosity
ρ=
density of liquid
τ0 =
shear stress at solid boundary
ξ = (ksie) Loss coefficient
µ=
contraction coefficient
[m2]
Natte doorsnede
[m]
Breedte
[m1/2/s]
Coefficient van Chezy
[-]
Snelheidscoëfficiënt
[-]
Contractiecoëfficiënt
[m]
Diameter
[m]
Gemiddelde hydraulische diepte
[J] =[Nm] Energie
[m]
Specifieke energie
[N]
Kracht
[-]
Getal van Froude
[m/s2]
Valversnelling
[m]
Energiehoogte
[m]
Energieverlies tgv wrijving
[m]
Lokaal energieverlies
[-]
Lokaal energieverlies coëfficiënt
[m]
Wandruwheid
[m]
Lengte
[kg]
Massa
[s/m1/3]
Coëfficiënt van manning
[N/m2]= [Pa] Piezometrische druk
[N/m2]
Druk
[m]
Natte omtrek
[m]
Stuwhoogte
[m3/s]
Debiet, afvoer
[m3/ms]
Debiet per m breedte
[m]
Straal
[m]
Hydraulische straal
[-]
Getal van Reynolds
[-]
Bodemverhang voor grenssnelheid
[-]
Energieverhang
[-]
Bodemverhang
[-]
Drukverhang, verhang water
[m/s]
Stroomsnelheid
[m/s]
Gemiddelde stroomsnelheid
[m3]
Volume
[m/s]
Gemiddelde stroomsnelheid
[m]
Waterdiepte
[m]
Kritische waterdiepte
[m]
Normale waterdiepte
[m]
Afstandshoogte
[m]
Dikte grenslaag
[m]
Wrijvingsfaktor
[kg/ms]=[N s/m2] Absolute viscositeit
[m2/s]
Kinematische viscositeit
[kg/m3]
Soortelijk gewicht
[N/m2]
Schuifspanning
[1]
Verliescoëfficiënt
[1]
Contractiecoëfficiënt
2
Fluid statics
F
General pressure intensity
p=
p = Pressure
𝐹 = Force
𝐴 = Area on which the force acts
[Pa=N/m2]
[N]
[m2]
Newton Force
F=m∙g
𝐹 = Force
𝑚 = Weight
𝑔 = earths gravity
[N]
[Kg]
[m/s2]
Fluid Pressure at a point
𝑝=
Pressure Head
𝑦=
p=
𝜌=
𝑔=
𝑦=
[Pa=N/m2]
[Kg/m3]
[m/s2]
[m]
Pressure
fluid density
earths gravity
distance surface to point
A
𝐹
𝐴
𝜌∙𝑔∙𝐴∙𝑦
𝐴
=𝜌∙𝑔∙𝑦
𝑝
𝜌∙𝑔
𝜌𝑓𝑟𝑒𝑠ℎ 𝑤𝑎𝑡𝑒𝑟 = 1000
𝜌𝑠𝑎𝑙𝑡 𝑤𝑎𝑡𝑒𝑟 = 1025
[Kg/m3]
[Kg/m3]
Potential Head
𝑧
𝑧=
[m]
height above datum
=
Piezometric Head
𝑧1 + 𝑦1 = 𝑧2 + 𝑦2
𝑧=
𝑦=
[m]
[m]
height above datum
distance surface to point
Velocity Head
𝑢 = Fluid Velocity
𝑔 = earths gravity
𝑢2
2∙𝑔
[m/s]
[m/s2]
3
Visualisation flow, streamlines, streaklines, streamtube
Flow rate / Discharge
𝑄 =𝑢∙𝐴
𝑄 = Flow rate
𝑢 = Fluid Velocity
𝐴 = Wetted Area
[m3/s]
[m/s]
[m2]
Wetted Area of a filled pipe
𝐴 = Wetted Area
𝐷 = Diameter pipe
A  1/ 4   D2
Continuity equation
(Principle of conservation of mass)
[m2]
[m2]
𝑢1 ∙ 𝐴1 = 𝑢2 ∙ 𝐴2 = 𝑢3 ∙ 𝐴3 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
Total Head or Bernoulli’s Equation
Total Head / Energy [m]
y=
p
ρ∙g
= Pressure Head
𝑧 = Potential Head
u2
2g
= Velocity Head
𝐻 =𝑦+𝑧+
𝑢2
2∙𝑔
[m]
[m]
[m]
Bernoulli’s Equation without head loss
𝑢12
𝑢22
𝑦1 + 𝑧1 +
= 𝑦2 + 𝑧2 +
= 𝐻 = 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡
2𝑔
2𝑔
Bernoulli’s Equation with head loss
𝑢12
𝑢22
𝑦1 + 𝑧1 +
= 𝑦2 + 𝑧2 +
+ ∆𝐻1−2
2𝑔
2𝑔
p
y = ρ∙g = Pressure Head
[m]
𝑧 = Potential Head
[m]
u
2
2g
= Velocity Head
∆𝐻 = Head Loss
𝑔 = earths gravity
[m]
[m]
[m/s2]
4
Momentum equation
𝐹𝑥 = 𝜌 ∙ 𝑄(𝑉2𝑥 − 𝑉1𝑥 )
𝐹=
𝜌=
𝑄=
𝑉1 =
𝑉2 =
[N]
[Kg/m3]
[m3/s]
[m/s]
[m/s]
Force
fluid density
Flow rate
Mean velocity before
Mean velocity after
Pitot
𝑢 = 2√2𝑔 ∙ ℎ
𝑢 = Fluid Velocity
𝑔 = earths gravity
h = Difference in pressure
[m/s]
[m/s2]
[m]
Discharge small orifice
𝑄 = 𝐶𝑣 ∙ 𝐶𝑐 ∙ 𝐴0 ∙ 2√2𝑔 ∙ ℎ
𝑄=
𝐴=
𝐶𝑣 =
𝐶𝑐 =
𝑔=
h=
[m3/s]
[m2]
[-]
[-]
[m/s2]
[m]
Flow rate
Wetted Area
velocity coefficient (0,97-0,99)
contraction coefficient (0,61-0,66)
earths gravity
Difference in pressure
2
3
3
Discharge large orifice
2
𝑄 = ∙ 𝑏 ∙ √2𝑔 ∙ (ℎ22 − ℎ12 )
𝑄=
𝑏=
𝑔=
h1 =
h2 =
[m3/s]
[m2]
[m/s2]
[m]
[m]
Flow rate
Width orifice
earths gravity
Difference in pressure from top
Difference in pressure from bottom
3
5
Turbulent and Laminar flow, Reynolds Number
𝜇
Kinematic viscosity
𝜐=
𝜌
𝜇 = Absolute viscosity
𝜐 = Kinematic viscosity
𝜌 = Density of liquid
[kg/ms]
[m2/s] water, 20°C= 1,00 ∙ 10−6
[kg/m3]
Reynolds Number
𝑅𝑒 =
𝜐=
𝜌=
𝑉=
R=
𝑅𝑒 =
[m2/s] water, 20°C= 1,00 ∙ 10−6
[kg/m3]
[m/s]
[m]
[1]
Kinematic viscosity
Density of liquid
Mean velocity
Hydraulic Radius = D/4
Reynolds Number
𝑉.4∙𝑅
𝜈
𝑅𝑒 > 4000 Turbulent flow
𝑅𝑒 < 2000 Laminar flow
𝐴
Hydraulic Radius
𝑅=
R = Hydraulic Radius
𝐴 = Wetted Area
𝑃 = Wetter Perimeter
[m]
[m2]
[m]
Hydraulic radius of a filled pipe
𝑅 = 𝑃 = 4 𝜋∙𝐷 = 14 ∙ 𝐷 [𝑚]
Hydraulic radius of a 50% filled pipe
𝑅=
Hydraulic Diameter
𝐷 =4∙𝑅
R = Hydraulic Radius
𝐷 = Hydraulic Diameter
[m]
[m]
𝑃
1
∙𝜋∙𝐷 2
𝐴
𝐴
𝑃
=
1
2
1
2
1
∙𝜋∙𝐷 2
∙4
𝜋∙𝐷
= 14 ∙ 𝐷 [𝑚]
6
Laminar flow in pipes and closed conduits
ℎ𝑓 =
Frictional head loss (laminar flow)
ℎ𝑓 =
𝜇=
𝐿=
𝑉=
D=
𝜌=
𝑔=
frictional head loss
Absolute viscosity
Length between the Head Loss
mean velocity
Hydraulic Diameter
Density of liquid
earths gravity
𝜌∙𝑔∙𝐷2
[m]
[kg/ms]
[m]
[m/s]
[m]
[kg/m3]
[m/s2]
4∙𝜇∙𝑉
Wall shear stress (laminar flow)
𝜏0=
τ0 =
𝜇=
𝑉=
R=
[N/m2]
[kg/ms]
[m/s]
[m]
shear stress at solid boundary
Absolute viscosity
mean velocity
Hydraulic Radius
32∙𝜇∙𝐿∙𝑉
𝑅
Turbulent flow in pipes and closed conduits
Head loss / Energy loss
∆𝐻
∆𝐻 = Head Loss
[m]
𝑢2
2g
= Velocity Head
𝜉 = Loss coefficient
𝑔 = earths gravity
2
𝑢
= 𝜉 ∙ 2𝑔
[m]
[1]
[m/s2]
Frictional head losses
𝐿
Darcy-Weisbach
∆𝐻𝑓 = 𝜆 ∙ ∙
∆𝐻𝑓 = Head Loss due to friction
𝜆 = Friction coefficient
[m]
[1]
𝑢2
2g
[m]
= Velocity Head
D = Hydraulic Diameter 4R
𝐿 = Length between the Head Loss
𝑔 = earths gravity
𝑢2
𝐷 2𝑔
= 𝜉𝑓 ∙
𝑢2
2𝑔
[m]
[m]
[m/s2]
7
1
Colebrook-White transition formula
𝜆=
D=
kS =
Friction coefficient
Hydraulic Diameter
surface roughness
√𝜆
= −2𝑙𝑜𝑔 (
𝑘𝑠
3,70∙𝐷
+
2,51
Re∙ √𝜆
)
[1]
[m]
[m]
Colebrook-White and Darcy Weisbach
𝑉 = −2√2𝑔 ∙ 𝐷 ∙ 𝑆𝑓 ∙ 𝑙𝑜𝑔 (
𝑉=
D=
kS =
𝜐=
Sf =
hf =
𝐿=
𝑔=
𝑘𝑠
3,70∙𝐷
mean velocity
Hydraulic Diameter
surface roughness
Kinematic viscosity
slope of hydraulic gradient
frictional head loss
Length between the Head Loss
earths gravity
+
2,51∙υ
D √2𝑔∙𝐷∙𝑆𝑓
[m/s]
[m]
[m]
[kg/ms]
[-]
[m]
[m]
[m/s2]
)
with 𝑆𝑓 =
ℎ𝑓
𝐿
water, 20°C= 1,00 ∙ 10−6
8
Local head losses
Sudden Pipe Enlargement
𝐴 2 𝑉2
∆𝐻𝑙 = (1 − 𝐴1 ) ∙ 2𝑔1
2
∆𝐻𝑙 =
𝜉𝑙 =
𝐴=
𝑉=
𝑔=
1=
2=
∆𝐻𝑙 =
(𝑉1 −𝑉2 )2
2𝑔
𝜉𝑙 = (1 −
𝐴1 2
)
𝐴2
Head Loss due to sudden pipe enlargement
Loss coefficient due to sudden pipe enlargement
Wetted Area
Mean Fluid Velocity
earths gravity
Before enlargement
After enlargement
Sudden Pipe Contraction
∆𝐻𝑙
[m]
[1]
[m2]
[m/s]
[m/s2]
2
2
𝑉2
= (𝐴𝐴1 − 1) ∙ 2𝑔
2
𝐴1 ≅ 0,6 ∙ 𝐴2
∆𝐻𝑙
= 0,44 ∙
∆𝐻𝑙 = Head Loss due to sudden pipe contraction
𝑉 = Mean Fluid Velocity after sudden pipe contraction
𝑔 = earths gravity
Tapered Pipe Enlargement
∆𝐻𝑡 =
𝜉𝑠 =
𝐴=
1=
2=
𝑛=
𝜉𝑡 = 𝑛 ∙ (
Head Loss due to tapered pipe enlargement
Loss coefficient due to tapered pipe enlargement
Wetted Area
Before enlargement
After enlargement
factor which depends on the widening angel α
𝑉22
2𝑔
[m]
[m/s]
[m/s2]
𝐴1
𝐴2
− 1)2
[m]
[1]
[m2]
9
∆𝐻𝑜 = 1 ∙
Submerged Pipe Outlet
𝑣1 2
2𝑔
∆𝐻𝑜 = Head Loss due to submerged pipe outlet
𝑣1 = Mean Fluid Velocity before pipe outlet
𝜉𝑜 = 1 Loss coefficient due to submerged pipe outlet
𝑔 = earths gravity
[m]
[m/s]
[1]
[m/s2]
𝑣2
Pipe Bends
∆𝐻𝑏 = 𝜉𝑏 ∙
∆𝐻𝑏 =
𝑣 = Mean Fluid Velocity
𝜉𝑏 = Loss coefficient due to pipe bend
𝑔 = earths gravity
Head Loss due to pipe bend
[m/s]
[1]
[m/s2]
2𝑔
[m]
Tabel 4.5 only applies for α = 90o and a smooth pipe.
With α = 90o and a rough pipe, increase ξ with 100%
With α = 45o use 75% ξ900
With α = 22,5o use 50% ξ900
Smooth and rough pipes are explained further on.
10
Partially full pipes
𝐴𝑝 (ℎ) =
1 2
2∙ℎ
𝐷
2
∙ 𝐷 ∙ arccos(1 −
) − ( − ℎ) ∙ √ℎ ∙ 𝐷 − ℎ2
4
𝐷
2
1 2
2∙ℎ
∙ 𝐷 ∙ arccos (1 − 𝐷 )
𝐷
4
𝑅𝑝 (ℎ) =
− ( − ℎ)
2
2
√ℎ ∙ 𝐷 − ℎ2
𝐴𝑝 =
𝑅𝑝 =
h=
D=
[m2]
[m]
[m]
[m]
Wetted Area partially filled pipe
Hydraulic radius partially filled pipe
water level partially filled pipe
Diameter pipe
Culverts
Culvert submerged 1
v 2c
2g
 tot  (ξ i  ξ w  ξ u  ....)
ΔΗ tot  ξ tot 
1

 i    1


w   
2
l
4R
u  1
∆𝐻𝑡𝑜𝑡 = Total Head Loss Culvert
𝜉𝑡𝑜𝑡 = Sum of Loss coefficients
𝑣𝑐 = Mean Fluid Velocity Culvert
𝜉𝑖 = Loss coefficient due to contraction
𝜉𝑤 = Loss coefficient due to friction
𝜉𝑜 = Loss coefficient due to outlet
𝜇 = Contraction coefficient
𝑔 = earths gravity
𝜆 = Friction coefficient
R = Hydraulic Radius
𝑙 = Length between the Head Loss
[m]
[1]
[m/s]
[1]
[1]
[1]
[1]
[m/s2]
[1]
[m]
[m]
11
Culvert submerged 2
q  m  Ac  2 g  H tot
m
q=
𝑚=
𝐴=
∆𝐻𝑡𝑜𝑡 =
𝜉𝑡𝑜𝑡 =
𝑔=
Flow rate Culvert
Discharge coefficient
Wetted Area Culvert
Total Head Loss Culvert
Sum of Loss coefficients
earths gravity
1
 tot
[m3/s]
[m]
[m2]
[m]
[1]
[m/s2]
Culvert partly submerged
Free flow broad crested weir
(Volkomen lange overlaat)
h3  23 H
3
qv  cv  b  H 2
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
 tot
q=
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)
12
Open channel flow
𝑉𝑎𝑣𝑒𝑟𝑎𝑔𝑒 =
𝑄 𝑉1 𝐴1 + 𝑉2 𝐴2 + 𝑉3 𝐴3
=
𝐴
𝐴1 + 𝐴2 + 𝐴3
Frictional head losses, turbulent flow
Mean boundary shear stress
𝜏0 = 𝜌 ∙ 𝑔 ∙ 𝑅 ∙ 𝑆0
τ0 = shear stress at solid boundary
R = Hydraulic Radius
𝑆0 = Slope of channel bed
[N/m2]
[m]
[1]
Chezy
𝑉 = 𝐶 ∙ √𝑅 ∙ 𝑆𝑓
𝑉 = Mean Fluid Velocity
R = Hydraulic Radius
𝑆𝑓 = Slope energy / total head
[m/s]
[m]
[1]
8𝑔
𝐶 = √ 𝜆 Chezy coefficient
[m1/2/s]
13
1
2
𝑉=
Manning
1
𝑄= ∙
𝑛
𝑉=
R=
𝑆𝑓 =
𝐴=
𝑃=
Mean Fluid Velocity
Hydraulic Radius
Slope Total Head
Wetted Area
Wetter Perimeter
𝑅 3 ∙𝑆𝑓2
𝑛
5
𝐴3
∙
2
𝑃3
1
𝑆𝑓2
[m/s]
[m]
[1]
[m2]
[m]
1
1/3
𝑛 = Mannings roughness coefficient
[s/m ] 𝐶 =
Specific energy
𝐻𝑠 = 𝑦 +
𝑉 = Mean Fluid Velocity
[m/s]
y=
p
ρ∙g
= Pressure Head / water depth
𝑅6
𝑛
𝑉2
2𝑔
[m]
3
𝑞2
Equilibrium / normal depth [m]
𝑦𝑛 = √ 2 2
𝑏 ∙𝐶 ∙𝑆
yn =
q=
b=
𝑆0 =
𝑆𝑓 =
normal depth, equilibrium depth
discharge
width
bed slope
Hydraulic gradient caused by friction
[m]
[m3/s]
[m]
[1]
[1]
𝐶 = √ 𝜆 Chezy coefficient
8𝑔
[m1/2/s]
Backwater, direct step method
∆𝑥 = ∆𝑦 ∙ (
Δx=
Δy=
Fr =
𝑆0 =
𝑆𝑓 =
[m]
[m]
[-]
[1]
[1]
horizontal distance from point
waterdepth
Froude number
bed slope
Hydraulic gradient caused by friction
𝑆0 = 𝑆𝑓
0
1−𝐹𝑟 2
𝑆0 −𝑆𝑓
)
14
Subcritical and Supercritical flow
𝑄2
3
Critical depth [m]
𝑦𝑐 = √
Critical velocity [m/s]
𝑉𝑐 = 2√𝑔 ∙ 𝑦𝑐
Froude Number
𝐹𝑟 =
yc =
Q=
B=
Vc =
V=
Fr =
[m]
[m3/s]
[m]
[m/s]
[m/s]
[-]
critical depth
discharge
width
critical velocity
actual velocity
Froude number
Subcritical flow
Supercritical flow
Fr < 1
Fr > 1
𝑔∙𝐵2
𝑉
2
√𝑔∙𝑦𝑐
=
𝑉
𝑉𝑐
V < Vc
V > Vc
Energy loss hydraulic jump
∆𝐸 =
ΔH = Energy loss hydraulic jump
y1= depth supercritical flow
Y2= depth subcritical flow
[m]
[m]
[m]
Critical bed slope
𝑆𝑐 =
Sc = critical bed slope
yc = critical depth
𝑛 = Mannings roughness coefficient
(𝑦2 −𝑦1 )3
4∙𝑦1 ∙𝑦2
𝑔∙𝑛2
1
𝑦𝑐3
[-]
[m]
[s/m1/3]
15
Hydraulic structures
Thin plate (sharp crested weirs)
2
𝑄𝑖𝑑𝑒𝑎𝑙
3
2
2
= ∙ 𝑏 ∙ √2𝑔 ∙ ℎ12
3
2
Rehbock formula Q= ∙ √2𝑔 ∙ (0.602 + 0.083 ∙
3
ℎ1
𝑃𝑠
3
) ∙ 𝑏 ∙ (ℎ1 + 0.0012)2
30𝑚𝑚 < ℎ1 < 750 𝑚𝑚, 𝑏 > 300 𝑚𝑚, 𝑃𝑠 > 100 𝑚𝑚, ℎ1 < 𝑃𝑠
Q=
b=
h1 =
Ps =
[m3/s]
[m]
[m]
[m]
discharge
width
pressure above crest
crest height
Vee weirs
𝑄 = 𝐶𝑑 ∙
Q=
𝐶𝑑 =
h1 =
θ=
[m3/s]
[-]
[m]
[°]
discharge
discharge coefficient
pressure above crest
angle vee
Rectangular broad crested weir
Ackers 𝐶𝑓 ≅ 0.91 + 0.21 ∙
0.45 <
Q=
b=
h1 =
Ps =
L=
Cf =
ℎ1
𝐿
8
15
5
2
𝜃
∙ √2𝑔 ∙ tan( ) ∙ ℎ12
2
θ=90°, Cd=0.59
3
2
𝑄𝑖𝑑𝑒𝑎𝑙 = 1.705 ∙ 𝑏 ∙ 𝐻1
+ 0.24 ∙ (
ℎ1
ℎ1 +𝑃𝑠
− 0.35)
ℎ1
ℎ1
< 0.8, 0.35 <
< 0.6
𝐿
ℎ1 + 𝑃𝑠
discharge
width
pressure above crest
crest height
length weir
friction coefficient
[m3/s]
[m]
[m]
[m]
[m]
[-]
3
2
Discharge broad-crested weir
2
2𝑔
𝑄 = 𝐶𝑑 ∙ 𝐶𝑣 ∙ ∙ √ ∙ 𝑏 ∙ ℎ2
Q = discharge
b = width
𝐶𝑣 = velocity coefficient )
𝐶𝑑 = discharge coefficient
h = water pressure above crest
[m3/s]
[m]
[-]
[-]
[m]
3
3
16
3
2
Venturi flume
𝑄 = 1.705 ∙ 𝑏2 ∙ 𝐶𝑑 ∙ 𝐶𝑣 ∙ 𝑦1
Q=
b=
𝐶𝑣 =
𝐶𝑑 =
y1 =
[m3/s]
[m]
[-]
[-]
[m]
discharge
width
velocity coefficient )
discharge coefficient
pressure above crest
Sewers
𝑄2
Filled pipes
∆𝐻 = 𝐿 ∙
∆𝐻 = Head Loss, energy loss
Q = discharge pipe
L = length of the pipe
C = Chezy coefficient
R = Hydraulic Radius
A = Wetted Area, flow surface
[m]
[m3/s]
[m]
[m1/2/s]
[m]
[m2]
Chezy coefficient
𝐶 = 18 ∙ 𝑙𝑜𝑔 [
C = Chezy coefficient
R = Hydraulic Radius
kS = surface roughness
[m1/2/s]
[m]
[m]
Energy Gradient [-]
𝑆𝑓 = 𝑖 =
Sf ,i = slope of hydraulic gradient
L = length
∆𝐻 = Head Loss, energy loss
[-]
[m]
[m]
𝐶 2 ∙𝑅∙𝐴2
𝑘
]
∆𝐻
𝐿
Overflow / weir sewer
𝑄 =𝑚∙𝐵∙𝐻
Q=
m=
B=
H=
[m3/s]
[m1/2/s]
[m]
[m]
discharge overflow
runoff coefficient (1,5 – 1,8)
Width crest overflow
Head at overflow
measured from top crest!!
12𝑅
3
2
17

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