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Hydraulics in urban water
Dette er mye repetisjon (for mye?) + noen ferske eksempler bruk
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Hydraulikk i Urban water networks
Fylte rør vs del-fylte rør Vann (fylte rør) (se Haestad bok!) Avløp (fylte rør, delfylte rør) Hydraulikken er lik uavhengig av vann eller avløp! (M, n, k varierer litt dog)
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Content 1 Basics 2 Steady 1D flow 3 General 1D flow Pipe flow
Open channel flow Special case: Steep channels 3 General 1D flow Unsteady free-surface flow: St. Venant equations Kinematic wave approximation Diffusive wave approximation
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1 Basics
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Flow models 3D, turbulent 1D, unsteady, non-uniform
Basic research Investigation of local processes 1D, unsteady, non-uniform Numerical models in urban hydrology Historical events Consequences of alternatives 1D, steady, uniform Easy to understand and handle Design and rough dimensioning
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Example 3D flow
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Forms of disharge in sewers
Designation Criterion Steady state d/dt = 0 Unsteady d/dt ≠ 0 Uniform d/dx = 0 Non- uniform d/dx ≠ 0 Continiuous q = 0 Discontiniuous q ≠ 0 Subcritical (*) Fr<1 Supercritical Fr>1 Laminar Re<2300 Turbulent Re>2300 Monophase wastewater Multiphase Wastewater+air
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Steady and uniform flow
Normal flow Steady Uniform Continuity
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Steady and uniform flow
Unique relation between flow rate and water depth
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Energy conservation Sf SPr v SP Reference horizon Dl
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2 Steady 1D Flow Pipe flow (fulle rør)
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Prandtl-Colebrook (Darcy-Weisbach)
Friction slope Friction loss Flow velocity
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Friction factor Smooth Transition Rough with Reynolds number
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Friction factor (or f) (Moody diagram)
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Friction factors, f(Re)
Re > 2300 : Colebrooke I Øving 1 kan begge disse benyttes (If…then..else)
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Forenklet formel for friksjon
Colebrooke tar lenger tid ved simuleringer av store nett Swamee- Jain: mye benyttet forenklet Ved falltapsmålinger/ruhetsmålinger på vannforsyningsledninger må Colebrooke benyttes!
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Roughness coefficient kS
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Equivalent Sand Roughness,
Equivalent Sand Roughness, Material (mm) Copper, brass 3.05x Wrought iron, steel 4.6x Asphalted cast iron Galvanized iron Cast iron Concrete Uncoated cast iron 0.226 Coated cast iron 0.102 Coated spun iron 5.6x10-2 Cement Wrought iron 5x10-2 Uncoated steel 2.8x10-2 Coated steel 5.8x10-2 Wood stave PVC 1.5x10-3 Compiled from Lamont (1981), Moody (1944), and Mays (1999 ks:
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Delfyllingskurve Kan også regnes ut matematisk (se ligninger for dette i Urban Drainage)
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How to calculate Q Colebrook (iteration) Swamee- Jain (cf. Exercise1)
Moody chart Capacity charts (pipe producers etc)
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Capacity of full pipe flow (*)
Velocity Flow rate with and
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Hentet fra NORVAR/NTNU/NLH kurs i flomberegning, 2004)
Hentet fra NORVAR/NTNU/NLH kurs i flomberegning, 2004). Illustrerer benevningsrot!
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Hazen-Williams Equation (US popular)
There are other empirical pipe-flow formulae, e.g. Velocity R = Hydraulic radius S = Slope C = Roughness coefficient 0.85 = Dimensional (!) coefficient
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Singular losses in valves and bens
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Local losses
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Example: CARE-S- singular losses from CCTV
CCTV inspections observe deteriorated pipes (intruding pipe etc) 3D hydraulic model used to used to estimate singular loss Hydraulic packages does not include singular losses between nodes, only in manholes How to adjust n (Mannings) to include singular losses between manholes?
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Modelling of the cross section in Mouse
Deteriorated sewers Modelling of the cross section in Mouse Modelling of the cross section in Mouse
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Hydraulic capacity of deterioted sewers – 3D analysis (Fluent)
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Conversion into 1D hydraulic roughness
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The effect of this - Application to 1D hydrodynamic sewer flooding modeling (MOUSE)
Results of analysis: Surface flooding which occurs with a 20 year return period a) Modeled without failures b) Failures included into model - there is problem with surface flooding - there is more flooding areas
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2 Steady 1D Flow Open channel flow
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Open channel flow Normal depth: Equilibrium between friction and fall Friction slope = Slope of pipe Hydraulic radius and relation to diameter for full pipe general
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Manning’s Equation Prerequisite: Normal flow conditions Velocity Flow rate If kS/D = to 0.01 Ackers (1958)
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Part-full flow Empirically determined Franke (1956) Velocity Flow rate
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Part-full flow with
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Operational roughness: kOp
kOp in mm
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Sub- and supercritical flow
Froude number F < 1 subcritical h > hC Information down- and upstream F > 1 supercritical h < hC Information only downstream F = 1 critical h = hC minimum energy supercritical subcritical hydraulic jump hC is independent from slope ! h > hC Flat channel h < hC Steep channel
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2 Steady 1D Flow Steep channels
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Air entrainment in steep channels
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Air entrainment in steep channels
Air volume fraction Boussinesq number Air-Water flow rate Air-Water velocity Flow cross section Design
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Fjerning av luft i trykkledninger
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Abrupt changes in slope
Commercial software does not model this!
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3 General 1D Flow Unsteady free-surface flow
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Continuity St. Venant equations “Mass” conservation V Q(x) h(t+t)
Q(x+x) x x x+x “Mass” conservation
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Continuity St. Venant equations
“Mass” conservation in differential form Divided by b·dx
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St. Venant: Momentum equation
Sf SP SP dx
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St. Venant: Momentum equation
Delete identical terms, cancel down dx with and acceleration
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St. Venant: Momentum equation
With flow cross section A and flow rate Q (1) Acceleration in time (2) Convective acceleration (3) Acceleration due to pressure gradient (4) Slope acceleration Numerical procedure to solve equations ! (time consuming…)
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Simplifications of St. Venant equations
Momentum equation Continuity Normal flow Kinematic wave approximation Diffusive wave approximation Dynamic wave equations (St. Venant equations)
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3 General 1D Flow Kinematic wave approximation
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Kinematic wave approximation
Continuity equation Momentum equation Solution for prismatic channel Unique relation between Q and h Only one boundary condition at upper boundary needed No upstream propagation of effects (waves)
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Kinematic wave approximation
Approximation valid if otherwise wave would break gradual rise of water table Unable to explain backwater effects ! Only apply to steep channels with supercritical flow
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3 General 1D Flow Diffusive wave approximation
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Diffusive wave approximation
Continuity equation Momentum equation Solution for prismatic channel or Wave-fronts propagate downstream and upstream (if flow is subcritical) Backwater effects can be described
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