Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Channel Design Hydromechanics VVR090 Design of Channels • lined channels – minimizing lining material costs • unlined channels – maximum permissible velocity and threshold of movement (stable hydraulic section) Concrete-lined channel Unlined channel 1 Objectives of Channel Design Transport water between two points in a safe and cost-effective manner. Includes economical, safety, and esthetics aspects. Here, mainly hydraulic aspects are considered. General observations: • Conveyance of a channel increases with the hydraulic radius (wetted perimeter deceases). From Manning’s formula. • The best hydraulic section is a semicircle (for a given area it has the minimum wetted perimeter). • For a specific cross section, the proportion that produce the best hydraulic section (maximum flow) might be derived. • The best hydraulic section might not be the best from an economical point of view. Best Hydraulic Cross Section I Maximize the flow for a given cross-sectional shape and area. From the Manning formula: Q= 1 AR 2 / 3 So1/ 2 n Qn = AR 2 / 3 So Thus: ⎛ Qn ⎞ max ⎜ = AR 2 / 3 ⎟ should be found for a given A ⎜ S ⎟ ⎝ o ⎠ 2 Best Hydraulic Cross Section II Optimum geometries for different cross sections. Rectangular cross section: best hydraulic cross section when the water depth is half the channel width. Practical Considerations for Best Hydraulic Section • The area needed to excavate the best hydraulic section might be larger than the area required to achieve the flow area • It may not be possible to construct a stable best hydraulic section in the natural material • The cost of excavation depends on other things than the amount of material removed (e.g., access to the site, cost of disposing material) • The slope of the channel must also be considered. 3 Channel Design – General Aspects 1. The minimum permissible velocity is the lowest velocity that will prevent sedimentation and vegetative growth (crude estimates: 0.6 – 0.9 m/s for sedimentation and 0.75 m/s for vegetation). 2. The side slopes depend primarily on the engineering properties of the material through which the channel is excavated. 3. The freeboard refers to the vertical distance between either the top of the channel or the top of the channel lining and the water surface (design flow at normal depth). Side Slopes 4 Channel Freeboard I Lined channel Channel Freeboard II Preliminary estimate of freeboard (unlined channel): F = Cy F: freeboard (ft) y: design depth (ft) C: coefficient (=1.5 for 20 ft3/s and 2.5 for 3000 ft3/s) 5 Flow Around a Channel Curve I Flow Around a Channel Curve II Rise in water surface at the outer bank: Δh = u 2b gR b: channel width R: distance from center of curve to centerline of channel Take into account velocity variation: Δh = 2.3 ⎛R ⎞ u2 log ⎜ o ⎟ g ⎝ Ri ⎠ Ri: inner radius Ro: outer radius 6 Use of Lined Channels Lined channels may be used: • to permit transmission of water at high velocities through areas of deep or difficult excavation • to permit transmission of water at a reduced construction cost • to decrease channel seepage • to reduce operation and maintenance costs • to ensure stability of the channel section Typical Lined Channel Cross Sections 7 Design Procedure for Lined Channels Minimize the cost of the lining material. Identical to find best hydraulic section, if uniform thickness of the lining material is employed. Design procedure: 1. Estimate n 2. Compute the value of the section factor 3. For appropriate expression for A and R, compute yN 4. Compute channel properties 5. Check minimum permissible velocity and Froude number 6. Estimate height of lining above surface and freeboard 7. Summarize results with dimensioned sketch Design for Different Lining Thickness or Material The base of the channel and the sides of the channel might be lined with different material of same material with different thickness. Design procedure involves minimizing the total cost (C) of the lining material: min ( C ) = min ( Cb + Cs ) Cb: material cost for channel base (per unit length) Cs: material cost of sides (per unit length) 8 Design of Stable, Unlined, Earthen Channels Find a stable cross section => One where neither scour nor deposition constitutes a problem. Three types of unstable sections: 1. Scouring occur but no deposition 2. Deposition occur but no scouring 3. Both scour and deposition occur Maximum Permissible Velocities 9 Theoretical Approach to Stable Cross Section I Bed shear stress (balance between gravity and force due to flow resistance): τo = ρgRSo For a wide channel, R ≈ y N , giving: τo = ρgy N So The shear stress is not uniformly distributed along the perimeter. Complex problem to determine correct distribution, but for trapezoidal cross section the following applies: τomax = ρgy N So τ max o = 0.76ρgy N So maximum along bottom maximum along side slopes Maximum Unit Tractive Force Channel side Channel bottom (in terms of ρgy N So ) 10 Theoretical Approach to Stable Cross Section II Study a particle in equilibrium, just before mobilization. Employ a balance between the shear force mobilizing the sediment and the resisting force depending on the material. Balance for particles along the bed: τ L Ae = Ws tan α τL = Ws tan α Ae Ae: effective area Ws: submerged particle weight a: angle of friction (= angle of repose) Angle of Respose The angle to the horizontal at which grains start to roll on a flat bed of sediment that is gradually tilted from the horizontal. A representative value on the angle of repose is 32 deg. 11 Theoretical Approach to Stable Cross Section III Balance for particles on the side slopes: ( τS Ae ) + (Ws sin Γ ) 2 2 = Ws cos Γ tan α Ws tan 2 Γ cos Γ tan α 1 − τS = tan 2 α Ae Tractive force ratio: K= τS tan 2 Γ sin 2 Γ = cos Γ 1 − = 1 − τL tan 2 α sin 2 α Angle of Repose tana 12 Maximum Unit Tractive Force for Different Materials non-cohesive cohesive Typical Channel Cross Sections 13 Design Procedure for Unlined Channels Find a stable cross section. Design procedure: 1. Estimate n 2. Estimate angle of repose for channel material 3. ... Consult French for the procedure. Example 14.265: Water Surface Profile Upstream an Obstruction Water flowing at the normal depth in a rectangular concrete channel that is 12 m wide encounters an obstruction (see figure), causing the water level to rise above the normal depth at the obstruction and for some distance upstream. The water discharge is 126 m3/s and the channel bottom slope is 0.00086. The depth of water just upstream from the obstruction is 4.55 m. Find the distance upstream to the point where the surface is at the normal depth. 14 Normal water depth: Q= 1 AR 2 / 3 S 1/ 2 n ⎛ 12 y N ⎞ 1 126 = 12 y N ⎜ ⎟ 0.013 ⎝ 12 + 2 y N ⎠ ⎛ 12 y N ⎞ 2.256 ⋅ ⎜ ⎟ ⎝ 12 + 2 y N ⎠ 2/3 − 2/3 0.000861/ 2 10.5 =0 yN y N = 2.95 m (by trial and error) Critical water depth: 1/ 3 ⎛ q2 ⎞ ycr = ⎜ ⎟ ⎝ g ⎠ 1/ 3 ⎛ 1 ⎛ 126 ⎞ 2 ⎞ =⎜ ⎜ 9.81 ⎜⎝ 12 ⎟⎠ ⎟⎟ ⎝ ⎠ = 2.24 m yN > ycr => subcritical flow at normal water depth Start at obstruction and calculate upstream through the step method. Δxi (y+u = 2 / 2g ) i +1 2 2 − ( y + u2 / 2g ) So − ( n u / R 4/3 ) i i +1/ 2 15 Table for Step Calculation 16