2.5 Design of Channels

Transcription

1 2.6 Design of Open Channels Open channels have uses in urban stormwater drainage urban sanitary-sewer systems irrigation delivery systems In the next few lectures, we'll discuss the design procedure for 3 types of open channels: lined (nonerodable channels) - are primarily used to maximize flow rates and minimize construction costs unlined (erodible channels) - are the least expensive and what is commonly found in nature, but often experience unacceptable levels of sediment transport and erosion. grass-lined -are commonly used to transmit intermittent irrigation and stormwater flows and to control erosion

2 Open Channel Design Fundamentals There are many design considerations which are common to all 3 types of channels. This includes the determination of the best hydraulic section. The best hydraulic section will accommodate the design flow at a reasonable cost and limit erosion/deposition of sediment and other material. Trapezoidal Channel To determine the optimum channel size for a trapezoidal channel, we need to determine the optimum channel width and side slope (to minimize excavation costs) which will maximize the flowrate. First recall Manning's Equation, Q= AR 3 So2 n Rearranging 3 2 Qn 5 5 A= P So Restating the above statement another way, we need to minimize A & P with respect to Q or 3 Qn 5 So Recall the area and wetted perimeter definitions for a trapezoidal cross section combining these two equations

3 substituting into Manning's equation The optimum perimeter/area is determined by differentiating with respect to y (while holding m constant). Similarly, the optimum channel side slope is determined by differentiating with respect to m (while holding y constant). Combing the optimum perimeter and optimum slope equations, we can determine the best hydraulic section for a trapezoid is This solution assumes the channel slope can be set to any value. Often this is not possible, in which case use a known (or specified) m value to determine the optimum perimeter. Please note, for lined channels, the side slope is often specified as being less than 33.7o. The best hydraulic section for a variety of channels is given in the below table

4 Other considerations The previous derivations does not consider the following points: freeboard (distance between water surface and top of channel) whether or not its possible to excavate the optimum channel cost of lining access to site Minimal permissible velocity m/s (2-3 ft/s) to prevent sedimentation 0.75 m/s (2.5 ft/s) to prevent vegetation growth Channel Slopes Longitudinal governed by topography (unless velocities are too low) Side Slopes a function of material

5 Other considerations (continued) Freeboard The freeboard is the vertical distance between the water surface and the top of the channel slope. Its basically a factor of safety to keep the channel from overflowing. For unlined channels, the freeboard is estimated with: F=0.55 Cy where F is the freeboard in meters y is the design flow depth C is a coefficient Generally, ASCE recommends a minimum freeboard of 30 cm. In sinuous channels, additional freeboard is need at he channel bends to account for the superelevation of the water surface. The superelevation in the vicinity of the bend is approximated with 2 hs = V T gr c where V is the average velocity T is the top width of the channel rc is the bend radius According to the USACE (1995) rc > 3T

6 Design of Lined Channels Lined (or rigid-boundary channels) are used in the following situations: transmit flow at high velocities decrease seepage losses decrease maintenance and operation costs ensure channel stability The design procedure for lined channels is as follows: 1. Estimate the roughness coefficient, n, and the freeboard coefficient, C, for the desired channel lining and flowrate 2. Compute the normal depth of flow, yn, with Manning's equation Q= AR 3 So2 n 3. Verify that: minimum permissible velocities are being met flow is subcritical, Fr < 0.8 maximum permissible velocities are being met V < 2.1 m/s unreinforced channel V < 5.5 m/s reinforced channels If necessary, choose new channel dimensions and recalculate yn to ensure above criteria are satisfied. 4. Calculate the required freeboard (including the additional freeboard at channel bends)

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8 Example~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Design lined channel to carry 20 m3/s on a longitudinal slope of The lining of the channel is to be reinforced float finished concrete. Consider a) the best hydraulic section and b) a section with side slopes of 1.5:1 (H:V). Solution~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ a) given: 1. Estimate the roughness coefficient, n, and the freeboard coefficient, C, for the desired channel lining and flowrate 2. Compute the normal depth of flow, yn, with Manning's equation determine best hydraulic section: Q= AR 3 So2 n b=2.4 m yn=2.09 m

9 3. Verify that: minimum permissible velocities are being met V=2.6 m/s flow is subcritical, Fr<0.8 Fr = 0.66 maximum permissible velocities are being met V < 5.5 m/s reinforced channels 4. Calculate the required freeboard (including the additional freeboard at channel bends) F=1.04 m The total channel depth (including freeboard) is 3.13 m. The channel is to have a bottom width of 2.4 m and side slopes is. 58:1 (H:V).

10 b) Step 1. Estimate the roughness coefficient, n, and the freeboard coefficient, C, for the desired channel lining and flowrate Step 2. Compute the normal depth of flow, yn, with Manning's equation determine the geometry given H:V=1.5: Q= AR 3 So2 n b=1.16 m yn=1.94 m

11 Step 3. Verify that: minimum permissible velocities are being met V=2.53 m/s flow is subcritical, Fr<0.8 Fr = 0.76 maximum permissible velocities are being met V < 5.5 m/s reinforced channels Step 4. Calculate the required freeboard (including the additional freeboard at channel bends) F=1.00 m The total channel depth (including freeboard) is 2.94 m. The channel is to have a bottom width of 1.16 m and side slopes is 1.5:1 (H:V).

12 Design of Unlined Open Channels In the design of unlined channels, it is necessary to assure that sediment won't be eroded from or deposited on channel beds. To design new (or analyze existing), we'll need to approximate the stress applied to the channel walls by the fluid and the stress required to move the sediment particles. Stress applied by the flow The stress applied by the flow is not applied evenly to the channel sidewalls and bottom walls. For the two conditions it is defined with bottom side Stress required to move sediment The stress required to move particles on the bottom is a function of the immersed particle weight, the particle geometry, and the applied friction between particles. Where the critical stress required for sediment to move becomes

13 For particles on the side slope of the channel, the relationship with gravity is more complicated. The above expression is modified, so the critical stress to move particles on the channel sides becomes where The ratio of the critical shear stress on the side to the critical shear stress on the bottom is defined as the tractive force ratio, K, with

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16 Design Procedure for Unlined Channels 1. Estimate the roughness coefficient, n, based on the perimeter characteristics (attached table) of the channel, and select the freeboard coefficient, C, based on the design flowrate in the channel. 2. Estimate the angle of repose (attached figure 4.31) 3. Estimate the channel sinuousness (attached table 4.15) 4. Specify a side slope angle (attached table 4.13) 5. Estimate the tractive force ratio 6. Estimate the permissible tractive force on the bottom and sides of the channel

17 7. Assume permissible shear stress on the channel sides is the limiting design factor, and determine the normal depth 8. Calculate the required bottom width, b, of the channel using Manning's equation 9. Compare the permissible tractive force on the bottom with the actual tractive force 10.Compare the permissible velocity and calculate the Froude number (verify that it is subcritical) 11. Estimate the required freeboard and super elevation if needed

18 Example~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Design a trapezoidal channel to carry 20 m3/s through a slightly sinuous channel on a slope of The channel is to be excavated in course alluvium with a 75-percentile diameter of 2 cm (.8 in) and with particles on the perimeter of the channel moderately rounded. Solution~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Step 1. Estimate the roughness coefficient, n, and select the freeboard coefficient, C, based on the design flowrate in the channel. Step 2. Estimate the angle of repose (attached figure 4.31) Step 3. Estimate the channel sinuousness (attached table 4.15) Step 4. Specify a side slope angle (attached table 4.13) Step 5. Estimate the tractive force Step 6. Estimate the permissible tractive force on the bottom and sides of the channel

19 Step 7. Determine the normal depth Step 8. Calculate the required bottom width, b, of the channel using Manning's equation Step 9. Compare the permissible tractive force on the bottom with the actual tractive force Step 10. Verify permissible velocity and that flow is subcritical Step 11. Estimate the required freeboard The total depth of the channel to be excavated (including the freeboard) is The channel is to have a bottom width of 24.2 and sides slopes of 2:1 (H:V).

20 Design of Grass-Lined Open Channels Grass-lined channels are often used to transmit intermittent irrigation and storm water flows. They are often preferable to lined channels because they provide increased storage, low velocities, and aesthetic benefits. The additional design considerations for grass-lined channels include: Channel Roughness- The Manning's n value is a function of the velocity and channel geometry, in addition to the roughness provided by the grass. The manning's n for the various retardances is given in the below figure.

21 Permissible Velocities -There exists a variety of considerations for determining the maximum permissible velocities in a grass-lined channel. Those include the table on the next page (table 4.18) and the below considerations from the soil conservation service Vmax(m/s) Conditions Sparse vegetive cover possible Vegetation to be established by seeding Dense sod (or temporarily diverted flow) Well-established sod Very special conditions???

22 Channel Cross Sections - In addition to the best hydraulic section and slope stability, grass lined channels may have to be designed to allow farm equipment to cross. Freeboard - The freeboard on a grass lined channel is defined with

23 Design Procedure for grass-lined channels This procedure must completed for BOTH upper and lower bound of retardance (ie. Mowed and unmowed conditions). Stage I - Lower Bound Retardance 1. Assume the value of the roughness coefficient and determine the value of VR 2. Select the maximum permissible velocity (attached table) 3. Using Manning's equation, compute new VR 4. Repeat until VR is converged 5. Determine A from design flow and maximum permissible velocity 6. Determine the channel proportions for the calculated values of R and A

24 Stage II - Upper Bound retardance 1. Assume a depth of flow for the channel assumed in Stage I and compute A and R. 2. Compute the averaged velocity 3. Compute VR using the results of 1 & 2 4. determine n from the attached figure using the upper bound retardance 5. Use the n from step 4, R from step 1, compute V from the manning's equation 6. Repeat 1-5 until V has converged 7. Add the appropriate freeboard and check the Froude number

25 Example~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ Design a triangular grass-lined channel to handle intermittent flow of 0.7 m3/s. The channel is to be excavated in an easily erodible soil, on a longitudinal slope of 2%, and lined with Bermuda grass. During the early stages of channel development, the height of the grass will be maintained at about 4 cm (a retardance of E); during the latter stages of development, the Bermuda grass is expected to be at a height of about 30 cm (a retardance of B). Solution~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ given: Q=0.7m3/s So=0.02 Bermuda grass easily eroded soil Stage I: E retardance Stage II: B retardance Stage I guess at n, determine VR determine Vmax from table compute R=VR/Vmax find new VR from Mannings eq repeat with adjusted n n VR (ft2/s) fig4.33 VR (m2/s) fig 4.33/10.76 R (m) VR/Vmax VR n (m2/s) =R5/3So1/2/n Find the Area (triangle)

26 Determine channel geometry from A and R m=1.71 y=0.48 m note: you'll get two solutions for m and have to choose the best Stage II: guess at y, from stage I from stage I geometry compute A & R from known geometry compute the average velocity (=Q/A) compute VR determine n (fig 4.33) compute V from manning's equation Y A R (m (m2) (m) ) VR/Vmax V VR (m/s (m2/s) ) =Q/ A VR n V V 2 (ft /s) (m/s) VR*10.76 fig4.33 =R2/3So1/2/ n Find the freeboard F=0.3

27 Check for subcritical flow Fr=0.38, subcritical, ok design The total depth of flow is 1.05 m. The triangular channel is to have side slopes of 1.7:1 (H:V).