LECTURE 9: Open channel flow: Uniform flow, best hydraulic sections, energy principles, Froude number

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1 LECTURE 9: Open channel flow: Uniform flow, best hydraulic sections, energy principles, Froude number Open channel flow must have a free surface. Normally free water surface is subjected to atmospheric pressure, which remains relatively constant througout the entire length of the channel. z V + P z V + P h L The pressure distribution in any section is directly proportional to the depth measured from the free water surface. In this case, the water surface lines corresponds to the hydraulic gradient line in pipe flow. z V + y z 2 V +y h L 1

2 9.1. Important terms in open channel flow: Top width of the channel = T, Width of the channel, W T W Wetted Perimeter (P) F Hydraulic Radius (R) W F A Hydraulic Depth (D) = T W Hydraulic radius for full flow R.D /.D = D, Full flow Hydraulic radius for half-full flow R.D //.D / = D Half - full Bottom Slope, S Side Slope (m) 2

3 2 common equations for the analysis of uniform open channel flow. Uniform = flow area does not change with the length of channel a) Chezy Equation b) Manning s Equation (derived from Chezy Equation) Chezy Equation: First Formula for uniform open channel flow. V C. R. S Velocity m/s Chezy s constant S = Slope of HGL for uniform flow S S C =./S/ /R../S C R R m = Depends on pipe material m = 0.35 for concrete pipe m = 0.25 for vitrified clay pipe Manning s Equation C. R/ Emprical relation for Chezy s constant V 1 n. R/. R /. S / 3

4 V 1 n. R/. S / Slope of EGL = L L Velocity m/s Manning s Constant Hydraulic Radius 9.2 Open Channel Flow Classification Space & Time Uniform Varied Flow Steady Flow Unsteady Flow Graduady Varied Flow Rapidly Varied Flow Uniform Flow: Water depth remains same along the channel length at a given time Varied Flow: Water depth or discharge changes along the length at a given time Steady Flow: The discharge and water depth at any section in the reach do not change with time. Unsteady Flow: The discharge and water depth at any section in the reach change with time. Uniform Flow => Water depth = Normal depth y 8.3. Uniform Flow in Open Channels 1. Water depth, flow area, discharge and velocity must remain unchanged in all sections of the entire channel. 2. EGL, the water surface, the channel bottom must be parellel to each other S S.. S Slope of EGL Slope water surface Slope of Channel 4

5 Example 9.1 (Ex. 6.1, Hwang, 4th Edition): A 3 m wide rectangular channel a discharge of 25 m /s at a uniform depth of 1.2 m. Determine the slope of the channel if n V. R /. S / y = 1.2 m Slope of EGL b = 3 m Area = b. y m ρ 2y b m R A/P 3.6 m/5.4 m 0.67 m Q A.R /. S / Q. A.R / S. /..... / S S

6 9.3 Energy Principles in Open Channel Flow Enerygy contained in a unit weight of water flowing in an open channel may also be measured in three basic forms: 1. Kinetic Energy 2. Pressure Energy 3. Elevation (Potential) Energy above a certain datum line. 1-) V Q A = V 2-) P = y Flow area in the channel Flow area in the channel Pressure energy in open channel flow is usually computed with reference to the free surface. If the free surface in a channel approximates a straight line slope, the pressure at any submerged point A is equal to the vertical distance measured from the free surface to the point. Figure 6.7 Flow over curved surfaces: (a) convex surface and (b) concave surface P y V P y V 6

7 Total energy in open channel H=z V + y Specific Energy (E) Energy with respect to channel bottom E y V E Q..A y For a given water area and discharge For a given discharge, Q, specific energy (E) at any section is a function of depth of flow only. Depth of flow Specific energy for a given discharge Vertex C on a specific enetgy curve represents the depth (yc) at which the discharge Q may be delivered through the section at minimum energy, E c. This depth is commonly known as the critical depth fort he discharge Q at a given section. The corresponding flow in the section is knwon as the critical flow. At a smaller depth the same dicharge can be delivered 7

8 only by a higher velocity and, hence, a higher specific energy. The state of rapid and shallow flow through a section is known as supercritical flow or rapid flow. At a larger depth the same discharge may be delivered through the section with a smaller velocity and a higher specific enerrgy than a critical depth. It is known as subcritical flow. For a given value of specific energy, E1, the discharge may pass through the channel section at either depth d1 (supercritical flow) or d2 (subcritical flow). These two depths known as alternate depths Froude Number At critical state the spefic energy of the flow takes a minimum value. This value can be computed by equating the first derivative of the specific energy with respect to the water depth zero. E. Q A. y Q. A.A +1 =0 A =Differantial water area=t Q.A. T +1 = 0 A/T = D Hydraulic Depth for rectangular sections D = y for rectangular cross sections Q.A. T = 0 V.D.A = 0 or V.D 1 V Froude number, N F =.D 8

9 F = 1 Flow is in critical state F < 1 Subcritical state F > 1 Supercritical state Q = A T D. A D = y A = b.y Q =. T y. A Q y. b q Q F Unit discharge y Q. y. 9