Half of top width must be equal to of one the sloping side of the channel Hydraulic radius must be equal to half of the depth of the flow Trapezoidal angle must be equal to 60. Trapezoidal cross section: The best cross section for a triangular channel occurs when the its sides make angle of 45 Sloping side = Area , A= Wetted Perimeter P= Triangular cross section: MOST ECONOMICAL SECTION OF CHANNELS Width of channel should be twice its depth of flow, b=2y Hydraulic radius is half the depth of flow, R=y/2 Rectangular cross section: Circular cross section: 45
Example 4.1 i can do it!!!! Water is flowing 0.3m deep in a 1m wide, open channel of rectangular cross section. as shown in the diagram below. The channel is made of concrete (made with steel forms), with a constant bottom slope of 0.003. Estimate the flow rate of water in the channel. Solution: Based on the description, this will be uniform flow, From table above, n = 0.011 The bottom slope is given as: S = 0.003 The cross-sectional area, A = 1m x 0.3m = 0.3m2 The wetted perimeter, P = 0.3 + 0.3 + 1 = 1.6m The hydraulic radius, R = A/P = 0.3/1.6 = 0.1875m Substituting values for all of the parameters into Manning Equation, the answer is: Q = 0.489 m3/s Example 4.2 Open channel of width = 3m as shown, bed slope = 1:5000, d = 1.5m. Find the flow rate using Manning equation, n = 0.025. Solution: Based on the description, this will be uniform flow, Coefficient, n = 0.025 The bottom slope is given as: S = 1:5000 The cross-sectional area, A = 9m2 The wetted perimeter, P = 9.708m The hydraulic radius, R = A/P = 0.927m Substituting values for all of the parameters into Manning Equation, the answer is: Q = 4.84 m3/s 46
i can do it!!!! Example 4.3 Calculate the best hydraulic rectangular cross-section to convey Q = 10m3/s discharge with n = 0.02 and bed slope = 0.0009. Solution: For the best rectangular hydraulic cross-section b = 2y Area, A = 2y x y = 2y2 R = y/2 answer: y = 1.87m b = 2 x 1.87 = 3.74m Example 4.4 Find the depth of flow for maximum velocity and maximum discharge in a circular sewer 1.50m diameter. Find the maximum velocity and maximum discharge through a circular sewer 0.75m radius given N=0.016 channel bed slope = 0.1 percent. 47
Example 4.5 i can do it!!!! A trapezoidal channel having side slope 2V: 3H,and bed slope 0.0004 is required to carry discharge 10 m3/s. Find the dimension of the channel for minimum cross-section. Take Manning’s as 0.014 Solution: Answer: y = 1.866m b = 0.606y = 0.606 x 1.866 = 1.131m Also known as bed slope Hydraulic gradient, S is the slope at the bottom or base of the channel Movement of fluid flow is also influenced by the bed slope Explain hydraulic gradient. Explain uniform flow in open channel. Explain the best hydraulic cross section. 48
GET NOW Non-uniform Open Channel PLO 2: PLO 1 PLO 9: : identify and analyze well-defined engineering problems reaching substantiated conclusions using codified method of analysis specific to their field of activity (DK1 to DK4) function effectively as an individual, and as a member in diverse technical teams apply knowledge of applied mathematics, applied science, engineering fundamentals and an engineering specialization as specified in DK1 to DK4 respectively to wide practical procedures and practices LIFE HAS TWO RULES: (1) NEVER QUIT (2) ALWAYS REMEMBER RULE 49
Steady flow and unsteady flow If the flow characteristics such as depth of flow, velocity of flow, rate of flow at any point in open channel flow do not change with respect to time, the flow is sad to be steady flow. Mathematically, steady flow is expressed as If at any point in open channel flow, the velocity of flow, depth of flow or rate of flow changes with respect to time, the flow is said to be unsteady flow. Mathematically, unsteady flow means Uniform Flow And Non-Uniform Flow If for a given length of channel, the velocity of flow, depth of flow, slope of the channel and crosssection remain constant, the flow is said to be uniform. On the other hand, if for a given length of the channel, the velocity of flow, depth of flow etc. do not remain constant, the flow is said to be non-uniform flow. CLASSIFICATION OF OPEN CHANNEL FLOWS Steady flow and unsteady flow Uniform flow and non-uniform flow Laminar flow and turbulent flow Sub-critical, critical and super critical flow Non-uniform flow in open channels is also called varied flow, which is classified in the following two types as: a. Rapidly varied flow (RVF) b. Gradually varied flow (GVF) Laminar flow and turbulent flow The flow is open channel is said to be laminar if the Reynold number (Re) is less than 500. Reynold number in case of open channel is defined as If the Reynold number is more than 2000, the flow is said to be turbulent in open channel flow. If Re lies between 500 to 2000, the flow is considered to be in transition state. 50
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The total energy of a flowing liquid per unit weight is given by, Specific energy, E of a flowing liquid is defined as energy per unit weight of the liquid with respect to the bottom of the channel. non-uniform flow SPECIFIC ENERGY, E 52
Consider a rectangular channel in which a steady but non-uniform flow is taking place. Let: Q = discharge through the channel b = width of the channel y = depth of flow q = discharge per unit width SPECIFIC ENERGY CURVE Critical depth (yc) , Critical velocity (Vc) & Minimum specific energy (Emin @ Ec) It is defined as the curve which shows the variation of specific energy with depth of flow. It is obtained as : 1 2 3 A= B= C= 53
Fr < 1.0 Fr = 1.0 Fr > 1.0 Streaming Flow or Sub-critical Flow or Tranquil Flow Super-critical Flow or Shooting Flow or Torrential Flow Alternate Depths @ Conjugate Depths SUB-CRITICAL, CRITICAL AND SUPER CRITICAL FLOW The flow in open channel is said to be sub-critical if the Froude number (Fr) is less than 1.0. Subcritical flow is also called tranquil or streaming flow. For sub-critical flow, Fr < 1.0. The flow is called critical if Fr = 1.0. And if Fr > 1.0 the flow is called super critical or shooting or rapid or torrential.The Froude number is defined as: Critical Depth Critical Velocity Minimum Specific Energy @ Sub-Critical Flow Max.Discharge Super-Critical Flow When the depth of flow in a channel is greater than the critical depth (yc) When the depth of flow in a channel is less than the critical depth (yc) For any other value of the specific energy, there are two depths, one greater than the critical depth and other smaller than the critical depth. 54
The height of water at the section 1-1 is small. As we move toward downstream, the height or depth of water increases rapidly over a short length of the channel. This is because at the section 1- 1, the flow is a shooting flow as the depth of water at section 1-1 is less than critical depth. Shooting flow is an unstable type of flow and does not continue on the downstream side. Then this shooting will convert itself into a streaming or tranquil flow and hence depth of water will increase. This sudden increase of depth of water is called a hydraulic jump or a standing wave. Thus hydraulic jump is defined as : “The rise of water level, which takes place due to the transformation of the unstable shooting flow (super-critical) to the stable streaming flow (sub-critical flow)." When the rapid change in the depth of flow is from a low stage to high stage, the result is usually and abrupt rise of water surface. This local phenomenon is known as the hydraulic jump. SPECIFIC ENERGY DIAGRAM HYDRAULIC JUMP 55
Expression of Hydraulic Jump Theoretically this would be a negative hydraulic jump, i.e. the fluid depth will decrease Only physically possible if some external force accelerates the fluid at that point Fr<1 @ no hydraulic jump The flow is uniform and pressure distribution is due to hydraulic before and after the jump Losses due to friction on the surface of the bed of the channel are small and hence neglected. The slope of the bed of the channel is small, so that the component of the weight of the fluid in the direction of flow is negligibly small. Before deriving an expression for the depth of hydraulic jump, the following assumptions are made : 1. 2. 3. Consider a hydraulic jump formed in a channel of horizontal bed is shown in Figure below. Consider two section 1 and 2 before and after hydraulic jump. A hydraulic jump occurs because of Fr changes: Depth of hydraulic jump in terms of upstream Froude Number Fr >1 and Fr <1 DDEEPPTTHH OOFF HHYYDDRRAAUULLIICC JJUUMMPP 1 2 56
Type of Jump Hydraulic Jump Expression of Jump Hydraulic jumps on horizontal floor are several distinct types. These types can be conveniently classified according to the Froude number F1 of the incoming flow as follows Undular Jump (1<Fr1<1.7) Slight undulation Two conjugate depths are close Transition is not abrupt - slightly ruffled water surface Characterized by: Weak Jump (1.7<Fr1<2.5) Eddies and rollers are formed on the surface Energy loss is small The ratio of final depth to initial depth is between 2.0 and 3.1 Characterized by: Oscillating Jump (2.5<Fr1<4.5) Jet oscillates from top to bottom - generating surface waves that persist beyond the end of the jump Ratio final depth to initial depth is between 3.1 to 5.0 To prevent destructive effects this type of jump should be avoided Characterized by: Steady Jump @ Stable Jump (4.5<Fr1<9) Position of jump fixed regardless of downstream conditions Good dissipation of energy (favored type of jump) Considerable rise in downstream water level Ratio of final to initial depth is between 5.9 and 12.0 Characterized by: Strong Jump (Fr1>9) Ratio of final to initial depth is over 12 and may exceed 20 Ability of jump to dissipate energy in massive Jump becomes increasingly rough Fr1 shuold not be allowed to exceed 12 Characterized by: 57
engineers design hydraulic jumps to reduce damage to structures and the streambed proper design can result in a 60-70% energy dissipation minimizes erosion and scouring due to high velocities dams, weirs and other hydraulic structures mix chemicals in water purification aerate water in water purification remove air pockets from water to prevent air locking in supply lines recover pressure head and to raise water levels downstream of canal maintain a high water level for irrigation or other water-distribution purposes at a control section critical flow conditions take place and this fixes a unique relationship between depth and discharge in the vicinity (e.g. sluice gate and weir) subcritical flows are controlled from downstream (e.g. reservoir) while supercritical flows have upstream control (e.g. spillway and weir) to control influences both the flows upstream and downstream of the control section; i.e. downstream flow control and upstream flow control, respectively 1. to dissipate energy 2. for chemical diffusion 3. for aeration 4. to increase flow level 5. to reduce uplift pressure Applications of hydraulic jump travelling down rivers/rapids kayaking and canoeing: playboat / surf hydraulic jumps For recreational applications characterized by a 4.5<Fr1<9 position of jump is fixed provides the most effective energy dissipation protects the structures and streambed by reducing velocity energy dissipation ranges from 45-70% Conclusion: An ideal design for energy dissipation would result in a "stable jump @ steady jump: Note 58
Energy loss, efficiency, power loss Energy Loss Although momentum is conserved throughout the hydraulic jump, the energy is not. There is an initial loss of energy when the flow jumps from supercritical to subcritical depths. The resulting loss of energy is equal to the change in specific energy across the jump and is given by the equation for ΔE below. The power dissipation Efficiency A more rational definition of the efficiency of hydraulic jump on a horizontal floor in a rectangular channel is given by the ratio of the energy actually dissipated in the jump to that required to be dissipated. Power Loss 59
Example 5.1 Find, E? Formula, E=? Given, Q=10m3/s y=3m b=5m V=? , = q/y Find, yc? Vc? Formula: yc = ? and Vc =? Given, Q=15m3/s b=5m q= Q/b ? Find, E? yc? Vc? Emin? Formula:E =? , yc =? , Vc =? and Emin =? Given, Q=15m3/s y=1.2m b=8m q= Q/b ? V = Q/A = Q/by ? Q1: Find the specific energy of flowing water through a rectangular channel of width 5 m when the discharge is 10 m3/s and depth of water is 3 m. Q2: Find the critical depth and critical velocity of the water flowing through a rectangular channel of width 5 m when discharge is 15 m3/s. Q3: The discharge of water through a rectangular channel of width 8 m is 15 m3/s when depth of flow of water is 1.2 m. Calculate: a. Specific energy of the flowing water b. Critical depth and critical velocity c. Value of minimum specific energy 60
Example 5.2 tips: for the given value of specific energy, the discharge will be maximum, when depth of flow is critical Find, yc = y =? Formula, yc =? Given, b=3m E=3m Qmax = area x velocity = (bxyc) x Vc Find, y1? y2? Formula: ? Given, b=5m E=4m Q= 20m3/s Find, y2 =? Formula: ? Given, Q=16m3/s y1=0.5m b=4m q= Q/b ? Q1: The specific energy for a 3 m wide channel is to be 3 kg-m/kg. What would be the maximum possible discharge? Q2: The specific energy for a 5 m wide rectangular channel is to be 4 Nm/N. If the rate of flow of water through the channel is 20 m3/s, determine the alternate depth of flow. Q3: The depth of flow of water, at a certain section of a rectangular channel of 4 m wide, is 0.5 m. This discharge through the channel is 16 m3/s. If a hydraulic jump takes place on the downstream side, find the depth of flow after the jump. 61
Example 5.3 tips: hydraulic jump will occur if the depth of flow on the upstream side is less than the critical depth on upstream side or if the Froude number on the upstream side is more than one. Find, yjump =? and energy loss =? Formula, ? Given, b=2m y1=0.3m Q=1.5m3/s Given, v1=10m/s y1=1m Given, v1=6m/s y1=0.4m b=8m Q1: The depth of flow of water, at a certain section of a rectangular channel of 2 m wide, is 0.3 m. The discharge through the channel is 1.5 m3/s. Determine whether a hydraulic jump will occur, and if so, find its height and loss of energy per kg of water. Q2: A sluice gate discharge water into a horizontal rectangular channel with a velocity of 10 m/s and depth of flow of 1 m. Determine the depth of flow after the jump and consequent loss in total head. Q3: A sluice gate discharge water into a horizontal rectangular channel with a velocity of 6 m/s and depth of flow is 0.4 m. the width of the channel is 8 m. Determine whether a hydraulic jump will occur, and if so, find its height and loss of energy per kg of water. Also determine the power lost in the hydraulic jump. 62
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