Critical Flow Computation Transitions Problems Transition Problem (By changing width) a. Compute the contraction for width of the channel for producing critical flow b. The depth and change in water level produced by i. A smooth contraction in width to 1.5 m ii. A smooth contraction in width to 0.75 m iii. A smooth expansion in width of 3.5 m Assume Q does not change, neglect head loss and α = 1.0 Transition Problem (Combinations of transitions) Water flows in a 6 m wide rectangular channel at a depth of 2 m and a velocity of 2 m/s. the channel is contracted to a width of 3 m. How much the channel bottom is to be simultaneously raised or lowered for the flow to be possible as specified? Neglect energy loss and take α = 1.0 Transition Problem The flow is taking part a section shown in Figure. The step height is 4.57 cm. The upstream depth 45.7 cm. The water surface drops by 7.63 cm from its original level on the step. Determine the discharge. Uniform Flow Introduction a flow is said to be uniform if its properties remain constant with respect to distance the depth of flow, flow velocity , discharge remain constant along the channel therefore, uniform flow is possible only in prismatic channels the slope of the energy line Sf , slope of the water surface Sw and bottom slope S0 will all be equal to each other Uniform Flow Formulas CHEZY EQUATION/ CHEZY FORMULA Uniform Flow Formulas CHEZY EQUATION/ CHEZY FORMULA there is no acceleration in uniform flow. By applying the momentum equation to a control volume encompassing Sections 1 and 2, distance L apart as P1 − W sin θ − Ff − P2 = M2 − M1 ………………………………………..(1) where P1 and P2 are the pressure forces and M1 and M2 are the momentum fluxes at Sections 1 and 2 respectively W = weight to fluid in the control volume and Ff = shear force at the boundary Uniform Flow Formulas CHEZY EQUATION/ CHEZY FORMULA Since the fl ow is uniform, P1 = P2 and M1 = M2 Also, W = γ AL and Ff = τ0 PL where τ0 = average shear stress on the wetted perimeter of length P and γ = unit weight of water. Replacing sinθ by S0 (= bottom slope Eq. (1) can be written as γ AL S0 = τ0 PL …….. (2) or τ0 = γ A/P S0 = γ R S0 ……….. (3) where R = A/P is defined as the hydraulic radius Uniform Flow Formulas CHEZY EQUATION/ CHEZY FORMULA Expressing the average shear stress τ0 as τ0 = k ρ V ^2 , where k = a coefficient which depends on the nature of the surface and flow parameters, Eq. (3) can be written as K ρ V ^2 = γ R S0 leading to V = C (R S)^(1/2) ……………………(4) where C = (γ /ρ * 1 /k ) ^ ½ = a coefficient which depends on the nature of the surface and the flow Uniform Flow Formulas CHEZY EQUATION/ CHEZY FORMULA Equation 4 is known as the Chezy formula after the French engineer Antoine Chezy, who is credited with developing this basic simple relationship in 1769 The dimensions of C are [L^1/2 T ^(−1)] and it can be made dimensionless by dividing it by g The coefficient C is known as the Chezy coefficient Uniform Flow Formulas DARCY–WEISBACH FRICTION FACTOR FORMULA Pipe Flow: A surface can be termed hydraulically smooth, rough or in transition depending on the relative thickness of the roughness magnitude to the thickness of the laminar sub-layer. The classification is as follows: Uniform Flow Formulas DARCY–WEISBACH FRICTION FACTOR FORMULA For pipe flow, the Darcy–Weisbach equation is where hf = head loss due to friction in a pipe of diameter D and length L; f = Darcy–Weisbach friction factor Uniform Flow Formulas DARCY–WEISBACH FRICTION FACTOR FORMULA For smooth pipes, f is found to be a function of the Reynolds number only For rough turbulent flows, f is a function of the relative roughness (εs/D) and type of roughness and is independent of the Reynolds number In the transition regime, both the Reynolds number and relative roughness play important roles Uniform Flow Formulas DARCY–WEISBACH FRICTION FACTOR FORMULA variation of f in various regimes of flow: It is usual to show the variation of f with Re and εs/D by a threeparameter graph known as the Moody chart Uniform Flow Formulas DARCY–WEISBACH FRICTION FACTOR FORMULA Open Channels: For purposes of flow resistance which essentially takes place in a thin layer adjacent to the wall, an open channel can be considered to be a conduit cut into two The hydraulic radius would then be the appropriate length parameter and prediction of friction factor f can be done by using eqn 1 to 4 Uniform Flow Formulas DARCY–WEISBACH FRICTION FACTOR FORMULA Darcy–weisbach equation can then be written for an open channel flow as which on rearranging gives Uniform Flow Formulas DARCY–WEISBACH FRICTION FACTOR FORMULA Noting that for uniform flow in an open channel hf / L = slope of the energy line =Sf = S0 , it may be seen that If f is involved on both sides of the equations. Simplified empirical forms of, which are accurate enough for all practical purposes, are given by Jain as follows: