Lecture 5 OPEN-CHANNEL FLOWFluid Mechanics: Fundamentals and ApplicationsThird Edition in SI Units Yunus A. Cengel, John M. CimbalaMcGraw-Hill, 2014Prof Taha TaherCopyright 2014 McGraw-Hill Education (Asia). Permission required for reproduction or display.

*135 UNIFORM FLOW IN CHANNELSFlow in a channel is called uniform flow if the flow depth (and thus the average flow velocity remains constant. Uniform flow conditions are commonly encountered in practice in long straight runs of channels with constant slope, constant cross section, and constant surface lining. The flow depth in uniform flow is called the normal depth yn, and the average flow velocity is called the uniform-flow velocity V0.In uniform flow, the flow depth y, the average flow velocity V, and the bottom slope S0 remain constant, and the head loss equals the elevation loss, hL = z1 - z2 = SfL = S0L.Chezy coefficientThe Chezy coefficient ranges from about 30 m1/2/s for small channels with rough surfaces to 90 m1/2/s for large channels with smooth surfaces.

*Manning coefficient n: It depends on the roughness of the channel surfaces.Manning equations (GaucklerManning equations)Critical Uniform FlowFlow through an open channel becomes critical flow when the Froude number Fr = 1 and thus the flow speed equals the wave speedFor film flow or flow in a wide rectangular channel with b >> yc,

*Superposition Method for Nonuniform PerimetersThe surface roughness and thus the Manning coefficient for most natural and some human-made channels vary along the wetted perimeter and even along the channel. A river, for example, may have a stony bottom for its regular bed but a surface covered with bushes for its extended floodplain.There are several methods for solving such problems, either by finding an effective Manning coefficient n for the entire channel cross section, or by considering the channel in subsections and applying the superposition principle.For example, a channel cross section can be divided into N subsections, each with its own uniform Manning coefficient and flow rate. When determining the perimeter of a section, only the wetted portion of the boundary for that section is considered, and the imaginary boundaries are ignored. The flow rate through the channel is the sum of the flow rates through all the sections.

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ExampleLink:lecture 5 example 13-3.xlsx*

Example*

Q= 1.5cmsb=1.2mn=0.014s=0.002Q (left)byApRnSQ (right)1.51.20.50.62.20.2730.0140.0020.811.51.20.60.722.40.3000.0140.0021.031.51.20.70.842.60.3230.0140.0021.261.51.20.80.962.80.3430.0140.0021.501.51.20.91.0830.3600.0140.0021.751.51.211.23.20.3750.0140.0021.991.51.21.11.323.40.3880.0140.0022.241.51.21.21.443.60.4000.0140.0022.501.51.21.31.563.80.4110.0140.0022.75

*136 BEST HYDRAULIC CROSS SECTIONSFor a given channel length, the perimeter of the channel is representative of the system cost, and it should be kept to a minimum in order to minimize the size and thus the cost of the system.The best hydraulic cross section for an open channel is a semicircular one since it has the minimum wetted perimeter for a specified cross-sectional area, and thus the minimum flow resistance.A rectangular open channel of width b and flow depth y. For a given cross-sectional area, the highest flow rate occurs when y = b/2.

*The flow rate increases as the flow aspect ratio y/b is increased, reaches a maximum at y/b=0.5, and then starts to decrease.We see the same trend for the hydraulic radius, but the opposite trend for the wetted perimeter p. These results confirm that the best cross section for a given shape is the one with the maximum hydraulic radius, or equivalently, the one with the minimum perimeter.

*Rectangular ChannelsTherefore, a rectangular open channel should be designed such that the liquid height is half the channel width to minimize flow resistance or to maximize the flow rate for a given cross-sectional area. This also minimizes the perimeter and thus the construction costs.

*Trapezoidal ChannelsParameters for a trapezoidal channel.

*The best hydraulic cross section for a circular channel of diameter D can be shown to be y = D/2.The best cross section for trapezoidal channels is half of a hexagon.

Discussion Note that the trapezoidal cross section is better since it has a smaller perimeter (3.37 m versus 3.68 m) and thus lower construction cost. This is why many man-made waterways are trapezoidal in shape. However, the average velocity through the trapezoidal channel is larger since Ac is smaller.

*Many man-made water channels are trapezoidal in shape because of low construction cost andgood performance.

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