Suppose that the trapezoidal channel of Fig. P10.17 contains...

Suppose that the trapezoidal channel of Fig. P10.17 contains sand and silt that we wish not to erode. According to an empirical correlation by A. Shields in 1936 , the average wall shear stress $\tau_{\mathrm{cni}}$ required to erode sand particles of diameter $d_{p}$ is approximated by $$\frac{\tau_{\mathrm{crit}}}{\left(\rho_{s}-\rho\right) g d_{p}} \approx 0.5$$ where $\rho_{s}=2400 \mathrm{kg} / \mathrm{m}^{3}$ is the density of sand. If the slope of the channel in Fig. $\mathrm{P} 10.17$ is 1: 900 and $n \approx 0.014$, determine the maximum water depth to keep from eroding particles of $1-\mathrm{mm}$ diameter.

Key Concepts

Manning’s Equation

Manning’s equation is a widely used formula in open channel hydraulics that relates the flow velocity to the channel geometry, roughness, hydraulic radius, and slope. It is instrumental in estimating the discharge and determining the water depth and shear stress distribution within the channel, which are essential for evaluating the risk of sediment erosion.

Open Channel Flow

Open channel flow refers to fluid flow with a free surface open to the atmosphere, commonly encountered in natural stream channels and engineered waterways. The analysis of such flow involves determining hydraulic parameters like velocity, shear stress, and water depth, which are crucial for designing channels to avoid undesirable erosion or sediment deposition.

Critical Shear Stress

Critical shear stress is the minimum shear force per unit area that must be exerted by moving water to dislodge sediment particles from the bed. It is a key parameter in determining whether the flow conditions will lead to sediment erosion and is directly influenced by the particle size, density difference between sediment and water, and gravitational forces.

Shields Criterion

The Shields criterion is an empirical method used in sediment transport to estimate the critical shear stress needed to initiate particle motion. It relates the shear stress on a bed to the weight and size of the sediment particles, providing a non-dimensional parameter that can be compared with experimental values to assess the likelihood of erosion.

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