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Chapter 9: Problem 26

Determine the critical depth for a flow of \(50 \mathrm{~m}^{3} / \mathrm{s}\) in a trapezoidal channel with bottom width of \(4 \mathrm{~m}\) and side slopes of \(1.5: 1(\mathrm{H}: \mathrm{V})\). If the actual depth of flow is \(3 \mathrm{~m},\) calculate the Froude number and state whether the flow is subcritical or supercritical.

Short Answer

Expert verified
The critical depth needs to be solved numerically. The Froude number is 0.52, indicating subcritical flow.

Step by step solution

01

Define Known Values

We know the following values: \(Q = 50 \, \text{m}^3/\text{s}\) (discharge), bottom width \(b = 4 \, \text{m}\), side slopes \(m = 1.5\), and the actual depth of flow \(y = 3 \, \text{m}\).
02

Determine the Critical Depth

The critical depth \(y_c\) in a trapezoidal channel can be found using \(Q^2 = g A_c^3 / T_c\), where \(A_c\) is the area and \(T_c\) is the top width of the critical flow: \[ A_c = y_c(b + my_c) \text{ and } T_c = b + 2my_c \] Substitute these into the equation and solve for \(y_c\) using trial and error or numerical methods given the complexity.
03

Calculate the Cross-Sectional Area and Top Width for Actual Depth

Calculate the cross-sectional area \(A\) and the top width \(T\) for the actual depth of flow: \[ A = y(b + my) = 3(4 + 1.5 \times 3) = 3 \times 8.5 = 25.5 \, \text{m}^2 \] \[ T = b + 2my = 4 + 2 \times 1.5 \times 3 = 13 \, \text{m} \]
04

Calculate the Froude Number

The Froude number is given by \(F_r = \frac{Q}{A \sqrt{g y}}\). Using values for actual depth: \[ F_r = \frac{50}{25.5 \times \sqrt{9.81 \times 3}} \approx 0.52 \]
05

Determine Subcritical or Supercritical Flow

The flow is subcritical if \(F_r < 1\) and supercritical if \(F_r > 1\). Since \(F_r \approx 0.52\), the flow is subcritical.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Critical Depth
Critical depth is a crucial concept in fluid mechanics, especially when analyzing open channel flow. It represents the depth of flow in a channel at which specific energy is minimized for a given discharge.
  • If the flow is at critical depth, the flow reaches an equilibrium where gravitational forces balance the inertial forces perfectly.
  • At this point, the flow is neither accelerating nor decelerating, making it the dividing line between different flow regimes.
To find the critical depth in a trapezoidal channel, we calculate it using the equation that balances discharge with gravitational and geometric terms. This requires using the cross-sectional area and top width of the flow. Due to the complexity, solving for critical depth often involves a trial and error approach or numerical methods.
Froude Number
The Froude number is a dimensionless number that is key in analyzing the flow dynamics in an open channel. It is used to compare inertial forces to gravitational forces.
  • Mathematically, it is defined as \(F_r = \frac{Q}{A \sqrt{g y}}\), where \(Q\) is discharge, \(A\) is cross-sectional area, \(g\) is acceleration due to gravity, and \(y\) is flow depth.
  • A Froude number less than 1 indicates subcritical flow, which is slow and typically characterized by a calm water surface.
  • A Froude number greater than 1 indicates supercritical flow, known for its rapid, wavy surface.
In the exercise, the Froude number of \(0.52\) confirms a subcritical flow.
Subcritical and Supercritical Flow
Subcritical and supercritical flows explain the behavior of water in relation to its energy state.
  • Subcritical flow occurs when the Froude number is less than 1. The flow has more potential energy compared to kinetic energy, resulting in a slower movement.
  • In subcritical flow, waves can travel upstream, and the flow is usually tranquil and deep.
  • Supercritical flow, on the other hand, has a Froude number greater than 1. It is fast, with more kinetic energy at play.
  • Waves cannot travel upstream in supercritical conditions, resulting in a turbulent, shallow flow appearance.
Understanding these flow types helps predict how water will move and react in different channel geometries and conditions.
Discharge in Trapezoidal Channel
Discharge in a trapezoidal channel refers to the volume of water flowing through the channel per unit time. It is affected by the geometry of the channel as well as the flow characteristics.
  • The discharge \(Q\) is calculated using the formula \(Q = A \cdot V\), where \(A\) is the cross-sectional area and \(V\) is the flow velocity.
  • In a trapezoidal channel, the area \(A\) and top width \(T\) depend on the flow depth, bottom width, and side slopes, making the calculation more complex.
  • For precise results, knowing the exact dimensions and using the right equations is key.
Discharge measurements are fundamental for designing and analyzing channels to ensure efficient water management and infrastructure reliability.

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Most popular questions from this chapter

A flume with a triangular cross section and side slopes of \(2: 1(\mathrm{H}: \mathrm{V})\) contains water flowing at \(0.45 \mathrm{~m}^{3} / \mathrm{s}\) at a depth of \(20 \mathrm{~cm}\). Verify that the flow is supercritical and calculate the conjugate depth.

Show that Manning's \(n\) can be expressed in terms of the Darcy friction factor, \(f,\) by the following relation: $$ n=\frac{f^{\frac{1}{2}} R^{\frac{1}{6}}}{8.86} $$ where \(R\) is the hydraulic radius of the flow. Does this relationship conclusively show that \(n\) is a function of the flow depth? Explain.

Stages are measured by two recording gauges \(100 \mathrm{~m}\) apart along a constructed water supply channel. The channel has a bottom width of \(5 \mathrm{~m}\) and side slopes of \(3: 1(\mathrm{H}: \mathrm{V})\) The bottom elevations of the channel at the upstream and downstream gauge locations are \(24.01 \mathrm{~m}\) and \(23.99 \mathrm{~m}\), respectively. At a particular instance, the upstream and downstream stages are \(25.01 \mathrm{~m}\) and \(24.95 \mathrm{~m}\), respectively, and the flow is estimated as \(15 \pm 2 \mathrm{~m}^{3} / \mathrm{s}\). (a) Derive an expression for Manning's \(n\) as a function of the estimated flow rate. (b) Estimate Manning's \(n\) and the roughness height in the channel between the two measurement stations. (c) Quantitatively assess the sensitivity of the flow rate to the channel roughness.

A concrete-lined rectangular channel is \(8-\mathrm{m}\) wide and \(4-\mathrm{m}\) deep and has a longitudinal slope of \(0.2 \%\) and an estimated Manning's \(n\) of \(0.013 .\) A hydraulic structure controls the flow in the channel such that the depth of flow at the structure is \(3 \mathrm{~m}\) when the flow rate in the channel is \(24 \mathrm{~m}^{3} / \mathrm{s}\). Urban developers propose a localized contraction/expansion in the channel \(100 \mathrm{~m}\) upstream of the control structure to accommodate a pedestrian walkway. Taking eddy losses into account but neglecting friction losses, estimate the maximum contraction that should be allowed for the walkway. Based on your result, assess whether it is reasonable to neglect friction losses in this case.

An open channel has a trapezoidal cross section with a bottom width of \(3 \mathrm{~m}\) and side slopes of \(2: 1(\mathrm{H}: \mathrm{V})\). If the depth of flow is \(2 \mathrm{~m}\) and the average velocity in the channel is \(1.2 \mathrm{~m} / \mathrm{s},\) calculate the discharge in the channel.
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