Explanation Given information: The width of the rectangular channel is b = 12 ft . The depth of the water flow is d 1 = 4 ft . The increased depth of water flow is d 2 = 5 ft . Calculation: Sketch the rectangular channel as shown in Figure 1. Refer to Figure 1. Calculate the area ( A ) as shown below. A = b d Here, d is the depth of the channel. Calculate the length of the slipstream ( Δ l ) as shown below. Δ l = v Δ t . Here, v is the velocity of the air and Δ t is the time. Calculate the flow rate ( Q ) as shown below. Q = A v Substitute bd for A . Q = b d v Calculate the mass rate of flow ( d m d t ) as shown below. mass = density × volume = density × area × length Δ m = ρ A Δ l Substitute v Δ t for Δ l and b d for A . Δ m = ρ b d v Δ t Δ m Δ t = ρ b d v 1 ρ d m d t = b d v Substitute Q for bdv . Q = 1 ρ d m d t For the condition of continuity of flow Q 1 = Q 2 = Q . Substitute bdv for Q . Q 1 = Q 2 = b d v Q 1 = b d 1 v 1 v 1 = Q 1 b d 1 Q 2 = b d 2 v 2 v 2 = Q 2 b d 2 Sketch the pressure acting on the channel as shown in Figure 2. Refer to Figure 2. Calculate the pressure ( p 1 and p 2 ) as shown below. p 1 = γ d 1 Here, γ is the unit weight of water. and p 2 = γ d 2 Sketch the force acting on the channel as shown in Figure 3. Refer to Figure 3. Calculate the resultant pressure forces ( F 1 and F 2 ) as shown below. Resultant pressure force F 1 . F 1 = 1 2 p 1 b d 1 Substitute γ d 1 for p 1 . F 1 = 1 2 γ d 1 b d 1 = 1 2 γ b d 1 2 Resultant pressure force F 2 . F 2 = 1 2 p 2 b d 2 Substitute γ d 2 for p 2 . F 2 = 1 2 γ d 2 b d 2 = 1 2 γ b d 2 2 The unit weight of water γ = ρ g . Sketch the Free Body Diagram of the water flowing as shown in Figure 4. Refer to Figure 4

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Chapter 14.3, Problem 14.85P

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Find the rate of flow in the channel.

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Chapter 14.3, Problem 14.84P

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Find the rate of flow in the channel.

Solution Summary: The author explains how to find the rate of flow in the channel. The width of the rectangular channel is b=12ft.

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