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Problem 5.1 When a liquid flows over a surface, the velocity may vary with position. The equation or graph that describes this variation is known as the velocity profile. A thin layer of liquid, draining from an inclined plane as shown in the figure, will have an analytic velocity profile described by the following equation. u = U₁ = U₁ [ ² (²) - (²] where U is the surface velocity, i.e., velocity of the water at the surface of the layer, u is the velocity of the water at any y-position in the layer, and h is the thickness of the layer. A graph of this velocity profile is shown in the figure. h u(y) Direction of gravity a) If the plane has constant width b into the paper, develop an expression for the volumetric flow rate in the film in terms of h, Uo, and b. [Hint: Use a differential area element dA = b dy and integrate from y = 0 to y = h.] b) Calculate the mass flow rate of liquid down the inclined plane, in kg/s, if the liquid has a specific gravity of 0.87 and b = 0.3 m, h = 5 mm, and U₁ = 0.2 m/s Ans: b) 0.10 kg/s ≤ m ≤ 0.2 kg/s

Problem 5.1 When a liquid flows over a surface, the velocity may vary with position. The equation or graph that describes this variation is known as the velocity profile. A thin layer of liquid, draining from an inclined plane as shown in the figure, will have an analytic velocity profile described by the following equation. u = U₁ = U₁ [ ² (²) - (²] where U is the surface velocity, i.e., velocity of the water at the surface of the layer, u is the velocity of the water at any y-position in the layer, and h is the thickness of the layer. A graph of this velocity profile is shown in the figure. h u(y) Direction of gravity a) If the plane has constant width b into the paper, develop an expression for the volumetric flow rate in the film in terms of h, Uo, and b. [Hint: Use a differential area element dA = b dy and integrate from y = 0 to y = h.] b) Calculate the mass flow rate of liquid down the inclined plane, in kg/s, if the liquid has a specific gravity of 0.87 and b = 0.3 m, h = 5 mm, and U₁ = 0.2 m/s Ans: b) 0.10 kg/s ≤ m ≤ 0.2 kg/s

Elements Of Electromagnetics
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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Problem 5.1 When a liquid flows over a surface, the velocity may vary with position. The equation or graph that describes this variation is known as the velocity profile. A thin layer of liquid, draining from an inclined plane as shown in the figure, will have an analytic velocity profile described by the following equation. u Uo = U. [2) -²] where U is the surface velocity, i.e., velocity of the water at the surface of the layer, u is the velocity of the water at any y-position in the layer, and h is the thickness of the layer. A graph of this velocity profile is shown in the figure. h u(y) Direction of gravity a) If the plane has constant width b into the paper, develop an expression for the volumetric flow rate in the film in terms of h, Uo, and b. [Hint: Use a differential area element dA = b dy and integrate from y = 0 to y = h.] b) Calculate the mass flow rate of liquid down the inclined plane, in kg/s, if the liquid has a specific gravity of 0.87 and b = 0.3 m, h = 5 mm, and U₁ = 0.2 m/s Ans: b) 0.10 kg/s ≤ m ≤ 0.2 kg/s
Transcribed Image Text:Problem 5.1 When a liquid flows over a surface, the velocity may vary with position. The equation or graph that describes this variation is known as the velocity profile. A thin layer of liquid, draining from an inclined plane as shown in the figure, will have an analytic velocity profile described by the following equation. u Uo = U. [2) -²] where U is the surface velocity, i.e., velocity of the water at the surface of the layer, u is the velocity of the water at any y-position in the layer, and h is the thickness of the layer. A graph of this velocity profile is shown in the figure. h u(y) Direction of gravity a) If the plane has constant width b into the paper, develop an expression for the volumetric flow rate in the film in terms of h, Uo, and b. [Hint: Use a differential area element dA = b dy and integrate from y = 0 to y = h.] b) Calculate the mass flow rate of liquid down the inclined plane, in kg/s, if the liquid has a specific gravity of 0.87 and b = 0.3 m, h = 5 mm, and U₁ = 0.2 m/s Ans: b) 0.10 kg/s ≤ m ≤ 0.2 kg/s
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