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9. A conical funnel of half-angle 0 is filled to some initial height họ with a fluid. At time t = 0 the drain plug is removed and the fluid drains through a hole (with area a) in the bottom of the funnel. Assume that the drain hole area is small compared to the fluid surface area.

9. A conical funnel of half-angle 0 is filled to some initial height họ with a fluid. At time t = 0 the drain plug is removed and the fluid drains through a hole (with area a) in the bottom of the funnel. Assume that the drain hole area is small compared to the fluid surface area.

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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Fluid mechanics problem

### Problem Statement 9. A conical funnel of half-angle θ is filled to some initial height h₀ with a fluid. At time t = 0, the drain plug is removed and the fluid drains through a hole (with area a) in the bottom of the funnel. Assume that the drain hole area is small compared to the fluid surface area. ### Questions **a. Fill in the following table** The dependent variable in this problem is the drain time, t_D: | Independent Variable | If the Independent Variable (with all other parameters held constant) | Your Prediction From Physical Intuition: Does t_D increase (↑) or decrease (↓) ? | Analysis Prediction: | Does Your Prediction Match the Analysis Prediction: | |---------------------------------|:----------------------------------------------------------------------:|:--------------------------------------------------------------------------------:|:----------------------:|:--------------------------------------------:| | Funnel half-angle θ | ↑ | | | | | Initial fluid height h₀ | ↑ | | | | | Drain hole area a | ↑ | | | | | Gravitational Acceleration g | ↑ | | | | **b. Find an expression for the fluid height h(t) as a function of time.** *Hint: The way in which this problem differs from the draining tank in class is that the tank area is no longer constant with height. You must find an equation for the tank area A = f(h).* **c. Find an expression for the funnel drain time.** ### Diagram The diagram illustrates a conical funnel with the following features: - \( \theta \): The half-angle of the funnel. - \( h \): The height of the fluid at any given time. - \( h = 0 \): The bottom level of the funnel. - \( h_0 \): The initial height of the fluid. ``` L / \ / \ \ θ \_____/ / h ``` Line Diagrams: - The left and right slants of the funnel are depicted. - A horizontal line marks \( h_0 \), the initial height. - \( h \) represents the height of the fluid column at an arbitrary time. - \( \theta \), the half-angle of the funnel, is denoted in the figure.
Transcribed Image Text:### Problem Statement 9. A conical funnel of half-angle θ is filled to some initial height h₀ with a fluid. At time t = 0, the drain plug is removed and the fluid drains through a hole (with area a) in the bottom of the funnel. Assume that the drain hole area is small compared to the fluid surface area. ### Questions **a. Fill in the following table** The dependent variable in this problem is the drain time, t_D: | Independent Variable | If the Independent Variable (with all other parameters held constant) | Your Prediction From Physical Intuition: Does t_D increase (↑) or decrease (↓) ? | Analysis Prediction: | Does Your Prediction Match the Analysis Prediction: | |---------------------------------|:----------------------------------------------------------------------:|:--------------------------------------------------------------------------------:|:----------------------:|:--------------------------------------------:| | Funnel half-angle θ | ↑ | | | | | Initial fluid height h₀ | ↑ | | | | | Drain hole area a | ↑ | | | | | Gravitational Acceleration g | ↑ | | | | **b. Find an expression for the fluid height h(t) as a function of time.** *Hint: The way in which this problem differs from the draining tank in class is that the tank area is no longer constant with height. You must find an equation for the tank area A = f(h).* **c. Find an expression for the funnel drain time.** ### Diagram The diagram illustrates a conical funnel with the following features: - \( \theta \): The half-angle of the funnel. - \( h \): The height of the fluid at any given time. - \( h = 0 \): The bottom level of the funnel. - \( h_0 \): The initial height of the fluid. ``` L / \ / \ \ θ \_____/ / h ``` Line Diagrams: - The left and right slants of the funnel are depicted. - A horizontal line marks \( h_0 \), the initial height. - \( h \) represents the height of the fluid column at an arbitrary time. - \( \theta \), the half-angle of the funnel, is denoted in the figure.
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