How Do You Calculate the Flow Rate from a Rectangular Opening in a Tank?

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To calculate the flow rate from a rectangular opening in a tank, one must apply Bernoulli's equation to determine the velocity of water at various depths, specifically H1 and H2. The challenge lies in integrating to find the average velocity across the opening, as the water does not exit uniformly. The discussion highlights the need to establish a method for calculating velocity at any arbitrary depth y within the tank. Assistance is sought in setting up the necessary integral for this calculation. Understanding these principles is crucial for accurately determining the volume of water that flows out per second.
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Homework Statement



A rectangular opening is cut into the side of a large open-topped water tank. The opening has width W and height H2 - H1, where H1 is the distance from the top of the tank to the top of the opening, and H2 is the distance from the top of the tank to the bottom of the opening. Determine the volume V of water that emerges from the opening per second.

You may assume that the surface area of the tank is extremely large compared to the area of the opening, but you should not assume that the water emerges from the opening with a single, uniform velocity.

Homework Equations



Bernoulli's equation

The Attempt at a Solution



So I first tried to get the velocity at H1 and H2 using Bernoulli's equation. Then I assume you have to integrate to account for the rate in between, but I'm having trouble setting up the integral. Help is appreciated, thanks.
 
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Can you find the velocity at any arbitrary depth y?
 

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