# Bernoulli's Equation (Water Outflow Speed of Tank)

• Von Neumann
In summary, the conversation discusses the problem of determining the speed of liquid emerging from a small hole at the base of a sealed and pressurized tank. Using Bernoulli's Equation and making certain assumptions, the solution is found to be \sqrt{\frac{2(P_{a}+ρgh)}{ρ}}. However, the book's solution is \sqrt{\frac{2P_{a}}{ρ+gh}}, indicating an error in the text. Upon further examination, it is determined that the units in the denominator do not match, making the text incorrect.
Von Neumann
Problem:

Suppose the top of a tank is sealed and pressurized to twice atmospheric pressure. What is the speed of the liquid emerging from a small hole at the base of the tank?

My Solution:

Using Bernoulli's Equation, and assuming v_top ≈ 0,

P_top + 1/2*ρ*v_top^2 + ρgh = P_hole + 1/2*ρ*v_hole^2 + ρg*0

2*P_a + ρgh = P_a + 1/2*ρ*v_hole^2

Solving for v_hole I get,

$\sqrt{\frac{2(P_{a}+ρgh)}{ρ}}$

While my book has,

$\sqrt{\frac{2P_{a}}{ρ+gh}}$

Anyone know where I went wrong? Or, less likely, if my text is incorrect? Thank you in advance.

The text is incorrect. The sum in the denominator is impossible, the units of the addends do not match.

Oh wow, I'm foolish for not thinking of that myself. My units come out to be the correct units of velocity.

## 1. What is Bernoulli's Equation?

Bernoulli's Equation is a fundamental equation in fluid dynamics that relates the pressure, velocity, and elevation of a fluid in motion. It states that the sum of the pressure, kinetic energy per unit volume, and potential energy per unit volume is constant along a streamline.

## 2. How is Bernoulli's Equation used to calculate the water outflow speed of a tank?

In the context of a tank with an opening at the bottom, Bernoulli's Equation can be used to determine the velocity of water flowing out of the tank. The equation takes into account the pressure at the surface of the water in the tank, the height of the water level, and the acceleration due to gravity.

## 3. What are the assumptions made when using Bernoulli's Equation for water outflow speed of a tank?

The most common assumptions are that the fluid is incompressible, the flow is steady, there is no friction or viscosity, and the cross-sectional area of the opening is constant. Additionally, the equation assumes that the fluid is flowing in a streamlined manner, with no turbulence or eddies.

## 4. How does the height of the water level in the tank affect the outflow speed?

According to Bernoulli's Equation, the velocity of the water flowing out of the tank is directly proportional to the square root of the height of the water level. This means that the higher the water level, the faster the water will flow out of the tank.

## 5. Can Bernoulli's Equation be used for any type of fluid?

Bernoulli's Equation is a fundamental law of fluid dynamics and can be applied to any fluid, as long as the assumptions mentioned in question 3 are satisfied. However, it is important to note that the equation may need to be modified for different types of fluids, such as compressible fluids or fluids with high viscosity.

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