Calculating Height Difference Using Bernoulli's Equation

How can we calculate the height difference between the water level in the piezometric tube and the surface of the reservoir?

Given a reservoir discharging at a rate of 1.1 m3/s through a circular pipe with a diameter of 1,180 mm, and no losses, what is the height difference, to 3 significant figures?

Step by Step Solution:

1. Calculate the velocity of the water flowing through the pipe.

2. Calculate the pressure difference between the water levels in the piezometric tube and the reservoir.

3. Apply Bernoulli's equation to find the height difference.

When calculating the height difference using Bernoulli's equation, we need to consider various factors. The first step is to determine the velocity of the water flowing through the pipe. Using the given flow rate and pipe diameter, we can calculate the velocity using the formula v = (4 * Q) / (π * d²).

Next, since there are no losses, the pressure difference between the water levels in the piezometric tube and the reservoir is zero, as both points are at atmospheric pressure. This simplifies the Bernoulli's equation to focus on kinetic and potential energy terms.

By rearranging the Bernoulli's equation and plugging in the calculated velocity, we can find the height difference h₂ between the water level in the piezometric tube and the surface of the reservoir. This height difference accounts for the energy changes in the fluid flow.

Understanding how to apply Bernoulli's equation in fluid dynamics helps in analyzing different flow scenarios and calculating important parameters like pressure, velocity, and height differences. This method provides a systematic approach to solving fluid mechanics problems.