Question

A water intake at a storage reservoir (see Fig. \(16-28\) ) has a cross- sectional area of \(7.60 \mathrm{ft}^{2}\). The water flows in at a speed of \(1.33 \mathrm{ft} / \mathrm{s}\). At the generator building \(572 \mathrm{ft}\) below the intake point, the water flows out at \(31.0 \mathrm{ft} / \mathrm{s} .(a)\) Find the difference in pressure, in \(\mathrm{lb} / \mathrm{in} .^{2}\), between inlet and outlet. (b) Find the area of the outlet pipe. The weight density of water is \(62.4 \mathrm{lb} / \mathrm{ft}^{3}\)

Short answer

a) The difference in pressure between the inlet and outlet is approximately 213 psi. b)The area of the outlet pipe is approximately \(0.328 \, \mathrm{ft}^{2}\).

Step-by-step solution

Understanding Bernoulli's Equation

Apply Bernoulli's equation: \( P_1 + \frac{1}{2} \rho v_1^2 + \rho gh_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho gh_2 \), where \(P_{1,2}\) denotes pressure, \(\rho\) is the fluid density, \(v_{1,2}\) the speed, and \(h_{1,2}\) the height at points 1(inlet) and 2(outlet). Here, the system is exposed to atmosphere, hence \(P_1\) and \(P_2\) can be taken as 0.

Calculate the difference in pressure

Substitute the given values into Bernoulli's equation and solve for the pressure difference: \( P_2 - P_1 = \frac{1}{2} \rho v_1^2 + \rho gh_1 - \frac{1}{2} \rho v_2^2 - \rho gh_2 \). Using the fact that \(P_1\) and \(P_2\) are zero and that the unit conversion from \(\mathrm{ft} \cdot \mathrm{lb} / \mathrm{s}^2\) to \(\mathrm{lb} / \mathrm{in}^2 (psi)\) is 1 psi = \(2.03602 \cdot 10^{-9} \mathrm{ft} \cdot \mathrm{lb} / \mathrm{s}^2\), plug in the given values to calculate the pressure difference.

Determine area of the outlet pipe

Use the continuity equation \(\rho A_1v_1 = \rho A_2v_2\) where \(A_{1,2}\) are areas and \(v_{1,2}\) velocities at points 1 and 2. Since the water is incompressible, the densities \(\rho\) cancel out. Rearrange for \(A_2\), the area of the outlet pipe: \(A_2 = A_1\frac{v_1}{v_2}\). Substitute the given values and solve for \(A_2\).

Key concepts

Fluid Dynamics

When we explore the behavior of liquids and gases in motion, we delve into the field of fluid dynamics. This branch of physics is essential in understanding how fluids behave under various forces and in different environments. Fluid motion is governed by various principles and equations, one of which is the renowned Bernoulli's equation. This equation illustrates the principle of conservation of energy for flowing fluids, indicating that as a fluid moves through a pipe or any conduit, the sum of its potential energy, kinetic energy, and pressure remains constant. The practical applications of fluid dynamics extend from the design of water supply systems and aircraft to predicting weather patterns and understanding the flow of blood in the human body.

In the given exercise, the water flowing from an intake to a generator building exhibits characteristics studied in fluid dynamics. The water's speed and the height difference between the intake and outlet points are factors central to determining the pressure difference and the outlet pipe area - all of which are assessed using the principles of fluid dynamics.

Pressure Difference

The concept of pressure difference is essential in both fluid dynamics and numerous real-world applications. It refers to the variance in pressure between two points within a fluid system. This difference in pressure is what drives the fluid from one location to another and is often in response to forces such as gravity, pumps, or atmospheric changes. In the context of our exercise, the pressure difference between the intake point and the generator building is what propels the water through the system.

To calculate the pressure difference using Bernoulli's equation, you can visualize the fluid having potential energy due to its elevation at the water intake and kinetic energy due to its motion. As the water descends to the generator building, some potential energy is converted into kinetic energy, which changes the pressure. Understanding how to calculate this pressure difference is vital for engineers when designing systems that rely on fluid flow, such as plumbing and irrigation systems.

Continuity Equation

The continuity equation is a fundamental concept in fluid dynamics, grounding itself in the principle of mass conservation. It states that within a closed system, the mass flow rate must remain constant. For incompressible fluids, like the water in our exercise, this means the product of cross-sectional area and velocity at any two points along the stream is constant. Mathematically, this is expressed as \( A_1v_1 = A_2v_2 \), where \( A \) represents the area and \( v \) the velocity at points 1 and 2 respectively.

In practice, the continuity equation allows us to solve problems like finding the outlet area of a pipe given the inlet area and the velocity of fluid at both ends. By understanding this relationship, one can infer various aspects of fluid behavior in conduits, such as how changes in pipe diameter affect flow velocity and pressure. This equation is crucial in the design of a myriad of systems, including water distribution networks, air ventilation systems, and even cardiovascular analyses in the medical field.