Question

At a certain point in a horizontal pipeline, the water's speed is 2.50 m/s and the gauge pressure is 1.80 \(\times\) 10\(^4\) Pa. Find the gauge pressure at a second point in the line if the cross-sectional area at the second point is twice that at the first.

Short answer

The gauge pressure at the second point is approximately 2.03 x 10^4 Pa.

Step-by-step solution

Understand the Problem

We need to find the gauge pressure at a second point in a horizontal pipeline where the cross-sectional area is twice of the first point.

Apply the Continuity Equation

According to the continuity equation, the product of the cross-sectional area and velocity is constant. Let \( A_1 \) and \( v_1 \) be area and velocity at the first point, and \( A_2 \) and \( v_2 \) at the second point. Since \( A_2 = 2A_1 \), by continuity: \[A_1 \cdot v_1 = A_2 \cdot v_2 \]\[A_1 \cdot 2.50 = 2A_1 \cdot v_2 \]Cancel \( A_1 \) from both sides and solve for \( v_2 \):\[2.50 = 2v_2 \Rightarrow v_2 = \frac{2.50}{2} = 1.25 \text{ m/s}\]

Apply Bernoulli's Equation

Bernoulli's equation relates pressures and velocities at two points in a fluid:\[P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2\] where \( \rho \) is the fluid density, \( v_1 = 2.50 \text{ m/s} \), and \( v_2 = 1.25 \text{ m/s} \). Solve for \( P_2 \) (the gauge pressure at the second point):\[1.80 \times 10^4 + \frac{1}{2} \cdot \rho \cdot (2.50)^2 = P_2 + \frac{1}{2} \cdot \rho \cdot (1.25)^2\]Simplify and solve for \( P_2 \):\[P_2 = 1.80 \times 10^4 + \frac{1}{2} \cdot \rho \cdot (2.50)^2 - \frac{1}{2} \cdot \rho \cdot (1.25)^2 \]

Simplify the Equation

Calculate the terms:\[\frac{1}{2} \cdot \rho \cdot (2.50)^2 - \frac{1}{2} \cdot \rho \cdot (1.25)^2 = \frac{1}{2} \cdot \rho \cdot (6.25 - 1.5625)\]\[= \frac{1}{2} \cdot \rho \cdot 4.6875\]Substitute this back into the expression for \( P_2 \):\[P_2 = 1.80 \times 10^4 + \rho \cdot 2.34375\]With \( \rho = 1000 \text{ kg/m}^3 \) (density of water):\[P_2 = 1.80 \times 10^4 + 1000 \cdot 2.34375 = 1.80 \times 10^4 + 2343.75 = 2.034375 \times 10^4 \text{ Pa}\]

Round Off the Answer

The calculated gauge pressure at the second point is \( 2.034375 \times 10^4 \text{ Pa} \), which can be rounded to significant figures traditionally provided by initial conditions; hence:\[P_2 \approx 2.03 \times 10^4 \text{ Pa}\]

Key concepts

Continuity Equation

The Continuity Equation is a fundamental principle in fluid dynamics stating that the mass flow rate of a fluid must remain constant from one cross-section of a pipe to another. This is expressed in the equation as:\[ A_1 \cdot v_1 = A_2 \cdot v_2 \]where \( A \) is the cross-sectional area and \( v \) is the velocity of the fluid. In the context of our exercise, this equation helps us understand how fluid velocity changes as the cross-sectional area of the pipe changes. If the area at the second point is twice that of the first, the fluid's velocity must decrease to half to maintain the same flow rate. By rearranging the continuity equation, we can determine the new velocity, \( v_2 \), which is found to be 1.25 m/s when the initial velocity \( v_1 \) is 2.50 m/s. Utilizing this equation is essential for solving problems where you need to ensure that the flow of fluid stays constant despite changes in pipe dimensions.

fluid dynamics

Fluid dynamics is the study of fluids (liquids and gases) in motion. It revolves around principles like the Continuity Equation and Bernoulli's Equation, both key to understanding how fluids behave in different scenarios. These principles allow us to predict and calculate the behavior of fluids in various applications, from natural phenomena to engineering systems.
In our horizontal pipeline example, fluid dynamics is utilized to model how water flows through the pipe. These calculations give insights into important parameters such as pressure and velocity at different points. It involves applying laws of physics to ensure efficient and accurate predictions of fluid behavior. Fluid dynamics also enables us to solve complex problems in design and analysis of pipelines, ensuring desired performance in systems like water supply networks. Understanding these concepts provides a solid foundation for applying physical laws to practical situations.

gauge pressure

Gauge pressure refers to the pressure of a fluid in a system above the atmospheric pressure. It is different from absolute pressure, as it does not consider atmospheric pressure in its measurement. Instead, it measures the pressure relative to atmospheric pressure.In the context of our pipeline problem, gauge pressure allows us to calculate changes in pressure due to the fluid's velocity as it moves through sections of the pipeline with different cross-sectional areas. By using Bernoulli's Equation, we can relate the gauge pressure at two points in the pipeline. Bernoulli's Equation is:\[ P_1 + \frac{1}{2}\rho v_1^2 = P_2 + \frac{1}{2}\rho v_2^2 \]Here, \( P_1 \) and \( P_2 \) are the gauge pressures at points 1 and 2, while \( \rho \) represents fluid density. Through solving this equation, we find the gauge pressure at the second point. This is important for ensuring safe and efficient operations in systems like pipelines or any fluid-handling equipment.

pipe flow

Pipe flow is the study of fluid movement within a closed conduit, typically a pipe. It is crucial in numerous applications ranging from household plumbing to extensive industrial and municipal water systems.
Pipe flow analysis helps determine how fluids travel through pipes, focusing on parameters like pressure, velocity, and flow rate. Bernoulli's and the Continuity Equations crucially aid in such analyses by providing a mathematical framework for predicting fluid behavior as conditions change along the pipe's length.
In the exercise, understanding how pipe diameter affects flow dynamics is essential. When the cross-sectional area doubles, the velocity decreases to maintain constant flow, leading to changes in pressure as described by Bernoulli's Equation. This emphasizes the interconnectedness of velocity, pressure, and area in pipe flows and underscores the importance of ensuring proper pipe design and operation for efficient fluid transport.