What is the gauge pressure at a point in one of the narrower branches of a split water line based on the given data?
The gauge pressure at point B in one of the narrower branches of the split water line is 11.5 KPa. This value is determined using the principle of continuity and Bernoulli's equation to analyze the fluid dynamics in pipes of different diameters and speeds.
Understanding the Principles of Fluid Dynamics
Fluid dynamics is the study of fluids in motion and the forces and pressures associated with their movement. In the scenario presented, the flow of water in pipes of varying diameters leads to changes in velocity and pressure at different points along the pipeline. To understand the gauge pressure at point B, we need to delve into two fundamental principles: the principle of continuity and Bernoulli's equation.
The Principle of Continuity
The principle of continuity states that for an incompressible fluid flowing through a pipe, the mass flow rate remains constant at different points along the pipeline. Mathematically, this can be expressed as A1V1 = A2V2, where A represents the cross-sectional area of the pipe and V represents the velocity of the fluid. In the given scenario, the larger diameter pipe splits into two narrower branches, resulting in an increase in fluid speed in the smaller pipes.
Bernoulli's Equation and Gauge Pressure
Bernoulli's equation is a fundamental principle in fluid dynamics that relates the pressure, velocity, and elevation of a moving fluid. It states that the total mechanical energy of the fluid remains constant along a streamline. By applying Bernoulli's equation to the scenario with pipes of different diameters and speeds, we can analyze the changes in pressure between the larger and narrower branches.
Calculation and Analysis
From the continuity equation, the velocity of water at point B in the narrower branch is calculated to be 9 m/s. By applying Bernoulli's equation and keeping the terms constant, we find that the gauge pressure at point B is 11.5 KPa. This result indicates that the gauge pressure decreases in the narrower branches compared to the initial pressure in the larger pipe due to the increase in fluid speed.
Implications and Conclusion
The analysis of fluid dynamics in split pipes highlights the interplay between pipe diameter, fluid velocity, and pressure distribution along the pipeline. Understanding the principles of continuity and Bernoulli's equation allows us to predict and calculate changes in pressure and velocity in fluid systems with varying geometries. In practical applications, this knowledge is essential for designing efficient and reliable fluid transport systems.