A jet of water strikes a splitter and splits into two stream of equal velocity but unequal thickness. All jets have a width w (into the paper). Friction forces of the water stream on the splitter are negligible. Ignore the weight of the splitter.
A jet of water strikes a splitter and splits into two stream of equal velocity but unequal thickness. All jets have a width w (into the paper). Friction forces of the water stream on the splitter are negligible. Ignore the weight of the splitter.
Elements Of Electromagnetics
7th Edition
ISBN:9780190698614
Author:Sadiku, Matthew N. O.
Publisher:Sadiku, Matthew N. O.
ChapterMA: Math Assessment
Section: Chapter Questions
Problem 1.1MA
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![### Water Jet Splitter Analysis
A jet of water strikes a splitter and divides into two streams of equal velocity but differing thickness. Each jet maintains a width denoted by \( w \) (perpendicular to the plane of the paper). For this analysis, friction forces between the water streams and the splitter are considered negligible, and the weight of the splitter is ignored.
**Hint:** Pressure forces can be ignored.
#### Problem Breakdown
**a) Determining the Thickness \( t \) in Terms of \( h \) and \( \alpha \)**
Using the integral mass conservation equation, derive an equation for the thickness \( t \) of the angled exit stream. Express \( t \) as a function of the incoming stream height \( h \) and the angle \( \alpha \).
\[ t = f(h, \alpha) \]
---
**b) Determining the Angle \( \theta \) in Terms of \( \alpha \)**
Apply the momentum equation in the vertical direction to derive an equation for the angle \( \theta \). Use the hint that the net vertical force on the splitter is zero to relate \( \theta \) and \( \alpha \).
\[ \theta = f(\alpha) \]
---
**c) Determining the Horizontal Force \( F_x \)**
Find an equation for the horizontal force \( F_x \) on the splitter as a function of the relevant parameters: fluid density \( \rho \), velocity \( V \), width \( w \), height \( h \), angle \( \alpha \), and angle \( \theta \).
\[ F_x = f(\rho, V, w, h, \alpha, \theta) \]
---
#### Diagram Explanation
The provided diagram illustrates the physical setup:
- The incoming water jet has a height \( h \) and width \( w \), moving at velocity \( V \) horizontally.
- Upon hitting the splitter, the water is divided into two streams:
- One stream exits vertically downward with a part of the initial height.
- The other stream exits at an angle \( \alpha \) with height \( t \), moving horizontally at velocity \( V \).
The splitter is positioned at the junction where the jet is divided. The drawing indicates the angles and dimensions that are relevant to the derived equations.
Use these specifications to solve for the necessary parameters and understand the behaviors of the water streams as influenced by the splitter.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F368d064c-ece9-4358-9be9-95ab5c57dddf%2Fc6508380-e393-4271-8b5e-83bb8a6d8cf3%2Fu6zvfc8.jpeg&w=3840&q=75)
Transcribed Image Text:### Water Jet Splitter Analysis
A jet of water strikes a splitter and divides into two streams of equal velocity but differing thickness. Each jet maintains a width denoted by \( w \) (perpendicular to the plane of the paper). For this analysis, friction forces between the water streams and the splitter are considered negligible, and the weight of the splitter is ignored.
**Hint:** Pressure forces can be ignored.
#### Problem Breakdown
**a) Determining the Thickness \( t \) in Terms of \( h \) and \( \alpha \)**
Using the integral mass conservation equation, derive an equation for the thickness \( t \) of the angled exit stream. Express \( t \) as a function of the incoming stream height \( h \) and the angle \( \alpha \).
\[ t = f(h, \alpha) \]
---
**b) Determining the Angle \( \theta \) in Terms of \( \alpha \)**
Apply the momentum equation in the vertical direction to derive an equation for the angle \( \theta \). Use the hint that the net vertical force on the splitter is zero to relate \( \theta \) and \( \alpha \).
\[ \theta = f(\alpha) \]
---
**c) Determining the Horizontal Force \( F_x \)**
Find an equation for the horizontal force \( F_x \) on the splitter as a function of the relevant parameters: fluid density \( \rho \), velocity \( V \), width \( w \), height \( h \), angle \( \alpha \), and angle \( \theta \).
\[ F_x = f(\rho, V, w, h, \alpha, \theta) \]
---
#### Diagram Explanation
The provided diagram illustrates the physical setup:
- The incoming water jet has a height \( h \) and width \( w \), moving at velocity \( V \) horizontally.
- Upon hitting the splitter, the water is divided into two streams:
- One stream exits vertically downward with a part of the initial height.
- The other stream exits at an angle \( \alpha \) with height \( t \), moving horizontally at velocity \( V \).
The splitter is positioned at the junction where the jet is divided. The drawing indicates the angles and dimensions that are relevant to the derived equations.
Use these specifications to solve for the necessary parameters and understand the behaviors of the water streams as influenced by the splitter.
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