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The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.?
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The velocity of flow through a pipe at the bottom of a tank depends on...
Understanding Flow Velocity in a Pipe
The velocity of flow through a pipe at the bottom of a tank can be influenced by various factors including water level and gravitational acceleration. To analyze this relationship, we can develop a non-dimensional functional relationship.
Key Variables
- Water Level (h): The height of water in the tank.
- Gravitational Acceleration (g): The acceleration due to gravity, typically 9.81 m/s².
- Flow Velocity (V): The velocity of water flowing through the pipe.
Non-dimensional Parameters
To create a non-dimensional relationship, we can define a dimensionless parameter:
- Froude Number (Fr): A key dimensionless number in fluid mechanics that represents the ratio of inertial forces to gravitational forces. It can be defined as:
Fr = V / √(g * h)
This shows the relationship between flow velocity, gravitational acceleration, and water level.
Functional Relationship
The non-dimensional relationship can be expressed as:
- V = k * √(g * h)
Where k is a non-dimensional constant that incorporates factors like pipe diameter and flow characteristics.
Conclusion
The relationship reveals that flow velocity increases with the square root of the water level and is directly influenced by gravitational acceleration. By using the Froude number and deriving a dimensionless equation, we can better understand the flow dynamics in various engineering applications.
This non-dimensional approach simplifies the analysis of fluid flow, allowing for easier comparison across different systems.
The velocity of flow through a pipe at the bottom of a tank can be influenced by various factors including water level and gravitational acceleration. To analyze this relationship, we can develop a non-dimensional functional relationship.
Key Variables
- Water Level (h): The height of water in the tank.
- Gravitational Acceleration (g): The acceleration due to gravity, typically 9.81 m/s².
- Flow Velocity (V): The velocity of water flowing through the pipe.
Non-dimensional Parameters
To create a non-dimensional relationship, we can define a dimensionless parameter:
- Froude Number (Fr): A key dimensionless number in fluid mechanics that represents the ratio of inertial forces to gravitational forces. It can be defined as:
Fr = V / √(g * h)
This shows the relationship between flow velocity, gravitational acceleration, and water level.
Functional Relationship
The non-dimensional relationship can be expressed as:
- V = k * √(g * h)
Where k is a non-dimensional constant that incorporates factors like pipe diameter and flow characteristics.
Conclusion
The relationship reveals that flow velocity increases with the square root of the water level and is directly influenced by gravitational acceleration. By using the Froude number and deriving a dimensionless equation, we can better understand the flow dynamics in various engineering applications.
This non-dimensional approach simplifies the analysis of fluid flow, allowing for easier comparison across different systems.
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The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.?
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The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.?.
The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.? for UPSC 2024 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.? covers all topics & solutions for UPSC 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The velocity of flow through a pipe at the bottom of a tank depends on the water level in the tank and the gravitational acceleration. Develop a non-dimensional functional relationship for the flow velocity.?.
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