Answered: 4.21 Water flows from a reservoir out a…

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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**4.21** Water flows from a reservoir out a pipe of diameter $ D $ and length $ L $. The describing equation is:

\[
0 = -\frac{1}{\rho} \frac{\partial p}{\partial z} + \nu \left( \frac{\partial^2 v_z}{\partial r^2} + \frac{1}{r} \frac{\partial v_z}{\partial r} \right)
\]

In this equation:
- $ \rho $ is the fluid density.
- $ \nu $ is the kinematic viscosity.
- $ p $ represents the pressure.
- $ v_z $ is the velocity in the $ z $ direction.
- $ r $ is the radial coordinate.

This equation is commonly associated with fluid mechanics, describing the flow dynamics in cylindrical coordinates for incompressible fluids within a pipe.

$**Boundary Conditions and Flow Analysis** The boundary conditions are $ v_z = 0 $ at $ r = D/2 $, $ v_z $ is finite at $ r = 0 $, and $ p = 0 $ at $ z = L $. The average velocity is $ V $. Normalize the equations, using $ V $ as the characteristic velocity, and select the appropriate characteristic lengths from the boundary conditions. A flow rate of 2 cfs is used during a particular study. The pipe is then doubled in length with the same diameter. Determine the new flow rate in order that identical dimensionless solutions result. **Diagram Explanation** - **Pipe Illustration:** The diagram shows a pipe with a length $ L $ and diameter $ D $, with fluid flow along the z-axis. - **Velocity $ V $:** Represents the average velocity of the fluid entering the pipe. - **Velocity Profile $ v_z $:** Indicates how velocity changes across the pipe's radius. It is zero at the pipe's wall ($ r = D/2 $) and has a finite value at the centerline ($ r = 0 $). - **Normalization Parameters:** Use $ V $ as the characteristic velocity and suitable characteristic lengths to achieve dimensionless solutions for various pipe configurations. - **Flow Rate Adjustment:** Calculate the necessary flow rate when the pipe length is doubled to maintain consistent dimensionless outcomes. This exercise demonstrates the importance of understanding and applying dimensionless analysis in fluid dynamics to predict and compare flow behavior in scaled systems.$

Transcribed Image Text:**Boundary Conditions and Flow Analysis** The boundary conditions are $ v_z = 0 $ at $ r = D/2 $, $ v_z $ is finite at $ r = 0 $, and $ p = 0 $ at $ z = L $. The average velocity is $ V $. Normalize the equations, using $ V $ as the characteristic velocity, and select the appropriate characteristic lengths from the boundary conditions. A flow rate of 2 cfs is used during a particular study. The pipe is then doubled in length with the same diameter. Determine the new flow rate in order that identical dimensionless solutions result. **Diagram Explanation** - **Pipe Illustration:** The diagram shows a pipe with a length $ L $ and diameter $ D $, with fluid flow along the z-axis. - **Velocity $ V $:** Represents the average velocity of the fluid entering the pipe. - **Velocity Profile $ v_z $:** Indicates how velocity changes across the pipe's radius. It is zero at the pipe's wall ($ r = D/2 $) and has a finite value at the centerline ($ r = 0 $). - **Normalization Parameters:** Use $ V $ as the characteristic velocity and suitable characteristic lengths to achieve dimensionless solutions for various pipe configurations. - **Flow Rate Adjustment:** Calculate the necessary flow rate when the pipe length is doubled to maintain consistent dimensionless outcomes. This exercise demonstrates the importance of understanding and applying dimensionless analysis in fluid dynamics to predict and compare flow behavior in scaled systems.

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