Answered: An incompressible fluid of density p…

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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$### Pipe Flow and Non-Dimensional Analysis #### Text Overview: An incompressible fluid of density $ \rho $ and viscosity $ \mu $ flows at average speed $ V $ through a long, horizontal section of a round pipe of length $ L $, inner diameter $ D $, and inner wall roughness height, $ \varepsilon $. The pipe is long enough that the flow is fully developed, meaning that the velocity profile does not change down the pipe. Pressure decreases (linearly) down the pipe in order to “push” the fluid through the pipe to overcome friction. Using the method of repeating variables (i.e., the Buckingham Pi method), develop a nondimensional relationship between pressure drop $ \Delta P = P_1 - P_2 $ and the other parameters in the problem. Be sure to modify your Pi groups as necessary to achieve established nondimensional parameters and name any that are common named parameters. For consistency, choose $ D $ rather than $ L $ or $ \varepsilon $ as one of your repeating parameters. #### Diagram Explanation: The accompanying diagram illustrates a horizontal pipe section with the following components: - **Pipe with Flow Indicators**: Arrows inside the pipe indicate fluid flow direction from left to right. - **Parameters Illustrated**: - $ \rho $ (density) and $ \mu $ (viscosity) of the fluid. - $ V $ as the average speed of flow. - $ L $ representing the length of the pipe. - $ D $ as the inner diameter of the pipe. - $ \varepsilon $ denoting the roughness height of the inner wall. - **Pressure Points**: - $ P_1 $: Pressure at the inlet. - $ P_2 $: Pressure at the outlet. - **Flow Characteristics**: - The linear drop in pressure along the pipe's length is shown to indicate the overcoming of frictional forces. This diagram and text provide the basic understanding necessary for conducting a dimensional analysis of fluid flow in the pipe.$

Transcribed Image Text:### Pipe Flow and Non-Dimensional Analysis #### Text Overview: An incompressible fluid of density $ \rho $ and viscosity $ \mu $ flows at average speed $ V $ through a long, horizontal section of a round pipe of length $ L $, inner diameter $ D $, and inner wall roughness height, $ \varepsilon $. The pipe is long enough that the flow is fully developed, meaning that the velocity profile does not change down the pipe. Pressure decreases (linearly) down the pipe in order to “push” the fluid through the pipe to overcome friction. Using the method of repeating variables (i.e., the Buckingham Pi method), develop a nondimensional relationship between pressure drop $ \Delta P = P_1 - P_2 $ and the other parameters in the problem. Be sure to modify your Pi groups as necessary to achieve established nondimensional parameters and name any that are common named parameters. For consistency, choose $ D $ rather than $ L $ or $ \varepsilon $ as one of your repeating parameters. #### Diagram Explanation: The accompanying diagram illustrates a horizontal pipe section with the following components: - **Pipe with Flow Indicators**: Arrows inside the pipe indicate fluid flow direction from left to right. - **Parameters Illustrated**: - $ \rho $ (density) and $ \mu $ (viscosity) of the fluid. - $ V $ as the average speed of flow. - $ L $ representing the length of the pipe. - $ D $ as the inner diameter of the pipe. - $ \varepsilon $ denoting the roughness height of the inner wall. - **Pressure Points**: - $ P_1 $: Pressure at the inlet. - $ P_2 $: Pressure at the outlet. - **Flow Characteristics**: - The linear drop in pressure along the pipe's length is shown to indicate the overcoming of frictional forces. This diagram and text provide the basic understanding necessary for conducting a dimensional analysis of fluid flow in the pipe.

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