Answered: The velocity profile in a circular pipe…

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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$The velocity profile in a circular pipe of radius $ R = 10 \, \text{cm} $ is given by: \[ U(r) = U_{\text{max}} \left[ 1 - \left( \frac{r}{R} \right)^7 \right] \] where $ U_{\text{max}} = 20 \, \text{m s}^{-1} $ and $ r $ is the radial coordinate, with $ r = 0 $ representing the center axis of the pipe. The working fluid is ethanol $(\mu = 0.001 \, 095 \, \text{Pa s}, \, \rho = 789 \, \text{kg m}^{-3})$. Determine the following: (a) The volume flow rate. $[0.489 \, \text{m}^3 \text{s}^{-1}]$ (b) The mass flow rate. $[386 \, \text{kg s}^{-1}]$ (c) The average velocity. $[15.6 \, \text{m s}^{-1}]$ (d) The shear stress at the wall of the pipe, if: \[ \tau(r) = \mu \frac{dU}{dr} \] $[-0.0767 \, \text{Pa}]$ (e) The dynamic pressure in the pipe, calculated using the average velocity. $[96 \, 006 \, \text{Pa}]$$

Transcribed Image Text:The velocity profile in a circular pipe of radius $ R = 10 \, \text{cm} $ is given by: \[ U(r) = U_{\text{max}} \left[ 1 - \left( \frac{r}{R} \right)^7 \right] \] where $ U_{\text{max}} = 20 \, \text{m s}^{-1} $ and $ r $ is the radial coordinate, with $ r = 0 $ representing the center axis of the pipe. The working fluid is ethanol $(\mu = 0.001 \, 095 \, \text{Pa s}, \, \rho = 789 \, \text{kg m}^{-3})$. Determine the following: (a) The volume flow rate. $[0.489 \, \text{m}^3 \text{s}^{-1}]$ (b) The mass flow rate. $[386 \, \text{kg s}^{-1}]$ (c) The average velocity. $[15.6 \, \text{m s}^{-1}]$ (d) The shear stress at the wall of the pipe, if: \[ \tau(r) = \mu \frac{dU}{dr} \] $[-0.0767 \, \text{Pa}]$ (e) The dynamic pressure in the pipe, calculated using the average velocity. $[96 \, 006 \, \text{Pa}]$

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