MMMU-Pro / validation_Energy_and_Power_28 / A pipe of radius R has a fully developed laminarflow of air at P0, T0 with a…

question

A pipe of radius R has a fully developed laminarflow of air at P0, T0 with a velocity profile of V = Vc[1 - (r/R)2], where Vc is the velocity on the center-line and r is the radius, as shown in <image 1>. Find the total mass flow rate and the average velocity, both as functions of Vc and R.

img type

• Diagrams

options

1. $\begin{aligned}V&=\frac{V_c}{2}\\\dot{m}&=\frac{\pi}{3\cdot\nu}\cdot V_c\cdot R^2\\\\\end{aligned}$
2. $\begin{aligned}V&=\frac{V_c}{4}\\\dot{m}&=\frac{\pi}{2\cdot\nu}\cdot V_c\cdot R^2\\\\\end{aligned}$
3. $\begin{aligned}V&=\frac{V_c}{2}\\\dot{m}&=\frac{\pi}{2\cdot\nu}\cdot V_c\cdot R^2\\\\\end{aligned}$
4. $\begin{aligned}V&=\frac{V_c}{4}\\\dot{m}&=\frac{\pi}{2\cdot\nu}\cdot V_c\cdot R^4\\\\\end{aligned}$
5. $\begin{aligned}V&=\frac{V_c}{2}\\\dot{m}&=\frac{\pi}{2\cdot\nu}\cdot V_c\cdot R^3\\\\\end{aligned}$
6. $\begin{aligned}V&=\frac{V_c}{5}\\\dot{m}&=\frac{\pi}{2\cdot\nu}\cdot V_c\cdot R^2\\\\\end{aligned}$
7. $\begin{aligned}V&=\frac{V_c}{2}\\\dot{m}&=\frac{\pi}{4\cdot\nu}\cdot V_c\cdot R^2\\\\\end{aligned}$
8. $\begin{aligned}V&=\frac{V_c}{3}\\\dot{m}&=\frac{\pi}{2\cdot\nu}\cdot V_c\cdot R^3\\\\\end{aligned}$
9. $\begin{aligned}V&=\frac{V_c}{3}\\\dot{m}&=\frac{\pi}{2\cdot\nu}\cdot V_c\cdot R^2\\\\\end{aligned}$

subject

Energy_and_Power