Fluid mechanics - Water flowing through a pipe

[theBEAST]

Homework Statement

The Attempt at a Solution

I am interested in why my method is flawed:

Alright I know how to do this problem. Essentially you solve mass flow rate = rho * double integral of u dA. HOWEVER, I decided to do this differently. I know that I can solve for the average u from the equation mass flow rate = rho * u_avg * A. I also know that the average u is just the integral of u over the integral of 1. I tried this and I get 1.528 m/s which is clearly not one of the answers. Why does my method not work?

 

[TSny]

In the expression: $\dot{m} = \rho u_{avg}A$,
$u_{avg}$ needs to be the average over the cross-sectional area of the pipe, not the average over the radius.

 

[theBEAST]

In the expression: $\dot{m} = \rho u_{avg}A$,
$u_{avg}$ needs to be the average over the cross-sectional area of the pipe, not the average over the radius.

Hmmm, according to the equation doesn't u only depend on r? In other words u is constant for some circular ring r = k?

 

[TSny]

Consider two intervals of radius, one near the center of the pipe and one out at the edge. Suppose the interval dr is the same for both. There are more patches of area in the same interval dr out near the edge. So, the value of the velocity out at the edge contributes more to the average over the cross–section than the value near the center.

 

[Nickie]

Your method is not taking into account the variation of velocity within the pipe. The average velocity you calculated is only valid for a uniform flow, which may not be the case in this problem. In fluid mechanics, it is important to consider the velocity profile and how it changes along the flow path. Therefore, using the mass flow rate equation is a more accurate approach as it takes into account the velocity distribution within the pipe.