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Q. 6.P.18

The flowrate of a fluid in a pipe is measured using a pitot tube, which gives a pressure differential equivalent to 40 mm of water when situated at the centre line of the pipe and 22.5 mm of water when midway between the axis and the wall. Show that these readings are consistent with streamline flow in the pipe.

Verified Solution

For streamline flow in a pipe, a force balance gives:

$-\Delta P \pi r^2=-\mu \frac{ d u}{ d r} 2 \pi r l$

and:                    $-u=\frac{-\Delta P}{2 \mu l} r$ and $-u=\frac{-\Delta P}{2 \mu l} \frac{r^2}{2}+$ constant.
When r = a (at the wall), u = 0, the constant $=-\Delta P a^2 / 4 \mu l$

and:                          $u=-\frac{\Delta P}{4 \mu l}\left(a^2-r^2\right)$

The maximum velocity, $u_{\max }=\frac{-\Delta P a^2}{4 \mu l}$

and:                                               $\frac{u}{u_{\max }}=1-\left(\frac{r}{a}\right)^2$

When $r=a / 2, u / u_{\max }=0.75$.

The pitot tube is discussed in Section 6.3.1 and:

$u=k \sqrt{h}$                (from equation 6.10)

At the centre-line, $u=u_{\max }$ and h = 40 mm.

∴                       $u_{\max }=K \sqrt{40}=6.32 K$

At a point midway between the axis and the wall, $u=u_{1 / 2}$ and h = 22.5 mm.

∴                       $u_{1 / 2}=K \sqrt{22.5}=4.74 K$

$u_{1 / 2} / u_{\max }=(4.74 K / 632 K )=0.75$

and hence the flow is $\underline{\underline{\text { streamline}}}$.