Question

While using Darcy-Wesibach equation for estimating head loss in a pipe flow, the friction factor was misjudged by +20% for this case, the error in estimating discharge is

• A. + 10%
• B. + 40%
• C. - 40%
• D. - 10%

Answer 1

The correct option is D - 10%

The head loss through the pipe is given by

$h_L=\frac{fL{Q}^2}{12.1D^5}$

If all the parameters remain same other than friction factor and discharge then

$f{Q}^2=constant$

Let this constant be C, then

$f{Q}^2=C$

$\Rightarrow Q^2=\frac{C}{f}....\left(i\right)$

Differentiating Q w.r.t f, we get

$2QdQ=C\times (-\frac{1}{f^2})df......\left(ii\right)$

Dividing (ii) by (i), we get

$\frac{2QdQ}{Q^2}=\frac{C\times (-\frac{1}{f^2})df}{\frac{C}{f}}$

$\Rightarrow \frac{2dQ}{Q}=-\frac{df}{f}$

$\Rightarrow \frac{dQ}{Q}=\frac{1}{2}\times \frac{df}{f}$

$\Rightarrow dQ=-\frac{Q}{2}\times \frac{df}{f}$

$\Rightarrow dQ=-\frac{100}{2}\times \frac{20}{100}$

$\Rightarrow dQ=-10\%$