Chapter 11, Problem 11.1.1P

To determine

The head loss due to friction in a long pipe using Manning’s equation and Hazen Williams equation.

Answer to Problem 11.1.1P

Manning’s equation: 0.522 m and 2.09 m

Hazen Williams equation: 0.638 m and 2.3 m

Explanation of Solution

Given:

$\begin{array}{l}Diameter=1m\\ Lengthofpipe=300m\\ n=0.013\\ C=100\\ Q=1m^3/sand2m^3/s\end{array}$

Formula used:

$\begin{array}{l}{h}_L=\left[\frac{n^{2}L}{{\left(A{R}^{2/3}\right)}^2}\right]Q^2\\ where,h_{L}isheadloss\\ nisManning\text{'}scoefficient\\ Lislengthofpipe\\ Aisarea\\ Rishydraulicradius\\ Qisdischarge\end{array}$

$\begin{array}{l}\frac{Q}{A}=0.849C{R}^{0.63}{\left(\frac{h_L}{L}\right)}^{0.54}\\ where,Qisdischarge\\ Aisarea\\ CisWilliamHazenscoefficient\\ Rishydraulicradius\\ h_{L}isheadloss\\ Lislengthofpipe\end{array}$

Using Manning’s equation, the head loss for the discharge of 1m³/s is

$\begin{array}{l}{h}_L=\left[\frac{n^{2}L}{{\left(A{R}^{2/3}\right)}^2}\right]Q^2\\ h_L=\left[\frac{n^{2}L}{{\left(\frac{\pi D^2}{4}\times {\left(\frac{D}{4}\right)}^{2/3}\right)}^2}\right]Q^2\\ h_L=\left[\frac{{0.013}^2\times 300}{{\left(\frac{\pi \times 1^2}{4}\times {\left(\frac{1}{4}\right)}^{2/3}\right)}^2}\right]\times 1^2\\ h_L=0.522m\end{array}$

The head loss for the discharge of 2m³/s is

$\begin{array}{l}{h}_L=\left[\frac{n^{2}L}{{\left(A{R}^{2/3}\right)}^2}\right]Q^2\\ h_L=\left[\frac{n^{2}L}{{\left(\frac{\pi D^2}{4}\times {\left(\frac{D}{4}\right)}^{2/3}\right)}^2}\right]Q^2\\ h_L=\left[\frac{{0.013}^2\times 300}{{\left(\frac{\pi \times 1^2}{4}\times {\left(\frac{1}{4}\right)}^{2/3}\right)}^2}\right]\times 2^2\\ h_L=2.09m\end{array}$

Using Hazen Williams equation, the head loss for the discharge of 1m³/s is

$\begin{array}{l}\frac{Q}{A}=0.849C{R}^{0.63}{\left(\frac{h_L}{L}\right)}^{0.54}\\ h_L=L{\left[\frac{1}{0.849CA{R}^{0.63}}\right]}^{1.85}{Q}^{1.85}\\ h_L=L{\left[\frac{1}{0.849C\left(\frac{\pi D^2}{4}\right)\times {\left(\frac{D}{4}\right)}^{0.63}}\right]}^{1.85}{Q}^{1.85}\\ h_L=300\times {\left[\frac{1}{0.849\times 100\times \left(\frac{\pi \times 1^2}{4}\right)\times {\left(\frac{1}{4}\right)}^{0.63}}\right]}^{1.85}\times 1^{1.85}\\ h_L=0.638m\end{array}$

The head loss for the discharge of 2m³/s is

$\begin{array}{l}{h}_L=L{\left[\frac{1}{0.849CA{R}^{0.63}}\right]}^{1.85}{Q}^{1.85}\\ h_L=L{\left[\frac{1}{0.849C\left(\frac{\pi D^2}{4}\right)\times {\left(\frac{D}{4}\right)}^{0.63}}\right]}^{1.85}{Q}^{1.85}\\ h_L=300\times {\left[\frac{1}{0.849\times 100\times \left(\frac{\pi \times 1^2}{4}\right)\times {\left(\frac{1}{4}\right)}^{0.63}}\right]}^{1.85}\times 2^{1.85}\\ h_L=2.3m\end{array}$

Conclusion:

The head loss due to friction in a long pipe using Manning’s equation are 0.522 m and 2.09 m and the head loss due to friction in a long pipe using the Hazen Williams equation are 0.638 m and 2.3 m.