Chapter 8, Problem 45P

A fluid is pumped into the network of pipes shownin Fig.P8.45. At steady state, the following flow balances must hold,

$\begin{array}{l}{Q}_1=Q_2+Q_3\\ Q_3=Q_4+Q_5\\ Q_5=Q_6+Q_7\end{array}$

where $Q_i=$ flow in pipe $i\left({\text{m}}^{\text{3}}\text{/s}\right)$. In addition, the pressure drops around the three right-hand loops must equal zero. The pressure drop in each circular pipe length can be computed with

$\Delta P=\frac{16}{{\pi }^2}\frac{fL\rho }{2D^5}{Q}^2$

where $\Delta P=$ the pressure drop $\left(\text{Pa}\right),f=$ the friction factor (dimensionless), $L=$ the pipe length $\left(\text{m}\right),\rho =$ the fluid density $\left({\text{kg/m}}^{\text{3}}\right)$, and $D=$ pipe diameter $\left(\text{m}\right)$. Write a program (or develop an algorithm in a mathematics software package) that will allow you to compute the flow in every pipe length given that $Q_1=1{\text{ m}}^{\text{3}}\text{/s}$ and $\rho =1.23{\text{ kg/m}}^{\text{3}}$. All the pipes have $D=500$ mm and $f=0.005$. The pipe lengths are: $L_3=L_5=L_8=L_9=2\text{ m;}$ $L_2=L_4=L_6=4\text{m};\text{ and }{L}_7=8\text{m}$.

FIGURE P8.45