Flow in Pipes: Systems & Networks Worked Examples

Pipe 1 has $L_1=100\text{ m}$, $D_1=0.200\text{ m}$, and Darcy $f_1=0.0200$. Pipe 2 has $L_2=200\text{ m}$, $D_2=0.300\text{ m}$, and Darcy $f_2=0.0150$. Both carry $Q=0.100\text{ m}^3/\text{s}$. Determine total major head loss.

Replace the two-pipe series system of $300\text{ m}$ total length by one pipe carrying $Q=0.100\text{ m}^3/\text{s}$. Require the same $h_f=6.184\text{ m}$ and, for this equivalent calculation, use fixed Darcy $f=0.0200$. Determine equivalent diameter.

Two parallel pipes connect the same nodes and carry $Q_T=0.100\text{ m}^3/\text{s}$. Pipe 1 has $L_1=200\text{ m}$, $D_1=0.200\text{ m}$, $f_1=0.0200$; Pipe 2 has $L_2=300\text{ m}$, $D_2=0.250\text{ m}$, $f_2=0.0220$. Treat the Darcy factors as fixed. Determine the branch flows.

Reservoir heads are $H_1=100\text{ m}$, $H_2=80.0\text{ m}$, and $H_3=60.0\text{ m}$. The branch resistance coefficients in $h=RQ|Q|$ are $R_1=2000$, $R_2=1000$, and $R_3=3000$. Take flow toward the junction as positive. Determine junction head and flow directions.

For one loop, take clockwise flow as positive. Three links obey $h=rQ|Q|$ with $(r,Q)$ values $(500,0.0400)$, $(800,0.0200)$, and $(300,-0.0100)$. Determine the first Hardy Cross correction.

Two parallel branches obey $h=RQ^2$ for positive flow, with $R_1=1000$ and $R_2=4000$. Determine equivalent resistance and branch flows when total discharge is $0.0900\text{ m}^3/\text{s}$.

A pump and system curve use $Q$ in $\text{L/s}$:

Determine the operating discharge and head.

A storage tank receives $0.0600\text{ m}^3/\text{s}$ and supplies $0.0450\text{ m}^3/\text{s}$ for a $30.0\text{ min}$ hydraulic time interval. If the tank plan area is constant at $120\text{ m}^2$, estimate the water-level change during this interval.

A water pipeline has $K=2.20\text{ GPa}$, $\rho=1000\text{ kg/m}^3$, $D=0.500\text{ m}$, pipe modulus $E=200\text{ GPa}$, wall thickness $e=10.0\text{ mm}$, and a simplified restraint condition matching the thin-wall celerity equation. For a $1000\text{ m}$ reach, estimate $a$ and $T_c=2L/a$.

Using the simplified pipeline in Problem 9 only as a numerical data set, take $a=1191\text{ m/s}$. If water velocity decreases rapidly by $2.00\text{ m/s}$ before a reflected wave returns, estimate the magnitude of the initial Joukowsky pressure rise.

A simple frictionless reservoir-pipe-valve teaching model has $L=500\text{ m}$, initial water velocity $V_0=2.00\text{ m/s}$, density $1000\text{ kg/m}^3$, and celerity $a=1000\text{ m/s}$. The valve closes approximately linearly in $T=2.00\text{ s}$. Compare the round-trip time and the simple slow-closure pressure estimate.