Four-Pipe Junction Network: 22-Agric-A4 Fluid Flow, December 2016, Q1

Problem 1: Four-Pipe Junction Network (equal value)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. Four identical horizontal cast-iron pipes (length $L=45\ \text{m}$, diameter $D=8\ \text{cm}$, relative roughness $\epsilon/D=0.00325$) radiate from reservoirs 1–4 and meet at a common junction a. Minor losses are neglected.

Quantity	Value
Pipe length, $L$ (each)	45 m
Pipe diameter, $D$ (each)	8 cm
Relative roughness, $\epsilon/D$	0.00325
$p_1,\,p_2,\,p_3,\,p_4$	950, 350, 675, 100 kPa
Water, 20°C	$\rho=998\ \text{kg/m}^3$, $\mu=0.001\ \text{kg/m}\cdot\text{s}$

Find. The flow rate $Q$ in each of the four pipes, including its direction relative to the junction.

Figure 1 — Four identical 45 m × 8 cm cast-iron pipes meet at junction a; the known reservoir pressures $p_1$–$p_4$ drive flow into a from the two higher-pressure ends and out toward the two lower-pressure ends.

Approach. Assume a trial junction pressure $p_a$; each branch is then a single Darcy–Weisbach pipe with a known pressure drop, so its velocity follows from an iterated Colebrook friction factor. Adjust $p_a$ until the four branch flows satisfy continuity at the junction (flow in = flow out), following the source's own guess-and-check hint.

Governing equations (identical form for every branch). For a branch with pressure drop $|\Delta p_i| = |p_i-p_a|$, $$\begin{aligned} |\Delta p_i| &= f_i\frac{L}{D}\frac{\rho V_i^2}{2} \\ \frac{1}{\sqrt{f_i}} &= -2\log_{10}\!\left(\frac{\epsilon/D}{3.7}+\frac{2.51}{Re_i\sqrt{f_i}}\right) \\ Re_i &=\frac{\rho V_i D}{\mu} \end{aligned}$$ with flow directed from the higher pressure toward the lower (into the junction if $p_i>p_a$, out if $p_i<p_a$).
Iterate on $p_a$ for junction continuity. Starting from the source's hint ($p_a\approx530$ kPa, $f_1\approx0.027$) and refining by bisection on $\sum Q_{\text{in}}-\sum Q_{\text{out}}=0$ converges to $$\boxed{p_a \approx 517.3\ \text{kPa}},$$ close to the hinted starting value, with $f_1=0.0270$ matching the hinted friction factor almost exactly.
Branch velocities and flow rates at the converged $p_a$. Applying Step 1 to each branch (Reynolds numbers all in the $3.6\times10^5$–$6.0\times10^5$ range, fully turbulent, $f\approx0.027$ for all four pipes since the roughness dominates): $$\begin{aligned} Q_1 &= V_1A = \boxed{136.8\ \text{m}^3/\text{h}}\ \text{(into a)} \\ Q_2 &= V_2A = \boxed{84.8\ \text{m}^3/\text{h}}\ \text{(out of a)} \\ Q_3 &= V_3A = \boxed{82.4\ \text{m}^3/\text{h}}\ \text{(into a)} \\ Q_4 &= V_4A = \boxed{134.3\ \text{m}^3/\text{h}}\ \text{(out of a)} \end{aligned}$$
Continuity check. Flow into the junction $Q_1+Q_3 = 136.8+82.4 = 219.2\ \text{m}^3/\text{h}$ equals flow out $Q_2+Q_4 = 84.8+134.3 = 219.2\ \text{m}^3/\text{h}$, confirming the converged $p_a$.

Pipe	Δp (kPa)	V (m/s)	f	Q (m³/h)	Direction
1 ($p_1=950$)	432.7	7.56	0.0270	136.8	into a
2 ($p_2=350$)	167.3	4.69	0.0271	84.8	out of a
3 ($p_3=675$)	157.7	4.55	0.0271	82.4	into a
4 ($p_4=100$)	417.3	7.42	0.0270	134.3	out of a
Junction pressure	$p_a\approx517.3$ kPa

Concept & Theory

Topic. Steady incompressible flow through a pipe network with a single unknown-pressure junction — a Hardy-Cross-style problem reduced to one iteration variable.

Key relations. Darcy–Weisbach $\Delta p = f(L/D)(\rho V^2/2)$; Colebrook–White implicit friction factor; junction continuity $\sum Q_{\text{in}}=\sum Q_{\text{out}}$.

Why this works. With four pipes of identical length, diameter and roughness meeting at one junction, the only unknown that couples all four branches is the junction pressure $p_a$: once it is guessed, every branch becomes an independent single-pipe problem, and continuity at the junction is the single scalar equation that pins $p_a$ down. This is the simplest case of the general pipe-network (Hardy-Cross) method, which iterates a correction to loop or junction heads until continuity is satisfied everywhere.

Common pitfall. Assuming every branch carries the same friction factor without checking — here it happens to converge to nearly the same $f\approx0.027$ for all four pipes because the flow is fully rough-turbulent and the relative roughness dominates, but that must be verified from each branch's own Reynolds number, not assumed.

Source. White, Fluid Mechanics, Ch. 6 (Viscous Flow in Ducts) — Pipes in Combination / Three-Reservoir and Pipe-Network Problems.

Question 1 of 4: Four-Pipe Junction Network

Problem 1: Four-Pipe Junction Network (equal value)