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Fluid Mechanics · Chapter 8 of 10 · Intermediate

Internal (Pipe) Flow

Friction makes pressure fall along every pipe. The friction factor turns that loss into a number, from the exact f = 64/Re of laminar flow to the Moody chart for turbulent flow.

Readiness check

This chapter quantifies friction in pipes. Tick only what you can do closed-notes.

Compute the Reynolds number and classify the flow.
Use the energy equation with a head-loss term.
Evaluate a base-10 logarithm.
Convert between head and pressure (ΔP = ρg h_L).
Read a value off a log-log chart.

0 or 1 weak itemsContinue with this chapter.

2 weak itemsReview the energy equation in Chapter 5 and Re in Chapter 1.

3 or more weak itemsRevisit head loss and flow regimes first.

The core idea

Friction in a pipe causes a head loss h_L = f(L/D)(V²/2g); the friction factor f is 64/Re in laminar flow and comes from the Moody chart (Colebrook equation) in turbulent flow.

h_L = f (L/D)(V²/2g)laminar: f = 64/Returbulent: 1/√f = −2 log(ε/D/3.7 + 2.51/Re√f)

Wall friction dissipates mechanical energy, so pressure drops steadily along a pipe. The Darcy-Weisbach equation packages this as a head loss proportional to length, velocity head, and a dimensionless friction factor f. In laminar flow f = 64/Re exactly. In turbulent flow f depends on both the Reynolds number and the relative roughness ε/D through the Colebrook equation, plotted as the Moody chart and approximated explicitly by the Haaland equation. Fittings and valves add minor losses on top.

The skill works when: you classify the flow by Re, pick the right friction-factor route, and add minor losses.

The skill breaks down when: a turbulent friction factor is taken as 64/Re, or roughness is ignored at high Re.

The concept. The Moody chart. In laminar flow f drops as 64/Re; past transition the turbulent curves depend on relative roughness, flattening to a constant at high Re where roughness dominates.

The skills, taught in order

Pipe flow is the friction factor applied through the energy equation. Five skills cover the regimes, head loss, laminar flow, turbulent flow, and minor losses.

8.1 Regimes and the entrance region

Pipe flow is laminar below Re ≈ 2300 and turbulent above ≈ 4000, with Re = ρVD/μ. Near the inlet the profile is still developing (the entrance region); downstream it is fully developed and the friction factor is constant. Most analysis assumes fully developed flow.

8.2 The friction factor and head loss

The Darcy-Weisbach equation gives the major head loss: h_L = f(L/D)(V²/2g). The pressure drop is ΔP = ρg h_L = f(L/D)(ρV²/2). Everything reduces to finding the dimensionless friction factor f.

8.3 Laminar flow

In laminar flow the friction factor is exactly f = 64/Re, with no roughness dependence (a smooth viscous profile rides over the bumps). This gives the Hagen-Poiseuille result ΔP = 32μLV/D², the only pipe case solvable in closed form.

8.4 Turbulent flow and the Moody chart

In turbulent flow f depends on Re and the relative roughness ε/D through the implicit Colebrook equation, read from the Moody chart. The explicit Haaland equation, 1/√f = −1.8 log[(ε/D/3.7)^1.11 + 6.9/Re], gives f directly to within a couple of percent.

Regime	Friction factor
Laminar (Re < 2300)	f = 64/Re (exact)
Turbulent	Colebrook / Moody chart, or Haaland
Fully rough (high Re)	f depends on ε/D alone

8.5 Minor losses

Entrances, bends, valves, and fittings each add a minor loss h_L = K_L(V²/2g), with a tabulated loss coefficient K_L. The total head loss is the sum of the major (pipe friction) and all the minor losses, which then drives the energy equation and the pump size.

Engineering connection: water and oil pipelines, HVAC ducts, fuel and hydraulic lines, and blood vessels are all sized by computing head loss and the pump or pressure needed to overcome it.

Worked example 1: laminar flow in a tube

Oil (ρ = 880 kg/m³, μ = 0.2 Pa·s) flows at 1 m/s through a 2 cm diameter tube, 10 m long. Find the Reynolds number, the friction factor, the head loss, and the pressure drop.

Figure 1. The low Reynolds number (88) confirms laminar flow, so f = 64/Re exactly. The pressure drop matches the Hagen-Poiseuille formula, the closed-form laminar result.

ProblemFind Re, f, h_L, and ΔP for the tube in Figure 1.
Given / findρ = 880 kg/m³, μ = 0.2 Pa·s, V = 1 m/s, D = 0.02 m, L = 10 m. Find Re, f, h_L, ΔP.
AssumptionsFully developed, steady, horizontal tube.
ModelClassify by Re, use f = 64/Re, then Darcy-Weisbach for head loss and ΔP = ρg h_L.
EquationsRe = ρVD/μ f = 64/Re h_L = f(L/D)(V²/2g)
SolveRe = (880 × 1 × 0.02)/0.2 = 88 (laminar). f = 64/88 = 0.727. h_L = 0.727 × (10/0.02) × (1²/19.62) = 18.5 m. ΔP = ρg h_L = 880 × 9.81 × 18.5 = 160 kPa.
CheckThe Hagen-Poiseuille formula ΔP = 32μLV/D² = 32 × 0.2 × 10 × 1/0.02² = 160 kPa, an exact match. Viscous oil in a small tube loses a lot of pressure even at modest speed.
ConclusionIn laminar flow the friction factor and pressure drop are exact, with no chart needed. High viscosity and small diameter make laminar losses large.

Result. Re = 88, f = 0.727, h_L = 18.5 m, ΔP = 160 kPa.

Worked example 2: turbulent flow in a steel pipe

Water at 20 °C (ν = 1.004×10⁻⁶ m²/s, ρ = 998 kg/m³) flows at 2 m/s through a 5 cm commercial-steel pipe (ε = 0.045 mm), 100 m long. Find the friction factor and the head loss.

Figure 2. The high Reynolds number and small roughness give a friction factor around 0.022 from the Moody chart (or Haaland). The flatter turbulent profile and roughness both raise the loss above the smooth-pipe value.

ProblemFind f and h_L for the steel pipe in Figure 2.
Given / findV = 2 m/s, D = 0.05 m, L = 100 m, ε = 0.045 mm, ν = 1.004×10⁻⁶ m²/s. Find f and h_L.
AssumptionsFully developed turbulent flow, commercial-steel roughness.
ModelCompute Re and ε/D, get f from the Haaland (or Colebrook) equation, then Darcy-Weisbach.
EquationsRe = VD/ν, ε/D = 0.045/50 = 0.0009 1/√f = −1.8 log[(ε/D/3.7)^1.11 + 6.9/Re] h_L = f(L/D)(V²/2g)
SolveRe = (2 × 0.05)/1.004×10⁻⁶ = 99,600 (turbulent). ε/D = 0.0009. Haaland gives f = 0.0216 (Colebrook 0.0218, within 1%). h_L = 0.0216 × (100/0.05) × (2²/19.62) = 0.0216 × 2000 × 0.2039 = 8.8 m.
CheckFor a smooth pipe at this Re, f would be about 0.018, so the roughness adds roughly 20%. The 8.8 m loss over 100 m (about 86 kPa) is typical for water mains, and sets the pump duty.
ConclusionTurbulent flow needs the Moody chart or an equation like Haaland, with both Re and roughness. The friction factor then feeds the same Darcy-Weisbach head loss as the laminar case.

Result. f ≈ 0.0216 (Haaland); head loss 8.8 m over 100 m (about 86 kPa).

Misconceptions and diagnostics

Mistake	Symptom	Diagnostic question	Correction
Using 64/Re in turbulent flow	Friction factor far too high	"Is Re below 2300?"	f = 64/Re is laminar only; use Moody or Haaland if turbulent.
Ignoring roughness at high Re	Head loss underestimated	"What is ε/D?"	At high Re, roughness sets f; include ε/D.
Forgetting minor losses	Pump undersized for a fitting-heavy line	"How many bends and valves?"	Add ΣK_L(V²/2g) to the major loss.
Wrong velocity head	Loss off by a square factor	"Did I use V²/2g with the pipe velocity?"	Use the average velocity in the pipe of interest.

Practice ladder

Level 1 · Direct skill

A flow has Re = 1500 in a 3 cm pipe. Find the friction factor.

Show answer

Re < 2300, so laminar: f = 64/Re = 64/1500 = 0.0427. No roughness needed.

Level 2 · Mixed concept

For the Worked Example 2 pipe, add a fully open gate valve (K_L = 0.2) and two elbows (K_L = 0.9 each). Find the total minor loss.

Show answer

ΣK_L = 0.2 + 2(0.9) = 2.0. h_L,minor = ΣK_L(V²/2g) = 2.0 × (4/19.62) = 0.41 m, small next to the 8.8 m major loss but not always negligible.

Level 3 · Independent problem

Water (ν = 1.0×10⁻⁶ m²/s) flows at 1.5 m/s in a 10 cm smooth pipe, 50 m long. Estimate Re, take f ≈ 0.018, and find the head loss.

Show answer

Re = VD/ν = 1.5 × 0.1/1.0×10⁻⁶ = 150,000 (turbulent). h_L = f(L/D)(V²/2g) = 0.018 × (50/0.1) × (1.5²/19.62) = 0.018 × 500 × 0.1147 = 1.03 m.

Level 4 · Transfer to real engineering

Find a real piping run (a home water line, a garden irrigation pipe, an HVAC duct). Estimate Re, the friction factor, and the head loss, and say what pump or pressure it demands.

What good work looks like

Re computed and the regime identified, f found by the correct route, head loss from Darcy-Weisbach plus minor losses, and a pump-head or pressure implication.

Working with AI, and proving it yourself

Use AI as an examiner, not a solver

"Check that I picked the friction-factor route that matches my Reynolds number."

"Give me five pipe flows; I will classify each and name how to get f."

"Compute the head loss." Finding f and applying Darcy-Weisbach yourself is the skill.

"What is the friction factor?" Reading the Moody chart or Haaland is the point.

Portfolio task

Analyse one pipe run: classify the flow, find f (64/Re or Moody/Haaland), compute the major and minor head losses, and state the pump head needed.

Must include: Re and regime, the friction factor with its method, and a total head loss.

Retrieval and spaced review

Closed notes. Answer out loud, then reveal.

1. Write the Darcy-Weisbach head loss.

h_L = f(L/D)(V²/2g); ΔP = ρg h_L.

2. Friction factor in laminar flow?

f = 64/Re, exact, with no roughness dependence.

3. How is f found in turbulent flow?

From the Colebrook equation (Moody chart), or explicitly from Haaland, using Re and ε/D.

4. What is a minor loss?

A loss at a fitting or valve: h_L = K_L(V²/2g).

5. What governs f at very high Re?

Relative roughness ε/D alone (the fully rough regime).

TodayFinish this quiz and Levels 1 and 2 of the ladder.

+1 dayRe-derive the laminar and turbulent losses from a blank page.

+3 daysOne laminar and one turbulent pipe problem.

+7 daysMove to external flow and drag, Chapter 9.

+30 daysCombine head loss with the pump energy equation of Chapter 5.

Textbook mapping

Item	Mapping
Primary source	Çengel and Cimbala, Fluid Mechanics: Fundamentals and Applications, Chapter 8 (Internal Flow)
Cross-reference	White, Ch. 6 · Munson, Ch. 8
Core topics	8.1 Regimes · 8.2 Friction factor and head loss · 8.3 Laminar flow · 8.4 Moody chart · 8.5 Minor losses
Engineering connection	Water and oil pipelines, HVAC ducts, and hydraulic lines.
Read next	Chapter 9: External Flow, Drag and Lift.