Turbulent pipe flow, Moody chart and minor losses

Why it matters

Almost every water main, cooling-water circuit, compressed-air line and large hydraulic return line runs turbulent. Sizing those pipes and choosing the pump depends on predicting the friction loss in straight runs and the extra losses in bends, valves, filters and fittings. Get the friction factor wrong and the pump is under-sized or the system wastes energy for its whole life.

Key ideas

Turbulence. Above a critical Reynolds number (about 2000 in pipes, fully turbulent beyond about 4000), small disturbances grow into irregular, three-dimensional eddies. The velocity at a point fluctuates about a time mean. The eddies mix momentum across the pipe much more strongly than molecular viscosity, which:

• flattens the mean velocity profile (roughly a 1/7-power law, u/u_max = (y/R)^(1/7); mean velocity ≈ 0.82 u_max);
• steepens the gradient at the wall, so wall shear stress and friction loss are much larger than for laminar flow at the same flow rate;
• creates "Reynolds stresses" (apparent stresses from velocity fluctuations), which cannot be predicted from first principles and are handled with empirical correlations.

Near-wall layers and roughness. Very close to the wall turbulence is damped and a thin viscous sublayer remains. If the wall roughness elements (height ε) are buried inside this sublayer, the pipe is hydraulically smooth and f depends on Re only. If they protrude well through it, the pipe is fully rough and f depends on ε/D only. In between is the transition zone where both matter. The same pipe can behave smooth at low Re and rough at high Re, because the sublayer thins as Re increases.

Darcy–Weisbach and the friction factor. Head loss in a straight pipe is h_f = f·(L/D)·V²/(2g) for laminar or turbulent flow. Dimensional analysis shows f = φ(Re, ε/D):

• Laminar: f = 64/Re.
• Turbulent, smooth: Blasius f = 0.316·Re^(−0.25) (valid for about 4000 < Re < 10⁵).
• Turbulent, general: the Colebrook equation (implicit) or the explicit Haaland approximation, which is within about 2 % of Colebrook.
• The Moody chart plots f against Re for curves of constant ε/D. Typical ε: drawn tubing about 0.0015 mm, commercial steel about 0.045 mm, cast iron about 0.26 mm (take values from your data book).
• Some texts use the Fanning factor f′ = f/4 with h_f = 4f′LV²/(2gD). Check which one your book or chart uses.

Three classic pipe problems. (1) Find the head loss for a given Q and D — direct. (2) Find Q for a given head loss and D — iterate (guess f, find V, update Re and f). (3) Find D for given Q and head loss — iterate on D.

Minor losses. Fittings disturb the flow, causing separation and mixing. Their loss is written h_m = K·V²/(2g), or as an equivalent length L_e = K·D/f of straight pipe. Typical K values (use your data book): sharp-edged entrance 0.5, well-rounded entrance about 0.04, exit to a reservoir 1.0, 90° standard elbow about 0.9, open gate valve about 0.2, open globe valve about 10. Two cases are derived from theory:

• Sudden expansion: h = (V₁ − V₂)²/(2g), from momentum plus energy (Borda–Carnot). The exit to a large tank is the limit V₂ → 0, giving K = 1.
• Sudden contraction: the jet forms a vena contracta and then re-expands, h = (1/C_c − 1)²·V₂²/(2g), roughly 0.5·V₂²/(2g) for a large area ratio. "Minor" losses are not minor in short, fitting-heavy systems such as hydraulic manifolds.

Pipes in series and parallel. In series the flow is the same in each pipe and the head losses add. In parallel the head loss is the same across every branch and the flows add. These are the electrical-circuit analogies used for pipe networks.

Formulas

Re = ρ·V·D / μ = V·D / ν

• Re (–), V = mean velocity (m/s), D = diameter (m), ν = kinematic viscosity (m²/s).

h_f = f · (L/D) · V² / (2g)

• h_f = friction head loss (m), f = Darcy friction factor (–), L = length (m). Δp = ρ·g·h_f (Pa).

f = 0.316 · Re^(−0.25) (Blasius, smooth, 4000 < Re < 10⁵)

1/√f = −2.0·log₁₀[ (ε/D)/3.7 + 2.51/(Re·√f) ] (Colebrook)

1/√f = −1.8·log₁₀[ ((ε/D)/3.7)^1.11 + 6.9/Re ] (Haaland, explicit)

• ε = absolute roughness (m).

h_m = K · V² / (2g), L_e = K·D / f

• K = loss coefficient (–), L_e = equivalent length (m).

h_exp = (V₁ − V₂)² / (2g) (sudden expansion)

Z₁ − Z₂ = (ΣK + f·L/D) · V² / (2g) (single pipe between two reservoirs)

• Z = free-surface elevations (m).

P = ρ·g·Q·h_L

• P = power dissipated or pumping power needed (W).

Worked examples

Example 1 — friction loss in a steel main (standard). Water (ν = 1.0 × 10⁻⁶ m²/s) flows at 0.05 m³/s through 300 m of 150 mm commercial steel pipe (ε = 0.045 mm). Find the head loss and the power dissipated.

1. Area A = π × 0.15²/4 = 0.01767 m²; V = 0.05/0.01767 = 2.829 m/s.
2. Re = V·D/ν = 2.829 × 0.15/10⁻⁶ = 4.24 × 10⁵ — turbulent. ε/D = 0.045/150 = 0.0003.
3. Haaland: 1/√f = −1.8·log₁₀[(0.0003/3.7)^1.11 + 6.9/424 400], giving f ≈ 0.0163. (Iterating Colebrook gives 0.0165; a Moody chart reading is about the same.)
4. h_f = f·(L/D)·V²/(2g) = 0.0165 × (300/0.15) × 2.829²/19.62 = 0.0165 × 2000 × 0.408 = 13.4 m.
5. Power: P = ρ·g·Q·h_f = 1000 × 9.81 × 0.05 × 13.4 = 6.6 kW. Answer: h_f ≈ 13.4 m; power ≈ 6.6 kW.

Example 2 — reservoir-to-reservoir flow with fittings (GATE level). Two reservoirs differ in level by 20 m and are joined by a 500 m long, 200 mm pipe with f = 0.02. Losses: sharp entrance K = 0.5, a partly open valve K = 5, exit K = 1. Find the flow rate.

1. Energy equation between the free surfaces (both at atmospheric pressure, negligible velocity): Z₁ − Z₂ = (ΣK + f·L/D)·V²/(2g).
2. ΣK = 0.5 + 5 + 1 = 6.5; f·L/D = 0.02 × 500/0.2 = 50. Total = 56.5.
3. V = √(2 × 9.81 × 20/56.5) = √6.945 = 2.635 m/s.
4. Q = V·A = 2.635 × π × 0.2²/4 = 0.0828 m³/s. Answer: Q ≈ 0.083 m³/s (83 L/s). The fittings account for 6.5/56.5 ≈ 12 % of the loss.

Example 3 — sudden expansion. Water at 4 m/s in a 100 mm pipe enters a 200 mm pipe. V₂ = 4 × (100/200)² = 1 m/s, so h = (4 − 1)²/19.62 = 0.459 m.

Common mistakes

• Using f = 64/Re for turbulent flow, or the Moody chart without first checking Re.
• Mixing Darcy and Fanning friction factors — the Darcy value is four times larger.
• Using ε in mm and D in m when forming ε/D.
• Forgetting the exit loss (K = 1) when a pipe discharges into a tank, or adding a velocity head at the outlet as well as the exit loss.
• Applying the velocity of the larger pipe to a sudden contraction; K is based on the downstream (smaller-pipe) velocity.
• In parallel pipes, adding the head losses instead of setting them equal.

For GATE ME

Expect Darcy–Weisbach calculations, flow between reservoirs with entrance, exit and valve losses, sudden-expansion loss derivation and numericals, equivalent pipe length for series pipes, flow split in parallel pipes, and power lost in friction. Conceptual questions test the dependence of f on Re and ε/D in each Moody-chart region and how h_f scales with D at constant Q (about D⁻⁵ in fully rough turbulent flow). Practise reading the Moody chart and doing one or two iterations quickly.

Quick check

1. On which parameters does the friction factor depend in the fully rough zone?
2. What is the Blasius friction factor at Re = 5000?
3. What is the loss coefficient for a pipe discharging into a large tank?
4. For pipes in parallel, which quantity is common to all branches?
5. Why can a pipe that is smooth at low Re behave as rough at high Re?

Answers: 1. relative roughness ε/D only; 2. 0.316 × 5000^(−0.25) ≈ 0.0376; 3. K = 1; 4. the head loss; 5. the viscous sublayer becomes thinner as Re increases, exposing the roughness elements.