Flow in Pipes: Fundamentals & Losses Worked Examples

These problems emphasize the Darcy friction-factor convention, explicit assumptions behind scaling relations, and the empirical limits of alternative pipe-flow formulas.

Problem 1: Reynolds Number in the Transition Region

Oil with kinematic viscosity ν=1.80×10−5 m2/s\nu=1.80\times10^{-5}\text{ m}^2/\text{s} flows at V=0.500 m/sV=0.500\text{ m/s} in a D=100 mmD=100\text{ mm} pipe. Determine Reynolds number and classify the regime using the conventional engineering ranges.

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Problem 2: Laminar Darcy Factor and Head Loss

Oil with ρ=900 kg/m3\rho=900\text{ kg/m}^3 and μ=0.250 Pa⋅s\mu=0.250\text{ Pa}\cdot\text{s} flows at Q=0.000500 m3/sQ=0.000500\text{ m}^3/\text{s} through a 50.0 mm50.0\text{ mm} diameter pipe 20.0 m20.0\text{ m} long. Determine ReRe, Darcy ff, head loss, and pressure drop.

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Problem 3: Correcting a Low-Reynolds-Number Friction-Factor Error

Glycerin has μ=1.50 Pa⋅s\mu=1.50\text{ Pa}\cdot\text{s} and ρ=1260 kg/m3\rho=1260\text{ kg/m}^3. It flows at 0.800 m/s0.800\text{ m/s} in a 20.0 mm20.0\text{ mm} tube. Determine the Reynolds number and Darcy friction factor.

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Problem 4: Darcy-Weisbach Major Loss

Water flows at Q=0.0500 m3/sQ=0.0500\text{ m}^3/\text{s} through a D=200 mmD=200\text{ mm} pipe L=300 mL=300\text{ m} long. Use Darcy f=0.0220f=0.0220. Determine mean velocity and major head loss.

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Problem 5: Turbulent Darcy Factor with Haaland

Water flow has Re=2.00×105Re=2.00\times10^5 in a 300 mm300\text{ mm} pipe with absolute roughness ϵ=0.150 mm\epsilon=0.150\text{ mm}. Estimate the Darcy factor using Haaland.

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Problem 6: Combined Major and Local Losses

A 150 mm150\text{ mm} pipe is 100 m100\text{ m} long and carries Q=0.0300 m3/sQ=0.0300\text{ m}^3/\text{s}. Use Darcy f=0.0200f=0.0200 and total local coefficient ∑K=6.50\sum K=6.50, all referenced to the pipe mean velocity. Determine total head loss.

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Problem 7: Equivalent Length of Fittings

Fittings in a D=200 mmD=200\text{ mm} pipe have combined K=8.00K=8.00, referenced to that pipe velocity. The Darcy factor is f=0.0200f=0.0200. Determine the equivalent straight-pipe length.

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Problem 8: Diameter from an Allowable Head Loss

A pipeline must carry Q=0.0400 m3/sQ=0.0400\text{ m}^3/\text{s} over L=500 mL=500\text{ m} with hf≤10.0 mh_f\le10.0\text{ m}. For a preliminary estimate, hold Darcy f=0.0200f=0.0200 constant. Determine the diameter implied by Darcy-Weisbach.

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Problem 9: Fixed-f Diameter Sensitivity

For a comparison only, hold QQ, LL, and Darcy ff fixed. If a pipe diameter is doubled, what is the ratio of new Darcy-Weisbach major loss to old loss?

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Problem 10: Sudden-Expansion Loss

A horizontal pipe expands suddenly from D1=100 mmD_1=100\text{ mm} to D2=200 mmD_2=200\text{ mm}. The upstream mean velocity is V1=4.00 m/sV_1=4.00\text{ m/s}. Determine downstream velocity and Borda-Carnot expansion loss.

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Problem 11: Wall Shear from Friction Slope

Water flows in a full circular pipe with D=0.500 mD=0.500\text{ m}. Measured major head loss is 2.00 m2.00\text{ m} per 100 m100\text{ m} of pipe. Determine average wall shear stress.

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Problem 12: Full-Pipe Manning and Hazen-Williams Checks

A full D=0.300 mD=0.300\text{ m} circular conduit is evaluated first with Manning n=0.0130n=0.0130 and slope S=0.00500S=0.00500. Separately, a Hazen-Williams check uses the same diameter, V=1.50 m/sV=1.50\text{ m/s}, and S=0.00300S=0.00300. Determine the Manning velocity and the Hazen-Williams coefficient required by the displayed SI relation.

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