10:2 Pipe Flow - Friction Factors, Moody Charts
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Overview
Derek Elsworth connects pipe-flow friction to dimensional similitude and derives the Darcy friction factor f = 64/Re for laminar flow from the Navier–Stokes solution for flow in a circular pipe. He explains how Moody charts relate friction factor to Reynolds number and relative roughness, then applies the chart in Bernoulli energy balances, where unknown flow velocity may require iteration.
Key takeaways
- For fully developed laminar flow in a circular pipe, the Darcy friction factor is f = 64/Re, derived from the parabolic Hagen–Poiseuille velocity profile.
- Relative roughness is ε/D, and Moody charts use it alongside Reynolds number to estimate friction factor in turbulent pipe flow.
- In the fully rough turbulent regime, friction factor is largely independent of Reynolds number, but in other turbulent conditions it can still depend on both Reynolds number and roughness.
- Pipe head loss follows hL = f(L/D)(V̄²/2g), so longer pipes and larger friction factors increase the energy required to maintain a given flow.
- When pipe velocity is unknown, solving the Bernoulli equation generally requires iteration because velocity determines Reynolds number, which determines friction factor.
- Dimensional similarity organizes pipe, external, and open-channel flow problems around geometry, flow regime, and a dimensionless measure of forces or resistance.
Chapters
- Geometric similitude matches model and prototype dimensions, Reynolds number matches flow regime, and Euler number relates pressure head to velocity head.
- A San Bruno gas-pipeline explosion illustrates how corrosion and embrittlement can weaken pipelines over time.
- The Lac-Mégantic rail disaster illustrates the risks of transporting volatile oil by train; pipelines are useful infrastructure but are not risk-free.
- The energy equation extends Bernoulli by including pump head added to the flow and head lost to friction.
- Bends and fittings create minor losses, while friction along a long pipe creates major losses that accumulate over distance.
- Pressure-drop measurements at known velocities provide Euler-number and Reynolds-number pairs that can be plotted to characterize pipe friction.
- The pipe-flow setup uses seven variables: length, diameter, roughness, velocity, density, viscosity, and pressure drop.
- With mass, length, and time as the three base dimensions, Buckingham Pi analysis gives four dimensionless groups.
- The groups represent pipe geometry, Reynolds number for kinematic similarity, and a pressure-to-velocity relationship for dynamic similarity.
- For fully developed laminar flow in a circular pipe, the velocity profile is parabolic, with maximum speed at the center and zero speed at the wall.
- The mean velocity follows the Hagen–Poiseuille relation: V̄ = D²ΔP/(32μL).
- Rewriting the pressure head loss in Darcy–Weisbach form gives f = 64/Re, where Re = ρV̄D/μ.
- The Moody chart plots friction factor against Reynolds number on logarithmic axes, with relative roughness ε/D distinguishing pipe surfaces.
- For laminar flow, f = 64/Re and roughness does not affect the friction factor; the transition region begins around Re = 2,000.
- In turbulent flow, the chart accounts for both Reynolds number and relative roughness; in the fully rough limit, friction factor becomes largely independent of Reynolds number.
- For a pipe connecting large tanks, the tank velocities are approximately zero and both free surfaces are at atmospheric pressure.
- The pipe-loss term is hL = f(L/D)(V̄²/2g), using the average velocity inside the pipe.
- If velocity is known, calculate Reynolds number, obtain f from the Moody chart, and evaluate the energy balance; if velocity is unknown, iterate because f depends on Reynolds number.
- The friction-factor relationship changes between laminar flow, transition, and turbulent flow, so selecting the appropriate Moody-chart region matters.
- Matching Reynolds number preserves the flow regime when comparing a scale model with a prototype.
- When solving for an unknown velocity, the corresponding Reynolds number and friction factor must be updated together until the energy equation is satisfied.
- External flow around a body uses the same similarity framework, with wind-tunnel force measurements related to flow velocity through dimensionless groups.
- Vortex shedding can produce oscillating forces; Elsworth links this mechanism to the Tacoma Narrows Bridge failure.
- Open-channel flow is driven by elevation and gravity rather than a pressure difference, and its resistance is commonly represented by Manning’s coefficient.
- The upcoming pipe-flow exercises will use Moody charts to calculate velocities and pressure losses in practical systems.
- Elsworth previews analyzing the umbilical tube supplying a jet motor, where frictional losses must be included in the energy balance.
- The lecture closes by connecting pipe-flow theory to everyday fuel delivery, including how gasoline reaches a car.
Transcript excerpt
0:00 And so we'll talk bit more about pipe flow today and also next week. These two weeks are devoted to flow in pipes. Talked last time about different regimes in terms of similitude. You have to have geometric similitude. A scale model if you're using it has to be scaled in all the dimensions. Dynamic kinematic similitude was that the flow regimes match up and we usually do that in terms of the Reynolds number to ensure that. >> [snorts] >> And then once we meet those two requirements, we then can look at dynamic similitude. And dynamic similitude's enforced through the Euler number which is the ratio of the pressure head to the velocity head.
0:47 And it'll it's the same number that we or same equation we use to calculate the force that's applied on the F-150 that sits on the runway at uh at SFO. The velocity is the velocity coming out of the jet engine and the pressure is the velocity being killed on the side of the truck. And that pressure gives you a a force which then you can use to resolve uh resolve forces. Uh fine. I won't put the sound on and I'll scoot through this a bit more. Actually not far away from San Francisco Airport, San Bruno, the airport we just flew
1:35 over. And I don't know exactly when this was, but this was a a gas pipeline explosion in a neighborhood [snorts] that obviously didn't do very much positive for the the neighborhood. I think I mentioned in this class that when the plan was to convert the coal plant at the corner of Burrows and College Avenue. Uh from coal to gas, it's used to create hot water that's used for the district heating on campus. [snorts] Uh when that was voiced, the plan was to
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Related notes
Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Derek Elsworth.