ME 310 Chap.9 Viscous Flow in Pipes
Viscous Flow in Pipes
Internal flow as an application
The syllabus places pipe and channel flow among the applications of the earlier theory, and the calendar assigns four lectures to viscous flow in pipes in early November. It combines the energy equation, the Navier-Stokes exact solution and dimensional analysis.
Laminar or turbulent
The Reynolds number decides the regime.
In round pipes, flow is usually laminar below about 2100 and turbulent above about 4000. Near the inlet the velocity profile develops; after the entrance length it stops changing and the flow is fully developed. Laminar fully developed flow has a parabolic profile and a flow rate proportional to the fourth power of diameter and to the pressure drop, so the friction factor is exactly 64 divided by the Reynolds number.
Turbulent flow has a blunter profile, much steeper near the wall, and a larger loss.
Head loss and friction factor
The Darcy-Weisbach equation writes the major head loss as friction factor times length over diameter times velocity head.
Dimensional analysis shows the turbulent friction factor depends on Reynolds number and relative roughness; the Colebrook relation and its chart, the Moody chart, give the values, and explicit approximations avoid iteration.
At very high Reynolds number the friction factor depends on roughness alone.
Minor losses and systems
Entrances, bends, valves and exits add losses proportional to velocity head through loss coefficients. A pumping problem then becomes one line of the energy equation: pump head equals elevation gain plus all losses.
Problems come in three types: finding the pressure drop is direct, while finding the flow rate or the diameter needs iteration because the friction factor depends on the unknown.
Reading a problem statement
Before any calculation, identify whether the unknown is the pressure drop, the flow rate or the diameter, list the minor-loss fittings, and note the pipe material so the roughness can be looked up.
What this chapter covers
- 01
Reynolds number and flow regime
- 02
Entrance length and fully developed flow
- 03
Laminar pipe flow and the fourth-power law
- 04
Darcy-Weisbach head loss
- 05
Friction factor, roughness and the Moody chart
- 06
Minor loss coefficients
- 07
Three types of pipe problem
Worked example · free
Pressure drop for laminar oil flow
- 1Average speed V = Q/A = 3.0 × 10⁻⁴ / 7.07 × 10⁻⁴ = 0.424 m/s.
- 2Re = ρVD/μ = 900 × 0.424 × 0.030 / 0.20 = 57, well inside the laminar range.
- 2Poiseuille: Δp = 128 μ ℓ Q / (π D⁴) = 128 × 0.20 × 20 × 3.0 × 10⁻⁴ / (π × 8.1 × 10⁻⁷).
- 2Δp = 60.4 kPa; the friction factor route with f = 64/57 gives the same value.
Key terms
- Friction Factor
- The friction factor is the dimensionless coefficient in the Darcy-Weisbach equation linking pipe head loss to velocity head and length over diameter.
- Relative Roughness
- Relative roughness is the average height of wall roughness divided by the pipe diameter.
- Minor Loss
- A minor loss is the head lost in a fitting, bend, valve, entrance or exit, expressed as a loss coefficient times the velocity head.
- Entrance Length
- The entrance length is the distance from a pipe inlet over which the velocity profile develops before becoming fully developed.
- Moody Chart
- The Moody chart is the plot of turbulent and laminar friction factor against Reynolds number for different relative roughness values.
Viscous Flow in Pipes FAQ
How do I decide between 64 over Re and the Moody chart?
Calculate the Reynolds number first. Below roughly 2100 in a round pipe the flow is laminar and 64 over Re is exact; above roughly 4000 it is turbulent and the friction factor must come from the Moody chart or a correlation.
Why do some pipe problems need iteration?
When the flow rate or the diameter is unknown, the Reynolds number is unknown too, and the turbulent friction factor depends on it. Guess a friction factor, solve, update Re and repeat until the values stop changing.
Are minor losses really minor?
Not always. In long straight pipes they are small next to friction, but in short systems with many fittings they can equal or exceed the pipe friction loss.
Why does halving the diameter matter so much in laminar flow?
At a fixed pressure drop the laminar flow rate is proportional to the fourth power of diameter. Halving the diameter therefore reduces the flow to one sixteenth, or requires sixteen times the pressure drop for the same flow.
Exam move
Work every pipe problem as the energy equation first, then fill in the loss terms; never start from a friction chart without knowing why you need it.
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