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Question pattern · Pipe flow
How pipe flow questions are usually set
Usually gives you- Pipe diameters at two sections
- A velocity or flow rate at one section
- A pressure at one section, sometimes heights
Usually asks for- Velocity and pressure at the other section
- Often the flow rate as well
Strategy
- 1Continuity first. A₁v₁ = A₂v₂ gives the missing velocity.
- 2Choose a datum. Decide where z = 0. A horizontal pipe means z₁ = z₂.
- 3Bernoulli between the sections. Rearrange for the unknown pressure.
- 4Flow rate. Q = Av at either section. Both should agree.
Common traps: Forgetting to square the diameter ratio. Mixing kPa and Pa in one equation. Dropping the z terms when the pipe slopes.
SomaReady · Thermodynamics & Fluid MechanicsSample page
Topic 2 · Continuity and Bernoulli
What goes in must come out
Water cannot pile up inside a full pipe. If the pipe narrows, the same flow has to squeeze through a smaller area, so it speeds up. That is the continuity equation.
Q = A₁v₁ = A₂v₂
Bernoulli's equation says the total energy along a streamline stays constant for ideal flow. When velocity goes up, pressure must come down.
p + ½ρv² + ρgz = constant
Exam habit: use pressures in Pa (not kPa) and ρ = 1000 kg/m³ for water, so every term is in the same units.
SomaReady · Thermodynamics & Fluid MechanicsSample page
Worked example 2.3
Flow through a pipe reducer
QuestionWater flows through a horizontal pipe that reduces from 200 mm to 100 mm diameter. In the larger section the velocity is 1.5 m/s and the pressure is 150 kPa. Find the velocity and pressure in the smaller section, and the flow rate.
- 1Continuity. v₂ = v₁(d₁d₂)² = 1.5 × (2)² = 6.0 m/s
- 2Bernoulli. p₁ + ½ρv₁² = p₂ + ½ρv₂² (z₁ = z₂)
- 3Pressure. p₂ = 150 000 + ½(1000)(1.5² − 6.0²) = 133 125 Pa
- 4Flow rate. Q = π × 0.2²4 × 1.5 = 0.0471 m³/s
Answerv₂ = 6.0 m/s, p₂ ≈ 133 kPa, Q ≈ 47.1 L/s
Sense check: the velocity went up, so the pressure went down. That is what Bernoulli predicts.
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Formula sheet
Thermo & Fluids essentials
| Flow rate | Q = Av | m³/s |
| Hydrostatic | p = ρgh | Pa |
| Bernoulli | p + ½ρv² + ρgz = C | ideal flow |
| Gas law | pV = mRT | R air = 287 J/kg·K |
| Combined | p₁V₁T₁ = p₂V₂T₂ | T in kelvin |
| Polytropic | pVn = C | |
| Polytropic work | W = p₁V₁ − p₂V₂n − 1 | J |
| Isothermal work | W = p₁V₁ ln V₂V₁ | J |
| First law | Q − W = ΔU | non-flow |