Fluid Mechanics — Flow Through Pipes and Hydraulic Machines
🟢 Lite — Quick Review (1h–1d)
Rapid summary for last-minute revision before your exam.
Laminar vs Turbulent Flow
- Reynolds number: Re = ρVD/μ = VD/ν
- Re < 2000 → Laminar; Re > 4000 → Turbulent; 2000–4000 → Transition
Pipe Flow Losses
- Darcy-Weisbach: h_f = f(L/D)(V²/2g)
- Major losses: f = 64/Re (laminar); Moody chart (turbulent)
- Minor losses: h_m = K(V²/2g) — entrances, exits, elbows, valves
Pipes in Series/Parallel
- Series: Q same; h_f_total = Σh_f; equivalent D from Hazen-Williams or Darcy
- Parallel: Head loss same; Q_total = ΣQ_i
Hydraulic Machines
- Turbines: Pelton (impulse), Francis (reaction mixed), Kaplan (reaction axial)
- Pumps: Centrifugal — head H = (p₂−p₁)/ρg + (V₂²−V₁²)/2g + z₂−z₁
Key Formulas Table
| Concept | Formula |
|---|---|
| Reynolds Number | Re = ρVD/μ = VD/ν |
| Darcy Friction | h_f = f(L/D)(V²/2g) |
| Minor Loss | h_m = K(V²/2g) |
| Pump Power | P = ρgQH/η |
| Turbine Power | P = ρgQH·η |
| Specific Speed (Turbine) | N_s = N√P/H^1.25 |
| Specific Speed (Pump) | N_s = N√Q/H^0.75 |
⚡ GATE Tip: Pipe network problems are common. Master the series/parallel rules and the Moody chart for friction factor.
🟡 Standard — Regular Study (2d–2mo)
Standard content for students with a few days to months.
Flow Classification and Reynolds Number
The Reynolds number classifies flow regime:
Re = ρVD/μ = VD/ν
where V = average velocity, D = pipe diameter, μ = dynamic viscosity, ν = kinematic viscosity.
- Laminar (Re < 2000): Viscous forces dominate; parabolic velocity profile; f = 64/Re
- Turbulent (Re > 4000): Inertial forces dominate; flat velocity profile; f from Moody chart
- Transition (2000–4000): Hysteresis region; behavior unpredictable
⚡ GATE Watch: Friction factor f is defined as the Darcy friction factor (fanning friction factor is 1/4 of Darcy). Verify which one is being used — most GATE problems use Darcy.
Darcy-Weisbach Equation
The head loss due to friction in a pipe of length L and diameter D is:
h_f = f × (L/D) × (V²/2g)
The friction factor f depends on:
- Laminar: f = 64/Re (theoretical, derived from Navier-Stokes)
- Turbulent: Use Moody chart or Colebrook-White equation
Colebrook-White Equation (Implicit)
1/√f = −2 log₁₀(ε/(3.7D) + 2.51/(Re√f))
where ε is the absolute roughness of the pipe inner surface.
Swamee-Jain equation (explicit approximation):
f = 0.25 / [log₁₀(ε/(3.7D) + 5.74/Re⁰·⁹)]² (valid for Re > 10⁵)
Minor Losses
Minor head losses occur at fittings, bends, expansions, contractions, and valves:
h_m = K × (V²/2g)
| Fitting | K value |
|---|---|
| Pipe entrance (sharp) | 0.5 |
| Pipe entrance (bell-mouth) | 0.05–0.1 |
| Pipe exit (sudden) | 1.0 |
| 90° elbow (standard) | 0.3–0.9 |
| 90° elbow (long radius) | 0.2 |
| Open globe valve | 6.0 |
| Open gate valve | 0.15 |
⚡ GATE Tip: For fully developed pipe flow over long distances, minor losses are often negligible compared to major (friction) losses. But for short pipes with many fittings, they matter.
Pipes in Series and Parallel
Series Connection
- Same discharge through all pipes: Q₁ = Q₂ = Q₃
- Total head loss: h_f,total = h_f₁ + h_f₂ + h_f₃
- Equivalent pipe: solve for D_eq such that h_f,eq = h_f,total for same Q
Parallel Connection
- Same head loss across each branch: h_f₁ = h_f₂
- Total discharge: Q_total = Q₁ + Q₂
- Flow splits according to resistance (Q proportional to 1/√f_i·L_i/D_i^5)
Equivalent Pipe Concept
For a pipe of length L_eq and diameter D_eq that gives the same head loss as the original system at the same flow rate:
h_f = f(L_eq/D_eq)(Q²/k²) where k = A√(2gD)
Hydraulic Machines — Turbines
Classification
| Type | Action | Head Range | Specific Speed |
|---|---|---|---|
| Pelton | Impulse (jet) | High (>300 m) | Low (10–50) |
| Francis | Reaction (mixed flow) | Medium (30–300 m) | Medium (50–300) |
| Kaplan | Reaction (axial) | Low (<30 m) | High (300–1000) |
Turbine Parameters
Power developed by turbine:
P = ρg Q H η_t
Specific speed (dimensionless form):
N_s = N√P / H^1.25 (turbines) N_s = N√Q / H^0.75 (pumps)
where N is operating speed (RPM), P is power (kW), H is head (m), Q is discharge (m³/s).
⚡ GATE Watch: Kaplan turbines look like propeller turbines but have adjustable runner blades — making them efficient across a range of flows. Francis turbines have a scroll casing and guide vanes.
Centrifugal Pumps
Operating Characteristics
- Head developed: H = (p₂−p₁)/ρg + (V₂²−V₁²)/2g + (z₂−z₁)
- Power input: P_in = ρgQH/η
- Suction head: NPSH available must exceed NPSH required (cavitation criterion)
Pump Affinity Laws (Similarity Laws)
For geometrically similar pumps operating at different speeds or sizes:
Q ∝ N × D³ H ∝ N² × D² P ∝ N³ × D⁵
⚡ Scaling: If speed changes from N₁ to N₂ at same diameter, Q₂/Q₁ = N₂/N₁, H₂/H₁ = (N₂/N₁)², P₂/P₁ = (N₂/N₁)³
Example Problem
Water (ρ = 1000 kg/m³) flows through a smooth pipe (D = 200 mm, L = 500 m) at 0.1 m³/s. Kinematic viscosity ν = 1×10⁻⁶ m²/s. Find head loss using Darcy-Weisbach (assume turbulent, f = 0.02).
Solution: V = Q/A = 0.1/(π×0.2²/4) = 0.1/0.0314 = 3.18 m/s Re = VD/ν = 3.18×0.2/1×10⁻⁶ = 6.36×10⁵ (clearly turbulent) h_f = f(L/D)(V²/2g) = 0.02×(500/0.2)×(3.18²/(2×9.81)) h_f = 0.02×2500×0.516 = 25.8 m of water
🔴 Extended — Deep Study (3mo+)
Comprehensive coverage for students on a longer study timeline.
Moody Chart Mastery
The Moody chart plots friction factor f versus Re for various relative roughness ε/D. Key zones:
- Laminar zone (Re < 2000): f = 64/Re (straight line, roughness irrelevant)
- Transition zone (2000–4000): Friction factor unreliable
- Hydraulically smooth zone (4000–10⁵): f depends mainly on Re; roughness negligible
- Transitional zone (10⁵–10⁶): f depends on both Re and ε/D
- Fully rough zone (Re > 10⁶): f depends mainly on ε/D; Re effect vanishes
⚡ GATE Strategy: For fully rough turbulent flow, the Colebrook equation simplifies to 1/√f ≈ 2 log₁₀(3.7D/ε). This is why head loss becomes independent of flow rate at very high Re.
Series-Parallel Pipe Networks
Hazen-Williams formula (empirical, used for water supply networks):
h_f = 10.67 × L × Q^1.852 / (C^1.852 × D^4.87)
where C is the Hazen-Williams roughness coefficient (120 for smooth pipes, 140 for new cast iron).
Pump Cavitation and NPSH
Net Positive Suction Head Available:
NPSH_A = (p_atm/ρg − p_v/ρg) − h_s + h_acc
where p_v = vapor pressure, h_s = suction lift, h_acc = acceleration head.
Cavitation occurs when NPSH_A < NPSH_required (specified by manufacturer).
Pump-Pipe System Curve
The system head curve (required by pump) consists of:
- Static head: H_s = z₂ − z₁
- Friction head: h_f ∝ Q² (Darcy friction)
- Velocity head differences
The operating point is where the pump curve intersects the system curve.
Turbine Governing
Pelton turbines use jet deflectors and spear valves to regulate flow. Francis turbines use guide vane angle adjustment. Kaplan turbines adjust both guide vanes and runner blade angle.
⚡ GATE Numerical Pattern: Pipe flow problems frequently combine continuity, Darcy-Weisbach, and Bernoulli in a single question. Start with what you know, identify unknowns, apply equations in sequence.
Water Hammer
Rapid valve closure creates a pressure surge (water hammer) traveling at speed a:
a = √(K/ρ) / √(1 + Kd/Et) (for thin-walled pipes)
Pressure rise: Δp = ρaΔV. This can cause significant pipe stress — addressed by surge tanks and relief valves.
Example — Pump Selection
A pump delivers 0.05 m³/s water to a height of 30 m through a pipe (D = 150 mm, L = 200 m, f = 0.015). Pump efficiency = 75%. Find pump power required.
Solution: V = Q/A = 0.05/(π×0.15²/4) = 2.83 m/s h_f = f(L/D)(V²/2g) = 0.015×(200/0.15)×(2.83²/19.62) = 0.015×1333×0.408 = 8.16 m Total head H = static + friction = 30 + 8.16 = 38.16 m P_hydraulic = ρgQH = 1000×9.81×0.05×38.16 = 18,710 W = 18.7 kW P_motor = P_hydraulic/η = 18.7/0.75 = 24.9 kW
GATE Previous Year Pattern
| Year | Topic | Marks |
|---|---|---|
| 2023 | Darcy-Weisbach + minor losses | 5 |
| 2022 | Centrifugal pump affinity laws | 2 |
| 2021 | Pipes in series/parallel network | 5 |
| 2020 | Specific speed — turbine selection | 2 |
| 2019 | Moody chart — turbulent friction | 5 |
⚡ Common Mistakes: (1) Confusing Darcy f with Fanning f (4× difference), (2) Wrong K value for minor loss coefficient, (3) Adding minor losses to major losses incorrectly, (4) Forgetting to include velocity head in pump head calculation.
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