Fluid Mechanics and Mechanical Operations
Fluid mechanics and mechanical operations is the most formula-dense part of GATE CH, and much of its marks come from one correct substitution into the right correlation. It covers statics, pipe flow and friction, pumps, metering, packed and fluidised beds, settling, filtration and size-reduction equipment.
Core rules
- Pipe flow. h_f = f(L/D)(v²/2g), where f is the Darcy factor (four times the Fanning factor). Laminar flow: f = 64/Re. At fixed volumetric flow, velocity scales as 1/D² and head loss as 1/D⁵.
- Energy equation. Bernoulli extended with pump head and friction: P₁/ρg + z₁ + v₁²/2g + H_p = P₂/ρg + z₂ + v₂²/2g + h_f.
- Pumps. Shaft power = ρgQH/η. NPSH_a = (P_atm − P_v)/ρg + z − h_f; cavitation begins when NPSH_a falls below NPSH_r. Affinity laws: Q ∝ N, H ∝ N², P ∝ N³.
- Meters. Orifice and venturi both give Q ∝ √ΔP; the orifice carries a permanent pressure loss that the venturi avoids.
- Packed bed. The Ergun equation adds a viscous term (∝1/Re) to an inertial term; both use superficial velocity.
- Settling. Stokes: v_t = d²(ρ_p − ρ)g/18μ, valid only while Re = ρvd/μ is below about 1.
- Filtration. Constant-pressure cake filtration gives V² ∝ t once medium resistance is negligible.
Worked example
A 0.1 mm sand particle (ρ_p = 2500 kg/m³) settles in water (ρ = 1000 kg/m³, μ = 10⁻³ Pa·s). Stokes gives v_t = (10⁻⁴)²(1500)(9.81)/(18×10⁻³) = 8.2×10⁻³ m/s. Check Re = 1000×8.2×10⁻³×10⁻⁴/10⁻³ ≈ 0.82, below 1, so Stokes was legitimate.
Traps
- Inserting Fanning f into a Darcy–Weisbach formula - a factor-of-four error.
- Applying Stokes to a particle with Re of 50.
- Confusing NPSH available with NPSH required.
- Using interstitial rather than superficial velocity in Ergun.