Water of density 1000 kg m⁻³ flows steadily through a horizontal pipe whose area narrows from 8.00 × 10⁻⁴ m² to 4.00 × 10⁻⁴ m². The upstream speed is 1.50 m s⁻¹ and pressure is 180 kPa. Model the flow as incompressible and non-viscous. Determine the volume flow rate, narrow-section speed and pressure. Then explain how a real pressure reading affected by viscosity would change the inferred flow rate, and outline a calibration procedure.

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Flow rate = 1.20 × 10⁻³ m³ s⁻¹, narrow speed = 3.00 m s⁻¹, and ideal narrow pressure = 176.6 kPa. Viscous loss enlarges the pressure drop and makes an ideal calibration overestimate flow. Continuity for steady incompressible flow requires A₁v₁ = A₂v₂. The volume rate is Q = (8.00 × 10⁻⁴)(1.50) = 1.20 × 10⁻³ m³ s⁻¹. The narrow speed is v₂ = Q/A₂ = (1.20 × 10⁻³)/(4.00 × 10⁻⁴) = 3.00 m s⁻¹. Equal heights allow P₁ + ½ρv₁² = P₂ + ½ρv₂². Hence P₂ = 180000 + 500(1.50²−3.00²) = 176625 Pa, or 176.6 kPa. Viscosity dissipates mechanical energy and produces an extra pressure loss beyond the ideal Bernoulli drop. Interpreting that larger drop with an ideal equation would infer a flow rate that is too high. Calibrate by collecting a measured volume for a measured time at several valve settings while recording pressure difference. Plot actual Q against √ΔP, repeat readings, include uncertainties, and use the empirical curve rather than assuming an ideal coefficient.

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