Introduction & Context

The mechanical-energy balance (an integrated form of the Bernoulli equation) is the standard tool for sizing centrifugal pumps in the process industries. It accounts for changes in elevation, static pressure, fluid velocity, and all irreversible losses (friction plus fittings). The result is the required pump head expressed in metres of flowing fluid, which can be converted to shaft power once an efficiency is assumed. The calculation is embedded in every hydraulic datasheet, P&ID review, and energy-audit checklist for water, chemicals, hydrocarbons, and food-grade fluids.

Methodology & Formulas

Control-volume definition
Station 1 (suction) → Station 2 (discharge) along a single pipe of length \(L\) and inner diameter \(D\).
Mechanical-energy balance (steady, incompressible, turbulent)
\[ \frac{p_1}{\rho g} + z_1 + \frac{v_1^2}{2g} + H_{\text{pump}} = \frac{p_2}{\rho g} + z_2 + \frac{v_2^2}{2g} + h_{\text{loss}} \] For the same pipe diameter \(v_1 = v_2\) and the kinetic terms cancel. Rearranging gives the head the pump must add: \[ H_{\text{pump}} = (z_2 - z_1) + \frac{p_2 - p_1}{\rho g} + h_{\text{loss}} \]
Head-loss decomposition
\[ h_{\text{loss}} = h_{\text{maj}} + h_{\text{min}} \]
Major (friction) loss – Darcy–Weisbach
\[ h_{\text{maj}} = f \frac{L}{D} \frac{v^2}{2g} \] with mean velocity \(v = \frac{4Q}{\pi D^{2}}\).
Friction factor for turbulent commercial steel pipes – Swamee–Jain explicit
\[ f = \frac{0.25}{\left[ \log_{10}\!\left( \dfrac{\varepsilon/D}{3.7} + \dfrac{5.74}{Re^{0.9}} \right) \right]^{2}} \]
Flow regime check

Regime Reynolds number

Laminar \(Re = \dfrac{\rho v D}{\mu} < 2300\)

Transitional \(2300 \le Re \le 4000\)

Fully turbulent \(Re > 4000\)
Minor (fitting) loss
\[ h_{\text{min}} = \sum K \frac{v^2}{2g} \] where \(\sum K\) is the sum of loss coefficients for valves, bends, entrances, exits, etc.
Application of safety margin
\[ H_{\text{req}} = H_{\text{pump}} \times \text{margin} \] Typical margins for fouling and uncertainty are 1.05–1.10.
Shaft power
\[ P_{\text{shaft}} = \frac{\rho g Q H_{\text{req}}}{\eta_{\text{pump}}} \] with \(\eta_{\text{pump}}\) supplied by the vendor or estimated from duty charts.

The Bernoulli equation is valid when the system can be treated as steady-state, incompressible, and with negligible viscous losses between the inlet and outlet reference points. For pump sizing, use it to calculate the ideal mechanical-energy increase (ΔH_ideal) the impeller must supply. Afterward, add empirical friction and minor losses (via K-values or equivalent lengths) and a vendor-supplied pump efficiency to obtain the actual head and brake horsepower. Do not rely on Bernoulli alone if the Reynolds number indicates turbulent flow in rough pipe; always correct with a Moody friction factor.

Choose one horizontal datum plane and stick to it for every calculation in the project. A common choice is the centerline of the pump suction nozzle. Express all elevations (z) relative to this plane; positive if above, negative if below. Consistency eliminates sign errors that can undersize or oversize the pump.

Yes, when the tank diameter is >10× the pipe diameter, the liquid level velocity is <1 % of the pipe velocity, so v²/2g is typically <0.005 m and may be dropped. Document the assumption; if future tank levels drop near the suction bell, re-check the term to avoid cavitation risk.

Regime	Reynolds number
Laminar	\(Re = \dfrac{\rho v D}{\mu} < 2300\)
Transitional	\(2300 \le Re \le 4000\)
Fully turbulent	\(Re > 4000\)

Application of Bernoulli Equation for Pump Sizing

Introduction & Context

Methodology & Formulas

Worked Example: Sizing a Cooling-Water Pump for a Heat-Exchanger Loop