Bernoulli's Equation

Bernoulli's Equation & Continuity — Detailed formulas, worked example and calculator

1) Fundamental formulas

Bernoulli (along a streamline, inviscid, steady):

$$\frac{P_1}{\rho g} \;+\; \frac{v_1^2}{2g} \;+\; h_1 \;=\; \frac{P_2}{\rho g} \;+\; \frac{v_2^2}{2g} \;+\; h_2$$

Continuity (incompressible):

$$Q = v_1 A_1 = v_2 A_2 \quad\text{with}\quad A=\frac{\pi d^2}{4}$$ $$\Rightarrow v_1 d_1^2 = v_2 d_2^2 \quad(\text{if circular pipe and same fluid})$$

Derived useful rearrangement (solve for \(P_2\))

From Bernoulli: $$\frac{P_1}{\rho g}+\frac{v_1^2}{2g}+h_1=\frac{P_2}{\rho g}+\frac{v_2^2}{2g}+h_2$$ Multiply by \(\rho g\) and rearrange: $$\boxed{P_2 = P_1 + \tfrac{1}{2}\rho\,(v_1^2 - v_2^2) + \rho g\,(h_1 - h_2)}$$ (This is the form used to compute pressure at point 2 given known \(P_1, v_1, v_2, h_1, h_2\).)

2) Inputs (change and press Compute)

Units: diameters in m, pressures in Pa (use 1e5 for 100 kPa), velocities in m/s, heights in m, density kg/m³, g m/s².

d₁ (diameter at section 1) d₂ (diameter at section 2) v₁ (velocity at section 1) P₁ (pressure at section 1) — in Pa h₁ (elevation of section 1) h₂ (elevation of section 2) Fluid density ρ (kg/m³) g (m/s²)

Quick results

Detailed step-by-step solution

Notes

• Bernoulli here neglects head loss (no friction) and assumes steady, incompressible flow along a streamline.
• If pipes have roughness/viscous losses, include head loss term \(h_L\) on RHS: add \(+h_L\) where appropriate.
• Pressures used above are absolute or gauge consistently. If P₁ is gauge and P₂ returned is absolute minus same atmospheric value—treat consistently.

Worked numerical example (with the values shown initially)

Given:

• \(d_1 = 0.100\ \mathrm{m},\ d_2=0.050\ \mathrm{m}\)
• \(v_1 = 2.00\ \mathrm{m/s}\)
• \(P_1 = 200{,}000\ \mathrm{Pa}\) (200 kPa)
• \(h_1 = 0.0\ \mathrm{m},\ h_2 = 1.5\ \mathrm{m}\)
• \(\rho = 1000\ \mathrm{kg/m^3},\ g = 9.81\ \mathrm{m/s^2}\)

How to interpret results: The calculator computes cross-sectional areas, flow rate, velocity at section 2 via continuity, velocity/pressure heads, and uses Bernoulli (no losses) to compute \(P_2\). Use consistent units; if you prefer to input pressures in kPa, multiply by 1000 for Pa.