Bernoulli Equation Visualizer
Bernoulli Equation Visualizer computes fluid flow properties and visualizes energy balance along a streamline.
Formulas Used in Bernoulli Equation Visualizer
The calculator applies the Bernoulli equation for steady, incompressible, inviscid flow along a streamline, representing the conservation of mechanical energy per unit volume:
\\[
P_1 + \frac{1}{2} \rho v_1^2 + \rho g h_1 = P_2 + \frac{1}{2} \rho v_2^2 + \rho g h_2
\\]
Solve for:
- Pressure (\\(P_2\\)): \\( P_2 = P_1 + \frac{1}{2} \rho (v_1^2 – v_2^2) + \rho g (h_1 – h_2) \\)
- Velocity (\\(v_2\\)): \\( v_2 = \sqrt{v_1^2 + \frac{2}{\rho} (P_1 – P_2 + \rho g (h_1 – h_2))} \\)
- Height (\\(h_2\\)): \\( h_2 = h_1 + \frac{P_1 – P_2 + \frac{1}{2} \rho (v_1^2 – v_2^2)}{\rho g} \\)
Variables:
- \\(P_1, P_2\\): Pressure at Points 1 and 2 (Pa), representing pressure energy.
- \\(\rho\\): Fluid density (kg/m³), affecting kinetic and potential terms.
- \\(v_1, v_2\\): Velocity at Points 1 and 2 (m/s), contributing to kinetic energy (\\(\frac{1}{2} \rho v^2\\)).
- \\(g\\): Acceleration due to gravity (9.80665 m/s²), used in potential energy (\\(\rho g h\\)).
- \\(h_1, h_2\\): Height at Points 1 and 2 (m), representing potential energy.
Example Calculation
Example: Water (\\(\rho = 1000 \, \text{kg/m}^3\\)), \\(P_1 = 101325 \, \text{Pa}\\), \\(v_1 = 2 \, \text{m/s}\\), \\(h_1 = 0 \, \text{m}\\), \\(v_2 = 5 \, \text{m/s}\\), \\(h_2 = 1 \, \text{m}\\), solve for \\(P_2\\)
\\[
P_2 = 101325 + \frac{1}{2} \times 1000 \times (2^2 – 5^2) + 1000 \times 9.80665 \times (0 – 1) \approx 89325 \, \text{Pa}
\\]