Bernoulli & Extended Energy Equation Calculator

Conservation of Energy in Pipes

The Bernoulli and Extended Energy equations govern the balance of pressure head, velocity head, and elevation head in a flowing fluid system. Real pipelines lose energy due to wall friction and components ($h_L$), and can gain or lose energy via mechanical pumps ($h_p$) and turbines ($h_t$).

• Energy Grade Line (EGL): Represents the total mechanical energy head ($H$) of the fluid system.
• Hydraulic Grade Line (HGL): Represents the sum of static pressure and elevation heads ($H - V^2/2g$).
• Continuity: Decreasing the pipe diameter increases the velocity head at the cost of static pressure head.

📝 Configuration

Execution Mode

Equation Solver Mode:

Solve For Unknown:

Fluid Properties

Fluid Density ($\rho$) [kg/m³]:

Point 1 (Upstream) Parameters

Pressure ($P_1$) [kPa]:

Velocity ($V_1$) [m/s]:

Elevation ($z_1$) [m]:

Point 2 (Downstream) Parameters

Pressure ($P_2$) [kPa]:

Velocity ($V_2$) [m/s]:

Elevation ($z_2$) [m]:

Continuity parameters (Optional)

Diameter 1 ($D_1$) [mm] (0 = skip):

Diameter 2 ($D_2$) [mm] (0 = skip):

Flow Rate ($Q$) [L/min]:

Pumps, Turbines & Losses

Pump Head added ($h_{pump}$) [m]:

Turbine Head extracted ($h_t$) [m]:

Friction Head Loss ($h_L$) [m]:

Quick Presets:

📊 Results & Grade Lines

Configure inputs and click Calculate Fluid Energy to view results.

📘 Calculation Methodology

Mathematical Model & Theory

The conservation of mechanical energy along a streamline in steady, incompressible, one-dimensional flow is described by the **Extended Energy Equation**:

$$\frac{P_1}{\rho g} + \frac{V_1^2}{2g} + z_1 + h_{pump} = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_{turb} + h_L$$

Where each term represents energy per unit weight (head, in meters):

Pressure Head ($P/\rho g$): Height of fluid column corresponding to static pressure.
Velocity Head ($V^2/2g$): Kinetic energy of the fluid per unit weight.
Elevation Head ($z$): Potential energy due to height above an arbitrary datum.
Pump Head ($h_{pump}$): Energy head added by mechanical pumps.
Turbine Head ($h_{turb}$): Energy head extracted by turbines.
Head Loss ($h_L$): Energy dissipated due to friction and fittings.

Academic References

Munson, B. R., Young, D. F., & Okiishi, T. H.: Fundamentals of Fluid Mechanics, John Wiley & Sons.
Cengel, Y. A., & Cimbala, J. M.: Fluid Mechanics: Fundamentals and Applications, McGraw-Hill, Ch. 5.

Worked Engineering Example

Problem Statement:
A pump lifts water ($\rho = 1000\text{ kg/m}^3$) from an open reservoir at $z_1 = 0\text{ m}$ to a tank at $z_2 = 15\text{ m}$ and $P_2 = 150\text{ kPa}$. The velocities are negligible at the reservoir surface and $V_2 = 2.0\text{ m/s}$ at the outlet. Pipe friction head loss is $h_L = 3.5\text{ m}$. Calculate the required pump head ($h_{pump}$).

Step-by-step Solution:
1. Apply energy equation ($P_1=0$, $V_1=0$, $h_{turb}=0$):
$$0 + 0 + z_1 + h_{pump} = \frac{P_2}{\rho g} + \frac{V_2^2}{2g} + z_2 + h_L$$ 2. Calculate heads:
- Pressure head: $\frac{150,000}{1000 \times 9.81} = 15.29\text{ m}$
- Velocity head: $\frac{2.0^2}{2 \times 9.81} = 0.20\text{ m}$
3. Solve for $h_{pump}$:
$$h_{pump} = 15.29 + 0.20 + (15 - 0) + 3.5 = 33.99\text{ m}$$
Final Result:
The pump must add 33.99 meters of head to lift the water.

🌀 Bernoulli & Extended Energy Equation Calculator