Bernoulli Equation Calculator (Solve for Mass and Volume Flow Rate)

| Added in Physics

What is the Bernoulli Equation Calculator?

The Bernoulli Equation Calculator computes volumetric and mass flow rates for fluid flow through pipes. Using the pipe diameter, flow velocity, and fluid density, this calculator determines how much fluid flows through the pipe per unit time.

Formulas

The calculator uses these fundamental flow rate equations:

$$Q = A \times v = \pi \left(\frac{d}{2}\right)^2 \times v$$

$$\dot{m} = Q \times \rho$$

Where:

  • Q = Volumetric flow rate (m³/s or ft³/s)
  • A = Cross-sectional area of pipe (m² or ft²)
  • d = Pipe diameter (m or ft)
  • v = Flow velocity (m/s or ft/s)
  • m-dot = Mass flow rate (kg/s or lb/s)
  • rho = Fluid density (kg/m³ or lb/ft³)

How to Use

  1. Select Unit System: Choose between metric (meters, kg) or imperial (feet, pounds)
  2. Enter Pipe Diameter: Input the internal diameter of the pipe
  3. Enter Flow Velocity: Input the average velocity of the fluid
  4. Enter Fluid Density: Input the density of the fluid (e.g., water is approximately 1000 kg/m³ or 62.4 lb/ft³)
  5. Calculate: Click the button to see volumetric and mass flow rates

Example Calculation

Example: Water flowing through a pipe with the following parameters:

  • Diameter: 0.1 m
  • Velocity: 2 m/s
  • Density: 1000 kg/m³ (water)

Solution:

First, calculate the cross-sectional area:
$$A = \pi \left(\frac{0.1}{2}\right)^2 = \pi \times 0.05^2 = 0.00785 \text{ m}^2$$

Volumetric flow rate:
$$Q = 0.00785 \times 2 = 0.0157 \text{ m}^3\text{/s}$$

Convert to m³/hr:
$$Q = 0.0157 \times 3600 = 56.55 \text{ m}^3\text{/hr}$$

Mass flow rate:
$$\dot{m} = 0.0157 \times 1000 = 15.7 \text{ kg/s}$$

Applications

  • Pipeline Design: Sizing pipes for required flow rates
  • Pump Selection: Determining pump capacity requirements
  • HVAC Systems: Calculating air or water flow in ducts and pipes
  • Process Engineering: Monitoring and controlling fluid flow in industrial processes
  • Water Distribution: Designing municipal water supply systems
  • Chemical Processing: Calculating reactant feed rates

Related Calculators

  • Continuity Equation Calculator
  • Pressure Drop Calculator
  • Reynolds Number Calculator
  • Pipe Flow Velocity Calculator

Frequently Asked Questions

Volumetric flow rate measures the volume of fluid passing through a pipe per unit time (e.g., m³/hr), while mass flow rate measures the mass of fluid per unit time (e.g., kg/s). Mass flow rate equals volumetric flow rate multiplied by fluid density.

The cross-sectional area of a circular pipe is calculated using A = pi times (d/2)², where d is the pipe diameter. This area is then multiplied by velocity to get volumetric flow rate.

Pure water at 20 degrees C has a density of approximately 998 kg/m³ or 62.4 lb/ft³. For practical calculations, 1000 kg/m³ is commonly used.

This calculator assumes ideal flow conditions without friction losses. For actual pipe systems, pressure drops due to friction should be calculated separately using the Darcy-Weisbach equation.