Bernoulli Equation Calculator | Pressure, Velocity & Height In Fluid Flow

Q: 2. What are the assumptions of the Bernoulli's equation?

Steady flow: The fluid properties at any point do not change with time. Incompressible fluid: The fluid density remains constant throughout the flow. Non-viscous flow: The effect of viscosity (internal friction) is neglected. Along a streamline: The equation applies only along a single streamline. No energy loss: There is no loss of energy due to friction, turbulence, or heat transfer.

In chemical engineering calculations,the Bernoulli principle is widely use to study fluid flow behavior. To learn this calculation, we have created the Bernoulli Equation Calculator, which will help process engineers and students to find pressure, velocity, or height between two points in a fluid system

Bernoulli Equation Calculator

Calculate pressure, velocity, and elevation relationships in fluid flow

P₁ + ½ρv₁² + ρgh₁ = P₂ + ½ρv₂² + ρgh₂

Point 1

Pressure (P₁) Pa (Pascals)

Velocity (v₁) m/s

Height (h₁) m (meters)

Point 2

Pressure (P₂) Pa (Pascals)

Velocity (v₂) m/s

Height (h₂) m (meters)

Fluid Properties

Fluid Density (ρ) kg/m³ (default: 1000 for water)

Gravity (g) m/s² (default: 9.81)

Results

How to use: Enter 5 out of 6 values (leave one blank to solve for it). The calculator will determine the missing value using the Bernoulli equation.

How to Use the Calculator:

Enter any five known values for pressure, velocity, or height at two points.
Leave the unknown value field blank.
Click the “Calculate” button.
The calculator will instantly show the missing value and the energy balance at both points.

What is Bernoulli Equation?

It is a fundamental principle in fluid mechanics that states that the total mechanical energy of a steady, incompressible, and frictionless fluid remains constant along a streamline. It combines pressure energy, kinetic energy, and potential energy of the fluid and is expressed as:

$P + \frac{1}{2} \rho v^2 + \rho g h = \text{constant}$

Example

A toluene storage tank (2.5 m diameter, 2.5 m liquid height) open to the atmosphere is drained through a 5 cm outlet at the bottom. The outlet is 2.5 m below the surface, and C_d=0.62. Find the outlet velocity and flowrate using Bernoulli’s equation.

Assume the density of toluene ρ=867 kg/m³

Height h = 2.5m, Outlet diameter D = 0.05 m, Discharge coefficient C_d= 0.62, Density (ρ) = 867 kg/m³

$v_2 = \sqrt{2 g (h_1 - h_2)}$

The ideal (Torricelli) velocity is:

$v_{\text{ideal}}=\sqrt{2 g h}$

Numerical:

$v_{\text{ideal}}=\sqrt{2\times 9.81 \times 2.5}=7.0036\ \mathrm{m/s}$

Apply the discharge coefficient to get actual exit velocity:

$v = C_d\,v_{\text{ideal}} = 0.62\times 7.0036 = 4.3422\ \mathrm{m/s}$

Outlet cross-sectional area:

$A=\frac{\pi d^2}{4}=\frac{\pi (0.05)^2}{4}=0.0019635\ \mathrm{m^2}$

Volumetric flowrate:

$Q = A\,v = 0.0019635 \times 4.3422 = 0.0085259\ \mathrm{m^3/s}$

Convert:

$Q = 0.0085259\ \mathrm{m^3/s} = 8.53\ \mathrm{L/s} = 30.69\ \mathrm{m^3/h}$

Mass flowrate (using $\rho=865\ \mathrm{kg/m^3}$

$\dot m = \rho\,Q = 865 \times 0.0085259 = 7.3749\ \mathrm{kg/s} \approx 26.55\ \mathrm{t/h}$

$v \approx 4.34\ \mathrm{m/s},\quad Q \approx 0.00853\ \mathrm{m^3/s}\ (8.53\ \mathrm{L/s}),\quad \dot m \approx 7.37\ \mathrm{kg/s}\ (26.55\ \mathrm{t/h})$

FAQ

1. What is Bernoulli's equation in simple terms?

It states that as the velocity of a fluid increases, its pressure decreases, and when the velocity decreases, the pressure rises, provided there is no energy loss due to friction.

2. What are the assumptions of the Bernoulli's equation?

Steady flow: The fluid properties at any point do not change with time.
Incompressible fluid: The fluid density remains constant throughout the flow.
Non-viscous flow: The effect of viscosity (internal friction) is neglected.
Along a streamline: The equation applies only along a single streamline.
No energy loss: There is no loss of energy due to friction, turbulence, or heat transfer.