Free Bernoulli Equation Calculator

Bernoulli Equation Calculator for Fluid Mechanics Analysis

This online fluid mechanics calculator harnesses the Bernoulli principle to solve for pressure, velocity, and elevation at any point along a streamline in an incompressible flow. It serves as a versatile Bernoulli equation solver and fluid pressure calculator, allowing you to compare two points on the same streamline and determine missing flow parameters instantly. Additionally, the tool can compute volumetric and mass flow rates, helping you select appropriate pipe diameters for steady flow conditions.

The Bernoulli Equation at a Glance

The Bernoulli equation describes the conservation of mechanical energy in a steady, inviscid, incompressible flow. For a fluid element moving along a streamline, the sum of its static pressure, kinetic energy per unit volume, and gravitational potential energy per unit volume remains constant:

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

Here, $p$ is the static pressure at the point of interest, $\rho$ is the fluid density (assumed constant), $v$ is the flow speed, $g$ is the acceleration due to gravity (typically $9.80665\ \mathrm{m/s}^2$ on Earth), and $h$ is the elevation relative to a reference level. The term $\frac{1}{2} \rho v^{2}$ represents the dynamic pressure (kinetic component), and $\rho g h$ is the hydrostatic pressure (gravitational component).

Comparing Two Points Along a Streamline

Because the total pressure remains constant, the equation can be written for two points along the same streamline:

$$p_1 + \frac{1}{2} \rho v_1^{2} + \rho g h_1 = p_2 + \frac{1}{2} \rho v_2^{2} + \rho g h_2$$

Given five of the six variables ($p_1, v_1, h_1, p_2, v_2, h_2$), the calculator solves for the sixth. For instance, suppose water ($\rho = 1000\ \text{kg/m}^3$) flows at point 1 with $p_1 = 1000\ \text{Pa}$, $v_1 = 2\ \text{m/s}$, and $h_1 = 3\ \text{m}$; at point 2 the pressure rises to $1200\ \text{Pa}$ with no change in elevation ($h_2 = 3\ \text{m}$). The calculator then yields $v_2 \approx 1.897\ \text{m/s}$, demonstrating how an increase in pressure reduces velocity in a horizontal flow.

Flow Rate Determination

The same Bernoulli equation solver can also compute flow rates. The volumetric flow rate $q$ (in $\text{m}^3/\text{h}$) through a pipe with diameter $d$ (in meters) is given by:

$$q = \pi \left( \frac{d}{2} \right)^{2} v \times 3600$$

where $v$ is the flow speed in $\text{m/s}$. Because the flow rate is constant along the streamline, for two points on the same streamline $q_1 = q_2$, which implies:

$$\pi \left( \frac{d_1}{2} \right)^{2} v_1 = \pi \left( \frac{d_2}{2} \right)^{2} v_2$$

The mass flow rate $\dot{m}$ is simply the product of volumetric flow rate and density:

$$\dot{m} = \rho q$$

Mass flow rate is a key specification for equipment such as fans and turbines.

Assumptions and Applicability

The Bernoulli equation as presented here relies on four key assumptions:

1. The flow is incompressible (density is constant).
2. The fluid is inviscid (viscous effects are negligible).
3. The flow is steady (properties do not change with time).
4. The analysis applies along a single streamline.

Under these conditions, the total pressure is conserved. For compressible gases, a modified form of the equation exists that accounts for specific heat, but the simple incompressible version is valid for gases at low speeds and moderate temperatures.

Practical Applications

This incompressible flow calculator is widely used in engineering to design water pumping systems, where controlling pressure variation at the pump suction is critical to avoid cavitation. The Bernoulli principle also explains everyday phenomena such as the lift on an airplane wing (faster flow over the curved upper surface creates lower pressure, generating upward lift), the Magnus effect in sports, and the operation of Venturi meters and Pitot tubes. By providing quick numerical solutions, this tool helps engineers and students verify design parameters and explore fluid behavior.

Whether you are sizing a pipe network, analyzing a pump system, or learning about fluid dynamics, this Bernoulli equation solver offers a convenient way to apply one of the most fundamental principles of fluid mechanics.