Bernoulli's Equation Calculator

Enter pressures, velocities, elevations, and fluid density to calculate the downstream pressure, Bernoulli constant, dynamic pressure head, and flow energy balance.

Created by Luis GonzalezLast updated: May 9, 2026

How to Use This Calculator

1
Enter Pressure at Point 1 (Pa)
Input the static pressure of the fluid at the initial point, measured in Pascals.
2
Enter Fluid Density (kg/m³)
Provide the density of the fluid, typically around 1000 kg/m³ for water at standard conditions.
3
Enter Velocity at Point 1 (m/s)
Provide the fluid's flow velocity at the first point, expressed in meters per second.
4
Enter Velocity at Point 2 (m/s)
Input the fluid's flow velocity at the second point of interest, in meters per second.
5
Enter Height at Point 1 (m)
Specify the elevation of the fluid at the initial point relative to a reference datum, in meters.
6
Enter Height at Point 2 (m)
Specify the elevation of the fluid at the second point relative to the same reference datum, in meters.
7
Enter Gravity (m/s²)
Input the acceleration due to gravity, which is approximately 9.81 m/s² on Earth.
8
Review Your Results
The calculator displays six result cards: Pressure at Point 2, Pressure Change ΔP, Bernoulli Constant, Dynamic Pressure at P2, Pressure Head at P2, and Velocity Ratio v2/v1.

Example Calculation

An engineer analyzes fluid flow through a pipe where velocity doubles and elevation rises by 1 meter.

Pressure 1 (Pa)

200000

Fluid Density (kg/m³)

1000

Velocity 1 (m/s)

Velocity 2 (m/s)

Height 1 (m)

Height 2 (m)

Gravity (m/s²)

9.81

Results

Pressure at Point 2

176690 Pa (Pressure decreased from point 1)

Pressure Change ΔP

-23310 Pa (Kinetic/potential gains consume static pressure)

Bernoulli Constant

204500 Pa (Moderate-energy streamline)

Dynamic Pressure at P2

18000 Pa (Static pressure dominates at point 2)

Pressure Head at P2

18.011 m (Moderate pressure head at point 2)

Velocity Ratio v2/v1

2.000 (Moderate velocity increase — pressure drops)

Tips

Maintain Consistent Units

Always ensure all inputs are in SI units (Pascals, meters, seconds, kilograms) to avoid errors. Mixing units like psi with meters will lead to incorrect calculations.

Account for Energy Losses

Bernoulli's Equation assumes an ideal, incompressible, non-viscous fluid with no energy losses. For real-world systems, factor in head losses due to friction or fittings, which can significantly reduce actual pressure at point 2.

Reference Datum Matters

The chosen reference datum for height measurements must be consistent for both points. A common practice is to set the lowest point in the system as the zero-height datum for simplicity.

Understanding Fluid Dynamics with Bernoulli's Equation

The Bernoulli's Equation Calculator helps engineers, physicists, and students analyze the relationship between fluid pressure, velocity, and elevation in a steady flow. This principle is a cornerstone of fluid dynamics, widely used in applications ranging from aircraft design to pipeline engineering, where understanding pressure variations can be critical for safety and efficiency. For instance, in a typical industrial pipe system, pressure drops can be hundreds of thousands of Pascals over just a few meters if velocities and height changes are significant.

The Mathematical Framework Behind Bernoulli's Equation

Bernoulli's Equation is derived from the principle of conservation of energy applied to fluid flow. It states that for an ideal fluid flowing along a streamline, the sum of its static pressure, kinetic energy per unit volume, and potential energy per unit volume remains constant. This means if one component (like velocity) increases, another (like pressure) must decrease to maintain the balance.

The core formula used by this calculator is:

Total Head = P1 + 0.5 × rho × v1^2 + rho × g × h1
P2 = Total Head - 0.5 × rho × v2^2 - rho × g × h2

Where:

P1: Pressure at point 1 (Pascals)
P2: Pressure at point 2 (Pascals)
rho: Fluid density (kilograms per cubic meter)
v1: Velocity at point 1 (meters per second)
v2: Velocity at point 2 (meters per second)
g: Gravitational acceleration (meters per second squared)
h1: Height at point 1 (meters)
h2: Height at point 2 (meters)

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Calculating Pressure at Point 2: A Worked Example

Consider an engineer evaluating a water pumping system where water flows from a lower point to a higher point.

Here are the known conditions:

Pressure 1 (P1): 200,000 Pa
Velocity 1 (v1): 2 m/s
Height 1 (h1): 10 m
Velocity 2 (v2): 4 m/s
Height 2 (h2): 15 m
Fluid Density (rho): 1000 kg/m³ (for water)
Gravity (g): 9.81 m/s²

Let's calculate the pressure at point 2:

Calculate the Total Head Constant at Point 1: Total Head = P1 + 0.5 × rho × v1² + rho × g × h1 Total Head = 200,000 + 0.5 × 1000 × (2)² + 1000 × 9.81 × 10 Total Head = 200,000 + 0.5 × 1000 × 4 + 98,100 Total Head = 200,000 + 2,000 + 98,100 = 299,100 Pa
Calculate Pressure at Point 2 (P2): P2 = Total Head - 0.5 × rho × v2² - rho × g × h2 P2 = 299,100 - 0.5 × 1000 × (4)² - 1000 × 9.81 × 15 P2 = 299,100 - 0.5 × 1000 × 16 - 147,150 P2 = 299,100 - 8,000 - 147,150 = 143,950 Pa

Therefore, the pressure at point 2 is 143,950 Pa. The total head constant is 299,100 Pa, and the pressure change is -56,050 Pa (143,950 - 200,000).

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Manual Calculation Walkthrough

Understanding the manual calculation of Bernoulli's Equation is crucial for grasping its underlying principles. Let's use the example values: Pressure 1 = 200,000 Pa, Velocity 1 = 2 m/s, Height 1 = 10 m, Velocity 2 = 4 m/s, Height 2 = 15 m, Fluid Density = 1000 kg/m³, and Gravity = 9.81 m/s².

First, calculate the total mechanical energy per unit volume (the total head constant) at Point 1. This involves summing the static pressure, dynamic pressure, and hydrostatic pressure. Total Head = P1 + (0.5 * rho * v1²) + (rho * g * h1) Total Head = 200,000 Pa + (0.5 * 1000 kg/m³ * (2 m/s)²) + (1000 kg/m³ * 9.81 m/s² * 10 m) Total Head = 200,000 + (0.5 * 1000 * 4) + (9810 * 10) Total Head = 200,000 + 2,000 + 98,100 = 299,100 Pa.

Next, use this constant to find the pressure at Point 2. Subtract the dynamic and hydrostatic pressure components at Point 2 from the total head. P2 = Total Head - (0.5 * rho * v2²) - (rho * g * h2) P2 = 299,100 Pa - (0.5 * 1000 kg/m³ * (4 m/s)²) - (1000 kg/m³ * 9.81 m/s² * 15 m) P2 = 299,100 - (0.5 * 1000 * 16) - (9810 * 15) P2 = 299,100 - 8,000 - 147,150 = 143,950 Pa.

The pressure at Point 2 is 143,950 Pa. This step-by-step breakdown ensures that each component of the energy equation is accounted for, leading to the final pressure determination.

How professionals interpret Bernoulli's Equation output

Professionals, particularly mechanical and civil engineers, interpret the output of Bernoulli's Equation to make critical design and operational decisions in fluid systems. When calculating the pressure at a second point (P2), they look for several key indicators. A significantly reduced P2 compared to P1 often signals a substantial increase in velocity or elevation, which could indicate a bottleneck or a need for a pump to maintain flow. For example, in a water distribution network, a predicted P2 below 200,000 Pa (approximately 2 bar) might be concerning for residential supply, as typical household pressure ranges from 275,000 to 550,000 Pa.

Conversely, a higher P2 could suggest a decrease in flow velocity, potentially leading to sedimentation in pipes or an undesirable buildup of static pressure. In aerospace engineering, understanding the pressure difference across an airfoil (which Bernoulli's principle helps explain) is crucial for generating lift. A pressure difference of 5,000-10,000 Pa between the upper and lower surfaces of a wing can be sufficient to generate significant lift for small aircraft. Engineers use these values to validate computational fluid dynamics (CFD) models and ensure their designs meet safety and performance standards, often requiring pressure values to remain within ±15% of ideal operational ranges.

Frequently Asked Questions

What is Bernoulli's principle used for?

Bernoulli's principle is fundamental in fluid dynamics, used to understand how fluid velocity, pressure, and height are related along a streamline. It's applied in designing aircraft wings, analyzing pipe flows, and even understanding blood circulation, often predicting pressure changes within ±10% in ideal scenarios.

When is Bernoulli's Equation applicable?

Bernoulli's Equation is applicable for steady, incompressible, non-viscous flow along a streamline. This means the fluid density doesn't change significantly, there's minimal internal friction, and the flow conditions remain constant over time. It's often accurate within 5% for low-viscosity fluids like water or air.

What does the 'total head constant' represent?

The total head constant in Bernoulli's Equation represents the total mechanical energy per unit volume of an ideal fluid along a streamline. It's the sum of the static pressure, dynamic pressure (due to velocity), and hydrostatic pressure (due to elevation). This constant typically ranges from thousands to millions of Pascals in engineering applications.

How does fluid density affect pressure calculations?

Fluid density is a critical factor in Bernoulli's Equation, directly impacting both the dynamic pressure and hydrostatic pressure terms. A denser fluid will exert greater pressure for the same velocity and height changes. For instance, water (approx. 1000 kg/m³) will show significantly larger pressure changes than air (approx. 1.2 kg/m³) under identical conditions.