🚀 Bernoulli Calculator & Simulator
- Set the pipe inlet (D1) and constricted throat (D2) diameters: Drag the sliders to specify the inlet diameter and the constricted throat diameter. (The throat diameter must always be smaller than the inlet diameter.)
- Set the inlet velocity and static pressure (P1): Adjust the average flow velocity (m/s) and initial static pressure (kPa) as the fluid enters the Venturi inlet.
- Observe real-time Venturi pressure drop: Witness the ‘Venturi Effect’ where particles entering the throat accelerate rapidly, and the height of the liquid manometer column at the constriction drops significantly.
- Utilize the static/dynamic pressure analysis report: Check in the real-time module whether the final pressure (P2) at the constricted cross-section enters negative pressure (vacuum), generating suction driving force for an injector.
📚 Mathematical Derivation of Bernoulli’s Equation and the Venturi Effect ▼
1. Basic Principles and Physical Definition of Bernoulli’s Theorem
The **Bernoulli’s Equation**, which is the most fundamental concept in fluid mechanics, is another formulation of the **law of conservation of energy** for a flowing fluid. When a frictionless, incompressible ideal fluid flows along a steady streamline, the sum of the **static pressure**, **dynamic pressure**, and **potential (elevation) energy** per unit volume remains constant at any point along the flow.
P + ½ × ρ × v2 + ρ × g × z = Constant
Where P is the static pressure of the fluid (Pa), ρ is the fluid density (kg/m3), v is the flow velocity (m/s), g is the acceleration due to gravity, and z is the elevation from the reference plane (m). In a horizontal pipe (z1 = z2), the potential energy terms cancel out, meaning that an increase in flow velocity must result in a decrease in static pressure, and conversely, a decrease in flow velocity leads to an increase in static pressure.
2. Velocity and Pressure Behavior in the Venturi Throat using the Continuity Equation
As a fluid passes through the constricted throat of a Venturi tube where the cross-sectional area decreases, mass conservation requires that the flow velocity increase in inverse proportion to the constriction ratio (Continuity Equation: A1v1 = A2v2).
v2 = v1 × (A1 / A2) = v1 × (D1 / D2)2
By substituting the accelerated velocity v2 into Bernoulli’s equation, we can derive the static pressure P2 at the throat.
P2 = P1 – ½ × ρ × (v22 – v12)
If the contraction ratio is extremely large and the accelerated velocity exceeds a certain threshold, the calculated pressure P2 can drop below atmospheric pressure, reaching a **vacuum pressure** region. This low-pressure vacuum creates a suction effect that draws in nearby fluids or gases, which serves as the core physical principle behind the design of spray guns, carburetors, and water ejectors.
3. Limiting Factors in Real Fluid Analysis (Loss and Cavitation)
Unlike ideal fluid calculations, when real viscous fluids flow, **frictional pressure loss (head loss)** occurs across the constriction due to internal wall friction and the formation of eddies in the expanding section. Furthermore, if the static pressure P2 drops below the **vapor pressure** of the fluid at its operating temperature, the fluid can boil and vaporize within the pipe, leading to cavitation where **cavitation bubbles** form. Since cavitation causes severe damage and erosion to the piping system, design engineers must carefully control the system to prevent the pressure from dropping below this critical threshold.