SV Technocrats India's made Bernoulli’s theorem, also known as Bernoulli’s principle, is a fundamental concept in fluid dynamics that describes the behavior of fluid flow. It states that for an incompressible, non-viscous fluid in steady flow, the sum of the pressure energy, kinetic energy, and potential energy per unit volume is constant along any streamline. This principle is named after the Swiss mathematician Daniel Bernoulli.
Bernoulli’s Equation
The Bernoulli equation can be expressed mathematically as:
P+12ρv2+ρgh=constantP + \frac{1}{2} \rho v^2 + \rho gh = \text{constant}P+21ρv2+ρgh=constant
where:
PPP is the static pressure of the fluid.
ρ\rhoρ is the density of the fluid.
vvv is the fluid velocity.
ggg is the acceleration due to gravity.
hhh is the height above a reference point.
Key Assumptions
Incompressible Fluid: The fluid density ρ\rhoρ remains constant.
Non-viscous Fluid: There is no internal friction (viscosity) within the fluid.
Steady Flow: The fluid properties at any given point do not change over time.
Along a Streamline: The equation applies to points along the same streamline.
Applications of Bernoulli’s Theorem
Venturi Effect: Used in devices like the Venturi meter to measure fluid flow rate by observing the pressure difference in a constricted section of a pipe.
Aerofoil and Lift: Explains how pressure differences around an aerofoil generate lift for airplanes.
Flow Measurement: Used in devices such as orifice plates and Pitot tubes to measure the velocity of fluid flow.
Hydraulic Structures: Helps in the design and analysis of spillways, weirs, and other hydraulic structures.
Experiment Procedure to Verify Bernoulli’s Theorem
Setup:
Use a horizontal pipe setup with varying cross-sectional areas (e.g., a Venturi tube).
Connect pressure taps at different points along the pipe to measure pressure.
Flow Adjustment:
Start the pump and regulate the flow to achieve a steady state.
Measure the flow rate using a flow meter.
Pressure Measurement:
Record the pressure readings at the different points along the pipe.
Measure the velocity of the fluid at different cross-sectional areas using the flow rate and the area of the sections.
Calculations:
Use Bernoulli’s equation to calculate the expected pressure, velocity, and height values at different points.
Compare the calculated values with the measured values to verify Bernoulli’s theorem.
Example Calculation
Suppose you have a horizontal pipe with a large diameter section (A1) and a small diameter section (A2). If the velocity in the large diameter section is v1v1v1 and the pressure is P1P1P1, in the small diameter section, the velocity v2v2v2 and pressure P2P2P2 can be found using Bernoulli’s equation and the principle of continuity (A1v1=A2v2A1v1 = A2v2A1v1=A2v2).
Using Bernoulli’s equation:
P1+12ρv12=P2+12ρv22P1 + \frac{1}{2} \rho v1^2 = P2 + \frac{1}{2} \rho v2^2P1+21ρv12=P2+21ρv22
From the continuity equation:
v2=A1A2v1v2 = \frac{A1}{A2} v1v2=A2A1v1
Substitute v2v2v2 into Bernoulli’s equation to solve for P2P2P2.
Safety Considerations
Secure Setup: Ensure all connections are tight and secure to prevent leaks.
Pressure Limits: Operate within the recommended pressure limits to avoid damaging the apparatus.
Electrical Safety: If using an electric pump, ensure proper grounding and insulation.
Bernoulli’s theorem is a cornerstone of fluid mechanics, providing insights into the relationship between pressure, velocity, and elevation in fluid flow. Its applications span across various fields, making it an essential principle in both theoretical and practical contexts.
