How To Derive Bernoulli's Equation

Deriving Bernoulli's Equation: A Step-by-Step Guide

Bernoulli's equation is a fundamental principle in fluid mechanics, describing the relationship between pressure, velocity, and elevation in a flowing fluid. Understanding its derivation is crucial for grasping its implications and applications in various fields, from aerodynamics to hydraulics. This comprehensive guide will walk you through the derivation of Bernoulli's equation, step-by-step, making it accessible even to those without an extensive background in physics. We'll explore the underlying principles and assumptions, providing a clear and insightful understanding of this vital equation.

Introduction: Understanding the Underlying Principles

Bernoulli's equation is a statement of the conservation of energy principle applied to a flowing fluid. It assumes that the fluid is incompressible (density remains constant), inviscid (no internal friction), and flows along a streamline (a path followed by a fluid particle). These assumptions simplify the analysis, allowing for a relatively straightforward derivation. While real-world fluids often deviate from these ideal conditions, Bernoulli's equation still provides a valuable approximation in many practical situations.

Step 1: Selecting a Control Volume and Applying the Conservation of Energy

Consider a streamline of an incompressible, inviscid fluid flowing through a pipe of varying cross-sectional area and elevation. We will analyze a small cylindrical control volume of fluid along this streamline. The energy of this fluid element consists of three primary components:

• Kinetic Energy: This energy is associated with the fluid's velocity. For a fluid element of mass dm, the kinetic energy is given by: dKE = (1/2) * dm * v², where v is the fluid velocity.

• Potential Energy: This energy is related to the fluid's elevation. For the fluid element at height z above a reference datum, the potential energy is: dPE = dm * g * z, where g is the acceleration due to gravity.

• Flow Work/Pressure Energy: This represents the work done on the fluid element by the pressure forces acting on it. For a cylindrical fluid element with cross-sectional area dA and length ds along the streamline, the work done is given by: dW = P * dA * ds = P * dV, where P is the pressure and dV is the volume of the fluid element.

According to the principle of conservation of energy, the total energy of the fluid element remains constant along the streamline. Therefore, the change in kinetic energy, potential energy, and flow work must sum to zero:

dKE + dPE + dW = 0

Step 2: Expressing Energy Changes in Terms of Infinitesimals

To make the equation more manageable, we express the changes in energy as infinitesimals. Substituting the expressions for kinetic energy, potential energy, and flow work from Step 1, we get:

(1/2) * dm * v² + dm * g * z + P * dV = 0

Step 3: Introducing the Continuity Equation

The continuity equation expresses the conservation of mass for an incompressible fluid. For a streamline, it states that the mass flow rate is constant:

ρ * A * v = constant

where ρ is the fluid density and A is the cross-sectional area of the pipe. This implies that any change in velocity is accompanied by a corresponding change in cross-sectional area. We don't need to explicitly use the continuity equation in the derivation but it's important to keep in mind the underlying principle of mass conservation, which implicitly influences the changes in velocity and pressure as the fluid flows along the streamline.

Step 4: Relating Mass and Volume

The mass of the fluid element, dm, can be expressed in terms of its volume, dV, and density, ρ:

dm = ρ * dV

Step 5: Substituting and Simplifying

Substituting dm = ρ * dV into the energy equation from Step 2, we have:

(1/2) * ρ * dV * v² + ρ * dV * g * z + P * dV = 0

Now, we divide the entire equation by ρ * dV:

(1/2) * v² + g * z + P/ρ = 0

Notice that we've cancelled out the dV term. This is a key step illustrating that the equation is independent of the size of the control volume we chose. The result is a statement about the properties at each point along the streamline.

Step 6: Integrating Along the Streamline

The equation derived in Step 5 represents the energy balance for an infinitesimal segment of the streamline. To obtain Bernoulli's equation for a finite segment of the streamline, we integrate along the streamline from point 1 to point 2:

∫₁² [(1/2) * v² + g * z + P/ρ] = constant

This integration simply implies that the quantity (1/2)v² + gz + P/ρ remains constant along the streamline. Thus, we arrive at Bernoulli's equation:

(1/2)ρv₁² + ρg z₁ + P₁ = (1/2)ρv₂² + ρg z₂ + P₂

or, more commonly written as:

P₁/ρ + (1/2)v₁² + gz₁ = P₂/ρ + (1/2)v₂² + gz₂ = constant

This equation states that the sum of pressure energy (P/ρ), kinetic energy ((1/2)v²), and potential energy (gz) remains constant along a streamline for an incompressible, inviscid fluid.

Explanation of Terms and Units

Let's break down the terms in Bernoulli's equation:

• P: Pressure (Pascals, Pa)
• ρ: Density (kilograms per cubic meter, kg/m³)
• v: Velocity (meters per second, m/s)
• g: Acceleration due to gravity (meters per second squared, m/s²)
• z: Elevation (meters, m)

Each term represents a specific form of energy per unit volume of the fluid. The equation shows how these energy forms are interconverted as the fluid flows. For example, if the velocity increases (kinetic energy increases), the pressure or elevation must decrease to maintain the constant total energy.

Applications of Bernoulli's Equation

Bernoulli's equation has wide-ranging applications in various fields:

• Aerodynamics: It explains lift generation on aircraft wings, the operation of carburetors, and the functioning of venturi meters. The higher velocity of air over the curved upper surface of a wing leads to lower pressure, creating a pressure difference that generates lift.

• Hydraulics: It is used in the design of pipelines, dams, and other hydraulic structures. It helps predict pressure drops along pipelines and the flow rates in various sections.

• Meteorology: It plays a role in understanding wind patterns and the formation of clouds. Pressure differences caused by variations in air velocity and elevation contribute to atmospheric circulation.

Limitations of Bernoulli's Equation

It's crucial to remember that Bernoulli's equation is based on several idealizations:

• Incompressible flow: The equation doesn't hold for compressible fluids like gases at high speeds where significant density changes occur.

• Inviscid flow: Real fluids have viscosity, resulting in energy losses due to friction. Bernoulli's equation doesn't account for these losses.

• Steady flow: The derivation assumes steady flow conditions, meaning that the fluid properties at any point don't change with time. The equation is not directly applicable to unsteady flows.

• Along a streamline: The equation applies specifically to flow along a single streamline. It doesn't necessarily hold between different streamlines.

Despite these limitations, Bernoulli's equation remains a powerful tool for understanding fluid flow in many practical situations. The approximations made in its derivation are often justified, leading to accurate predictions in many applications.

Frequently Asked Questions (FAQ)

Q: What is the difference between Bernoulli's equation and the Navier-Stokes equations?

A: Bernoulli's equation is a simplified form of the Navier-Stokes equations, applicable only under specific conditions (incompressible, inviscid, steady flow along a streamline). The Navier-Stokes equations are much more general and account for viscosity and unsteady flow, but they are significantly more complex to solve.

Q: Can Bernoulli's equation be used for turbulent flow?

A: Bernoulli's equation is strictly valid only for laminar (smooth) flow. While it can sometimes provide a rough approximation for turbulent flow, it's generally not reliable. More sophisticated methods are required for accurate analysis of turbulent flows.

Q: How does Bernoulli's equation relate to the conservation of energy?

A: Bernoulli's equation is a direct consequence of the conservation of energy principle applied to fluid flow. The equation expresses the constancy of the total energy (pressure, kinetic, and potential) along a streamline.

Q: What are some real-world examples where Bernoulli's equation is applied?

A: Numerous examples exist: aircraft wing lift, carburetor operation, venturi meters, blood flow in arteries, and the design of spillways in dams.

Conclusion: A Powerful Tool for Understanding Fluid Mechanics

The derivation of Bernoulli's equation, while seemingly complex at first, is built upon fundamental principles of energy conservation and fluid mechanics. Understanding its derivation provides a deeper appreciation for its applications and limitations. While the idealizations involved restrict its direct applicability to all fluid flows, Bernoulli's equation remains a crucial and powerful tool for understanding and analyzing a wide range of fluid mechanical phenomena. Its simplicity and elegance make it a cornerstone concept in the study of fluid dynamics. By mastering the derivation and understanding its underlying assumptions, you can confidently apply it to various problems and gain a more comprehensive understanding of fluid behavior.