3.3 Conservation of energy

Energy conservation is a cornerstone of fluid dynamics, governing how energy transfers and converts in fluid systems. It applies to both steady and unsteady flows, allowing us to analyze fluid properties and system performance in various scenarios.

The energy equation accounts for kinetic, potential, and internal energy in fluids. It helps us understand phenomena like flow in pipes, energy extraction in turbines, and pump performance. Practical applications include designing efficient hydraulic systems and preventing issues like cavitation.

Principle of conservation of energy

• States that energy cannot be created or destroyed, only converted from one form to another
• Fundamental principle in fluid dynamics used to analyze energy transfer and conversion in fluid systems
• Applies to both steady and unsteady flows, allowing for the determination of fluid properties and system performance

Types of energy in fluids

Kinetic energy of fluids

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• Energy associated with the motion of a fluid
• Depends on the fluid's velocity and mass
• Expressed as $KE = \frac{1}{2}mv^2$, where $m$ is the mass and $v$ is the velocity
• Examples: flowing water in a pipe, wind energy harnessed by turbines

Potential energy of fluids

• Energy stored in a fluid due to its position or configuration
• Gravitational potential energy depends on the fluid's height and mass
• Expressed as $PE = mgh$, where $m$ is the mass, $g$ is the gravitational acceleration, and $h$ is the height
• Examples: water stored in an elevated tank, hydraulic systems

Internal energy of fluids

• Energy associated with the molecular motion and intermolecular forces within a fluid
• Depends on the fluid's temperature and pressure
• Consists of sensible and latent heat
• Important in compressible flows and heat transfer processes
• Examples: thermal energy in a hot gas, phase change in boiling or condensation

Energy equation for steady flows

Derivation of steady flow energy equation

• Obtained by applying the principle of conservation of energy to a control volume
• Accounts for energy entering, leaving, and stored within the system
• Assumes steady flow conditions, where fluid properties are independent of time
• Derived by considering work, heat transfer, and changes in kinetic, potential, and internal energy

Terms in steady flow energy equation

• Kinetic energy term: $\frac{v^2}{2}$, represents the energy due to fluid motion
• Potential energy term: $gz$, represents the energy due to fluid elevation
• Flow work term: $\frac{P}{\rho}$, represents the work done by pressure forces
• Heat transfer term: $\dot{Q}$, represents the heat added or removed from the system
• Shaft work term: $\dot{W}_s$, represents the work done by or on the fluid by external devices (pumps or turbines)

Assumptions and limitations

• Assumes steady flow conditions, where fluid properties are independent of time
• Neglects viscous effects and assumes no friction losses
• Assumes no chemical reactions or phase changes within the system
• Limited to systems with single inlet and outlet streams
• Requires knowledge of fluid properties and system geometry

Energy equation for unsteady flows

Time-dependent energy equation

• Accounts for changes in fluid properties and system energy over time
• Includes local and convective acceleration terms to capture transient effects
• Expressed as $\frac{\partial}{\partial t}(\rho e) + \nabla \cdot (\rho e \vec{V}) = -\nabla \cdot \vec{q} - \nabla \cdot (P\vec{V}) + \rho \vec{f} \cdot \vec{V}$
• Allows for the analysis of time-varying flows and system transients

Local and convective acceleration terms

• Local acceleration term: $\frac{\partial}{\partial t}(\rho e)$, represents the rate of change of energy at a fixed point
• Convective acceleration term: $\nabla \cdot (\rho e \vec{V})$, represents the rate of change of energy due to fluid motion
• Capture the temporal and spatial variations in fluid properties and energy
• Essential for analyzing transient flows and system dynamics

Transient flow examples

• Startup and shutdown of pumps or turbines
• Valve opening and closing in pipelines
• Pressure surge or water hammer in hydraulic systems
• Pulsating flows in blood vessels or industrial processes

Application of energy equation

Bernoulli's equation for incompressible flow

• Simplified form of the steady flow energy equation for incompressible fluids
• Relates pressure, velocity, and elevation along a streamline
• Expressed as $\frac{P}{\rho} + \frac{v^2}{2} + gz = constant$
• Widely used in fluid mechanics for analyzing flow in pipes, channels, and around objects

Restrictions on Bernoulli's equation

• Applies only to incompressible fluids (constant density)
• Assumes steady, frictionless flow along a streamline
• Neglects viscous effects and energy losses
• Limited to regions without significant heat transfer or shaft work
• Requires continuous and non-vortical flow

Examples of Bernoulli's equation

• Calculating pressure differences in a venturi meter or orifice plate
• Determining the velocity of a fluid flowing out of a tank
• Analyzing lift force on airfoils or wind turbine blades
• Estimating the flow rate in a pipe with varying cross-section

Energy losses in fluid systems

Major and minor losses

• Major losses: caused by friction in straight pipes or ducts
• Minor losses: caused by flow through fittings, valves, bends, and other obstructions
• Both contribute to the overall pressure drop and energy dissipation in fluid systems
• Accounted for using loss coefficients or equivalent length methods

Friction losses in pipes

• Caused by fluid viscosity and wall roughness
• Depend on the Reynolds number and relative roughness of the pipe
• Calculated using friction factor correlations (Moody diagram or equations like Darcy-Weisbach)
• Expressed as pressure drop or head loss
• Examples: flow through long pipelines, oil and gas transportation, water distribution networks

Local losses in fittings and valves

• Caused by flow separation, recirculation, and turbulence in fittings and valves
• Characterized by loss coefficients (K-factors) specific to each type of fitting or valve
• Pressure drop calculated as $\Delta P = K \frac{\rho v^2}{2}$, where $K$ is the loss coefficient
• Examples: elbows, tees, contractions, expansions, valves, and other flow obstructions

Pumps and turbines

Pump work and efficiency

• Pumps add energy to a fluid to increase its pressure or velocity
• Pump work is the product of the pressure rise and the volumetric flow rate
• Expressed as $\dot{W}_p = Q\Delta P$, where $Q$ is the flow rate and $\Delta P$ is the pressure rise
• Pump efficiency is the ratio of the useful hydraulic power to the input shaft power
• Affected by factors such as pump design, operating conditions, and fluid properties

Turbine work and efficiency

• Turbines extract energy from a fluid to produce mechanical work
• Turbine work is the product of the pressure drop and the volumetric flow rate
• Expressed as $\dot{W}_t = Q\Delta P$, where $Q$ is the flow rate and $\Delta P$ is the pressure drop
• Turbine efficiency is the ratio of the output shaft power to the available hydraulic power
• Affected by factors such as turbine design, operating conditions, and fluid properties

Net positive suction head (NPSH)

• Measure of the pressure available at the inlet of a pump to prevent cavitation
• NPSH available (NPSHA) depends on the system design and fluid properties
• NPSH required (NPSHR) is a characteristic of the pump and varies with flow rate
• For safe operation, NPSHA must be greater than NPSHR to avoid cavitation
• Insufficient NPSH can lead to pump damage, reduced efficiency, and system failure

Cavitation and its effects

Causes of cavitation

• Occurs when the local pressure in a fluid drops below its vapor pressure
• Can be caused by high fluid velocities, abrupt changes in geometry, or excessive suction in pumps
• Influenced by factors such as fluid temperature, dissolved gases, and surface roughness
• Examples: propellers, hydraulic turbines, control valves, and pumps

Consequences of cavitation

• Formation and collapse of vapor bubbles in the fluid
• Generates high-pressure shock waves and localized high temperatures
• Causes erosion, pitting, and damage to surfaces exposed to cavitation
• Leads to reduced efficiency, increased vibration and noise, and premature failure of components
• Can also affect the performance and stability of fluid machinery

Prevention of cavitation

• Ensure sufficient NPSH by proper system design and operation
• Maintain fluid pressure above the vapor pressure, especially at critical locations
• Use cavitation-resistant materials or coatings for exposed surfaces
• Employ flow control devices (orifices, valves) to regulate pressure and minimize pressure drops
• Monitor and maintain proper fluid quality (temperature, dissolved gases, cleanliness)

Numerical problems and solutions

Problem-solving strategies

• Identify the given information, unknowns, and governing equations
• Determine the appropriate assumptions and simplifications for the problem
• Apply the relevant equations (energy equation, Bernoulli's equation, loss calculations)
• Solve for the unknown variables using algebraic manipulation or iterative methods
• Verify the results using dimensional analysis and physical intuition

Common pitfalls and misconceptions

• Neglecting energy losses or assuming frictionless flow in real systems
• Misapplying Bernoulli's equation to compressible or unsteady flows
• Incorrectly using loss coefficients or friction factors
• Ignoring the limitations and assumptions of the governing equations
• Failing to consider the effects of cavitation or NPSH in pump systems

Practice problems and answers

• Worked examples covering various aspects of energy conservation in fluids
• Problems involving the application of the energy equation, Bernoulli's equation, and loss calculations
• Scenarios related to pumps, turbines, and cavitation
• Step-by-step solutions demonstrating problem-solving techniques and common calculations
• Emphasis on developing a systematic approach to problem-solving in fluid mechanics