3.1 Conservation of mass

Conservation of mass is a fundamental principle in fluid dynamics, stating that mass cannot be created or destroyed within a closed system. This concept is crucial for understanding fluid behavior and forms the basis for analyzing mass flow in various applications.

The principle is applied through mass balance analysis, considering inflow, outflow, and potential sources or sinks within a defined control volume. This approach enables engineers to predict and model fluid systems' behavior in steady and unsteady flow conditions across diverse fields.

Principle of mass conservation

• Fundamental principle in fluid dynamics states mass cannot be created or destroyed within a closed system
• Total mass of a fluid system remains constant over time, regardless of any processes occurring within the system
• Applies to both compressible and incompressible fluids, as well as single-phase and multiphase systems

Mass balance in fluid systems

• Analysis of mass flow into and out of a defined control volume or across a control surface
• Accounts for accumulation, generation, or consumption of mass within the system
• Essential for understanding and predicting the behavior of fluid systems in various applications

Steady vs unsteady flow

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• Steady flow occurs when fluid properties (velocity, pressure, density) at any point do not change with time
• Unsteady flow is characterized by time-dependent changes in fluid properties
• Mass balance equations differ for steady and unsteady flow conditions

Inflow vs outflow

• Inflow represents the mass entering the control volume through its boundaries
• Outflow represents the mass leaving the control volume through its boundaries
• Net mass flow rate is the difference between inflow and outflow rates

Sources and sinks

• Sources introduce additional mass into the control volume (chemical reactions, phase change)
• Sinks remove mass from the control volume (chemical reactions, phase change)
• Must be accounted for in the mass balance equations

Control volumes

• Defined region in space chosen for analysis of a fluid system
• Can be fixed or moving, and may have arbitrary shapes and boundaries
• Selection of appropriate control volume is crucial for accurate mass balance analysis

Fixed vs moving boundaries

• Fixed control volumes have stationary boundaries and are commonly used for steady flow analysis
• Moving control volumes have boundaries that move with the fluid or a specific reference frame
• Moving control volumes are used for unsteady flow and problems involving moving objects (vehicles, turbomachinery)

Selection of control surfaces

• Control surfaces are the boundaries of the control volume across which mass flow is analyzed
• Proper selection of control surfaces simplifies the mass balance equations and reduces computational complexity
• Control surfaces should be chosen to align with the flow direction and minimize the need for complex integrations

Continuity equation

• Mathematical expression of the principle of mass conservation in fluid systems
• Relates the rate of change of mass within a control volume to the net mass flow across its boundaries
• Can be expressed in differential or integral form, depending on the problem requirements

Differential form

• Continuity equation expressed in terms of partial derivatives of density and velocity components
• Applicable to infinitesimal fluid elements and provides a point-wise description of mass conservation
• Differential form: $\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$

Integral form

• Continuity equation expressed in terms of surface and volume integrals of mass flow rates
• Applicable to finite control volumes and provides a global description of mass conservation
• Integral form: $\frac{\partial}{\partial t} \int_{CV} \rho dV + \int_{CS} \rho \vec{V} \cdot d\vec{A} = 0$

Simplifications and assumptions

• Assumptions such as steady flow, incompressible flow, or constant density can simplify the continuity equation
• Simplifications lead to reduced forms of the equation, making analytical or numerical solutions more tractable
• Examples of simplified continuity equations:
  • Steady, incompressible flow: $\nabla \cdot \vec{V} = 0$
  • Steady, one-dimensional flow: $\rho_1 A_1 V_1 = \rho_2 A_2 V_2$

Applications of mass conservation

• Mass conservation principle is applied to various types of fluid systems and flow conditions
• Understanding the specific characteristics of each application is essential for accurate modeling and analysis

Incompressible flow

• Fluid density remains constant throughout the flow field
• Applicable to liquids and gases at low Mach numbers (typically < 0.3)
• Simplified continuity equation: $\nabla \cdot \vec{V} = 0$

Compressible flow

• Fluid density varies significantly with pressure changes
• Applicable to gases at high Mach numbers (> 0.3) and in systems with large pressure gradients
• Requires the use of the full continuity equation, often coupled with energy and momentum equations

Multiphase systems

• Involve the presence of multiple fluid phases (gas-liquid, liquid-liquid, gas-solid)
• Mass conservation equations must account for the exchange of mass between phases (evaporation, condensation, dissolution)
• Requires additional equations and models to describe the interactions between phases

Conservation of mass vs momentum

• Mass conservation is one of the fundamental principles in fluid dynamics, along with momentum and energy conservation
• Momentum conservation deals with the balance of forces acting on a fluid and the resulting changes in velocity
• Mass and momentum conservation equations are often solved together to fully describe the behavior of a fluid system

Numerical methods for mass conservation

• Analytical solutions to the continuity equation are often difficult or impossible for complex flow problems
• Numerical methods are employed to discretize the equation and solve for the flow field variables

Finite volume method

• Divides the flow domain into small control volumes and applies the integral form of the continuity equation
• Mass balance is enforced for each control volume, ensuring global conservation
• Widely used in computational fluid dynamics (CFD) software packages

Finite element method

• Discretizes the flow domain into elements and applies the continuity equation in its weak form
• Approximates the flow variables using shape functions and solves for their nodal values
• Provides high accuracy and flexibility in handling complex geometries and boundary conditions

Experimental verification

• Experimental techniques are used to validate the predictions of mass conservation models and simulations
• Flow visualization and mass flow rate measurements provide valuable data for comparison and model refinement

Flow visualization techniques

• Qualitative methods to observe the flow patterns and identify regions of mass accumulation or depletion
• Examples include smoke injection, dye injection, and particle image velocimetry (PIV)
• Help in understanding the overall flow behavior and detecting anomalies

Mass flow rate measurements

• Quantitative methods to determine the mass flow rates at specific locations in a fluid system
• Examples include orifice plates, Venturi meters, and Coriolis mass flow meters
• Provide accurate data for validating the mass balance calculations and assessing the accuracy of numerical models