Glossary

3.1 Conservation of mass

5 min read•august 20, 2024

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 analysis, considering , , 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

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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  • occurs when fluid properties (velocity, pressure, ) 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 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

  • have stationary boundaries and are commonly used for steady flow analysis
  • 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

  • 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 , depending on the problem requirements

Differential form

  • 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
  • : ∂ρ∂t+∇⋅(ρV⃗)=0\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: ∂∂t∫CVρdV+∫CSρV⃗⋅dA⃗=0\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, , 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: ∇⋅V⃗=0\nabla \cdot \vec{V} = 0
    • Steady, one-dimensional flow: ρ1A1V1=ρ2A2V2\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: ∇⋅V⃗=0\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 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

Key Terms to Review (29)

∫ρdv = constant: The equation ∫ρdv = constant represents the principle of conservation of mass in fluid dynamics. It signifies that the total mass of a fluid system remains unchanged over time, as long as no mass enters or leaves the control volume. This relationship is crucial for analyzing fluid flow and understanding how changes in density and volume affect mass continuity.
A1v1 = a2v2: The equation $$a_1v_1 = a_2v_2$$ represents the principle of continuity in fluid dynamics, illustrating the conservation of mass in a flowing fluid. This equation states that the product of the cross-sectional area (a) and the fluid velocity (v) at one point in a flow must equal the product at another point, ensuring that mass remains constant along a streamline. This fundamental concept connects various aspects of fluid behavior, including flow rates, pipe dimensions, and velocity changes.
Bernoulli's Principle: Bernoulli's Principle states that in a flowing fluid, an increase in the fluid's speed occurs simultaneously with a decrease in pressure or potential energy. This principle helps explain various phenomena in fluid dynamics, linking pressure and velocity changes to the behavior of fluids in motion, and is foundational for understanding how different factors like density and flow type influence the overall dynamics of fluid systems.
Compressible flow: Compressible flow refers to the fluid flow in which the density of the fluid changes significantly due to variations in pressure and temperature. This behavior is especially important in high-speed flows, where the changes in density cannot be ignored, such as in gases moving at speeds close to or greater than the speed of sound. Understanding compressible flow is crucial for analyzing systems where mass conservation, momentum transfer, and energy interactions play key roles.
Conservation of mass: Conservation of mass is a fundamental principle stating that mass cannot be created or destroyed in an isolated system. This principle is crucial in fluid dynamics, as it helps to understand how mass flows through different regions and the relationships between various properties of fluids under different conditions.
Continuity Equation: The continuity equation is a fundamental principle in fluid dynamics that describes the conservation of mass in a flowing fluid. It states that the mass flow rate must remain constant from one cross-section of a flow to another, meaning that any change in fluid density or velocity must be compensated by a change in cross-sectional area. This concept connects various aspects of fluid motion, including flow characteristics and the behavior of different types of flows.
Continuity Principle: The continuity principle states that mass cannot be created or destroyed in a closed system, implying that the mass flow rate must remain constant throughout the system. This principle is crucial in fluid dynamics as it describes how fluids behave when they move through varying cross-sectional areas, leading to changes in velocity and pressure while maintaining a consistent mass flow.
Control Surfaces: Control surfaces are defined as specific boundaries within a fluid system that help in analyzing the flow of fluid and the transfer of mass, momentum, and energy. These surfaces can be fixed or moving and are crucial for applying fundamental principles such as the conservation of mass, allowing for the identification of inflow and outflow rates across the boundaries.
Control Volume Analysis: Control volume analysis is a method used in fluid dynamics to study the behavior of fluid flow by defining a specific region in space, called the control volume, where mass and energy transfers can be analyzed. This approach helps in applying fundamental principles like conservation of mass, momentum, and energy to understand how fluids interact with their surroundings. It is essential for solving problems related to fluid flow in various engineering applications.
Density: Density is a physical property defined as the mass of a substance per unit volume, typically expressed in units like kilograms per cubic meter (kg/m³) or grams per cubic centimeter (g/cm³). It is essential for understanding how substances behave in various contexts, including their buoyancy in fluids, their conservation in mass flow, and their influence on pressure and velocity in fluid dynamics.
Differential Form: A differential form is a mathematical object that generalizes the concepts of functions and differentials in calculus, allowing for the integration and differentiation of quantities in a rigorous way. In the context of fluid dynamics, differential forms provide a powerful framework for expressing conservation laws, such as mass, momentum, and energy, in a way that is coordinate-free and emphasizes the geometric nature of these laws.
Experimental Verification: Experimental verification is the process of confirming theoretical predictions or hypotheses through controlled experiments and empirical observations. This process is crucial in validating concepts like buoyancy and Archimedes' principle, as well as ensuring that the conservation of mass holds true in various fluid dynamics scenarios.
Fixed Control Volumes: Fixed control volumes are designated regions in space through which fluid can flow, but the volume itself remains stationary over time. This concept is crucial for analyzing fluid behavior and properties within a defined space, allowing for the application of conservation principles, particularly conservation of mass.
Flow Rate Conservation: Flow rate conservation refers to the principle that, in a closed system, the mass flow rate of a fluid remains constant over time, assuming no mass is added or removed from the system. This principle is essential in fluid dynamics as it ensures that the amount of fluid entering a certain area is equal to the amount exiting, which relates directly to the conservation of mass and continuity equation.
Flow visualization techniques: Flow visualization techniques are methods used to make the flow of fluids visible, allowing for a better understanding of fluid behavior and properties. These techniques help in observing flow patterns, identifying areas of turbulence, and analyzing the effects of different forces acting on a fluid, ultimately aiding in the study of conservation of mass and other fundamental principles of fluid dynamics.
Incompressible Flow: Incompressible flow refers to the condition in fluid dynamics where the fluid density remains constant throughout the flow field, regardless of pressure variations. This simplification is particularly useful for analyzing liquids and low-speed gas flows, as it enables the use of simplified equations and models, making it easier to predict fluid behavior in various applications.
Inflow: Inflow refers to the movement of fluid into a specified region or system. This term is crucial in understanding how fluids interact with their surroundings and how they are conserved within a defined volume. It plays a key role in fluid dynamics, particularly when applying principles like conservation of mass, as it affects how mass and volume are accounted for in any analysis involving fluid movement.
Integral Form: Integral form refers to the mathematical representation of physical laws that relate the quantity of interest over a finite region to the behavior of that quantity at the boundaries of that region. This approach is essential for understanding conservation principles, as it expresses how quantities like mass, momentum, and energy are conserved within a control volume. Integral form provides a clear and practical way to analyze fluid motion and changes in flow properties across a defined area.
Laminar Flow: Laminar flow is a smooth, orderly flow of fluid characterized by parallel layers that slide past one another with minimal mixing. This type of flow occurs at low velocities and is primarily influenced by viscosity, allowing for predictable and stable movement that contrasts sharply with chaotic turbulent flow.
Mass balance: Mass balance refers to the principle that, in a closed system, the mass of a substance must remain constant over time, accounting for all inputs, outputs, and changes within the system. This concept is crucial in fluid dynamics as it ensures that mass is conserved throughout processes such as fluid flow, chemical reactions, and heat transfer, linking it directly to the broader principle of conservation of mass.
Mass Flow Rate: Mass flow rate is the measure of the mass of a substance that passes through a given surface per unit time, typically expressed in kilograms per second (kg/s). This concept is crucial in understanding fluid motion and behavior, as it connects to the conservation of mass, ensuring that mass is neither created nor destroyed in a flowing system. It also plays a significant role in analyzing flow fields and stream functions, which help visualize the movement and distribution of fluids in various applications.
Mass flow rate measurements: Mass flow rate measurements quantify the amount of mass passing through a given point in a system per unit time. This measurement is crucial in fluid dynamics, as it helps to analyze and control various processes, ensuring that the principle of conservation of mass is upheld, which states that mass cannot be created or destroyed in an isolated system.
Moving Control Volumes: Moving control volumes refer to a specific approach in fluid dynamics where the control volume itself is not fixed in space but rather moves with the flow of the fluid. This concept is crucial in analyzing fluid motion and the associated conservation laws, as it allows for a clearer understanding of how mass, momentum, and energy transfer occur within a dynamic system.
Multiphase systems: Multiphase systems refer to physical systems that involve two or more distinct phases of matter, such as solid, liquid, and gas, coexisting and interacting within a single system. These systems are crucial in various fields, including engineering and natural sciences, as they often display complex behaviors influenced by the interactions among the different phases. Understanding the dynamics of multiphase systems is essential for accurately applying the principles of conservation of mass, which helps predict how mass is distributed and conserved across different phases.
Outflow: Outflow refers to the movement of fluid out of a defined system or control volume, playing a crucial role in understanding fluid dynamics and the conservation of mass. This concept highlights how mass is not lost but rather moves from one area to another, ensuring that the total mass within a closed system remains constant. The balance between inflow and outflow is essential for analyzing fluid behavior in various applications, including engineering and environmental science.
Sources and Sinks: Sources and sinks refer to the processes that add or remove mass from a control volume within a fluid system. In fluid dynamics, a source is where fluid or mass enters the system, while a sink is where fluid or mass exits. Understanding these concepts is crucial for analyzing the conservation of mass, as they directly affect how mass balances are established in various applications, such as in natural environments or engineered systems.
Steady Flow: Steady flow refers to a condition in fluid dynamics where the fluid's velocity at a given point does not change over time. This concept is crucial as it allows for the simplification of analyses in various fluid systems, ensuring that parameters like pressure and density remain consistent as the fluid moves.
Turbulent Flow: Turbulent flow is a type of fluid motion characterized by chaotic changes in pressure and flow velocity. In this state, the fluid experiences irregular fluctuations and eddies, leading to increased mixing and energy dissipation compared to smooth, laminar flow.
Velocity Field: A velocity field is a mathematical representation that describes the velocity of fluid particles at every point in a given space. It captures how the speed and direction of the fluid flow change over that space, which is crucial for understanding the behavior of fluids in various contexts, such as flow patterns, vorticity, and conservation laws. Understanding velocity fields helps in analyzing how fluid moves and interacts with boundaries and forces, forming the foundation for studying fluid dynamics in diverse scenarios.
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