The conservation of mass principle is fundamental in fluid mechanics. It states that mass can't be created or destroyed, leading to the continuity equation. This equation balances the rate of mass change within a control volume with the net mass flow into it.

For steady-state, incompressible flows, the continuity equation simplifies. It equates volume flow rates at inlets and outlets, allowing us to relate velocities and cross-sectional areas. This concept is crucial for understanding flow behavior in pipes, nozzles, and other fluid systems.

Conservation of Mass (Continuity Equation)

Conservation of mass equation

• Conservation of mass principle states mass cannot be created or destroyed
• Control volume defines a fixed region in space through which fluid flows
  • Fluid enters and exits the control volume through its control surface (pipe, duct, or channel)
• Mass balance for a control volume:
  • Rate of change of mass within the control volume equals the net rate of mass flow into the control volume
  • Mathematical expression: $\frac{\partial}{\partial t} \int_{CV} \rho dV + \int_{CS} \rho \vec{V} \cdot \vec{n} dA = 0$
    • $\rho$ represents fluid density (kg/m³)
    • $\vec{V}$ represents velocity vector (m/s)
    • $\vec{n}$ represents unit normal vector pointing outward from the control surface
    • $dV$ represents differential volume element (m³)
    • $dA$ represents differential area element (m²)

Continuity equation simplification

• Steady-state flow occurs when flow properties do not change with time at any point
  • Mathematically, $\frac{\partial}{\partial t} \int_{CV} \rho dV = 0$
• Incompressible flow occurs when fluid density remains constant
  • Density $\rho$ is constant throughout the flow
• Simplified continuity equation for steady-state, incompressible flow:
  • $\int_{CS} \vec{V} \cdot \vec{n} dA = 0$
  • For a control volume with one inlet and one outlet:
    • Volume flow rate at the inlet $Q_{in}$ equals volume flow rate at the outlet $Q_{out}$
    • $A_{in}V_{in} = A_{out}V_{out}$, where $A$ is cross-sectional area (m²) and $V$ is average velocity (m/s)

Mass flow rate calculations

• Mass flow rate $\dot{m}$ is the mass of fluid passing through a cross-section per unit time (kg/s)
  • Calculated using $\dot{m} = \rho Q = \rho AV$, where $\rho$ is density, $Q$ is volume flow rate, $A$ is cross-sectional area, and $V$ is average velocity
• For steady-state, incompressible flow with multiple inlets and outlets:
  • Sum of mass flow rates at inlets equals sum of mass flow rates at outlets
    1. $\sum \dot{m}{in} = \sum \dot{m}{out}$
    2. $\sum \rho A_{in}V_{in} = \sum \rho A_{out}V_{out}$
• Solving for unknown velocities or areas using given mass flow rates or volume flow rates
  • Apply the simplified continuity equation
  • Use given information (density, area, velocity) to solve for the unknown variable
• Relating changes in velocity to changes in cross-sectional area
  • For steady-state, incompressible flow: $A_1V_1 = A_2V_2$ (flow through a pipe or nozzle)
  • As cross-sectional area decreases, velocity increases to maintain constant volume flow rate (venturi meter)