1. Principle
For any system of fixed mass: \[ \frac{D m_{\text{system}}}{D t} = 0 \] where \(D/Dt\) is the material (Lagrangian) derivative following the system.
2. Control volume formulation
Using the Reynolds Transport Theorem, the conservation of mass for a control volume (CV) with control surface (CS) is: \[ \frac{d}{dt} \int_{CV} \rho \, dV + \oint_{CS} \rho\, \vec{V} \cdot \vec{n} \, dA = 0 \] where:
- \(\rho\): fluid density (kg/m\(^3\))
- \(\vec{V}\): velocity vector (m/s)
- \(\vec{n}\): outward unit normal to the surface
The first term is the rate of change of mass inside the CV; the second term is the net mass flux out through the CS.
3. Differential (continuity) equation
Applying the divergence theorem and shrinking the CV to a point yields: \[ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0 \]
- For incompressible flow (\(\rho = \text{const}\)): \[ \nabla \cdot \vec{V} = 0 \]
4. Steady-flow form
For steady flow (\(\partial/\partial t = 0\)), the integral form reduces to: \[ \sum_{\text{inlets}} \dot{m} = \sum_{\text{outlets}} \dot{m} \] where \(\dot{m} = \rho V_n A\) is the mass flow rate through each inlet/outlet, \(V_n\) being the normal velocity component.
5. Worked examples
Example 1: Steady incompressible flow in a pipe junction
Two inlet pipes (1 and 2) merge into one outlet pipe (3). Given: \(\rho = 1000\ \text{kg/m}^3\), \(Q_1 = 0.02\ \text{m}^3/\text{s}\), \(Q_2 = 0.01\ \text{m}^3/\text{s}\). Find \(V_3\) if \(D_3 = 0.1\ \text{m}\).
Solution: \[ Q_3 = Q_1 + Q_2 = 0.03\ \text{m}^3/\text{s} \] Cross-sectional area: \[ A_3 = \frac{\pi D_3^2}{4} \approx 7.854\times 10^{-3}\ \text{m}^2 \] Velocity: \[ V_3 = \frac{Q_3}{A_3} \approx 3.82\ \text{m/s} \]
Example 2: Tank filling
A rigid tank of volume \(V_t = 2\ \text{m}^3\) initially contains air at \(\rho_i = 1.2\ \text{kg/m}^3\). Air enters at \(\dot{m}_{in} = 0.5\ \text{kg/s}\) with no outflow. Find the density after 10 s.
Mass balance: \[ \frac{d}{dt}(\rho V_t) = \dot{m}_{in} \] Integrating: \[ \rho(t) = \rho_i + \frac{\dot{m}_{in}}{V_t} t \] \[ \rho(10) = 1.2 + \frac{0.5}{2} \times 10 = 3.7\ \text{kg/m}^3 \]
6. Applications
- Flow measurement and metering
- Design of piping networks and ducts
- CFD simulations (continuity equation)
- Environmental modeling (pollutant transport)
- Combustion and chemical reactor analysis
7. Notes
- Mass conservation applies to all physical processes, including multiphase and reacting flows (with chemical mass conservation for each species).
- In relativistic physics, mass-energy conservation generalizes the principle.