6.2 Conservation of Mass (Continuity Equation)

Conservation of Mass (Continuity Equation)

The conservation of mass principle gives you a way to track fluid as it moves through a system. Because mass can't be created or destroyed, any fluid entering a control volume must either accumulate inside it or leave through an outlet. The continuity equation is the mathematical statement of this balance, and it's one of the most-used tools in fluid mechanics for analyzing pipes, nozzles, ducts, and branching flow networks.

Conservation of Mass Equation

A control volume is a fixed region in space that you define for analysis. Fluid flows into and out of this region through its boundary, called the control surface. Think of it like drawing an invisible box around a section of pipe: you don't track individual fluid particles, you just monitor what crosses the boundaries.

The mass balance for a control volume says: the rate of change of mass stored inside the control volume, plus the net mass flux leaving through the control surface, equals zero.

$\frac{\partial}{\partial t} \int_{CV} \rho \, dV + \int_{CS} \rho \vec{V} \cdot \vec{n} \, dA = 0$

Here's what each term means:

• $\rho$ — fluid density (kg/m³)
• $\vec{V}$ — velocity vector of the fluid (m/s)
• $\vec{n}$ — outward-pointing unit normal vector on the control surface
• $dV$ — differential volume element (m³)
• $dA$ — differential area element on the control surface (m²)

The first integral tracks how much mass is accumulating (or depleting) inside the control volume over time. The second integral sums up the mass flux across every part of the control surface. The dot product $\vec{V} \cdot \vec{n}$ is positive where flow leaves (outflow) and negative where flow enters (inflow), which is why this single integral captures the net mass flow.

Continuity Equation Simplification

Two assumptions dramatically simplify the general equation:

Steady-state flow means flow properties at any fixed point don't change with time. The storage term drops out:

$\frac{\partial}{\partial t} \int_{CV} \rho \, dV = 0$

Incompressible flow means density $\rho$ is constant throughout the fluid. Since $\rho$ doesn't vary, you can pull it out of the surface integral and then cancel it entirely.

With both assumptions, the continuity equation reduces to:

$\int_{CS} \vec{V} \cdot \vec{n} \, dA = 0$

For a control volume with one inlet and one outlet where velocity is uniform across each cross-section, this becomes:

$A_{in} V_{in} = A_{out} V_{out}$

where $A$ is the cross-sectional area (m²) and $V$ is the average velocity (m/s). Both sides represent the volume flow rate $Q$ (m³/s), so you can also write this as $Q_{in} = Q_{out}$.

This relationship is why fluid speeds up when it enters a narrower section of pipe. If the area shrinks by half, the velocity must double to maintain the same volume flow rate. A venturi meter exploits exactly this effect to measure flow speed.

Mass Flow Rate Calculations

The mass flow rate $\dot{m}$ is the mass of fluid passing through a cross-section per unit time (kg/s):

$\dot{m} = \rho Q = \rho A V$

For steady, incompressible flow with multiple inlets and outlets (like a branching pipe junction), conservation of mass requires:

$\sum \dot{m}_{in} = \sum \dot{m}_{out}$

Since $\rho$ is constant and cancels, this is equivalent to:

$\sum A_{in} V_{in} = \sum A_{out} V_{out}$

Solving a typical problem follows these steps:

1. Sketch the control volume and identify all inlets and outlets.
2. Write the continuity equation: $\sum Q_{in} = \sum Q_{out}$.
3. Substitute $Q = AV$ for each inlet and outlet.
4. Plug in known values (densities, areas, velocities) and solve for the unknown.

The key takeaway for area-velocity relationships: in steady, incompressible flow, $A_1 V_1 = A_2 V_2$. A decrease in cross-sectional area always produces a proportional increase in velocity, and vice versa. This inverse relationship between area and velocity shows up constantly in nozzle design, pipe sizing, and flow measurement devices.