3.1 Conservation of mass

Conservation of mass states that mass cannot be created or destroyed within a closed system. In fluid dynamics, this principle lets you track how fluid moves through pipes, channels, and complex systems by ensuring that every kilogram entering a region is accounted for. It's the starting point for nearly all fluid analysis, and the continuity equation that comes from it will show up constantly throughout this course.

Principle of mass conservation

The total mass of a fluid system remains constant over time, no matter what processes happen inside it. This holds for compressible and incompressible fluids alike, and it applies to single-phase flows (just water in a pipe) as well as multiphase systems (steam and liquid water coexisting in a boiler).

The key idea: if you draw an imaginary boundary around a region of fluid, any change in the mass inside that boundary must be explained by mass flowing in or out across the boundary.

Mass balance in fluid systems

A mass balance tracks the flow of mass into and out of a defined control volume (a region you choose for analysis) across its control surface (the boundary of that region). The general statement is:

Steady vs. unsteady flow

• Steady flow: fluid properties (velocity, pressure, density) at every point stay constant in time. The accumulation term drops to zero, so mass in equals mass out.
• Unsteady flow: properties change with time, so you must keep the time-derivative (accumulation) term. Examples include filling a tank or a pressure transient in a pipeline.

Inflow vs. outflow

• Inflow is mass crossing the control surface into the control volume.
• Outflow is mass crossing the control surface out of the control volume.
• The net mass flow rate is inflow minus outflow. For steady flow with no sources or sinks, this net rate must be zero.

Sources and sinks

Sometimes mass appears or disappears within the control volume relative to a particular phase or species. A chemical reaction can generate a product (source) or consume a reactant (sink). Phase changes like evaporation add vapor mass while removing liquid mass. These terms must be included in the mass balance whenever they're present.

Control volumes

A control volume is simply the region in space you choose to analyze. Picking the right one can make a problem straightforward or painfully complicated.

Fixed vs. moving boundaries

• Fixed control volumes have stationary boundaries. These are the most common choice and work well for steady-flow problems like flow through a stationary pipe junction.
• Moving control volumes have boundaries that translate or deform with time. You'd use these for problems involving moving objects, such as a rocket expelling exhaust or flow through a turbomachine rotor.

Selection of control surfaces

Good control surface choices simplify your math:

• Align surfaces perpendicular to the flow direction so that velocity vectors are normal to the surface. This avoids dealing with dot-product angles.
• Place surfaces where you already know (or can easily measure) flow properties like velocity or pressure.
• Choose surfaces at inlets and outlets where the flow is approximately uniform, reducing complex integrations to simple multiplications.

Continuity equation

The continuity equation is the mathematical form of mass conservation applied to a fluid. It relates the rate of change of mass inside a control volume to the net mass flux across its boundaries.

Differential form

The differential form applies at a single point in the flow field and describes mass conservation for an infinitesimally small fluid element:

$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$

Here, $\rho$ is the fluid density and $\vec{V}$ is the velocity vector. The first term is the local rate of change of density; the second term represents the net rate of mass flux out of the element per unit volume. Together they say: any decrease in local density must be compensated by a net outflow, and vice versa.

Integral form

The integral form applies to a finite control volume and is what you'll use most often for engineering problems:

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

• The first integral is the time rate of change of total mass inside the control volume.
• The second integral sums the mass flow rate across the entire control surface, where $d\vec{A}$ points outward by convention. Outflow contributes positively; inflow contributes negatively.

Simplifications and assumptions

Depending on the problem, you can often simplify the full continuity equation significantly:

• Steady flow (properties don't change with time): the time-derivative term drops out.
• Incompressible flow (constant density): density cancels, giving $\nabla \cdot \vec{V} = 0$ in differential form.
• Steady, one-dimensional flow: the integral form reduces to the familiar relation

$\rho_1 A_1 V_1 = \rho_2 A_2 V_2$

This says the mass flow rate $\dot{m} = \rho A V$ is the same at every cross-section. For an incompressible fluid ($\rho_1 = \rho_2$), it simplifies further to $A_1 V_1 = A_2 V_2$. This is why water speeds up when you squeeze a garden hose nozzle: the area decreases, so velocity must increase.

Applications of mass conservation

Incompressible flow

When density is effectively constant, the continuity equation simplifies considerably. This assumption works well for all liquids and for gases flowing at low Mach numbers (typically $Ma < 0.3$, which corresponds to about 100 m/s in air at sea level). The simplified form $\nabla \cdot \vec{V} = 0$ means the velocity field is divergence-free: fluid elements can change shape but not volume.

Compressible flow

At higher speeds or in systems with large pressure gradients, density changes become significant and can't be ignored. Gas flowing through a converging-diverging nozzle at $Ma > 0.3$ is a classic example. Here you need the full continuity equation, and it's typically solved simultaneously with the momentum and energy equations because density, pressure, velocity, and temperature are all coupled.

Multiphase systems

When multiple phases coexist (gas-liquid in a boiler, oil-water in a separator, gas-solid in a fluidized bed), you write a separate mass conservation equation for each phase. These equations include inter-phase mass transfer terms to account for evaporation, condensation, dissolution, or other phase-change processes. Additional closure models are needed to describe how the phases interact.

Conservation of mass vs. momentum

Mass conservation tells you how much fluid goes where. Momentum conservation (Newton's second law applied to fluids) tells you why it goes there by balancing forces and velocity changes. In practice, you almost always solve the continuity and momentum equations together as a coupled system. Energy conservation adds a third equation when temperature or compressibility effects matter. These three conservation laws form the foundation of the Navier-Stokes equations.

Numerical methods for mass conservation

For anything beyond simple geometries and flow conditions, analytical solutions to the continuity equation aren't feasible. Numerical methods discretize the domain and solve the equations computationally.

Finite volume method

1. Divide the flow domain into a mesh of small, non-overlapping control volumes.
2. Apply the integral form of the continuity equation to each control volume.
3. Approximate the fluxes across each face of each control volume.
4. Solve the resulting algebraic system iteratively.

Because the method directly enforces mass balance on every cell, it guarantees global mass conservation. This is why the finite volume method is the backbone of most commercial CFD software (ANSYS Fluent, OpenFOAM, etc.).

Finite element method

1. Discretize the domain into elements (triangles, tetrahedra, etc.).
2. Express the continuity equation in its weak (variational) form.
3. Approximate flow variables within each element using shape functions.
4. Solve for the unknown nodal values.

The finite element method handles complex geometries and irregular boundaries more naturally than structured finite volume meshes. It's especially common in multiphysics problems where fluid flow is coupled with structural deformation.

Experimental verification

Models and simulations are only useful if they match reality. Experimental techniques provide the data needed to validate mass conservation predictions.

Flow visualization techniques

These qualitative methods let you observe flow patterns and spot regions where mass might be accumulating or depleting unexpectedly:

• Dye injection: colored dye is introduced into a liquid flow to trace streamlines. Common in water tunnel experiments.
• Smoke injection: similar concept for gas flows, often used in wind tunnels.
• Particle image velocimetry (PIV): a laser sheet illuminates tracer particles, and high-speed cameras capture their displacement between frames. PIV gives you a full 2D (or even 3D) velocity field, making it possible to check whether $\nabla \cdot \vec{V} = 0$ holds in incompressible regions.

Mass flow rate measurements

These quantitative methods measure $\dot{m}$ at specific locations:

• Orifice plates: a plate with a hole restricts the flow, creating a measurable pressure drop proportional to flow rate. Simple and cheap, but introduces a permanent pressure loss.
• Venturi meters: a gradually converging-diverging section measures flow rate via pressure difference with less energy loss than an orifice plate.
• Coriolis mass flow meters: the fluid flows through a vibrating tube, and the Coriolis force causes a phase shift proportional to mass flow rate. These measure $\dot{m}$ directly (not volume flow rate), making them highly accurate regardless of fluid properties.

Comparing measured mass flow rates at inlets and outlets against model predictions is one of the most direct ways to verify that a simulation correctly enforces mass conservation.