18.2 : Conservation of Mass in Fixed, Nondeforming Control Volume

The principle of conservation of mass is fundamental in fluid dynamics and is crucial for analyzing flow within fixed control volumes, such as pipes or ducts. This principle states that the total mass within a control volume remains constant unless altered by the inflow or outflow of mass through the control surfaces. This results in a vital relationship for steady, incompressible flow where the mass entering a system equals the mass leaving it.

In the case of a sewer pipe, which can be modeled as a fixed, nondeforming control volume, mass conservation dictates that the inlet's mass flow rate must equal the outlet's mass flow rate. The control surfaces in this scenario are at the inlet and outlet of the pipe, enclosing the flow region. Given that the flow is incompressible, the density of the fluid remains constant. This means that the relationship between velocity and cross-sectional area becomes critical.

Mass Flow Rate and Cross-Sectional Area

The mass flow rate (ṁ) is expressed in terms of fluid density (Ρ) and flow rate (Q). Here, the flow rate is determined by multiplying the velocity (V) by the cross-sectional area (A).

For steady flow, where the density remains constant, this equation simplifies to indicate that any change in velocity must correspond to a change in the pipe's cross-sectional area. When the pipe's cross-sectional area decreases, as might happen at the outlet of a sewer pipe, the velocity of the fluid must increase to conserve mass flow rate. The continuity equation describes this phenomenon:

Where V₁ and A₁ denotes the velocity and cross-sectional area at the inlet and V₂ and A₂ at the outlet. This relationship predicts flow behavior and design systems that maintain appropriate flow rates while adhering to the principle of mass conservation.