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
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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: ∂t∂∫CVρdV+∫CSρV⋅ndA=0
ρ represents fluid density (kg/m³)
V represents velocity vector (m/s)
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, ∂t∂∫CVρdV=0
Incompressible flow occurs when fluid density remains constant
Density ρ is constant throughout the flow
Simplified continuity equation for steady-state, incompressible flow:
∫CSV⋅ndA=0
For a control volume with one inlet and one outlet:
Volume flow rate at the inlet Qin equals volume flow rate at the outlet Qout
AinVin=AoutVout, where A is cross-sectional area (m²) and V is average velocity (m/s)
Mass flow rate calculations
Mass flow rate m˙ is the mass of fluid passing through a cross-section per unit time (kg/s)
Calculated using m˙=ρQ=ρAV, where ρ 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
∑m˙in=∑m˙out
∑ρAinVin=∑ρAoutVout
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: A1V1=A2V2 (flow through a pipe or nozzle)
As cross-sectional area decreases, velocity increases to maintain constant volume flow rate (venturi meter)