Control volumes are essential tools in fluid mechanics, simplifying analysis by defining boundaries for fluid flow. They allow us to apply conservation laws to complex systems, making it easier to study flow through pipes, ducts, and open channels.

The Reynolds Transport Theorem is a powerful mathematical framework that connects changes in extensive properties within a control volume to the net flux across its surface. It's crucial for applying conservation laws to mass, momentum, and energy in fluid systems.

Control Volumes and the Reynolds Transport Theorem

Control volumes in fluid mechanics

• Control volume represents an imaginary boundary or region in space chosen for analysis of fluid flow
  • Can be fixed in space, moving with constant velocity, or deforming over time
  • Fluid can enter or leave the control volume by crossing the control surface which is the boundary of the control volume
• System refers to a specific amount of matter that can be solid or fluid and its mass remains constant
• Control volumes simplify analysis of fluid flow problems by allowing application of conservation laws
  • Useful in analyzing flow through pipes, ducts (HVAC systems), and open channels (rivers, canals)

Derivation of Reynolds Transport Theorem

• Extensive properties depend on the amount of matter in the system and include mass, momentum, and energy
  • Expressed as $B = \int_{CV} b \rho dV$ where $b$ is the intensive property, $\rho$ is density, and $dV$ is the differential volume element
• Reynolds Transport Theorem relates the change of an extensive property in a control volume to the rate of change within the control volume and the net flux across the control surface
  • Mathematical expression: $\frac{DB_{sys}}{Dt} = \frac{\partial}{\partial t} \int_{CV} b \rho dV + \int_{CS} b \rho (\vec{V} \cdot \vec{n}) dA$
    • $\frac{DB_{sys}}{Dt}$ represents the rate of change of the extensive property $B$ in the system
    • $\frac{\partial}{\partial t} \int_{CV} b \rho dV$ represents the rate of change of $B$ within the control volume
    • $\int_{CS} b \rho (\vec{V} \cdot \vec{n}) dA$ represents the net flux of $B$ across the control surface
  • $\vec{V}$ is the velocity vector of the fluid
  • $\vec{n}$ is the unit normal vector pointing outward from the control surface
  • $dA$ is the differential area element on the control surface

Applications of Reynolds Transport Theorem

1. Conservation of mass: Set $b = 1$ in the Reynolds Transport Theorem

   • Resulting equation: $\frac{\partial}{\partial t} \int_{CV} \rho dV + \int_{CS} \rho (\vec{V} \cdot \vec{n}) dA = 0$
   • For steady flow, $\frac{\partial}{\partial t} \int_{CV} \rho dV = 0$, and the equation reduces to $\int_{CS} \rho (\vec{V} \cdot \vec{n}) dA = 0$

2. Conservation of momentum: Set $b = \vec{V}$ in the Reynolds Transport Theorem

   • Resulting equation: $\sum \vec{F} = \frac{\partial}{\partial t} \int_{CV} \vec{V} \rho dV + \int_{CS} \vec{V} \rho (\vec{V} \cdot \vec{n}) dA$
   • $\sum \vec{F}$ represents the sum of all external forces acting on the control volume such as pressure, gravity, and shear

3. Conservation of energy: Set $b = e + \frac{V^2}{2} + gz$ in the Reynolds Transport Theorem

   • $e$ is the specific internal energy, $\frac{V^2}{2}$ is the specific kinetic energy, and $gz$ is the specific potential energy
   • Resulting equation: $\dot{Q} - \dot{W} = \frac{\partial}{\partial t} \int_{CV} (e + \frac{V^2}{2} + gz) \rho dV + \int_{CS} (e + \frac{V^2}{2} + gz) \rho (\vec{V} \cdot \vec{n}) dA$
   • $\dot{Q}$ represents the rate of heat transfer into the control volume
   • $\dot{W}$ represents the rate of work done by the control volume on its surroundings