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Reynolds Transport Theorem (RTT): A Fundamental Relation Between System and Control Volume Analyses

Reynolds Transport Theorem (RTT): A Fundamental Relation Between System and Control Volume Analyses

September 21, 2024

The Reynolds Transport Theorem is a foundational principle in fluid mechanics and thermodynamics that provides a bridge between the Lagrangian (system) and Eulerian (control volume) descriptions of fluid motion. It allows us to relate the rate of change of an extensive property for a system (a fixed quantity of matter) to the behavior of that property within a control volume (a specified region in space) and across its control surfaces.

Statement of the Reynolds Transport Theorem

For a general extensive property B of a system, the Reynolds Transport Theorem is expressed as:

$$ \frac{dB_{\text{sys}}}{dt} = \frac{\partial}{\partial t} \int_{\text{CV}} b , \rho , dV + \int_{\text{CS}} b , \rho , (\mathbf{V} \cdot \mathbf{n}) , dA $$

Where:

  • $\left( \frac{dB_{\text{sys}}}{dt} \right)$ = Rate of change of the extensive property B for the system.
  • $B_{\text{sys}} = \int_{\text{system}} b dm $ = Total amount of B in the system.
  • $\left( b = \frac{dB}{dm} \right)$ = Intensive property per unit mass (property density).
  • $ \rho $ = Fluid density (mass per unit volume).
  • $ \mathbf{V} $ = Velocity vector of the fluid at each point.
  • $ \mathbf{n} $ = Outward-pointing unit normal vector on the control surface.
  • $ \mathbf{V} \cdot \mathbf{n} $ = Component of velocity normal to the control surface.
  • $ \text{CV} $ = Control Volume (the region in space under consideration).
  • $ \text{CS} $ = Control Surface (the boundary of the control volume).
  • $ dV $ = Differential volume element within the control volume.
  • $ dA $ = Differential area element on the control surface.

Physical Meaning of the Terms

1. Left-Hand Side: System Perspective

$\left(\frac{dB_{\text{sys}}}{dt}\right)$: Rate of Change of B for the System

  • System Definition: A system refers to a specific, identifiable collection of matter that moves and deforms as it flows. It is a Lagrangian perspective, tracking individual fluid particles over time.
  • Physical Meaning: This term represents the total rate at which the extensive property B changes within the system due to all possible effects (e.g., accumulation, generation, depletion).
  • Example Properties: Total mass, momentum, energy, or any other conserved property.

2. First Term on the Right-Hand Side: Time Rate of Change within the Control Volume

$\left(\frac{\partial}{\partial t} \int_{\text{CV}} b , \rho , dV\right)$: Accumulation or Depletion of B within the Control Volume

  • Control Volume (CV): A specified region in space through which fluid flows. The CV can be stationary or moving but is fixed in space for the analysis. This is an Eulerian perspective, focusing on specific locations in space rather than individual particles.
  • Physical Meaning:
  • Accumulation: If more of the property B is entering the CV than leaving, or if it is being generated within the CV, the amount of B within the CV increases over time.
  • Depletion: If more of B is leaving the CV than entering, or if it is being consumed within the CV, the amount decreases over time.
  • Interpretation: This term quantifies how the amount of B inside the CV changes solely due to time-dependent processes within the volume.

3. Second Term on the Right-Hand Side: Net Flux Across the Control Surface

$\left(\int_{\text{CS}} b , \rho , (\mathbf{V} \cdot \mathbf{n}) , dA \right)$: Net Rate of Flow of B Across the Control Surface

  • Control Surface (CS): The boundary that encloses the control volume through which fluid can flow in or out.
  • $\mathbf{V} \cdot \mathbf{n}$ : The dot product represents the component of the fluid velocity normal (perpendicular) to the control surface at each point.
  • Positive Value: Fluid (and property B) is flowing out of the CV.
  • Negative Value: Fluid (and property B) is flowing into the CV.
  • Physical Meaning:
  • Influx (Inflow): The amount of B carried into the CV by the fluid.
  • Efflux (Outflow): The amount of B carried out of the CV by the fluid.
  • Net Flux: The difference between efflux and influx gives the net rate at which B is transported across the control surface.
  • Interpretation: This term accounts for the transport of B due to fluid motion across the boundaries of the CV.

Connecting the Perspectives

The RTT effectively states that:

Total Rate of Change of B for the System = Rate of Change of B within the Control Volume + Net Rate of Flow of B Across the Control Surface

  • System Approach (Lagrangian): Considers the property changes for a moving collection of matter.
  • Control Volume Approach (Eulerian): Focuses on fixed locations in space and how properties change within and across those locations.

The theorem allows us to apply conservation laws (mass, momentum, energy) within a control volume framework, which is immensely practical for engineering analyses where tracking individual fluid particles is not feasible.

Applications of the Reynolds Transport Theorem

  1. Conservation of Mass (Continuity Equation):
  • By letting $B = m$ (mass), and $b = 1$ (since, $b = \frac{dB}{dm}$ ):

$$ \frac{dm_{\text{sys}}}{dt} = \frac{\partial}{\partial t} \int_{\text{CV}} \rho , dV + \int_{\text{CS}} \rho , (\mathbf{V} \cdot \mathbf{n}) , dA $$

  • For incompressible steady flow, this reduces to the familiar continuity equation:

$$ \int_{\text{CS}} \rho , (\mathbf{V} \cdot \mathbf{n}) , dA = 0 $$

  1. Conservation of Momentum:
  • By letting ( B = \mathbf{P} = m \mathbf{V} ) (linear momentum), and ( b = \mathbf{V} ):

$$ \frac{d\mathbf{P}{\text{sys}}}{dt} = \frac{\partial}{\partial t} \int{\text{CV}} \mathbf{V} , \rho , dV + \int_{\text{CS}} \mathbf{V} , \rho , (\mathbf{V} \cdot \mathbf{n}) , dA $$

  • This form is used to analyze forces in fluid systems, such as jet propulsion or flow through pipes.
  1. Conservation of Energy:
  • By letting ( B = E ) (total energy), and ( b = e ) (specific energy):

$$ \frac{dE_{\text{sys}}}{dt} = \frac{\partial}{\partial t} \int_{\text{CV}} e , \rho , dV + \int_{\text{CS}} e , \rho , (\mathbf{V} \cdot \mathbf{n}) , dA $$

  • Essential for analyzing energy transfer processes, heat exchangers, turbines, etc.

Key Insights

  • Extensive vs. Intensive Properties:

    • Extensive Property B: Depends on the amount of matter (e.g., mass, momentum, energy).
    • Intensive Property b: Independent of the amount of matter, property per unit mass (e.g., velocity, specific energy).

  • Flux Representation:

    • The surface integral represents the flow of the property due to the movement of fluid across the control surface.

  • Selection of Control Volume:

    • The choice of CV is strategic, often chosen to simplify the problem by exploiting symmetries or areas where certain effects can be neglected.
    • The CV can be fixed, moving, or deforming, depending on the problem.

  • Time Dependence:

    • The partial derivative $ \frac{\partial}{\partial t} $ indicates that the control volume integral considers only the explicit time dependence within the CV, not the movement of the CV itself (if any).

Conclusion

The Reynolds Transport Theorem is a powerful tool that generalizes conservation laws for any extensive property in fluid systems. By analyzing the behavior of properties within a control volume and their flow across control surfaces, we can effectively study and solve complex fluid mechanics problems that involve the transport of mass, momentum, energy, and other quantities.

Understanding each term’s physical significance in the RTT provides deep insights into the underlying physics of fluid flow and is essential for proficiently applying conservation principles in engineering analyses and design.