Energy Equation

Derivation of Energy Equation from Navier-Stokes

• From the Navier-Stokes equation derived previously, there are cases requiring solving it with the energy equation.
• Example: When density is a function of temperature and pressure, or in natural convection where the density gradient is created by a temperature gradient.
• Energy and Navier-Stokes equations need to be dealt with together in certain situations.
• For constant density ρ, the velocity field can be used separately in the energy equation to solve for temperature field.
• Variability in ρ means the equations need to be coupled.

Reynolds Transport Theorem

• Use the Reynolds transport theorem (RTT) to derive the energy equation.
• Total energy (N) includes internal energy, kinetic energy, and potential energy: E = i + V²/2 + gz.
• Apply RTT for total energy of the system.
• RTT Equations for non-deformable and stationary control volume leading to simplified forms involving divergence theorem.

System vs. Control Volume

• First law of thermodynamics applied to a system: dE/dt (system) = Q (heat) - W (work).
• Energy balance: dE/dt total change = heat input - work done.
• Conversion from system to control volume through RTT.

Heat Transfer and Work Done in Control Volume

• Heat transfer (Q): Combines volumetric and surface terms.
  • Volumetric: Q''' (rate of heat generation per unit volume).
  • Surface: q'' (heat flux vector) adjusted by the unit normal vector η.
• Work Done (W): Includes body force and surface force considerations.
  • Body force: b_i u_i dV (dot product with velocity).
  • Surface force: T_i u_i dA, using tau.η and divergence theorem for simplification.

First Law within Control Volume

• Heat transfer and work done in control volume converted to volumetric integrals.
• Algebraic sign adjustments for work done by/against force.
• Energy balance and transformation to conservative forms using continuity equation.

Energy Balance and Internal Energy

• Subtract mechanical energy equation from total energy equation to isolate internal energy conservation.
• Viscous dissipation represented as τ_ij (viscous stress tensor) terms.
• Surface forces responsible for heating, derived from Newtonian and Stokesian fluid assumptions.

Energy Equation in Terms of Internal Energy and Enthalpy

• Conversion of internal energy to enthalpy (h = i + p/ρ).
• Generalised form: ρ Dh/Dt - Dp/Dt = Q''' - Δ.q + μΦ (viscous dissipation).

Integration of Heat and Work Terms

• Derive terms related to heat generation and flux, work done by body and surface forces.
• Application of divergence theorem and thermodynamic principles.

Conversion to Practical Forms

• Internal energy: i + kinetic energy simplified.
• Introduction of enthalpy (h) to derive in terms of measurable quantities (temperature).
• Future work: Transform enthalpy (h) equation to temperature (T) depending on thermodynamic properties.

Summary

• Energy equation from Navier-Stokes and RTT for various cases.
• Important thermodynamic and fluid mechanics interactions considered.
• Viscous dissipation and work done impacts on internal energy.
• Next steps involve deriving temperature-based energy equation from enthalpy.