ME 525: Combustion Lecture 11: Momentum and Energy Conservation

1. ME 525: CombustionLecture 11: Momentum and Energy Conservation • Review of the momentum equation in one and in three • dimensions. • Role of momentum transfer in combustion- laminar • flow for most of this class, but with references to • turbulent mixing and combustion to connect with • applications. • Energy conservation equation for reacting flows in • different coordinate systems. • Relationship between species and energy conservation • equations. • Non-dimensional numbers.

2. Momentum Equation Momentum equation represents balance of forces in 3 dimensions: 3 equations representing 3 components of the forcevector.Forces in each direction include pressure, shear and body forces. Visualizing in 2 dimensions for simplicity x

3. Momentum Equation Momentum equation represents balance of forces in 3 dimensions: 3 equations representing 3 components of the forcevector. Forces in each direction include pressure, shear and body forces

4. Momentum Equation The x-component of the momentum equation is: For steady 1-D flow, neglecting friction

5. Momentum Equation: Cylindrical Coordinates for Jet Flows The x-component of the momentum equation is: A B C D E A: x momentum flow by axial convection per unit volume B: x momentum flow by radial convection per unit volume C: Viscous forces per unit volume D: Approximate pressure gradient in the x direction E: Body force in the x direction

6. Control Volume for Steady State Energy Equation . QRad”’

7. Energy Conservation – One-Dimensional Form Assumptions: • Steady state • No radiation heat transfer • No shaft work or viscous dissipation • Potential energy is negligible • Constant area duct With these assumptions, the 1-D energy equation is:

8. The Energy Flux Vector

9. Conservation of Energy The advection-diffusion balance is: The second term on RHS includes energy change because of advection of the mixture as well as diffusion of each species within the mixture.

10. Conservation of Energy and Species • Species conservation: • Combine conservation of energy and species to get equation 7.55 from Turns • We will now write forms of the energy equation for low-speed flames referred to as “deflagrations.” For these flames, we can neglect KE (different story for detonations, to be discussed later).

11. Convenient forms of the Energy Equation • Eqn. 7.55 applies to multi-component as well as binary systems, no assumptions is made with regard to physical properties k, r, cP, etc. • Convenient forms recognizing definitions and assumptionscan be written as:

12. Energy Conservation Equation: Characteristic Lengths and Times