Lecture 20: Laminar Non-premixed Flames – Introduction, Non-reacting Jets, Simplified Description of Laminar Non- premixed Flames Yi versus f Experimental Data Qualitative characteristics of laminar non-premixed or diffusion (of fuel and oxidizer) flames. Review of conserved scalar concept. Role of the momentum equation in deflagration regime: Non-reacting jet mixing solution. Simplified theoretical description of a laminar non-premixed (or diffusion) flame.

Non premixed Flame Configurations Vertical wall fire Horizontal wall fire Inclined wall fire Upward flame spread Downward flame Corner fire Beam, column fires Pool fires Forest fires Platform fire Combination fires Spherical stagnation pt. flame • Opposed jet nonpremixed flame Air Fuel Stagnation point flow non- premixed flame Swirling flow flame with cross fuel injection

Laminar Jet Diffusion Flames (Non-premixed Jet Flames)

Nonpremixed flames Fuel (F) and oxidizer (O) are stored apart. When combustion is desired, F and O must come together at the molecular level. How many molecules of each decides interim and final products and their temperature. Staged pre or post reaction mixing and rich and lean reactions all lead to different products. Specific strategies such as R-Quench-L, Lean Direct Injection, Direct Injection-Spark Ignition have emerged. Learning the non-premixed flame regime is important. Learning about equipment for specific strategies is critical

Mixture Fraction, Mixedness, Progress Variables, Reaction Fraction, and Reactedness Concept of a conserved scalar is very useful for nonpremixed flames. A conserved scalar is a quantity defined such that there are no sink or source terms in the conservation equation for that quantity. (Sink and source terms result from reactions, heat transfer, and work transfer) Quantification of how non-(pre)mixed and when Total Energy is a conserved scalar in the absence of net heat loss to or work done on boundaries. Elemental Mass Fractions, Fraction of Mass that originated in the fuel stream(s) and Fraction of Mass that originated in the oxidizer stream(s) are all conserved scalars.

Review of Conserved Scalar, Definition of Mixture Fraction In nonpremixed flames species mass fractions very continuously as mixing at the molecular level and chemical reaction occurs. Definition of mixture fraction f: Independent of the progress of reaction. That means CH4+O2 have the same mixture fraction as CO+H2O+H2 Both are f=16/(16+32)=(12+2+2)/(28+18+2) =16/48= 1/3 Fuel Mass + Mass in products that came from fuel f stoich= 16/16+64= 16/80=0.2

Review of Conserved Scalar, Definition of Mixture Fraction Consider the three-"species" reaction: For this system the mixture fraction will be:

Conservation equation for the mixture fraction. Assume all species diffuse at the same rate:

Review of Conserved Scalar, Definition of Mixture Fraction Divide the product species conservation equation by and add to the fuel species conservation equation: Substituting for the mixture fraction we obtain:

Review of Conserved Scalar, Definition of Mixture Fraction When kinetic energy is neglected (along with potential energy, thermal radiation, viscous dissipation, differential diffusion) we can write a similar equation for the absolute enthalpy: These conserved scalar equations, in cylindrical coordinates, will be very useful for our study of laminar non-premixed flames.

Solution for a Non-reacting, Constant Density Laminar Jet Consider first the case of a laminar jet of fuel issuing into air with no chemical reaction. Assuming the air and fuel have the same density, there is an analytical solution for the flow field away from the potential core region of the jet.

Solution for a Non-reacting, Constant Density Laminar Jet

Assumptions: Non-reacting, Constant Density Laminar Jet - MW(jet fluid) = MW(air), ideal gases. - Constant P, T, and r throughout the flow field. - Steady state. - Fick's law applies. - Equal species and momentum diffusivities, Sc = n /D = 1. - Neglect axial diffusion of momentum and species. - Solution applies downstream of the jet core region. 14/36

Conservation Equations: Non-reacting, 𝜌=C Laminar Jet Mass Axial Momentum Species

Boundary Conditions for a Non-reacting, 𝜌=C Laminar Jet Along the jet centerline: Far from the jet: At the jet exit plane , r ≤ R: At the jet exit plane, r > R:

Solution for a Non-reacting Constant Density Laminar Jet The solution to this problem can be found in Schlichting Boundary Layer Theory for the region of the flow beyond the jet core where the flow is similar. The solution is given by:

Axial velocity distribution: Fuel mass fraction distribution (assuming Sc = n/D = 1):

Infinitely Fast Kinetics Infinitesimal Flame Sheet Approximation for Nonpremixed Flames Law, Combustion Physics, 2006 Finite-rate Kinetics Infinitely Fast Kinetics

Simplified Theoretical Description of Laminar Jet Diffusion Flame • Assume: 1. Laminar, steady, axisymmetric flow 2. Three "species": fuel, product, oxidizer 3. Flame (reaction) sheet assumption, infinitely fast chemical kinetics 4. Equal species diffusivities 5. Le = 1 6. No radiation transport 7. Axial diffusion is neglected 8. Vertical flame axis

Conservation Equations: Cylindrical Coordinates, Thin Flame Conservation of Mass Conservation of Axial Momentum Conservation of Fuel Mass Fraction (Inside the Flame Sheet) Conservation of O2 Mass Fraction (Outside the flame sheet) Conservation of Product Mass Fraction (Everywhere)

Simplified Theoretical Description of Laminar Jet Diffusion Flame Conservation of Species Mass Fraction

Conserved Scalar Equations for Laminar Jet Flame Boundary Conditions At the jet exit plane

Non-dimensional Laminar Jet Diffusion Flame A dimensionless enthalpy is defined: The non-dimensional conservation equations and boundary conditions for h* and f are identical, and therefore h* = f. The non-dimensional conservation equations and boundary conditions for h* and f are identical, and therefore h* = f.

Description of Global Fast Chemistry Assume that the reaction kinetics are described by a single-step, three-species reaction: Inside the flame sheet Outside the flame sheet

State Relationship for Temperature: Fuel Side Assume constant heat capacities, and that for the jet fluid and oxidizer far from the jet, T = 298 K. Inside the flame sheet: Oxidizer = air Solve the h* = f equation for T :

State Relationship for Temperature: Oxidizer Side

Experimental Support for State Relationships for Major Species