1. 1 MAE 5310: COMBUSTION FUNDAMENTALS Introduction to Laminar Diffusion Flames: Non-Reacting Constant Density Laminar Jets Mechanical and Aerospace Engineering Department Florida Institute of Technology D. R. Kirk

2. 2 LAMINAR DIFFUSION FLAME OVERVIEW Subject of lots of fundamental research –Applications to residential burners (cooking ranges, ovens) –Used to develop an understanding of how soot, NO 2, CO are formed in diffusion burning –Mathematically interesting: transcendental equation with Bessel functions (0 th and 1 st order) Introduce concept of conserved scalar (very useful in various aspects of combustion and introduced here) Desire to understand flame geometry (usually desire short flames) –What parameters control flame size and shape –What is the effect of different types of fuel –Arrive at useful (simple) expression for flame lengths for circular-port and slot burners CO 2 production in diffusion flame

3. 3 LAMINAR DIFFUSION FLAME OVERVIEW (LECTURE 1) Reactants are initially separated, and reaction occurs only at the interface between fuel and oxidizer (mixing and reaction taking place) Diffusion applies strictly to molecular diffusion of chemical species In turbulent diffusion flames, turbulent convection mixes fuel and air macroscopically, then molecular mixing completes the process so that chemical reactions can take place Orange Blue Full range of  throughout reaction zone

4. 4 NON-REACTING CONSTANT DENSITY LAMINAR JETS Examine non-reacting laminar jet of fluid (fuel) issuing into a infinite reservoir of quiescent fluid (oxidizer) –Why? Simpler case to develop understanding of basic flow field Physical description of jet (Reference picture on next slide) –Potential core: effects of viscous shear and diffusion have yet to be felt Both the velocity profile and nozzle-fluid mass fraction remain unchanged from their nozzle-exit values and are uniform in this region Similar to developing pipe flow, except that in a pipe conservation of mass requires uniform flow to accelerate –Between potential core and jet ‘edge’, both velocity and fuel concentration (mass fraction) decrease monotonically to zero at edge of jet Beyond potential core (x > x c ), effects of viscous shear and mass diffusion are active across whole width of jet –Initial jet momentum is conserved through entire flow field Jet momentum flow at any x, J = momentum flow issuing from nozzle, J e

5. 5 NON-REACTING CONSTANT DENSITY LAMINAR JETS Processes that control velocity field (convection and diffusion of momentum) are similar to processes that control fuel concentration field (convection and diffusion of mass) Distribution of Y F (r,x) similar to distribution of u x (r,x)/u e Because of high concentration of fuel in center of jet, fuel molecules diffuse radically outward in accordance with Fick’s law (see assumptions page) Effect of moving downstream is to increase time available for diffusion to take place –Width of the region containing fuel molecules grows with axial distance, x, and centerline fuel concentration decays Centerline velocity decay Radial velocity decay

6. 6 DETAILED ANALYSIS: NON-REACTING LAMINAR JETS Assumptions –Jet velocity profile is uniform at tube exit (r ≤ R) –Molecular weights of jet and reservoir fluid are equal (MW fuel =MW air ), constant T and P, ideal gas, constant  –Species molecular transport is by binary diffusion governed by Fick’s law –Momentum and species diffusivities are constant and equal Schmidt, Sc = /D = 1 (Recall Le =  /D) –Only radial diffusion of momentum and species is important Axial diffusion is neglected Implies that solution only applies some distance downstream of nozzle exit since near exit axial diffusion is quite important

7. 7 GOVERNING EQUATIONS AND BOUNDARY CONDITIONS Boundary Layer Equations (see Schlichting or White) Conservation of mass Momentum Boundary Conditions: Along the jet centerline (r = 0) No sources of sinks of fluid along axis Symmetry At large radii (r → ∞) At jet exit (x =0) axial velocity and fuel mass fraction are uniform Everywhere else they are zero

8. 8 FLOW FIELD RESULTS: SIMILARITY SOLUTION The velocity field can be obtained by assuming the profiles to be similar. The idea of similarity is that the intrinsic shape of the profile is the same everywhere in the flow field For this problem implication is that radial distribution u x (r,x), when normalized by local centerline velocity u x (0,r), is a function that depends only on similarity variable, r/x Solution for axial velocity Solution for radial velocity  contains similarity variable, r/x Axial velocity in dimensionless form Dimensionless centerline velocity

9. 9 CENTERLINE VELOCITY DECAY FOR LAMINAR JETS Velocity decays inversely with axial distance and is directly proportional to jet Reynolds number, Re j Solution is not valid near nozzle Decay is more rapid with lower Re jets –As Re is decreased, relative importance of initial jet momentum becomes smaller in comparison with viscous shearing action, which slows the jet Figure also represents decay of centerline mass fraction, Y F (see next slides) Re j =2.97 Re j =29.7 Re j =297

10. 10 SPREADING RATE, SPREADING ANGLE, JET HALF WIDTH Other parameters are frequently used to characterize jets –Jet half-width, r 1/2 Radial location where jet velocity has decayed to 1/2 of centerline value –Spreading rate Ratio of the jet half-width to the axial distance, x, –Spreading angle,  Angle whose tangent is the spreading rate High Reynolds number jets are narrow Low Reynolds number jets are wide Consistent with Reynolds number dependence of velocity decay

11. 11 CONCENTRATION FIELD SOLUTION AND RESULTS Solution of concentration field is mathematically similar to governing equation for momentum conservation If /D = 1 (Lewis number unity), function form of solution for Y F is identical to what for u x /ue Q F is volumetric flow rate from nozzle Written with Re j as controlling parameter Centerline expression Solutions can only be applied far from nozzle

12. 12 EXAMPLES: NON-REACTING LAMINAR JETS Part 1 –A jet of ethylene (C 2 H 4 ) exits a 10 mm diameter nozzle into still air at 300 K, and 1 atm. –Compare spreading angles and axial location where jet centerline mass fraction drops to stoichiometric value –Initial jet velocities of 10 cm/s and 1 cm/s,  ethylene at 300 K is 1.023x10 -9 N s/m 2 –Answer comment: Low-velocity jet is much wider Fuel concentration of low-velocity jet decays to same value as high-velocity jet in 1/10 th distance Part 2 –Using 1 cm/s as a baseline case, determine what nozzle exit radius is required to maintain same flow rate if exit velocity is increased by a factor of 10 to 10 cm/s Part 3 –Determine axial location for Y F,0 = Y F,stoichiometric for condition in Part 2 and compare with baseline –Answer comment: The distance calculated in Part 3 is identical to the 1 cm/s case in Part 1 Spatial fuel mass-fraction distribution depends on initial volumetric flow rate, Q, for a given fuel ( =  /  = constant) Problem #4-40 from F. White, Viscous Fluid Flow: –Air at 20 °C and 1 atm issues from a circular hole and forms a round laminar jet. At 20 cm downstream of the hole the maximum jet velocity is 35 cm/s. Estimate, at this position (a) the 1% jet thickness, (b) the jet mass flow, and (c) an appropriate Reynolds number for the jet

13. 13 LOOK AGAIN AT BUNSEN BURNER Fuel-rich pre-mixed inner flame Secondary diffusion flame Results when CO and H products from rich inner flame encounter ambient air What determines shape of flame? (velocity profile, flame speed, heat loss to tube wall) Under what conditions will flame remain stationary? (flame speed must equal speed of normal component of unburned gas at each location) What factors influence laminar flame speed and flame thickness ( , T, P, fuel type) How to characterize blowoff and flashback Most practical devices (Diesel-engine combustion) has premixed and diffusion burning