Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

1. Advanced Transport Phenomena Module 7 Lecture 32 Similitude Analysis: Flame Flashback, Blowoff & Height Dr. R. Nagarajan Professor Dept of Chemical Engineering IIT Madras

2. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Flashback of a Flame in a Duct: • Depends on existence of region near duct where local streamwise velocity < prevailing laminar flame speed, Su • No flame can propagate closer to wall than “quenching distance” dq, given by: a mixture thermal diffusivity 2

3. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Flashback of a Flame in a Duct: • Critical condition for flashback is of “gradient” form, i.e., U/d ̴Su/dq, or: • Multiplying both sides by d2/au leads to correlation law of Peclet form: • Basis for accurate flashback predictions in similar systems

4. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Flashback of a Flame in a Duct: Correlation of flashback limits for premixed combustible gases in tubes (after Putnam and Jensen (1949))

5. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • “Blow-Off” from Premixed Gas Flame “Holders”: • Oblique premixed gas flames can be stabilized in ducts even at feed-flow velocities >> Su • Anchor is well-mixed zone of recirculating reaction products • e.g., found immediately downstream of bluff objects (rods, disks, gutters), in steps of ducts • Sharp upper limit to feed-flow velocity above which blow-out or extinction occurs  Ubo

6. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS Schematic of an oblique flame “ anchored” to a gutter-type (2 dimensional) flame-holder (stabilizer) in a uniform stream of premixed combustible gas (Su<U<Ubo)

7. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • “Blow-Off” from Premixed Gas Flame “Holders”: • Su measure of reaction kinetics • Similitude to GT combustor efficiency example yields: where

8. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • “Blow-Off” from Premixed Gas Flame “Holders”: • Based on SA of Su data: • Solving for Peclet number at blow-off:

9. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • “Blow-Off” from Premixed Gas Flame “Holders”: • Experimentally:

10. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS “Blow-Off” from Premixed Gas Flame “Holders”: Test of proposed correlation of the dimensionless "blow-off” velocity, for a flame stabilized by a bluff body of transverse dimension L in a uniform, premixed gas stream ( adapted from Spalding (1955))

11. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • “Blow-Off” from Premixed Gas Flame “Holders”: • Alternative approach: recirculation zone likened to WSR • 3D stabilizer of transverse dimension L exhibits recirculation zone with effective volume given by:

12. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • “Blow-Off” from Premixed Gas Flame “Holders”: • Fuel-flow rate into recirc/ reaction zone can be written as: • Blow-out occurs when corresponding volumetric fuel consumption rate is near

13. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • “Blow-Off” from Premixed Gas Flame “Holders”: • Rearranging: • Additional insight: blow—off velocity scales linearly with transverse dimension of flame stabilizer, at sufficiently high Re • Experimentally verified

14. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Laminar Diffusion Flame Height: • Buoyancy and fuel-jet momentum contribute to height, Lf, of fuel-jet diffusion flame • Simple model: relevant groupings of variables • Beyond realm of ordinary dimensional analysis • Treat hot “flame sheet” region as cause of natural convective inflow of ambient oxidizer

15. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Laminar Diffusion Flame Height: • For any fuel/ oxidizer pair, when buoyancy dominates: • If fuel-jet momentum dominates, at constant Re: • Rj = ½ dj • Fr  Froude number:

16. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS Correlation of laminar jet diffusion flame lengths ( adapted from Altenkirch et al. (1977))

17. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Fuel Droplet Combustion at High Pressures: • Burning time, tcomb, depends on pressure • In rockets, aircraft gas turbines, etc., pressures > 20 atm may be reached • Based on diffusion-limited burning-rate theory & available experimental data at p ≥ 1 atm: • K  burning rate constant • Dependent on fuel type & environmental conditions • Independent of droplet diameter

18. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS • Fuel Droplet Combustion at High Pressures: • Each fuel also has a thermodynamic critical pressure, pc, to which prevailing pressure, p, may be compared: • Hence, following correlation may be obtained: • Corresponding-states analysis for high-pressure droplet combustion is reasonably successful • Allows estimation of burning times where data are not available

19. PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS Fuel Droplet Combustion at High Pressures: Approximate correlation of fuel-droplet burning rate constants at elevated pressures ( based on data of Kadota and Hiroyasu (1981))

20. ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS • Configurational Analysis (Becker, 1976): • Establishing conditions of similarity by forming a sufficient set of eigen-ratios • Leads to similitude criteria in the form of dimensionless ratios of: • Inventories • Source strengths • Fluxes • Lengths • Time, etc. • Can start analysis from macroscopic or microscopic CVs

21. ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS • But, interpretation not unique: • All may be regarded as ratios of same type, e.g., characteristic times • Relevant even to SS problems • e.g., length ratio in forced convection system  fluid transit-time ratio

22. ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS • e.g., momentum-flux ratio (Re)  ratio of times required for momentum diffusion & convection through common area, L2 • e.g., Pr  ratio of times governing diffusive decay of nonuniformities of energy (L2/a) and momentum (L2/n)

23. ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS • Characteristic Times: • e.g., fluid-phase, homogeneous-reaction Damkohler number  ratio of characteristic flow time to characteristic chemical reaction time • e.g., surface (heterogeneous) Damkohler number  ratio of characteristic reactant diffusion time across boundary layer to characteristic consumption time on surface

24. ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS • Characteristic Times: • e.g., Stokes number (governing dynamical non-equilibrium in two-phase flows)  ratio of particle stopping time to fluid transit time (L/U) • Attractive way of dealing with complex physicochemical problems, e.g.: • Gas-turbine spray combustor performance • Coal-particle devolatilization

25. ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS • Further simplifications possible through: • rational parameter groupings • Dropping parameters via sensitivity analysis • Make maximum use of available insight & data– analytical & experimental

26. BENEFITS OF SIMILITUDE ANALYSIS • Integrates results of: • Scale-model testing • Full-scale testing • Mathematical modeling • Allows judicious blend of all three • Based on fundamental conservation principles & constitutive laws • Maximum insight with minimum effort

27. BENEFITS OF SIMILITUDE ANALYSIS • Yields set-up rules for designing scale-model experiments amenable to quantitative use • Reflects relative importance of competing transport phenomena • Leads to smaller set of relevant parameters compared to “dimensional analysis” • Sensitivity-analysis can enable “approximate” or “partial” similarity analysis

28. OUTLINE OF PROCEDURE FOR SA • Write necessary & sufficient equations to determine QOI in most appropriate coordinate system • PDEs • bc’s • ic’s • Constitutive equations • Introduce nondimensional variables • Use appropriate reference lengths, times, temperature differences, etc. • Normalize variables to range from 0 to 1 • Make suitable (defensible) approximations • Drop negligible terms

29. OUTLINE OF PROCEDURE FOR SA • Express dimensionless QOI in terms of dimensionless variables & parammeters • Inspect result for implied parametric dependence • This constitutes “similitude relation” sought • Stronger than conventional dimensional analysis • Fewer extraneous criteria

30. SIMILITUDE ANALYSIS: REMARKS • Apparently dissimilar physico-chemical problems lead to identical dimensionless equations • Establishing useful analogies (e.g., between heat & mass transfer) • Dimensionless parameters will represent ratios between characteristic times in governing equations • Problem may simplify considerably when such ratios  0 or  ∞

31. SIMILITUDE ANALYSIS: REMARKS • Approximate similitudes possible for complicated problems • Some conditions may be “escapable” • Resulting simplified equations may have invariance properties • Allow further reduction in number of governing dimensionless parameters • Allow extraction of functional dependencies, simplify design of experiments