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

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

3.  Flashback of a Flame in a Duct:  Critical condition for flashback is of “gradient” form, i.e., U/d ̴ S u /  q, or:  Multiplying both sides by d 2 /  u leads to correlation law of Peclet form:  Basis for accurate flashback predictions in similar systems PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 3

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

5.  “Blow-Off” from Premixed Gas Flame “Holders”:  Oblique premixed gas flames can be stabilized in ducts even at feed-flow velocities >> S u  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  U bo PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 5

6. Schematic of an oblique flame “ anchored” to a gutter-type (2 dimensional) flame-holder (stabilizer) in a uniform stream of premixed combustible gas (S u <U<U bo ) 6

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

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

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

10. “Blow-Off” from Premixed Gas Flame “Holders”: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 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)) 10

11.  “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: PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 11

12.  “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 PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 12

13.  “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 PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 13

14.  Laminar Diffusion Flame Height:  Buoyancy and fuel-jet momentum contribute to height, L f, 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 PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 14

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

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

17.  Fuel Droplet Combustion at High Pressures:  Burning time, t comb, 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 PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 17

18.  Fuel Droplet Combustion at High Pressures:  Each fuel also has a thermodynamic critical pressure, p c, 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 PARTIAL MODELING OF CHEMICALLY REACTING SYSTEMS 18

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

20.  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 ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS 20

21.  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 ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS 21

22.  e.g., momentum-flux ratio (Re)  ratio of times required for momentum diffusion & convection through common area, L 2  e.g., Pr  ratio of times governing diffusive decay of nonuniformities of energy (L 2 /  ) and momentum (L 2 / ) ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS 22

23.  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 ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS 23

24.  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 ALTERNATIVE INTERPRETATIONS OF DIMENSIONLES GROUPS: “EIGEN-RATIOS 24

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

26.  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 BENEFITS OF SIMILITUDE ANALYSIS 26

27.  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 BENEFITS OF SIMILITUDE ANALYSIS 27

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 28

29.  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 OUTLINE OF PROCEDURE FOR SA 29

30.  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  ∞ SIMILITUDE ANALYSIS: REMARKS 30

31.  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 SIMILITUDE ANALYSIS: REMARKS 31