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

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 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 LAMINAR DIFFUSION FLAME OVERVIEW (LECTURE 1) Reactants are initially separated, and reaction occurs only at 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 JET FLAME PHYSICAL DESCRIPTION Much in common with isothermal (constant  ) jets As fuel flows along flame axis, it diffuses radially outward, while oxidizer diffuses radially inward Flame surface is defined to exist where fuel and oxidizer meet in stoichiometric proportions –Flame surface ≡ locus of points where  –Even though fuel and oxidizer are consumed at flame,  still has meaning since product composition relates to a unique value of  Products formed at flame surface diffuse radially inward and outward For an over-ventilated flame (ample oxidizer), flame length, L f, is defined at axial local where  (r = 0, x = L f ) = 1 Region where chemical reactions occur is very narrow and high temperature reaction region is annular until flame tip is reached In upper regions, buoyant forces become important: –Buoyant forces accelerate flow, causing a narrowing of flame –Consequent narrowing of flame increases fuel concentration gradients, dY F /dr, which enhanced diffusion –Effects of these two phenomena on L f tend to cancel (from circular and square nozzles) –Simple theories that neglect buoyancy do a reasonable job

5 REACTING JET FLAME PHYSICAL DESCRIPTION Figure from “An Introduction to Combustion”, by Turns Flame surface = locus of points where  =1

6 SOOT AND SMOKE FORMATION For HC flames, soot is frequently present, which typically is luminous in orange or yellow Soot is formed on fuel side of reaction zone and is consumed when it flows into an oxidizing region (flame tip) Depending on fuel and  res, not all soot that is formed may be oxidized Soot ‘wings’ may appear, which is soot breaking through flame Soot that breaks through called smoke

7 FLAME LENGTH, L f Relationship between flame length and initial conditions For circular nozzles, L f depends on initial volumetric flow rate, Q F = u e  R 2 –Does not depend independently on initial velocity, u e, or diameter, 2R, alone Recall Still ignoring effects of heat release by reaction, gives a rough estimate of L f scaling and flame boundary –Y F = Y F,stoich –r = 0, so  = 0 L f is proportional to volumetric flow rate L f is inversely proportional to stoichiometric fuel mass fraction –This implies that fuels that require less air for complete combustion produce shorter flames Goal is to develop better approximations for L f

8 PROBLEM FORMULATION: ASSUMPTIONS 1.Flow conditions –Laminar –Steady –Axisymmetric –Produced by a jet of fuel emerging from a circular nozzle of radius R –Burns in a quiescent infinite atmosphere 2.Only three species are considered: (1) fuel, (2) oxidizer, and (3) products –Inside flame zone, only fuel and products exist –Outside flame zone, only oxidizer and products exist 3.Fuel and oxidizer react in stoichiometric proportions at flame –Chemical kinetics are assumed to be infinitely fast (Da = ∞) –Flame is represented as an infinitesimally thing sheet (called flame-sheet approximation) 4.Species molecular transport is by binary diffusion (Fick’s law) 5.Thermal energy and species diffusivities are equal, Le = 1 6.Only radial diffusion of momentum, thermal energy, and species is considered –Axial diffusion is neglected 7.Radiation is neglected 8.Flame axis is oriented vertically upward

9 GOVERNING CONSERVATION PDES Axisymmetric continuity equation Axial momentum conservation Equation applies throughout entire domain (inside and outside flame sheet) with no discontinuities at flame sheet Species conservation Flame-sheet approximation means that chemical production rates become zero All chemical phenomena are embedded in boundary conditions If i is fuel, equation applies inside boundary If i is oxidizer, equation applies outside boundary Energy conservation: Shvab-Zeldovich form Production term becomes zero everywhere except at flame boundary Applies both inside and outside flame, but with a discontinuity at flame location Heat release from reaction enters problem formulation as a boundary condition at flame surface

10 MATHEMATICALLY FORMIDABLE EQUATION SET 5 conservation equations 1.Mass 2.Axial momentum 3.Energy 4.Fuel species 5.Oxidizer species 5 unknown functions 1.v r (r,x) 2.u x (r,x) 3.T(r,x) 4.Y F (r,x) 5.Y Ox (r,x) Problem is to find five functions that simultaneously satisfy all five equations, subject to appropriate boundary conditions This is much more complicated that it already appears! –Some of boundary conditions necessary to solve fuel and oxidizer species and energy equation must be specified at flame –Location of flame is not known until complete problem is solved –Not only is solving 5 coupled PDEs formidable, but would require iteration to establish flame front location for application of BC’s Recast equations to eliminate unknown location of flame sheet → conserved scalars

11 CONSERVED SCALAR APPROACH Mixture fraction Single mixture fraction relation replaces two species equations Involves no discontinuities at flame Symmetry No fuel in oxidizer Square exit profile Absolute enthalpy With given assumptions replace S-Z energy equation, which involves T(r,x), with conserved scalar form involving h(r,x) No discontinuities in h occur at flame Mass and momentum equations remain unchanged and use BC for velocity as non-reacting jet

12 NON-DINEMSIONAL EQUATIONS Gain insight by non-dimensionalizing governing PDEs –Identification of important dimensionless parameters Characteristic scales: –Length scale, R –Nozzle exit velocity, u e Dimensionless axial distance Dimensionless radial distance Dimensionless axial velocity Dimensionless radial velocity Dimensionless mixture enthalpy At nozzle exit, h = h F,e and, this h * = 1 At ambient (r → ∞), h = h ox,∞, and h * = 0 Dimensionless density ratio Note: mixture fraction, f, is already dimensionless, with 0 ≤ f ≤ 1

13 NON-DINEMSIONAL EQUATIONS Continuity Axial momentum Mixture fraction Enthalpy (energy) Dimensionless boundary conditions Interesting features: Mixture fraction and enthalpy have same form Do not need to solve both since h * (r *,x * ) = f(r *,x * )

14 FROM 3 EQUATIONS TO 1 If we can neglect buoyancy, RHS of axial momentum equation = 0 General form is now same as mixture fraction and dimensionless enthalpy equation Can simplify even further if assume mass and momentum diffusivity equal (Sc = 1) Single conservation equation replaces individual axial momentum, mixture fraction (species mass), and enthalpy (energy) equations!

15 STATE RELATIONSHIPS Generic variable, , for u x *, f, h * –Continuity still couples  * and u x * –f and h * are coupled with  * through state relationships To solve jet flame problem, need to relate  * to f –Employ equation of state –Requires a knowledge of species mass fraction and temperature Step 1: relate Y i and T as functions of mixture fraction, f Step 2: arrive at relationship for  =  (f) Stoichiometric mixture fraction Inside flame (f stoic < f ≤ 1) At flame (f = f stoic ) Outside flame (0 ≤ f < f stoic )

16 SIMPLIFIED MODEL OF JET DIFFUSION FLAME

17 STATE RELATIONSHIPS To determine mixture temperature as a function of f, requires calorific equation of state To simplify the problem more 1.Assume constant and equal specific heats between fuel, oxidizer and products 2.Enthalpies of formation of oxidizer and products are zero –Result is that enthalpy of formation of fuel is equal to its heat of combustion Calorific equation of state Substitute calorific equation of state into definition of dimensionless enthalpy, h *, and note that h * = f Definitions Note that Turns takes T ref =T ox,∞ Solve dimensionless enthalpy for T provides a general state relationship, T = T(f) Remember that Y F is also a function of f

18 STATE RELATIONSHIPS Comments –Temperature depends linearly on f in regions inside and outside flame, with maximum at flame –Flame temperature ‘At the flame’ is identical to constant P, adiabatic flame temperature calculated from 1 st Law for fuel and oxidizer with initial temperatures of T F,e and T ox,∞ –Problem is now completely specified: with state relationships Y F (f), Y ox (f), Y Pr (f), and T(f), mixture density can be determined solely as function of mixture fraction using ideal gas equation Inside the flame: At the flame: Outside the flame:

19 BURKE-SCHUMANN SOLUTION (1928) Earliest approximate solution to laminar jet flame problem –Circular and 2D fuel jets –Flame sheet approximation –Assumed that a single velocity characterized flow (u x = u, v r = 0) Continuity requires that  u x = constant No need to solve axial momentum equation, inherently neglects buoyancy Variable density conservation equation Mixture fraction definition Use of reference density and diffusivity, assumed to be constant Final differential equation Transcendental equation for L f J 0 and J 1 are 0 th and 1 st order Bessel functions, m defined by solution to J 1 ( m R 0 )=0 S is molar stoichiometric ratio of oxidizer to fuel

20 ROPER/FAY SOLUTION (1977) Characteristic velocity varies with axial distance as modified by buoyancy If density is constant, solution is identical to non-reacting jet, with same flame length Variable density solution Buoyancy is neglected I(  ∞ /  f ) is a function obtained by numerical integration as part of solution Recast equation with volumetric flow rate Laminar flame lengths predicted by variable density theory are longer than those predicted by constant density theory by a factor

21 FLAME LENGTH CORRELATIONS Circular Port: S: molar stoichiometric oxidizer-fuel ratio D ∞ : mean diffusion coefficient evaluated for oxidizer at T ∞ T F : fuel stream temperature T f : mean flame temperature Square Port: Inverf: inverse error function Theoretical Experimental Theoretical Experimental

22 EXAMPLE 9.3 It is desired to operate a square-port diffusion flame burner with a 50 mm high flame. –Determine the volumetric flow rate required if the fuel is propane. –Determine the heat release of the flame. –What flow rate is required if methane is substituted for propane? To solve this problem in class, make use of Roper’s experimental correlation

23 FLOW RATE AND GEOMETRY Figure compares L f for a circular port burner with slot burners having various exit aspect ratios h/b, all using CH 4 All burners have same port area, which implies that mean exit velocity is same for each configuration Essentially a linear dependence of Lf on flow rate for circular port burner Greater than linear dependence for slot burners Flame Froude numbers (Fr = ratio of initial jet momentum to buoyant forces) is small: flames are dominated by buoyancy As slot burners become more narrow (h/d increasing), L f becomes shorter for same flow rate h b

24 FACTORS AFFECTING STOICHIOMETRY Recall that stoichiometric ratio, S, used in correlations is defined in terms of nozzle fluid and surrounding reservoir –S = (moles ambient fluid / moles nozzle fluid) stoic –S depends on chemical composition of nozzle and surrounding fluid –For example, S would be different for pure fuel burning in air as compared with a nitrogen diluted fuel burning in air Influence of fuel types, general HC: C n H m Plot of flame lengths relative to CH 4 Circular port geometry Flame length increases as H/C ratio of fuel decreases Example: Propane (C 3 H 8 : H/C=2.66) flame is about 2.5 times as long as methane (CH 4 : H/C=4) flame

25 FACTORS AFFECTING STOICHIOMETRY Primary aeration –Many gas burning applications premix some air with fuel gas before it burns as a laminar jet diffusion flame Called primary aeration, which is typically on order of percent of stoichiometric air requirement –This tends to make flames shorter and prevents soot from forming –Usually such flames are distinguished by blue color –What is maximum amount of air that can be added? If too much air is added: –rich flammability limit may be exceeded –implies that mixture will support a premixed flame Depending on flow and burner geometry, flame may propagate upstream (flashback) If flow velocity is high enough to prevent flashback, an inner premixed flame will form inside the diffusion flame envelope (similar to Bunsen burner)

26 FACTORS AFFECTING STOICHIOMETRY Oxygen content of oxidizer –Amount of oxygen has strong influence on flame length –Small reductions from nominal 21% value for air, result in greatly lengthened flames Fuel dilution with inert gas –Diluting fuel with an inert gas also has effect of reducing flame length via its influence on the stoichiometric ratio –For HC fuels –Where  dil is the diluent mole fraction in the fuel stream