MAE 5310: COMBUSTION FUNDAMENTALS

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

LAMINAR DIFFUSION FLAME OVERVIEW • Subject of fundamental research • Applications to residential burners (cooking ranges, ovens) • Used to develop an understanding of how soot, NO2, CO are formed in diffusion burning • Mathematically interesting: transcendental equation with Bessel functions (0th and 1st 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 CO2 production in diffusion flame

LAMINAR DIFFUSION FLAME OVERVIEW • 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 f throughout reaction zone

LOOK AGAIN AT BUNSEN BURNER • 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 (f, T, P, fuel type) • How to characterize blowoff and flashback • Most practical devices (Diesel-engine combustion) has premixed and diffusion burning Secondary diffusion flame Results when CO and H products from rich inner flame encounter ambient air Fuel-rich pre-mixed inner flame

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 first 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 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 > xc), 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, Je

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 YF(r,x) similar to distribution of ux(r,x)/ue • 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 region containing fuel molecules grows with axial distance, x, and centerline fuel concentration decays Radial velocity decay Centerline velocity decay

NON-REACTING LAMINAR JETS: ASSUMPTIONS • Jet velocity profile is uniform at tube exit (r ≤ R) • Molecular weights of jet and reservoir fluid are equal (MWfuel=MWair), constant T and P, ideal gas, constant r • Species molecular transport is by binary diffusion governed by Fick’s law • Momentum and species diffusivities are constant and equal • Schmidt, Sc = n/D = 1 (Recall Le = a/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

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

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 ux(r,x), when normalized by local centerline velocity ux(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

CENTERLINE VELOCITY DECAY FOR LAMINAR JETS • Velocity decays inversely with axial distance and is directly proportional to jet Reynolds number, Rej • 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, YF (see next slides) Rej=297 Rej=29.7 Rej=2.97

SPREADING RATE, SPREADING ANGLE, JET HALF WIDTH • Other parameters are frequently used to characterize jets • Jet half-width, r1/2 • 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, a • 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

CONCENTRATION FIELD SOLUTION AND RESULTS Solution of concentration field is mathematically similar to governing equation for momentum conservation If n/D = 1 (Lewis number unity), function form of solution for YF is identical to what for ux/ue QF is volumetric flow rate from nozzle Written with Rej as controlling parameter Centerline expression Solutions can only be applied far from nozzle

EXAMPLES: NON-REACTING LAMINAR JETS • Part 1 • A jet of ethylene (C2H4) 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, methylene at 300 K is 1.023x10-9 N s/m2 • Answer comment: • Low-velocity jet is much wider • Fuel concentration of low-velocity jet decays to same value as high-velocity jet in 1/10th 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 YF,0 = YF,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 (n = m/r = 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

JET FLAME PHYSICAL DESCRIPTION • Much in common with isothermal (constant r) 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 f = 1 • Even though fuel and oxidizer are consumed at flame, f still has meaning since product composition relates to a unique value of f • Products formed at flame surface diffuse radially inward and outward • For an over-ventilated flame (ample oxidizer), flame length, Lf, is defined at axial local where f(r = 0, x = Lf) = 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, dYF/dr, which enhanced diffusion • Effects of these two phenomena on Lf tend to cancel (from circular and square nozzles) • Simple theories that neglect buoyancy do a reasonable job

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

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 tres, 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

FLAME LENGTH, Lf • Relationship between flame length and initial conditions • For circular nozzles, Lf depends on initial volumetric flow rate, QF = uepR2 • Does not depend independently on initial velocity, ue, or diameter, 2R, alone • Recall • Still ignoring effects of heat release by reaction, gives a rough estimate of Lf scaling and flame boundary • YF = YF,stoich • r = 0, so x = 0 • Lf is proportional to volumetric flow rate • Lf 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 Lf

PROBLEM FORMULATION: ASSUMPTIONS • 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 • 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 • 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) • Species molecular transport is by binary diffusion (Fick’s law) • Thermal energy and species diffusivities are equal, Le = 1 • Only radial diffusion of momentum, thermal energy, and species is considered • Axial diffusion is neglected • Radiation is neglected • Flame axis is oriented vertically upward

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

MATHEMATICALLY FORMIDABLE EQUATION SET • 5 conservation equations • Mass • Axial momentum • Energy • Fuel species • Oxidizer species • 5 unknown functions • vr(r,x) • ux(r,x) • T(r,x) • YF(r,x) • YOx(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

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

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

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*)

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 equationreplaces individual axial momentum, mixture fraction (species mass), and enthalpy (energy) equations!

STATE RELATIONSHIPS • Generic variable, z, for ux*, f, h* • Continuity still couples r* and ux* • f and h* are coupled with r* through state relationships • To solve jet flame problem, need to relate r* to f • Employ equation of state • Requires a knowledge of species mass fraction and temperature • Step 1: relate Yi and T as functions of mixture fraction, f • Step 2: arrive at relationship for r = r(f) Stoichiometric mixture fraction Inside flame (fstoic < f≤ 1) At flame (f = fstoic) Outside flame (0 ≤ f < fstoic)

SIMPLIFIED MODEL OF JET DIFFUSION FLAME

STATE RELATIONSHIPS • To determine mixture temperature as a function of f, requires calorific equation of state • To simplify the problem more • Assume constant and equal specific heats between fuel, oxidizer and products • 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 Tref=Tox,∞ Solve dimensionless enthalpy for T provides a general state relationship, T = T(f) Remember that YF is also a function of f

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 1st Law for fuel and oxidizer with initial temperatures of TF,e and Tox,∞ • Problem is now completely specified: with state relationships YF(f), Yox(f), YPr(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:

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 (ux = u, vr = 0) • Continuity requires that rux = 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 Lf J0 and J1 are 0th and 1st order Bessel functions, lm defined by solution to J1(lmR0)=0 S is molar stoichiometric ratio of oxidizer to fuel

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(r∞/rf) 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

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

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

FLOW RATE AND GEOMETRY Figure compares Lf for a circular port burner with slot burners having various exit aspect ratios h/b, all using CH4 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), Lf becomes shorter for same flow rate b h

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: CnHm Plot of flame lengths relative to CH4 Circular port geometry Flame length increases as H/C ratio of fuel decreases Example: Propane (C3H8: H/C=2.66) flame is about 2.5 times as long as methane (CH4: H/C=4) flame

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 40-50 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)

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 cdil is the diluent mole fraction in the fuel stream

MAE 5310: COMBUSTION FUNDAMENTALS