MAE 5310: COMBUSTION FUNDAMENTALS

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MAE 5310 COMBUSTION FUNDAMENTALS

• Introduction to Laminar Diffusion Flames
• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk

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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,
  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
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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 f throughout reaction zone
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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

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LOOK AGAIN AT BUNSEN BURNER
Secondary diffusion flame Results when CO and
H products from rich inner flame encounter
ambient air
Fuel-rich pre-mixed inner flame

• 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

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REACTING JET FLAME PHYSICAL DESCRIPTION
Flame surface locus of points where f 1
Figure from An Introduction to Combustion, by
Turns
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SOOT AND SMOKE FORMATION

• For HC flames, soot is frequently present, which
  typically is luminous in orange or yellow
• Soot is fomred 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

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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
• In lectures to come, we will develop better
  approximations for Lf

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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 8)
• Flame is represented as an infinitesimally thing
  sheet (called flame-sheet approximation)
• Species molecular transport is by binary
  diffusion (Ficks 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

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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
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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 the boundary conditions necessary to
  solve the fuel and oxidizer species and energy
  equation must be specified at the 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 BCs
• Recast equations to eliminate unknown location of
  flame sheet ? conserved scalars

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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
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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 Dimensi
onless mixture enthalpy At nozzle exit, h hF,e
and, this h 1 At ambient (r ? 8), h hox,8,
and h 0 Dimensionless density ratio Note
mixture fraction, f, is already dimensionless,
with 0 f 1
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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)
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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!
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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 lt f 1) At flame (f
fstoic) Outside flame (0 f lt fstoic)
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SIMPLIFIED MODEL OF JET DIFFUSION FLAME
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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
TrefTox,8 Solve dimensionless enthalpy for T
provides a general state relationship, T
T(f) Remember that YF is also a function of f
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STATE RELATIONSHIPS
Inside the flame At the flame Outside
the flame

• 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,8
• 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

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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
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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(r8/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
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FLAME LENGTH CORRELATIONS
Circular Port S molar stoichiometric
oxidizer-fuel ratio D8 mean diffusion
coefficient evaluated for oxidizer at T8 TF fuel
stream temperature Tf mean flame
temperature Square Port Inverf inverse
error function
Theoretical Experimental Theoretical
Experimental
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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
  Ropers experimental correlation

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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 Essential
ly 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
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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/C2.66) flame is about 2.5 times as long
as methane (CH4 H/C4) flame
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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)

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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