MAE 5310: COMBUSTION FUNDAMENTALS

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MAE 5310 COMBUSTION FUNDAMENTALS

• Laminar Diffusion Flame Solution and Applications
• November 26, 2007
• Mechanical and Aerospace Engineering Department
• Florida Institute of Technology
• D. R. Kirk

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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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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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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, QF
• Lf is inversely proportional to stoichiometric
  fuel mass fraction
• This implies that fuels that require less air for
  complete combustion produce shorter flames
• GOAL 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)
• Chemical production rates become zero in species
  conservation and S-Z equation
• Species molecular transport is by binary
  diffusion (Ficks law)
• Thermal and species diffusivities are equal (Le
  a/D 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 Note that White
B-3 is incompressible Axial momentum
conservation Equation applies throughout entire
domain (inside and outside flame sheet) with no
discontinuities at flame sheet, compare to White
B-8 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
Equation of state to relate density and
temperature
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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 boundary conditions necessary to solve
  fuel and oxidizer species and energy equation
  must be specified at flame
• Location of flame (f1 contour) 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
• Only require boundary conditions

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CONSERVED SCALAR APPROACH (CHAPTER 7)
Mixture fraction Single mixture fraction, f,
relation replaces two species equations Involves
no discontinuities at flame Note that once f
is known, location of flame is known since
ffstoic at this location
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 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