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

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, 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 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

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 throughoutreaction 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, Lf, is

defined at axial local where (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

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, Lf

• Relationship between flame length and initial conditions

• For circular nozzles, Lf depends on initial volumetric flow rate, QF = ueR2

– 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 = 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

stoichF

Ff

eF

FF

DY

QL

RuQ

Dx

QY

,

2

22

8

3

41

8

3

8

PROBLEM FORMULATION: ASSUMPTIONS1. 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 = 16. Only radial diffusion of momentum, thermal energy, and species is considered

– Axial diffusion is neglected7. Radiation is neglected8. Flame axis is oriented vertically upward

9

GOVERNING CONSERVATION PDES

RT

PMW

r

r

dTcDr

r

dTcvr

x

dTcur

YYY

rrY

Dr

rr

Yvr

rx

Yur

r

gr

ru

r

rr

vur

rx

uur

r

r

vr

rx

u

mix

P

PrPx

OxF

i

irix

x

rxxx

rx

0

1

0111

111

01

Pr

Axisymmetric continuity equation

Axial momentum conservationEquation applies throughout entire domain (insideand outside flame sheet) with no discontinuitiesat flame sheet

Species conservationFlame-sheet approximation means that chemicalproduction rates become zeroAll chemical phenomena are embedded inboundary conditionsIf i is fuel, equation applies inside boundaryIf i is oxidizer, equation applies outside boundary

Energy conservation: Shvab-Zeldovich formProduction term becomes zero everywhere exceptat flame boundaryApplies both inside and outside flame, but with adiscontinuity at flame locationHeat release from reaction enters problemformulation as a boundary conditionat 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. vr(r,x)

2. ux(r,x)

3. T(r,x)

4. YF(r,x)

5. 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

11

CONSERVED SCALAR APPROACH

0

0

rrh

Dr

r

hvr

x

hur

rrf

Dr

r

fvr

x

fur

rx

rx

Ox

F

Ox

hRrh

hRrh

hxh

xr

h

Rrf

Rrf

xf

xr

f

0,

0,

,

0,0

10,

10,

0,

0,0Mixture fractionSingle mixture fraction relation replaces two species equationsInvolves no discontinuities at flame

Symmetry

No fuel in oxidizer

Square exit profile

Absolute enthalpyWith 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, ue

e

oxeF

ox

e

rr

e

xx

hh

hhh

u

vv

u

uu

R

rr

R

xx

*

,,

,*

*

*

*

* Dimensionless axial distance

Dimensionless radial distance

Dimensionless axial velocity

Dimensionless radial velocity

Dimensionless mixture enthalpyAt nozzle exit, h = hF,e and, this h* = 1At ambient (r → ∞), h = hox,∞, and h* = 0

Dimensionless density ratio

Note: mixture fraction, f, is already dimensionless, with 0 ≤ f ≤ 1

13

NON-DINEMSIONAL EQUATIONS

00,10,10,1

10,10,10,1

0,0,0,0

0,,,

0,0

0

0

01

*****

*****

**

**

**

*

*

*****

**

*

**

*****

*****

*

**

****

****

*

**2*

**

*****

*****

*

***

*****

rhrfru

rhrfru

xr

hx

r

fx

r

u

xhxfxu

xv

r

hr

Ru

D

rhvr

rhur

x

r

fr

Ru

D

rfvr

rfur

x

ru

gR

r

ur

Rurvur

ruur

x

ux

vrrr

x

x

x

x

r

eerx

eerx

ee

x

eerxxx

xr

Continuity

Axial momentum

Mixture fraction

Enthalpy (energy)

Dimensionless boundaryconditions

Interesting features:Mixture fraction and enthalpy have same formDo not need to solve both since h*(r*,x*) = f(r*,x*)

14

FROM 3 EQUATIONS TO 1

0Re

1

1

0

0

0

**

****

****

*

*

**

*****

*****

*

**

****

****

*

*

**

*****

*****

*

rr

rvr

rur

x

DDSc

r

hr

Ru

D

rhvr

rhur

x

r

fr

Ru

D

rfvr

rfur

x

r

ur

Rurvur

ruur

x

rx

eerx

eerx

x

eerxxx

If we can neglect buoyancy, RHSof axial momentum equation = 0General 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 ux*, f, h*

– Continuity still couples * and ux

*

– 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 Yi and T as functions of mixture fraction, f

• Step 2: arrive at relationship for = (f)

stoic

stoicox

F

ox

F

stoic

ox

stoic

stoicF

stoic

f

fY

f

fY

Y

Y

Y

Y

f

fY

Y

f

ffY

f

Pr

Pr

Pr

1

0

1

0

0

1

1

0

1

1

1

Stoichiometric mixture fraction

Inside flame (fstoic < f ≤ 1)

At flame (f = fstoic)

Outside flame (0 ≤ f < fstoic)

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

,,,

,,

,,

,,

,*

oxoxeFp

cF

refeFpceF

refoxpox

oxeFPc

oxPcF

refPcFii

TTTfc

hYfT

TTchh

TTch

fTTch

TTchYh

TTchYhYh Calorific equation of state

Substitute calorific equation of state into definition of dimensionless enthalpy, h*, and note that h* = f

DefinitionsNote 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

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 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

,,,

,,,

,,, 11

oxoxeFp

c

oxoxeFp

cstoic

cpstoic

stoicox

p

c

stoic

stoicoxeF

TTTc

hfT

TTTc

hfT

hcf

fT

c

h

f

fTTfTInside 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 (ux = u, vr = 0)

• Continuity requires that ux = constant

• No need to solve axial momentum equation, inherently neglects buoyancy

1

02

11

2exp

1

1

01

01

20

2

12

00

1

,

RJ

R

SR

RL

D

RJ

RJ

r

Yr

rrD

x

Yu

DD

f

ffY

r

YDr

rrx

Yu

m

fm

m mm

m

Fref

Frefx

refref

stoic

stoicF

iix

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, m defined by solution to J1(mR0)=0S is molar stoichiometric ratio of oxidizer to fuel

20

ROPER/FAY SOLUTION (1977)

f

ref

F

stoicF

F

reff

f

refref

F

stoicFf

stoicF

Ff

IY

Q

DL

I

m

YL

Y

Q

DL

11

8

3

11

8

3

1

8

3

2,

,

,

Characteristic velocity varies with axial distanceas modified by buoyancyIf density is constant, solution is identical tonon-reacting jet, with same flame length

Variable density solutionBuoyancy is neglectedI(∞/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

f

ref

F

I

12

21

FLAME LENGTH CORRELATIONS

25.0

67.0

25.0

67.0

11045

116

11ln

1330

11ln4

Sinverf

TT

Q

L

T

T

SinverfD

TT

Q

L

S

TT

Q

L

T

T

SD

TT

Q

L

FF

f

f

FF

f

FF

f

f

FF

f

Circular Port:

S: molar stoichiometric oxidizer-fuel ratioD∞: mean diffusion coefficient evaluated for oxidizer at T∞

TF: fuel stream temperatureTf: 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 GEOMETRYFigure compares Lf for a circular portburner with slot burners having variousexit aspect ratios h/b, all using CH4

All burners have same port area,which implies that mean exit velocityis same for each configuration

Essentially a linear dependence of Lf on flow rate for circular port burnerGreater than linear dependence for slot burners

Flame Froude numbers (Fr = ratio of initial jet momentum to buoyant forces) is small: flames are dominated by buoyancyAs slot burners become more narrow (h/d increasing), Lf 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: CnHmPlot of flame lengths relative to CH4

Circular port geometry

Flame length increases as H/C ratio offuel decreases

Example: Propane (C3H8: H/C=2.66) flame is about 2.5 times as long as methane (CH4: H/C=4) flame

S

TT

Q

L FF

f 11ln

1330

2

4

O

mn

S

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 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)

purepri

pri

S

S1

1

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

211

4

Odil

mn

S