Lecture 20: Laminar Non-premixed Flames – Introduction, Non-reacting Jets, Simplified Description of Laminar Non-

premixed Flames Yi versus f Experimental Data

• Qualitative characteristics of laminar non-premixed or diffusion (of fuel and oxidizer) flames.

• Review of conserved scalar concept.

• Role of the momentum equation in deflagration regime: Non-reacting jet mixing solution.

• Simplified theoretical description of a laminar non-premixed (or diffusion) flame.

• Opposed jet nonpremixed flame

x

y

z

X

YZ

Infrared camera

d

• Stagnation point flow non- premixed flame

Air

Fuel

• Swirling flow flame with cross fuel injection

• Spherical stagnation pt. flame

• Vertical wall fire

• Horizontal wall fire

• Inclined wall fire

• Upward flame spread

• Downward flame spread

• Corner fire

• Beam, column fires

• Pool fires

• Forest fires

• Platform fire

• Combinationfires

Non premixed Flame Configurations

Laminar Jet Diffusion Flames (Non-premixed Jet Flames)

• Fuel (F) and oxidizer (O) are stored apart.• When combustion is desired, F and O must come

together at the molecular level. • How many molecules of each decides interim and

final products and their temperature.• Staged pre or post reaction mixing and rich and lean

reactions all lead to different products. • Specific strategies such as R-Quench-L, Lean Direct

Injection, Direct Injection-Spark Ignition have emerged.

• Learning the non-premixed flame regime is important.

• Learning about equipment for specific strategies is critical

Nonpremixed flames

• Concept of a conserved scalar is very useful for nonpremixed flames.

• A conserved scalar is a quantity defined such that there are no sink or source terms in the conservation equation for that quantity. (Sink and source terms result from reactions, heat transfer, and work transfer) Quantification of how non-(pre)mixed and when

• Total Energy is a conserved scalar in the absence of net heat loss to or work done on boundaries.

• Elemental Mass Fractions, Fraction of Mass that originated in the fuel stream(s) and Fraction of Mass that originated in the oxidizer stream(s) are all conserved scalars.

Mixture Fraction, Mixedness, Progress Variables,Reaction Fraction, and Reactedness

• In nonpremixed flames species mass fractions very continuously as mixing at the molecular level and chemical reaction occurs.

• Definition of mixture fraction f:

mass of material from fuel stream =

mass of mixture in differential volume elementf

Fuel Mass

+ Mass in products that came from fuel

Independent of the progress ofreaction. That means

CH4+O2 have the same mixturefraction as CO+H2O+H2

Both are f=16/(16+32)=(12+2+2)/(28+18+2)=16/48= 1/3f stoich= 16/16+64= 16/80=0.2

Review of Conserved Scalar, Definition of Mixture Fraction

• Consider the three-"species" reaction:

1 1kg fuel kg oxidizer kg products

• For this system the mixture fraction will be:

1

1F Prf Y Y

Review of Conserved Scalar, Definition of Mixture Fraction

• Assume all species diffuse at the same rate:

1

2

F FF

Pr PrPr

d Y d Ydm m

dx dx dx

d Y d Ydm m

dx dx dx

D

D

Conservation equation for the mixture fraction.

1 1

0 31

F Pr F Pr

PrF

d Y Y d Y Ydm

dx dx dx

mm

D

0d f d d f

m Ddx dx dx

• Divide the product species conservation equation by

and add to the fuel species conservation equation:

• Substituting for the mixture fraction we obtain:

1 1

0 31

F Pr F Pr

PrF

d Y Y d Y Ydm

dx dx dx

mm

D

0d f d d f

m Ddx dx dx

Review of Conserved Scalar, Definition of Mixture Fraction

• When kinetic energy is neglected (along with potential energy, thermal radiation, viscous dissipation, differential diffusion) we can write a similar equation for the absolute enthalpy:

• These conserved scalar equations, in cylindrical coordinates, will be very useful for our study of laminar non-premixed flames.

0,

0

ref

T

i f i pTi

h Y h c dT

dh d dhmdx dx dx

D

Review of Conserved Scalar, Definition of Mixture Fraction

• Consider first the case of a laminar jet of fuel issuing into air

with no chemical reaction. Assuming the air and fuel have the

same density, there is an analytical solution for the flow field

away from the potential core region of the jet.

Solution for a Non-reacting, Constant Density Laminar Jet

Solution for a Non-reacting, Constant Density Laminar Jet

- MW(jet fluid) = MW(air), ideal gases.

- Constant P, T, and r throughout the flow field.

- Steady state.

- Fick's law applies.

- Equal species and momentum diffusivities, Sc = n /D

= 1.

- Neglect axial diffusion of momentum and species.

- Solution applies downstream of the jet core region.

14/36

Assumptions: Non-reacting, Constant Density Laminar Jet

vv 10rx

r

x r r

Axial Momentum

v v v1v vx x xx r rx r r r r

Mass

Conservation Equations: Non-reacting, 𝜌=C Laminar Jet

Species

1v v

, 1

F F Fx r

F Ox F

Y Y Yr

x r r r r

In this case f Y Y Y

D

Along the jet centerline:

vv 0, 0, 0, 0x Fr

Yx x x

r r

Far from the jet:

v , , 0x Fx Y x

Boundary Conditions for a Non-reacting, 𝜌=C Laminar Jet

At the jet exit plane , r ≤ R:

v , 0 v ,0 1x e Fr Y r At the jet exit plane, r > R:

v , 0 0 ,0 0x Fr Y r

• The solution to this problem can be found in Schlichting

Boundary Layer Theory for the region of the flow beyond the

jet core where the flow is similar. The solution is given by: 22

314

22

3v 1

8 4

3 1v

161

4

ex

er

e

J

x

J

x

Solution for a Non-reacting Constant Density Laminar Jet

2 22

3v

16e e

e e e

J rJ R

x

Axial velocity distribution:

22v v0.375 1

v 4

vRe

v ( 0)0.375 Re

v

x e e

e

e ej

xj

e

R R

x

R

r R

x

Fuel mass fraction distribution (assuming Sc = n/D = 1):

22v0.375 Re 1

v 4x

F je

RY f

x

Law, Combustion Physics, 2006

Finite-rate Kinetics

Infinitely Fast Kinetics

Infinitesimal Flame Sheet Approximation for Nonpremixed Flames

• Assume:

1. Laminar, steady, axisymmetric flow2. Three "species": fuel, product, oxidizer3. Flame (reaction) sheet assumption, infinitely

fast chemical kinetics4. Equal species diffusivities5. Le = 16. No radiation transport7. Axial diffusion is neglected8. Vertical flame axis

Simplified Theoretical Description of Laminar Jet Diffusion Flame

Conservation of Mass

v v10x r r

x r r

Conservation of Axial Momentum

v1 1 1v v v v xx x x rr r r g

r x r r r r r

Conservation Equations: Cylindrical Coordinates, Thin Flame

1 1 1v v 0Fx F r F

Yr Y r Y r

r x r r r r r

D

Conservation of Fuel Mass Fraction (Inside the Flame Sheet)

1 1v v 0Oxx Ox r Ox

Yr Y r Y r

r r r r r

D

Conservation of O2 Mass Fraction (Outside the flame sheet)

Pr 1 F OxY Y Y Conservation of Product Mass Fraction (Everywhere)

Conservation of Species Mass Fraction

Pr

1 1 1v v 0

1 1 1v v 0

1

Fx F r F

Oxx Ox r Ox

F Ox

Yr Y r Y r

r x r r r r r

Yr Y r Y r

r x r r r r r

Y Y Y

inside flame sheet

outside flame sheet

everywhere

D

D

Simplified Theoretical Description of Laminar Jet Diffusion Flame

1 1 1v v 0x r

fr f r f r

r x r r r r r

D

Conserved Scalar Equations for Laminar Jet Flame

1 1 1v v 0x r

hr h r h r

r x r r r r r

D

• Boundary Conditions

vv 0, 0, 0, 0, 0xr

f hx x x x

r r r

v , , 0 ; ,x Oxx f x h x h At the jet exit plane

v , 0 v ; ,0 1; ,0 ; / 2x e Fr f r h r h o r d

v , 0 0; ,0 0; ,0 ; / 2x Oxr f r h r h d r

• A dimensionless enthalpy is defined:

,* * *

, ,

; 1, , 0Ox

F e Ox

h hh h h

h h

at jet exit far from jet

Non-dimensional Laminar Jet Diffusion Flame

• The non-dimensional conservation equations and boundary conditions for h* and f are identical, and therefore h* = f.

• The non-dimensional conservation equations and boundary conditions for h* and f are identical, and therefore h* = f.

• Inside the flame sheet

1, 1stoich F Prf f Y Y :

1 1; ; 0 ;

1 1 1stoich

stoich F Ox Prstoich stoich

f f ff Y Y Y

f f

11 1

1F Prkg F kg Ox kg Pr f Y Y

• Assume that the reaction kinetics are described by a single-step, three-species reaction:

Description of Global Fast Chemistry

, 1stoich Ox Prf f Y Y :• Outside the flame sheet

0; 1 ;F Ox Prstoich stoich

f fY Y Y

f f

• Assume constant heat capacities, and that for the jet fluid and oxidizer far from the jet, T = 298 K.

• Inside the flame sheet:

0 0, ,

0 0 0, , ,,*

0 0, , , ,

F F Pr Pr F f F Pr f Pr P ref

F f F Pr f Pr P ref f OxOx

F e Ox f F f Ox

h Y h Y h Y h Y h c T T

Y h Y h c T T hh hh f

h h h h

State Relationship for Temperature: Fuel Side

• Oxidizer = air Solve the h* = f equation for T : 0, 0f Oxh

0 0, ,

0,

0 0 0, , ,

0 0 0, , ,

1

1 1

1 1

F f F Pr f Pr P ref

f F

ref f F F f F Pr f PrP

stoichref f F f F f Pr

P stoich stoich

Y h Y h c T Tf

h

T T f h Y h Y hc

f f fT f h h h

c f f

0 0, ,

0 0 0, , ,,*

0 0, , , ,

Ox Ox Pr Pr Ox f Ox Pr f Pr P ref

Ox f Ox Pr f Pr P ref f OxOx

F e Ox f F f Ox

h Y h Y h Y h Y h c T T

Y h Y h c T T hh hh f

h h h h

State Relationship for Temperature: Oxidizer Side

0,

0,

0 0 0 0, , , ,

*

1 1

Pr f Pr P ref

f F

ref f F Pr f Pr ref f F f PrP P stoich

Y h c T Tf h

h

fT T f h Y h T f h h

c c f

Experimental Support for State Relationships for Major Species