ME 525: Combustion Lecture 27: Carbon Particle Combustion Carbon surface reactions. One-film model. Two-film model

ME 525: CombustionLecture 27: Carbon Particle Combustion

ME 525: CombustionLecture 27: Carbon Particle Combustion

• Carbon surface reactions.

• One-film model.

• Two-film model.

Combusting Carbon Particle: Surface Reactions

Combusting Carbon Particle: Surface Reactions

1( ) 2( ) 2( )

2( ) 2( ) ( )

3( ) 2( ) ( )

4( ) 2 ( ) ( ) 2( )

2 2

2

s g g

s g g

s g g

s g g g

kC O CO

kC O CO

kC CO CO

kC H O CO H

Combusting Carbon Particle: One-Film ModelCombusting Carbon Particle: One-Film Model

• Temperature and Species Profiles, One-Film Model

Combusting Carbon Particle: One-Film Model Combusting Carbon Particle: One-Film Model

Assume:

1. Particle burns in quiescent, infinite medium containing O2

and inert N2 initially.

2. Burning process is quasi-steady.

3. Reaction 1 is dominant at the surface (not a good

assumption).

4. Particle has uniform temperature, radiates as a gray body.

5. No diffusion of gas-phase species into particle.

6. Le = 1 in the gas phase

7. kg, cPg, Dg, rg are constants evaluated at some mean

temperature.


rs

r

( ) 2( )

2( )

1

1

s I g

I g

kg C kg O

kg CO

2,O im 2COm r

2,CO sm

netm r 2Om r

Cm


• Following the same procedures as for the evaporating droplet :

2, 2, 2,

2,2,

,

2, 2,,

2,

1 /4 ln 4 ln 1

1 /

4 ln 1

O I O O sC s s

I O sO s I

C s O m

O O sO m

I O s

Y Y Ym r r

YY

m r B

Y YB

Y

D D

D

2,

2

2

1

32 /12 2.664

4

O i I C

CO I C

I

i i

m m

m m

m r m


rs

Ts

YO2,∞

T∞

YO2,s

YCO2,s

• From a consideration of the chemical kinetics at the carbon particle surface we obtain:

21 2

2, 2,2

2

4C s Cs

CO s O smixs

u s O u s

m r k O MW

x P Y PMWO

R T MW R T

21 2,

2

2,

4 C mixC s O s

O u s

C kin O s

MW MW Pm r k Y

MW R T

m K Y

Combusting Carbon Particle: One-Film Model, Diffusion and Kinetic Resistances


• Burning of the carbon particle is controlled by matching of the reaction rate and diffusion rate of oxygen to the surface.

• To analyze these effects we develop expressions for the kinetic and diffusive resistances

,

2, 2,,

2,

4 ln 1C s O m

O O sO m

I O s

m r B

Y YB

Y

D

• The kinetic resistance is : 1

2, 21 2

2

14O s C mix

C kin skin O u s s

Y MW MW Pm R r k

R MW R T r



• For the diffusive resistance, note that the typical value of the transfer number BO,m is typically <<1 [ln (1+x) ≈ x for x<<1]:

2, 2,,

2,

2, 2, 2, 2,

2, 4

O O sO m

I O s

O O s O O sC

diffI O s s

Y YB

Y

Y Y Y Ym

RY r

D



• Combining these two relations:

2, 2, 2, 2,2,

2,2,

2,

O s O O s kin OC O s

kin diff kin diff

kin OO

kin diff OC

diff kin diff

Y Y Y R Ym Y

R R R R

R YY

R R Ym

R R R

Combusting Carbon Particle: Two-Film ModelCombusting Carbon Particle: Two-Film Model

• Temperature and Species Profiles, Two-Film Model

Combusting Carbon Particle: Two-Film Model Combusting Carbon Particle: Two-Film Model

Assume:

1. Particle burns in quiescent, infinite medium containing O2

and inert N2 initially.

2. Burning process is quasi-steady.

3. Reaction 3 is dominant at the surface.

4. Particle is surrounded by a flame sheet.

5. At the flame sheet, CO reacts in stoichiometric proportion

with O2 to produce CO2.

6. Particle has uniform temperature, radiates as gray body.

7. No diffusion of gas-phase species into particle.

8. Le = 1 in the gas phase

9. kg, cPg, Dg, rg are constants evaluated

at some mean temperature


rs

rf

2,CO im2,CO om

COm 2,O imCm


( ) 2( ) ( )1 1s s g s gkg C kg CO kg CO

2,

2,

2, 1

44 /12 3.67

CO i s C

C CO CO i

CO C CO i s C

s

m m

m m m

m m m m

• At the particle surface:


( ) 2( ) 2( )1 1g f g f gkg CO kg O kg CO

• At the flame sheet:

2, 2 2,

2

2, 2,

1

1 1

16 / 28 0.57

C CO o O CO CO i

O f CO f s C

CO o C O i f s C

f

m m m m m

m m m

m m m m


In this problem analysis:

Knowns:

Unknowns:

rf

rs

TsTf

YO2,∞

T∞

YCO2,s

YI,∞

YCO2,f

YI,f

2, ,, , ,s O Ir T Y Y

2, 2, ,, , , , , ,C CO s CO f I f f f sm Y Y Y T r T


• Regarding the surface temperature Ts and flame temperature

Tf as known quantities for the moment, we obtain the following

four relations by considering the species balances:

, 2,

2,

2,

1

1 /4 ln

1 /

I f CO f

CO f ss fC

f s CO s s

Y Y

Yr rm

r r Y

D

2,

, ,

4 ln 1 /

exp / 4

C f CO f s

I f I C f

m r Y

Y Y m r

D

D


rf

rs

Ts

Tf

YO2,∞

T∞

YCO2,s

YI,∞

YCO2,f

YI,f

Rearranging these expressions gives us a relation between the burning rate and

Cm 2,CO sY

2,

2, 2,2,

2,

4 ln 1

2 1 /

1 1 /

C s CO m

O CO s s sCO m

s CO s s s

m r B

Y YB

Y

D


• Another relation between and is provided by the

chemical kinetic equation for the surface reaction: Cm 2,CO sY

23 2

2, 2,2

2

4C s Cs

CO s CO smixs

u s CO u s

m r k CO MW

x P Y PMWCO

R T MW R T

2,23

2

2,

4 CO sC mixC s

CO u s

C kin CO s

Y PMW MWm r k

MW R T

m K Y


• The rate coefficient k3 is given by

83 34.016 10 exp 29,790 / ,s s

mk T k T K

s

• For reaction 3 the enthalpy of combustion is given by

0 0 0, ( ) , 2( ) , ( )

4

( )

1

1800 ,

396,332 / 117,3800 3.66 4.66

44 / 30

1.47 10

c f C s s s f CO g s s f CO g s

s

c

cs

h h T h T h T

at T K

kJ kmolh

kg kmol

kJh

kg C


rf

rs

Ts

Tf

YO2,∞

T∞

YCO2,s

YI,∞

YCO2,f

YI,f

The enthalpy of combustion

is negative - the reaction is

endothermic! The energy

needed to drive the reaction

comes from the reaction of

CO and O2 at the flame

sheet.


• Often assumed that the particle burning process is in the

diffusion-controlled regime:

• In the diffusion-controlled regime, surface chemical kinetics

are fast compared to diffusion times and the CO2 reacts as

soon as it gets to the particle surface.

• For burning carbon particles, the process will be diffusion

controlled at high pressures, when the surface temperature is

high, and/or when the particle is large.

2, 0CO sY


• Up to now we have assumed that the surface and flame

temperatures are known. Two more equations to determine

these temperatures are provided by energy balances at the

flame sheet and at the particle surface. Solution for the whole

problem is found by iteration in the quantities

2,, , ,C CO s s fm Y T and T


rs

C Cm h

CO COm hs fQ

radQs iQ

• At the particle surface:

2

2 4 4

4

4

C c s f rad s g

s

s s s

dTm h Q Q r k

dr r r

r T T

2 2CO COm h


• The temperature gradient is found from the same temp profile expression that we obtained for the burning liquid droplet:

exp exp exp

( )

exp exp

T C T C T Cs f f s

s f

T C T C

s f

Z m Z m Z mT T T T

r r rT r

Z m Z m

r r

2

exp /

exp / exp /

4

s

s f T C sT C

r r s T C s T C f

pgT

g

T T Z m rZ mdT

dr r Z m r Z m r

cZ

k


2 4 4

exp /

exp / exp /

4

pg C s f T C s

C c

T C s T C f

s s s

c m T T Z m rm h

Z m r Z m r

r T T

• The energy equation at the surface becomes:

Combusting Carbon Particle: Two-Film Model, Diffusion and Kinetic Resistances


• We can develop expressions for the diffusive and kinetic resistances for the two-film model in analogy with the one-film model. Recall the two expressions that we developed previously for the burning rate:

2,

2, 2,2,

2,

4 ln 1

2 1 /

1 1 /

C s CO m

O CO s s sCO m

s CO s s s

m r B

Y YB

Y

D

2,23 2,

2

4 CO sC mixC s kin CO s

CO u s

Y PMW MWm r k K Y

MW R T



• The kinetic resistance is found immediately: 1

2, 23 2

2

14CO s C mix

C kin skin CO u s s

Y MW MW Pm R r k

R MW R T r

• For the diffusive resistance, we note that the typical value of the transfer number BCO2,m is typically 0.1-0.2:

2, 2,

2, 2,

2, 2, 2,

44 /12 3.67

2 0.73

2.67 0.73

s

CO s O

CO m O

O CO s O

Y YB Y

Y Y Y



• The diffusive resistance is thus given by:

2,

2, 2, 2,

2, 2,

2.67 0.73 1

4 2 0.73

OC

diff

O CO s O

diffss CO s O

diffs

kin

Ym

R

Y Y YR

rr Y Y

Rr

R

D

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ME 525: Combustion Lecture 27: Carbon Particle Combustion Carbon surface reactions. One-film model. Two-film model