Modeling of Combustion Systems A Practical Approach 7 pot

Then we may write a law of mass action as G.2 If the sign is negative, then the reaction consumes the species with time rather than produces them.. We may also define the reaction rate f

Appendix G

Kinetics Primer

Consider a general reaction:

or equivalently,

(G.1)

comprising r i moles of reactants R i and p k moles of products P k Then we may

write a law of mass action as

(G.2)

If the sign is negative, then the reaction consumes the species with time rather than produces them We may also define the reaction rate for species

k as rr k:

(G.3)

where is the molar volume [L3/N] and the reaction rate has units of [N/L3] For a constant volume (density) reaction, the equation reduces to

(G.4)

where the brackets indicate the molar concentration of the enclosed species

r1R1+r2R2+ ↔$ p1 1P +p2 2P +$

j

m

k k k

n

R

1 1

P

−1 = −1 = 1 = 1

1 2 1 2

r

dN

dN

dN

dN dt

R1 R2$ P1 P2$

rr

r V

dN

dN dt

k

= −1 1ˆ Rk = 1 1ˆ Pk

ˆ

V

rr r

d

d dt

k k k k k

= −1 ⎡⎣ ⎤⎦ =R 1 ⎡⎣ ⎤⎦P

614 Modeling of Combustion Systems: A Practical Approach

Usually, one defines a reaction coordinate known as the conversion (x k),

having the property that for species k the reaction starts at x k = 0 and ends

at x k = 1 The general definition is

(G.5)

where N k,0 is the starting number of moles of species k, and N k is the

con-centration at some particular conversion of interest Thus, N k,0 is a constant

and N k is a variable We may also write

(G.6)

For constant density, we have

,

where [k] is the concentration of species k, and [k0] is the starting concentra-tion We may write the conversion for any particular species and relate it to any other species according to

(G.7)

Or in terms of a single conversion (say, xR1), we may write

(G.8)

For constant density, we may write

(G.9)

(G.10)

N

k

k k k

= , − , 0 0

N k= −(1 x N k) k,0

k

k=⎡⎣ ⎤⎦ − ⎡⎣ ⎤⎦

⎡⎣ ⎤⎦

0 0

⎡⎣ ⎤⎦ = −(1 )⎡⎣ ⎤⎦0

N

N

N p

R

R R R

P

1 0

1

1

2 0 2 2

1 0 1 , , ,

⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ = =$

⎛⎛

⎝⎜

⎞

⎠⎟ =

⎛

⎝⎜

⎞

⎠⎟ =

P P P 1

2 0 2 2 , $

N

r

N N

r

N

R

R R R

P R P 1

2 0

1 0 1 2 2

1 0

1 0 1 1 1

= , = = =

,

, ,

R P

2 0

1 0 1 2 2 , ,

N

r

R R

1 0

1

1

2 0 2 2 , ,

⎡⎣ ⎤⎦

⎛

⎝

⎜ ⎞

⎠

⎟ =⎛⎡⎣ ⎤⎦

⎝

⎜ ⎞

⎠

⎟ = =

1 1

2 0 2 2 , ,

⎡⎣ ⎤⎦

⎛

⎝

⎜ ⎞

⎠

⎟ =⎛⎡⎣ ⎤⎦

⎝

⎜ ⎞

⎠

⎟ =

P

$

R R

R R

P R

1

2 0

1 0 1 2 2

1 0

1 0

= ⎡⎣ ⎤⎦

⎡⎣ ⎤⎦ = = ⎡⎡⎣ ⎤⎦

, ,

, ,

$

⎣⎣ ⎤⎦ =⎡⎣⎡⎣ ⎤⎦⎤⎦ =

r

r

1 1 1

2 0

1 0 1 2 2

P P

P R

, ,

$

Kinetics Primer 615

We may also substitute mole fractions for concentrations using

(G.11)

For combustion in furnaces, the ideal gas law applies:

(G.12)

where are the total moles of the reaction This gives

(G.13)

We may also write as a function of conversion:

(G.14)

Typically, we use Equation G.8 or Equation G.10 to recast Equation G.14

in terms of a single conversion

RT

k

⎡⎣ ⎤⎦ =

N V

P RT

k

ˆ

N k

∑

ˆ

= ∑

N k

∑

∑ ∑= (1− ) ,0