Theory of the Linear Pyrolysis of Condensed Materials - Fast Reactions in Energetic Materials - page 10

⎛

1 + 1 + 4Bi / M 2
2

⎞
⎠ ( T S −

2 + M 2

T ∞ ) 2

q
λ

Bi
δ

⎝

T ∞ ) 2

2 −

− ( T 1 −

2

δ

1 + 1 + 4Bi / M 2
2

T ∞ ) 2 ⎛

⎞ 2

= M 2
δ

⎝ ⎠

2 ( T 1 −

1 + 1 + 4Bi / M 2
2

⎡ ⎤

2M 2
δ

Q
c

Q
2 c

⎣ ( T S − ⎦ .

∓ T ∞ ) ∓ (1.27)

2

Substitution of Eq. (1.27) into Eq. (1.24) gives

1 + 1 + 4Bi / M 2
2

T ∞ ) 2 ⎛

⎞

2 x = x 1

2

dT
dx

M 2
δ

2M 2 Q
δ

⎝ ⎠

− ∓

=

2 ( T 1

2 c

1 + 1 + 4Bi / M 2
2

⎡ ⎤

(1.28)

Q
2 c

⎣ ( T S − ⎦

× T ∞ ) ∓

exp

.

2 Qk 0 R T S
acE

E
R T S

± −

Equation (1.7) with boundary conditions Eq. (1.4) and T | x = x 1 = T 1 is the energy

equation for zone II. The solution of this equation is

1 + 1 + 4Bi / M 2
2

⎛ ⎞

x − x 1

⎝ − ⎠ ,

T = T ∞ +( T 1 − T ∞ ) exp M (1.29)

δ

from which

1 + 1 + 4Bi / M 2
2

x = x 1

dT
dx

M
δ

= − ( T 1 − T ∞ ) .

If the temperature gradients on the border between the two zones are equal,

1 + 1 + 4Bi / M 2
2

1 + 1 + 4Bi / M 2
2

T ∞ ) 2 ⎛

⎞

T ∞ ) 2 ⎛

⎞

2 2

M 2
δ

= M 2
δ

⎝ ⎠ ⎝ ⎠

2 ( T 1 − 2 ( T 1 −

1 + 1 + 4Bi / M 2
2

⎡ ⎤

exp

.

2M 2
δ

2 Qk 0 R T S
acE

Q
c

Q
2 c

E
R T S

⎣ ( T S − ⎦ ±

∓ T ∞ ) ∓ −

2

(1.30)