Table 1 Thermal degradation modes proposed for gas-solid reactions [32,33,34]
Degradation mode | Code | Differential form: f(α) | Integral form: g(α) |
|---|---|---|---|
Diffusion | |||
One-way transport | D1 | 1 / (2α) | α2 |
two-way transport, Valensi-Barrer | D2 | -1 / Ln(1-α) | α + (1-α)Ln(1-α) |
three-way transport, Jander | D3 | 1,5(1-α)2/3 / [1-(1-α)1/3] | [1-(1-α)1/3]2 |
Ginstling-Brounshtein | D4 | 1,5 / [(1-α)-1/3–1] | 1-2α/3-(1-α)2/3 |
Zhuravlev | D5 | 1,5(1-α)2/3 / [1 / (1-α)1/3–1] | [1 / (1-α)1/3–1]2 |
Anti-Jander | D6 | 1,5(1 + α)2/3 / [(1 + α)1/3–1] | [(1 + α)1/3–1]2 |
Kroger-Ziegler | D7 | [1,5(1-α)2/3 / [1-(1-α)1/3]] / t | [1-(1-α)1/3)]2 – log(t) |
Two dimensions, Jander | D8 | (1-α)1/2 / [1-(1-α)1/2] | [1-(1-α)1/2)]2 |
Two dimensions, Anti-Jander | D9 | (1 + α)1/2 / [(1 + α)1/2–1] | [(1 + α)1/2–1]2 |
Interfacial transfer | D10 | 3(1-α)4/3 | [1/ (1-α)1/3–1] |
Transfer and diffusion | D11 | 3 / [(1-α)-4/3 - (1-α)−1] | 1/(1-α)1/3–1 + 1/3Ln(1-α) |
Diffusion with two directions | D12 | 3 / [(1-α)-8/3 - (1-α)-7/3] | 1/5(1-α)-5/3 – ¼(1-α)-4/3 + 1/20 |
Random nucleation and nuclei growth | |||
Avrami-Erofeev [36] n = 1, 2, 3, 4 et 5 | An | x(1-α)[−Ln (1-α)]y x = 4, 2, 3, 4/3 and 3/2 y = 3/4, 1/2, 2/3, 1/4 and 1/3 | [−Ln(1-α)]z z = 1/4, 1/2, 1/3, 3/4 and 2/3 |
Chemical reactions | |||
Zero order | F0 | Constant | α |
First order | F1 | 1-α | -Ln(1-α) |
Second order | F2 | (1-α)2 | (1-α)−1–1 |
Contraction (surface, volume and interface respectively for n = 2, 3 and 4) | Rn | x(1-α)y x = 2, 3 et 3/2. y = 1/2, 2/3 and 1/3 | 1-(1-α)z z = 1/2, 1/3 and 2/3 |
Power / Exponential | |||
Low power (half, third and quarter respectively for n = 2, 3 and 4) | Pn | nαx x = 1/2, 2/3 and ¾ | αy y = 1/2, 1/3 et ¼ |
Exponential | E1 | α | Ln(α) |