Thermogravimetric and kinetic analysis to discern synergy during the co-pyrolysis of microalgae and swine manure digestate

Arun K. Vuppaladadiyam; Hao Liu; Ming Zhao; Abdul F. Soomro; Muhammad Zaki Memon; Valerie Dupont

doi:10.1186/s13068-019-1488-6

Table 1 Common solid-state reaction mechanisms [44, 45]

From: Thermogravimetric and kinetic analysis to discern synergy during the co-pyrolysis of microalgae and swine manure digestate

Reaction mechanisms	Differential form f(α)	Integral form g(α)
A₂—Nucleation and nuclei growth (Avrami Eq. 1)	$ 2(1 - \alpha )[ - \ln (1 - \alpha )]^{1/2} $	$ [ - \ln (1 - \alpha )]^{1/2} $
A₃—Nucleation and nuclei growth (Avrami Eq. 2)	$ 3(1 - \alpha )[ - \ln (1 - \alpha )]^{3/2} $	$ [ - \ln (1 - \alpha )]^{1/3} $
A₄—Nucleation and nuclei growth (Avrami Eq. 3)	$ 4(1 - \alpha )[ - \ln (1 - \alpha )]^{3/4} $	$ [ - \ln (1 - \alpha )]^{1/4} $
R₂—Phase boundary controlled reaction (contracting area)	$ 2(1 - \alpha )^{1/2} $	$ [1 - (1 - \alpha )]^{1/2} $
R₃—Phase boundary controlled reaction (contracting volume)	$ 3(1 - \alpha )^{2/3} $	$ [1 - (1 - \alpha )]^{1/3} $
D₁—One-dimensional diffusion	$ (1/2)\alpha $	$ \alpha^{2} $
D₂—Two-dimensional diffusion (Valensi equation)	$ [ - \ln (1 - \alpha )]^{ - 1} $	$ (1 - \alpha )\ln (1 - \alpha ) + \alpha $
D₃—Three-dimensional diffusion (Jander equation)	$ (3/2)[1 - (1 - \alpha )^{1/3} ]^{ - 1} (1 - \alpha )^{2/3} $	$ [1 - (1 - \alpha )^{1/3} ]^{2} $
D₄—Three-dimensional diffusion (Ginstling–Brounshtein equation)	$ (3/2)[1 - (1 - \alpha )^{1/3} ]^{ - 1} $	$ [1 - (2/3)\alpha )] - (1 - \alpha )^{2/3} $
F₁—Random nucleation with one nucleus on the individual particle	$ 1 - \alpha $	$ - \ln (1 - \alpha ) $
F₂—Random nucleation with two nuclei on the individual particle	$ (1 - \alpha )^{2} $	$ 1/(1 - \alpha ) $
F₃—Random nucleation with three nuclei on the individual particle	$ (1/2)(1 - \alpha )^{3} $	$ 1/(1 - \alpha )^{2} $
P₁—Mampel power law $ \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) $	$ 2\alpha^{1/2} $	$ \alpha^{1/2} $
P₂—Mampel power law $ \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right) $	$ 3\alpha^{2/3} $	$ \alpha^{1/3} $
P₃—Mampel power law $ \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right) $	$ 4\alpha^{3/4} $	$ \alpha^{1/4} $

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Reaction mechanisms	Differential form f(α)	Integral form g(α)
A₂—Nucleation and nuclei growth (Avrami Eq. 1)	\( 2(1 - \alpha )[ - \ln (1 - \alpha )]^{1/2} \)	\( [ - \ln (1 - \alpha )]^{1/2} \)
A₃—Nucleation and nuclei growth (Avrami Eq. 2)	\( 3(1 - \alpha )[ - \ln (1 - \alpha )]^{3/2} \)	\( [ - \ln (1 - \alpha )]^{1/3} \)
A₄—Nucleation and nuclei growth (Avrami Eq. 3)	\( 4(1 - \alpha )[ - \ln (1 - \alpha )]^{3/4} \)	\( [ - \ln (1 - \alpha )]^{1/4} \)
R₂—Phase boundary controlled reaction (contracting area)	\( 2(1 - \alpha )^{1/2} \)	\( [1 - (1 - \alpha )]^{1/2} \)
R₃—Phase boundary controlled reaction (contracting volume)	\( 3(1 - \alpha )^{2/3} \)	\( [1 - (1 - \alpha )]^{1/3} \)
D₁—One-dimensional diffusion	\( (1/2)\alpha \)	\( \alpha^{2} \)
D₂—Two-dimensional diffusion (Valensi equation)	\( [ - \ln (1 - \alpha )]^{ - 1} \)	\( (1 - \alpha )\ln (1 - \alpha ) + \alpha \)
D₃—Three-dimensional diffusion (Jander equation)	\( (3/2)[1 - (1 - \alpha )^{1/3} ]^{ - 1} (1 - \alpha )^{2/3} \)	\( [1 - (1 - \alpha )^{1/3} ]^{2} \)
D₄—Three-dimensional diffusion (Ginstling–Brounshtein equation)	\( (3/2)[1 - (1 - \alpha )^{1/3} ]^{ - 1} \)	\( [1 - (2/3)\alpha )] - (1 - \alpha )^{2/3} \)
F₁—Random nucleation with one nucleus on the individual particle	\( 1 - \alpha \)	\( - \ln (1 - \alpha ) \)
F₂—Random nucleation with two nuclei on the individual particle	\( (1 - \alpha )^{2} \)	\( 1/(1 - \alpha ) \)
F₃—Random nucleation with three nuclei on the individual particle	\( (1/2)(1 - \alpha )^{3} \)	\( 1/(1 - \alpha )^{2} \)
P₁—Mampel power law \( \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 2}}\right.\kern-0pt} \!\lower0.7ex\hbox{$2$}}} \right) \)	\( 2\alpha^{1/2} \)	\( \alpha^{1/2} \)
P₂—Mampel power law \( \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}} \right) \)	\( 3\alpha^{2/3} \)	\( \alpha^{1/3} \)
P₃—Mampel power law \( \left( {n \, = \,{\raise0.7ex\hbox{$1$} \!\mathord{\left/ {\vphantom {1 4}}\right.\kern-0pt} \!\lower0.7ex\hbox{$4$}}} \right) \)	\( 4\alpha^{3/4} \)	\( \alpha^{1/4} \)