Table 3.

Transport equations for dynamic multiphasic fluidized bed reaction system.

No.	Type of equations	Equations
1	General transport equation	$\frac{d\left(\alpha \rho \varphi \right)}{dt}+\nabla .\left(\alpha \rho \overrightarrow{\upsilon }\varphi \right)=\nabla .{\tau }^{″}+S\varphi$
2	The volume fraction dual-phase density	$\rho ={\alpha }_2{\rho }_2+\left(1-{\alpha }_2\right){\rho }_1$
3	Momentum Equation	$\frac{d}{dt}\left(\rho \overrightarrow{\upsilon }\right)+\nabla .\left(\rho \overrightarrow{\upsilon }\overrightarrow{\upsilon }\right)=$−$\nabla p+\nabla .\mu [(\nabla .\overrightarrow{\upsilon }+{\overrightarrow{\upsilon }}^T)+\rho \overrightarrow{g}+\overrightarrow{F}$
4.	The energy equation shared between the phases	$\frac{d}{dt}\left(\rho E\right)+\nabla .\left(\overrightarrow{\upsilon }(\rho E+p\right)=\nabla .k_{eff}\nabla T)+S_h$
5.	Inter-phase species transport equations	$\frac{d}{dt}\left(\rho Y_i\right)+\nabla .(\rho \overrightarrow{\upsilon }{Y}_i)=$−$\nabla .{\overrightarrow{J}}_i+R_i+S_i$
6.	Mass transfer in bubble phase	$S_b=R_{bi}({}_{i}M{}_b\pm {}_{i}M{}_{bp})$
7.	Mass transfer in emulsion phase	$S_{eu}=R_{ei}({}_{i}M{}_e\pm {}_{i}M{}_{ep})$
8.	The net velocity of the reactants	${\overrightarrow{u}}_{net}=\frac{{\displaystyle \sum }r{\gamma }_j^r{M}_j^r{\overrightarrow{u}}_{r,j}}{{\displaystyle \sum r{\gamma }_j^r{M}_j^r}}$
9.	Momentum transfer for the bubble phase	$S_{bu}^{\overrightarrow{U}}={\Re }_{i,p}\left({}_{i}M{}_b{\overrightarrow{U}}_{net}-{}_{i}M{}_{bp}{\overrightarrow{U}}_i\right)$
10	Momentum transfer for the emulsion phase	$S_{eu}^{\overrightarrow{U}}={\Re }_{i,p}\left({}_{i}M{}_{eu}\overrightarrow{U}-{}_{i}M{}_{eu}{\overrightarrow{U}}_i\right)$