${\Re }_{\mathrm{r}}$	instantaneous consumption rate of monomer (kmol/s)
$k_{\mathrm{f},\mathrm{r}}$	Forward rate coefficient for reaction, rth
${\displaystyle {\prod }_{ct}}$	molar concentrations of surface-adsorbed species on the catalyst active site
${\eta }_{\mathrm{i},\mathrm{g},\mathrm{r}}^{\prime }{}^{\prime }$	rate exponent for the ith gas phase reactant
${\eta }_{\mathrm{j},\mathrm{s},\mathrm{r}}^{\prime }$	rate exponent for the jth gas phase reactant
$C_{\mathrm{i}}$	gas phase species
$S_{\mathrm{J}}$	solid phase species
${\mathrm{R}}_{ i,j(bu)}$	net molar rate of production in bubble phase
${\mathrm{R}}_{ i,j(eu)}$	net molar rate of production in emulsion phase
${\mathrm{R}}_{ i,j(ct)}$	net molar rate of production of catalyst active side for any phase
U0	Superficial gas velocity m/s
$B_{\mathrm{i},\mathrm{r}}^{\prime }$	stoichiometric coefficient for each reactant species in bubble phase
$B_{\mathrm{i},\mathrm{r}}^{″}$	stoichiometric coefficient for produced PP in bubble phase
$E_{\mathrm{i},\mathrm{r}}^{″}$	stoichiometric coefficient for produced PP in emulsion phase
$E_{\mathrm{i},\mathrm{r}}^{\prime }$	stoichiometric coefficient for each reactant species in emulsion phase
$C_{\mathrm{i},\mathrm{r}}^{\prime }$	stoichiometric coefficient for each reactant species on catalyst active site
$C_{\mathrm{i},\mathrm{r}}^{″}$	stoichiometric coefficient for produced PP on catalyst
${\widehat{\Re }}_{\mathrm{i},\mathrm{p}}$	monomer conversion rate
$N_{\text{as}}$	number of active site types
$M_{\mathrm{i}}$	i-th component
$k_{\mathrm{p}}$	propagation rate coefficient for a site of type j with terminal
$V_{\mathrm{R}}$	reactor volume(m3)
${\mathsf{\epsilon }}_{\min }$	void fraction of the bed at minimum fluidization
$U_0$	superficial gas velocity (m/s)
$A$	cross sectional area of the reactor(m2)
${\left[M_i\right]}_{in}$	concentration of component i in the inlet gaseous stream
$R_{\mathrm{v}}$	volumetric polymer phase out flow rate from the reactor (m3/s)
$R_{\mathrm{i},\mathrm{p}}$	instantaneous rate of reaction for monomer i (kmol/s)
$C_{\mathrm{pi}}$	specific heat capacity of component i (J/kg K)
${\mathsf{\rho }}_{\text{pp}}$	density of mixed gas density(kg/m3)
$C_{\mathrm{p},\mathrm{sol}}$	specific heat capacity of solid product (J/kg K)
$T_{\text{in}}$	temperature of the inlet gaseous stream (K)
$T_{\mathrm{r}}$	Average temperature in the bed at (K) during ${\mathsf{\epsilon }}_{\min }$
$R_{\mathrm{p}}$	production rate (kg/s)
${\mathsf{\rho }}_{\mathrm{eu}}$	Average density of emulsion phase
$\mathsf{\phi }$	phase variable
$\mathsf{\alpha }$	phase volume fraction
$\mathsf{\rho }$	the phase density
$\overrightarrow{\upsilon }$	phase velocity
${\tau }^{″}$	the diffusion term
$S\mathsf{\phi }$	the source term
E	Phase energy
$k_{\text{eff}}$	effective thermal conductivity
$S_{\mathrm{h}}$	volumetric heat sources
$R_{\mathrm{i}}$	net rate of production of species i
$S_{\mathrm{i}}$	rate of i, species transport
j	Jth item (either a reactant or a product).
$H_{\stackrel{\rightharpoonup }{\mathrm{b}}\mathrm{e}}$	bubble to emulsion heat transfer coefficient(W/m3 K)
$U_{\mathrm{b}}$	bubble velocity (m/s)
${\mathsf{\rho }}_{\mathrm{pol}}$	polymer density(kg/m3)
$C_{\mathrm{p},\mathrm{sol}}$	specific heat capacity of solid product (J/kg K)
${\mathsf{\epsilon }}_{\mathrm{bu}}$	void fraction of bubble for Geldart B particles
${\mathsf{\rho }}_{\mathrm{pol}}$	polymer density(kg/m3)
$C_{\mathrm{p},\mathrm{pol}}$	specific heat capacity of solid product (J/kg K)
$\Delta H_{\mathrm{R}}$	heat of reaction (J/kg)
${\mathsf{\alpha }}_{\mathrm{bu}}$	volume fraction of bubbles in the bed
$\left[M_i\right]$	“I” monomer concentration (molMe/m3)
P0	active sites concentration (kmol/m3)
N(0, j)	uninitiated site of type j produced by formation reaction
N(r,j)	living polymer molecule of length r, growing at an active site of type j, with terminal monomer M
NH(0,j)	uninitiated site of type j produced by transfer to hydrogen reaction
r	number of units in polymer chain
kfm (j)	transfer rate coefficient for a site of type j with terminal monomer M reacting with monomer M
kfcat (j)	transfer rate coefficient for a site of type j with terminal monomer M reacting with catalyst
kh (j)	rate coefficient for reinitiating of a site of type j by monomer M
ki (j)	rate coefficient for initiation of a site of type j by monomer M
kp (j)	propagation rate coefficient for a site of type j with terminal monomer M reacting with monomer M
PBM	Population Balance Model
PMLM	Polymeric Multilayer Model
Subscripts and superscripts
bu	bubble phase
eu	emulsion phase
g	gas mixture property
i	component type number
in	inlet
j	active site type number
pol	polymer
ref	reference condition