In the following the list of the equations solved for each reactor model
$$\frac{\partial ω_{i}}{\partial t} = \frac{MW_{i}{R_{i}}^{hom}}{ρ} + \frac{αMW_{i}{R_{i}}^{het}}{ρ} - \frac{ω_i}{m}\frac{\partial m}{\partial t}$$
$$\frac{\partial m}{\partial t} = αV\sum_{i}^{NS}{MW_{i}R_{i}^{het}}$$
$$\frac{\partial θ_{j}}{\partial t} = \frac{{R_{j}}^{het}}{Γ}$$
$$\frac{\partial T}{\partial t} = \frac{Q^{hom}}{ρc_{p}} + \frac{αQ^{het}}{ρc_{p}}$$
$$\frac{\partial ω_{i}}{\partial t} = \frac{\dot{m}(ω^0 - ω_i )}{Vρ} + \frac{MW_{i}{R_{i}}^{hom}}{ρ} + \frac{αMW_{i}{R_{i}}^{het}}{ρ}$$
$$\frac{\partial θ_{j}}{\partial t} = \frac{{R_{j}}^{het}}{Γ}$$
$$\frac{\partial T}{\partial t} = \frac{\dot{m}(T^0 - T)}{Vρ} + \frac{Q^{hom}}{ρc_{p}} + \frac{αQ^{het}}{ρc_{p}}$$
1-D Pseudo-Homogeneous Plug Flow Reactor: Steady-State $$\frac{\partial ω_{i}}{\partial z} = \frac{MW_{i}A({R_{i}}^{hom} + α{R_{i}}^{het})}{\dot{m}}$$
$$0 = \frac{{R_{j}}^{het}}{Γ}$$
$$\frac{\partial T}{\partial z} = \frac{A(Q^{hom} + αQ^{het})}{\dot{m}c_{p}}$$
1-D Pseudo-Homogeneous Plug Flow Reactor: Transient $$\frac{\partial ω_{i}}{\partial t} = -\frac{\dot{m}}{A}\frac{\partial ω_{i}}{\partial z} + D^{mix}_{i}\frac{\partial^2 ω_{i}}{\partial^2 z} + \frac{MW_{i}{R_{i}}^{hom}}{ρ} + \frac{αMW_{i}{R_{i}}^{het}}{ρ}$$
$$\frac{\partial θ_{j}}{\partial t} = \frac{{R_{j}}^{het}}{Γ}$$
$$\frac{\partial T}{\partial t} = -\frac{\dot{m}}{Aρ}\frac{\partial T}{\partial z} + \frac{k^{gas}_{mix}}{ρc_{p}}\frac{\partial^2 T}{\partial^2 z} + \frac{Q^{hom}}{ρc_{p}} + \frac{αQ^{het}}{ρc_{p}}$$
1-D Heterogeneous Plug Flow Reactor: Steady-State $$0 = -\frac{\dot{m}}{Aρ}\frac{\partial ω_{i}}{\partial z} + D^{mix}_{i}\frac{\partial^2 ω_{i}}{\partial^2 z} -\frac{A_{s}K_{mat}}{ε}(ω_{i} - ω^S_{i})+ \frac{MW_{i}{R_{i}}^{hom}}{ρ}$$
$$0 = A_{s}K_{mat}ρε(ω_{i} - ω^S_{i}) + εαMW_{i}{R_{i}}^{het}$$
$$0 = \frac{{R_{j}}^{het}}{Γ}$$
$$0 = -\frac{\dot{m}}{Aρ}\frac{\partial T}{\partial z} + \frac{k^{gas}_{mix}}{ρc_{p}}\frac{\partial^2 T}{\partial^2 z} + \frac{Q^{hom}}{ρc_{p}} - \frac{A_{s}h(T - T^s)}{ρc_{p}ε}$$
$$0 = \frac{k^S}{ρ^Sc_{p}^S}\frac{\partial^2 T^S}{\partial^2 z} + \frac{αQ^{het}}{ρ^Sc_{p}^S(1-ε)} + \frac{A_{s}h(T - T^s)}{ρ^Sc_{p}^S(1-ε)}$$
1-D Heterogeneous Plug Flow Reactor: Transient $$\frac{\partial ω_{i}}{dt} = -\frac{\dot{m}}{Aρ}\frac{\partial ω_{i}}{\partial z} + D^{mix}_{i}\frac{\partial^2 ω_{i}}{\partial^2 z} -\frac{A_{s}K_{mat}}{ε}(ω_{i} - ω^S_{i})+ \frac{MW_{i}{R_{i}}^{hom}}{ρ}$$
$$0 = A_{s}K_{mat}ρε(ω_{i} - ω^S_{i}) + εαMW_{i}{R_{i}}^{het}$$
$$\frac{\partial θ_{j}}{\partial t} = \frac{{R_{j}}^{het}}{Γ}$$
$$\frac{\partial T}{\partial t} = -\frac{\dot{m}}{Aρ}\frac{\partial T}{\partial z} + \frac{k^{gas}_{mix}}{ρc_{p}}\frac{\partial^2 T}{\partial^2 z} + \frac{Q^{hom}}{ρc_{p}} - \frac{A_{s}h(T - T^s)}{ρc_{p}ε}$$
$$\frac{\partial T^S}{\partial t} = \frac{k^S}{ρ^Sc_{p}^S}\frac{\partial^2 T^S}{\partial^2 z} + \frac{αQ^{het}}{ρ^Sc_{p}^S(1-ε)} + \frac{A_{s}h(T - T^s)}{ρ^Sc_{p}^S(1-ε)}$$
Here is the symbols meaning:
Symbol
Meaning
Unit dimension
$i$ Gas specie index
$-$
$j$ Coverage specie index
$-$
$α$ Catalytic load
$\frac{1}{m}$
$ε$ Reactor void fraction
$-$
$ω$ Gas mass fraction
$-$
$ω^0$ Gas mass fraction at initial conditions
$-$
$ω^S$ Gas mass fraction in the solid phase
$-$
$ρ$ Gas density
$\frac{kg}{m^3}$
$ρ^S$ Solid density
$\frac{kg}{m^3}$
θ
Coverage fraction
$-$
$Γ$ Site density
$\frac{kmol}{m^2}$
$A$ Reactor cross section area
$m^2$
$A_{s}$ Reactor specific area
$\frac{1}{m}$
$c_{p}$ Gas specific heat
$\frac{J}{kgK}$
$c_{p}^S$ Solid specific heat
$\frac{J}{kgK}$
$D^{mix}$ Mixture diffusion coefficient
$\frac{m^2}{s}$
$h$ Gas-to-solid heat transfer coefficient
$\frac{W}{m^2K}$
$k^{gas}_{mix}$ Mixture thermal conductivity
$\frac{W}{mK}$
$k^S$ Solid thermal conductivity
$\frac{W}{mK}$
$K_{mat}$ Gas-to-solid mass transfer coefficient
$\frac{m}{s}$
$m$ Total mass
$kg$
$\dot{m}$ Inlet mass flow rate
$\frac{kg}{m^3s}$
$MW_{i}$ Gas specie molecular weight
$\frac{kg}{kmol}$
$Q^{hom}$ Heat of reaction from homogeneous reactions
$\frac{J}{m^3s}$
$Q^{het}$ Heat of reaction from heterogeneous reactions
$\frac{J}{m^2s}$
${R_{i}}^{hom}$ Gas specie reaction rate from homogeneous reactions
$\frac{kmol}{m^3s}$
${R_{i}}^{het}$ Gas specie reaction rate from heterogeneous reactions
$\frac{kmol}{m^2s}$
${R_{j}}^{het}$ Coverage reaction rate
$\frac{kmol}{m^2s}$
$t$ Time
$s$
$T$ Gas temperature
$K$
$T^S$ Solid temperature
$K$
$V$ Reactor volume
$m^3$
$z$ Reactor lenght
$m$