List of Symbols

A

Annular cross section area between reactor tube and particles feeder tube (m²)

a

Cross section area of solid particles feeder tube (m²)

C_c

Cunningham correction factor, which defined in reference [23]

C_µ, C_1ε, C_2ε

Constant (0.09, 1.44 and 1.92, respectively)

D

Inner diameter of reactor tube (m)

D_e

Effective diffusivity in the solid particles (m².s^–1)

D

Outer diameter of solid feeder tube (m)

d_p

Particle diameter (m)

e_H2

Percent of excess hydrogen gas (%)

F_D

Drag force factor, which was calculated as follow [23]: (s^–1)FD=18μgCcρFe3O4dp2

g

Gravity acceleration (m.s^–2)

H

Height of the reaction zone (m)

h

Height of the particle feeder tube (m)

K

Equilibrium constant for reaction (11), was calculated by HSC software [20]

k

Turbulent kinetic energy (m².s^–2)

k_g

Mass transfer coefficient of reducing gas in the gaseous film around the particles, was calculated from reference [21] (m.s^–1)

k_s

Intrinsic reaction rate constants for reaction (11), was presented in reference [16] as below: (m³.kmol^–1.s^–1)ks=9.9×104exp−4.63×108RT

L

Height of the reactor tube (m)

MFe,MFe3O4,MH2,MH2O,MN2

Molecular weight of iron, magnetite, hydrogen, water vapor and Nitrogen, respectively (kg.kmol^–1)

m^._s

Particles feed rate (kg.s^–1)

P

System total pressure (pa)

PH20,PN20

Initial partial pressures of hydrogen and nitrogen, respectively (pa)

Q⁰

Initial flow rate of entering gas mixture to the annulus space at top of the reactor (m³.s^–1)

QH20,QN20

Initial flow rate of entering hydrogen and nitrogen to the annulus space at top of the reactor tube, respectively (m³.s^–1)

QH2req.

Required hydrogen flow rate to achieve mentioned excess hydrogen for reaction (11), can be calculated by following equation [16]: (m³.s^–1)QH2req.=4MH2m.1+1KρH2MFe3O4×1+eH2100

q

Flow rate of solid particles carrier gas (m³.s^–1)

Rˉ

Universal Gas Constant (pa. m³.kmol^–1.K^–1)

T

Temperature (K)

t

Time (s)

t_end

Residence time of the particles in the reaction zone (s)

u

Velocity of gas phase in the x direction (m.s^–1)

V

Velocity vector of gas phase (m.s^–1)

V⁰

Initial velocity vector of entering gas mixture to the annulus space at top of the reactor tube (m.s^–1)

V_p

Velocity vector of the particles (m.s^–1)

v

Velocity of gas phase in the y direction (m.s^–1)

v⁰

Initial velocity vector of the particles in the injected gas (m.s^–1)

W

Weight of the particles (N)

w

Velocity of gas phase in the z direction (m.s^–1)

X

Conversion of the particles

x

x position in the reactor (m)

x_p

Position of the particle (m)

yH20,yN20

Initial molar fractions of hydrogen and nitrogen at the top of reaction zone, respectively

yH2,yH2O,yN2

Molar fractions of hydrogen, water vapor and nitrogen at each height of the reaction zone, respectively

y

y position in the reactor (m)

y_p

y position of the particles (m)

z

z position in the reactor (m)

z_p

z position of the particles (m)

Greek symbols

δ_ij

Kronecker delta: δij=1i=j0i≠j

σ_k, σ_ε

Turbulent Prandtl numbers (1 and 1.3, respectively)

Ε

Turbulent dissipation (m².s^–3)

θ

Dimensionless temperature: θ=T−1.6×103T+1.6×103

ε

Turbulent kinetic energy dissipation rate (J. kg^–1)

ε_p

Porosity of reduced ash layer in the particle, which was calculated by the following equation:εp=1−ρFe3O4MFe3ρFeMFe3O4

ρg

Density of the gas mixture, which was calculated using ideal gas mixture assumption (kg. m^–3)

ρp (ρFe,ρFe3O4)

Densities of iron and magnetite particles, respectively (kg. m^–3)

ρH2,ρH2O,ρN2

Densities of gases hydrogen, water vapor and nitrogen, respectively, which were calculated using ideal gas assumption (kg. m^–3)

μg

Viscosity of the gas mixture, which was calculated by Wilke method (pa.s)

μH2,μH2O,μN2

Viscosities of gases hydrogen, water vapor and nitrogen, respectively, which were calculated using the Sutherland equations [17] (pa.s)

µ_t

Turbulent viscosity of the flow based on the k-ε model (Pa.s):μt=ρgCμk2ε

τ_react, τ_ash, τ_film

Time constants of chemical reaction, diffusion in the ash layer and diffusion in the gaseous film, respectively, which were calculated using the following equations [19, 21]: (s)

τash=ρFe3O4MH2dp26ρH2MFe3O4De,τfilm=2ρFe3O4MH2dp3ρH2MFe3O4kg,τreact=2ρFe3O4MH2dpρH2MFe3O4ks1−1K