A transient computational fluent dynamic (CFD) simulation applying a pressure-based solver was employed to solve the density weighted Reynolds-averaged Navier-Stokes (RANS) equations, as follows [12]:

{ ∂ ρ ˜ u i ∂ x i = 0 ∂ ρ ˜ u i u j ∂ x i = ∂ ρ ˜ u ″ i u ″ j ∂ x i − ∂ p ∂ x + ∂ τ i j ∂ x i + ρ ˜ g i (2)

ρ: Density (kg/m³)

u_i: Favre-averaged velocity in tensor notation

x, y, z: Direction of coordinate axes

i, j, k: Tensorial indices

': Fluctuations with respect to a Reynolds averaging

'': Fluctuations with respect to a Favre averaging

g = Gravity (m/s²)

u_i, u_j: Reynolds stress (kg/m∙s²)

u_i: Favre-averaged velocity in tensor notation

u_i: Resolved fluctuating velocity components

p: Pressure (Pa)

∂ ρ ˜ h u j ∂ x i = ∂ ρ ˜ h ″ u ″ j ∂ x i − ∂ p ˜ ∂ x − ∂ J i Z ˜ ∂ x i + τ i j ∂ u i ∂ x i + q i + q r , (3)

With:

h: Enthalpy (j/kg)

J i Z : Diffusive flux of chemical species

q_i, q_r: The generic source term and reaction heat term

∂ ρ ˜ Y z u i ∂ x i = ∂ ρ ˜ Y ″ z ∂ x i − ∂ J i Z ˜ ∂ x i + R ˙ z , (4)

Y: Mass fraction of species

R_z: Production rate of zth component

Where, the stress tensor, τ i j , is given by [12]:

τ i j = [ u ( ∂ u i ∂ x j + ∂ u j ∂ x i ) ] − 2 3 u ∂ u i ∂ x j δ i j , (5)

The species product by diffusive flux of combustion is defined by the Fick’s law:

J i Z ˜ = u t S C k ∂ Y z ∂ x i , (6)

u_t: Turbulent eddy viscosity (kg∙m/s).

Sc: Schmidt number.

To solve the problem in the present work, we use the discrete ordinates (DO) model of the radiative transfer Equation (RTE), considering the absorption and emission effects [12]:

d I ( r , s ) d s + ( a + σ s ) I ( r , s ) = a n 2 σ T 4 π + σ s 4 π ∫ 0 4 π ( r , s ) ( r , s ) Φ ( r , s ) d Ω , (7)

I: I is the radiation intensity and

( a + σ s ) : The optical thickness of the medium

a: Absorption coefficient

r : Position vector

Φ : Porosity

s : Direction vector

n: Refractive index

The S2S model assumes any absorption, emission, or scattering of radiation by the medium can be ignored; as a result, reducing the computational cost by only considering the surface-to-surface radiation. In this model, the radiation heat transfer to a surface from another surface is a direct function of the surface- to-surface view factor. Therefore, the radiation energy balance for each surface follows the equation below [12]

q o u t , k = ε k σ T k 4 + ρ k ∑ j = 1 N F k j q o u t , j , (8)

F k j : The view factor between surface k and surface j.

Combustion inside the burner and furnace domains was modeled using the eddy-dissipation model (EDM). It should be noted that, because in the gas-fired furnace the burners create high velocity combustion products, the Arrhenius chemical kinetic calculations were not considered thereby the computational cost is significantly reduced. In EDM, the species transport formulation for the local mass fraction species (Y_z), for the zth species, was solved assuming that the reaction rates were dominated by turbulence. Therefore, the species transport equation becomes as follows [13]:

∂ ∂ t ( ρ Y z ) + ∇ ⋅ ( ρ υ ¯ Y z ) = − ∇ ⋅ J z ¯ + R ˙ z + S ˙ z , (9)

S ˙ z : Source term of zth component.

The standard k-ε model uses model transport equations to obtain dissipation rate k and turbulence kinetic energy k. Hence, the model transport equation for k can be obtained by applying the precise equation, whereas the model transport equation for k can be determined through the application of physical reasoning. Thus, to derive the k-ε model, assuming a complete turbulent flow with an insignificant molecular viscosity, making this k-ε model only applicable in flows which are fully turbulent. Hence, alterations are needed to improve both the model’s applicability and its performance. Two model variants that use ANSYS Fluent are the k-e model and realizable k-e model, as formulated below [14]

∂ ∂ t ( ρ k ) + ∂ ∂ x i ( ρ k u i ) = ∂ ∂ x j [ ( u + u i σ k ) ∂ k ∂ x i ] + G k + G b − ρ ε − T M + S k (10)

∂ ∂ t ( ρ ε ) + ∂ ∂ x i ( ρ ε u i ) = ∂ ∂ x j [ ( u + u t σ ε ) ∂ ε ∂ x j ] + C 1 ε ε k ( G k + C 3 ε G b ) − C 2 ε ρ ε 2 k + S ε (11)

where:

G k = production of turbulence kinetic energy due to velocity gradients.

G b = generation of turbulence kinetic energy due to buoyancy.

T M = contribution of the fluctuating dilatation in compressible turbulence to the overall dissipation rate.

C 1 ε , C 2 ε , C 3 ε : constants.

σ k , σ ε : turbulent Prandtl numbers for k and, respectively.

S k , S ε : the sources terms defined by the users.

Here, we will present the erosion model using the experimental formula below [15]:

R e r o s i o n = ∑ p = 1 N p a r t i c l e m ˙ p C ( d p ) f ( α ) ω b ( ω ) A f a c e , (12)

C(d_p) = function of particle

α = impact angle of the particle path with the wall face

f(α) = function of impact angle

ω = relative velocity of particle

b(ω) = function of relative particle velocity

A_face = area of the cell face at the wall

Default values are: C = 1.8 × 10⁻⁹, f = 1 and b = 0

Using the Tulsa Angle Dependent Model, Equation (30), can be rewrite as describe below:

E R = 1559 e − 6 B − 0.59 F s ω 1.75 f ( α ) (13)

The Equation (13) will rewrite by making substitutions as follows:

ω 1.73 = ω b ( ω )

1559 e − 6 B − 0.59 F = C ( d p )

To make these simulations, some assumptions and boundary conditions are required, such as:

The boiler is considered as a cube box;

We consider the a single layer graphene sheet with the size: 500 mm (long) and 250 mm (large);

The temperature into the boiler is considered variable;

We considered the pressure into boiler at 4 Mpa;

The graphene sheet is applied inside of wall of rectangular cube, meaning it already coated;

For our research we consider mechanical exfoliation as the production method of graphene and Chemical Vapor Deposition as method for coating and

The substrate is the steel.

Following elements will be used for the simulation:

Chemical elements: sodium (Na), sulfur (S), chloride (Cl) and Phosphorous (P)

Gases and composite elements: Sodium Chloride (NaCl), Hydrogen Chloride (HCl), Dioxide of Carbone (CO₂) and Dioxide of Sulfur (SO₂).

We choice to simulate the behavior of graphene in front of these elements because they are the dominants elements met into WTE.

See Table 3 and Table 4.

For this simulation we still using, the geometry and mesh at the point 4.1.1. We will simulate the erosion behavior of graphene coated the boiler walls in presence of gases and one simple chemical element such as: HCl, SO₂, KCl, P and CO₂.

Corrosion is a chemical phenomenon which ANSYS cannot do. Therefore, we will proceed by analytic method to solve this issue. That method is a microscopic analysis aspect of graphene:

➢ Firstly, we analyze the pore diameter (Table 7) of the carbon ring in terms of the electron density is smaller than the kinetic diameter of various gases and chemicals elements, such as: He, H₂, CO₂, O₂, N₂, and CH₄. For that we can take an example of the pore diameter of an octagon ring (considering electron density) is only 1.5 Å. Only large vacancies with a size above 5Å, that is, two lattice parameters, can be penetrated by the gas molecule.

➢ Secondly, we refer to the relation (1), the initial energy to form vacancy in the graphene structure is approximately to 7.5 eV [16] and according to the Raman Spectra of temperature evolution (Figure 5), and graphene conserve that energy for the temperature under 700˚C.

The hexagonal pore lattice diameter of graphene is 0.246 nm of and with a measured C-C bond length of 0.14 nm, the Graphene is considering the nuclei of the carbon atoms. When the Van der Waals radii (0.11 nm) pore diameter is reduced to 0.064 nm the carbon atoms are considered. The graphene ensures minimal permeability when the pore lattice present a small geometric and that even for smaller atoms such as helium. Moreover, the dense and delocalized electron cloud of π-conjugated carbon network in Graphene blocks the Class within its close packed aromatic rings and poses a repelling field to the reactive atom or molecule, consequently providing a physical separation between the refined metal surface and environmental reactants.

This energy barrier for a single layer coating of graphene is high enough to block the diffusion of oxygen to the underlying metal interface. The energy barrier magnitude of varies with molecular permission through the graphene lattice and the path of the atomic, suggesting that graphene can be the thinnest ever known corrosion barrier.