2.4.1. Turbulence

ε model is based on the equation model and introduce ε mod *ε* uous phase of Euler's method and discretized using P1+P1 The motion of gas is simplified to the impressible and isothermal flow according to the process conditions and gas characteristics. The k-ε model is based on the equation model and introduced an equation about the turbulent dissipation rate. The calculation is small but there is more data accumulation. It has a wide range of industrial applications with the good accuracy and good convergence. The Reynolds-Averaged Navier-Stokes (RANS) k-ε model is used to simulate and analyze the turbulence. The equations are solved on the basis of RANS equations for conservation of momentum and the continuity equation for conservation of mass in the *spf* interface. The effects of the turbulence flow are modeled through two equations—turbulent kinetic energy *k* and dissipation rate ε with the reliability constraints. The flow near wall is described by the wall functions. Gas is represented by a continuous phase of Euler's method and discretized using P1+P1 method. Two advanced algorithms of the streamline diffusion and crosswind diffusion are applied to the Navier-Stokes equations and turbulence equations to converge the model easily. The pseudo-time stepping algorithm is used to solve the stationary equation. The velocity of the CFL digital expression of the turbulence variable ratio parameter is 1 m/s and the length scale factor is 0.035. The specific parameters of

<sup>μ</sup> *σ σ*

the turbulence model are *C*e1 = 1.44, *C*e2 = 1.92, *C*<sup>µ</sup> = 0.09, σ<sup>k</sup> = 1, σ<sup>c</sup> = 1.3, κ<sup>v</sup> = 0.41, *B* = 5.24. These parameters are RANS k-ε model constants. The governing equations of the turbulent flow are as follows.

Incompressible flow equation in steady state:

$$\rho \left( \mathbf{U} \cdot \mathbf{V} \right) \mathbf{U} = \nabla \cdot \left[ -\mathbf{P} \mathbf{I} + (\mu + \mu\_T) \left( \mathbf{V} \mathbf{U} + \left( \mathbf{V} \mathbf{U} \right)^T \right) \right] + \mathbf{F} \tag{1}$$

$$
\rho \nabla \cdot (\mu) = 0 \tag{2}
$$

The transport equation of *k*:

$$
\rho \rho (\mu \cdot \nabla) k = \nabla \cdot \left[ (\mu + \frac{\mu\_T}{\sigma\_k}) \nabla k \right] + P\_k - \rho \varepsilon \tag{3}
$$

where the production term and the turbulent viscosity are listed as following:

$$p\_k = \mu\_T [\nabla \mathcal{U} : (\nabla \mathcal{U} + (\nabla \mathcal{U})^T)] \quad \mu\_T = \rho \mathbb{C}\_u \frac{k^2}{\varepsilon} \tag{4}$$

The transport equation of ε:

$$\rho(\mathcal{U}\cdot\nabla)\varepsilon = \nabla \cdot \left[ (\mu + \frac{\mu\_T}{\sigma\_\varepsilon}) \nabla \varepsilon \right] + \mathcal{C}\_{\varepsilon 1} \frac{\varepsilon}{k} P\_k - \mathcal{C}\_{\varepsilon 2} \rho \frac{\varepsilon^2}{k} \quad \varepsilon = ep \tag{5}$$

where µ*T*, µ, ρ, *P*, *U*, *I*, *F*, *k*, ε, σ*<sup>k</sup>* , and *P<sup>k</sup>* , stand for the eddy viscosity, viscosity, density, pressure, velocity vector, unit matrix, volume force vector, turbulent kinetic energy, turbulent dissipation rate, respectively. The SI unit is used.

## 2.4.2. Erosion

The sulfur particles are considered as the discrete phases in the *fpt* interface during simulation. The movement of particles under the framework of Lagrange is governed by Newton's second law (6) and affected by the drag force, gravity, and brown force. The drag force (7) is generated by the speed difference between gas and particles and controlled by stokes' law (8). In addition, the body force (9) formed by the effect of accelerated or decelerated particles on the movement of gas is obtained through gas-particles interaction in the *fpi* interface, and then the gas velocity in the *spf* interface and particles velocity in the *fpt* interface are coupled by gas-particles interaction. Through sampling and analysis of the pipe ash, and in order to simplify the particle model, all sulfur particles, releasing 500 particles per 0.08 s at the boundary of inlet, are introduced into the tube by high-speed gas and there are the same physical properties of density (2360 kg/m<sup>3</sup> ), diameter (50 µm), and shape (sphere). The turbulent dispersion model of particles adopts discrete random walk, the variable time step method. The turbulent kinetic energy and turbulent dissipation rate of particles are coupled to those of gas in the *spf* interface. In addition, there is still no recognized universally applicable theoretical model due to the complexity of the erosion behavior of material. To assess the interaction between particles and pipes, Finnie erosion model is used to explain the rule of the particle erosion of plastic materials at low impact angles, thus the classical Finnie erosion model (10) is used to describe the impact of particles and the count method is used to count the quantities of particles hitting the wall considering the shape of the pipeline model. The ratio of the normal and tangential force and number multiplication factor are set to 1. The surface hardness of wall is 640 N/mm<sup>2</sup> and the surface mass density is 7.98 g/cm<sup>3</sup> respectively.

Newton's second law and drag force equation:

$$\frac{d(m\_p v)}{dt} = F\_D + F\_{\mathcal{S}} + F\_{\text{brown}} \tag{6}$$

*Lubricants* **2020**, *8*, 92

$$F\_D = (\frac{1}{\pi\_P}) m\_P (u - v) \tag{7}$$

where, *F<sup>D</sup>* is Drag force, *F<sup>g</sup>* is gravity, and *Fbrown* is brown force. The *m<sup>p</sup>* is the particle mass (SI unit: kg), τ*<sup>p</sup>* is the particle velocity response time (SI unit: s), *V* is the velocity of particle (SI unit: m/s), and *U* is the fluid velocity (SI unit: m/s).

The stokes drag law for the particle response time is defined as:

$$
\tau\_P = \frac{\rho\_P d\_P^2}{18\mu} \tag{8}
$$

where, µ is the fluid viscosity (SI unit: Pa·s), ρ*<sup>p</sup>* is the particle density (SI unit: kg/m<sup>3</sup> ), and *d<sup>p</sup>* is the particle diameter (SI unit: m).

Body force of particle-to-syngas calculation equation:

$$F\_{V,j} = -\frac{1}{V\_j} \sum\_{i=1}^{N} n\_i F\_{D,i} \int \delta(r - q\_i)dV \tag{9}$$

where, *Fv,j* is the average volume force, a mesh element *j* with volume *V<sup>j</sup>* , δ is the Dirac delta function, *FD,i* is the drag force exerted on the *i*th particle, *n<sup>i</sup>* is the force multiplication factor of the *i*th model particle, and *N* is the total number of particles.

The Finnie erosion equation is listed [6].

$$\begin{aligned} V &= \frac{cMl^2}{4p(1+\frac{mr^2}{l})} [(\cos \alpha)^2] \quad \tan \alpha > \frac{P}{2} \\ V &= \frac{cMl^2}{4p(1+\frac{mr^2}{l})} \frac{2}{p} [\sin(2\alpha) - 2\frac{(\sin \alpha)^2}{p}] \quad \tan \alpha \le \frac{P}{2} \end{aligned} \tag{10}$$

where, *c* (dimensionless) is the fraction of particles cutting in an idealized manner; *M* (SI unit: kg) is the total mass of eroding particles; *U* (m/s) is the magnitude of the incident particle velocity; *p* (Pa) is the Vickers hardness of the material; *m* (SI unit: kg) is the mass of an individual particle hitting the surface; *r* (SI unit: m) is the average particle radius; *I* (SI unit: kg/m<sup>2</sup> ) is the moment of inertia of an individual particle about its center of mass. For an isotropic sphere, *I* = 2 mr<sup>2</sup> /5; α (rad) is the angle of incidence, with α = 0 tangent to the surface and α = π/2 normal to the surface; *P* is a dimensionless parameter, defined as *P* = *K*/(1 + 2*mr*/*I*) and *K* (dimensionless) is the ratio of vertical and horizontal forces action on the particle. The mass loss per unit area vs. time is chosen to evaluate the erosion rate of steel.

## 2.4.3. Chemical Reaction and Electrochemical Corrosion

The ionization reactions of carbonic acid are generated into the hydrogen ion and bicarbonate ions, in the *chem* interface, and the distribution of the substance concentration is affected by diffusion and convection in the *tds* interface [7]. Table 2 shows reaction equilibrium constant, positive reaction rate, and diffusion coefficient. The convection is coupled by the turbulent mixing and affected by the motion of gas depending on the mass balance Equation (11). Turbulent kinematic viscosity of substance under turbulence is governed by that of turbulent gas, and turbulent Schmidt number is 0.71.

Electrochemical corrosion is described by the reactions of dissolved iron and hydrogen evolution in the *siec* interface. The iron dissolution reaction (Fe2<sup>+</sup> + 2e<sup>−</sup> → Fe) governed by anode Tafel Equation (12) occurred on the wall, and the hydrogen evolution reaction (2H<sup>+</sup> + 2e<sup>−</sup> → H2) controlled by concentration-dependent kinetic Equation (13) occurred on the inner surface of the tube. The interface of metal/electrolyte is considered to be an electrode-electrolyte coupled wall to complete charge transfer and charge conservation between ions and electrons. The electrolyte conductivity is 2.5 × 10−<sup>3</sup> S/m. The temperature in the pipeline is 250 ◦C. The concentration of Fe2<sup>+</sup> is 1 × 10−<sup>9</sup> mol/L. So, the concentration of H<sup>+</sup> is calculated by the Nernst equation, the H<sup>+</sup> concentration value is <sup>1</sup> <sup>×</sup> <sup>10</sup>−<sup>6</sup> mol/L. According to calculation, the initial potential of electrolyte is <sup>E</sup>eq,*Fe* = <sup>−</sup>0.9068 V, and the initial potential of electrode is 0 V. The exchange current density and Tafel slope of iron dissolved reaction are 10−<sup>3</sup> A/m<sup>2</sup> and 40 mV per decade [8], respectively. Normally, anodic oxidation of iron presents a Tafel slope of less than 60 mV. The equilibrium potential depends on Equation (14). The exchange current density of hydrogen evolution reaction is 1.1 × 10−<sup>2</sup> A/m<sup>2</sup> . The equilibrium potential is −0.3112 V.

Mass balance equation is:

$$\frac{\partial c\_{i}}{\partial t} + \nabla \cdot (-D\nabla c\_{i}) + \mathcal{U} \cdot \nabla c\_{i} = \mathcal{R}\_{i} \tag{11}$$

where, *C<sup>i</sup>* is the concentration of the species (SI unit: mol/m<sup>3</sup> ), *D<sup>i</sup>* is the diffusion coefficient (SI unit: m<sup>2</sup> /s), *R<sup>i</sup>* is a reaction rate expression for the species (SI unit: mol/(m<sup>3</sup> ·s)), and *U* is the velocity vector (SI unit: m/s).

Anode Tafel equation:

$$i\_{l\alpha} = i\_0 \times 10^{\frac{\eta}{A\_d}} \tag{12}$$

where, *iloc* denotes the local charge transfer current density, *i*<sup>0</sup> denotes the exchange current density, and *A<sup>a</sup>* denotes the Tafel slope.

Concentration-dependent kinetics:

$$i\_{l\alpha} = i\_0 [\mathbf{C}\_R \exp(\frac{\alpha\_d F \eta}{RT}) - \mathbf{C}\_O \exp(\frac{-\alpha\_d F \eta}{RT})] \tag{13}$$

where, *iloc* denotes the local charge transfer current density, *i*<sup>0</sup> is the exchange current density, C<sup>R</sup> and C<sup>O</sup> are dimensionless expressions, describing the dependence on the reduced and oxidized species in the reaction.

$$\mathrm{E\_{eq,Fe}} = -0.44 + 2.303 \frac{\mathrm{RT}}{2 \mathrm{F}} \times \log(10^{-9}) \tag{14}$$

where, R is the gas constant, T is the absolute temperature in Kelvin, F is Faraday's constant. The Fe2<sup>+</sup> concentration value of 1 × 10−<sup>9</sup> mol/L and H<sup>+</sup> concentration value of 1 × 10−<sup>6</sup> mol/L are measured and calculated by coal chemical companies.

**Table 2.** Constant for the model.


Note: In the table, *T<sup>f</sup>* is temperature in degrees Fahrenheit, *T* is absolute temperature in Kelvin, *T<sup>c</sup>* is temperature in degrees Celsius, *P* is the absolute pressure, *D* is the diffusion coefficient, equilibrium (*K*) and forward (*k<sup>f</sup>* ) reaction rate coefficients.
