**4.2 Simulation conditions**

#### **4.2.1 Geometric model and grid**

As we all know, in numerical simulation process, the grid structure has a greatly effect on the calculation precision. Mesh with bad structure may lead to the enlargement of relative error and stability degradation or even simulation procedure divergent.

Fig. 5. Diagram of grid for geometric model.

Meanwhile, the grid formation technique has become a critical part in modern computational fluid dynamics. The mesh formation method can be divided two ways. One is algebraic method and the other is differential method. Thereinto, differential method can be used to produce smooth grid to suit complex flow domain. Of course, we can adjust the mesh degree of closeness by changing the control function. And if more accurate solution needed, mash must be thicker.

In this study, horizontal pipe section with 1 meter length, 0.08 meter diameter and range from 127.7 to 128.7m were chosen as research object. In this pipe section, fly ash was conveyed by compressed gas. Fig.5 gave the pipe geometric model with full grid.

#### **4.2.2 Boundary conditions**

384 Computational Simulations and Applications

*e g*

 <sup>2</sup> 0 <sup>4</sup> <sup>1</sup> 3 *s s sp*

It must be noted that, in the equations above, the symbol of e stands for granule collision

*e*

As we all know, in numerical simulation process, the grid structure has a greatly effect on the calculation precision. Mesh with bad structure may lead to the enlargement of relative

*e e*

 

*dg e*

<sup>0</sup> ,max 1 /*s s g* 

> 

> >

<sup>1</sup> 1/3

(37)

(39)

(38)

(40)

The expression of granule phase pressure is given as below.

<sup>0</sup> 1 21 *s ss <sup>s</sup> p*

When e= 1, it means elastic collision, which is in case of no energy loss.

When 0 1 *e* , it means energy will diffuse in form of elastic collision.

error and stability degradation or even simulation procedure divergent.

The total viscosity of granule phase is the below equation.

recovery coefficient. And it obeys the following rule.

When e=0, it means complete inelastic collision.

Fig. 5. Diagram of grid for geometric model.

**4.2 Simulation conditions 4.2.1 Geometric model and grid** 

	- Inlet boundary for gas

There are the assumptions that the gas axial velocity cross-section of the entrance with the fully developed turbulent flow of smooth pipe, radial velocity is zero, given the pressure of the entrance, turbulent kinetic energy expression is

$$k = \frac{\Im}{2} \left(\mu\_{\mathcal{S}} I\_{\mathcal{S}}\right)^2 \tag{41}$$

Turbulent dissipation rate can be expressed by

$$
\omega = \mathcal{C}\_{\mu}^{0.75} \frac{k^{1.5}}{l} \tag{42}
$$

Where Ig is gas turbulent intensity, to the fully arisen turbulent flow, then,

$$I\_{\mathcal{g}} = 0.16 \left( \text{Re}\_{\mathcal{g}D\_H} \right)^{1/8} \tag{43}$$

To the equation, *DH* is hydraulic diameter. From the equation above, it can be concluded, the Reynolds number in the gas turbulent intensity equation is regard hydraulic diameter as characteristic length.

*L* is length dimension, to circle pipe,

$$I = 0.07L\_{\nu}$$

*L* is pipe diameter.

The gas velocity of inlet is set 9.9m/s.

Outlet boundary for gas

The assumption that the fully developed conditions of the pipe flow, namely the normal derivative of the variables solved is zero, given the export pressure.

$$\frac{\partial \phi}{\partial \mathbf{x}\_i} = 0 \quad \text{ (} \boldsymbol{\phi}\_{\mathcal{S}} = \boldsymbol{\mu}\_{\mathcal{S}^i}, \boldsymbol{k}\_\star \text{\textquotedblleft} \text{\textquotedblright} \end{bmatrix} \tag{44}$$

#### Pipe wall surface

In the study, no-slip condition was adapted. Each parameter near the pipe wall was considered as zero. And wall function method was applied. So

Numerical Simulation of Dense Phase Pneumatic Conveying in Long-Distance Pipe 387

It must be noted that, in this paper, the turbulent kinetic energy and turbulent dissipation

In this paper, Finite Volume Method was utilized to discrete the governing equations above. The selected pipe section was divided into many non-concurrent domains which was called calculating grid. And then, each nodal point of stationary divided domain and its controlled volume were confirmed. In the process of discretization, the physical quantity of this

In this study, we used SIMPLE method to carry out two phase flow simulation. And the gas solid two phase flow were coupled each other. First of all, on the basis of initial condition and boundary condition, pure gas phase governing equation can be solved. And then, we can resolve granule phase governing equations which are based on gas flow characteristics. The last step was to gain the gas and solid flow field respectively by combining with this

Solid phase gas phase pipe ρs=770kg·m-3 ρg=2.03kg·m-3 D=0.08m ds=60μm μg=1.85×10-5 L=1m

rate near the pipe wall were defined zero.

**4.2.3 Basic parameter in process of simulation** 

Table 2. Basic parameter used for simulation.

**4.3 Numerical simulation process 4.3.1 Equation discretization** 

**4.3.2 Numerical calculation method** 

**4.3.3 Numerical calculation circuitry** 

two single flow and coupling effect between two phases.

The numerical calculation circuitry is shown as below.

Table 2 gave the basic parameter in process of numerical simulation

controlled volume were defined and stored in the determined nodal point.

$$\pi\_w = \begin{cases} \frac{\overline{\mu\_g} k C\_\mu^{0.25} \rho\_g k^{0.5}}{\ln \left( E Y^+ \right)} & Y^+ \ge 11.63 \\\\ \frac{\overline{\mu\_g} \mu\_g}{\delta\_g} & Y^+ \le 11.63 \end{cases} \tag{45}$$

Where *ugi* stands for the gas velocity which is parallel to the pipe axis near the pipe wall. / 5.025.0 *<sup>g</sup> <sup>y</sup> kCY* , *δy* is the distance between calculated nodes and pipe wall. *E*=9.793. 2. Boundary condition for solid phase

Inlet boundary for solid

Homogeneous inlet conditions are set and the volume fraction of particles is given. The expressions of turbulent kinetic energy and turbulent dissipation rate are set as following.

$$k\_s = \frac{\mathfrak{B}}{2} \left(\mu\_s I\_s\right)^2, \mathcal{E}\_s = \mathbb{C}\_{\mu}^{3/4} \frac{k\_s^{3/2}}{l} \tag{46}$$

Where

$$I\_s = 0.16 \left(\text{Re}\_{sD\_H}\right)^{1/8} \tag{47}$$

The solid velocity is set 4.3 m/s based on experiment data.

Outlet boundary for solid

The assumption that the solid phase was the fully developed conditions of the pipe flow, namely the normal derivative of the variables solved is zero.

$$\frac{\partial \phi\_s}{\partial \mathbf{x}\_i} = \mathbf{0} \left( \phi\_s = \mathbf{u}\_{si}, k\_{s\prime}, \mathbf{z}\_s \right) \tag{48}$$

Pipe wall surface

To granule phase, the velocity doesn't agree with no-slip condition, thus the velocity value can't be equal to zero. According to particles collision near pipe wall research, the granule phase normal velocity can be given as following.

$$\left(\left(\lambda\_1-\lambda\_2\right)\left(u\_{si}\right)\_w+\lambda\_1 h \text{Kn}\left(\frac{\partial u\_{si}}{\partial \mathbf{x}\_i}\right)\_w\right)\_w=0\tag{49}$$

where,

$$
\mathcal{A}\_1 = \left(1 - \frac{e^2}{2}\right)^{1/2}, \mathcal{A}\_2 = \left[\frac{e(1-e)}{1+e}\right]^{1/2}
$$

h means the distance between the center point of the first control bulk and the pipe wall. Kn is Knudsen number, which can be given as following.

$$\text{Kn} = \pi\_s \left| \mu\_{gi} - \mu\_{si} \right|\_w = \pi\_s \left| \mu\_{si} \right|\_w \tag{50}$$

It must be noted that, in this paper, the turbulent kinetic energy and turbulent dissipation rate near the pipe wall were defined zero.

## **4.2.3 Basic parameter in process of simulation**

Table 2 gave the basic parameter in process of numerical simulation


Table 2. Basic parameter used for simulation.

### **4.3 Numerical simulation process 4.3.1 Equation discretization**

386 Computational Simulations and Applications

*Y*

*l*

(46)

0.16 Re *s sDH I* (47)

(48)

(49)

(50)

*Y*

11.63

(45)

11.63

0.25 0.5

Where *ugi* stands for the gas velocity which is parallel to the pipe axis near the pipe wall.

*<sup>g</sup> <sup>y</sup> kCY* , *δy* is the distance between calculated nodes and pipe wall. *E*=9.793.

Homogeneous inlet conditions are set and the volume fraction of particles is given. The expressions of turbulent kinetic energy and turbulent dissipation rate are set as following.

> 3/2 <sup>3</sup> <sup>2</sup> 3/4 , <sup>2</sup> *<sup>s</sup> s ss s <sup>k</sup> k uI C*

The assumption that the solid phase was the fully developed conditions of the pipe flow,

0 ,, *<sup>s</sup> s si s s*

To granule phase, the velocity doesn't agree with no-slip condition, thus the velocity value can't be equal to zero. According to particles collision near pipe wall research, the granule

> 12 1 0 *si si <sup>w</sup> <sup>i</sup> <sup>w</sup> <sup>u</sup> u hKn*

> > 1 , 2 1 *e e e*

h means the distance between the center point of the first control bulk and the pipe wall. Kn

*<sup>s</sup> gi si s si <sup>w</sup> <sup>w</sup> Kn u u u*

 

 

 

1 2

 *u k*

*x*

1/2 1/2 <sup>2</sup>

1

*e*

*i*

*x* 

 

ln *gi g*

*u kC k EY*

 

*gi g y*

*u*

 

2. Boundary condition for solid phase Inlet boundary for solid

Outlet boundary for solid

Pipe wall surface

1/8

The solid velocity is set 4.3 m/s based on experiment data.

namely the normal derivative of the variables solved is zero.

phase normal velocity can be given as following.

is Knudsen number, which can be given as following.

 / 5.025.0

Where

where,

*w*

In this paper, Finite Volume Method was utilized to discrete the governing equations above. The selected pipe section was divided into many non-concurrent domains which was called calculating grid. And then, each nodal point of stationary divided domain and its controlled volume were confirmed. In the process of discretization, the physical quantity of this controlled volume were defined and stored in the determined nodal point.
