**2. The Eulerian two-fluid model**

Most literatures give tribute to the paper by Davidson (1961) as the first to apply the concept of hydrodynamic model in fluidized beds in his analysis of a single isolated bubble rising in an unbounded fluidized bed. However, Anderson & Jackson (1967) were the first to formulate the complete CFD TFM for gas-solid multiphase flows in the mid 60's. Since then many have made significant efforts to develop detailed microbalance models to study the complex hydrodynamics of gas-fluidized beds (e.g., Gidaspow, 1994; Enwald et al., 1996; Kuipers & van Swaaij, 1998).

Owing to the continuum description of the particulate suspension, the TFM requires additional closure laws for the solid rheology. Two of the most important transport

Numerical Simulation of Dense

The volume fractions are related as:

డ൫ఌఘ൯

**3.2 Interphase momentum transfer** 

Gidaspow (1994) was used in this work.

ߚ ൌ <sup>ଷ</sup> <sup>ସ</sup> ܥௗ

Where,

ߚ ൌ ͳͷͲ ൫ଵିఌ൯

The particle Reynolds number is given by:

**3.3 Kinetic theory of granular flow** 

మ ఌ

ఌ൫ଵିఌ൯ ௗ

ܥௗ ൌ ൝ ଶସ ோ

ఓ ൫ௗ൯

<sup>మ</sup> ͳǤͷ൫ͳ െ ߝ൯

ߝ௦หݑ െ ݑหߩ

ቂͳ ͲǤͳͷ൫ܴ݁൯

ఘ ௗ

ͲǤͶͶǡ ܴ݁ ͳͲͲͲ

ܴ݁ ൌ ఌఘห௨ି௨ೞหௗ ఓ

Almost all recent TFM of gas-solid systems used the Kinetic Theory of Granular Flow (KTGF) principle to derive the constitutive equations to describe the rheology of the particulate phase, i.e., the particulate phase viscosity and the particulate phase pressure gradient. The KTGF is basically an extension of the classical kinetic theories of non-uniform gases as described by Chapman & Cowling (1970). It was first applied to granular flows by Jenkins & Savage (1983) and Lun et al. (1984). Later Sinclair & Jackson (1989) applied this theory to model gas-solid flow in a pipe. The model was further developed and applied to dense gas-solid fluidized beds by Ding & Gidaspow (1990) and Gidaspow (1994). The

Ǥ଼ቃǡ ܴ݁ ͳͲͲͲ

డሺఌೞఘೞೞሻ

Gas-Solid Multiphase Flows Using Eulerian-Eulerian Two-Fluid Model 239

In the TFM the two phase are coupled through the interphase momentum transfer, hence it is one of the most important and dominant force in modelling gas-solid systems. The drag force acting on a particle in fluid–solid systems is generally represented by the product of a momentum transfer coefficient and the slip velocity (ug−us) between the two phases. Numerous empirical correlations for calculating the momentum transfer coefficient, β, of gas-solid systems have been reported in the literature. These have been compared and validated by different researchers before, e.g. van Wachem et al. (2001), Taghipour et al. (2005), and Vejahati et al. (2009). All these researchers reported that the different drag models available gave quantitatively similar predictions of the macroscopic bed characteristic and bubble properties. As a result, the most commonly used drag model of

(4) ߩߝ௦൯ െ ൫ߚ െ ܲߘߝ൯െ൫ ή ߘ ൌ ൯ߩߝ൫ ή ߘ డ௧

డ௧ ߘή ሺߝ௦ߩ௦௦௦ሻ ൌߘή ሺ௦ሻ െ ߝ௦ߘܲ െ ߘܲ௦ ߚ൫ െ ௦൯ߝ௦ߩ௦) 5 (

The conservation of momentum for the gas and the solids phase are described by:

ߝ௦ ߝ ൌ ͳ (3)

หݑ െ ݑ௦หǡ݂݅ߝ ͲǤͺ (6)

(9)

(8)

ିଶǤହǡ݂݅ߝ ͲǤͺ (7)

variables that appear in the momentum equation of the solid are the solids stress tensor and solids pressure. These variables depend strongly on the collisional behaviour of the individual particles, hence difficult to express. So far, there are two types of approaches to treat these variables. The first one is commonly known as the Constant Viscosity Model (CVM) and was applied by many of the early TFM computer simulations (e.g. Gidaspow & Ettehadieh, 1983; Tsuo & Gidaspow, 1990; Kuipers et al., 1992, 1993; Enwald et al., 1996). This approach assumes a constant value for the solids viscosity obtained from some experimental and empirical correlations. The solids phase pressure, which prevents particles from reaching impossibly low values of void fraction, was assumed to depend on the solid volume fraction and it is determined from experiments. The advantage of this model is its simplicity and thus easy to implement in a computer codes. However, it is difficult to take into account the underlying characteristics of the solid phase rheology due to mutual particle collisions. The second approach makes use of the Kinetic Theory of Granular Flow (KTGF) and develops some relations as a function of the particle velocity and position (Ding & Gidaspow, 1990). One of the advantages of the KTGF is that it can give a more fundamental insight of the particle-particle interactions. Detail explanation of this model is presented in section 3.3 below.

Over the past 20 years, a large number of researchers have devoted significant effort to apply and validate the TFM based on the KTGF for different flow regimes and particle classes. For example, Boemer et al. (1997, 1998), van Wachem et al. (1998, 1999, 2001), Hulme et al. (2005), Patil et al. (2005), Lindborg et al. (2007) have been devoted to validate the model for bubble behaviour in gas-solid fluidized beds. The majority of these and other validation works are only relevant for beds without immersed tubes and to date little has been done to validate the TFM for fluidized beds with immersed obstacles. Those who performed numerical simulation using the TFM for beds with immersed tubes are mainly limited to beds with single or few tubes. Moreover, their validations involved mainly qualitative comparisons such as voidage distribution and solid circulation near the tube surface in an attempt to investigate the heat transfer coefficient or erosion characteristics of the tubes (Bouillard et al., 1989; Gamwo et al., 1999; Gustavson & Almstedt, 2000; Yurong et al., 2004; Schmidt & Renz, 2005; Gao et al., 2007). There are also attempts in validating the TFM using time-averaged bubble properties (Das Sharma & Mohan, 2003; Asegehegn et al., 2011a). Nevertheless, these are limited to only few immersed horizontal tubes.

In this chapter of the book, numerical simulations of gas-solid fluidized beds were performed using the granular TFM for beds without and with dense immersed tubes. The results of bubble properties were thoroughly analyzed and validated with experimental results obtained from pseudo-2D bed. Moreover, comparisons between 2D and 3D simulations were performed.

#### **3. Numerical modelling using the Granular two-fluid model**

#### **3.1 Governing equations**

The conservation of mass for both the gas and the solids phase can be written as:

$$\frac{\partial (\varepsilon\_g \rho\_g)}{\partial t} + \nabla \cdot \left(\varepsilon\_g \rho\_g \mathbf{u}\_g\right) = 0 \tag{1}$$

$$\frac{\partial(\varepsilon\_s \rho\_s)}{\partial t} + \nabla \cdot (\varepsilon\_s \rho\_s \mathbf{u}\_s) = 0 \tag{2}$$

The volume fractions are related as:

238 Computational Simulations and Applications

variables that appear in the momentum equation of the solid are the solids stress tensor and solids pressure. These variables depend strongly on the collisional behaviour of the individual particles, hence difficult to express. So far, there are two types of approaches to treat these variables. The first one is commonly known as the Constant Viscosity Model (CVM) and was applied by many of the early TFM computer simulations (e.g. Gidaspow & Ettehadieh, 1983; Tsuo & Gidaspow, 1990; Kuipers et al., 1992, 1993; Enwald et al., 1996). This approach assumes a constant value for the solids viscosity obtained from some experimental and empirical correlations. The solids phase pressure, which prevents particles from reaching impossibly low values of void fraction, was assumed to depend on the solid volume fraction and it is determined from experiments. The advantage of this model is its simplicity and thus easy to implement in a computer codes. However, it is difficult to take into account the underlying characteristics of the solid phase rheology due to mutual particle collisions. The second approach makes use of the Kinetic Theory of Granular Flow (KTGF) and develops some relations as a function of the particle velocity and position (Ding & Gidaspow, 1990). One of the advantages of the KTGF is that it can give a more fundamental insight of the particle-particle interactions. Detail explanation of this model is

Over the past 20 years, a large number of researchers have devoted significant effort to apply and validate the TFM based on the KTGF for different flow regimes and particle classes. For example, Boemer et al. (1997, 1998), van Wachem et al. (1998, 1999, 2001), Hulme et al. (2005), Patil et al. (2005), Lindborg et al. (2007) have been devoted to validate the model for bubble behaviour in gas-solid fluidized beds. The majority of these and other validation works are only relevant for beds without immersed tubes and to date little has been done to validate the TFM for fluidized beds with immersed obstacles. Those who performed numerical simulation using the TFM for beds with immersed tubes are mainly limited to beds with single or few tubes. Moreover, their validations involved mainly qualitative comparisons such as voidage distribution and solid circulation near the tube surface in an attempt to investigate the heat transfer coefficient or erosion characteristics of the tubes (Bouillard et al., 1989; Gamwo et al., 1999; Gustavson & Almstedt, 2000; Yurong et al., 2004; Schmidt & Renz, 2005; Gao et al., 2007). There are also attempts in validating the TFM using time-averaged bubble properties (Das Sharma & Mohan, 2003; Asegehegn et al., 2011a).

In this chapter of the book, numerical simulations of gas-solid fluidized beds were performed using the granular TFM for beds without and with dense immersed tubes. The results of bubble properties were thoroughly analyzed and validated with experimental results obtained from pseudo-2D bed. Moreover, comparisons between 2D and 3D

(1) ൯ൌͲߩߝ൫ ή ߘ డ௧

డ௧ ߘή ሺߝ௦ߩ௦௦ሻ ൌ Ͳ (2)

Nevertheless, these are limited to only few immersed horizontal tubes.

**3. Numerical modelling using the Granular two-fluid model** 

డ൫ఌఘ൯

డሺఌೞఘೞሻ

The conservation of mass for both the gas and the solids phase can be written as:

presented in section 3.3 below.

simulations were performed.

**3.1 Governing equations** 

$$
\varepsilon\_s + \varepsilon\_g = 1 \tag{3}
$$

The conservation of momentum for the gas and the solids phase are described by:

$$\frac{\partial(\varepsilon\_{g}\rho\_{g}\mathbf{u}\_{g})}{\partial t} + \nabla \cdot \left(\varepsilon\_{g}\rho\_{g}\mathbf{u}\_{g}\mathbf{u}\_{g}\right) = \nabla \cdot \left(\mathbf{r}\_{g}\right) - \varepsilon\_{g}\nabla P - \beta\left(\mathbf{u}\_{g} - \mathbf{u}\_{s}\right) + \varepsilon\_{g}\rho\_{g}\mathbf{g} \tag{4}$$

$$\frac{\partial(\varepsilon\_{s}\rho\_{s}\mathbf{u}\_{s})}{\partial t} + \nabla \cdot (\varepsilon\_{s}\rho\_{s}\mathbf{u}\_{s}\mathbf{u}\_{s}) = \nabla \cdot (\mathbf{r}\_{s}) - \varepsilon\_{s}\nabla P - \nabla P\_{s} + \beta \left(\mathbf{u}\_{g} - \mathbf{u}\_{s}\right) + \varepsilon\_{s}\rho\_{s}\mathbf{g} \tag{5}$$

#### **3.2 Interphase momentum transfer**

In the TFM the two phase are coupled through the interphase momentum transfer, hence it is one of the most important and dominant force in modelling gas-solid systems. The drag force acting on a particle in fluid–solid systems is generally represented by the product of a momentum transfer coefficient and the slip velocity (ug−us) between the two phases. Numerous empirical correlations for calculating the momentum transfer coefficient, β, of gas-solid systems have been reported in the literature. These have been compared and validated by different researchers before, e.g. van Wachem et al. (2001), Taghipour et al. (2005), and Vejahati et al. (2009). All these researchers reported that the different drag models available gave quantitatively similar predictions of the macroscopic bed characteristic and bubble properties. As a result, the most commonly used drag model of Gidaspow (1994) was used in this work.

$$\beta = 150 \frac{\left(1 - \varepsilon\_{\theta}\right)^{2}}{\varepsilon\_{\theta}} \frac{\mu\_{\theta}}{\left(d\_{p}\right)^{2}} + 1.75 \{1 - \varepsilon\_{\theta}\} \frac{\rho\_{\theta}}{d\_{p}} \left|u\_{\theta} - u\_{s}\right|, \quad \text{if} \quad \varepsilon\_{\theta} \le 0.8 \tag{6}$$

ߚ ൌ <sup>ଷ</sup> <sup>ସ</sup> ܥௗ ఌ൫ଵିఌ൯ ௗ ߝ௦หݑ െ ݑหߩ ିଶǤହǡ݂݅ߝ ͲǤͺ (7)

Where,

$$\mathcal{C}\_d = \begin{cases} \frac{24}{Re\_p} \left[ 1 + 0.15 \left( Re\_p \right)^{0.697} \right] & \text{ , } Re\_p \le 1000\\ 0.44 & \text{ , } Re\_p > 1000 \end{cases} \tag{8}$$

The particle Reynolds number is given by:

$$Re\_p = \frac{\varepsilon\_g \rho\_g |u\_g - u\_s| d\_p}{\mu\_g} \tag{9}$$

#### **3.3 Kinetic theory of granular flow**

Almost all recent TFM of gas-solid systems used the Kinetic Theory of Granular Flow (KTGF) principle to derive the constitutive equations to describe the rheology of the particulate phase, i.e., the particulate phase viscosity and the particulate phase pressure gradient. The KTGF is basically an extension of the classical kinetic theories of non-uniform gases as described by Chapman & Cowling (1970). It was first applied to granular flows by Jenkins & Savage (1983) and Lun et al. (1984). Later Sinclair & Jackson (1989) applied this theory to model gas-solid flow in a pipe. The model was further developed and applied to dense gas-solid fluidized beds by Ding & Gidaspow (1990) and Gidaspow (1994). The

$$
\Theta = \frac{1}{3}u'^2\tag{10}
$$

$$\frac{1}{2}\frac{3}{2}\left(\frac{\partial(\varepsilon\_{s}\rho\_{s}\theta)}{\partial t}+\nabla\cdot(\varepsilon\_{s}\rho\_{s}u\_{s}\Theta)\right)=(-P\_{\sf s}I+\tau\_{\sf s})\colon\nabla u\_{\sf s}-\nabla\cdot q-\chi\_{\sf s}-f\_{\sf s} \tag{11}$$

$$0 = \{-P\_{\rm S}I + \tau\_{\rm s}\}; \mathcal{V}u\_{\rm s} - \chi\_{\rm s} \tag{12}$$

$$\boldsymbol{\sigma}\_{g} = -\boldsymbol{\varepsilon}\_{g} \left[ \left( \boldsymbol{\xi}\_{g} - \frac{2}{3} \boldsymbol{\mu}\_{g} \right) \left( \boldsymbol{\nabla} \cdot \boldsymbol{\mathfrak{u}}\_{g} \right) \boldsymbol{I} + \boldsymbol{\mu}\_{g} \left( \left( \boldsymbol{\nabla} \boldsymbol{\mathfrak{u}}\_{g} \right) + \left( \boldsymbol{\nabla} \boldsymbol{\mathfrak{u}}\_{g} \right)^{\boldsymbol{\tau}} \right) \right] \tag{13}$$

$$\sigma\_s = -\varepsilon\_s \left[ \left( \xi\_s - \frac{2}{3} \mu\_s \right) (\nabla \cdot \mathbf{u}\_s) I \right.\\ \left. + \mu\_s \left( (\nabla \mathbf{u}\_s) + (\nabla \mathbf{u}\_s)^{\mathsf{T}} \right) \right] \tag{14}$$

$$
\xi\_s = \frac{4}{3} \varepsilon\_s \rho\_s d\_p g\_o (1+e) \sqrt{\theta/\pi} \tag{15}
$$

$$
\mu\_{\rm s,KTGF} = \mu\_{\rm s,col} + \mu\_{\rm s,kin} \tag{16}
$$

$$
\mu\_{s,col} = \frac{4}{5} \varepsilon\_s \rho\_s d\_p g\_o (1+e) \sqrt{\Theta/\pi} \tag{17}
$$

$$
\mu\_{\rm s,kin} = \frac{1}{15} \sqrt{\theta \pi} \rho\_{\rm s} d\_{\rm p} g\_0 \varepsilon\_{\rm s}^2 (1+e) + \frac{1}{16} \sqrt{\theta \pi} \rho\_{\rm s} d\_{\rm p} \varepsilon\_{\rm s} + \frac{10}{96} \sqrt{\theta \pi} \frac{\rho\_{\rm s} d\_{\rm p}}{g\_0 (1+e)} \tag{18}
$$

$$\gamma\_s = 12(1 - e^2) \frac{\varepsilon\_s^2 \rho\_s g\_0}{d\_p \sqrt{\pi}} \Theta^{3/2} \tag{19}$$

$$g\_0 = 1 + 4\varepsilon\_s \left\{ \frac{1 + 2.5\varepsilon\_s + 4.5904\varepsilon\_s^2 + 4.515439\varepsilon\_s^3}{\left[1 - \left(\frac{\varepsilon\_s}{\varepsilon\_{s,\max}}\right)^3\right]^{0.67002}} \right\} \tag{20}$$

$$P\_{\rm s,KTGF} = \varepsilon\_{\rm s} \rho\_{\rm s} \Theta + 2g\_o \varepsilon\_{\rm s}^2 \rho\_{\rm s} \Theta (1+e) \tag{21}$$

$$
\mu\_{\rm s} = \mu\_{\rm s,KTGF} + \mu\_{\rm s,f} \tag{22}
$$

$$P\_S = P\_{S,KTGF} + P\_{S,f} \tag{23}$$

$$
\mu\_{s,f} = \frac{p\_s \sin \phi}{2\sqrt{I\_{2D}}} \tag{24}
$$

$$P\_{s,f} = Fr \frac{\left(\varepsilon\_s - \varepsilon\_{s,m\&n}\right)^n}{\left(\varepsilon\_{s,max} - \varepsilon\_s\right)^p} \tag{25}$$

$$
\pi\_{\mathcal{W}} = \frac{\pi}{6} \sqrt{3} \phi' \frac{\varepsilon\_{\varepsilon}}{\varepsilon\_{\varepsilon, \max}} \rho\_{\text{s}} g\_0 \sqrt{\Theta} u\_{\text{slip}} \tag{26}
$$


Numerical Simulation of Dense

Fig. 2. Bubble dimensions.

**5. Results and discussions** 

by the CFD software, Fig. 3.

The bubble aspect ratio, AR, is defined as:

Gas-Solid Multiphase Flows Using Eulerian-Eulerian Two-Fluid Model 245

The rise velocity was calculated from the difference in the vertical-coordinate of the centriod between consecutive time frames and dividing by the time interval between the frames.

Where yg is the vertical component of the centre of gravity of the bubble, t is the time and t is the time delay between consecutive frames of the images, 1/50 s in this case. The velocity

Once the instantaneous bubble properties at each section of the bed are calculated, a number averaging was used to calculate the time-averaged bubble properties with bed height.

Where is any of the bubble property such as aspect ratio, diameter, rise velocity, and N is total number of bubble properties recorded during the total averaging time considered.

All simulations were performed for 20 s of real flow time and the first 5 s were neglected to reduce the start-up effect. Thus, the results reported were averaged over the last 15 s of real flow time. Bubble properties were calculated from the volume fraction contours produced

These volume fraction contours were then analysed by the in-house code. The first step in analysing bubble properties is to discriminate the bubble from the rest of the bed. This was done by setting a solid volume fraction cut-off point to produce discriminated volume

� ���� ��� �

Where dy and dx are the vertical and horizontal extremes shown in Fig. 2.

**dB**

The bubble diameter was calculated from the area equivalent AB as:

is attributed to the mean vertical height according to:

�� � d�⁄d� (27)

�� � ����⁄� (28)

�� � ����t � �t� � ���t�� �t ⁄ (29)

� � ����t � �t� � ���t�� �⁄ (30)

��� ⁄� (31)

Fig. 1. Bed geometries: left - without immersed tubes (NT) and right with staggered tube arrangement (S6). All dimensions are in mm.
