3.1.2. Model formulation

As the temperature increases, the effect of adhesion force becomes dominant due to surface softening, resulting in a quick defluidization. The adhesive force associated with the plasticviscous flow mechanism can be described by:

$$F\_{\rm ad} = \pi b^2 \sigma \tag{1}$$

particles was considered as the characteristic residence time for which particles within a

Mechanism and Prevention of Agglomeration/Defluidization during Fluidized-Bed Reduction of Iron Ore

where β is a proportional coefficient; Ug is the operating gas velocity; and Umf is the minimum

The surface viscosity of solid is a function of temperature and is assumed be estimated by the

Es

μ<sup>s</sup> ¼ μs<sup>0</sup> exp

where Es is the activation energy for the surface viscosity and T the absolute temperature.

Fd ¼ αCd

To accurately predict the segregate force, this model employs the drag force acting on particles to represent the force against agglomeration, which is related to the effect of the particle size,

> π 8 d2 pρgU<sup>2</sup>

Re <sup>ð</sup><sup>1</sup> <sup>þ</sup> <sup>0</sup>:173Re<sup>0</sup>:<sup>657</sup>Þ þ <sup>0</sup>:<sup>413</sup>

Re <sup>¼</sup> dpρgUg μg

where α is a proportional coefficient, representing the unknown errors in this equation. Cd is the drag coefficient; Re is Reynolds number; ρ<sup>g</sup> and μ<sup>g</sup> are the gas density and viscosity,

> ¼ αCd π 8 d2 pρgU<sup>2</sup>

In this work, we defined two number groups, Na and Nd, representing the adhesion force and

π 8 d2 pρgU<sup>2</sup>

Na <sup>¼</sup> <sup>π</sup>

Nd ¼ Cd

<sup>ð</sup>Ug � <sup>U</sup>mf<sup>Þ</sup> <sup>ð</sup>3<sup>Þ</sup>

http://dx.doi.org/10.5772/intechopen.68488

RT <sup>ð</sup>4<sup>Þ</sup>

<sup>g</sup> ð5Þ

<sup>1</sup> <sup>þ</sup> 16300Re�1:<sup>09</sup> <sup>ð</sup>6<sup>Þ</sup>

<sup>g</sup> ð8Þ

<sup>μ</sup>sdpðUg � <sup>U</sup>mf<sup>Þ</sup> <sup>ð</sup>9<sup>Þ</sup>

<sup>g</sup> ð10Þ

ð7Þ

113

fluidized bed remain in contact with each other as reported by Seville et al. [13, 25]:

fluidization velocity.

respectively.

the drag force, respectively:

Arrhenius' expression [25]:

gas velocity, and gas properties. The expression is [26]:

Cd <sup>¼</sup> <sup>24</sup>

If the adhesive force equals the drag force, the bed is defluidized:

Therefore, the defluidization criterion (Eq. (8)) can be expressed as:

πAβb<sup>2</sup> μsdpðUg � UmfÞ

<sup>t</sup> <sup>¼</sup> <sup>β</sup>

where σ represents the tensile stress of the agglomerate and b is the radius of the connection between the particles.

According to Benson et al. [24], the tensile stress of the agglomerate is

$$
\sigma = \frac{At}{\mu\_s d\_p} \tag{2}
$$

where A is a constant; t represents the connect time of two particles; dp is the mean size of the particle; and μ<sup>s</sup> is the surface viscosity of the particle materials.

In a fluidized bed where particles are intermittently mixed, the contact time of particles is dependent on the bubbles motion. The residence time is required to be sufficiently long for particle connection to form agglomerates. Therefore, in this study, the connect time of two particles was considered as the characteristic residence time for which particles within a fluidized bed remain in contact with each other as reported by Seville et al. [13, 25]:

evolution of forces acting on particles with temperature based on the surface viscosity and bubble motion. By analyzing the experimental data with a statistical regression, a force balance model is developed to describe the defluidization processes in a fluidized bed, by which the

The fluidization behavior of bed particles depended on the forces acting on them. Therefore, this model employed the balance of cohesive and segregate forces to simulate agglomeration/ defluidization and predict the defluidization temperature. Taking account of particle moving, colliding, coalescing, and breaking in a fluidized-bed system, the following assumptions are made to describe the main characteristics of the defluidization phenomena based on the

2. The fluidizing gases do not react with the bed particles, and no coating layer form on the

3. The adhesive force between two particles arises from surface viscosity and is determined

4. The force against agglomeration is the drag force acted on particles due to bubble motion. 5. If the adhesive force equals or exceeds the segregation force, the bed defluidization appears.

As the temperature increases, the effect of adhesion force becomes dominant due to surface softening, resulting in a quick defluidization. The adhesive force associated with the plastic-

<sup>F</sup>ad <sup>¼</sup> <sup>π</sup>b<sup>2</sup>

where σ represents the tensile stress of the agglomerate and b is the radius of the connection

<sup>σ</sup> <sup>¼</sup> At μsdp

where A is a constant; t represents the connect time of two particles; dp is the mean size of the

In a fluidized bed where particles are intermittently mixed, the contact time of particles is dependent on the bubbles motion. The residence time is required to be sufficiently long for particle connection to form agglomerates. Therefore, in this study, the connect time of two

According to Benson et al. [24], the tensile stress of the agglomerate is

particle; and μ<sup>s</sup> is the surface viscosity of the particle materials.

σ ð1Þ

ð2Þ

temperature dependence of the defluidization behavior is predicted.

3.1. Modeling defluidization phenomena

previously described experimental results:

by the plastic-viscous flow mechanism.

viscous flow mechanism can be described by:

1. Bed material particles are spherical and in uniform size.

3.1.1. Assumptions

112 Iron Ores and Iron Oxide Materials

surface.

3.1.2. Model formulation

between the particles.

$$t = \frac{\beta}{(\mathcal{U}\_{\\$} - \mathcal{U}\_{\text{mf}})} \tag{3}$$

where β is a proportional coefficient; Ug is the operating gas velocity; and Umf is the minimum fluidization velocity.

The surface viscosity of solid is a function of temperature and is assumed be estimated by the Arrhenius' expression [25]:

$$
\mu\_s = \mu\_{s0} \exp\left(\frac{E\_s}{RT}\right) \tag{4}
$$

where Es is the activation energy for the surface viscosity and T the absolute temperature.

To accurately predict the segregate force, this model employs the drag force acting on particles to represent the force against agglomeration, which is related to the effect of the particle size, gas velocity, and gas properties. The expression is [26]:

$$F\_d = \alpha \mathbb{C}\_d \frac{\pi}{8} d\_p^2 \rho\_g \mathcal{U}\_g^2 \tag{5}$$

$$\mathbf{C}\_d = \frac{24}{\text{Re}} (1 + 0.173 \text{Re}^{0.657}) + \frac{0.413}{1 + 16300 \text{Re}^{-1.09}} \tag{6}$$

$$\text{Re} = \frac{d\_p \rho\_g \mathcal{U}\_\mathcal{S}}{\mu\_\mathcal{g}} \tag{7}$$

where α is a proportional coefficient, representing the unknown errors in this equation. Cd is the drag coefficient; Re is Reynolds number; ρ<sup>g</sup> and μ<sup>g</sup> are the gas density and viscosity, respectively.

If the adhesive force equals the drag force, the bed is defluidized:

$$\frac{\pi A \beta b^2}{\mu\_s d\_p (\mathcal{U}\_\mathcal{g} - \mathcal{U}\_{\text{mf}})} = \alpha \mathcal{C}\_d \frac{\pi}{8} d\_p^2 \rho\_g \mathcal{U}\_\mathcal{g}^2 \tag{8}$$

In this work, we defined two number groups, Na and Nd, representing the adhesion force and the drag force, respectively:

$$N\_a = \frac{\pi}{\mu\_s d\_p (\mathcal{U}\_\mathcal{S} - \mathcal{U}\_{\text{mf}})} \tag{9}$$

$$N\_d = \mathbb{C}\_d \frac{\pi}{8} d\_p^2 \rho\_g \mathcal{U}\_\mathcal{g}^2 \tag{10}$$

Therefore, the defluidization criterion (Eq. (8)) can be expressed as:

$$N\_a = \mathbf{K} \cdot \mathbf{N}\_d \tag{11}$$

Figure 8, at a constant fluidizing velocity, both the adhesion force and the drag force increase with increasing temperature. However, the increase of adhesion force is much more rapid than that of drag force, especially at the temperatures above the initial sintering temperature. Therefore, at a given temperature, namely, the defluidization temperature, the adhesion force begins to be greater than the drag force, and thus the defluidization appears. This explained the temperature dependence of defluidization behavior. On the other hand, as the gas velocity increases, the drag force of the particles increases, whereas the adhesion force decreases at a constant temperature. As a result, the state of fluidized particles gets out of the defluidization region because the drag force is greater than the adhesion force. Therefore, the temperature to reach defluidization is delayed by increasing the gas velocity. Comparing the calculated defluidization temperature with the experimental data in Figure 9(a), both the tendencies are in a good agreement, although the calculated values are to some extent lower than the

Mechanism and Prevention of Agglomeration/Defluidization during Fluidized-Bed Reduction of Iron Ore

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115

Figure 9(b) presents the effect of gas type on defluidization temperature. According to the calculated results, the defluidization temperature decreases when using the gas with greater viscosity and density as a fluidizing agent. As seen in Figure 9(b), at a constant gas velocity the adhesion force for different gases almost has no change, whereas the drag force is strongly dependent on the gas properties and increases with increasing the gas viscosity. Comparing the three fluidizing gases, the defluidization temperatures are in the following sequence: H2< N2< Ar. This was because the fluidizing gas with greater viscosity can produce a stronger drag force to resist agglomeration, which was in accord with the experimental results [21].

The calculated defluidization temperatures were in a good agreement with the experimental results in all experiment conditions, and thus confirmed the predicted modeling. The model successfully described the defluidization temperature as a function of gas velocity and gas property. According to the results above, the fluidizing phase diagram was obtained as shown in Figure 10, which was divided into the stable fluidization and the defluidization region. The fluidization state was maintained below the curve intersection of Na and K∙Nd, while the bed was defluidized above the intersection. This suggested that at a certain operating parameter,

Figure 9. Comparison of calculated defluidization temperature with experimental data: (a) Influence of gas velocity; and

experimental ones.

3.2.2. Influence of gas properties

(b) influence of gas properties.

$$K = \frac{\alpha}{A b^2 \beta} \tag{12}$$

where K is a regressive constant, representing the unknown errors in this equation. The variables in the model are as a function of temperature, and the correlations are nonlinear. Therefore, by combining Eqs. (9)–(12), the temperature to reach defluidization was obtained by a numerical method.

#### 3.2. Modeling results and comparison with experimental data

#### 3.2.1. Influence of gas velocity

Figure 8 presents the results obtained at different gas velocities. According to the definition of defluidization criterion, the temperature corresponding to the intersection of the curves of Na and K∙Nd is the defluidization temperature. As it can be seen, the temperature to reach defluidization increases with increasing the gas velocity for all the fluidizing gases. In previous studies [27–29], the generation of agglomeration and defluidization depended on the balance of the cohesive and breaking forces. And if the adhesive force between particles exceeded the breakage force, agglomeration and defluidization in the bed probably occur. As shown in

Figure 8. The variation of the calculated values of Na and K∙Nd with temperature: (a) N2; (b) Ar; and (c) H2.

Figure 8, at a constant fluidizing velocity, both the adhesion force and the drag force increase with increasing temperature. However, the increase of adhesion force is much more rapid than that of drag force, especially at the temperatures above the initial sintering temperature. Therefore, at a given temperature, namely, the defluidization temperature, the adhesion force begins to be greater than the drag force, and thus the defluidization appears. This explained the temperature dependence of defluidization behavior. On the other hand, as the gas velocity increases, the drag force of the particles increases, whereas the adhesion force decreases at a constant temperature. As a result, the state of fluidized particles gets out of the defluidization region because the drag force is greater than the adhesion force. Therefore, the temperature to reach defluidization is delayed by increasing the gas velocity. Comparing the calculated defluidization temperature with the experimental data in Figure 9(a), both the tendencies are in a good agreement, although the calculated values are to some extent lower than the experimental ones.
