*2.3. Hydrodynamics*

For modeling purposes, the fluidized bed was split into two phases, a solid-free bubble phase (fraction: <sup>ε</sup>*b*) and a solid-loaded dense phase (fraction: 1-<sup>ε</sup>*b*), as shown in Figure 3.

**Figure 3.** Simplified reactor scheme of the modeled process.

## 2.3.1. Dense Phase

In the model, the dense phase is considered to be a perfused pack with porosity ε*d* and gas velocity *ud*. The calculation of *ud* and ε*d* is performed following the methods presented by Hilligardt [26]. The minimum fluidization gas velocity, *umf*, is determined using the Ergun equation (Equation (1)), which is formulated based on the definitions of the Archimedes and Reynolds numbers, as well as the

Sauter diameter (Equations (2)–(4)). In the present work, a sphericity of ψ = 0.75 [27] and a porosity at minimum fluidization <sup>ε</sup>*mf* = 0.45 [27] are assumed.

$$Ar = 150 \frac{1 - \varepsilon\_{mf}}{\psi^2 \cdot \varepsilon\_{mf}^3} Re\_{mf} + \frac{1.75}{\psi^\* \varepsilon\_{mf}^3} Re\_{mf}^2 \tag{1}$$

$$Ar = \frac{\mathcal{S} \cdot d\_{sv}^3}{\nu\_{\mathcal{S}}^2} \frac{\rho\_{\mathcal{P}} - \rho\_{\mathcal{S}}}{\rho\_{\mathcal{S}}} \tag{2}$$

$$Re\_{mf} = \frac{d\_{sv} \cdot u\_{mf}}{\nu\_{\mathcal{S}}} \tag{3}$$

$$d\_{\rm sv} = \sqrt{\varphi} \cdot \overline{d\_p} \tag{4}$$

Analysis of the fluidized bed inventory showed a mean particle size, *dp*, of 350μm [11]. According to Hilligardt [26], the real velocity in the dense phase is higher than the calculated minimum fluidization gas velocity, *umf*, and it can be estimated with the following empirical equation:

$$\|u\_d\|\_{\mathbb{H}=0} = \|u\_{mf} + \frac{1}{4}(u\_{empty}|\_{\mathbb{H}=0} - |u\_{mf})\tag{5}$$

For the remaining reactor heights *ud*(*h*) is determined by the continuity equation. The porosity ε*d* is calculated, as proposed by Richardson and Zaki [28], as

$$
\varepsilon\_d(h) = \varepsilon\_{mf} \left(\frac{\mu\_d(h)}{\mu\_{mf}}\right)^{\frac{1}{\mu\_{Rx}}}.\tag{6}
$$

For the parameter *nRz*, Richardson and Zaki [28] provided the empirical equation:

$$m\_{Rz} = \begin{cases} 4.65 & \text{if } \operatorname{Re}\_{\sharp} \le 0.2\\ 4.4 \cdot \operatorname{Re}\_{\sharp}^{-0.03} & \text{if } 0.2 \le \operatorname{Re}\_{\sharp} \le 1\\ 4.4 \cdot \operatorname{Re}\_{\sharp}^{-0.1} & \text{if } 1 \le \operatorname{Re}\_{\sharp} \le 500\\ 2.4 & \text{if } \operatorname{Re}\_{\sharp} > 500 \end{cases} \tag{7}$$

where*Res* is the Reynolds number from the rate of descent of a single particle. In this work, the parameter *nRz* is used to adjust the calculated bed height to the experimental values. With *nRz* = 5.5, the model could be fitted to the real bed height determined experimentally.

## 2.3.2. Bubble Phase

Werther [29] developed a model (Equation (8)) to determine the bubble diameter, *db*, depending on the height of the fluidized bed, taking into account the coalescence and separation of the bubbles:

$$\frac{d(d\_b)}{dh} = \left(\frac{2\varepsilon\_b}{9\pi}\right)^{1/3} - \frac{d\_b}{3 \cdot 280 \frac{u\_{mf}}{\mathcal{S}} \cdot u\_b^\*(h)}\tag{8}$$

At the position *h* = 0, the initial bubble diameter is calculated using the correlation *db* = 1.3 .*<sup>V</sup>*2*steam*/*g*0.2, according to Tepper [27] and Davidson [30]. The initial bubble diameter depends on the volume flow of steam, . *Vsteam*, through an orifice of a gas distributor. The ascent velocity, *u*∗*b*, of a bubble can be determined by Equations (9)–(11) from Hilligardt [26] and Tepper [27]:

$$
u\_b^\* = \sqrt{\frac{4gd\_b}{3C\_{D,b}}}\tag{9}$$

$$C\_{D,b} = \frac{16}{Rc\_b} + 2.64\tag{10}$$

$$Re\_b = \frac{d\_b \cdot u\_b^\*}{\nu\_d} \text{ with } \nu\_d = 60 \cdot d\_{sv}^{3/2} \cdot \text{g}^{1/2}. \tag{11}$$

The parameter *CD,b* is the drag coefficient of a single bubble and ν*d* is the viscosity of the suspension phase. However, the ascent velocity *u*∗*b* does not equal the gas velocity in the bubble phase *ub*, as the bubbles are additionally perfused by gas streams coming from the suspension phase [26,27]. Following the method proposed by Hilligardt [26], the gas velocity *ub* can be determined using the following empirical correlation:

$$u\_b = u\_b^\* + 2.7u\_d\tag{12}$$

#### 2.3.3. Fluidized Bed Height

With a defined inventory of the fluidized bed, *MFluidB*, the height of the fluidized bed, *HFluidB*, can be calculated by integrating the solid mass along the axial co-ordinate *h* [27]:

$$M\_{\rm FluidB} = \int\_0^{H\_{\rm fluidB}} (1 - \varepsilon\_b)(1 - \varepsilon\_d)\rho\_p A\_d d\mathbf{h} \tag{13}$$

## 2.3.4. Elutriation Rate

When gas bubbles rise to the surface of the fluidized bed and break, solid particles are thrown into the freeboard and entrained by the upward gas volume flow [31]. While the major fraction of these particles fall back into the fluidized bed, small particles whose terminal velocity is lower than the gas velocity are elutriated from the freeboard [31].

In the 200 kW facility, a cyclone is installed at the exit of the freeboard to reduce the extraction of bed inventory by leading the particles back into the fluidized bed [11]. It is assumed that there is an additional CO2 capture effect in the freeboard: Due to lower freeboard temperatures, the position of the chemical equilibrium can be changed, which enables a further carbonation reaction. However, as bigger particles fall or are transferred back into the fluidized bed by the cyclone, it is assumed that only fine particles have a contribution to the additional carbonation reaction. According to [31], the elutriation rate of particles (in g/s) is described by

.

$$M\_{dlat} = k\_{elut} x\_{fime} \tag{14}$$

Herein, *kelut* is the elutriation rate constant and the weight fraction of fine particles, *xfine*, present in the bed was identified by measurements at the 200 kW facility. With a secondary cyclone, a mean particle size *dp* of 25 μm of elutriated particles was found. Considering the particle size distribution of the raw limestone, it can be derived that fine particles do not originate from the make-up of the raw material. Thus, the source of these particles must be attrition or fragmentation effects. From experiments, a correlation was derived to calculate the weight fraction of the fine particles:

$$x\_{fine} = a\_1 \tanh\left(\left(\mu - a\_2\right)/a\_3\right) + a\_4. \tag{15}$$

with the parameters *a*1 = −0.12697, *a*2 = 0.71214, *a*3 = −0.01191, and *a*4 = 0.12807.
