*2.4. Gasifier Dimensions*

In Figure 4a, a schematic diagram of the bubbling fluidized bed gasifier facility is shown, including the inlet/outlet gas and solid flows, as well as details of their axial position (in mm). These data were used to parametrize the simulation model. Here, . *MRegOut* is the mass flow at the outlet of the regenerator to the gasifier. By tuning this mass flow rate, the desired gasification temperature

can be achieved. The height of the fluidized bed, *HFluidB*, depends on the fluidization velocity and, hence, the distance between the inlet of . *MRegOut* and *HFluidB* is variable.

**Figure 4.** (**a**) Scheme of bubbling fluidized bed gasifier with levels (in mm) of inlet/outlet flows; Cell model of the fluidized bed, (**b**) Gaseous flows, and (**c**) Solid flows.

## *2.5. Mass Balance*

The gasifier is discretized along the reactor height. This includes the bubbling bed as well as the freeboard area. Figure 4b,c show how the calculation with axial discretization in cells proceeds through the fluidized bed for both gaseous and solid components. In each calculation cell of this 1d model, solid and gas components are considered to be fully mixed.

## 2.5.1. Fluidized Bed

According to the discretization shown in Figure 4b, the mass balance for each gaseous component in both dense phase (*P* = *d*) and bubble phase (*P* = *b*) is stated in Equation (16):

$$\frac{dM\_{P,j}^{n}}{dt} = 0 = \dot{M}\_{P,j}^{n-1} - \dot{M}\_{P,j}^{n} + \dot{M}\_{A,P,j}^{n} + \mathcal{M}\mathcal{W}\_{\dot{f}} \cdot \sum\_{I\_{P}} \nu\_{i,\dot{f}} \mathcal{R}\_{P,\dot{I}}^{n} + \dot{\mathcal{M}}\_{in,P,j}^{n} \tag{16}$$

where . *MP*,*<sup>j</sup>* is the convective gas mass flow and . *MA*,*P*,*<sup>j</sup>* is the exchange mass between bubble and suspension phases of component *j* in phase P; *Ri* is the reaction rate of reaction *i* and <sup>ν</sup>*i,j* is the stoichiometric coefficient of component *j* in reaction *i*; *IP* describes the maximum number of reactions taking place in the phase and, with the term . *Min*,*P*,*j*, external inflows (e.g., from a secondary steam inlet) can be considered. Convective mass flows between adjacent calculation cells are defined by [27]

$$\dot{M}\_{d,j}^{n} = \rho\_d^n \mu\_d^n \mathbf{x}\_{d,j}^n \varepsilon\_d^n A\_c^n \tag{17}$$

$$\dot{M}\_{b,j}^{n} = \rho\_b^n \mu\_b^n x\_{b,j}^n \left(1 - \varepsilon\_d^n\right) \mathcal{A}\_c^n \tag{18}$$

The exchange between bubble and suspension phases inside the cell *n* is defined by [27]

$$\dot{M}\_{A,d,j}^{n} = K\_{db}^{n} A\_{db}^{n} \left( \rho\_{b}^{n} \mathbf{x}\_{b,j}^{n} - \rho\_{d}^{n} \mathbf{x}\_{d,j}^{n} \right) \tag{19}$$

*Appl. Sci.* **2020**, *10*, 6136

$$\dot{M}\_{A,b,j}^{n} = K\_{db}^{n} A\_{db}^{n} \left( \rho\_{d}^{n} \mathbf{x}\_{d,j}^{n} - \rho\_{b}^{n} \mathbf{x}\_{b,j}^{n} \right) \tag{20}$$

According to Hilligardt [26], the mass transfer coefficient between the bubble and suspension phase is calculated as *Kndb* = 2.7*und* 4 , and the mass exchange area over all bubbles in the cell *n* is *Andb* = <sup>6</sup>ε*nb* ·*Ac*·*dh*/*dnb* . Additionally, overall mass balances for the suspension (*P* = *d*) and bubble (*P* = *b*) phases are developed in Equation (21), including the molar weight *MWj* of each component *j*:

$$\frac{dM\_P^n}{dt} = 0 = \dot{M\_P}^{n-1} - \dot{M\_P}^n + \sum\_{\uparrow} \dot{M\_{A,P,j}} + \sum\_{\uparrow} \sum\_{l\_P} \text{MW}\_{\uparrow} \nu\_{i,j} \text{R}\_i + \dot{M\_{in,P}}.\tag{21}$$

Beside a balance for the gaseous components, a separate balance equation (Equation (22)) exists for the solids. According to Figure 4c, the amount of each solid component *k* in a calculation cell *n* is considered:

$$\frac{dM\_k^n}{dt} = 0 = \dot{M}\_{dr,k}^{n-1} + (1-a)\dot{M}\_{w,k}^{n-1} + \dot{M}\_k^{n+1} - \dot{M}\_{dr,k}^n - \dot{M}\_{w1,k}^n - \dot{M}\_k^n + M\mathcal{W}\_j \cdot \sum\_{l\_d} \nu\_{i,k} R\_{d,k}^n + \dot{M}\_{in,k}^n \tag{22}$$

The cell adjacent to the freeboard (*n* = *N*) additionally includes a sink term for elutriated fine particles . *Melut*. It is important to split the mass balance into a description of gaseous and solid components as, in real plant operations, there exists a downward flow from the surface of the fluid bed (*HFluidB*) to the bottom (leaving the gasifier through the loop seal) and an upward flow due to the wake and drift of each rising gas bubble. Here, the term wake ( . *Mw*) describes solids that fasten to the bubbles on their way upwards and drift ( . *Mdr*) refers to solids that are loosely drawn upwards through the bubble movement. The solid transport by bubbles, together with the conical asymmetric cross section, see Figure 4a, in the lower part of the fluidized bed, leads to the strong exchange and motion of solids. To consider these effects, a model was developed in which the drift part of a rising bubble is fully mixed in each cell, whereas the proportion of wake that is mixed with or bypasses each cell can be chosen using the parameter α. The Term ⎛⎜⎜⎜⎜⎝*MWj*· *Id* <sup>ν</sup>*i*,*<sup>k</sup>Rnd*,*<sup>k</sup>*⎞⎟⎟⎟⎟⎠ considers chemical reactions and with the term ( . *M n in*,*<sup>k</sup>*), inflows (e.g., from solid circulation) are included in the equation. The amount of solids which are transported with each bubble can be described by the empirical Equations (23) and (24) mentioned in [27]:

$$\dot{M}\_{\text{\textquotedblleft}P} = A\_{\text{\textquotedblleft}P}(1 - \varepsilon\_d) \varepsilon\_b u\_{b^\dagger}^\* \left[ 0.59 - 0.046 \ln(Ar) \right] \tag{23}$$

$$\dot{M}\_{dr} = A\_{\varepsilon} \rho\_P (1 - \varepsilon\_d) \varepsilon\_b \cdot 0.38 u\_b^\* \cdot [1.5 - 0.135 \ln(Ar)] \tag{24}$$

In Figure 4c, the concept is illustrated to adapt the 1d model for gasification experiments with this highly three-dimensional behavior of the real system, through the adjustable parameter α.
