# A Detailed One-Dimensional Hydrodynamic and Kinetic Model for Sorption Enhanced Gasification
## 2 SEG Process Description and Model Development
### Mass Balance
#### 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):  
\[\frac{dM_{P,j}^{n}}{dt}=0=\dot{M}_{P,j}^{n-1}-\dot{M}_{P,j}^{n}+\dot{M}_{A,P,j} ^{n}+MW_{j}\sum_{Ip}\
u_{ij}R_{P,i}^{n}+\dot{M}_{in,P,j}^{n} \tag{16}\]  
where \(\dot{M}_{P,j}\) is the convective gas mass flow and \(\dot{M}_{A,P,j}\) is the exchange mass between bubble and suspension phases of component \(j\) in phase P; \(R_{i}\) is the reaction rate of reaction \(i\) and \(\
u_{ij}\) is the stoichiometric coefficient of component \(j\) in reaction \(i\); \(I_{P}\) describes the maximum number of reactions taking place in the phase and, with the term \(\dot{M}_{in,P,j}\) external inflows (e.g., from a secondary steam inlet) can be considered. Convective mass flows between adjacent calculation cells are defined by [27]  
\[\dot{M}_{d,j}^{n}=\rho_{d}^{n}u_{d}^{n}x_{d,j}^{n}e_{d}^{n}A_{c}^{n} \tag{17}\]  
\[\dot{M}_{b,j}^{n}=\rho_{b}^{n}u_{b}^{n}x_{b,j}^{n}\Big{(}1-\varepsilon_{d}^{n} \Big{)}A_{c}^{n} \tag{18}\]  
The exchange between bubble and suspension phases inside the cell \(n\) is defined by [27]  
\[\dot{M}_{A,d,j}^{n}=K_{db}^{n}A_{db}^{n}\Big{(}\rho_{b}^{n}x_{b,j}^{n}-\rho_{d }^{n}x_{d,j}^{n}\Big{)} \tag{19}\]  
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. \[\dot{M}_{A,b,j}^{n}=K_{db}^{n}A_{db}^{n}\left(\rho_{d}^{n}x_{d,j}^{n}-\rho_{b}^{n }x_{b,j}^{n}\right) \tag{20}\]  
According to [PERSON] (2016), the mass transfer coefficient between the bubble and suspension phase is calculated as \(K_{db}^{n}=\frac{2.7u_{d}^{n}}{4}\), and the mass exchange area over all bubbles in the cell \(n\) is \(A_{db}^{n}=6\varepsilon_{b}^{n}.A_{c}.dh/d_{b}^{n}\). Additionally, overall mass balances for the suspension (\(P=d\)) and bubble (\(P=b\)) phases are developed in Equation (21), including the molar weight \(MW_{j}\) of each component \(j\):  
\[\frac{dM_{P}^{n}}{dt}=0=\dot{M}_{P}^{n-1}-\dot{M}_{P}^{n}+\sum_{j}\dot{M}_{A,P,j}^{n}+\sum_{j}\sum_{l_{P}}MW_{j}v_{i,j}R_{i}+\dot{M}_{in,P}^{n}. \tag{21}\]  
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:  
\[\frac{dM_{k}^{n}}{dt}=0=\dot{M}_{dr,k}^{n-1}+(1-\alpha)\dot{M}_{w,k}^{n-1}+ \dot{M}_{k}^{n+1}-\dot{M}_{dr,k}^{n}-\dot{M}_{w1,k}^{n}-\dot{M}_{k}^{n}+MW_{j} \sum_{l_{L}}v_{i,k}R_{d,k}^{n}+\dot{M}_{in,k}^{n} \tag{22}\]  
The cell adjacent to the freeboard (\(n=N\)) additionally includes a sink term for elutriated fine particles \(\dot{M}_{elut}\). 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 (\(H_{FluidB}\)) 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 (\(\dot{M}_{w}\)) describes solids that fasten to the bubbles on their way upwards and drift (\(\dot{M}_{dr}\)) 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 \(\alpha\). The Term (\(MW_{j}\cdot\sum\limits_{l_{d}}v_{i,k}R_{d,k}^{n}\)) considers chemical reactions and with the term (\(\dot{M}_{in,k}^{n}\)), inflows (e.g., from solid circulation) are included in the equation.