## 2 Numerical Model
### Model Setup
#### 2.2.1 Sub-Model of the Reactive-Bed Domain  
The reactive bulk of solid particles is assumed as a porous medium at the the scale of a representative elementary volume (REV) [33; 34]. Balance equations are formulated for each chemical component, that is, CaO, Ca(OH)2 and H2O in terms of molar fractions for water, and volume fractions for the solid components, respectively. By balancing both solid components separately, the model is able to represent the change in solid volume and, thus, in porosity due to the chemical reaction. This is in contrast to the model described in Reference [20], where the solid volume is constant. The importance of taking the solid-volume change into account is described in detail in Reference [28] for a  
Figure 1: Reactor setup; **Left**: Experimental setup described in Reference [32]. The solid bulk is kept between filter plates. Water is provided at constant pressure in the surrounding casing. Air flows through the heat transfer channel (HTF) to provide/remove heat. Thermocouples are placed at different locations, here T1, T3 and T7. For better visibility, the figure is not true to scale. The dimensions are given in mm. **Top right**: In the model, the setup consists of two separate domains, the reactive bulk and the (HTF). The two domains are coupled by a conductive heat flux. Due to symmetry, only half of the bulk domain is simulated. **Bottom right**: Domain coupling; a heat flux is calculated between the cell centers of the two-dimensional reaction bulk and the 1D HTF channel. directly heated fixed-bed reactor. The chemical reaction is included by reaction kinetics as source/sink terms. The gaseous component is transported via advection due to pressure gradients. The mass balance equation of the gaseous component considers changes in the gas storage due to changes in density, advection, and sources/sinks:  
\[\frac{\partial(\varrho_{g}\dot{\varphi}_{g})}{\partial t}-\
abla\cdot\left( \varrho_{g}\mathbf{v}\right)\,=\,(1-\dot{\varphi}_{g})\cdot\frac{dX}{dt}, \tag{2}\]  
with gas density \(\varrho_{g}\), porosity \(\dot{\varphi}_{g}\), velocity \(\mathbf{v}\) and reaction rate \(\frac{dX}{dt}\). The factor \((1-\dot{\varphi}_{g})\) takes into account that the reaction occurs only in the solid volume fraction. The gas velocity \(\mathbf{v}\) is determined by [PERSON]'s law:  
\[\mathbf{v}=\frac{\mathbf{K}}{\mu_{g}}(\text{grad}\,p), \tag{3}\]  
with the permeability \(\mathbf{K}\), the gas viscosity \(\mu_{g}\), and pressure \(p\). The solid bulk is considered as rigid. Thus, the solid mass balance equations contain only terms for storage and reaction, but no advection term:  
\[\frac{\partial(\varrho_{\lambda}\dot{\varphi}_{\lambda})}{\partial t}=\pm(1- \dot{\varphi}_{g})\cdot\frac{dX}{dt},\qquad\lambda\in\{\text{CaO},\text{Ca}( \text{OH})_{2}\}, \tag{4}\]  
with the solid volume fractions \(\dot{\varphi}_{\lambda}\). Due to stoichiometry, the amount of reaction rate \(\frac{dX}{dt}\) is the same for CaO and Ca(OH)\({}_{2}\) but with different signs. The bulk energy balance equation considers heat flux by gas advection and heat conduction in the porous bulk. Alterations in pressure, temperature, and presence of phases change the energy stored in the system. The chemical reaction releases/consumes reaction enthalpy \(\Delta h_{R}\). Assuming local thermal equilibrium, see also Reference [35], the energy balance equation is written as:  
\[\frac{\partial}{\partial t}\left(\dot{\varphi}_{g}\varrho_{g}u_{g }\right)+(1-\dot{\varphi}_{g})\varrho_{s}c_{ps}\frac{\partial T}{\partial t}+ \
abla\cdot\left(\varrho_{g}\mathbf{w}h_{g}\right)-\
abla\cdot\left(\lambda_{ \text{eff}}\
abla T\right)=\] \[(1-\dot{\varphi}_{g})\cdot\frac{dX}{dt}\bigg{(}\Delta h_{R}- \frac{\dot{\varphi}_{g}}{1-\dot{\varphi}_{g}}\frac{p}{\varrho_{g}}\bigg{)}, \tag{5}\]  
with the specific internal energy of the gas \(u_{g}\), the solid heat capacity \(c_{ps}\), the specific enthalpy of the gas \(h_{g}\), and the effective heat conductivity \(\lambda_{\text{eff}}\).