# Assessment of Appropriate Geometry for Thermally Efficient CO\({}_{2}\) Adsorption Beds
## 2 Adsorption Beds and Model Description
### Governing Equations  
To simulate adsorption, flow, and energy balances, the following governing equations are used. The overall mass balance is given by [31]:  
\[\epsilon\frac{\partial(\rho)}{\partial t}+\
abla\cdot\left(\rho\overrightarrow{ v}\right)=-(1-\epsilon)\rho_{p}\sum_{i}M_{i}\frac{\partial q_{i}}{\partial t} \tag{1}\]  
Figure 1: Adsorption beds investigated in this study. where \(\rho\) is the gas mixture density, \(\varepsilon\) is the bed porosity, \(v\) is the velocity vector, \(M\) is the gas molar mass, \(q\) is the adsorption uptake, and \(t\) is the time. The mass balance of each gas can be given by:  
\[\varepsilon\frac{\partial(\rho y_{i})}{\partial t}+\
abla\cdot\left(\rho\overset {\rightarrow}{v}y_{i}\right)=-\
abla\left(\varepsilon\rho D_{disp,i}\
abla y _{i}\right)-(1-\varepsilon)\rho_{P}M_{i}\frac{\partial q_{i}}{\partial t} \tag{2}\]  
Here, \(y\) is the molar fraction of species. The quantity \(\partial q_{i}\partial t\) represents the adsorption kinetic, which is expressed from the linear driving force (LDF) model [32,33]:  
\[\frac{\partial q_{i}}{\partial t}=k_{L,i}(q_{i}^{*}-q_{i}) \tag{3}\]  
Here, \(q_{i}\)* is the equilibrium uptake of species \(i\), \(q_{i}\) is the actual uptake of species \(i\), \(k_{L,i}\) is the kinetic constant. \(q_{i}\)* is estimated using the dual-site Langmuir model [16]:  
\[q_{i}^{*}=\frac{q_{m,i1}K_{eq,i1}y_{i}P}{1+K_{eq,i1}y_{i}P}+\frac{q_{m,i2}K_{ eq,i2}y_{i}P}{1+K_{eq,i2}y_{i}P} \tag{4}\]  
where \(q_{m,i1}\) and \(q_{m,i2}\) are the uptake limits and \(K_{eq,i1}\) and \(K_{eq,i2}\) are constants. \[K_{eq,i}=k_{0}e^{-(\frac{\Delta H_{i}}{\kappa l})} \tag{5}\]  
Here, \(k_{0}\) is the temperature-independent constant, \(\Delta H_{i}\) is adsorption heat, \(R\) (8.314 J/mol/K) is the gas constant, and \(T\) is the tested gas temperature (\(K\)). The momentum equation is given by [31]:  
\[\varepsilon\frac{\partial\left(\rho\overset{\rightarrow}{v}\right)}{ \partial t}+\
abla\cdot\left(\rho\overset{\rightarrow}{v}\overset{ \rightarrow}{v}\right)=-\
abla p+\
abla\cdot\overset{\rightarrow}{\tau}+ \rho\overset{\rightarrow}{g}+S_{momentum} \tag{6}\]  
Here, \(\overset{\rightarrow}{\tau}\) is the stress tensor, \(\overset{\rightarrow}{g}\) is the gravitational acceleration vector, and \(S_{momentum}\) is the momentum source term that can be calculated based on the Ergun equation [31]. \[S_{momentum}=-\left(\frac{\mu}{\kappa}v_{i}+C_{2}\frac{1}{2}\rho\left| \overset{\rightarrow}{v}\right|v_{i}\right) \tag{7}\]  
where \(\mu\) is the dynamic viscosity, \(\kappa\) is the bed permeability, and \(C_{2}\) is the inertia resistance. For the energy equation, adsorption bed and external solid wall energy conservation equations are expressed as:  
\[\frac{\partial}{\partial t}\left[\varepsilon\rho E_{g}+(1-\varepsilon)\rho_{b }E_{s}+\
abla\cdot\left(\overset{\rightarrow}{v}(\rho E+P)\right)\right]= \
abla\cdot\left(k_{eff}\
abla T-\sum_{i}h_{i}\overset{\rightarrow}{\bar{f}_ {i}}+\overset{=}{\bar{\tau}}\cdot\overset{\rightarrow}{v}\right)+(1- \varepsilon)\rho_{b}\underset{i}{\sum}\Delta H_{i}\frac{\partial q_{i}}{ \partial t} \tag{8}\]  
\[\frac{\partial}{\partial t}(\rho_{w}C_{w}T_{w})=\
abla(k_{w}\
abla T_{w}) \tag{9}\]  
where \(E_{g}\) is the total gas energy, \(E_{s}\) is the total adsorbent energy, \(\rho_{b}\) is the density of the bed, \(h_{i}\) is the enthalpy of sensible heat, \(\overset{\rightarrow}{J}\) is the diffusion flux, \(k_{eff}\) is the bed's effective thermal conductivity (as given in the below Equation [18]), \(\rho_{w}\) is the wall density, \(C_{w}\) is the wall specific heat capacity, \(k_{w}\) is the wall thermal conductivity, and \(T_{w}\) is the local wall temperature. Figure 1:0  
\(k_{g}\) is the gas thermal conductivity, and \(k_{s}\) is the adsorbent thermal conductivity.