## 2 Dynamic Model Configuration for the CO2 Capture Process
Similarly, the mass and energy balance of the vapor bulk can be modeled as [2]:  
\[0=\frac{-1}{L\cdot A}\frac{\partial F_{i}^{V}}{\partial y}-N_{i}\cdot Sp\cdot \omega\cdot MW_{i} \tag{3}\]  
\[0=\frac{-1}{L\cdot A}\frac{\partial F_{H}^{V}}{\partial y}+Sp\cdot\omega\left( H_{cap}^{cond}+H_{cap}^{conv}\right) \tag{4}\]  
where \(F_{i}^{V}\) is the vapor component mass flow along the axis of the column, and \(H_{cap}^{cond}\) and \(H_{vap}^{conv}\) are the heat fluxes of vapor conduction and convection at the vapor film-vapor bulk interface, respectively. The mass transfer rates in the vapor and liquid films can be determined using the Maxwell-Stefan Formulation:  
\[\frac{1}{\delta}\frac{\partial x_{i}^{M}}{\partial z}=\frac{1}{c_{t}}\sum_{k= 1}^{nc}\left(\frac{x_{i}^{M}N_{k}-x_{k}^{M}N_{i}}{\chi_{i,k}}\frac{\mu^{R}}{ \mu}\frac{T}{298.15}\right),\ k\
eq i \tag{5}\]  
where \(\delta\) is the film thickness, \(x_{i}^{M}\) is the molar fraction of the \(i\)-th component, \(c_{t}\) is the total molar concentration, \(nc\) is the number of components, \(\chi_{i,k}\) is the diffusivity, \(\mu\) and \(\mu^{R}\) are the viscosity and reference viscosity, respectively, and \(T\) is the temperature. The interface model is formulated as:  
\[f_{i}^{L}x_{i}^{M,L}=f_{i}^{V}x_{i}^{M,V} \tag{6}\]  
where \(f_{i}^{L}\) and \(f_{i}^{V}\) are the liquid and vapor fugacity coefficients, and \(x_{i}^{M,L}\) and \(x_{i}^{M,V}\) are the equilibrium molar components in the vapor and liquid phases. The overall reaction for carbonate formation can be given as [37]:  
\[CO_{2}+MEA+B\rightleftharpoons MEACOO^{-}+BH^{+} \tag{7}\]  
The reaction rate of CO\({}_{2}\) can be calculated as [37]:  
\[r_{CO_{2}}=\left\{k_{MEA}^{T}[MEA]+k_{H_{2}O}^{T}[H_{2}O]\right\}[MEA][CO_{ 2}]-\left(\frac{k_{f}^{T}}{K_{sq}^{T}}\right)\left[MEAH^{+}\right]\left[MEACOO^{-}\right] \tag{8}\]  
where \(B=MEA\) and/or \(H_{2}O\), \(k_{MEA}^{T}\) and \(k_{H_{2}O}^{T}\) are the third order kinetic rate constant of \(MEA\) and \(H_{2}O\), respectively, \(k_{f}^{T}\) is the forward reaction rate, \(k_{sq}^{T}\) is the equilibrium constant, and the superscript \(T\) represents the temperature dependence. The physical properties of the solvents in the capture process are calculated using the go-statistical associating fluid theory (gSAFT) package (Process System Enterprise, London, UK), which is also developed by the PSE Company based on the Statistical Association Fluid Theory [38]. The steady-state process conditions of the nominal working point, that is 100% power plant load and 80% CO\({}_{2}\) capture rate, are shown in Table 1, based on reference [13]. Within the CO\({}_{2}\) capture system, there are many process variables which should be well controlled, such as the liquid level of the absorber sum, the reboiler pressure, etc. Among these variables, the control of the CO\({}_{2}\) capture rate and reboiler temperature are of the highest importance because the CO\({}_{2}\) emission level must be controlled to satisfy the environmental demand and the temperature in the reboiler should be maintained to keep the balance between energy supply and consumption. The lean solvent flow rate and extracted steam flow rate have the most significant influence on the two key controlled variables, as demonstrated in references [3, 7, 12, 30, 39]. For these reasons, the dynamic relationship between these four key variables is studied in this paper, which results in a 2 \(\times\) 2 model structure for the nonlinearity analysis and the construction of the multi-linear model. The other process variables are controlled to the given set point by well-tuned decentralized proportional integral (PI) controllers, and hence the control of these variables is not considered in the advanced control structure. The entire control structure of the CO\({}_{2}\) capture process is presented in Figure 2.0