# Modeling and Simulation of IGCC Considering Pressure and Flow Distribution of Gasifier
## 3 IGCC System Modeling
### Gasifier Model and Volume-Resistance Method  
Each gasifier compartment is separated into two sub-modules. One is the volume sub-module and the other is the resistance sub-module, as shown in Figure 3.  
Figure 1: Schematic diagram of the compartment method. Figure 2: The layout of the integrated gasification combined cycle (IGCC) system. The dashed line in the control section represents that this control volume has both volume and resistance properties. We can assume that the pressure change occurs only in the volume sub-module due to the flow rate difference, while the flow rate change occurs only in the resistance sub-module due to the pressure difference. We will deal with the pressure and flow distribution based on this assumption. The V-R method doubles the number of compartments with new volume and resistance sub-modules, as shown in Figure 4.  
Based on the above idea, using the mass conservation theorem, the state equation of pressure in the volume sub-module was derived [17]. The equation can be written as the follows:  
\[\frac{dP_{i}}{d\tau}=\frac{R_{g}T_{i}}{V_{i}}\sum\frac{dm_{i,j}}{d\tau} \tag{1}\]  
where \(P\) is the state parameter pressure and \(\tau\) is the time; \(R_{g}\) is the ideal gas constant; \(T_{i}\) and \(V_{i}\) stand for the temperature and the volume of compartment \(I\), respectively; \(m_{i,j}\) stands for the mass of component \(j\) in compartment \(i\), which is calculated through the chemical reaction. The state equation of mass was derived from Reference [18] and can be written as:  
\[\frac{dn_{g,i}Y_{j,i}}{d\tau}=\dot{n}_{j,i-1}+R_{j,i}-\dot{n}_{j,i} \tag{2}\]  
where \(n_{g,i}\) stands for the total mole number of syngas in compartment \(i\) and \(Y_{j,i}\) stands for the molar fraction of component \(j\) in compartment \(i\); \(\dot{n}_{j,i-1}\) stands for the inlet molar flow rate of component  
Figure 4: Diagram of the gasifier compartment with the volume-resistance (V-R) model. Figure 3: Diagram of the module structure in one compartment. from compartment \(i\); \(G_{j,i}\) stands for the outlet molar flow rate of component \(j\) of compartment \(i\); and \(R_{j,i}\) stands for the reaction rate of component \(j\) in compartment \(i\). The homogeneous and heterogeneous chemical reactions are contained in each compartment, and the reaction rate is shown as follows. For the homogeneous reaction [18],  
\[k_{i}=k_{0}exp\left(-\frac{E}{RT_{g}}\right)C_{A}C_{B} \tag{3}\]  
For the heterogeneous reaction [19],  
\[R_{i}=\frac{1}{\left[\frac{1}{k_{diff}}+\frac{1}{k_{s}\gamma^{2}}+\frac{1}{k_{ dash}}\left(\frac{1}{Y}-1\right)\right]} \tag{4}\]  
where \(k_{i}\) and \(R_{i}\) are the homogeneous and heterogeneous reaction constants, respectively; \(k_{0}\), \(E\), \(C_{A}\), \(C_{B}\), \(k_{diff}\), \(k_{s}\), \(k_{dash}\) are all coefficients and can be found in References [18; 19]. It should be noted that Equation (4) is related to the mole number while Equation (1) requires the mass. Thus, molar mass should be considered to solve the pressure equation. Using the momentum conservation theorem, the equation of the flow rate in the volume sub-module can be written as follows,  
Figure 1:0  
where \(G_{i}\) is the mass flow rate of compartment \(i\); \(A\) and \(x\) are the sectional area and the length of each compartment; \(U\) and \(\sigma\) are the wet perimeter and friction resistance, respectively. The flow rate equation can be simplified to reduce the quantity of the state parameters,  
Figure 1:1  
Besides, the reaction heat and heat loss of slag must be considered to build the energy equation. Because of the flow rate and reaction rate, the temperature is also a state parameter. Therefore, we can build the state equation for the temperature of each compartment [18]. \[ \begin{split}\left(n_{c,i}c_{p,s}+\sum\limits_{j=1}^{N_{g}}n_{g,i}Y_{ij}c_{p,g,j}\right)\frac{dT_{i}}{d\tau}\\ =G_{c,i-1}c_{p,s}T_{i-1}-G_{c,i}c_{p,s}T_{i}+T_{i-1}\sum\limits_{ j=1}^{N_{g}}G_{g,i-1}Y_{i-1,j}c_{p,g,j}\\ -T_{i}\sum\limits_{j=1}^{N_{g}}G_{g,i}Y_{ij}c_{p,g,j}+Q_{CH}-Q_{s,i }\end{split}