# Dynamic Modeling and Validation of Peak Ability of Biomass Units
## 3 Dynamic Modeling of Biomass Unit Peak Ability
### Feed-Heat Model  
The particle size of biomass fuel is uneven, the density distribution is different, and the fuel composition is complex [18]. It is difficult to detect the feeding amount of the boiler online, so it is necessary to estimate the feeding amount of the boiler. In addition, the transmission of feed on the belt has certain delay characteristics, so the transmission process can be approximated by a pure delay link. Assuming the type and density of fuel remain unchanged, the fuel quantity \(F_{b}\) (kg/s) in the boiler is  
\[F_{b}(t)=K_{1}\mu_{B}(t-\tau), \tag{1}\]  
where \(\mu_{B}(t)\) (kg/s) is the fuel quantity entering the furnace at a certain time. \(K_{1}\) is the proportional coefficient, and \(K_{1}=1\) is taken for simple calculation. The delay time \(\tau\) is related to the belt structure. The fuel fed into the grate boiler needs to be gradually distributed over the moving grate and combusted in stages and zones, rather than being burned immediately [19]. Therefore, there is a certain amount of fuel stored on the combustion grate, and the newly added fuel and the fuel consumed by combustion reach a new balance. According to the law of the conservation of mass,  
\[c_{1}\frac{\mathrm{d}B(t)}{\mathrm{d}t}=F_{b}(t)-B_{r}(t)(1-A_{ar})-D_{ash}, \tag{2}\]  
where \(c_{1}\) is the dynamic parameter to be identified. \(B(t)\) (kg) is the current total fuel quantity of the grate. \(B_{r}(t)\) (kg/s) is the combustion rate of the furnace fuel. \(A_{ar}\) (%) is the mass fraction of the received base ash of biomass fuel, and \(A_{ar}=13.73\%\) is taken for simple calculation. \(D_{ash}\) (kg/s) is the amount of slag discharged. Due to the low ash content of biomass fuel and the small amount of residual residue after fuel combustion [20], the effect  
Figure 2: Production process flowchart of biomass grate furnace. of the slag discharge amount \(D_{ash}\) will not be considered in this model. Equation (2) can be simplified as  
\[c_{1}\frac{\mathrm{d}B(t)}{\mathrm{d}t}=F_{b}(t)-B_{r}(t)(1-A_{ar}). \tag{3}\]  
The fuel combustion rate \(B_{r}(t)\) is affected by the total amount of fuel on the grate, the oxygen concentration and the furnace temperature [21]. \(B_{r}(t)\) is decided by  
\[B_{r}(t)=k_{r}C_{O_{2}}B(t)k_{s}, \tag{4}\]  
where \(k_{r}\) is the combustion coefficient. \(C_{O_{2}}\) (mol/m\({}^{3}\)) is the oxygen concentration. \(k_{s}\) is the combustion reaction rate coefficient, which is a function of the oxygen diffusion rate \(k_{g}\), fuel chemical reaction rate \(k_{c}\) and ash layer diffusion mass transfer resistance coefficient \(k_{a}\)[22]. In order to simplify the calculation, the influence of \(k_{a}\) on \(k_{s}\) is not considered temporarily, that is  
\[k_{s}=\frac{1}{(\frac{1}{k_{g}}+\frac{1}{k_{c}})}=\frac{1}{(\frac{d_{c}}{shD_ {g}}+\frac{1}{k_{c}})}, \tag{5}\]  
where \(sh\) is the Sherwood number. \(D_{g}\) (m\({}^{2}\)/s) is the oxygen diffusion coefficient and \(d_{c}\) (m) is the average particle size. In practical application, \(k_{s}\) can be determined through the empirical formula proposed by [PERSON] [23],  
\[k_{c}=ATE^{-E/(RT)}, \tag{6}\]  
where \(A\) is the frequency factor. \(T\) (K) is the furnace temperature. \(E\) (kJ/mol) is the apparent activation energy. \(R\) (kJ/(mol\(\cdot\)K)) is the gas constant. The oxygen concentration \(C_{O_{2}}\) is proportional to the total air supply volume \(A_{ir}\). The unit load will affect the total air volume and oxygen concentration in the furnace, so the oxygen concentration calculation formula is  
\[C_{O_{2}}=K_{O_{2}}A_{ir}=k_{O_{2}}N_{e}(t)A_{ir}, \tag{7}\]  
where \(K_{O_{2}}\) is the relationship coefficient between oxygen concentration and total air supply volume. \(A_{ir}\) (m\({}^{3}\)/s) is the total air supply volume under the standard state, which is composed of primary air and secondary air. \(k_{O_{2}}\) is the correction factor. \(N_{e}(t)\) (MW) is the unit load.