## 3 Optimal Design of Pulse Dust Collection System
### Working Principles of Pulse Dust Collector with Multiple Filter Groups  
In order to fully describe the changes in pressure drop in the three stages of filter instability, filter stability, and filter aging during filtration, this study proposes a new model that is different from other studies. The dust collector is a structure of multiple filter groups (composed of N = 9 filter groups), and each group is composed of two filters in series. The effective filtering area of each filter group is \(A_{g}=2\times A_{e}=5.934\) m\({}^{2}\) and the total effective filtering area is \(A=N\times A_{g}=53.406\) m\({}^{2}\)[33]. Therefore, the parameters of the analysis model include filtering speed, the pressure drop of the temporary powder cake, the pressure drop of the residual powder cake, the pressure drop of the filter, and the total pressure drop of the filter, as shown in Figure 3.  
Figure 1: The pulse dust collection system. Figure 2: The pulse dust collection system. (1)  
Filtration speed  
We divide the state of the filter into a filtering and a cleaning period. During the filtering period, the flow \(V_{f}\) according to the flow, speed, and area can be calculated as shown in Equation (8), where N is the number of filter groups, \(A_{g}\) is the filter area of each filter group, and \(Q\) is the gas flow of the filter [34]. \[V_{f}=\frac{Q}{N\times A_{g}},\ \forall t\in[0,T_{w})\cup[t_{j}+T_{p},\ t_{j}+T_{w}),\ j\geq 2 \tag{8}\]  
(2)  
According to [PERSON]'s law, the overall pressure drop of the temporary powder cake is shown in Equation (9), where \(W\) is dust cake mass, \(V\) is filtration speed, and \(K_{2}\) is the resistance coefficient of the temporary powder cake. Because the dust used in the simulation test is granular coal ash powder, we use the equation proposed by [PERSON], [PERSON] et al. [2] to describe the resistance coefficient \(K_{2}\) of the temporary powder cake, as shown in Equation (10), where \(\kappa\) is the dynamic shape factor, \(d_{g}\) is the average diameter, \(\sigma_{g}\) is the standard deviation, \(\rho_{p}\) is the dust density, \(\mu\) is the air viscosity, and \(v(\varepsilon_{o})\) is the void function, which depends on the porosity \(v(\varepsilon_{o})\) of the total dust cake. \[P_{c}=K_{2}\times W\times V \tag{9}\]  
\[K_{2}=\frac{18\kappa}{d_{g}^{2}exp(4 ln^{2}\sigma_{g})}\frac{v(\varepsilon_{o} )}{\varepsilon_{o}{}^{2}}\times\frac{\mu}{\rho_{p}} \tag{10}\]  
(3)  
After pulse cleaning, there will be a certain percentage mass of the residual powder cake as \(\Delta W_{rj}\), and this will form the pressure drop of the residual powder cake as \(\Delta P_{rj}^{*}\), with gradual accumulation and \(U\) for unit step functions, as shown in Equation (11). When the pressure drop of the residual powder cake \(P_{rj}^{*}\) is greater, the cleaning efficiency \(\eta_{j}^{*}\) of the filter will also be greater, as shown in Equation (12). \(\lambda_{1}\) is the characteristic value  
Figure 3: The model of filter elements in pulse dust collector. of cleaning efficiency, and \(P^{*}_{r_{j-1}}\) is the pressure drop of the residual powder cake in the previous filtration cycle. \[P^{*}_{r_{j-n}}=\sum_{j=2}^{m}\Delta P^{*}_{r_{j}}U\big{(}t-t_{j}\big{)},\ \forall m\in\mathbb{N},[2,\infty] \tag{11}\]  
\[\eta^{*}_{j}=\Big{(}1-e^{-\lambda_{1}\times P^{*}_{r_{j-1}}}\Big{)} \tag{12}\]  
(4) The pressure drop of the filter  
The pressure drop of the filter is caused by the residual powder cake blocking the filter element, and the result is a decrease in the air permeability of the filter element. According to [PERSON]'s law, the filter pressure drop \(P^{*}_{f}\) can be expressed as Equation (13), where \(K^{*}_{1}\) is the resistance coefficient of the filter, \(k_{f}\) is the air permeability of the filter, and \(\lambda_{2}\) is the characteristic value with the air permeability of the filter due to dust blockage.