## 2 Modeling and Optimization Method
### BCT Model and Pressure Drop Calculation
#### 2.1.3 Modeling for a Comprehensive-Type Variable Cross-Sectional Structure  
The comprehensive type of circular tube with a variable cross-sectional structure is a reduction-increase or increase-reduction structure, as shown in Figure 4. For a Newtonian fluid, the structure has a related mature formula for calculating the pressure drop [38; 39].  
In Figure 4, \(\mathbf{L}_{d}\) and \(\mathbf{L}_{D}\) are the length of the reduction section and increase section (m), respectively. For the pipe diameter reduction-increase structure [35; 40; 41] (as shown in Figure 4a), the pressure drop calculation formula is  
\[\Delta p_{\mathbf{sk}}=\mathbf{K}_{\mathbf{d}}\cdot\left(\Delta p_{\mathbf{s}}+\Delta p_{\mathbf{ k}}\right) \tag{9}\]  
\[\mathbf{K}_{\mathbf{d}}=1.01+15.46\cdot\mathbf{e}^{\frac{-\mathbf{s}_{\mathbf{d}}}{0.069}}+0.73 \cdot\mathbf{e}^{\frac{-\mathbf{s}_{\mathbf{d}}}{0.22}} \tag{10}\]  
In these formulas: \(\Delta p_{\mathbf{sk}}\) is the reduction-increase pipe diameter structure pressure drop; \(\mathbf{K}_{\mathbf{d}}\) is the influence coefficient of the pressure drop of the increased structure; and \(\mathbf{n}_{d}=\mathbf{L}_{d}/\mathbf{D}\), where \(0.04\leq\mathbf{n}_{d}\leq 0.2\).  
For the increase-reduction pipe diameter structure [35; 40; 41] (as shown in Figure 4b), the pressure drop calculation formula is  
\[\Delta p_{\mathbf{ks}}=\mathbf{K}_{\mathbf{D}}\cdot\left(\Delta p_{\mathbf{s}}+\Delta p_{\mathbf{ k}}\right) \tag{11}\]  
\[\mathbf{K}_{\mathbf{D}}=2.573-2.565\cdot\mathbf{e}^{\frac{-\mathbf{n}_{\mathbf{D}}}{7.291}} \tag{12}\]  
In these formulas: \(\Delta p_{\mathbf{ks}}\) is the pressure drop of the increase-reduction pipe diameter structure; \(\mathbf{K}_{\mathbf{D}}\) is the influence coefficient of the pressure drop of the increased structure; and \(\mathbf{n}_{\mathbf{D}}=\mathbf{L}_{\mathbf{D}}/\mathbf{D}\), where \(0.15\leq\mathbf{n}_{\mathbf{D}}\leq 0.37\).  
Combined with the geometric characteristics of the BCT, the pressure drop of the reduction-increase and increase-reduction pipe diameter structures was further simplified and expressed as  
\[\mathbf{n}_{\mathbf{d}1}=\frac{\mathbf{s}}{\mathbf{d}+2\cdot\mathbf{h}} \tag{13}\]  
Figure 4: Schematic diagram of pipe diameter reduction–increase and increase–reduction structures. (**a**) reduction–increase structure; (**b**) increase–reduction structure.  
\[\mathbf{n}_{D1}=\frac{\mathbf{w}}{d+2\cdot\mathbf{h}} \tag{14}\] \[\Delta\mathbf{p}_{sk1}=\left(1.01+15.46\cdot e^{\frac{-s}{000\cdot(d+2 \cdot h)}}+0.73\cdot e^{\frac{-s}{0212\cdot(d+2\cdot h)}}\right).\] (15) \[\left\{\rho\cdot\left\{3-5\cdot\frac{d^{2}}{(d+2\cdot h)^{2}}+2 \cdot\left[\frac{d^{2}}{(d+2\cdot h)^{2}}\right]^{2}\right\}\cdot\frac{v_{2}^{2 }}{2}\right\}\] \[\Delta\mathbf{p}_{ks1}=\left(2.573-2.565\cdot e^{\frac{-s}{021\cdot(d +2\cdot h)}}\right).\] \[\left\{\rho\cdot\left\{3-5\cdot\frac{d^{2}}{(d+2\cdot h)^{2}}+2 \cdot\left[\frac{d^{2}}{(d+2\cdot h)^{2}}\right]^{2}\right\}\cdot\frac{v_{2}^ {2}}{2}\right\} \tag{16}\]  
In these formulas: \(\Delta\mathbf{p}_{sk1}\)--reduction-increase BCT pipe diameter structure pressure drop (MPa); \(\Delta\mathbf{p}_{ks1}\)--increase-reduction BCT pipe diameter structure pressure drop (MPa); \(\mathbf{s}\)--slot pitch (m); \(\mathbf{d}\)--trail diameter (m); \(\mathbf{h}\)--groove depth (m); \(\mathbf{w}\)--groove width (m); \(\rho\)--liquid density (kg/m\({}^{3}\)).