## 2 Modeling and Optimization Method
### BCT Model and Pressure Drop Calculation
#### 2.1.2 Modeling for a Single-Type Variable Cross-Sectional Structure  
A single type of circular tube with a variable cross-sectional structure has a reduced or increased diameter, as shown in Figure 3. In a Newtonian fluid, the structure has a mature formula for calculating the pressure drop [34].  
In Figure 3, \(\mathbf{D}\) and \(\mathbf{d}\) are the large and small diameters (m), respectively; \(\mathbf{L}_{\mathbf{D}}^{\mathrm{s}}\) and \(\mathbf{L}_{\mathbf{d}}^{\mathrm{s}}\) are the effective lengths of the flow passage generated by the pressure drop in the reduced structure (m), respectively; \(\mathbf{L}_{\mathbf{D}}^{\mathrm{k}}\) and \(\mathbf{L}_{\mathbf{d}}^{\mathrm{k}}\) are the effective lengths of the flow passage generated by the pressure drop in the increased structure (m), respectively. For the reduced pipe diameter structure [35; 36; 37] (as shown in Figure 3a), the pressure drop calculation formula is:  
\[\Delta\mathbf{p}_{\mathrm{s}}=\rho\cdot\mathbf{g}\cdot\left(\frac{\mathbf{v}_{\mathbf{d}}^{2} -\mathbf{v}_{\mathbf{D}}^{2}}{2\cdot\mathbf{g}}+\mathbf{h}_{\mathbf{js}}\right) \tag{1}\]  
\[\mathbf{h}_{\mathbf{js}}=\frac{0.5\cdot\left(1-\mathbf{d}^{2}/\mathbf{D}^{2}\right)^{2}\cdot \mathbf{v}_{\mathbf{d}}^{2}}{2\cdot\mathbf{g}} \tag{2}\]  
For the increased pipe diameter structure [35; 36; 37] (as shown in Figure 3b), the pressure drop calculation formula is:  
\[\Delta\mathbf{p}_{\mathrm{k}}=\rho\cdot\mathbf{g}\cdot\left(\frac{\mathbf{v}_{\mathbf{D}}^{2} -\mathbf{v}_{\mathbf{d}}^{2}}{2\cdot\mathbf{g}}+\mathbf{h}_{\mathbf{jk}}\right) \tag{3}\]  
\[\mathbf{h}_{\mathbf{jk}}=\frac{\left(\mathbf{D}^{2}/\mathbf{d}^{2}-1\right)^{2}\cdot\mathbf{v}_{ \mathbf{D}}^{2}}{2\cdot\mathbf{g}} \tag{4}\]  
In these formulas: \(\Delta\mathbf{p}_{\mathrm{s}}\) and \(\Delta\mathbf{p}_{\mathrm{k}}\)--reduced or increased pipe diameter structure pressure drop (MPa), respectively; \(\mathbf{D}\) and \(\mathbf{d}\)--large diameter and small diameter (m), respectively; \(\mathbf{v}_{\mathbf{D}}\) and \(\mathbf{v}_{\mathbf{d}}\)--mean velocity of the large diameter and small diameter cross-sections (m/s), respectively; \(\rho\)--liquid density (kg/m\({}^{3}\)).  
When the fluid density is constant, the velocity is inversely proportional to the cross-sectional area. The fluid flow through the contraction channel will accelerate and the flow through the expansion channel will decelerate. Combined with the continuous equation of fluid mechanics, the pressure drops of the pipe diameter reduction or increased structure were further simplified and expressed as:  
\[\mathrm{A}_{\mathbf{D}}\cdot\mathbf{v}_{\mathbf{D}}=\mathrm{A}_{\mathbf{d}}\cdot\mathbf{v}_{\mathbf{ d}} \tag{5}\]  
\[\mathbf{D}=\mathbf{d}+2\cdot\mathbf{h} \tag{6}\]  
Figure 3: Schematic diagram of the pipe structure with a reduced or increased diameter. (**a**) reduced structure; (**b**) increased structure.  
\[\Delta p_{\mathbf{k}1}=\rho\cdot\frac{\mathbf{d}^{2}}{\left(\mathbf{d}+2\cdot\mathbf{h}\right)^{2}} \cdot\left(\frac{\mathbf{d}^{2}}{\left(\mathbf{d}+2\cdot\mathbf{h}\right)^{2}}-1\right) \cdot v_{\mathbf{d}}^{2} \tag{7}\]  
\[\Delta p_{\text{s1}}=\rho\cdot\left\{3-5\cdot\frac{\mathbf{d}^{2}}{\left(\mathbf{d}+2 \cdot\mathbf{h}\right)^{2}}+2\cdot\left[\frac{\mathbf{d}^{2}}{\left(\mathbf{d}+2\cdot\mathbf{h }\right)^{2}}\right]^{2}\right\}\cdot\frac{v_{\mathbf{d}}^{2}}{2} \tag{8}\]  
In these formulas: \(\Delta p_{\mathbf{k}1}\) and \(\Delta p_{\text{s1}}\)--simplified increased and reduced pipe diameter structure pressure drops (MPa), respectively; \(\mathbf{d}\)--trail diameter (m); \(\mathbf{h}\)--groove depth (m); \(\mathbf{v}_{D}\) and \(\mathbf{v}_{d}\)--mean velocities of large diameter and small diameter cross-sections (m/s), respectively; A\({}_{D}\) and A\({}_{d}\)--the cross-sectional areas of the large and small paths (m\({}^{2}\)), respectively; \(\rho\)--liquid density (kg/m\({}^{3}\)).