# A Review: Factors Affecting Internal Two-Phase Flow-Induced Vibrations
## 5 Comparison of Dimensionless Forces  
[PERSON] et al. [48] proposed an empirical correlation for dimensionless forces as shown in Equation (13). This correlation was based on experimental data for 20 mm internal diameter U-bends and tees, and appear to provide better results for void fraction values in the range of 55 to 98%. This correlation was developed on the base of model [72] for external cross flow. To develop this model, the Weber, Reynolds, and Froude numbers were considered, as shown below:  
\[\overline{F_{RMS}}=\frac{F_{RMS}}{\rho_{I}V_{m}^{2}\left(\pi\frac{D^{2}}{4} \right)}=CWe^{-0.4} \tag{16}\]  
The dashed line shown in Figure 6 is drawn using Equation (16) with \(C\) of 10. This was proposed as a reasonable approximation of dimensionless forces. The dash dotted line shown in Figure 6 is drawn using \(C\) of 3.5 using Equation (16) and it is the best fit curve for slug flow regime. [PERSON] et al. [99] proposed that to be conservative, an upper bound of dimensionless forces should be defined and from their experiments on 12 to 52 mm internal diameter U-bends, \(C_{\text{max}}\) of 25 was proposed. This corresponds to the dotted line shown in Figure 6. The experimental results for 12 and 52 mm U-bends are included here, which shows that \(C_{\text{max}}\) = 25 is a good approximation for upper limit. The dimensionless forces due to slug flow from experiments conducted by [PERSON] et al. [33; 34], in 6 mm internal diameter horizontal and vertical elbows for bend radii of 16.5 and 25 and the results for 25.4 mm (1 inch) internal diameter elbow, are also recorded in Figure 6. Notably, both horizontal and vertical elbows are subjected to the horizontal two-phase flow in these experiments. The overall forces on vertical elbow are higher than the horizontal elbow, which was thought to be due to gravity. Notably, forces in vertical elbow are also higher than the maximum limit introduced by [PERSON] et al. [69] at C of 25. The forces due to the 16.5 bend radius are lower than the 25 mm bend radius for the same diameter pipe, which shows that the bend radius affects the forces. Moreover, it can be observed that the forces in 25.4 mm (1 inch) pipe are higher than the 6 mm experiments. This behavior was unusual, considering that the 25.4 mm (1 inch) experiments contained more gas than the 6 mm experiments. [PERSON] et al. [76] reported that this can also be due to the difference of stiffness of material used (glass vs. perspex). Overall, the results of [PERSON] experiments show higher forces than the [PERSON] experiments and results for 25.4 mm bend are higher than \(C\) = 10 line. Experimental results for 20 mm internal diameter tee [42], for elbow with 70 mm diameter under the horizontal flow [98], and for 6 to 25 mm diameter vertical U-tubes [8], are also reported. All these results are in agreement with [PERSON] et al. [48] approximation at \(C\) of 10. Results [66] for 52.5 mm vertical elbow are also presented and these results are spread over a wide range from \(C\) of 3.51 to 25 lines. This can be attributed to the fact that 0 to 100% void fraction was considered, covering a wide range of flow conditions. The simulation results of [PERSON] et al. [65] for vertical elbow of 52.5 mm diameter are presented and they mostly lie between \(C\) of 10 and 25 lines. Figure 6 can serve as a database for comparison and can also be used to validate the simulation results.