## 3 Results and Discussion
### Computational Fluid Dynamics Simulations
#### 3.2.4. CFD Model Validation with Experiments
However, for the volume mean and Sauter mean diameter, the disparities were higher, falling within the range of 14-27%. This observation highlights larger variations between the experimental and simulated results, particularly for the volume mean and Sauter mean diameter metrics. The arithmetic mean exhibited a better fit than the Sauter mean diameter when comparing the experimental and CFD simulation results. This is because the arithmetic mean was less affected by extreme values. The arithmetic mean simply calculated the average of all droplet sizes, whereas the Sauter mean was more influenced by the ratio of the volume-to-surface area, making it more sensitive to variations in the distribution's range. From Table 3, it is evident that the experimental mean was consistently larger than the CFD mean. This suggests that the CFD simulations tended to underestimate the droplet diameter. The observed disparity between the CFD simulation and experimental results could be attributed to the use of a fitted log-normal distribution in the simulations. This distribution, while providing a continuous representation, introduced errors and may not have accurately captured the real droplet size distribution, which potentially included larger droplets. In a continuous size distribution, there tends to be less coalescence due to the smoother variation in droplet velocities. Conversely, experiments with a discontinuous distribution may exhibit more significant differences in droplet velocities, potentially leading to increased coalescence. One potential enhancement for the model could involve using the actual population of droplets instead of a fitted distribution. However, implementing this approach becomes challenging in practical scenarios with numerous nozzle holes. Additionally, the assumption in the CFD simulation that droplets are perfectly spherical reduces the likelihood of collisions and coalescence, given the minimized collision surface area. In addition to the assumptions related to droplet size distribution and spherical shapes, another significant assumption in the CFD simulation is the uniform injection velocity for all droplets. While a size distribution was considered, an injection velocity distribution was not incorporated. This assumption implies that all droplets share the same initial velocity upon injection, contributing to an underestimation of collisions in the simulation. In reality, droplets often exhibit a range of velocities, and neglecting this variation in the CFD model leads to an oversimplified representation. The consequence is a diminished prediction of collisions, as the uniform velocity assumption diminishes the potential for differences in droplet velocities that would contribute to collision events. Addressing this limitation by incorporating a more realistic distribution of injection velocities could enhance the accuracy of the CFD model in predicting collision and coalescence phenomena. Figures 10-13 depict the experimental and CFD droplet size frequency distributions, accompanied by fitted normal distribution curves and linear fits of R-squared values for \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline
**Run** & & & & & & & & & & & \\
**Case** & & & & & & & & & & & \\
**Exp.** & **CFD** & **Diff.** & **Exp.** & **CFD** & **Diff.** & **Exp.** & **CFD** & **Diff.** & **Exp.** & **CFD** & **Diff.** \\ \hline N1H1 & 393 & 352 & 10 & 411 & 360 & 12 & 428 & 367 & 14 & 464 & 382 & 17 \\ N1H2 & 461 & 400 & 13 & 491 & 414 & 15 & 517 & 429 & 17 & 575 & 459 & 20 \\ N2H1 & 470 & 406 & 13 & 497 & 416 & 16 & 526 & 427 & 18 & 590 & 450 & 23 \\ N2H2 & 444 & 385 & 13 & 479 & 394 & 17 & 513 & 404 & 21 & 588 & 424 & 27 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Comparison of diameter means (\(d_{10}\), \(d_{20}\), \(d_{30}\), and \(d_{32}\)) between experimental data and CFD simulations for the respective run cases. runs N1H1 to N2H2, respectively. There is a significant difference in the standard deviation of droplet sizes between the experimental results and the CFD simulations.