## 4 Discussion  
Regarding the development of the three numerical methods performed to solve the presented system, all of them showed reliable numerical stability and quick converged time despite that the topology of the derived system of equations from the application of NLS is highly nonlinear. In the case of [PERSON]'s method, the best performance was obtained by simulation 3 (6 iterations), getting a high rate of converged. In this case, the selection of the initial vector is a crucial aspect of obtaining reliable outcomes. [PERSON]'s method showed excellent results for all three simulations. Analyzing in detail, simulations 2 and 3 posted the best performance solving with a similar computational time (4.950 and 4.263 s) and the number of iterations (6 iterations). Otherwise, LMM showed higher computational times for simulations 2 and 3 (94.312 and 4.263 s), indicating that its converged rate was lower than the other two algorithms. In this case, both [PERSON] and [PERSON] methods presented high conversion rates, which is an expected result considering that these algorithms are very similar in their mathematical structure. Otherwise, the damping parameter inserted in the Broyden cycle helps to improve the  
Figure 10: Behavior of discrepancy error and solution vector for Levenberg–Marquardt method-simulation 3. (a) Behavior of discrepancy error respect to iterations. (b) Behavior of values for solution vector respect to iterations.  
Figure 9: Behavior of discrepancy error and solution vector for Levenberg–Marquardt method-simulation 2. (a) Behavior of discrepancy error respect to iterations. (b) Behavior of values for solution vector respect to iterations.  
converge rate for those initial vectors that are far from the real solution. This fact is evidenced by comparing the results for simulation 1 for both methods, which ran with the same initial vector, but the obtained outcomes for these were quite different (divergence for [PERSON]'s method, while convergence in 21 iterations for [PERSON]'s method). Once the initial vector is close enough to the final solution, these methods behave similarly. Finally, we implemented the Levenberg-Marquardt method for the problem addressed in this study. It showed very different behavior compared with the simulations made by [PERSON] and [PERSON] methods. In this case, this algorithm solves simulations 2 and 3, with a low rate of convergence and it took several iterations (1370 and 1667, respectively) to get a final vector under the minimum acceptable discrepancy error. The above means that for this case study, it is not recommended the use of the Levenberg-Marquardt algorithm and through the implementation of Newton-based methods, we can get better convergence rates and more stability in the simulations. It is worth mentioning that this is an unexpected result, taking into account that the LMM is more robust than the NRM, but in some cases, the Levenberg-Marquardt can be a little bit slower than the Newton algorithm. For this problem, the lower performance may be related to the selection and optimization of its damping value [29], which in the case of [PERSON]'s method worked very well, as the simulations demonstrated. The obtained results can be useful for further development of calculation in the design of separation processes. The above is relevant, considering that there were efforts to model this system or similar ones, but applying different equation models such as NRTL [30]. One of the main outcomes of these findings is the further use of the data for modeling separation systems, which involve the acetone-butanol mixture [31]. An example of this is the design of distillation columns for the purification stage of the acetone-butanol-ethanol (ABE) fermentation. That reaction generates a complex mixture formed by ABE components, water and other substances [32].