Eng 2023 , 4
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spacings, the dimension of the spacings and the number of plies, so that the partition of the paper roll partition by the consumer is neither easy nor too hard [3–5]. This balance is called the perforation efficiency and can be determined accordingly to the standard [6] by Equation (1): E p = 100 1 − S p S np (1) where E p is the perforation efficiency (%), S p is the average tensile strength of perforated papers (N/m) and S np is the average tensile strength of unperforated papers (N/m). During the tissue paper manufacturing process, raised up cellulosic fibers are found on the sheet surface, which help in consumer hygiene, but which in excess can form agglomerates, impairing the quality of the final product. To reduce the loss of cellulosic fibers on the paper surface, it is desirable that the perforation blade have relatively thin teeth [3,4]. Thus, the proper geometry of the blade must be considered. The perforator is also responsible for the visual appearance of the free edge of the remaining paper roll. The consumer wants an aesthetically pleasing free edge (smoother and less irregular between the cut and uncut areas) after tearing off the desired amount of paper [3,4]. The geometric discontinuity of the perforation line will affect the existing stress field in this area, thus affecting the stress concentration factor and consequently the final efficiency. The ratio between the highest value in a geometric discontinuity and the nominal stress in the minimum cross section is called the stress concentration factor [7]. In a previous work develop by Vieira et al. [8], they concluded that in toilet paper samples with a stress concentration factor above 0.11, a tear occurs at other locations away from the perforation line. On the other hand, toilet papers with a stress concentration factor below 0.11 tear along the perforation line. Another study carried out by Vieira et al. [9] showed that the perforation efficiency increases with an increase in the cut distance, stabilizing with a cut distance of 6 mm. The predicted differences of numerical simulations, when compared to experimental tests, decreases from 27% to 4% with a cutting distance ranging from 2 mm to 8 mm. However, the numerical simulations shown a trend in terms of the stabilization of the perforation efficiency for a cutting distance of 6 mm. The current study aimed to verify if the perforation line at 0 ◦ is the best solution to maximize the perforation efficiency. To carry out this study, four commercial two-ply toilet papers were tested with the line of perforation at several angles. The perforation efficiency was evaluated at each angle. According to the authors’ knowledge, there are limited studies on this subject.
2. Simulation–Materials and Methods 2.1. Optimization
The optimization of a constrained problem, using discrete variables, is better per- formed using the genetic algorithm (GA) [10] than using gradient-based methods, with the use of the GA avoiding the trap of local minima [11]. For this problem, the objective was to find the minimum force necessary to detach the toilet paper service by optimizing the angle α and the blank distance d of the paper cuts (see Figure 1), where the cut distance was maintained constant in all simulations ( c = 3 mm). Additionally, a second optimization was performed regarding only the angle α by maintaining the blank distance d =1.0mm. As usual, the design variables were coded as genes (coded as integer numbers) grouped into chromosomes (strings). The chromosomes were weighted as the fitness function (minimum force), representing the chromosome phenotype. Populations of pos- sible optimal values were generated considering their probabilistic characteristics, which evolved over generations through reproductions. To avoid local minima, it is necessary to use enough search points within the design variables space [10]. The GA algorithm begins with a random population and assesses the fitness function. Reproduction is carried out by selecting the best individuals and generating the offspring. During reproduction, the genes can be exchanged by the crossovers [11].
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