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ESTATISTICA APLICADA E PROBABILIDADE PARA ENGENHEIROS MONTGOMERY PDF

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Estatistica Aplicada E Probabilidade Para Engenheiros Montgomery Pdf

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Villasana et al. Schneider and Neege 24 examined the hydrocyclone simulation for the treatment of non-metallic minerals. Menezes et al. Boylu 12 conducted studies with sodic bentonite for optimization of hydrocyclone, and concluded that the best results were obtained with 4 bars of pressure, vortex opening of In this study, the average diameter of smectite clays was evaluated in terms of hydrocyclone pressure factors, apex diameter and vortex diameter.

A complete factorial design with addition at the central points was used to model the hydrocycloning process, as optimizing factorial design with contour surface can be successfully applied in processing clays in order to evaluate the optimum factor points, simulate the best responses, and evaluate the effects of the main factors, in addition to evaluating the interactions between the adopted factors.

According to 21 , the chemical composition of clay was composed of: A schematic of this equipment can be found elsewhere The process variables used in this study were: pressure 3.

After this period, the dispersion was pumped into the hydrocyclone for centrifugation with pressures ranging from 3. Physical, chemical and mineralogical characterization of the sample in natura and after hydrocycloning were performed using: granulometric analysis by laser diffraction Cilas, mod.

In cases where the objective is to reach a target value, the transformation formulation stops being unilateral and becomes bilateral. The bilateral formulation, represented by equation 5, occurs when the interest response has two restrictions: one maximum and the other one minimum. Materials and methods 3. Material, factors selection and experimental organization The material used was SAE steel wire, with diameter gauges 2.

The steel wire diameter was also considered as an important factor, for there was assumption that its mass could influence results of investigated mechanical properties. Nevertheless, in this research, it was used the blocks analysis methodology, that is, for block 1, it was allocated experiments related only to diameter 2.

Factorial Matrix 23 68 Design of Experiments - Applications Factors such as speed, lead temperature and polymer concentration were tested by means of the factorial planning, using the matrix The result was divided by half the amplitude R between the minimum and maximum values of factors adjustment. Thus, the reduced variables dimensionality was restricted to the range [-1 to 1], according to equation 6 and Table 2.

Transformation of physical variables to reduced variables 4. Results and discussion 4.

Sequence of experiments and statistical analysis In the experiments, all replicas related to block 1 were initially carried out, and then the ones corresponding to block 2. Six replicas were used for each experimental condition.

This experimental sequence is displayed in parentheses and in subscript format next to values obtained from mechanical properties as displayed in Tables 3, 4 and 5.

This analysis was carried out separately so that factors significance for each response of studied mechanical properties could be verified, as shown in Tables 6, 7 and 8. Terms Effect Coefficient T p Constant Statistical modeling for multiple responses Using coefficients calculated using the significance test, by means of the Minitab Statistical Software, it was possible to build statistical models which represent the relationship between process input variables factors and output variables mechanical properties.

Such statistical models are defined in equations 7, 8 and 9.

To this, blocks were analyzed separately, that is, the response variables were optimized primarily for the wire diameter 2. Specifications minimum, nominal and maximum concerning the diameter the 2. In that case, one seeks nominal values target for mechanical properties such as traction resistance limit and hardness and, for the mechanical property called yield, one seeks to maximization, for the higher the value, the better the product itself.

The composite desirability D is the overall index calculated from combination of each response variables processed through a geometric mean and this index is responsible for showing the best condition to optimize all responses variables at the same time. Specifications for 2. Figure 2. Desirability function applied in multiple responses Minitab Statistical Software Values obtained for desirability D and individual desirability d , show that the process was well optimized, since these indices are found to be very close to the optimum condition 1.

By analyzing Figure 2, it was found that speed factor, when increased, also causes increased amounts of response variables tensile strength MPa and hardness BH.

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A concern in using 2 k planning is the linearity assumption on the effects of factors. Therefore, the addition of CtPt is used to increase the number of levels in 2 k , meaning, from 2 to 3 levels. The central points were added in triplicate, since they do not have repercussions on estimates of the effects on planning 28 , Statistical analysis of data: The procedure was used to infer whether such effects really exist, to determine the level of confidence In this case, the idea was to perform a breakdown of the variation into the variation attributed to differences between the experimental units mean square of the residue - MSR and the variation attributed to differences between the experimental units and attributed to the differences caused by treatments or factors mean square of treatments - MS treat.

The test is based on two hypotheses: The F-test was used to compare means. In this case, the controllable variables influenced the response variable. Table II shows the configuration for Anova analysis of variance. The coefficient of determination R 2 was calculated using Eq.

It represents the explained variation of the model.

The correlation coefficient R was calculated using Eq. B which represents the correlation between the variables. R values range between -1 and 1, and the closer to 1 means better results.

Value -1 indicates a negative linear correlation, and value 1 indicates a positive linear correlation.

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The maximum percentage of explainable variation was calculated using the Eq. C, which represents the maximum variation that the model explains. This value was compared with the value explained by the model R 2 max. In the factorial design 2 3 with addition of 3 CtPt central points , the adopted model is presented in Eqs. It was observed that the sample presented the following mineralogical compositions: These results showed that the clay used in this study had elevated levels of accessory minerals such as kaolinite, mica, quartz, feldspar and calcite.

These results are similar to preliminary results found in Figure 1: X-ray diffraction patterns for the AM1 sample in natura, with and without ethylene glycol. Figura 1: Difratogramas de raios X para a amostra AM1 in natura com e sem etilenoglicol.

It was observed that the clay in natura presented bimodal behavior with wide distribution range varying from 0. The clay fraction corresponded to Figure 2: Particle size distribution of the sample in natura. Figura 2: Table III contains the results of the average diameter of particle sizes. Regarding the hydrocyclone process variables, from in natura clay particle size distribution Fig. In studies 21 , 30 on the use of hydrocycloning for low pressures below 3 bar, it was found that the clay fraction concentrate values were lower than those obtained in this study.

On the other hand, the apex, vortex and pressure factors influenced the response when analyzed individually. The higher response values for the average diameters of the apex, vortex and pressure occurred when the level was at the lowest value: Figure 3: Values of main effects for the average particle diameter.

Figura 3: In Table IV , the analysis of variance showed that a second-order model can possibly be adopted, as there is curvature in the evaluated region. However, since the obtained model Eq. A can explain In Eq. F the principal factors apex, vortex and pressure were significant and appear in the equation with high coefficients, being respectively: The positive sign that precedes the factors indicates the importance of the factor in hydrocyclone.

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Regarding the interaction, the significant factors influencing the response were: The model explains Figure 4: Adjusted values x residues: Figura 4: Regarding optimization, it was possible to obtain the optimal or stationary point of the adopted factors. In this case, the optimum point was equal to 4.

This value represents the optimal response of initially adopted factors in the experimental design. The interactions of the vortex and apex factors shown in Fig. From the result obtained for the optimum point 4. Figure 5: Optimized response of apex, vortex and pressure factors. Figura 5: Figure 6: Surface contour and optimized response of: Figura 6: Table V contains the simulated values in the optimum point region of the significant factors.

First, by varying the pressure and by fixing the apex and vortex simulations 1 to 10 ; then, by varying all factors in the region near the optimum point simulations 11 to The lowest values for the average diameter was achieved when pressure increased.

This value was obtained in the simulations 1 to 10, wherein the diameter of the apex was set at 4 mm, the vortex at 5 mm optimal points , and the pressure in the optimum region varied between 8 to The best result was obtained with a pressure equal to Next, simulations 11 to 20 were performed by varying all the factors in the regions close to the optimum point; the lowest value for the average diameter was obtained in simulation 16, wherein the value for the average diameter was equal to 4. After studying hydrocyclone process variables for treating smectite clays using modeling, simulation and optimization, it can be concluded that: Paulo Silva, J.

Costa, R. Menezes, H.Figure 3: Golyk, S. Harvard Business Review, Pereira, V.

Design of experiments is a systematic approach to experimental design and analysis that has the potential to improve the efficiency and effectiveness of wind tunnel testing. These clays are constituted by lamella formed by an octahedral sheet of alumina Al 2 O 3 between two tetrahedral silica sheets SiO 2.

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