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Vol. 8. Issue 1.
Pages 871-875 (January - March 2019)
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Vol. 8. Issue 1.
Pages 871-875 (January - March 2019)
Original Article
DOI: 10.1016/j.jmrt.2017.10.009
Open Access
Blocking of factorial experiments in galvanized wire zinc weight test
Lucas Sonego Fernandes
Corresponding author

Corresponding author.
, José Demétrio Leite de Figueiredo, Oscar Olímpio de Araújo Filho, Edval Gonçalves de Araújo
Federal University of Pernambuco, Mechanical Engineering Department, Recife, Pernambuco, Brazil
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Tables (4)
Table 1. Results for zinc layer test for each factor level.
Table 2. Results for zinc layer test for each factor level.
Table 3. Zinc layer test with blocking applied.
Table 4. ANOVA applied in zinc layer test with blocking.
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Industrial competitivity has been, in last years, the business stress aggravated by the strong advance of globalization, with huge effects in commercial world. The run for lower operational costs and for the resources optimization forces the serious organizations to use statistical methods to guarantee the most different development and improvement projects in its processes, for the effect of an error in these areas can lead to significant losses of company image, revenue and market share. The objective of this paper is to contextualize the factorial experiments with blocking method in industrial applications, giving practical examples for method applications and, with a real example, to apply the blocking tool in order to evidentiate its benefits.

Statistical methods
Factorial experiments
Zinc weight
Full Text

In any manufacturing process, there are important variables that impact directly or indirectly on the final product. The influence of these variables on the result, on the product, may or may not be controlled [1]. The experience of long-time operators working with a production process may be sufficient to avoid the need for total control of all the variables in a production chain or to allow the control of fewer, more important variables. However, some cost, market and even survival needs require more bold projects, developments and improvements, with changes of levels [2]. For these, a more accurate control, with more advanced methods and experimental analyzes to guarantee the effect, the desired change within the specifications of the product/service becomes almost mandatory. It is worth emphasizing that the scope of this paper is not restricted only to the industrial production chain, but to any research that contemplates quantitative or qualitative values, in the most varied areas of science.

Common errors in research projects, developments, and process improvements are seen to cause certain discomforts in the conclusions of statistical analyzes or, often, to hinder conclusions about variables and trends. Such errors are basically concentrated before the application of statistical analysis methods; they are born in data collection/sample collection. The sequence of steps of a well-defined project is practically a change of step, new analysis, new hypotheses, different sciences being applied, etc. So, if there is not a thorough knowledge of the process in which the data are being investigated, the chance of having the wrong conclusions is high. This knowledge is not only about the variables of influence, but also how each behaves, the cause of it, etc. Often, some interconnected factors in the data collection and acquisition of values (quantitative or qualitative) are known, but are difficult to incorporate into the analyzes. An example to illustrate what has been said is the acid concentration analysis in a bath. There are reliable gauges in the market for acid solutions, but not all companies own or have the capital to invest in such a device. Therefore, the laboratory analysis methods use the glassware with various solutions to the operator, at a given time, to wait for the color change of a reagent to from the volume swept in the test tube go to a table to set the concentration. Some non-controllable variables can then be evaluated in this example: reagent staining, and volume swept in the test tube. There is not enough confidence to state that all operators would evaluate reagent staining in the same way, or that the swept volume would be read by each one by the same value. Uncertainty calculations used for instruments would not suffice in this example, except for an in-depth study on the subjectivity of the operators to the color changes of the reagent, which for most institutions is not feasible.

Within this context, the blocking method is highlighted, where all known and uncontrolled parameters that are of considerable influence are blocked, to reduce their erroneous effects on the conclusions of the analyzes. They are extremely important premises, minimizing noise and knowing the factors that have been blocked. A technique that minimizes errors and optimizes time, but does not eliminate process knowledge needs [3].

2Material and methods

There are three important principles in an experimental design: replication, randomness and blocking. Replication has its importance in obtaining the experimental errors and the randomness in obtaining more precise estimates of values. Blocking, on the other hand, aims to increase the precision of the experiment. They are factors that disturb the system, but that there is no interest, for several reasons, in studying them. If the diameter measurement of a drawing wire is made by two people leading to a possible inhomogeneity of the data, each person is treated as a block. This is the fundamental concept of blocking and can be applied in a similar way to the previous example.

Generally, experiments involving more than one variable aim to study the interactions between these variables to make decisions about processes. The analysis of variance (ANOVA) is a widely-used approach in these cases because it allows conclusions regarding the interaction of the variables, since the verification of the difference between means is performed. It is necessary, then, a better detail of the analysis of variance [4].

In any experiment, there are factors and its levels to consider to a specific response. Generalizing, we have yij to the experimental values of the response variable, yit to the sum of the observations for line i (i the treatment), corresponding to the averages of observations for line i, ytt to the sum of all observations and to the average of all observations. Therefore, we have Eqs. (1) and (2) below.

By grouping the different factor levels and number of replicates in an orderly way, adding the columns of totals and averages, we have the experiment data. As the interest is in verifying the difference between the means to, then, to conclude that there is influence or not of the factors in the response of interest, the hypotheses are: equal means (H0) and different means (H1). In Eq. (2), the method basically consists of the division of total variability into its components, expressed in terms of the total quadratic sum.

Further on in Eq. (3), we have the quadratic sum of the differences between the means of the treatments and the global mean and the quadratic sum of the differences between the differences of the observations within the treatments and the average of the treatments – random errors. Eq. (3) can be expressed as.

where SQT is the quadratic total sum, SQT is the quadratic sum of the treatments and SQE is the quadratic sum of the random errors. The degrees of freedom of this sum are respectively a1 and Na. Using the degrees of freedom to express the quadratic mean with the same indexes as above, one can arrive at the F statistical hypothesis test.
where Fo is the calculated F statistic, it is the quadratic mean of the treatments, and is the quadratic mean of the error. Therefore, if Fo is less than F tabulated (considering the level of significance and degrees of freedom), it concludes that the null hypothesis is true. Otherwise, we conclude that the hypothesis H1 is true, the means being different from each other. When the calculated F is greater than the tabulated F (critical), we conclude that the null hypothesis (H0) is rejected, if the hypothesis H1 is true: different means from each other.

In the above example, the way the samples were collected, and the traction test method was not mentioned. As stated in the introduction of this work, the step of obtaining the samples is extremely important for the conclusions of the analyzes of an experiment, which may or may not embody serious errors that may lead to false assertions. In industrial processes and important research, an uncontrolled and unrecognized error can ruin a research project or prevent a change in the threshold of a product, a service. At this stage, the preliminary analyzes are fundamental and this is where the concept of blocking fits, in many cases, in a positive way.

In a process of continuous galvanization of wires, there are more than 7 steps to obtain a galvanized wire, covered with a protective layer of zinc, greatly increasing its life time. The entrance material for continuous galvanizing lines are drawn wires. Due to its high level of dislocations, the galvanizing process already begins with the annealing stage for partial recrystallization of the grains, changing the tensile strength limit of the wires to the ranges pre-established by the customers. At the recrystallization temperature and exposed to atmospheric air, there is immediate oxidation of the wires forcing the need for a surface cleaning with chemical surface treatments, giving the next five steps of the galvanization process. Finally, the wire is continuously submerged in a zinc bath between 450°C and 500°C where the reaction occurs between the iron present in the wire steel and the zinc, forming an intermetallic alloy called the zinc iron alloy [5]. For this reaction, some factors are more important, such as wire temperature, zinc bath temperature, wire surface cleaning and immersion time (as a function of wire speed) [6]. The intermetallic alloy formed by the reaction of the iron and the zinc have different layers, the closer to the wire the higher the iron content and the closer to the pure zinc layer the higher the zinc content [7]. The greater detail of the process will not be approached, since the already explained concept allows the application of the blocking method to continue.

The experiment to be carried out is in function of the immersion time of the wire in the zinc bath to evaluate the zinc layer formed, in unit g/m2.

3Results and discussion

For the velocity factor, related to the immersion time, 3 levels are taken, and through laboratory tests, the measurements of the resulting layers are made for each sample. The test used will be discussed later (Table 1).

Table 1.

Results for zinc layer test for each factor level.

Operator  40 (m/min)  Operator  60 (m/min)  Operator  80 (m/min) 
83  60  13  41 
105  70  14  61 
94  66  15  83 
92  10  75  16  50 
72  11  73  17  77 
70  12  102  18  90 
Mean (g/m286    74.3    67 

Applying the ANOVA method to evaluate the difference between the means obtained, we have Table 2.

Table 2.

Results for zinc layer test for each factor level.

Group  Count  Sum  Mean  Variance 
40 (m/min)  516  86  184.4 
60 (m/min)  446  74.33  212.3 
80 (m/min)  402  67  377.2 
Variability source  SQ  gl  MQ  F  P-value  F critic 
Between groups  1101.77  550.88  2.13  0.15  3.68 
Within groups  3869.33  15  257.95       
Total  4971.1  17         

Therefore, we can assume the null hypothesis as true, as we have seen previously. Even if the means are different from one another, the ANOVA test does not allow this hypothesis due to the analysis of the variances. According to the literature, the influence of the wire speed on the formation of the zinc layer is considerable, that is, it shows that some error built up in the collection of samples or in the tests prevented this assertion from being true, according to Eq. (6).

where Y is the thickness of the total zinc layer, K is the layer growth constant, t is the reaction time and n is the time constant of layer formation.

To evaluate the reason why the ANOVA test did not allow the Eq. (6) to be observed, we return to the data collection, to collect the samples and the tests. Laboratory tests to measure zinc layer are adjacent to production lines and in recent years there has been a radical shift in large organizations to increase the operational multifunctionality of employees, accumulating functions to increase productivity and, at the same time, competitiveness. The test is based on two methods: gravitational or volumetric [8,9]. The volumetric test is based on the dissolution of zinc in an acid bath generating hydrogen in the gaseous state, where the volume of the test tube is changed, and the quantity of zinc consumed is read from a graduated scale. The gravimetric test, in turn, is based on measurements of weight and diameter of the galvanized sample and, after chemical etching, the procedure is repeated with the pickled sample, thus having the total zinc weight. Through geometric relations, one has the zinc layer in g/m2. The most important factor in each of the tests is the subjectivity of the operator in deciding that the reaction between acid and zinc has ceased, because that is where the biggest mistakes happen. For it to be known that the reaction has ceased, one must observe the stagnation of the generation of gases in the sample and this is often at the discretion of the operator. There are more precise but much more expensive methods on the market and most companies are not willing to buy them until then.

If the error of subjectivity in the decision of the laboratory analysis is known [10], it is clear it should be controlled, since the ANOVA test itself did not satisfy the Eq. (6) very widespread in the literature. In the previous planning, the choice of the operators was of random form, which can be said that there was no planning in this question. As mentioned, this error is totally connected to the operator, therefore, one can apply the blocking method, creating for each operator a separate block. As before there were 18 operators in the sample ratio, it is decided to reduce that number to 6 people and each performing the tests at all levels of the speed factor. We have, therefore, Table 3.

Table 3.

Zinc layer test with blocking applied.

Operator  40 (m/min)  60 (m/min)  80 (m/min) 
87  75  65 
86  76  62 
88  73  64 
85  74  66 
83  78  63 
88  73  63 
Mean (g/m286.2  74.8  63.8 

Note the formation of blocks between operators, each with sample results for each level of the speed factor. With this, the errors are blocked, reducing the randomness previously evidenced. If its value is large, it can be inferred that the block has a great influence in the reduction of noise (variability), being useful then to improve the precision of the comparison of means of the treatments.

Therefore, once again applying the ANOVA test to the data in Table 3, we have Table 4.

Table 4.

ANOVA applied in zinc layer test with blocking.

Resume  Count  Sum  Mean  Variance 
87  140  70  50 
86  138  69  98 
88  137  68.5  40.5 
85  140  70  32 
83  141  70.5  112.5 
88  136  68  50 
60 (m/min)  449  74.83  3.76 
80 (m/min)  383  63.83  2.16 
Source of variability  SQ  gl  MQ  F  P-value  F critic 
Lines  9.66  1.93  0.48  0.77  5.05 
Columns  363  363  90.75  0.00  6.60 
Error  20       
Total  392.66  11         

It is observed in the ‘columns’ line that the critical F is smaller than the calculated F, assuming the region of rejection of the null hypothesis, asserting, then, that the hypothesis of different means is true. Thus, Eq. (6) is satisfied.


It is interesting to analyze that the same process generated different responses, only by varying the data collection, organizing them in such a way as to minimize known but uncontrolled errors. In this paper, blocking method using ANOVA was used and was successful, allowing the nuisance factors and levels to not disturb the response.

This approach can be applied to any science, if all the points mentioned here are observed and, in all cases, it is essential to know the process to make any decision. Not all the factors of interest can be controlled, but the expertise applied to the process, planning, testing and data analysis is vital to the success of the experiment. Data alone does not mean anything, but when understood and treated can lead processes, products and services to higher levels, increasing productivity and competitiveness in the globalized market.

Conflicts of interest

The authors declare no conflicts of interest.


The authors are grateful for the postgraduate program of the mechanical engineering department of UFPE for supporting the project.

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