Getting a Grip on Scrap

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Getting a Grip on Scrap

Applying Probability and Statistics in Analyzing Scrap and Steel Composition Data from Electrical Steel

Production

Seyed Mohamad Seyedali

February 2013

Master thesis

School of Industrial Engineering and Management Department of Material Science and Engineering

Royal Institute of Technology SE-100 44 Stockholm

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Master of Science Thesis MMK 2013:x MKN yyy

Getting a Grip on Scrap: Applying Probability and Statistics in Analyzing Scrap and Steel Composition

Data from Electrical Steel Production

SeyedMohamadSeyedali

Approved Examiner

Pär Jönsson

Supervisor

Patrik Ternstedt

Commissioner

Kobolde & Partners AB

Contact person

Rutger Gyllenram

Abstract

This study intends to better control the final composition of steel by trying to have a better knowledge of elements including copper, nickel, molybdenum, manganese, tin and chromium in the scrap. This objective was approached by applying probability and statistical concepts such as normal distribution, multiple linear regression and least square and non-negative least square concepts.

The study was performed on the raw materials’ information of Ovako Smedjebacken and Ovako Hofors, two steel production plants in Sweden. The information included but were not limited to the amount of the different scrap types used in the charge, total weight of the charge and the final composition of the produced steel.

First, the concept of normal distribution was used as to consider the variations of the alloying elements between the estimated and measured alloy contents. The data were then used to consider a model for distribution factor of the studied elements. Also, an estimation of the alloy contents in the scrap type given the final steel composition was carried out using the concept of probability and statistics. At the end, a comparison of the results from the different concepts was done.

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This thesis is dedicated to the loving memory of my

grandfather, the man I admired the most.

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Acknowledgment

First, I would like to deeply thank Ovako Smedjebacken and in particular Anders Gustafsson and Ovako Hofors and specially Ola Stüffe since this study was not possible without charge data and their support.

I would then like to express my greatest gratitude to my supervisor at Kobolde & Partners AB, Patrik Ternstedt, for his excellent support, guidance and patience during this study. Moreover, I extend my deepest feelings of appreciation to Rutger Gyllenram for his immense help from the beginning to the end of the study. Furthermore, I would like to endlessly thank Olle Westerberg for providing me the opportunity to write this thesis at Kobolde & Partners AB and for his fruitful comments. I am also honoured and privileged to have Prof. Pär Jönsson of KTH, as my support and examiner for this thesis. At the end, I would like to thank all the members of Kobolde & Partners AB for providing me with a friendly and professional environment to work in.

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1. Introduction

According to Ovako’s 2011 annual report, this company is the biggest Swedish scrap buyer [1].

Plants located at Hofors, Hällefors, Smedjebacken and Boxholm all in Sweden, along with Imatra in Finland form the backbone of the company’s steel production. The data analyzed in this report is obtained from Hofors and Smedjebacken.

This data is being used in order to perform a series of analyses based on probability theories and statistics such as the concept of normal distribution and multiple linear regression in order to better control the final composition of steel by trying to have a better knowledge of elements in the scrap. This data contains the record of different charges fed into the Electric Arc Furnaces.

These records basically include but are not limited to the amount of each scrap type used in the charge, the total weight of charge, and the final composition of the produced steel. The difference between the record received from Hofors and Smedjebacken is that the latter also contains an estimation of the amount (percentage) of alloying elements in every single scrap type used in each charge. These percentages are obtained by a series of measurements carried out at the plant by a separate investigation and provided to this study. The data from Hofors on the other hand; contains this estimation only for ten of the scrap types out of twenty.

The elements considered in this analysis are Cu, Mn, Mo, Cr, Ni and Sn and they have been analyzed individually. The reason for choosing these elements is that they can play different positive and negative roles during steel production. For instance, copper is in most cases considered an undesirable tramp element and you generally try to keep it below 0.2% and Sn is generally an unwanted element in steel production processes [2].

It is also worth mentioning that most of these alloying elements can not be easily removed if entered in the steel production cycle and even if it is achievable, it will incur high costs for removal [2]. Also, the fact that some of these elements and in particular nickel are expensive, makes it necessary to pay a particular attention to their amount, addition and removal during the smelting processes [2].

2. Method

The basic probability concepts which need to be explained in this report can be mentioned as follows;

2.1. Variance, Standard Deviation and Normal Distribution

Normal distribution is one of the central concepts in statistics. The main reasons for that can be listed as follows [3];

1) A high analytical tractability.

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2) The symmetrical shape of the graphs (known informally as bell curve) makes it a popular choice for population models. Even though other distribution models are also available, most of them do not have the analytical tractability of normal distribution.

3) According to the Central Limit Theorem normal, a normal distribution can be a suitable choice for approximating large variety of distributions in large samples.

Mean and variance are the two parameters of normal distribution that are noted as µ and σ respectively. Variance can be stated as the second central moment of one random variable X and it provides a measure of the degree of spread of a distribution around its mean value [3]. It is obvious that a zero variance indicates no variation in the random variable X [3]. The concept of variance gives rise to a standard deviation (SD). A standard deviation, on the other hand, can be described as the positive square root of variance of random variable X [4]. Small values for standard deviations mean that X is less variable compared to the cases with higher values [4].

A variance describes the average of all possible outcomes, weighted according to the probability of their occurrence. The standard deviation of X, shortly written as sd (X), can be expressed as [4]:

( ) √ ( )

where, var (X) is the short term for variance of X which can be expressed as [4]:

( ) E (X-µ)² σ ²

where, X is a random variable, µ=E(X) is the expected value (here a mean value) of a set of data [4].

The noted equations of (2-1) and (2-2) will be used to generate a normal distribution bell curve.

A normal distribution function can be expressed mathematically as [4]:

( )

√ ( ^{( )} )

The function produces a bell curve graph and probability mass will be put symmetrically around the mean value [4].

2.2. Multiple Linear Regression

Linear regression is a technique which is frequently used to make credit-scoring models from data [4]. The linear probability model can be expressed as follows [4]:

( | ) ( | ) 0 1 1 k k

Where y is dependent variable, x_kis the independent variable and k is a coefficient. The idea of using linear regression is to generate a function with multiple independent variables, which have (2-1)

(2-2)

(2-3)

(2-4)

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an effect on a dependent variable. This function enables the prediction of behavior of y given x.

In case of this study, y is the percentage of a specific alloying element in the final composition, x is the amount of scrap type k and k is the content of the specific element in scrap type k. 0 is the base coefficient which in this case is equal to zero. The reason is that ₀states the amount of the specific element entering the process from the sources other than scrap. In this thesis, it was assumed that scrap is the only source of alloying elements. It is obvious that by having the values of x_k and y, it is possible to calculate _k. Later, by applying the concept of a standard deviation, it is possible to see how reliable the obtained values of k are.

2.3. Least Square Method and Non-Negative Least Square Method

Another concept applied in this study is the least square method. It can be explained as drawing a line which passes through a number of data points and is as close as possible to all of them. If the linear equation of this line is y = c + dx, the Residual Sum of Squares (RSS) will be [3];

∑ ( i – (c +dxi))²

RSS measures the vertical distance from each data point to the line c + dx and then sums the squares of these. The least square method estimates the intercept a and slope b through linear regression, such that the line a + bx minimizes RSS [3];

∑ ( i – (c +dxi))² = ∑ ( i – (a + xi))² The calculations were carried out as follows;

∑

Where mj is the mass of scrap type j, c’is the content of a specific element (e. g. Cu) in scrap type j, mi is the total mass of the charge i and c’fin is the concentration of the mentioned specific element in the final steel analysis (produced by charge i). This can be expressed as in the following matrices which will be used in the least square method as;

[ ] [ ] [ ] To be more specific about each matrix it must be noted that;

[

] [ ] [

( ) ( ) ]

a) A is an i × j matrix of i number of charges and j number of scrap types. (e. g. mij array is the mass of scrap type j in charge i)

(2-5)

(2-6)

(2-7)

(2-8)

(2-9)

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b) B is i × 1 matrix in which m_totand c_fin represent the total mass and percentage of a specific element (e. g. Cu) in the final composition for charge i respectively.

c) X is 1 × j matrix in which c’ is the concentration of the element in matrix B in scrap type j. Subsequently X can be calculated by the least square method as follows [5];

X = (A^TA)^-1 A^TB

The arrays in this matrix are the concentration of the mentioned specific element (e. g.

Cu) in every single scrap type.

As will be discussed in results and discussion section, the results obtained using this method contained negative concentration. Since there is no such thing as negative concentration, application of an alternative method was necessary. For this reason, the concept of Non Negative Least Square (NNLS) was applied. In many real life situations, non-negativity might be favorable in order to avoid the physically impossible and interpretable results. Non negative least square inverse problem can be expressed as [6];

²; [ i]> 0, ∀i

where A is m × n matrix, b is a real vector of length m and x is a real vector of length n.

represents Euclidean 2-norm and [.]i is the ith entry of the vector [6].

In this thesis, GNU Octave Software was used for solving the NNLS problems [7]. The command used for this purpose is known as lsqnonneg [8]. This command uses an algorithm which is mentioned in literature (see [9] and [10]).

2.4. Analysis of the Data

The data provided by Smedjebacken was used for two major estimations. First, the concept of normal distribution was used as to consider the variations of alloying elements. Second, the data were used to consider a model for distribution factors of the elements studied in this report.

For the first estimation, records of 972 charges were analyzed by the normal distribution method.

The amount of each type of scrap in each charge was given in the record along with the total weight of the charge. The amounts (percentages) of considered elements (Cu, Ni, Mo, Cr, Sn and Mn) in each scrap type were also given. Equation (2-12) summarizes the calculation of the total percentage of considered elements in each charge:

∑ ( _∀ ) = cki

Where mj is the mass of scrap type j, ck is the measured content of element k in scrap type j, wi is the total weight of charge i and c_kiis the percentage of element k in the steel produced by charge i. The resulting figures where later subtracted from the measured final composition which was (2-12) (2-10)

(2-11)

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also included in the records of the company. The result was then used to obtain the standard deviation and later for calculation of a normal distribution bell curve. The same mathematical process was performed for five steel types (chosen randomly out of 972 steel charges). A normal distribution curve was plotted for them so that it would be possible to compare the different steel types and whole steel types all together. The five steel grades which have been chosen were 98580, 13127, 98177, 92920 and 96760. The variations of the final compositions between these five grades are relatively minor (the same is true for the compositions of the scraps types that were used to produce these grades) and the only basis for selecting these grades is that they are mentioned more frequently in 972 charges.

Normal distribution was plotted for two different cases. The first case is that all the existing elements in the scrap will end up in the melt. The second condition is that not all the Cr and Mn content end up in melt. The reason is that the concentrations of these two elements are related to the carbon content, while this effect is not significant for the rest of the elements. In the first case, the normal distribution curve for Cr and Mn will not be centered around zero as will be shown in the results section.

For the second condition therefor, a distribution factor needed to be estimated. In order to do so, the reaction of the elements with oxygen needed to be taken into account. It was important to note that according to Ellingham diagram, some of the elements which are placed at the top of the diagram have a smaller oxygen affinity than those located at the lower parts. So it is easier for elements such as Mn and Cr to form oxides compared to Cu and Ni [11].

Also, it is important to notice that some elements such as Cr and Mn compete with C for oxygen in the reactions such as (2-13) and (2-14) [11];

2Cr(s) + 3CO(g) = Cr2O3(s) + 3C(s)

Mn(s) + CO(g) = MnO(s) + C(s)

In the above reactions, Cr and Mn will be oxidized and go to slag while the reverse reaction occurs simultaneously. This will result in the distribution factors of these elements being a function of carbon concentration. Therefore, it creates the need to consider the effect of C content in the melt on the concentration of other elements and to see that how much of these elements will end up in the slag as the result of these reactions.

For calculating a distribution factor with regard to the carbon content in the melt the following formula was used:

∑ _∀ = dk

where m_kji is the mass of element k in the scrap type j in charge i, m’_i is the content of element k in the final production from charge i and dk is distribution factor for element k. Then dk was (2-15) (2-13) (2-14)

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plotted against carbon concentration in the final steel composition (c_C) for all the considered elements which can be seen in results section. By drawing the linear regression line and calculating its equation, it was possible to obtain a function for the distribution factor of a specific element. This function depended on the carbon content.

By inserting the carbon content (measured and given in the aim composition for each charge) in the function, dividing ( ) in equation (2-9) by the resulting value and defining A and X matrices mentioned in (2-9), non-negative least square calculations were done with Octave to obtain an estimation of the amount of a specific element in each scrap type.

The same distribution factor function obtained from Smedjebacken was used for the Hofors charge records to estimate the amount of elements in each scrap type. Furthermore, it was used to test the accuracy of the obtained distribution factor from Smedjebacken.

3. Results and Discussion

3.1. Normal Distribution for Testing the Estimated Scraps’ Compositions

Plots of the normal distribution curves for each element in 5 different scrap types (separately and together) can be seen in the following graphs. These graphs show the Smedjebacken data set.

Figures (3.1) to (3.6) show the normal distribution for the difference between the measured and calculated (measured – calculated) Cu, Ni, Mo, Sn, Cr and Mn contents in the final compositions respectively.

Figures (3.1) to (3.4) show that the resulting difference is distributed closely around zero, which shows that the calculated values are relatively close to the measured ones. Only slight shifts towards the positive (for the Cu and Sn) and negative (for the Ni and Mo) can be seen in the curves. A shift in the positive direction shows that the measured values are slightly higher than the calculated ones, while the negative shifts state the opposite.

However, as figures (3.5) and (3.6) are suggesting, the shifts are more pronounced for Cr and Mn compared to Cu, Ni, Sn and Mo. The reason for the shifts is probably rooted in the relation between the amount of Cr and Mn with the carbon content in the melt, as mentioned earlier in the section (2.4). So it was necessary to find a distribution factor. The dependency of the considered elements on carbon content is discussed in the section (3.2). The two figures also show that the shift towards the negative is greater for Mn than Cr. This is probably due to the huge difference between the amount of Mn in the scrap type (average 0.76%) and the Mn content in the final composition (average 0.09%).

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Figure 3.1: Normal distribution curve for Cu. y axis indicates density of function for distribution of Cu while x indicates difference between measured and calculated Cu content (measured – calculated).

Figure 3.2: Normal distribution curve for Ni. y axis indicates density of function for distribution of Ni while x indicates difference between measured and calculated Ni content (measured – calculated).

-5 0 5 10 15 20 25 30

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Deansity of Distribution

Difference (Measured - Calculated)

Normal Distribution Curve for Cu

All 98580 13127 98177 92920 96790

-5 0 5 10 15 20 25 30 35

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Deansity of Distribution

Normal Distribution Curve for Ni

All 98580 13127 98177 92920 96790

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Figure 3.3: Normal distribution curve for Mo. y axis indicates density of function for distribution of Mo while x indicates difference between measured and calculated Mo content (measured – calculated).

Figure 3.4: Normal distribution curve for Sn. y axis indicates density of function for distribution of Sn while x indicates difference between measured and calculated Sn content (measured – calculated).

0 10 20 30 40 50 60 70 80 90

-0.08 -0.06 -0.04 -0.02 0 0.02 0.04 0.06 0.08

Density of Distribution

Normal Distribution Curve for Mo

All 98580 13127 98177 92920 96790

-50 0 50 100 150 200 250

-0.03 -0.02 -0.01 0 0.01 0.02 0.03

Density of Distribution

Normal Distribution Curve for Sn

Series1 98580 13127 98177 92920 96790 All

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Figure 3.5: Normal distribution curve for Cr without applying distribution factor.

Figure 3.6: Normal Distribution curve for Mn without applying distribution factor.

-5 0 5 10 15 20 25

-0.4 -0.2 0 0.2 0.4

Normal Distribution Curve for Cr

Series1 98580 13127 98177 92920 96790 All

0 2 4 6 8 10 12 14

-1 -0.8 -0.6 -0.4 -0.2 0

Normal Distribution Curve for Mn

All 98580 13127 98177 92920 96790

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The results for modeling of the distribution factors for the considered elements are presented in figures (3.7) to (3.12). These distribution factors were calculated for the data obtained from Smedjebacken and then tested on Hofors data when necessary.

As figure (3.7) to (3.10) show, Cu, Ni, Mo and Sn do not indicate any relation with the carbon content. The slope of the trend lines are relatively low and most of the data points are scattered around 1 along the y axis which represents the ratio between the element content in the charge and the element content in the final steel. This suggests that almost all elements in the charge will end up in the final composition regardless of the initial carbon content.

Figure 3.7: Ratio between estimated Cu in 972 charges and Cu in the final composition is presented on the y axis plotted versus carbon content in the final steel produced in each charge in x axis.

Figure 3.8: Ratio between estimated Ni in 972 charges and Ni in the final composition plotted versus carbon content in each charge.

y = 0.4529x + 0.9929 0

1 2

0 0.1 0.2 0.3 0.4 0.5

Distribution

%C

Distribution Factor for Cu

y = 0.2817x + 0.9644 0

2 4

0 0.1 0.2 0.3 0.4 0.5

Distribution

%C

Distribution Factor for Ni

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Figure 3.9: Ratio between estimated Mo in 972 charges and Mo in the final composition plotted versus carbon content in each charge.

Figure 3.10: Ratio between estimated Sn in 972 charges and Sn in the final composition plotted versus carbon content in each charge.

Figure 3.11: Ratio between estimated Mn in 972 charges and Mn in the final composition plotted versus carbon content in each charge.

y = 0.3919x + 0.9424 0

2 4 6

0 0.1 0.2 0.3 0.4 0.5

Distribution

%C

Distribution Factor for Mo

y = -0.013x + 1.0979 0

2 4

0 0.1 0.2 0.3 0.4 0.5

Distribution

%C

Distribution Fcator for Sn

y = 0.9158x + 0.0389 0

0.2 0.4 0.6 0.8

0 0.1 0.2 0.3 0.4 0.5

Distribution

%C

Distribution Factor for Mn

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Figure 3.12: Ratio between estimated Cr in 972 charges and Cr in the final composition plotted versus carbon content in each charge.

According to figures (3.11) and (3.12), the Mn and Cr contents show a relation to the C content.

Mn is distributed approximately between 0 and 0.4 meaning that ratio between the Mn in the final composition and the Mn in scrap is relatively low. The trend line also has a greater slope compared to the other elements except for Cr. Cr also has a relatively high slope and a wide distribution around 0 and 1.

The trend lines for the graph (3.11) and (3.12) show the following function for the Cr and Mn content as a function of the C content;

dCr= 2.6 cC +0.17 dMn = 0.9 cC + 0.03

where d_Cr and d_Mn are the distribution factor for Cr and Mn respectively. Furthermore, c_Cis the amount of C in the aim composition for each charge. As mentioned earlier, cC was replaced by the carbon content in the final composition. The result was multiplied by the left side of the equation (2-12). By using the obtained values to plot the normal distribution curve between measured and calculate values, an adjusted graph for Cr and Mn was obtained thus can be seen in figures (3.13) and (3.14).

As can be seen in the figure (3.13) and (3.14), applying the distribution factor for Cr and Mn resulted in the plot to be distributed approximately around 0 for both cases.

Also for further investigations, ( ) in equation (2.9) was divided by d_Cr and d_Mn. Then, the same matrix calculations which were explained in the penultimate paragraph of section (2.4) were done with Octave to calculate non negative least square. The result was an estimation of the amount of Cr in each scrap type that can be seen in the table (1), (2) and (3) and as will be discussed in section (3.3).

y = 2.651x + 0.1774 0

1 2 3

0 0.1 0.2 0.3 0.4 0.5

Distribution

%C

Distribution Factor for Cr

(3.1) (3.2)

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Figure 3.13: Normal distribution curve for Mn with applying distribution factor.

Figure 3.14: Normal distribution curve for Cr with applying distribution factor.

3.3. Least Square Method

Table (1) shows the results obtained through the least square method for the concentration of Cr in two different data sets from Hofors and Smedjebacken. These results contain negative values.

-5 0 5 10 15 20 25 30 35

-0.2 -0.1 0 0.1 0.2

Difference (Measured-Calculated)

Normal Distribution Curve for Mn

All Scraps 98580 13127 98177 92920 96790

-5 0 5 10 15 20 25 30 35 40

-0.15 -0.1 -0.05 0 0.05 0.1 0.15

Normal Distribution for Cr

All Scraps 98580 13127 98177 92920 96790

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As can be seen in table (1), a negative concentration for Coke for Smedjebacken and for A333 for Hofors is calculated. These two values are being calculated as 0 when using the NNLS method. This is due to one disadvantage of the LS method, for which some values of x will result in negative probabilities larger than 1 and smaller than 0 [4].

Table 1: Comparison between least square (LS) method and non-negative least square method (NNLS) in terms of estimation of negative concentration. The data are from Smedjebacken and Hofors.

Raw Material (Smedjebacken)

Composition (LS)

Composition (NNLS)

Raw Material (Hofors)

Composition (LS)

Composition (NNLS)

W11B 0.099 0.097 A101 0.153 0.153

W741 0.331 0.331 A103 0.318 0.318

W711 0.282 0.28 A107 0.276 0.276

W721 0.349 0.348 A212 0.635 0.635

W27 0.201 0.198 A333 -0.003 0

W28 0.239 0.236 A402 0.301 0.301

W11 0.129 0.127 A407 0.3 0.3

W12 0.212 0.21 A409 0.503 0.503

W22 0.423 0.421 A412 0.753 0.753

W633 0.314 0.311 A416 0.182 0.182

W117 0.174 0.172 A452 1.061 1.06

W31 0.18 0.177 A455 0.021 0.022

W37T 0.283 0.282 A462 1.169 1.169

W38 0.134 0.131 A466 0.352 0.352

W36 0.115 0.112 A472 1.162 1.162

W37 0.106 0.103 A492 0.597 0.597

W100 0.096 0.094 A604 0.498 0.498

Wcoke -0.69 0 A607 0.151 0.151

W12B 0.165 0.163 A686 0.152 0.152

W29 0.419 0.416 A698 0.372 0.371

W177 0.1 0.099 - - -

W721B 0.252 0.251 - - -

W27B 0.087 0.085 - - -

W742 0.337 0.336 - - -

3.4. Cr Composition Estimation by Applying the NNLS Method and Distribution Factor

By applying the obtained distribution factor and the NNLS method, an estimation of the Cr content in each scrap type was done as presented in table (2). Also, the difference between

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measured and calculated Cr content was calculated along with the standard deviation for the amount of scrap type on loading (SD) and the portion that each scrap type consist in the whole charge. Figure (3.15) shows the calculated Cr content (obtained by distribution factor form Smedjebacken data set and NNLS method) plotted against measured Cr content in each scrap type for Smedjebacken and Hofors data sets respectively. The Hofors data points are located under the linear regression. The reason is that these data points are only for 10 of the scrap types out of 20. These scrap types are usually loaded in lower amounts compared to the other ten scrap types. The data for these figures are presented in ables (2) and (3) respectively.

As can be seen in the table (2), the absolute value of difference for Smedjebacken is varying between 0.015 for W27B and 0.415 for W29. By considering the values for the difference it is possible to explain the important factors which may contribute to an increase or a decrease of the match between the calculated and measured Cr content.

For evaluation of the results, two factors have been taken into consideration. These are (a) the SD of each scrap type on loading and (b) the portion (amount used) of each scrap type in the total charge. SD is the variation of weight of each scrap type between charges. It is worth mentioning that the higher SDs are favorable. A higher SDs indicate that the data points that were used for drawing least square line were more spread out (e. g. forming a linear cloud), which gives a better basis for calculating the line compared to the case where the data points are gathered (e. g. forming a circular cloud). This spread of data points will result in a more accurate line as the outcome.

It can be seen that in most scrap types with relatively high SD (greater than average), the difference is lower. In cases were both SD and portion are high, some of the lowest differences are recorded.

Table (3) shows the difference between the measured and calculated Cr content in the Hofors charges with the same distribution function as obtained at Smedjebacken.

The data from Hofors contained measured Cr contents for half of the scrap types. Therefore, it is difficult to be deterministic about the results. As can be seen in the table (3) the scrap types have mostly high amounts of Cr. The average amount of Cr in the aim composition for about 560 charges in the Hofors case is approximately 0.25 and the standard deviation for these charges is about 0.13 percent. With regard to these percentages and the differences between measured and calculated values, the accuracy of the results can be questioned. However, given the fact that the considered scrap types are just above 30% of the total charge and given that not all the scrap types were considered (due to lack of data), this inaccuracy can be explained. The accuracy of the data provided by both companies for this study is also another important factor for evaluating the results. It is important to know the sensitivity of the companies to variations in the concentrations of each element to be able to judge on the results. This is usually available in the internal analysis book of the steel companies.

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By looking at the difference between measured and calculated data from Hofors, it can be seen that the lowest differences occurred when the standard deviation for the amount of scrap type on loading is relatively high. Also by looking at the scrap types with the same Cr content, it can be seen that the scrap type with a lower difference has a higher standard deviation and portion.

Table2: Calculated Cr content can be compared by considering standard deviation and portion of scrap types during each charge (Smedejebacken date). *AV stands for Absolute Value.

Scrap type Measured Cr

Calculated Cr

Difference (Msrd-Calc)

SD on Loaded Scrap Type

Portion (%)

W11B 0.054 0.09687 -0.04287 8.614397 3.754208

W741 0.242 0.33051 -0.08851 0.788558 0.152605

W711 0.487 0.28036 0.20664 3.227225 1.927393

W721 0.506 0.34764 0.15836 3.067583 1.280705

W27 0.07 0.19762 -0.12762 2.756695 2.071704

W28 0.144 0.23632 -0.09232 5.393928 9.86453

W11 0.1 0.12676 -0.02676 12.94501 27.79137

W12 0.278 0.20982 0.06818 2.735445 0.859582

W22 0.353 0.42142 -0.06842 4.670838 8.185773

W633 0.451 0.31116 0.13984 2.026119 0.837485

W117 0.242 0.17201 0.06999 6.642248 22.9592

W31 0 0.17744 -0.17744 1.727309 0.368524

W37T 0.501 0.28226 0.21874 1.992091 1.588864

W38 0.079 0.13124 -0.05224 1.883025 0.455466

W36 0 0.11248 -0.11248 4.094383 2.371768

W37 0 0.10311 -0.10311 1.561047 0.25522

W100 0.042 0.09413 -0.05213 7.329845 4.233637

W12B 0.18 0.16281 0.01719 9.157388 4.435711

W29 0 0.41596 -0.41596 0.631401 0.138485

W177 0.038 0.09861 -0.06061 1.972277 0.173655

W721B 0.376 0.25077 0.12523 3.550397 1.876252

W27B 0.101 0.08536 0.01564 3.321176 2.539976

W742 0.498 0.33647 0.16153 2.930315 1.877895

Average 0.2 0.2 0.11 (AV*) 4.04 4.34

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Table3: Calculated Cr content can be compared by considering standard deviation and portion of scrap types during each charge (Hofors data). *AV stands for Absolute Value.

Scrap Type Measured Cr

Calculated Cr

Difference (Msrd-Calc)

Portion SD on Loaded Scrap Type

A212 1.42 0.63 0.79 2.51643 6.70934

A402 0.21 0.3 -0.09 3.511196 8.026488

A412 1.41 0.75 0.66 10.07388 12.07652

A416 1.41 0.18 1.23 3.601431 5.554658

A452 1.59 1.06 0.53 3.734261 9.050677

A455 1.59 0.02 1.57 1.122883 3.243887

A462 1 1.17 -0.17 1.782141 6.678897

A466 1 0.6 0.4 0.769127 3.140641

A472 1.28 0.5 0.78 2.103569 8.229245

A492 1.28 0.15 1.13 1.524769 2.513744

Average 1.2 0.53 0.68 (AV*) 5 6.4

Figure 3.15: Measured Cr Vs. calculated Cr content for Smedjebacken data set (left) and measured Cr Vs.

calculated Cr content for Hofors data set (right).

4. Conclusion

As mentioned in this text, the purpose of this study was to better control the final composition of steel by trying to have a better knowledge of elements in the scrap. According to the results and discussion section it is possible to conclude that:

 there is relatively a good match between the measured and calculated contents of considered elements. This was suggested by normal distribution curves.

0 0.1 0.2 0.3 0.4 0.5

0 0.2 0.4

Measured Cr

Calculated Cr

Measured Vs. Calculated Cr

0 0.5 1 1.5

Measured Cr

Calculated Cr

Measured Vs. Calculated Cr

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 the distribution factors obtained by this study provided a reasonable estimation of the relation between the Cr and Mn content and C content.

 the conditions for the methods to work are;

1) Using NNLS method instead of LS method.

2) Correction for oxidation of Cr and Mn which depends on C content.

 As it will be discussed in the future work section, along with verification of the results, more data on scrap analysis and accuracy of the received data are also needed to be able to judge the results obtained by NNLS method for estimation of the elements’ content in the scrap types. However the method is possibly validated if data set meets the following conditions;

a) High standard deviations for the amount of scrap types on loading can be the most important factor which might contribute to lower difference between the measured and calculated Cr content.

b) High portion of scrap types also can contribute to decrease the difference between the measured and calculated Cr content, especially when the standard deviation is also high.

5. Future Work

The verification of the results is probably the most important part for the future work. For this verification, this study suggests the following;

a) An X-Ray Fluorescence (XRF) which is an appropriate method for a spectrochemical determination of the elements existing in a sample [12]. Therefore, an on-site evaluation of the content of the elements considered in this study is needed in order to be able to evaluate the accuracy of the results.

b) Plotting normal distribution curves for the calculated Cr contents by NNLS method might be of interest. With this, it is possible to see how contents obtained by NNLS deviate from the actual Cr contents obtained by measurement of the analysis of the final steel product.

c) Obtaining more data on scrap compositions from Ovako Hofors seems necessary. The data used in this study lacked measured Cr contents for some of the scraps. Thus it is difficult to make determinations on the results obtained from the Hofors data.

d) It is of interest to investigate how accurate the Smedjebacken and Hofors data (which were used in this study) are.

e) The accuracy of the data received for the measured Mn content from Smedjebacken need to be investigated and measurements probably need to be redone. In fact, most of the scrap types were reported to have exactly 0.8% of Mn. This value is unusually low and with no variation. Lack of reliable measured Mn contents made it impossible for this study to calculate the Mn content in scrap types with the NNLS method.

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f) It might be of interest to know the acceptable range of composition deviation in products (i. e. how flexible are Smedjebacken and Hofors when it comes to the accuracy of the composition of each steel grade). For instance, in the Smedjebacken data, the average amount of Cr in the aim steel is approximately 0.084 for all charges and the standard deviation for the same data set is 0.033%. It is important to know how sensitive the Smedjebacken data is to higher (or lower) standard deviations than 0.033%. This data might also improve the knowledge of how much the final composition will be affected by the analytical accuracy of the estimations.

g) It might be important to know the exact composition of each scrap type provided for this study. This makes it possible to consider the effect of elements on each other and on the variations in the steel composition.

h) It is important to do an investigation on the scrap which has the highest and the lowest difference between the measured and calculated Cr content. These differences are presented in Table (2) and (3).

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References

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Young AB, Stockholm, Sweden, (8 May 2012).

2. Gyllenram, R., Ekerot, S., Jönsson, P., Lubricating the Recycling Machine, Royal Institute of Technology (KTH), Revue de Métallurgie © EDP Sciences, DOI:

10.1051/metal/2012017, Stockholm, Sweden, (4 April 2012), metal120010.

3. Casella, G., Berger, R.L., Statistical Inference, Second Edition, Duxbury - Thomson Learning, Inc., CA, USA, 2002.

4. Rudas, T., Handbook of Probability: Theory and Applications, SAGE Publications, CA, USA, (2008).

5. Golub, G., Numerical Methods for Solving Least Squares Problems, Numer. Math. 7 (1965) 206-216.

6. Chen, J., Richard, C., Bermudez, J.C.M., Honeine, P., Nonnegative Least-Mean-Square Algorithm, IEEE Transactions on Signal Processing, Vol. 59, NO. 11, (November 2011).

7. http://www.gnu.org/software/octave/about.html (Last viewed on 15 November 2012).

8. http://www.mathworks.se/help/matlab/ref/lsqnonneg.html (Last viewed on 15 November 2012).

9. Lawson, C.L., Hanson, R.J., Solving Least Squares Problems, Prentice-Hall, Englewood Clis, NJ, 1974; reprinted by SIAM, Philadelphia, PA, 1995.

10. Santiago, C.P., On the nonnegative Least Square Algorithm, Georgia Institute of Technology – School of Industrial and Systems Engineering, Georgia, USA (7 August 2009).

11. Gaskell, D.R., Introduction to the Thermodynamics of Materials, Fourth Edition, Taylor

& Francis Books, Inc., London, Great Britain, (2003).

12. http://webh01.ua.ac.be/mitac4/micro_xrf.pdf (Last viewed on 15 November 2012).

Getting a Grip on Scrap