S i l v i u- G ab r ie l S T R OE , G he o r ghe G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r am e t e r s , Fo od a nd E nvir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 176 journal homepage: www.fia.usv.ro/fiajournal Journal of Faculty of Food Engineering, Ştefan cel Mare University of Suceava, Romania Volume XII, Issue 2 – 2013, pag. 169 - 175 S TA T IS TI CAL S TUDY OF TH E DE PENDE NC E B ETW EE N CONCE NT RAT ION OF ME TA LL IC E L EM ENTS M IGRA TED FRO M S TA INL ESS S TE E L GR AD E AIS I321 AND WORKI NG PARAM ETE RS *Silviu-Gabriel STROE1, Gheorghe GUTT1 1Faculty of Food Engineering, Stefan cel Mare University of Suceava, Universitatii str. 13, Suceava, Romania, silvius@fia.us.ro; g.gutt@fia.usv.ro * Corresponding author Received April 27th 2013, accepted May 25th 2013 Abstract: One of the main problems concerning food safety is the possible migration of ions in metallic materials intended to come into contact with food. The aim of this paper is to find and apply the mathematical modeling of experimental data which should describe as accurately as possible the dependence between the variables used in the experimental plan. The research made and presented in this study also aims to create a real possibility for rapid intervention in the process control when one of the parameters cannot be maintained at a predetermined value. In this paper we used some experimental data obtained by testing the migration of metal ions from austenitic stainless steel grade AISI 321 samples in solutions with concentrations of 3%, 6% and 9% acetic acid. To find an accurate mathematical model describing the phenomena of diffusion, ANOVA method, known as the variance analysis, was used. In order to obtain the mathematical model we used a polynomial model with independent variables: corrosive environment temperature (X1), exposure time (X2), stirring environment (X3) and the dependent variables Y - concentrations of elements Ti, Cr, Mn, 56Fe and Ni found in solutions. Values of the regression coefficients very close to the value 1 (the dependent variables coefficients are valid) were obtained, which demonstrates the validity of the applied method. Keywords: austenitic stainless steel, mathematical modeling, ANOVA method, dependent and independent variables, coded values, validity of the model. 1. Introduction Stainless steels are commonly used as metallic materials intended to come into contact with food environments. This widespread use of stainless steels led to increase the importance of studying the diffusion processes of metal ions in food environments, research on the interaction between corrosive environments and stainless steels being the subject of scientific research for a long time [1], [2], [3], [4]. In this context, one of the most used grade of stainless steel is 300 series such as AISI 321, both containing 11wt% Ni [5]. Diffusion of metal ions in foodstuffs is a slow process that occurs in different types of environments, often in acidic environments [6], [7], [8], [9], [10]. In the last decade, based on several scientific researches, the modeling of migration metallic constituents of materials in contact with foodstuffs has played an important part in the quality assurance system in the food industry. By classical Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o rg he G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nv ir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 177 methods or by using specialized software, mathematical modeling has become a relatively inexpensive tool for scientists aiming to minimize the number of experiments and to determine the influence of various parameters influencing the safety and quality processes [11-13]. Since the beginning we must analyze if the problem is correctly defined and consistent, if its solution to provide useful information for the study and influence factors correspond to the characteristics that must be satisfied. To find an accurate mathematical model describing the phenomena of diffusion, the ANOVA method, known as the analysis of variance, was used. ANOVA is widely used because it is recommended when studying a larger number of levels of the independent variables, making possible to observe with greater accuracy the effect of the independent variables on the dependent ones and their combined effect. The aim of this work was to apply the ANOVA method for modeling the experimental data obtained from the migration test, which describes exactly the dependence between the variables used in the experimental plan. 2. Experimental 2.1. Materials and samples preparation In this study metal samples of AISI 321 stainless steel grade were used. The chemical composition of the steel is given in Table 1 (EN 10088-2:2005). Table 1 Chemical composition of austenitic stainless stell AISI 321 (wt %) Fe C Mn P S Si Cr Ni Ti 68 0.08 2 0.045 0.03 1 18 11 0.4 Metal samples were cut from unused sheet, free from deformation or scratches. Sample sizes were established according to the D.M. 03/21/1973, which stipulates that the ratio of the exposed surface of the sample and the volume of solution should be between 0.5 ... 2. Sample sizes were 40 x 40 x 2.5 mm. Acetic acid is recommended to test metal alloys in contact with foodstuffs. Acetic acid solutions 3%, 6% and 9%, were used as corrosive environments. To avoid contamination of corrosive solutions with foreign compounds, the surface of metal samples was washed with a detergent solution at 40°C and rinsed in double distilled water. Ultrasonic cleaning was performed at 45°C for 15 minutes in the ultrasonic. The samples were dried in oven at 50°C. Profile roughness evaluation, complying with EN ISO 4287, was determined with an optical profilometer (Nano Focus µscan) for contactless 2D and 3D measurement of microscopic surface structures. The surface roughness was Ra - 0.7816±0.019µm. 2.2. Migration test In the experimental studies the following working parameters were used:  Testing temperature - T [ °C]  Stirring environment - n [rot/min]  Exposure time - t [min] Variation levels of working parameters are given in Table 2. Table 2 Variation levels of working parameters Working parameters Minimum value Central value Maximum value Testing temperature 22 28 34 Stirring environment 0 125 250 Exposure time 30 60 90 In the case of the experiments made in Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o rg he G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nv ir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 178 stationary environment and at different temperatures of the environment, the samples were maintained in the oven while for those requiring temperatures of 28°C and 34°C and stirring environment, a magnetic heating stirrer Heidolph MR Hei- Tec was used. 2.3. Chemical analysis of corrosive environments The concentrations of metal ions Ti, Cr, Mn, 56Fe and Ni from acid solutions were analyzed using mass spectrometry and inductively coupled plasma ICP-MS - Agilent 7500 model. 3. Results and Discussion The concentrations of metallic elements Ti, Cr, Mn, 56Fe and Ni obtained by mass spectrometry and inductively coupled plasma were used in mathematical modeling of diffusion phenomena. Since the literature has not provided a sufficiently accurate mathematical apparatus to describe the interdependencies mentioned we chose to develop a mathematical model with statistical means by ANOVA method. To achieve this mathematical model Design Expert® software and a polynomial model with multiple variables were used:  Independent variables: testing temperature (X1), exposure time (X2), stirring environment (X3);  Dependent variables (response functions) Y1, Y2, Y3, Y4 and Y5: concentrations expressed in ppb of elements Ti, Cr, Mn, 56Fe and Ni. 3.1. Mathematical modeling of concentration of metallic elements migrated in 3% acetic acid solution. After having made the analysis of variance ANOVA, a polynomial model with the lowest value of the critical probability P for all dependent variables: Ti, Cr, Mn, 56Fe and Ni, was chosen. It is known that the null hypothesis of the dependent variables is rejected if the value of the variables of P is less than the chosen significance threshold (α=0.05). The statistical summary of mathematical models found to describe dependent variables Ti, Cr, Mn, 56Fe and Ni is given in Table 3. The mathematical models are presented in equations (1), (2), (3), (4) and (5) and comparative graphical representation of measured values and the values obtained by modeling ANOVA are shown in Figures 1-5. Table 3. Model summary Statistics Dependent variable P Model Standard deviation [σ] R-Squared [R2] Adjusted R-Squared [R2 adjusted] Y1 0.033 Cubic 0.24 0.9905 0.9440 Y2 0.016 Quartic 1.62 0.9925 0.9512 Y3 0.015 Quadratic 1.20 0.9878 0.8872 Y4 0.046 Cubic 91.33 0.9898 0.8876 Y5 0.025 Quartic 4.13 0.9903 0.9812 The mathematical model of the response function Y1 (Ti concentration) is given in the equation (1): Y = 1.71 + 0.97 ∙ + 0.94 ∙ X − 0.50 ∙ X − 0.22 ∙ − 0.094 ∙ X − 0.19 ∙ X − 0.17 ∙ ∙ X − 0.19 ∙ ∙ X − 0.40 ∙ X ∙ X − 0.33 ∙ ∙ X − 0.17 ∙ X ∙ X − 0.23 ∙ ∙ X − 0.36 ∙ X ∙ X − 0.05 ∙ X ∙ X − 0.13 ∙ X ∙ X + 0.05 ∙ ∙ X ∙ X (1) Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o r ghe G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nvir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 179 To check the validity of the mathematical model obtained, the experimental values vs model values were plotted (Figure 1). Figure 1. Validity check of the mathematical model obtained The mathematical model of the response function Y2 (Cr concentration) is given in equation (2): Y = 7.26 + 3.72 ∙ X + 1.44 ∙ X + 2.67 ∙ X + 3.61 ∙ X + 8.11 ∙ X + 2.11 ∙ X + 0.25 ∙ X ∙ X + 2.50 ∙ X ∙ X − 2.00 ∙ X ∙ X + 1.58 ∙ X ∙ X + 0.50 ∙ X ∙ X − 0.58 ∙ X ∙ X + 1.17 ∙ X ∙ X + 2.50 ∙ X ∙ X + 1.83 ∙ X ∙ X + 0.75 ∙ X ∙ X ∙ X − 6.92 ∙ X ∙ X + 6.50 ∙ X ∙ X ∙ X − 0.17 ∙ X ∙ X − 1.00 ∙ X ∙ X ∙ X + 1.00 ∙ X ∙ X ∙ X − 0.67 ∙ X ∙ X (2) The validity of the mathematical model obtained for Cr has been plotted by experimental values vs model values (Figure 2). Figure 2. Validity check of the mathematical model obtained for chromium The mathematical model of the response function Y3 (Mn concentration) is given in equation (3): Y = 8.57 + 1.90 ∙ X + 2.40 ∙ X − 0.92 ∙ X − 0.20 ∙ X + 0.33 ∙ X − 1.78 ∙ X + 1.68 ∙ X ∙ X − 0.60 ∙ X ∙ X − 0.24 ∙ X ∙ X (3) The validity of the mathematical model obtained for Mn has been plotted by experimental values vs model values (Figure 3). Figure 3. Validity check of the mathematical model obtained for manganese The mathematical model of the response function Y4 (56Fe concentration) is given in equation (4): Y = 861.85 + 293.33 ∙ X + 145.56 ∙ X − 50,00 ∙ X − 108.89 ∙ X − 3.89 ∙ X − 310.56 ∙ X + 67.50 ∙ X ∙ X − 9.17 ∙ X ∙ X + 28.33 ∙ X ∙ X + 54.17 ∙ X ∙ X + 107.50 ∙ X ∙ X − 27.50 ∙ X ∙ X − 157.50 ∙ X ∙ X + 5.00 ∙ X ∙ X − 43.33 ∙ X ∙ X + 37.50 ∙ X ∙ X ∙ X (4) The validity of the mathematical model obtained for 56Fe has been plotted by experimental values vs model values (Figure 4). R² = 0.9785 0.00 0.50 1.00 1.50 2.00 2.50 3.00 3.50 4.00 0.00 1.00 2.00 3.00 4.00 M od el v al ue , T i[ pp b] Experimental value, Ti [ppb] R² = 0.9925 0.00 5.00 10.00 15.00 20.00 25.00 30.00 35.00 40.00 0.00 10.00 20.00 30.00 40.00 M od el v al ue , C r [p pb ] Experimental value, Cr [ppb] R² = 0.908 0.00 5.00 10.00 15.00 20.00 0.00 5.00 10.00 15.00 20.00M od el v al ue , M n [p pb ] Experimental value, Mn [ppb] Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o r ghe G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nvir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 180 Figure 4. Validity check of the mathematical model obtained for 56Fe The mathematical model of the response function Y5 (Ni concentration) is given in equation (5): Y = 35.56 + 9.00 ∙ X + 6.33 ∙ X + 3.44 ∙ X − 3.89 ∙ X + 0,11 ∙ X − 7.89 ∙ X + 7.50 ∙ X ∙ X − 2.50 ∙ X ∙ X + 0.002 ∙ X ∙ X + 4.00 ∙ X ∙ X − 1.17 ∙ X ∙ X + 0.003 ∙ X ∙ X − 3.00 ∙ X ∙ X − 1.17 ∙ X ∙ X + 2.00 ∙ X ∙ X + 1.25 ∙ X ∙ X ∙ X + 0.83 ∙ X ∙ X + 0.50 ∙ X ∙ X ∙ X + 7.83 ∙ X ∙ X + 2.00 ∙ X ∙ X ∙ X − 6.50 ∙ X ∙ X ∙ X + 0.83 ∙ X ∙ X (5) The validity of the mathematical model obtained for Ni has been plotted by experimental values vs model values (Figure 5). Figure 5. Validity check of the mathematical model obtained for nickel 3.2. Mathematical modeling of concentration of metallic elements migrated in 6% acetic acid solution. Statistical summary of mathematical models found to describe dependent variables Ti, Cr, Mn, 56Fe and Ni in 6% acetic acid solutions is given in Table 4. The mathematical models are presented in equations (6), (7), (8), (9) and (10) and comparative graphical representation of measured values and the values obtained by modeling ANOVA are shown in Figures 6-10. Table 4. Model summary Statistics Dependent variable P Model Standard deviation [σ] R-Squared [R2] Adjusted R-Squared [R2 adjusted] Y1 0.0001 Cubic 0.23 0,9931 0,9820 Y2 0.0061 Cubic 1.29 0,9803 0,9488 Y3 0.0670 Quartic 0.61 0,9949 0,9669 Y4 0.0145 Quartic 13.79 0,9988 0,9921 Y5 0.0516 Quartic 28.65 0,9952 0,9688 The mathematical model of the response function Y1 (Ti concentration) is given in equation (6): Y = 1.44 + 1.03 ∙ X + 0.58 ∙ X + 0.31 ∙ X + 1.25 ∙ X + 0.017 ∙ X − 0.05 ∙ X + 0,42 ∙ X ∙ X + 0.75 ∙ X ∙ X + 0.45 ∙ X ∙ X + 0.85 ∙ X ∙ X + 0.47 ∙ X ∙ X − 0.06 ∙ X ∙ X − 0,12 ∙ X ∙ X − 0,03 ∙ X ∙ X + 0.81 ∙ X ∙ X ∙ X (6) The validity of the mathematical model obtained for Ti has been plotted by experimental values vs model values (Figure 6). R² = 0.9568 0.00 200.00 400.00 600.00 800.00 1000.00 1200.00 1400.00 1600.00 0.00 500.00 1000.00 1500.00 M od el v al ue ,5 6 F e [p pb ] Experimental value, 56Fe [ppb] R² = 0.9803 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 10.00 20.00 30.00 40.00 50.00 60.00 M O de l v al ue , N i [ pp b] Experimental value, Ni [ppb] Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o r ghe G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nvir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 181 Figure 6. Validity check of the mathematical model obtained for titanium The mathematical model of the response function Y2 (Cr concentration) is given in equation (7): Y = 10.33 + 3.83 ∙ X + 8.06 ∙ X + 2.83 ∙ X − 0.17 ∙ X + 0.17 ∙ X − 0.83 ∙ X − 0,75 ∙ X ∙ X + 0.083 ∙ X ∙ X + 1.08 ∙ X ∙ X − 3.08 ∙ X ∙ X − 1.75 ∙ X ∙ X − 0.75 ∙ X ∙ X − 0,25 ∙ X ∙ X + 1.25 ∙ X ∙ X − 1.58 ∙ X ∙ X − 0.25 ∙ X ∙ X ∙ X (7) The validity of the mathematical model obtained for chromium has been plotted by experimental values vs model values (Figure 7). Figure 7. Validity check of the mathematical model obtained for chromium The mathematical model of the response function Y3 (Mn concentration) is given in equation (8): Y = 6.46 + 1.36 ∙ X + 1.57 ∙ X − 0.17 ∙ X − 0.39 ∙ X − 0.99 ∙ X − 0.29 ∙ X + 1.37 ∙ X ∙ X − 1.07 ∙ X ∙ X − 0.075 ∙ X ∙ X + 0.68 ∙ X ∙ X + 1.37 ∙ X ∙ X + 0.84 ∙ X ∙ X − 0.058 ∙ X ∙ X + 0.23 ∙ X ∙ X + 0.77 ∙ X ∙ X + 1.13 ∙ X ∙ X ∙ X + 1.06 ∙ X ∙ X + 1.72 ∙ X ∙ X ∙ X + 0.21 ∙ X ∙ X + 0.40 ∙ X ∙ X ∙ X + 0.075 ∙ X ∙ X ∙ X + 1.16 ∙ X ∙ X (8) The validity of the mathematical model obtained for manganese has been plotted by experimental values vs model values (Figure 8). Figure 8. Validity checkof the mathematical model obtained for manganese The mathematical model of the response function Y4 ( 56Fe concentration) is given in equation (9): Y = 298.89 + 120.56 ∙ X + 108.89 ∙ X + 93.33 ∙ X + 11.67 ∙ X − 3.33 ∙ X + 66.67 ∙ X + 47.50 ∙ X ∙ X + 2.50 ∙ X ∙ X + 5.00 ∙ X ∙ X + 19.17 ∙ X ∙ X − 32.50 ∙ X ∙ X + 14.17 ∙ X ∙ X − 30.83 ∙ X ∙ X + 5.00 ∙ X ∙ X − 8.33 ∙ X ∙ X + 37.50 ∙ X ∙ X ∙ X + 42.50 ∙ X ∙ X + 27.50 ∙ X ∙ X ∙ X − 67.50 ∙ X ∙ X + 27.50 ∙ X ∙ X ∙ X − 25.00 ∙ X ∙ X ∙ X + 0.001 ∙ X ∙ X (9) The validity of the mathematical model obtained for 56Fe has been plotted in (Figure 9). R² = 0.9931 0.00 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 0.00 2.00 4.00 6.00 8.00 10.00 M od el v al ue , T i[ pp b] Experimental value, Ti [ppb] R² = 0.9803 0.00 5.00 10.00 15.00 20.00 25.00 0.00 5.00 10.00 15.00 20.00 25.00 M od el v al ue , C r [p pb ] Experimental value, Cr [ppb] R² = 0.9949 0.00 5.00 10.00 15.00 20.00 25.00 0.00 5.00 10.00 15.00 20.00 25.00 M od el v al ue , M n [p pb ] Experimental value, Mn [ppb] Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o r ghe G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nvir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 182 Figure 9. Validity check of the mathematical model obtained for 56Fe The mathematical model of the response function Y5 (Ni concentration) is given in equation (10): Y = 603.04 + 151.11 ∙ X + 83.33 ∙ X + 117.22 ∙ X − 95.56 ∙ X + 54.44 ∙ X − 100.56 ∙ X + 22.50 ∙ X ∙ X + 30.00 ∙ X ∙ X + 7.50 ∙ X ∙ X − 30.00 ∙ X ∙ X − 85.83 ∙ X ∙ X − 1.67 ∙ X ∙ X + 0.83 ∙ X ∙ X + 16.67 ∙ X ∙ X + 10.00 ∙ X ∙ X − 18.75 ∙ X ∙ X ∙ X − 46.67 ∙ X ∙ X − 21.25 ∙ X ∙ X ∙ X + 135.83 ∙ X ∙ X − 6.25 ∙ X ∙ X ∙ X − 1.25 ∙ X ∙ X ∙ X − 26.67 ∙ X ∙ X (10) The validity of the mathematical model obtained for nickel has been plotted in (Figure 10). Figure 10. Validity check of the mathematical model obtained for nickel 3.3. Mathematical modeling of concentration of metallic elements migrated in 9% acetic acid solution. Statistical summary of mathematical models found to describe dependent variables Ti, Cr, Mn, 56Fe and Ni in 9% acetic acid solutions is given in Table 5, the mathematical models are presented in equations (11), (12), (13), (14) and (15), and comparative graphical representation of measured values and the values obtained by modeling ANOVA are shown in Figures 11-15. Table 5. Model summary Statistics Dependent variable P Model Standard deviation [σ] R-Squared [R2] Adjusted R-Squared [R2 adjusted] Y1 0.042 Quartic 0.032 0.9920 0.9948 Y2 0.017 Cubic 3.78 0.9850 0.9746 Y3 0.009 Quartic 0.11 0.9996 0.9975 Y4 0.052 Cubic 23.76 0.9921 0.9794 Y5 0.048 Quadratic 4.56 0.9885 0.9365 The mathematical model of the response function Y1 (Ti concentration) obtained using ANOVA methodis is given in equation (11): Y = 1.39 + 0.36 ∙ X + 0.43 ∙ X + 0.24 ∙ X − 0.072 ∙ X + 0.028 ∙ X + 0.028 ∙ X + 0,10 ∙ X ∙ X − 0.075 ∙ X ∙ X + 0.075 ∙ X ∙ X − 0.12 ∙ X ∙ X − 0.058 ∙ X ∙ X − 0.017 ∙ X ∙ X − 0,042 ∙ X ∙ X + 0.042 ∙ X ∙ X + 0.001 ∙ X ∙ X ∙ X − 0.017 ∙ X ∙ X − 0.025 ∙ X ∙ X ∙ X + 0.008 ∙ X ∙ X + 0.10 ∙ X ∙ X ∙ X + 0.025 ∙ X ∙ X ∙ X − 0.092 ∙ X ∙ X (11) R² = 0.9988 0.00 100.00 200.00 300.00 400.00 500.00 600.00 700.00 800.00 0.00 200.00 400.00 600.00 800.00 M od el v al ue , 5 6 F e [p pb ] Experimental value, 56Fe [ppb] R² = 0.9952 250.00 350.00 450.00 550.00 650.00 750.00 850.00 200.00 400.00 600.00 800.00 M od el v al ue , N i[ pp b] Experimental value, Ni [ppb] Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o r ghe G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nvir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 183 The validity of the mathematical model obtained for titanium has been plotted in (Figure 11). Figure 11. Validity check of the mathematical model obtained for titanium The mathematical model of the response function Y2 (Cr concentration) obtained using ANOVA methodis is given in equation (12): Y = 13.93 + 2.39 ∙ X + 13.17 ∙ X + 0.50 ∙ X − 4.44 ∙ X + 4.06 ∙ X − 0.11 ∙ X + 4.75 ∙ X ∙ X + 4.67 ∙ X ∙ X + 3.42 ∙ X ∙ X − 4.75 ∙ X ∙ X + 7.00 ∙ X ∙ X + 3.42 ∙ X ∙ X − 0,33 ∙ X ∙ X + 2.75 ∙ X ∙ X − 2.25 ∙ X ∙ X + 2.75 ∙ X ∙ X ∙ X (12) To check the validity of the mathematical model obtained, the experimental values vs model values were plotted (Figure 12). Figure 12. Validity check of the mathematical model obtained for chromium The mathematical model of the response function Y3 (Mn concentration) obtained using ANOVA methodis is given in equation (13): Y = 5.64 − 0.78 ∙ X + 1.83 ∙ X + 0.46 ∙ X − 1.73 ∙ X + 1.12 ∙ X + 0.37 ∙ X + 0,57 ∙ X ∙ X − 0.53 ∙ X ∙ X + 0.10 ∙ X ∙ X + 0.30 ∙ X ∙ X + 0.48 ∙ X ∙ X + 0.75 ∙ X ∙ X + 0.40 ∙ X ∙ X + 0.058 ∙ X ∙ X − 0.17 ∙ X ∙ X − 0.41 ∙ X ∙ X ∙ X + 0.42 ∙ X ∙ X + 0.31 ∙ X ∙ X ∙ X − 0.23 ∙ X ∙ X − 0.24 ∙ X ∙ X ∙ X + 0.24 ∙ X ∙ X ∙ X − 0.46 ∙ X ∙ X (13) To check the validity of the mathematical model obtained for the Y3 variable, the experimental values vs model values were plotted (Figure 13). Figure 13. Validity check of the mathematical model obtained for manganese The mathematical model of the response function Y4 ( 56Fe concentration) obtained using ANOVA methodis is given in equation (14): Y = 522.96 + 123.33 ∙ X + 58.33 ∙ X + 96.67 ∙ X − 103.89 ∙ X − 13.89 ∙ X − 12.78 ∙ X + 16.67 ∙ X ∙ X − 5.83 ∙ X ∙ X + 6.67 ∙ X ∙ X + 5.00 ∙ X ∙ X + 12.50 ∙ X ∙ X + 0.001 ∙ X ∙ X + 42.50 ∙ X ∙ X − 15.00 ∙ X ∙ X − 15.00 ∙ X ∙ X − 17.50 ∙ X ∙ X ∙ X (14) R² = 0.9992 0.50 0.70 0.90 1.10 1.30 1.50 1.70 1.90 2.10 2.30 2.50 0.50 1.00 1.50 2.00 2.50 M od el v al ue ,T i[ pp b] Experimental value, Ti [ppb] R² = 0.9645 -10.00 0.00 10.00 20.00 30.00 40.00 50.00 60.00 0.00 20.00 40.00 60.00 M od el v al ue ,C r [p pb ] Experimental value, Cr [ppb] R² = 0.9996 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 2.00 4.00 6.00 8.00 10.00 M od el v al ue , M n [p pb ] Experimental value, Mn [ppb] Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o r ghe G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nvir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 184 The validity of the mathematical model obtained has been plotted by experimental values vs model values (Figure 14). Figure 14.Validity check of the mathematical model obtained for 56Fe The mathematical model of the response function Y5 (Ni concentration) obtained using ANOVA method is given in equation (15): Y = 21.29 + 12.72 ∙ X + 10.72 ∙ X + 10.78 ∙ X + 3.72 ∙ X + 2.72 ∙ X + 3.22 ∙ X + 3.25 ∙ X ∙ X + 5.50 ∙ X ∙ X + 5.83 ∙ X ∙ X (15) To check the validity of the mathematical model obtained for Y5 variable, the experimental values vs model values were plotted (Figure 15). Figure 15. Validity check of the mathematical model obtained for nickel 4. Conclusions The aim of this study was to find and apply a method for modeling the experimental data obtained by migration test, which describes how exactly the dependence between variables used in the experimental design is. In order to establish the dependences between the working parameters of migration test and the metal concentrations of Ti, Cr, Mn, 56Fe and Ni, found in corrosive solutions using mass spectrometry ICP-MS inductively coupled plasma, the ANOVA method was used; By utilizing this statistical method to obtain mathematical models that gives dependence of the working parameters and Ti, Cr, Mn, 56Fe and Ni concentrations, these models can be used to estimate the objective function values. So, the simulation process may be obtained by using mathematical models, but they are produced with certain approximations and assumptions, allowing however, a description of processes for a wide range of values; For the mathematical models we obtained values of the regression coefficients very close to the value 1 (the dependent variables coefficients are valid), which demonstrates the validity of the applied statistical method; The comparative graphical representation of the experimental values vs values obtained by modeling shows that the deviation is less than 5%. These small differences give a real opportunity to use mathematical models in real intervention in production processes. 5. References [1]. ASAMI K., HASHIMOTO K., Surface analytical study of atmospheric corrosion of stainless steel, Proc. 13th International Corrosion Congress, paper 038, Melbourne (1996); [2]. BEDDOES J., BUCCI K., The Influence of Surface Condition on the Localized Corrosion of 316L Stainless Steel Orthopaedic Implants, R² = 0.9921 100.00 200.00 300.00 400.00 500.00 600.00 700.00 800.00 100.00 300.00 500.00 700.00 900.00 M od el v al ue , 5 6 F e [p pb ] Experimental value, 56Fe [ppb] R² = 0.9585 0.00 10.00 20.00 30.00 40.00 50.00 60.00 70.00 80.00 90.00 0.00 20.00 40.00 60.00 80.00 M od el v al ue , N i[ pp b] Experimental value, Ni [ppb] Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava Volume XII, Issue 2 – 2013 S i l v i u- G ab r ie l S T R OE , G he o rg he G U TT , St a t i s t i ca l s t u d y of t he d e pe nd en c e be t we en c o nc e nt r at i on o f m e t a l li c e l e m e n ts m i g r at ed f r om s t a i n le ss s t e e l g r ad e AI S I 3 21 a nd wo r k i n g p a r a m e t e r s , Fo od a nd E nv ir o nm e nt Sa f e t y, V o l ume X I I, Is s ue 2 – 2 0 13 , p a g. 1 76 -1 85 185 Journal of Materials Science: Materials in Medicine, 10, 389 (1999); [3]. 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