233 
 

Journal homepage: www.fia.usv.ro/fiajournal 
Journal of Faculty of Food Engineering,  

Stefan cel Mare University of Suceava, Romania  
Volume XIV, Issue 2 – 2015, pag.  233 - 240 

 
 

 
 

MATHEMATICAL MODELLING OF THE CONSTITUENTS’ CONCENTRATIONS 

OF AISI 304 STAINLESS STEEL SAMPLES THAT DIFFUSE INTO SIMULATED 

ACIDIC ENVIRONMENTS 

 
*Silviu-Gabriel STROE1 

 
1Faculty of Food Engineering, Stefan cel Mare University of Suceava,  

13 Universitatii Street, 720229, Suceava, Romania,  
silvius@fia.usv.ro 

*Corresponding author 
Received May 16th 2014, accepted June 7th 2014 

 
 
Abstract: The purpose of this research was to study the mathematical modelling of dependence 
between testing parameters and some metallic elements diffused into  simulated acidic environments. 
Diffusion processes occurring at the contact between AISI 304 stainless steel samples and simulated 
acidic environments were analyzed to fulfill this goal. In order to process the experimental data by 
statistical and mathematical methods, the following steps were taken: diffusion testing of Cr, Mn, 56Fe 
and Ni elements from AISI304 stainless steel samples into solutions with concentration of 3%, 6% and 
9% acetic acid; chemical analysis of corrosive solutions using mass spectrometry and inductively 
coupled plasma method (ICP-MS); the results were processed using ANOVA method and thus resulting 
the mathematical models that describe the dependence between variables. A polynomial model with 
independent variables was used to obtain the mathematical models: temperature of acidic simulated 
solutions (X1), testing time (X2), stirring grade of environment (X3) and the dependent variables 
(response function): Y1 - concentration of chromium (Cr), Y2 - concentration of manganese (Mn), Y3 - 
concentration of iron (56Fe) and Y4 - concentration of nickel (Ni) found in corrosive environments. 
After having made the analysis of variance ANOVA, a model having the lowest P value (critical 
probability) for all variables was chosen. The regression coefficient was determined to verify the 
validity of each mathematical model.  
 
Keywords: stainless steel, simulated environments, diffusion, ANOVA method, mathematical model.   
 
 
1. Introduction  
 
It is known that food raw materials have 
their own natural metal content. In case 
when there are additions to this natural 
content, the diffusion from metal surfaces 
coming into direct contact can cause 
serious health problems or a change 
unwanted organoleptic characteristics of 
the finished product. These metals are 
found in absolutely all foodstuffs in lower 
or higher concentrations, depending on 

various circumstances [1]. A mathematical 
model that describes the physical processes 
which occur at the contact of some 
material from the food processing chain is 
considered a very important tool that can 
replace, at least partially, the experimental 
investigations which are expensive and 
require longer time. The models thus 
obtained aimed at complying with the 
conformity assessment of specific 
regulations concerning migration limits 
and at describing the time variation of the 

http://www.fia.usv.ro/fiajournal
mailto:silvius@fia.usv.ro


Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - 
SuceavaVolume XIV, Issue 2 – 2015 

 

S i l v i u- G ab r ie l S T R OE ,  Mathematical modelling of the constituents’ concentrations of AISI 304 stainless steel samples 
that diffuse into simulated acidic environments,  Fo od  a nd E nvir o nme nt  Sa f e t y,  V o l u me  X IV ,  Is s ue  2 – 20 1 5,  p a g  
2 3 3-2 4 0  

234 
 

values of parameters. Having in view the 
advantages of this tool, in recent years, 
numerous studies have been undertaken for 
this purpose [2-7]. The research described 
in the literature highlights a relatively 
limited approach of the diffusion systems 

consisting of materials intended to come 
into contact with foodstufs.  
Firstly, it is useful to present a schematic 
illustration of the mathematical 
formulation stages of this problem and the 
finding of adequate solutions is shown in 
Figure 1 [8]. 

 

 
Fig. 1. The problem formulating stages and finding solutions [8] 

 
 

The general mathematical model solutions 
thus obtained provide a quantitative and 
qualitative perspective on how the 
parameters affect the phenomenon; further 
research could be focused on the 
application of these basic equations [8]. 
The aim of this work was to study the 
mathematical modelling of dependence 
between the constituents’ concentrations of 
AISI 304 stainless steel samples that 
diffuse into simulated acidic solutions and 
the testing parameters. Similar researches 
have been conducted for advanced 
characterization behavior of AISI321 
stainless steel samples in acidic 
environments [4]. 
 
2. Matherials and methods 
 
2.1. Metallic samples  
In this research metallic samples made of 
AISI304 stainless steel grade were used. 
The samples sizes were of 400.5 x 400.5 
x 1 mm and they were established by the  
Ministerial Decree of  21.03.1973, which 
stipulates that the ratio of exposed surface 
of the stainless steel samples and solution 
volume should be between 0.5 ... 2 [9].  
The chemical composition of metallic 
samples (according to the SR EN 10088-
2:2005 Romanian standard) is shown in 
Table 1.  

 

Table 1 
Chemical composition of AISI304 stainless steel 

samples (wt %) 
 

Fe C Mn P S Si Cr Ni 
67 0.07 1-2 0.045 0.03 0-1 17-19 8-10.5 

 
A systeM μScan (manufactured by 
NanoFocus - Germany) was used to 
determine roughness of the metallic 
samples. Measurement and calculation of 
usual surface parameters were made 
according to DIN EN ISO 4287 and DIN 
EN ISO 4288 standards. The surface mean 
roughness of AISI304 stainless steel 
samples was = 0.5988750.0125	 .  
 
2.2. Corrosive environments 
All solutions were freshly prepared with 
quality analytical chemical reagents. Given 
the fact that acidic environments are  one 
of the most aggressive environments in the 
food processing industry, within the 
experiment we used 3%, 6% and 9%  
CH3COOH in double distilled water 
(according to Italian D. M. of 21-03-1973) 
[9]. Glacial acetic acid (Sigma-Alorich, 
Germany) was used to prepare the 
experimental solutions.  
 
2.3. Testing method 
To cover a domain as broad as that of 
using AISI304 stainless steel, three testing 

Solutions Solving Mathematizatio
n 

Initial 
assumption

s 

Problem 
formulation 

Identificatio
n 

phenomenon 



Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - 
SuceavaVolume XIV, Issue 2 – 2015 

 

S i l v i u- G ab r ie l S T R OE ,  Mathematical modelling of the constituents’ concentrations of AISI 304 stainless steel samples 
that diffuse into simulated acidic environments,  Fo od  a nd E nvir o nme nt  Sa f e t y,  V o l u me  X IV ,  Is s ue  2 – 20 1 5,  p a g  
2 3 3-2 4 0  

235 
 

parameters were chosen. To determine the 
variation levels of testing parameters, the 
usual values encountered in the processing 
industry were considered. 
The levels of the three parameters used in 
the experiment are shown in Table 2. 
 

Table 2 
Variation levels of testing parameters 

Levels 
 

Parameters 
Minimum 

level 
Central 

level 
Maximum 

level 
Temperature 
solution, [°C] 22 28 32 

Testing time, 
[min] 30 60 90 

Stirring 
environment, 

[r/min] 
0 125 250 

 
The samples were kept in the oven to make 
the experiments in stationary environment 
and at temperatures different from the 
room temperature one (22°C). A magnetic 
heat stirrer Heidolph MR Hei-Tec 
(Heidolph, Germany) was used to make 
the experiments at the temperature of 
28°C, 34°C respectively and in stirred 
environment. 
 
2.4. Chemical analysis of corrosive 
environments 
Inductively coupled plasma mass spectrometry 
(ICP-MS) was used to analyze the chemical 
composition of corrosive environments, both 
before having used them in the corrosion tests, 
and after having made them, taking into 
consideration the adavantages of this method 
as compared to other analytical ones. 
 
2.5. Mathematical modelling of the 
constituents’ concentrations 
ANOVA method, known as the variance 
analysis was appealed to in order to find a 
mathematical model as accurate as 
possible which describe the diffusion 
phenomena.  
This technique was chosen due to the fact 
that it is highly recommended when 
studying a larger number of levels of 

independent variables, providing higher 
accuracy of the effect that the independent 
variables have upon the dependent ones as 
well as of the their joint effect.    
Having in view that some practical values 
(minimum values) of responses are to be 
determined, it is necessary to establish 
some interdependences capable of 
describing both the nature and the extent of 
the influences taken into consideration (see 
also [4]).  
To develop the predictive-mathematical 
models the Design Expert software (trial 
version) was used. 
In order to obtain the mathematical models 
a polynomial model with independent 
variables was used: acidic simulated 
solutions temperature (X1), testing time 
(X2), stirring grade of environment (X3) 
and the dependent variables (response 
function): Y1 - concentration of chromium 
(Cr), Y2 - concentration of manganese 
(Mn), Y3 - concentration of iron (56Fe) and 
Y4 - concentration of nickel (Ni) found in 
corrosive environments. After having made 
the analysis of variance ANOVA, a model 
having the lowest P value (critical 
probability) for all variables was chosen. 
The regression coefficient was determined 
to verify the validity of each mathematical 
model.  
 
3. Results and discussion 
 
The chemical analysis of corrosive 
environments was intended to determine 
the concentration of Cr, Mn, Fe and Ni 
elements migrated from AISI304 stainless 
steel samples.  
 
3.1. Mathematical modelling of metallic 
constituent concentrations migrated into 
3% CH3COOH solutions  
The statistical summary of mathematical 
models is shown in Table 3.  
 



Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - 
SuceavaVolume XIV, Issue 2 – 2015 

 

S i l v i u- G ab r ie l S T R OE ,  Mathematical modelling of the constituents’ concentrations of AISI 304 stainless steel samples 
that diffuse into simulated acidic environments,  Fo od  a nd E nvir o nme nt  Sa f e t y,  V o l u me  X IV ,  Is s ue  2 – 20 1 5,  p a g  
2 3 3- 24 0  

236 
 

Table 3 
Summary statistic of mathematical models proposed for the dependent variables 

 
Dependent 

variable P Model 
Standard deviation 

[σ] 
R-Squared 

[R2] 
Adjusted R-Squared 

 [R2 adjusted] 
Y1 0.0070 Quintic 8.59 0.9967 0.9783 
Y2 0.0190 Quintic 1.71 0.9964 0.9765 
Y3 0.0490 Quintic 323.04 0.9910 0.9300 
Y4 0.0099 Quintic 7.82 0.9964 0.9765 

 
The mathematical model that defines the 
variation of chromium concentration 
(dependent variable) depending on the 
testing parameters (independent variables) 
is shown in Equation 1. 
 

= 7.96 + 3.83 ∙ X − 3.72 ∙ X − 1.28 ∙ X +
3.06 ∙ X + 6.06 ∙ X + 3.06 ∙ 	X + 17.75 ∙ X ∙
X − 45.25 ∙ X ∙ X − 0.25 ∙ X ∙	X + 25.08 ∙
X ∙ X + 	48.92 ∙ X ∙	X + 15.25 ∙ X ∙ X +
9.58 ∙ 	X ∙ X − 9.50 ∙	X ∙ X ∙ X + 	5.42 ∙ X ∙
X + 30.25 ∙ X ∙	X ∙ X + 43.42 ∙ X ∙ X +
21.00 ∙ X ∙ X ∙		X − 26.25 ∙ X ∙ X ∙ X −
9.58 ∙ X ∙	X                (1) 
 
In order to check the validity of the 
mathematical model proposed, the 
experimental values vs model values were 
plotted (Figure 2). 
 

 
Fig. 2. Validity check of the Y1 (chromium 

concentration) mathematical model 
 
The mathematical model of the Y2 
response function is shown in Equation 2. 
 

= 4.04 − 0.59 ∙ X − 0.35 ∙ X2 + 1.39 ∙ X +
0.38 ∙ X + 0.68 ∙ X2 + 0.18 ∙ X + 3.18 ∙ X ∙
X2 − 8.20 ∙ X ∙ X + 0.30 ∙ X2 ∙ X + 4.18 ∙ X ∙
X2 + 8.98 ∙ X ∙ X + 4.24 ∙ X ∙ X2 − 7.88 ∙
X ∙ X − 0.47 ∙ X2 ∙ X + 1.95 ∙ X2 ∙ X − 0.63 ∙

X ∙ X2 ∙ X + 0.83 ∙ X ∙ X2 + 4.02 ∙ X ∙ X2 ∙
X + 8.35 ∙ X ∙	X + 3.43 ∙ X ∙ X2 ∙ X −
2.73 ∙ X ∙ X2 ∙ X − 0.40 ∙ X2 ∙ X                    (2) 
 
The validity of the Y2 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 3). 
 

Fig.3. Validity check of the Y2 (manganese 
concentration) mathematical model  

 
The mathematical model of the Y3 
response function is shown in Equation 3. 
 

= 394.07 − 88.33 ∙ X − 1.67 ∙ X2 + 116.67 ∙
X − 56.11 ∙ X + 123.89 ∙ X2 − 1.11 ∙ X +
437.50 ∙ X ∙	X2 − 887.50 ∙ X ∙ X + 0.002 ∙ X2 ∙
X + 372.50 ∙ X ∙ X2 + 1012.50 ∙ X ∙ X +
637.50 ∙ X ∙ X2 − 762.50 ∙ X ∙ X − 115 ∙ X2 ∙
X + 55 ∙ X2 ∙ X + 430 ∙ X ∙ X2 ∙ X + 194.17 ∙
X ∙ X2 + 305 ∙ X ∙ X2 ∙ X + 1019.17 ∙ X ∙
X + 772.5 ∙ X ∙ X2 ∙ X + 52.50 ∙ X ∙ X2 ∙
X − 253.3 ∙ X2 ∙	X                              (3) 
 
The validity of the Y3 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 4). 
 

R² = 0.9967

0.00

50.00

100.00

150.00

200.00

250.00

0.00 50.00 100.00 150.00 200.00 250.00M
od

el
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Experimental value, Cr [ppb]

R² = 0.996

0.00

10.00

20.00

30.00

40.00

50.00

0.00 10.00 20.00 30.00 40.00 50.00

M
od

el
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[p
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Experimental value, Mn [ppb]



Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - 
SuceavaVolume XIV, Issue 2 – 2015 

 

S i l v i u- G ab r ie l S T R OE ,  Mathematical modelling of the constituents’ concentrations of AISI 304 stainless steel samples 
that diffuse into simulated acidic environments,  Fo od  a nd E nvir o nme nt  Sa f e t y,  V o l u me  X IV ,  Is s ue  2 – 20 1 5,  p a g  
2 3 3- 24 0  

237 
 

Fig. 4. Validity check of the Y3 (iron 
concentration) mathematical model 

 
The mathematical model of the Y4 
response function is shown in Equation 4. 
 

= 20.44 ∙ X − 8.39 ∙ X2 + 2.00 ∙ X − 7.56 ∙
X − 0.86 ∙ X2 + 0.64 ∙ X + 15.25 ∙ X ∙ X2 −
24.75 ∙ X ∙ X − 2.25 ∙ X2 ∙ X + 26.83 ∙ X ∙
X2 + 37.50 ∙ X ∙ X + 8.60 ∙ X ∙ X2 − 37.40 ∙
X ∙ X + 9.00 ∙ X2 ∙ X + 10.83 ∙ X2 ∙ X −
13.63 ∙ X ∙ X2 ∙ X + 14.23 ∙ X ∙ X2 + 28.88 ∙
X ∙ X2 ∙ X + 34.23 ∙ X ∙ X + 2.63 ∙ X ∙ X2 ∙
X − 23.88 ∙ X ∙ X2 ∙ X + 4.03 ∙ X2 ∙ X    (4) 
 

The validity of the Y4 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 5). 
 

Fig. 5. Validity check of the Y4 (nickel 
concentration) mathematical model  

 
3.2. Mathematical modelling of metallic 
constituent concentrations migrated into 
6% CH3COOH solutions  
Statistical summary of mathematical 
models to describe the found dependent 
variables Cr, Mn, 56Fe and Ni is shown in 
Table 4. 

Table 4 
Summary statistic of mathematical models proposed for the dependent variables 

Dependent 
variable P Model 

Standard deviation 
[σ] 

R-Squared 
[R2] 

Adjusted R-Squared 
 [R2 adjusted] 

Y1 0.0400 Cubic 2.67 0.9618 0.9006 
Y2 0.0480 Quintic 0.19 0.9999 0.9978 
Y3 0.0015 Quintic 17.25 0.9979 0.9866 
Y4 0.0018 Quintic 27.76 0.9890 0.9553 

 
The mathematical models that define the 
studied response functions (dependent 
variables) depending on testing parameters 
(independent variables) are presented in 
the following equations. 
 

= 19.67 + 2.50 ∙ X + 7,72 ∙ X + 4,28 ∙ X −
4.33 ∙ X + 2.00 ∙ X + 0.17 ∙ X + 0.083 ∙ X ∙
X 1.50 ∙ X ∙ X + 3.00 ∙ X ∙ X − 3.58 ∙ X ∙ X −
0.67 ∙ X ∙ X + 3.25 ∙ X ∙ X − 1.00 ∙ X ∙ X +
2.83 ∙ X ∙ X − 0.33 ∙ X ∙ X + 2.13 ∙ X ∙ X ∙
X                 (5) 
 
The validity of the Y1 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 6). 

Fig. 6. Validity check of the Y1 (chromium 
concentration) mathematical model 

 
The mathematical model of the Y2 (Mn 
concentration) response function is shown 
in Equation 6. 
 

R² = 0.9964

0.00

50.00

100.00

150.00

200.00

250.00

0.00 50.00 100.00 150.00 200.00 250.00

M
od

el
 v

al
ue

, N
i [

pp
b]

Experimental value, Ni [ppb]

R² = 0.9965

0.00

10.00

20.00

30.00

40.00

0.00 10.00 20.00 30.00 40.00M
od

el
 v

al
ue

, C
r 

[p
pb

]

Experimental value, Cr [ppb]

R² = 0.996

0.00

10.00

20.00

30.00

40.00

50.00

0.00 10.00 20.00 30.00 40.00 50.00

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 - 
SuceavaVolume XIV, Issue 2 – 2015 

 

S i l v i u- G ab r ie l S T R OE ,  Mathematical modelling of the constituents’ concentrations of AISI 304 stainless steel samples 
that diffuse into simulated acidic environments,  Fo od  a nd E nvir o nme nt  Sa f e t y,  V o l u me  X IV ,  Is s ue  2 – 20 1 5,  p a g  
2 3 3- 24 0  

238 
 

= 3.87 − 1.60 ∙ X + 4.70 ∙ X + 0.75 ∙ X +
0.54 ∙ X + 3.04 ∙ X − 1.11 ∙	X + 	0.30 ∙ X ∙
X + 3.05 ∙ X ∙ X + 1.92 ∙ X ∙ X − 4. 25 ∙ X ∙
X − 3.10 ∙ X ∙ X + 1.75 ∙ X ∙ X − 1.55 ∙
X ∙ X + 1. 72 ∙ X ∙ X − 1.73 ∙	 X ∙ X +
1.83 ∙ X ∙ X ∙ X − 4.77 ∙ X ∙ X − 0.32 ∙ X ∙
X ∙ X + 2.83 ∙	X ∙ X − 1.42 ∙ X ∙ X ∙ X +
1.15 ∙ X ∙ X ∙	X − 0.49 ∙ X ∙ X + 2.68 ∙ X ∙
X ∙ X + 3.00 ∙ X ∙ X ∙ X + 2.85 ∙ X ∙ X ∙
X                                                            (6) 
 
The validity of the Y2 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 7). 
 

Fig. 7. Validity check of the Y2 (manganese 
concentration) mathematical model 

 
The mathematical model of the Y3 (56Fe 
concentration) response function is shown 
in Equation 7. 
 

= 167.41 − 16.11 ∙ X + 143.33 ∙ X + 80.00 ∙
X + 123.89 ∙ X + 128.89 ∙	X + 18.89 ∙ X −
70 ∙ X ∙ X − 72.50 ∙ X ∙ X − 12.50 ∙ X ∙ X −
90 ∙ X ∙ X + 57.50 ∙	 X ∙ X + 31.67 ∙
X ∙ X − 0.83 ∙ X ∙ X − 2.50 ∙ X ∙ X −
22.50 ∙ X ∙ X + 0.002 ∙ X ∙ X ∙	 X − 88.33 ∙
X ∙ X + 10 ∙ X ∙ X ∙ X − 5.83 ∙ X ∙ X +
152.50 ∙ X ∙ X ∙ X − 5.00 ∙ X ∙ X ∙ X − 5.83 ∙
X ∙ X                                                    (7) 
 
The validity of the Y3 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 8). 
 

Fig. 8. Validity check of the Y3 (iron 
concentration) mathematical model 

 
The mathematical model of the Y4 (Ni 
concentration) response function is shown 
in Equation 8. 
 

= 311.44 − 24.67 ∙ X − 181.50 ∙ X + 41.17 ∙
X + 7.50 ∙ X + 24.33 ∙ X − 8.83 ∙ X −
69.25 ∙ X ∙ X + 6.33 ∙ X ∙ X − 15.67 ∙ X ∙ X +
97.25 ∙ X ∙	X − 1.50 ∙ X ∙ X + 35.75 ∙
X ∙ X − 8.00 ∙ X ∙ X + 7.50 ∙ X ∙ X +
12.50 ∙ X ∙ X − 7.38 ∙ X ∙	X ∙	X              (8) 
 
The validity of the Y4 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 9). 
 

Fig. 9. Validity check of the Y4 (nickel 
concentration) mathematical model 

 
3.3. Mathematical modelling of metallic 
constituent concentrations migrated into 
9% CH3COOH solutions  
Statistical summary of mathematical 
models to describe the found dependent 
variable Cr, Mn, 56Fe and Ni is shown in 
Table 5. 
 

R² = 0.9981

0.00
2.00
4.00
6.00
8.00

10.00
12.00

0.00 5.00 10.00M
od

el
 v

al
ue

, M
n 

[p
pb

]

Experimental value, Mn [ppb]

R² = 0.9958

0.00
200.00
400.00
600.00
800.00

1000.00
1200.00

0 500 1000 1500M
od

el
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al
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,5
6 F

e
[p

pb
]

Experimental value, 56Fe [ppb]

R² = 0.9623

0.00

5.00

10.00

15.00

20.00

2.00 7.00 12.00 17.00

M
od

el
 v

al
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, N
i [

pp
b]

Experimental value, Ni [ppb]



Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - 
SuceavaVolume XIV, Issue 2 – 2015 

 

S i l v i u- G ab r ie l S T R OE ,  Mathematical modelling of the constituents’ concentrations of AISI 304 stainless steel samples 
that diffuse into simulated acidic environments,  Fo od  a nd E nvir o nme nt  Sa f e t y,  V o l u me  X IV ,  Is s ue  2 – 20 1 5,  p a g  
2 3 3- 24 0  

239 
 

Table 5 
Summary statistic of mathematical models proposed for the dependent variables 

 
Dependent 

variable P Model 
Standard deviation 

[σ] 
R-Squared 

[R2] 
Adjusted R-Squared 

 [R2 adjusted] 
Y1 0.010 Quintic 1.06 0.9965 0.9771 
Y2 0.020 Quintic 0.21 0.9981 0.9876 
Y3 0.035 Quintic 41.82 0.9960 0.9730 
Y4 0.040 Cubic 0.89 0.9823 0.9220 

 
The mathematical models that define the 
studied response functions are presented in 
the following equations. 
 

= 8.67 + 0.33 ∙ X + 2.17 ∙ X + 0.17 ∙ X −
6.00 ∙ X − 2.50 ∙ X − 0.50 ∙ X + 2.00 ∙ X ∙
X + 0.25 ∙ X ∙ X + 1.50 ∙ X ∙ X + 1.00 ∙ X ∙
X + 1.25 ∙ X ∙ X + 4.00 ∙	X ∙ X + 0.75 ∙
X ∙ X + 3.00 ∙ X ∙ X + 6.00 ∙ X ∙ X + 3.00 ∙
X ∙ X ∙	X + 6.00 ∙ X ∙ X + 1.25 ∙ X ∙ X ∙
X + 0.75 ∙ X ∙ X + 3.50 ∙ X ∙ X ∙ X + 2.00 ∙
X ∙ X ∙ X + 1.50 ∙ X ∙ X                        (9) 
 
The validity of the Y1 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 10). 
 

Fig. 10. Validity check of the Y1 (chromium 
concentration) mathematical model 

 
The mathematical model of the Y2 (Mn 
concentration) response function is shown 
in Equation 10. 

= 1.46 − 0.00005 ∙ X + 1.21 ∙ X + 0.13 ∙
X − 0.22 ∙ X + 0.88 ∙ X − 0.12 ∙ X + 1.15 ∙
X ∙ X + 0.025 ∙ X ∙ X + 0.63 ∙ X ∙ X − 0.33 ∙
X ∙ X + 0.26 ∙ X ∙ X + 0.93 ∙ X ∙ X + 0.11 ∙
X ∙ X + 0.91 ∙ X ∙ X + 0.042 ∙ X ∙ X +
0.93 ∙ X ∙ X ∙	X + 0.25 ∙ X ∙ X + 0.20 ∙ X ∙

X ∙ X + 0.17 ∙ X ∙ X + 0.90 ∙ X ∙ X ∙ X +
0.10 ∙ X ∙ X ∙ X + 0.025 ∙ X ∙ X          (10) 
The validity of the Y2 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 11). 
 

Fig. 11. Validity check of the Y2 (manganese 
concentration) mathematical model 

 
The mathematical model of the Y3 (56Fe 
concentration) response function is shown 
in Equation 11. 
 

= 147.78 − 18.89 ∙ X + 190.00 ∙ X + 57.78 ∙
X − 66.67 ∙ X + 83.33 ∙ X − 16.67 ∙ X +
157.50 ∙ X ∙ X − 47.50 ∙ X ∙ X + 100.00 ∙ X ∙
X − 42.50 ∙ X ∙ X + 40.83 ∙ X ∙ X + 180.83 ∙
X ∙ X − 9.17 ∙ X ∙ X + 53.33 ∙ X ∙ X −
15.00 ∙ X ∙ X + 105.00 ∙ X ∙ X ∙	X + 12.50 ∙
X ∙ X − 7.50 ∙ X ∙ X ∙ X + 77.50 ∙ X ∙
	X + 162.50 ∙ X ∙ X ∙ X + 7.50 ∙ X ∙ X ∙
X − 20.00 ∙ X ∙ X                              (11) 
 
The validity of the Y3 mathematical model 
proposed was plotted by experimental 
values vs model values (Figure 12). 
 
 

R² = 0.9965

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 = 0.9981

0.00
1.00
2.00
3.00
4.00
5.00
6.00
7.00
8.00
9.00

10.00

0.00 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 - 
SuceavaVolume XIV, Issue 2 – 2015 

 

S i l v i u- G ab r ie l S T R OE ,  Mathematical modelling of the constituents’ concentrations of AISI 304 stainless steel samples 
that diffuse into simulated acidic environments,  Fo od  a nd E nvir o nme nt  Sa f e t y,  V o l u me  X IV ,  Is s ue  2 – 20 1 5,  p a g  
2 3 3- 24 0  

240 
 

Fig. 12. Validity check of the Y3 (iron 
concentration) mathematical model 

 
The mathematical model of the Y4 (Ni 
concentration) response function is shown 
in Equation 12. 
 

= 8.44 + 1.09 ∙ X + 1.28 ∙ X + 1.74 ∙ X −
1.49 ∙ X − 0.23 ∙ X + 0.37 ∙ X + 1.07 ∙ X ∙
X + 0.59 ∙ X ∙	 X + 0.76 ∙ X ∙ X − 0.47 ∙ X ∙
X − 0.21 ∙ X ∙ X + 0.0083 ∙ X ∙ X + 0.91 ∙
X ∙ X + 0.24 ∙ X ∙ X + 0.72 ∙ X ∙ X + 1.18 ∙
X ∙ X ∙	X                                               (12) 
 
The validity of the Y4 mathematical model 
proposed was been plotted by experimental 
values vs model values (Figure 13). 
 

Fig. 13. Validity check of the Y4 (nickel 
concentration) mathematical model 

 
4. Conclusions 
 
ICP-MS method and ANOVA method 
were used to establish the relationships 
between migration test parameter values 
and Cr, Mn, 56Fe and Ni concentrations 
found in solutions. 
Besides the fact that these instrumental 
analysis techniques provide the checking 

of mathematical models’ validity for each 
dependent variable, we can also draw the 
conclusion that they are highly performing 
and accurate instruments that can be used 
in  estimating the values of objective 
function (see also [4]). 
 
5. References 
 
[1]. BARNES, K. A., SINCLAIR, C. R., 
WATSON, D.H., Chemical migration and food 
contact materials, Woodhead Publishing Limited, 
ISBN-13: 978-1-84569-029-8, England, (2007); 
[2]. SANCHES SILVA A., CRUZ J. M., 
SENDOR GARCIA R., FRANZ R., PASEIRO 
LOSADA, P., Kinetic migration studies from 
packaging films into meat products, ScienceDirect, 
Meat Science 77, 238-245, (2007); 
[3]. SOOJIN J., IRUDAYARAJ J.M., Food 
Processing operations modeling - Design and 
Analysis, Second Edition, CRC Press-
Taylor&Francis Group, 1-2, (2009); 
[4]. STROE S. G., GUTT G., Statistical study of 
the dependence between concentration of metallic 
elements migrated from stainless steel grade 
AISI321 and working parameters, Food and 
environment safety, Faculty of Food Engineering, 
Volume 12, Issue 2, (2013); 
[5]. TEHRANY E. A., DESOBRY S., Partition 
coefficient of migrants in food stimulants/polymers 
systems, ScienceDirect, Food Chemistry 101, 1714-
1718, (2007); 
[6]. AMANI S. ALTURIQI,  LAMIA A. 
ALBEDAIR, The Egyptian Journal of Aquatic 
Research, ScienceDirect, Volume 38, Issue 1, 45–
49, (2012); 
[7]. KAMAL J. ELNABRIS, SHAREEF K. 
MUZYED, NIZAM M. EL-ASHGAR, Heavy 
metal concentrations in some commercially 
important fishes and their contribution to heavy 
metals exposure in Palestinian people of Gaza Strip 
(Palestine), Journal of the Assoc. of Arab 
Universities for Basic and Applied Sciences, 
Volume 13, Issue 1, April 2013, Pages 44–51, 
(2013); 
[8]. DATTA A. K., Biological and 
Bioenvironmentat Heat and Mass Transfer, Cornell 
University Ithaca, New York, Marcel Dekker Inc., 
ISBN: 0-8247-0775-3, 11-12, (2002); 
[9]. D.M. 21-03-1973, Italian law text, Decreto 
Ministeriale del 21/03/1973 - Disciplina igienica 
degli imballaggi, recipienti, utensili, destinati a 
venire in contatto con le sostanze alimentari o con 
sostanze d'uso personale, (1973); 
 

R² = 0.9958

0.00

500.00

1000.00

1500.00

0 500 1000 1500M
od

el
 v

al
ue

,5
6 F

e 
[p

pb
]

Experimental value, 56Fe [ppb]

R² = 0.9623

0.00

5.00

10.00

15.00

20.00

2.00 7.00 12.00 17.00M
od

el
 v

al
ue

, N
i[

pp
b]

Experimental value, Ni [ppb]