ЗВІТ  З  НДР  29-81  ЗА  2007 – 2009 Р


   
 

71 
 

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

Ştefan cel Mare University of Suceava, Romania  
Volume XV, Issue 1 - 2016, pag. 71  - 75 

 

MATHEMATICAL MODELING OF PREPARING  

 "OIL / WATER" EMULSIONS 
*Oksana LUGOVSKA1, Vasilij SIDOR1 

 
1National University of Food Technologies, Volodimirska 68, Kiev, Ukraine,  

oksana.lugovska320@gmail.com 
*Corresponding author 

Received March 6th 2016, accepted March 29th 2016 
 

Abstract: Past studies on the effect of the number of emulsifiers in emulsion characteristics obtained 
showed that the change in the fate of particle distribution depended on the particle size emulsion close 

to Gaussian distribution. Therefore, the designed factor model depends on the particle size emulsion of 

their number. 
 
Keywords: emulsion, phase, starch, Arabic gum, mathematical modeling 
 
 

1. Introduction 
 

Nowadays, the emulsion is widely used in 
different sectors of the food industry. 
Getting a stable emulsion system is an 
important and promising issue.  
As the stabilizing and emulsifying 
ingredient in the manufacture of scented 
oil emulsions using gum Arabic (E 414) 
and starch (E 1450). Starch - is one of the 
most widely used thickeners and 
emulsifier. Gum Arabic provides better 
emulsion stability [1]. 
Found that a stable, emulsions are closely 
associated with the mechanism of 
dispersion and depends on many factors, 
such as oil content, type and concentration 
of emulsifier, the route of administration 
phases, time and intensity and degree of 
dispersion and temperature. Study of 
factors that ensure stability of emulsion, 
led to the conclusion that the critical 
degree of dispersion [2–3]. 
Experiments found that for each type of 
emulsifier it has its own optimum 
concentration that provides the highest 
resistance obtained emulsions [4]. For an 
introduction to emulsify oils (for each 
concentration of emulsifier) is also 
optimum in which the most stable 

emulsion is obtained, that are determining 
the optimal ratio between the aqueous and 
oil phases.  
For each emulsifier has its own optimal 
concentration for the stability of the 
emulsion. [5-6]. 
The optimum concentrations of emulsifiers 
for certain ratios of the phases in obtaining 
stable emulsions are not fixed and depend 
on the degree of dispersion [7]. To 
determine the most effective parameters of 
technological processes of production 
EMULSIONS to build a mathematical 
model of conduct and its optimization of 
defined parameters. 
Model - a conventional image of the object 
which displays its most significant 
characteristics required for the study. 
• Any model performs predictive function 

without which it would build inappropriate 
for theory and practical use. 
• Experimental models based only on the 

laws of probab 
ility theory. 
• In constructing these models studied 

processes conventionally regarded as 
deterministic, but the model is introduced 
elements of evaluating the probability of 
obtaining a certain result. 

http://www.fia.usv.ro/fiajournal
mailto:oksana.lugovska320@gmail.com


Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava 
Volume XV, Issue 1  – 2016 

 

Oksana LUGOVSKA, Vasilij SIDOR, Mathematical modeling of preparing "oil / water" emulsions, Food and 
Environment Safety, Volume XV, Issue 1 – 2016, pag. 71  – 75 
 

72 
 

Determined factor analysis -it analysis of 
factors resulting indicator can be presented 
as a product, part or the algebraic sum of 
factors [8]. 
Past studies on the effect of the number of 
emulsifiers in emulsion characteristics 
obtained showed that changing the fate of 
particle distribution depending on the 
particle size emulsion close to Gaussian 
distribution - probability distribution, 
which is set in one-dimensional case, the 
probability density function. 

2

2

2
)(

2
1

)( 









x

exf  

 
where μ - expectation (average), median 
and mode of distribution, and the 
parameter  
σ - standard deviation (σ ² - variance) 
distribution. 
If the variable has a normal distribution, 
then it there (end) points for all values such 
that in different parts of the domain, it can 
be given different analytical expressions or 
piecewise given function. 
Piecewise given function is a function that 
is defined on the set of real numbers given 
on each of the intervals, the domain of the 
components, a separate formula. In general 
terms: 

















33

22

11

),(
),(
),(

bxaifx
bxaifx
bxaifxf

y


  

where - function defined on a certain 
interval - the lower limit of the range - 
upper limit of the range, 
The analysis of the experimental data 
shows that the value of the particle size can 
be determined at such intervals (0, 0.1); 
[0.1, 1) and (1, 10). In the interval (0, 0.1) 
function is a linear regression on the 
interval (0.1, 1) and (1, 10) function is 
polynomial model n- order. If we denote 
the fate of particle distribution variable y, a 
particle size emulsion x, then the general 
model takes the form. 

















.101,
;11,0,

;1,00,

5
5

4
4

3
3

2
210

5
5

4
4

3
3

2
210

xifxbxbxbxbxbb
xifxaxaxaxaxaa

xifc
y

 
Calculation of the model is the method of 
least squares (MLS). The method is based 
on minimizing the sum of squared 
deviations of certain functions of the 
unknown variables. It is used to "solve" 
overriding systems of equations (equations 
when the number exceeds the number of 
unknowns), to find a solution in the case of 
conventional systems of nonlinear 
equations to approximate point values of 
some function. MLS is a basic regression 
analysis methods for the estimation of the 
unknown parameters of regression models 
for the sample data.Objective Research to 
conduct mathematical modeling study of 
the influence quantity of emulsifier gum 
arabic or starch and the oil phase to final 
figures emulsions "oil / water" 
 

2. Material and methods 
 
We built a model of distribution of 
particles emulsions containing emulsifiers 
equal amount of gum arabic (5%) and 
different amounts of oil phase.For oil 
content of 14.0% phase model is: 



























101,0001,0002,0
017,0041,0308,0867,1

11,0,381,78106,299
517,339261,97381,27974,3

1,00,0

54

32

54

32
1

xifxx
xxx

xifxx
xxx

xif

y
 

For each oil phase built their system, 
describing the experimental data for 
predicting the future of individual particle 
size 
Similarly, processed research data at 
constant oil phases and changes in the 
content and formed stabilizer system of 
mathematical models describing the 
formation of particles of different size 
stabilizer. 



Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava 
Volume XV, Issue 1  – 2016 

 

Oksana LUGOVSKA, Vasilij SIDOR, Mathematical modeling of preparing "oil / water" emulsions, Food and 
Environment Safety, Volume XV, Issue 1 – 2016, pag. 71  – 75 
 

73 
 

Similarly, there was a process of modeling 
the distribution of the fate of particle size 
emulsion particles at different pressure 
homogenization.  
 
 
 

3. Results and discussion 
Figure 1 shows the experimental and 
calculated curves of particle size 
distribution with stable emulsions 
containing emulsifier gum arabic - 5% oil 
content and a different phase (8.0%; 
10.0%; 11.0%; 12.0%; 14 0%).

a)                                                                            b) 
Fig. 1. Curves size distribution of particles emulsions with constant content of emulsifier gum arabic - 5% 

and oil content varying phases: 1 - 8.0%; 2 - 10.0%; 3 - 11.0%; 4 - 12.0%; 5 - 14.0%: 
a) experimental data; b) estimates 

 
Construct models of the size distribution of 
particles emulsions with constant 
emulsifier starch content (12%) and 
different content oil phase. 
Figure 2 shows the experimental and 
calculated curves of size distribution of 

particles emulsions with constant 
emulsifier starch content of 12% oil 
content and a different phase (8.0%; 
10.0%; 11.0%; 12.0%; 14 0%. 

 
a)                                                                     b) 

 
Fig.2. Curves size distribution of particles emulsions with constant emulsifier starch content of 12.0% and 

oil content varying phases: 1 - 8.0%; 2 - 10.0%; 3 - 11.0%; 4 - 12.0%; 5 - 14.0%:  
a) experimental data; b) estimates 



Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava 
Volume XV, Issue 1  – 2016 

 

 
 
Oksana LUGOVSKA, Vasilij SIDOR, Mathematical modeling of preparing "oil / water" emulsions, Food and 
Environment Safety, Volume XV, Issue 1 – 2016, pag. 71  – 75 

74 
 

Construct models of the size distribution of 
particles emulsions with a constant number 
of oil phase - 10.0% and different content 
emulsifier gum arabic. 
Figure 3 shows the experimental and 
calculated curves of size distribution of 

particles emulsions with a constant number 
of oil phase - 10.0% and different content 
emulsifier gum arabic (4.0%; 5.0%; 5.5%; 
6.0%; 7.0%. 

 
 

a)                                                                                  b) 
Fig. 3. Curves size distribution of particles emulsions with a constant number of oil phase - 10.0% and 

different content emulsifier gum arabic 6 - 4.0%; 7 - 5.0%; 8 - 5.5%; 9- 6.0%; 10 - 7.0%:  
a) experimental data; b) estimates 

 
Construct models of the size distribution of particles emulsions with a constant number of oil 
phase - 11.0% and different content emulsifier starch. For example, the starch content of 8.0% 
emulsifier model is: 

 
 

a)                                                                                  b) 
Fig. 4. Curves size distribution of particles emulsions with a constant number of oil phase - 11.0% and 

different content emulsifier starch 6 - 8.0%; 7 - 10.0%; 8 - 11.0%; 9 - 12.0%; 10 - 14.0%:  
a) experimental data; b) estimates 



Food and Environment Safety - Journal of Faculty of Food Engineering, Ştefan cel Mare University - Suceava 
Volume XV, Issue 1  – 2016 

 

 
 
Oksana LUGOVSKA, Vasilij SIDOR, Mathematical modeling of preparing "oil / water" emulsions, Food and 
Environment Safety, Volume XV, Issue 1 – 2016, pag. 71  – 75 

75 
 

 
 
 
4. Conclusion 
 
The best result of research in starch 
emulsion – is to obtain the maximum 
number of particles of about 1 micron.  
In order to determine the most effective 
process parameters of food production to 
build a mathematical model of 
optimization of certain parameters. 
Analysis of a priori information and 
experience on current technology and 
conditions of production of aromatic 
emulsions allow for process modeling 
approaches applied regression analysis and 
choose the regression equation of the 
second order. 
The results of computational experiments 
show that almost all indicators emulsion 
stabilizer content is more important. 
The relative error of calculations does not 
exceed 3%. For all models Fisher criterion 
settlement does not exceed the tabular 
value that indicates the adequacy of 
research models and can be used for 
decision making. 
Unlike emulsion stabilizer - acacia, 
modeling parameters emulsions with starch 
show that oil phase is more influentual. 
The relative error of calculations does not 

exceed 4.5%, and the estimated Fisher 
criterion is less than the table and all 
models also adequately describe the study 
process for all models of criterion Fisher 
estimated value does not exceed tabular 
Ftabl. = 4.96/. Therefore all the models 
adequately describe the study process and 
can be used for decision-making). 
 

5. References 
 
[1]. PHILLIPS G.О., WILLIAMS P.A., (Eds.) 
Handbook ofHydrocolloids. Cambridge: Wood 
Head Publishing, 156 с, (2000). 
[2]. McKENNA В.M. (Ed.), Texture in Food — 
Vol. 1: Semi-Solid Foods. — Cambridge: 
Woodhead Publishing, 480 с, (2003). 
[3]. STEPHEN P., U.S. Patent for invention № 
6576285, Cholesterol lowering beverage, Bader, 
Fowler, 10.06.2003. 
[4]. IMESONA. (Ed.), Thickening and Gelling 
Agents for Food. 2ndEd., London: Blackie 
Academic and Professional, 408 с, (1999). 
[5]. WHISTLERR.L.B.,MILLER J.N., 
PASCHALL E.F., (Eds.), Starch Chemistry and 
Technology. 2ndEd. Orlando, FL: Academic Press, 
1984, 3508. Galliard T. (Ed.) Starch; Properties and 
Potential /Society of Chemical Industry. Chichester, 
UK: John Wiley and Sons, 210 с (1987). 
[6]. ATWELLW.A. THOMAS D.J. Starches. — 
St. Paul, MN: American Association of Cereal 
Chemists, 150 с. (1997). 
[7]. A. AYMESON, Food thickeners, stabilizers, 
gelling agents, "Profession" Publishers, St. 
Petersburg, 2012.- Art. 24-44