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ISSN 2522-1841 (Online) AZERBAIJAN CHEMICAL JOURNAL № 3 2023 61
ISSN 0005-2531 (Print) UDC 622.276.72
EVOLUTION OF THE DISTRIBUTION FUNCTION OF DROPS IN AN OIL EMULSION
M.R.Manafov\ S.R.Rasulov2, F.R-Shikhieva1*, V.LKerimli1
1M.Nagiyev Institute of Catalysis and Inorganic Chemistry, Ministry of Science and Education of
the Republic of Azerbaijan 2Azerbaijan State Oil and Industrial University
mmanafov@gmail.com sakit.rasulov@asoiu.edu.az miracle1990@list.ru Corrospendence: fatmashikhieva9981@gmail.com
Received 13.01.2023 Accepted 08.01.2023
Oil emulsions are polydisperse media and have a droplet size in the macro and micro range. Macro and microemulsions can be found in both water-in-oil emulsion types and oil-in-water emulsion types. Taking into account the size distribution of water-oil emulsion droplets has a great influence on the accuracy of calculations when modeling the process of oil dehydration. Droplet size and droplet size distribution are important for the stability and viscosity properties of the emulsion. In the article, based on the rate of coalescence and fragmentation of particles, an expression is proposed for the rate of change in the number and size of particles per unit volume. On the basis of the Fokker-Planck equation, an expression for the distribution function of water droplets in an oil emulsion is found and, on its basis, an estimate of the distribution is investigated.
Keywords: oil emulsions, Fokker-Planck equations, distribution function evolution, coalescence, coagulation, droplet size, rheological parameters.
doi.org/10.32737/0005-2531-2023-3-61-69 Introduction
Currently, the production of high-viscosity and heavy oils is growing rapidly and increasingly leads to the formation of very stable water-oil emulsions [1]. For the mining and oil refining industry, this is a serious problem. At the moment, the water cut of many large fields reaches 80% or more, this is due to the use of waterflooding methods to increase their oil recovery. The main part of the water extracted together with the oil passes into the emulsion. To prepare such oils for processing at a refinery, special equipment and chemicals are needed. Therefore, the possibility of rapid separation of water-oil emulsions is the main issue in the preparation of oil [2-5].
Every year, millions of tons of oil are produced and processed in the world. An important stage in this technological chain is the high-quality preparation of oil. A complex mixture is extracted from oil wells, consisting of oil, formation water containing mineral salts, associated petroleum gas and mechanical impu-
rities (sand, drilling mud, etc.). In this form, the transportation of oil through the main pipelines is not economically feasible, therefore, after the wells, the oil is sent through the pipeline to the oil treatment unit. The preparation of oil directly at the field occupies an important place in the chain associated with the production, collection, and transportation of commercial oil for further processing. Oil coming from the field to the oil treatment plant is an emulsion, which, as a rule, is never monodisperse, since it contains water droplets of different sizes.
Settling of an oil-water emulsion is a technological operation used for phase separation, i.e. precipitation of water in an oil-water emulsion. This operation is the main stage in the process of destruction of oil emulsions. As a rule, it is preceded by the processes of processing the emulsion with a demulsifier and preparing it for separation.
In the preparation of product oil, highly efficient settling apparatuses are used, which
combine the processes of separation of petroleum gas and oil dehydration.
For the destruction of already formed oil emulsions, demulsifiers are widely used - surfactants, which, unlike natural emulsifiers, contribute to a significant decrease in the stability of oil emulsions. Demulsifiers for the destruction of oil emulsions should be able to penetrate the oil-water interface, cause flocculation and coalescence of water droplets, and well wet the surface of mechanical impurities [6].
The nature of water deposition in an emulsion differs sharply from the nature of the deposition of a single particle, because the parameters of the medium in which water particles are deposited are constantly changing.
Oil emulsions are a mechanical mixture of oil and formation water, insoluble in each other and in a finely dispersed state. They are polydisperse media with water droplet sizes of 1-150 micron, although they contain coarse (150-1000 micron) and colloidal (0.001-1 micron) particles. Such a scatter of sizes has a significant effect on the mechanism of structure formation, structure destruction, separation, and settling of droplets in oil emulsions.
The mechanism of destruction of oil emulsions can be divided into three elementary stages: collision of water globules; merging them into larger drops; precipitation of drops or separation in the form of a continuous aqueous phase.
There are three types of emulsions: wa-ter-in-oil (W/O), oil-in-water (O/W) [7] and complex emulsions such as water-in-oil-in-water. A complex emulsion is also known as a multiple emulsion [8]. Figure 1 shows these types of emulsion.
Fig. 1. Classification of oil emulsions [9] In the field treatment of oil, the technological parameters of the processes of separation, dehydration and desalting provide the required
quality of oil supplied for further processing. Optimal process parameters can be determined using computer simulation systems (MS).
A water-in-oil emulsion is a type of emulsion in which the continuous phase is usually hydrophobic materials such as oil and the dispersed phase is water [8]. More than 95% of the oil emulsion formed in the field are of the W/M type [9]. W/O emulsions contain three substances such as; solvent, surfactant and water.
In the process of drop formation, water droplets of a certain size are formed. Depending on the size of the droplets, stratification and destruction of the system occurs. Thus, the kinetic or sedimentation stability reflects the ability of the system to maintain the same distribution of particles of the dispersed phase per unit volume of the dispersion medium for a certain time, the same at each point. A large spread in the size of water droplets, taking into account their coalescence and crushing, is characterized by a distribution function in size and residence time, which forms the basis for the evolution of the state of an oil emulsion. The use of the evolution of the size distribution function of water droplets in technical calculations of the separation of oil emulsions makes it possible to interpret the complete picture of the state of the system at any time. Coalescence and crushing of drops significantly change the dispersion of oil emulsions, which is characterized by the evolution of the probability distribution function over time and size, described by the Boltzmann kinetic equation and the stochastic Fokker-Planck equation [10-14]. Changes in the size and shape of water droplets in an oil emulsion as a result of their coalescence, deformation, and crushing significantly affect the rheological parameters, in particular, the effective viscosity of the emulsion. Structural viscosity of a dispersed medium, associated with the content of dispersed water droplets, as well as with various physical phenomena of interaction between particles, varies from the value of the molecular viscosity of a Newtonian fluid in the absence of dispersed particles to shear or bulk viscosity at high concentrations of particles. Moreover, the structural viscosity of oil emulsions depends on the con-
tent of water, asphalt-resinous substances and paraffins, the high concentration of which complicates the degree of interaction between drops.
At the moment, quite a lot of formulas are known for calculating the size of droplets [15, 16].
Taking into account the size distribution of water-oil emulsion droplets has a great influence on the accuracy of calculations when modeling the process of oil dehydration.
It should be noted that the use of different methods for measuring the particle size distribution can lead to very different results. A compilation of information from two measurement methods to determine the particle size distribution (PSD) with improved accuracy compared to two separate methods [17].
Droplet size distribution (DSD) is an important property of an oil emulsion because it affects the viscosity and stability of the emulsion and can be used to classify emulsions [10, 18-21]. Each processing method works in a range of droplet sizes, so DSD can indicate the best method, such as gravity separation and advanced process, among others [22].
Control, monitoring and correction of equipment sizes can be carried out using DSD data, which helps to reduce the cost of transporting and treating oil [23].
Particle size analysis provides useful information about the structure and stability of multiple emulsions, which are important characteristics of these systems. It also makes it possible to observe the process of growth of particles dispersed in multiple emulsions and, accordingly, the change in their size over time [24].
The average droplet size, the difference between the maximum and minimum droplet diameters of the dispersed phase, and the degree of their dispersity are considered as significant parameters characterizing a given emulsion [25].
Particle size analysis provides useful information about the structure and stability of multiple emulsions, which are important characteristics of these systems. It also makes it possible to observe the process of growth of particles dispersed in multiple emulsions and, accordingly, the change in their size over time [26-29].
Evolution of the distribution function of drops in an oil emulsion
The processes of coalescence and crushing of water droplets in emulsions can proceed simultaneously. Then the rate of change in the number and size of particles per unit volume is determined by the rates of their coalescence and crushing
f = U' - U< (1)
where is the current number of drops in the volume, the rate of coalescence and fragmentation of drops. At a fast droplet crushing rate, the particle distribution function is asymmetric with respect to the maximum and is characterized by one maximum independent of the shear rate, although at slow coagulation, the particle size distribution function can have several maxima and minima, i.e., be multimodal. Moreover, each maximum will characterize the primary, secondary, etc. coagulation of particles of the dispersed medium.
With slow coagulation of solid particles, it is important to build the evolution of the residence time and size distribution function, which gives a complete picture of the change in the number and size of particles over time. The Fokker-Planck equation was used to construct the evolution of the particle distribution function [30, 31]. The stochastic Fokker-Planck equation describes dispersed systems with a continuous change in the properties of the medium and the size of dispersed inclusions. Although the processes of coalescence and fragmentation are characterized by an abrupt change in the properties of particles (sizes), in principle, for a sufficiently long period of time, the change in the average properties can be assumed to be quasi-continuous with an infinitely small jump. In particular, it can be assumed that the average size of drops and bubbles changes continuously with time and obeys the equation for the change in the average mass of particles over time
dm , s
— = ®{a)m (2)
Many experimental studies on fragmentation and coagulation of particles in a turbulent flow
show that the average particle size is set at the minimum or maximum level, which corresponds to the aggregative stability of the dispersed medium. Given the above, in equation (2) should be considered as the reduced mass relative to the extreme values of the particles,
i.e. m = ^Pd (a - amin)3 - for crushing and 6
ft ( \s m = —pd (amax - a ) - for coalescence. In par-
6
ticular, based on equation (2), we obtain an expression for changing the size of droplets during their crushing in the form
da
— = -K (co, a)(a - amin) = m0- m±a = f(a)
m0 = Kamin, m1 = K, t = 0, a = a0 (3)
which is a time-continuous process (where
m = ftPd (a - amm )3 for a p°wer-law non-6
Newtonian fluid and m = —pd (amax - a)3 for a
6
viscous-plastic fluid).
Thus, considering the change in particle size as a continuous function, the Fokker-Planck equation in the simplest case, taking into account (3), can be written as below [12,13,14]
t = 0, P(a,0) = P0(a); a ^ 0, P(a,t) ^ 0
(4)
here B - stochastic diffusion coefficient.
The solution of this equation presents great difficulties associated with setting the form of the function f (a), although some particular analytical solutions to equation (4), depending on the nature of the function setting, can be found in [3-5].
The solution of equation (4) by the method of separation of variables will be represented as
P(r, t) =
roexp (^tl) ^ Cn exp(-2knt)
(5)
here 0 = mRa°/
m^a,2 - B
, a =
2B
L(a]- La-
guerre functions
0+1 œ
0
J ^c (r W
C„ =
fkal r ^2B ,
dr
0-1
0+1
2 2 rl n + 0+1 lm 0 n!
(6)
Solutions (5) and (6) characterize the evolution of the droplet probability density distribution function over size and time. The asymptotic value of the distribution at t ^ ^ is obtained from the solution (5), taking into account the properties of the Laguerre function, in the form
Pœ(r) = CRrdexp ( -
ka^r2 2B
= CPRa6exp(-ba2)
(7)
9 + 1
CpR = 2a-°(£)' , b = k/.
2 B
Having introduced some simplifications, taking into account the initial distribution in the form of a lognormal function
P (a) = A (a) exp (-m0 (ln a - a0 )2)
and the limiting distribution (7), taking into account the experimental data for a family of distribution curves with the number m , with some assumptions in a more simplified form, we obtain
œ i-
P(a,t) = XAn(t)exP -mn(t)(lna-asn)
(8)
here a = ln as -is a parameter corresponding
to the logarithm of the maximum value of each extremum.
On Figure 1 shows the evolution of the distribution function during the crushing of a non-Newtonian viscoelastic drop (oil) in an aqueous medium.
The evolution of the distribution function i s describ ed by the following semi-empirical e qu ati on
c
n=0
P(a, t) = A1aexp[-52(ln a — ax)2] + A2aexp[—20.8(ln a — a2)2] + A3aexp[—15(ln a — a3)2] + A4aexp[—11.6(ln a — a4)2] (9)
here A, A, A, A
- coefficients
depending on the time and speed of the stirrer.
The spectrum of large and small drops is practical l y bound to shi ft rel ative to each other (Figure 2).
Fig. 2. The characteristic distribution of the distribution function during the crushing of drops in size and time, equal to: a - 0.5; b-2.0; c - 15; d - 60 sec.
It is important to note that the fluctuation of the distribution function on the left side of the curve indicates secondary, tertiary, etc. the nature of the fragmentation of drops, and on the right side - about their multiple coalescence. However, after some time, when the resources of the large-drop or small-drop spectrum are exhausted, the spectrum begins to behave like a single-hump spectrum. In practice, the behaviour of multihump distributions in the model representation is confirmed when the distribution is represented by the sum of two or more distribution functions. The nature of the evolution of the distribution function and the change in the coefficient of turbulent diffusion can also be significantly affected by the settling of particles from a turbulent flow. The distribution spectrum changes significantly with a change in the rate of droplet deposition. In conclusion, we note that the phenomena of coalescence and fragmentation of droplets are of a spasmodic
nature. In the case of small jumps, such processes are satisfactorily described by the Fok-ker-Planck equation. Obviously, the jumps must become smaller and smaller and more likely, so the diffusion process can always be approximated by a jump process, but not vice versa.
If in equation (2) instead of particle sizes, we use the content of asphaltenes, resins and paraffins in oil, then we obtain their distribution according to their concentrations
f KC
P (C) = AC exp
2B
(10)
e+1
here A = 21
2m
kg, K - coagulation and
k
destruction coefficients, m = k / K, d = —,
g B
B — stochastic diffusion coefficient.
It should be noted that the distribution of
concentrations of asphaltenes, resins, and paraf-
fins is an important factor in the separation of oil emulsions since their content determines the rheological properties of oil. High content of asphaltenes, resins, and paraffins corresponds to a high value of the effective viscosity of the oil emulsion and a low value of the drop settling rate. In addition, the high content of asphaltenes and resins corresponds to a large thickness of the adsorption layer on the droplet surface, which significantly increases the resistance of the system to coalescence and droplet enlargement.
Below in Figure 3 shows the nature of the values of the distribution function of the concentration of asphaltenes, resins, and paraffins calculated according to equation (10) and comparison with experimental data for various oil fields. All calculations were performed using the OptimMe software package [32, 33]. A
large scatter of experimental points does not allow one to uniquely determine the coefficients included in Eq. (10).
The distribution parameters for each curve are shown in Table below.
Numerous experimental and practical studies have shown that the distributions of as-phaltene and resin particles during their crushing and coagulation in a turbulent flow are multimodal (double-humped or multi-humped) in nature, associated with the presence of secondary, tertiary, etc. in the physical system. phenomena of coagulation and fragmentation of particles. Moreover, due to the multi-stage and reusable collision and fragmentation in the flow, a specific interaction of two distribution humps is practically observed (associated with a change in the values of the maximum and coordinates).
14|
C, % m air.
Fig. 3. Distribution curves of asphaltenes, resins and paraffins in Devonian oils: 1 - asphaltenes; 2 - paraffins; 3 - resins. (dots-experiment [34]).
Coefficients to distribution curves of asphaltenes, resins and paraffins in Devonian oils
Ao 0 m - Zib n
asphaltenes 19.5 1 0.035 2
resins 2.6 2 0.0075 3
paraffins 0.38 2 0.00035 3
Results and discussion
The content of various particles of the dispersed phase in the composition of crude oil significantly affects the rheological parameters of the liquid. The main phenomena in the processes of coalescence of water droplets in oil emulsions are the destruction of the adsorption film on the surface due to asphalt-tar substances, the thinning and rupture of the interfacial film between the drops, and the coalescence of the drops.
Based on the Fokker-Planck equation, the evolution of the distribution function of water droplets in an oil emulsion as a function of size and time (4) and (5) has been studied.
The intensification of the processes of flow and separation of oil emulsions is associated primarily with the rheological properties of the oil emulsion and flow turbulence. High-frequency turbulent pulsations contribute to the mechanical weakening of the adsorption and interfacial film and intermolecular bonds between its components, decrease in strength and destruction of the film as a result of their deformation (tension, compression), and improve the conditions for mutual effective collision (increase in the frequency of collisions) and coalescence.
Conclusion
1. The use of the evolution of the size distribution function of water droplets in the technical calculations of the separation of oil emulsions makes it possible to interpret the complete picture of the state of the system at any time. At a fast droplet crushing rate, the particle distribution function is asymmetric with respect to the maximum and is characterized by one maximum independent of the shear rate, although at slow coagulation, the particle size distribution function can have several maxima and minima, i.e., be multimodal. With slow coagulation of solid particles, it is important to build the evolution of the residence time and size distribution function, which gives a complete picture of the change in the number and size of particles over time. The Fokker-Planck equation is solved by the method of
separation of variables, the solution of which characterizes the evolution of the distribution function of the probability density of drops in size and time. The evolution of the distribution function is described by a semi-empirical simple equation.
2. In the well-known equation for the change in the average mass of particles over time, instead of particle sizes, the content of asphaltenes, resins and paraffins in oil is used, then their distribution by their concentrations is obtained. Studies have shown that the distributions of asphaltene and resin particles during their crushing and coagulation in a turbulent flow are multimodal (double-humped or multi-humped) in nature, associated with the presence of secondary, tertiary, etc. in the physical system. phenomena of coagulation and fragmentation of particles.
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NEFT EMULSÍYASINDA SU DAMCILARININ PAYLANMA FUNKSÍYASININ TOKAMÜLÜ
M.R.Manafov, S.RRasulov, F.R.§rnyeva, V.LKarimli
Neft emulsiyalan polidispers mühitdir va makro va mikro diapazonda damci ölgüsüna malikdir. Makro va mikro emulsiyalar ham yagda su emulsiyasinda, ham da suda yag emulsiyasinda tapila bilar. Neftin susuzla§dirilmasi prosesinin modella§dirilmasi zamani hesablamalarin düzgünlüyüna su-neft emulsiya damcilarinin ölgülarina göra paylanmasinin nazara alinmasi böyük tasir göstarir. Damci ölgüsü va damci ölgüsü paylanmasi emulsiyanin sabitlik va özlülük xüsusiyyatlari ügün vacibdir. Maqalada hissaciklarin birla§ma va pargalanma süratina asaslanaraq, vahid hacmda hissaciklarin sayi va ölgüsünün dayi§ma sürati ügün ifada taklif olunur. Fokker-Plank tanliyi asasinda neft emulsiyasinda su damcilarinin paylanma funksiyasimn ifadasi tapilir va onun asasinda paylanmamn qiymatlandirilmasi öyranilir.
Agar sözlar: neft emulsiyalari, Fokker-Plank t3nlikhri, paylanma funksiyasmm tskamülü, koalessensiya, koaqulyasiya, damci ölgüsü, reoloji parametrbr.
ЭВОЛЮЦИЯ ФУНКЦИИ РАСПРЕДЕЛЕНИЯ КАПЕЛЬ В НЕФТЯНОЙ ЭМУЛЬСИИ
М.Р.Манафов, С.Р.Расулов, Ф.Р.Шихиева, В.И.Керимли
Нефтяные эмульсии полидисперсные среды и имеют размер капель в диапазоне макро и микро. Макро- и микроэмульсии можно найти как в типах эмульсий вода-в-масле, так и в типах эмульсий масло-в-воде. Учёт распределения капель водонефтяной эмульсии по размерам оказывает большое влияние на точность расчётов при моделировании процесса обезвоживания нефти. Размер капель и распределение капель по размерам важны для стабильности и вязкостных свойств эмульсии. В статье, на основе скорости процессов коалесценции и фрагментации частиц предложено выражение скорости изменения числа и размеров частиц в единице объема. На основе уравнения Фоккера-Планка найдено выражение функции распределения капель воды в нефтяной эмульсии и на его основе исследована оценка распределения.
Ключевые слова: нефтяные эмульсии, уравнения Фоккера-Планка, эволюция функции распределения, коалесценция, коагуляция, размер капель, реологические параметры.
