modelingandsimulationinengineeringmodelingofthe...

Research ArticleModeling and Simulation in Engineering Modeling of theElectrocoagulation Processes in Nonisothermal Conditions

Andrii Safonyk and Olena Prysiazhniuk

Department of Automation Electrical Engineering and Computer-Integrated TechnologiesNational University of Water and Environmental Engineering Rivne 33028 Ukraine

Correspondence should be addressed to Andrii Safonyk safonikukrnet

Received 6 April 2019 Revised 14 September 2019 Accepted 10 October 2019 Published 3 November 2019

Academic Editor Jing-song Hong

Copyright copy 2019 Andrii Safonyk andOlena Prysiazhniukis is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

e paper suggests an approach to modeling the electrocoagulation process that is based on the generalization of the equations ofincompressible fluid flow in nonisothermal conditions In the model was taking into account the ratio between the values of theparameters which characterize the domination of convective and mass-exchange components of the process over diffusion Anasymptotic approximation of solutions of corresponding boundary value problems is constructed Based on the received solutionswe conducted a computer simulation of the process of distribution of iron concentration inside the reactor that allows predictingvarious hydrodynamic phenomena such as internal recirculation and dead zones that affect the formation of a coagulant einfluence of current strength on the concentration of the target component at the exit from the reactor was investigated using thedeveloped mathematical model In addition our findings also show the effect of the rate of heat formation from the electrodes onthe efficiency of obtaining of coagulant

1 Introduction

Reducing the volume of discharges of pollutants to the waterreservoirs and transition of enterprises to work according tothe scheme of the closed cycle of water use is the maindirection of protection of the water environment For thetreatment of sewage of textile production the reagentmethod is most often used

It requires the use of a significant number of additionalchemicals one of which is divalent iron Different tech-nologies are used for its receipt but the most resource-saving is electrocoagulation is method has several ad-vantages such as versatility absence of additional salinewater pollution in the process of purification small size andcompactness of installations ease of management no needfor reagents easy maintenance of the equipment weaksensitivity to changes in the conditions of the process andobtaining sludge with better structural and mechanicalproperties Among the main reasons hindering the wide-spread introduction of electrocoagulation into productionare the technological complexity of the process and relatively

high cost of electricity In this regard the actual direction ofscientific research is the development of electrocoagulationtechnology for wastewater treatment which will allow thecreation of small-waste closed-end systems of water con-sumption at the enterprise and will be environmentally safeand economically feasible

At present many scientific studies are devoted to thesimulation of electrocoagulation wastewater treatment[1ndash3] However the proposed mathematical models gen-erally do not describe the kinetics of the process inside theelectrocoagulator or do not take into account a number ofparameters for example the effect of stomp force andchanges in water temperature on the rate of flow of theprocess

It is necessary to create source-saving technologies forwater purification erefore the purpose of this work is todevelop a mathematical model of electrocoagulation pro-cessing which takes into account the process of formation ofdivalent iron from a solution of the electrolyte the influenceof technical characteristics (current strength flow rate inletconcentration of suspended particles and temperature) on

HindawiModelling and Simulation in EngineeringVolume 2019 Article ID 9629643 6 pageshttpsdoiorg10115520199629643

the kinetics of the process of mass and heat transfer in theelectrocoagulation installation and also the study of theinfluence of these parameters on the efficiency of the for-mation of a coagulant

is method of sewage treatment is based on elec-trolysis which uses metal anodes which are exposed toelectrolytic dissolution As a result of the dissolution of theanodes the water is enriched with the corresponding ionsforming in the neutral or low-alkaline medium the hy-droxide of iron Flakes of hydroxides of metal collide withgas bubbles connect with them and expose them to thesurface of the liquid after which they are separatedmechanically e electrocoagulation efficiency is influ-enced by the material of the electrodes the distance be-tween them the velocity of the water flow between theelectrodes the temperature and composition of the waterthe voltage and the current density e application ofelectrochemical methods is advisable at relatively highelectrical conductivity of sewage due to the presence ofinorganic acids alkalis or salts in them (with a minimumconcentration of salts equal to 05 gl)

11 Statement of the Problem Modeling of processes isexecuted based on equations which describe the motion ofan incompressible fluid between the electrodes estructure of the flow in the electrocoagulation installation(see Figure 1) can be either laminar or turbulent Incomplex geometry the speed field includes a randomturbulent component that generates streams and turbulentvortices

Continuous flow equations include equations of fluiddynamics mass pulse and energy and provide a basis formodeling the process ese equations have many com-mon features so that a common variable can be used todescribe the traditional forms of fluid flow equationsincluding scalar quantities such as temperature andconcentration is general equation is written in thefollowing form [3]

z(ρQ)

zt+ nabla(ρvQ) nabla DQnablaQ1113872 1113873 + SQ (1)

where the variable Q may be the velocity in the direction xy and z temperature or concentration ρ is liquid densitykgm3 v is average velocity vector DQ is diffusion co-efficient m2s and SQ is source of temperature or con-centration For electrochemical systems equation (1) issimplified through the equation of mass continuity and theNavierndashStokes equation [4]

e basic equations for an incompressible turbulent flowcan be formulated as follows [5]

ρ(vnabla)v minus nablaP + nabla μ + μT( 1113857 nablav +(nablav)T

1113872 11138731113872 1113873 (2)

nabla(ρv) 0 (3)

where P is pressure Pa and μ is dynamic viscosity Pamiddots eReynolds stress can be expressed in terms of turbulentviscosity μT according to the standard model [6]

In a nonisothermal turbulent flow for the case whendensity and thermal conductivity are constant values ap-plying the averaging rule to the heat equation we obtain[7ndash9]

zT

zt+(v middot nabla)T anabla2T +

qV

ρcρ (4)

where a kρcρ is thermal diffusivity m2s k is thermalconductivity W(mmiddotK) cρ is specific heat capacity J(kgmiddotK)ρ is density kgm3 and qV is the intensity of internal heatsources Wm3 e amount of heat released during theelectrode heating of the liquid is proportional to the currentstrength and its time of passage and the voltage drop isqv I middot U middot t where U is applied voltage V and I is currentA

e change in concentration S is described by means ofthe scalar transport equation with the contributions ofturbulent [10]

zS

zt minus vnablaS + nabla(DnablaS) + SC (5)

where D D + Dturb is the total diffusion coefficient D isthe coefficient of molecular diffusion and Dturb is the co-efficient of turbulent diffusion which depends on and theSchmidt turbulent number ScT (according to the KaysndashCrawford model [5])

e effectiveness of the formation of flocs (coagulant)to a large extent depends on the size of the formed bubbles[11] Electroflotation produces a large number of fine gasbubbles the size of which varies from 5 to 90 μmdepending on the state of electrolysis Smaller bubble sizescreate the best kinetics of flotation due to the high ratio ofsurface area to volume Furthermore small bubbles aremore likely to get smaller contact angle in the system forthree-phase (gas-liquid-solid) than larger bubbles esmaller contact angle produces more stable units In ad-dition the time of detention of small bubbles in the flo-tation unit is longer than the time of the detention of largerbubbles since they have a lower velocity is contributesto increase the probability of collision of gas bubbles andflocs Conversely the shear forces of the larger bubbles arehigh due to the high velocity of motion which can causethe floc to break

In [12] deposition of Brownian particles to hydrogenbubbles is describede velocity of the flotation componentof the electroflotation process is quantified as follows

zS

zt η

3RgTI

8FdbAspa

1113888 1113889C (6)

where Rg is ideal gas constant pa is its atmosphericpressure As is cross-sectional area of the chamber and η isefficiency of accumulation with one bubble which is definedas the proportion of the pollutant in the path of the bubblethat actually adheres to the bubble In [13] numerical ex-pressions are also proposed for the calculation of η

us in order to find the distribution of the coagulantconcentrationS and temperatureT in the electrocoagulatorwe obtained the system of two questions

2 Modelling and Simulation in Engineering

zS

zt minus vnablaS + nabla(D(T)nablaC) + η

3RgTI

8FdbAspa

1113888 1113889C

zT


IUt

ρcρ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(7)

with the following conditions

C(x y 0) C0(x y)

T(x y 0) T0(x y)

C(0 y t) Clowast(y t)

T(0 y t) Tlowast(y t)

C(L y t) Clowast(y t)

T(L y t) Tlowast(y t)

zC

zy

11138681113868111386811138681113868111386811138681113868y0 0

zC

zy

11138681113868111386811138681113868111386811138681113868yylowast 0

zT

zy

11138681113868111386811138681113868111386811138681113868y0 minus α T(x 0 t) minus T

lowastlowast( 1113857

zT

zy

11138681113868111386811138681113868111386811138681113868yylowast minus α T x y

lowast t( 1113857 minus T


(8)

2 Methods

Taking into account the relationship between the values ofparameters that characterize certain components of theprocess (in particular the dominance of convective andmass-exchange factors over diffusion) leads to the compli-cation of the mathematical model of the process esesingularities are generated by the presence of small pa-rameters One of the effective ways of solving the corre-sponding problems in the case of the predominance ofconvective factors of the process over diffusion during fil-tration of aqueous solutions in the model regions limited by

equipotential or quasi-equipotential lines and flow lines is asfollows phased fixation of the characteristics and compo-nents of the process and environment solving filtrationproblems using conformal or quasi-conformal mappings ofa complex potential or quasi-potential area on a physicalarea and transition in equations of convective diffusion andboundary and initial conditions from physical variables tothe coordinates of the region of complex potential or quasi-potential which greatly simplifies their recording andprovides the possibility of autonomous research parallelcomputing

Taking into account that the problem is to find thevelocity field (2) and (3) with the given boundary condi-tions [5] is resolved [12] in particular calculated velocityfield and a number of other variables such as filtrationconsumption and replaced the variables x x(φψ) y

y(φψ) in the system (7) and conditions (8) the corre-sponding ldquodiffusion problemrdquo [14 15] was received esolutions with precision O(δn+1) (δ -small parametercharacterizing the predominance of the convective andmass-exchange components of the process of mass transferD(T) δ d(T) |v|gt vlowast ≫ δ) was found in the form ofasymptotic series [16]

C(φψ t) 1113944n

i0δi

Ci(φψ t) + 1113944n+1

i0δi

Ci(ξψ t)

+ 11139442n+1

i0δi2Ci(φ ζ t) + 1113944

2n+1

i0δi2 Ci(φϕ t)

+ RC(φψ t δ)

(9)

T(φψ t) 1113944n

i0δi

Ti(φψ t) + 1113944n+1

i0δi

Ti(ξψ t)

+ 11139442n+1

i0δi2Ti(φ ζ t) + 1113944

2n+1

i0δi2 Ti(φ ϕ t)

+ RT(φψ t δ)

(10)

where RC andRT are residual members of the schedulesCi(φψ t) and Ti(φψ t) are regular parts of the asymptoticseries Ci(ξψ t) and Ti(ξψ t) are functions of the type ofadjoining layer at the output of the filtration flow from theregion

Ci(φ ζ t)

Ti(φ ζ t) Ci(φ ϕ t) and Ti(φ ϕ t) are

functions of the type of adjoining layer in the windows of theside walls of the electrocoagulation installation and

(a) (b)

Figure 1 Scheme of electrocoagulation installation

Modelling and Simulation in Engineering 3

ξ (L minus x) middot δminus 1 ζ y middot δminus 12 and ϕ (ylowast minus y) middot δminus 12 arecorresponding regularizing transformations

Expressions for regular parts of the asymptotic solutionsare found as a result of applying the procedure of

substitution of the series (9) and (10) and the equation ofcoefficients for the same powers of the small parameter andsolving the corresponding problems

Ti(φψ t)

1113946t

01113957Ti f

minus 1(1113957t minus t + f(φψ)ψ)1113957t1113872 1113873d1113957t + τlowastlowasti(φψ) tlef(φψ)

1113946φ

φlowast

1113957Ti(1113957φ f(1113957φψ) + t minus f(φψ))

v(1113957φψ)d1113957φ + τlowasti (φψ) tgtf(φψ) i 0 n

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S0(φψ t)

e1113938

φ

φlowastg0(~φf(~φψ)minus f(φψ)+t)( )(v(~φψ))d~x

S0(t minus f(φψ)) tgtf(φψ)

e1113938

t

0g0 fminus 1(~t+f(φψ)minus t)ψ~t( )d~t

Slowast fminus 1(f(φψ) minus tψ)( 1113857 tlef(φψ)

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Si(φψ t)

e1113938

φ

φlowastgi(φψf(~φψ)minus f(φψ)+t)( )(v(~φψ))d~x

1113946φ

φlowast

hi(φψ f(1113957φψ) minus f(φψ) + t)

v(1113957φψ)

times eminus 1113938

φ

φlowastgi(~~φψf(~~φψ)minus f(~φψ)+t)d~~φ( )(v(~~φψ))

d~φ tgtf(φψ)

e1113938

t

0gi fminus 1(~t+f(φψ)minus tψ)~t( )d~t

1113946t

0hi f

minus 1(1113957t + f(φψ) minus tψ)1113957t1113872 1113873e

minus 1113938~t

0gi fminus 1(~~t+f(x)minus tψ)~~t( 1113857d~~td1113957t tlef(φψ) i 1 n

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(11)

where f(φψ) 1113938φφlowast

(d1113957φ)(v(1113957φψ)) 1113957T0(φψ t) (IUtcρ)τlowast0(φψ) Tlowast(fminus 1(f(φψ) minus tψ)) τlowastlowast0(φψ) T0(tminus f(φ

ψ)ψ) 1113957Ti(φψ t) dT((z2Timinus 1zφ2) + (z2Timinus 1zψ2)) dT

aλmδλ τlowasti (φψ) τlowastlowasti(φψ) 0 middot (i 1 n) gi(φψ t)

η(3RgITi(φ ψ t))(8FdbAsP) hi(φ ψ t) diC((z2Ciminus 1zφ2) + (z2Ciminus 1zψ2)) and diC are known functions that arethe sum of the products of the members of the series (9) and(10) their partial derivatives and the coefficients of thecorresponding powers of the small parameter in the de-composition of the corresponding functions d(L minus ξδψ t)

in the Taylor series in the vicinity φ L Functions of thetype of adjoining layer are found similarly to [14ndash16]

3 Results of Numerical Calculations

e computer simulation was carried out at the followingvalues of the parameters η 08 c 331 middot 103 Jkg ρ

1060 kgm3 λ 02 W(mmiddots) a 139 middot 10minus 8 m2sRg 831 J(molmiddotK) F 965 middot 104 Cmol As 15 m2 db

15 middot 10minus 6 m D 10minus 9 m2s and Tlowastlowast 294 K Provided thatthe current strength is equal to 70 A the concentration ofiron ions at the exit from the electrocoagulator is obtained165mgl at moments of 90 minutes 684mgl at 180min

and 1018mgl at 240min [2] e results of the experimentand the calculation of the model problem (7) and (8) arepresented in Figure 2

In practice current is the only operating parameter afterother operating parameters such as pH volume flow ofwater and design dimensions of the reactor and others areregulated which affects the kinetics of the coagulant for-mation processe distribution of the concentration of ironand water temperature in the electrocoagulation installationat the initial time and after 240min is shown in Figures 3 and4 As expected the concentration of iron increases in thespace between the electrodes which conventionally wearsout to the exit of the electrocoagulation installation whereits concentration is highest Similarly the temperature of theelectrolyte increases uniformly along the length of theelectrocoagulation installation and due to the convectivetransfer of the entrance to the cold water and cooling by thewalls we obtain the maximum temperature values near theelectrodes and at the outlet of the electrocoagulation in-stallation (Figure 4)

Analysis of the distribution of iron concentration insidethe reactor allows predicting various hydrodynamic phe-nomena such as internal recirculation and dead zones thataffect the formation of a coagulant


4 Conclusions

A mathematical model describing the laws of the course ofheat and mass transfer processes in an electrocoagulationplant was developed e solution of the corresponding

model problem with the use of the asymptotic approxi-mation of the solution of the boundary value problem isfound and the results of calculations of the distribution ofthe concentration of iron and water temperature in theelectrocoagulation installation and the output of the

0 50 100 150t (min)

C (x

t)

(mg

l)

200

20

40

60

80

100

Figure 2 Distribution of the concentration of divalent iron over time at the exit of the coagulator

0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)

Figure 3 e distribution of the concentration of iron at the initial time (a) and after 240min (b)

0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)

Figure 4 e distribution of the water temperature at the initial time (a) and after 240min (b)


coagulator depending on the current strength are given eproposed methodology for calculating the concentrationdistribution can be used to analyze the influence of heat andmass transfer in the electrolyte and the kinetics of the re-action on the electrodes as well as the basis for experimentaland theoretical studies of optimization and automation ofthe process of formation of coagulant by the method ofelectrocoagulation

Data Availability

Previously reported experimental data were used to supportthis study and are available at DOI 101109STC-CSIT20188526661 ese prior studies (and datasets) arecited at relevant places within the text as references [2]

Conflicts of Interest

e authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] J N Hakizimana B Gourich M Chafi et al ldquoElectro-coagulation process in water treatment a review of electro-coagulation modeling approachesrdquo Desalination vol 404pp 1ndash21 2017

[2] A Safonyk A Bomba and I Tarhonii ldquoModeling and au-tomation of the electrocoagulation process in water treat-mentrdquo Advances in Intelligent Systems and Computingvol 871 pp 451ndash463 2018

[3] V Khandegar and A K Saroha ldquoElectrocoagulation for thetreatment of textile industry effluent - a reviewrdquo Journal ofEnvironmental Management vol 128 pp 949ndash963 2013

[4] H Versteeg and W Malalasekera An Introduction to Com-putational Fluid Dynamics e Finite Volume MethodVol 503 Pearson Education London UK 2007

[5] M A Sandoval R Fuentes F C Walsh J L Nava andC P de Leon ldquoComputational fluid dynamics simulations ofsingle-phase flow in a filter-press flow reactor having a stack ofthree cellsrdquo Electrochimica Acta vol 216 pp 490ndash498 2016

[6] R Enciso L Padilla C Ojeda et al ldquoComputational fluiddynamics characterization of a rotating cylinder electro-chemical reactor using an RANS-RNG turbulence modelrdquoInternational Journal of Electrochemical Science vol 7pp 12181ndash12192 2012

[7] V Kulinchenko and S Tkachenko ldquoHeat-transfer with ele-ments of mass-transfer (theory and practice of process)rdquoFeniks pp 1ndash917 2014

[8] A P Vlasyuk and V V Zhukovskii ldquoMathematical simu-lation of the migration of radionuclides in a soil mediumunder nonisothermal conditions with account for catalyticmicroparticles and nonlinear processesrdquo Journal of Engi-neering Physics and ermophysics vol 90 no 6 pp 1386ndash1398 2017

[9] P M Martyniuk M T Kuzlo S K Matus andT P Tsvietkova ldquoMathematical model of nonisothermalmoisture transference in the form of water and vapor in soilsin the case of chemical internal erosionrdquo Far East Journal ofMathematical Sciences (FJMS) vol 102 no 12 pp 3211ndash32212017

[10] A S Naje S Chelliapan Z Zakaria M A Ajeel andP A Alaba ldquoA review of electrocoagulation technology for

the treatment of textile wastewaterrdquo Reviews in ChemicalEngineering vol 33 pp 263ndash292 2017

[11] R Alam and J Q Shang ldquoElectrochemical model of electro-flotationrdquo Journal of Water Process Engineering vol 12pp 78ndash88 2016

[12] X Chen and G Chen ldquoElectro-flotationrdquo in Electrochemistryfor the Environment C Comninellis and G Chen Edspp 263ndash277 Springer Science + Business Media LLC BerlinGermany 2010

[13] Y Fukui and S Yuu ldquoCollection of submicron particles inelectro-flotationrdquo Chemical Engineering Science vol 35 no 5pp 1097ndash1105 1980

[14] A Safonyk S Martynov S Kunytsky and O PinchukldquoMathematical modelling of regeneration the filtering mediabed of granular filtersrdquo Advances in Modelling and Analysis Cvol 73 no 2 pp 72ndash78 2018

[15] A Y Bomba and A P Safonik ldquoMathematical simulation ofthe process of aerobic treatment of wastewater under con-ditions of diffusion and mass transfer perturbationsrdquo Journalof Engineering Physics and ermophysics vol 91 no 2pp 318ndash323 2018

[16] A Bomba Y Klymiuk I Prysiazhniuk O Prysiazhniuk andA Safonyk ldquoMathematical modeling of wastewater treatmentfrom multicomponent pollution by using microporous par-ticlesrdquo AIP Conference Proceedings vol 1773 pp 1ndash11 2016


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the kinetics of the process of mass and heat transfer in theelectrocoagulation installation and also the study of theinfluence of these parameters on the efficiency of the for-mation of a coagulant

is method of sewage treatment is based on elec-trolysis which uses metal anodes which are exposed toelectrolytic dissolution As a result of the dissolution of theanodes the water is enriched with the corresponding ionsforming in the neutral or low-alkaline medium the hy-droxide of iron Flakes of hydroxides of metal collide withgas bubbles connect with them and expose them to thesurface of the liquid after which they are separatedmechanically e electrocoagulation efficiency is influ-enced by the material of the electrodes the distance be-tween them the velocity of the water flow between theelectrodes the temperature and composition of the waterthe voltage and the current density e application ofelectrochemical methods is advisable at relatively highelectrical conductivity of sewage due to the presence ofinorganic acids alkalis or salts in them (with a minimumconcentration of salts equal to 05 gl)

11 Statement of the Problem Modeling of processes isexecuted based on equations which describe the motion ofan incompressible fluid between the electrodes estructure of the flow in the electrocoagulation installation(see Figure 1) can be either laminar or turbulent Incomplex geometry the speed field includes a randomturbulent component that generates streams and turbulentvortices

Continuous flow equations include equations of fluiddynamics mass pulse and energy and provide a basis formodeling the process ese equations have many com-mon features so that a common variable can be used todescribe the traditional forms of fluid flow equationsincluding scalar quantities such as temperature andconcentration is general equation is written in thefollowing form [3]

z(ρQ)

zt+ nabla(ρvQ) nabla DQnablaQ1113872 1113873 + SQ (1)

where the variable Q may be the velocity in the direction xy and z temperature or concentration ρ is liquid densitykgm3 v is average velocity vector DQ is diffusion co-efficient m2s and SQ is source of temperature or con-centration For electrochemical systems equation (1) issimplified through the equation of mass continuity and theNavierndashStokes equation [4]

e basic equations for an incompressible turbulent flowcan be formulated as follows [5]

ρ(vnabla)v minus nablaP + nabla μ + μT( 1113857 nablav +(nablav)T

1113872 11138731113872 1113873 (2)

nabla(ρv) 0 (3)

where P is pressure Pa and μ is dynamic viscosity Pamiddots eReynolds stress can be expressed in terms of turbulentviscosity μT according to the standard model [6]

In a nonisothermal turbulent flow for the case whendensity and thermal conductivity are constant values ap-plying the averaging rule to the heat equation we obtain[7ndash9]

zT


qV

ρcρ (4)

where a kρcρ is thermal diffusivity m2s k is thermalconductivity W(mmiddotK) cρ is specific heat capacity J(kgmiddotK)ρ is density kgm3 and qV is the intensity of internal heatsources Wm3 e amount of heat released during theelectrode heating of the liquid is proportional to the currentstrength and its time of passage and the voltage drop isqv I middot U middot t where U is applied voltage V and I is currentA

e change in concentration S is described by means ofthe scalar transport equation with the contributions ofturbulent [10]

zS

zt minus vnablaS + nabla(DnablaS) + SC (5)

where D D + Dturb is the total diffusion coefficient D isthe coefficient of molecular diffusion and Dturb is the co-efficient of turbulent diffusion which depends on and theSchmidt turbulent number ScT (according to the KaysndashCrawford model [5])

e effectiveness of the formation of flocs (coagulant)to a large extent depends on the size of the formed bubbles[11] Electroflotation produces a large number of fine gasbubbles the size of which varies from 5 to 90 μmdepending on the state of electrolysis Smaller bubble sizescreate the best kinetics of flotation due to the high ratio ofsurface area to volume Furthermore small bubbles aremore likely to get smaller contact angle in the system forthree-phase (gas-liquid-solid) than larger bubbles esmaller contact angle produces more stable units In ad-dition the time of detention of small bubbles in the flo-tation unit is longer than the time of the detention of largerbubbles since they have a lower velocity is contributesto increase the probability of collision of gas bubbles andflocs Conversely the shear forces of the larger bubbles arehigh due to the high velocity of motion which can causethe floc to break

In [12] deposition of Brownian particles to hydrogenbubbles is describede velocity of the flotation componentof the electroflotation process is quantified as follows

zS

zt η

3RgTI

8FdbAspa

1113888 1113889C (6)

where Rg is ideal gas constant pa is its atmosphericpressure As is cross-sectional area of the chamber and η isefficiency of accumulation with one bubble which is definedas the proportion of the pollutant in the path of the bubblethat actually adheres to the bubble In [13] numerical ex-pressions are also proposed for the calculation of η

us in order to find the distribution of the coagulantconcentrationS and temperatureT in the electrocoagulatorwe obtained the system of two questions


zS


3RgTI

8FdbAspa

1113888 1113889C

zT


IUt

ρcρ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(7)


C(x y 0) C0(x y)

T(x y 0) T0(x y)





zC

zy

11138681113868111386811138681113868111386811138681113868y0 0

zC

zy

11138681113868111386811138681113868111386811138681113868yylowast 0

zT

zy



zT

zy




(8)

2 Methods





C(φψ t) 1113944n

i0δi

Ci(φψ t) + 1113944n+1

i0δi

Ci(ξψ t)

+ 11139442n+1

i0δi2Ci(φ ζ t) + 1113944

2n+1

i0δi2 Ci(φϕ t)

+ RC(φψ t δ)

(9)

T(φψ t) 1113944n

i0δi

Ti(φψ t) + 1113944n+1

i0δi

Ti(ξψ t)

+ 11139442n+1

i0δi2Ti(φ ζ t) + 1113944

2n+1

i0δi2 Ti(φ ϕ t)

+ RT(φψ t δ)

(10)


Ci(φ ζ t)



(a) (b)






Ti(φψ t)

1113946t

01113957Ti f


1113946φ

φlowast



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S0(φψ t)

e1113938

φ



e1113938

t



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Si(φψ t)

e1113938

φ


1113946φ

φlowast


v(1113957φψ)


φ


d~φ tgtf(φψ)

e1113938

t


1113946t

0hi f


minus 1113938~t


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(11)















4 Conclusions



0 50 100 150t (min)

C (x

t)

(mg

l)

200

20

40

60

80

100


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)




Data Availability




References





















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


zS


3RgTI

8FdbAspa

1113888 1113889C

zT


IUt

ρcρ

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

(7)


C(x y 0) C0(x y)

T(x y 0) T0(x y)





zC

zy

11138681113868111386811138681113868111386811138681113868y0 0

zC

zy

11138681113868111386811138681113868111386811138681113868yylowast 0

zT

zy



zT

zy




(8)

2 Methods





C(φψ t) 1113944n

i0δi

Ci(φψ t) + 1113944n+1

i0δi

Ci(ξψ t)

+ 11139442n+1

i0δi2Ci(φ ζ t) + 1113944

2n+1

i0δi2 Ci(φϕ t)

+ RC(φψ t δ)

(9)

T(φψ t) 1113944n

i0δi

Ti(φψ t) + 1113944n+1

i0δi

Ti(ξψ t)

+ 11139442n+1

i0δi2Ti(φ ζ t) + 1113944

2n+1

i0δi2 Ti(φ ϕ t)

+ RT(φψ t δ)

(10)


Ci(φ ζ t)



(a) (b)






Ti(φψ t)

1113946t

01113957Ti f


1113946φ

φlowast



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S0(φψ t)

e1113938

φ



e1113938

t



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Si(φψ t)

e1113938

φ


1113946φ

φlowast


v(1113957φψ)


φ


d~φ tgtf(φψ)

e1113938

t


1113946t

0hi f


minus 1113938~t


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(11)















4 Conclusions



0 50 100 150t (min)

C (x

t)

(mg

l)

200

20

40

60

80

100


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)




Data Availability




References





















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Ti(φψ t)

1113946t

01113957Ti f


1113946φ

φlowast



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

S0(φψ t)

e1113938

φ



e1113938

t



⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Si(φψ t)

e1113938

φ


1113946φ

φlowast


v(1113957φψ)


φ


d~φ tgtf(φψ)

e1113938

t


1113946t

0hi f


minus 1113938~t


⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

(11)















4 Conclusions



0 50 100 150t (min)

C (x

t)

(mg

l)

200

20

40

60

80

100


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)




Data Availability




References





















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


4 Conclusions



0 50 100 150t (min)

C (x

t)

(mg

l)

200

20

40

60

80

100


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)


0 20 40 60 80

1009080706050403020100

(a)

0 20 40 60 80

1009080706050403020100

(b)




Data Availability




References





















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Data Availability




References





















RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


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